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Dimensionless Physical Quantities in Scienceand Engineering
Dimensionless PhysicalQuantities in Scienceand Engineering
Josef KunesDepartment of PhysicsUniversity of West BohemiaPlzenCzech Republic
AMSTERDAM � BOSTON � HEIDELBERG � LONDON � NEW YORK � OXFORD
PARIS � SAN DIEGO � SAN FRANCISCO � SINGAPORE � SYDNEY � TOKYO
Elsevier
32 Jamestown Road, London NW1 7BY
225 Wyman Street, Waltham, MA 02451, USA
First edition 2012
Copyright r 2012 Elsevier Inc. All rights reserved
No part of this publication may be reproduced or transmitted in any form or by any means,
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This book and the individual contributions contained in it are protected under copyright by
the Publisher (other than as may be noted herein).
Notices
Knowledge and best practice in this field are constantly changing. As new research and
experience broaden our understanding, changes in research methods, professional practices,
or medical treatment may become necessary.
Practitioners and researchers must always rely on their own experience and knowledge in
evaluating and using any information, methods, compounds, or experiments described
herein. In using such information or methods they should be mindful of their own safety and
the safety of others, including parties for whom they have a professional responsibility.
To the fullest extent of the law, neither the Publisher nor the authors, contributors, or
editors, assume any liability for any injury and/or damage to persons or property as a matter
of products liability, negligence or otherwise, or from any use or operation of any methods,
products, instructions, or ideas contained in the material herein.
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Cataloging-in-Publication Data
A catalogue record for this book is available from the Library of Congress
ISBN: 978-0-12-416013-2
For information on all Elsevier publications
visit our website at elsevierdirect.com
This book has been manufactured using Print On Demand technology. Each copy is
produced to order and is limited to black ink. The online version of this book will show
color figures where appropriate.
I dedicate this book to the memory of my teachersProf. Vladimır Marcelli
andProf. Josef Hosek.
They took me and my workplace,half a century ago,
on the path of modelling and experiment.
PrefaceAll things are numbers.
Pythagoras of Samos (570�495 BC)
The century of informatics, characterized by the remarkable growth of information,
is closely connected with the increasing importance of the model experiment. The
specific important has computer modelling, in practically all scientific spheres.
Modelling is a powerful tool not only in terms of the scale assignment of a model
and its original, as it was for a long time in the case of physical models, but also in
terms of the compression of the information flow on a model and by the processing
of the modelling results. In this sense, the importance of dimensionless quantities �and especially of similarity criteria � has, in fact, increased. The dominant opinion,
still held by many even today, that these quantities are connected primarily with
physical modelling, is mistaken. On the contrary, their importance has grown
significantly in today’s dominant mathematical and computer modelling. The
dimensionless formulation of a mathematical or computer model does not lose its
physical importance; on the contrary, it is magnified. Simplification and generaliza-
tion of modelling results to the dimensionless forms are also other significant attri-
butes. One example is the generation of experimental mathematical models used
whenever it is not possible to use an exact mathematical model. It follows that
dimensionless quantities and similarity theory have an even wider importance in the
contemporary development of modelling than they did in the past.
The use of dimensionless quantities in science matches only the growth of
globalization in the world. Its use in science is, above all, in interlinking dimen-
sional physical and other quantities in dimensionless groups and sinks so growth of
information flow in modelling.
Dimensionless Physical Quantities in Science and Engineering presents in nine
chapters approximately 1200 dimensionless quantities from several types of fields
in which modelling plays an important role. It is probably the most extensive col-
lection of these quantities involving both classic and newly developing fields. In
addition to traditional fields like fluid mechanics and heat transfer, in which using
dimensionless quantities has long been common, this book includes many others,
in particular newly developing fields such as solid phase mechanics, electromagne-
tism, physical macro- and nanotechnology, technology and mechanical engineering,
geophysics and ecology. Each dimensionless quantity is presented with both its
physical characteristics and its significance in the relevant field.
This book is not only a simple summary of these quantities, but also features
a clarification of their physical principles, areas of use and other specific
properties. The book also facilitates the retrieval of dimensionless quantities for
practical use. Furthermore, it provides citations to important sources, also facilitat-
ing the use of dimensionless quantities in their appropriate fields.
The wide range of different spheres in which dimensionless quantities have
extraordinary importance presented considerable difficulty. First of all, for their
specific dissimilarities in such physically different spheres. Moreover, there are
many problems with terminology, and the unsystematic designation of single non-
traditional dimensionless quantities by various authors. The book presents the
established or most often used designations, and recommends preferred
designations.
The book is an attempt to systematically present dimensionless physical quanti-
ties from different scientific and engineering fields. It concerns especially the fields
and processes related to physical substance. Therefore, this book does not include
some fields, such as economics, in which modelling has considerable importance,
but in which the use of dimensionless quantities is out of physical line. The use of
dimensionless quantities in medicine, biology, physiology and other spheres is
increasing along with the fast development of modelling in those fields. However,
this book is mostly concerned with typical dimensionless quantities or their modifi-
cations. A similar situation exists in other fields.
In this book, the presented bibliography is divided into four parts, including a
part discussing books and proceedings (A) and section on articles in journals (B).
Many information sources are presented on Internet sites (C), and there is also a
list (D) of some simple calculators for determination of dimensionless quantities,
and information related to simple software.
A mutual connection of relevant dimensionless quantities by means of cross-
references is an important advantage of this book.
Josef Kunes
x Preface
Foreword
The most fascinating feature of the book Dimensionless Physical Quantities in
Science and Engineering is the presentation of about 1200 dimensionless quantities
relevant for modelling in several research areas, both the traditional and newly
developing ones, which hopefully will contribute to the further development of
model experiments. For this reason, we are indeed fortunate to have Professor
Kunes to guide the thoughts and activities of many scientists and students.
The reason for presenting this relatively great number of dimensionless quanti-
ties in this book is their extraordinary significance for modelling. They are relevant
not only to scale models but also to almost all types of models, including computer
models. The prevalent opinion that these quantities are primarily connected to
physical modelling is erroneous. On the contrary, we cannot ignore the fact that
their importance continues to grow, especially in today’s predominant mathemati-
cal and computer modelling. With the dimensionless expressions used in a mathe-
matical or computer model, the physical meaning is not lost, but is, in fact,
intensified. Moreover, the process investigated becomes more transparent, and the
number of variable quantities and the intensity of the flow of information are sig-
nificantly reduced. This is also true of the number of output quantities according to
which the modelling results are generalized. The dimensionless quantities have fun-
damental significance for experimental mathematical models (phenomenological
mathematical models) generally, when more exact asymptotic mathematical models
cannot be used.
The English edition of this book will thus certainly make a significant contribu-
tion to possible future achievements in the relevant fields of research.
Professor J. Vlcek
Head of the Department of Physics
University of West Bohemia
Plzen, Czech Republic
1 Introduction
The main goal of physics is to describe a maximum of phenomena with a minimum
of variables.CERN Courier [A40]
The dimensionless quantity expresses either a simple ratio of two dimensionally
equal quantities (simple) or that of dimensionally equal products of quantities in the
numerator and in the denominator (composed). The dimensionless quantities can
be divided into several groups. The most important group consists of the physical
similarity criteria obtained by some of the similarity theory methods. They are also
called generalized variable quantities. The dimensionless physical constants belong
to another group. In addition, the approximate ratio quantities can also be included
among the dimensionless quantities. They usually come from experimental results
and from the experimenter’s intuition, i.e. without using any of the similarity theory
methods. Usually, the extent of the validity of these quantities is limited only to
a certain area. Other dimensionless ratio quantities can be created as well, which do
not have any full-value importance from the modelling point of view.
Each of the similarity criteria can be expressed in the form of a mutual relation
between, for example, two forces, momentums or energies acting in a process.
Therefore, by observing the size of the criterion, an idea can be obtained from the
character of the investigated process. This fact is well known, for example, with
the Reynolds number Re, which expresses the dynamic-to-viscous force ratio and
characterizes viscous fluid flow. According to the value of the Re number, the flow
can be distributed into three fundamental characteristic types: laminar, transient
and turbulent. This is similar to the use of the Weber number for single-phase and
two-phase fluids, where this number expresses the ratio of the surface tension force
to the inertial one. More details are in [A23] and [A24], where examples using the
Weber number are given for condensation, boiling and motion of gas bubbles in a
fluid or interaction of a drop with a warm wall.
Sometimes, however, it seems as though the transition from dimensional physical
quantities to dimensionless ones would obscure the view of the investigated process. In
fact, the contrary is true because the reduced number of variable quantities � expressed
in the dimensionless way � enables one to understand the mutual physical contexts in
the investigated process more deeply. A good example is the Fourier number, which
is very often used to express dimensionless time. In fact, it is used in all unsteady
processes occurring in various fields. For example, in thermomechanics � in the
case of heat conduction � it expresses the coupling of time with the characteristic
Dimensionless Physical Quantities in Science and Engineering. DOI: 10.1016/B978-0-12-416013-2.00001-4
© 2012 Elsevier Inc. All rights reserved.
geometrical dimension and the thermal diffusivity. It is described by a single
dimensionless variable which is expressive of the influence of all three dimensional
quantities on the temperature field.
Because of a misunderstanding of the physical significance of the similarity
criteria and the functions thereof as generalized variables, dimensional quantities
are often applied even if an experiment or other research should result in an expres-
sion of the information obtained in a condensed and generalized way, so that it can
be utilized for all physically similar systems. With dimensional quantities, on the
contrary, the immediate perception of the process is advantageous in measurement
and identification of an investigated system. The dimensionless similarity criteria
impose a limitation on repeating solutions for tasks of similar character.
Physical similarity criteria are divided into composed criteria and simple (parame-
trical) criteria. Among other things, the importance of similarity criteria is that they
keep a deep physical significance despite the fact that they are outwardly dimension-
less. That is to say, they express the ratios of diverse physical quantities such as forces,
energies and momentums, which especially enable one to understand the acting mech-
anism of individual quantities in analyzing complex physical processes. The physical
similarity criteria can be obtained either by the dimensional analysis method or by the
similarity analysis method, either of a phenomenological physical model or by exact
mathematical model analysis. These methods have been described in the author’s
book Similarity and Modeling in Science and Engineering (CISP, Cambridge, 2012)
in more detail.
This book is primarily focused on physical similarity criteria in various fields
which are characteristics of the development of contemporary science and engi-
neering. Eight chapters summarize about 30 independent fields or spheres in which
modelling plays an important role. The most widespread application of similarity
criteria has already occurred in the era of pre-computer modelling in fluid mechan-
ics and in heat transfer. The present number of criteria in these fields corresponds
to this, and so do the systematic descriptions and surveys presented in the literature,
such as [B11] and [B12]. Of course, the origin of new fields and the increasing
importance of existing ones, together with the entrance of computer modelling, has
resulted in the emergence of many other similarity criteria and modifications of
original ones. However, the literature lacks both adequate descriptions and analyses
of criteria in individual fields and magnetism are partial exceptions. A survey of
the similarity criteria of several fields is given in [A32] and [A33].
The dimensionless quantities from tens of fields are summarized in eight chapters.
Among them, new original and modified dimensionless quantities are presented,
which have been introduced and used in the workplaces of the author.
This especially concerns chapters 5, 6, 7 and 8. Included in each of the chapters
are brief profiles of many important scientists and engineers who worked in the fields
surveyed and have similarity criteria named after them. This should contribute to
the recognition that the dimensionless physical quantities have a human intellectual
dimension in addition to their physical significance.
2 Dimensionless Physical Quantities in Science and Engineering
2 Physics and Physical Chemistry
The basic laws of physics and chemistry are like each other.Dmitri Ivanovich Mendeleev (1834�1907)
2.1 Physics, Mathematics and Geometry
In physics, dimensionless physical quantities and constants have been widely used,
in thermodynamics, optics, radiation and other spheres of physics, especially in appli-
cations in various natural scientific and technical branches, and have become
an important tool in their development. In mathematics, dimensionless quantities
have their theoretical base in the theory of groups and also in linear algebra and
matrix calculus. At the same time, the fundamental theorem to determine the simi-
larity criteria is the dimensional homogeneity of equations of mathematical physics
as defined by Fourier. The similarity criteria are practically important in numerical
mathematics and computer modelling, e.g. not only in generalized dimensionless
expressions of numerical solution stability of mathematical physics equations
but also in other spheres of mathematics. Among the best known physical dimen-
sionless quantities are the following numbers: Abbe, Fresnel and Snellen numbers in
optics; Bejan, Boyle, Carnot, Gay-Lussac, Pitzer and Van der Walls numbers in ther-
modynamics; and the Planck number for radiation.
In mathematics, for example, diverse dimensionless numbers express the
stability conditions of the numerical solution, such as the Courant, Damkohler,
Neumann and Peclet numbers and other mathematical and geometrical dimen-
sionless numbers.
2.1.1 Abbe Number V
V 5nD 21
nF 2 nC
nD, nF, nC (�) � refractive indices of the material at the wavelength of the
Fraunhofer D-, F- and C spectral lines (589.2, 486.1 and 656.3 nm, respectively).
It is used to classify the glasses in the dispersion measurement in the visible radia-
tion band. Low-fracture glasses have high values of V, e.g. for lead crystal glass it is
V, 50, whereas for crown glass it is V. 50. For heavy flint glasses, the common
extent of V is about 20. Very light crown glasses have values of V up to 60.
Info: [C2].
Dimensionless Physical Quantities in Science and Engineering. DOI: 10.1016/B978-0-12-416013-2.00002-6
© 2012 Elsevier Inc. All rights reserved.
Ernst Abbe (23.1.1840�14.1.1905), German physicist and
astronomer.
He was engaged in physics, mathematics and meteorol-
ogy but in optics and astronomy above all. He deduced the
mathematical theory of a light microscope. He designed and
fabricated high-quality lenses for scientific purposes. He
manufactured special instruments. He was the co-founder
of the Carl Zeiss works in Jena.
2.1.2 Energy Accommodation Coefficient rE, σ, α
rE 5Ein 2Ere
Ein 2EW
Ein, Ere (J) � incident and reflected energy flux; EW (J) � reflected energy flux
obtained if the molecules are in thermal equilibrium.
It characterizes the mutual energetic effects of gas molecules with a solid body
surface with heat passing in diluted gases. It expresses the energy of that part
of the total number of gas molecules which come in contact with the surface
and the energy which, after rebound or reemission, is reduced because of being
accommodated to the surface temperature. Besides, it is the measure of the
thermal energy transfer perfection. With the complete transfer, rE5 1 is valid;
and with a complete (mirrored) reflection of the energy, rE5 0. Its size depends
on the physical properties of the surroundings and usually does not differ very
much from the number one.
Info: [A33].
2.1.3 Avogadro Number NA
NA 5 6:02203 1023 mol21
The Avogadro number expresses the number of particles (e.g. atoms, molecules
or ions) in a chemically homogeneous body with the substance quantity of one
mole (mol). The mole is the substance quantity of the set which contains exactly
as many elementary individual units (e.g. atoms, molecules or other particles) as
the atoms in 0.012 kg of the nuclide of the carbon isotope 126C: It is widely applied
in physics, chemistry and other branches.
Info: [C5].
4 Dimensionless Physical Quantities in Science and Engineering
Lorenzo Romano Amedeo Carlo Avogadro (9.8.1776�9.7.1856), Italian mathematician, physicist and chemist.
He was engaged in statistics, physics and chemistry, and
in meteorology as well. In the year 1811, he published the
hypothesis known later as the Avogadro law, which expresses
the fact that equal gas volumes contain equal numbers of
molecules under equal temperature and pressure. However,
Avogadro could not prove the hypothesis precisely by experi-
ment and did not live to see its acceptance. To honour his
work in the molar mass theory, the number of molecules in
one mole of particles was named after him.
2.1.4 Bejan Thermodynamical Number Be
Be5S1
S1 1 S2
S1 (J K21) � entropy generation contribution by heat transfer; S2 (J K21) � entropy
generation contribution by fluid friction.
It expresses the ratio of heat transfer unreturnability to the total unreturnability
caused by heat transfer and fluid friction.
Info: [C6]
Adrian Bejan (born 24.9.1948), American engineer of
Romanian origin.
His research activity involves a wide area of thermal
engineering and thermodynamics. He was engaged in
the entropy minimization problem; the energy conversion
analysis (exergy); natural convection; convection in porous
materials; heat and mass transfer; and problems of turbu-
lence, melting, solidification, condensation, contamination,
solar energy conversion, cryogenic engineering, applied
superconductivity and tribology. He created the constructive
form and structure theory in nature.
2.1.5 Boltzmann Distribution, Boltzmann Factor NB, Pn
NB 5Ni
N5 e2
γBhkT
Ni (m23) � number of states having energy Ei; N (m23) � total number of particles;
γ (�) � magnetogyroscopic ratio; B (T) � magnetic induction; h (J s) � Planck
constant; k (J K21) � Boltzmann constant; T (K) � sample temperature.
5Physics and Physical Chemistry
In physics, it represents the prediction of the function of particle distribution
of which each one has the energy Ei. Alternatively, it expresses the volume magne-
tizing vector as well.
Info: [C8].
Ludwig Boltzmann (p. 205).
2.1.6 Boyle Number Bo
Bo5TB
Tcrit
TB (K) � Boyle temperature; Tcrit (K) � critical temperature.
This number expresses the ratio of the thermodynamic temperature of Boyle’s
point, corresponding to the zero isobar, to the critical temperature. BoA 2:3; 3h i:Info: [C11].
Robert Boyle (25.1.1627�30.12.1691), Irish chemist and
mathematician.
He is called the father of chemistry. He applied experi-
mental and quantitative methods. He was the first to deliver
the modern definition of chemical elements, and he used
it to measure the acidity of colour indicators. He discovered
the indirect proportionality between gas pressure and volume
under constant temperature. This is Boyle’s law.
2.1.7 Bulk Concentration Nbc
Nbc 5cb
R
cb (kg m23) � concentration of bulk particles; R (kg m23) � liquid density.
It characterizes the relative concentration of solid particles in a solution. Filtration.
2.1.8 Carnot Number Ca
Ca5T2 2 T1
T2
T1, T2 (K) � absolute temperatures.
6 Dimensionless Physical Quantities in Science and Engineering
It characterizes the theoretical efficiency of the Carnot circulation occurring
between two thermal states, limited by the thermodynamic temperatures T1 and T2.
Thermodynamics.
Info: [B20],[C16].
Nicolas Leonard Sadi Carnot (1.6.1796�24.8.1832), French
physicist.
He was engaged in thermodynamics above all. In the year
1824, he elaborated the thermal machines theory in his work
Reflections sur la Puissance Motrice de la Feu (Considerations
on the Driving Power of Fire). He designed the Carnot reversal
thermal circulation and found that thermal machine efficiency
depends only on the inlet and outlet temperatures.
2.1.9 Coefficient of Variation C
C5σμ
σ (�) � standard deviation2; μ (�) � mean value.
In probability and statistical theories, it expresses the dispersion size of the
probability distribution. It is often used to evaluate the normal distribution with
a positive mean value and with a standard deviation which is less than the mean
deviation expressively. In compliance with the distribution character of the standard
deviation, the variation coefficient size can be greater or less than one. Mathematics,
statistics.
Info: [C21].
2.1.10 Compressibility Factor Z
Z5pv
r Tð1Þ; Zcrit 5Wa21 ð2Þ
p (Pa) � pressure; v (m3 kg21) � specific volume; r (J kg21 K21) � specific gas
constant; T (K) � temperature; Wa (�) � Van der Waals number (1.) (p. 32).
This factor characterizes the mutual molecular coupling in a substance for a
certain thermodynamic state. At the thermodynamic critical point, the relation (2)
is valid. In ideal gases with Z5 1, the deviations from this value express the size of
the intermolecular coupling.
Info: [C83].
7Physics and Physical Chemistry
2.1.11 Courant Numerical Number Cou, CFL, ν
Cou5wΔτΔx
,C; where C# 1 ð1Þ;
Cou5wxΔτΔx
1wyΔτΔy
,C ð2Þ;
Cou5Δτ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwx
Δx
� �2
1wy
Δy
� �2s
# 1 ð3Þ;
CFL5ΔτΔx4
,C ð4Þ
w (m s21 � velocity; Δτ (s) � time step; Δx, Δy (m) � mesh increment in the
x and y axis; C (�) � constant (time stepping parameter) depends on the particular
equation to be solved and not on Δτ and Δx; wx, wy (m s21) � flow (fluid) velocity
component in the x and y axis.
It expresses the stability criterion of a numeric process solution of one-dimen-
sional fluid flow (1), two-dimensional (2) fluid flow and two-dimensional flow
through a porous material (3). For certain fourth-order partial differential equations,
the form (4) can be used. The CFL condition can be a very limiting constraint on
the time step Δτ. Numerical mathematics.
Info: [B114].
2.1.12 Courant Wave Number Cou
Cou5w2Δτ2
Δ2#
1
2; if Δ5Δx5Δy
w (m s21) � velocity of wave propagation;Δτ (s) � time step;Δ,Δx,Δy (m) � mesh
increment.
It expresses the numerical solution stability condition for wave propagation.
Numerical mathematics.
Info: [A11]
2.1.13 Courant�Friedrichs�Lewy Numerical Number CFL, NCFL, Cou
CFL5Δτk
w
ΔxwBL ð1Þ;
CFL5Δτk
wx
Δx1
wy
Δy1
wz
Δz
� �wBL ð2Þ
8 Dimensionless Physical Quantities in Science and Engineering
Δτ (s) � time step; k (m s21) � filtration coefficient; w, wx, wy, wz (m s21) � flow
velocity and its components; Δx, Δy, Δz (m) � space step; wBL (m s21) � saturation
velocity.
This number represents the numerical criterion to determine optimal time
step in a non-stationary, two-phase, one-dimensional (1) filtering flow or in a
three-dimensional (2) one. The stability condition is CFL# 1.
Info: [B114].
2.1.14 Crank�Nicolson Parameter μC
μC 5 0:5 ð1Þ; μC 5 0 ð2Þ; μC 5 1 ð3Þunj (�) � dimensionless-dependent variable; ΔFo (�) � Fourier difference number
(p.13); j (�) � geometric nodal point; n (�) � time level in the numeric solution.
It occurs in the Crank�Nicolson differential diagram for a one-dimensional
solution of the parabolic Fourier’s equation:
un11j 2 unj 5μCΔFo ðun11
j11 2 2un11j 1 un11
j21 Þ1ðunj112 2unj 1 unj21Þh i
:
In case relation (1) is valid, the Crank�Nicolson solution is stable unconditionally.
With relation (2) holding, the explicit solution (FTCS) is obtained. In the case of (3),
the implicit solution is gained.
Info: [C24].
John Crank (6.2.1916�3.10.2006), English mathematician.
Together with the mathematician Phyllis Nicolson, he
worked in the sphere of the numerical solution of second-
order partial differential equations � especially those of
unsteady heat conduction � and the solution thereof by the
finite differences method. They were engaged in the problem
of numerical solution instability and of the related choice
of optimal geometric and time steps. The Crank�Nicolson
method is stable numerically, but a simple system of linear
equations must be solved at every time level.
Phyllis Nicolson (21.9.1917�6.10.1968), English mathematician.
Together with the mathematician John Crank, she worked
on the solution of second-order partial differential equations
numerically. As a mathematician, she was engaged in the
magnetron theory and its interpretation.
9Physics and Physical Chemistry
2.1.15 Damkohler Numerical Number Danum
Danum 5kΔx
w
k (s21)� disintegration unit rate;Δx (m)� mesh increment; w (m s21)� flow velocity.
This number expresses the stability criterion in a digital process of the one-
dimensional modelling of a chemical conversion in fluid flow. It characterizes
the hydrodynamic influence in chemical reactions.
Info: [C98].
Gerhard Friedrich Damkohler (1908�1944), German physical chemist.
2.1.16 Dimensionless Heat Capacity NC
NC 5C
nR5
C
Nk
C (J K21) � heat capacity of a body; n (mol) � amount of matter in the body;
R (J mol21 K21) � molar gas constant; N (�) � number of molecules in the body;
k (J K21) � Boltzmann constant.
It expresses the body heat capacity in a dimensionless shape. Thermomechanics.
Info: [C123].
2.1.17 Eddington Number Ed
Ed5 1363 2256 � 1:5753 1079
It expresses the exact number of protons in the Universe, where 136 represents
the inverse value of the fine structure constant α (p. 13) as it could be stated by
measurement in the given time.
Info: [C43].
Arthur Stanley Eddington (28.12.1882�22.11.1944),
American physicist and astronomer.
Probably, he is the most prominent astrophysicist of the
twentieth century. He examined the stellar system interior. The
basic contribution consisted in the verification of gravitational
curvature influence on the bending of rays around the Sun.
He explained the pulsation of stars theoretically. He calculated
the Sun’s nucleus temperature in millions of Kelvin. He was
one of the first who defined Einstein’s relativity theory with
more precision.
10 Dimensionless Physical Quantities in Science and Engineering
2.1.18 Eddington�Dirac Number EdD
EdD � 1040
This hypothetical number follows from the fact that: (i) the ratio of the mutual
force of the electron and the proton equals 2.273 1039, (ii) the ratio of the elemen-
tary length (radius of the electron) to the radius of the Universe is 33 1040 and
(iii) the ratio of the elementary time (electron radius to light velocity) to the
Universe’s age is 63 1040. It represents only the mysterious dimensionless number
in the extent of 1040 approximately, which was presented by Dirac with respect to
digital relations between the microscopic and macroscopic scales and the effects
of various forces. Considering Dirac’s hypothesis of extensive numbers, obviously
this numerical relation between very different phenomena was given a deeper
cosmologic significance. At present, this idea is not set greater physical store
mostly.
Info: [C129].
Arthur Stanley Eddington (see above).
2.1.19 Energy Efficiency ηt
ηt 5W
E; ηtAh0; 1i
W (J) � mechanical work or energy released by the process; E (J) � quantity of work
or energy used as input to run process.
It is an important technical indicator for economic utilization of processes
and facilities. According to thermodynamic law, the efficiency ηt5 1 cannot be
reached.
Info: [C46].
2.1.20 Entropy Generation Number NS
NS 5L2T0EG
λðTw 2 T0Þ2;
where EG 5λT20
ðrxTÞ2 1 ðryTÞ2� �
1ηT0
1 ðryuÞ2
L (m) � characteristic length (wall thickness); T0 (K) � input fluid temperature;
EG (W m23 K21) � entropy change (volume density of heat flux) by tempera-
ture change 1 K; λ (W m21 K21) � wall thermal conductivity; Tw (K) � wall
11Physics and Physical Chemistry
temperature; η (Pa s) � dynamic viscosity of the fluid; rxT, ryT (K m21) � tempera-
ture gradient in the direction of x and y axis; ryu (s21) � velocity gradient in the
direction of y axis.
It characterizes the fluid entropy change in laminar streaming of viscous
incompressible fluid through an inclined canal with isothermic walls. It was
determined from the analysis of the second law of thermodynamics. Fluid
mechanics.
Info: [B76].
2.1.21 Feigenbaum Delta δ
δ5 limn-N
Tn 2 Tn21
Tn11 2 Tn
Tn (s) � value of the nth bifurcations period.
It characterizes the convergence velocity of the cascade redoubling period. It
is determined experimentally and has the value δ. 1. It is used in the dissipation
theory of non-linear systems in the phase transfer measurement � for example,
in electronic circuits, lasers, chemical reactions and in fluid mechanics � if
approaching the turbulent state.
For example, Feigenbaum showed that all non-linear dynamic systems, showing
periodic doubling, tend towards chaos and usually have a value of δ5 4.669.
Chaos theory.
2.1.22 Fermi’s Paradox K
K5 eTτ � 10433 106
T (year) � age of the Universe (T5 1010); τ (year) � specific time of exponential
development of our civilization (τ5 102 years).
This gigantic dimensionless number, exceeding the framework of theoretical
physics � see Eddington number Ed (p. 10)� characterizes the growth of technologi-
cal civilization during the Universe’s existence. This number is so large that the total
number of elementary particles in the Universe is very small as compared to it.
According to Fermi, ‘If there were any civilisations in the Universe, their spaceship
would have been in our Solar system long ago’. The absence probability of ‘space
miracles’ is 102433 106 in our Universe, or virtually zero. Unfortunately, nobody
has discovered them. Fermi’s paradox consists in the idea that our miracleless world
is fantastic and does exist.
Info: [C54].
12 Dimensionless Physical Quantities in Science and Engineering
2.1.23 Fine Structure Constant, Sommerfeld Fine-Structure Constant α
α5e2
4 π ε0h c
e (C) � elementary charge (1.602193 10219 C); ε0 (F m21) � permittivity of vacuum;
h (J s) � Planck constant h5 h (2π)21 (1.05458873 10234 J s); c (m s21) � speed
of light.
It is the basic dimensionless physical constant characterizing the electromagnetic
interaction intensity. It was introduced into physics by A. Sommerfeld in the year
1916. It is used in analyzing Feynman’s quantum electrodynamic diagrams. Its
exact value, determined on a physical basis is α215 137 but α215 137.0399976
if determined by experimental procedure.
Info: [A29].
Arthur Stanley Eddington (p. 10).
2.1.24 Fourier Difference Number ΔFo, Fomesh
ΔFo5Fomesh 5aΔτðΔxÞ2 ð1Þ; ΔFo, 0:5 ð2Þ;
ΔFo#1
4ð3Þ; ΔFo5
1
6ð4Þ
a (m2 s21) � thermal diffusivity; Δτ (s) � time step; Δx (m) � finite increase of
distance coordinate x.
In equation (1), it expresses the relation between the amount of time and the
geometric steps in the numerical solution of the parabolic Fourier equation. For
the explicit 1-D task, condition (2) is valid for solution stability. For the 2-D task,
condition (3) is valid; for the 3-D task, condition (4) is valid. Numerical mathematics.
The final difference method.
Info: [A10].
Jean Baptiste Joseph Fourier (p. 175).
2.1.25 Fresnel Number F
F5L2
λs
L (m) � characteristic size (radius) of the aperture; λ (m) � wavelength; s (m) �distance of the screen from the aperture.
13Physics and Physical Chemistry
It characterizes Frauenhofer diffraction equation (F{1) and that of Fresnel
(F$ 1). Optics.
Info: [C59].
Augustin Jean Fresnel (10.5.1788�14.7.1827), French
physicist and mathematician.
Due to his works in optics, he became one of the foun-
ders of light wave theory. He showed the reason for optical
diffraction, consisting in transversal light undulation. He
created the mathematical theory of refraction and polariza-
tion in anisotropic materials. From this theory, conical
refraction was predicted and discovered soon afterwards.
Joseph von Fraunhofer (6.3.1787�7.6.1826), German
physicist.
In the year 1814, he investigated the solar spectrum and
discovered dark spectral lines called the Fraunhofer lines.
He is well known for his work on light diffraction in systems
with small Fresnel numbers. This is called Fraunhofer’s
diffraction to honour him.
2.1.26 f-Stop Number Nf
Nf 5f
D
f (m) � focal length of the lens or mirror; D (m) � aperture diameter;
In optics, the Nf 5f2; f3; . . . is used currently. In film exposition, it is P~ 1
N2f
:Optics, film, photography.
2.1.27 Gay-Lussac Number Gc
Gc51
βΔT
β (K21) � coefficient of bulk expansion; ΔT (K) � temperature difference.
In the dimensionless form, it characterizes the relative thermal volume expansi-
bility of substances.
Info: [A29].
14 Dimensionless Physical Quantities in Science and Engineering
Joseph Louis Gay-Lussac (1778�1850), French chemist
and physicist.
For ideal gases, he expressed the law of gas pressure
dependence on temperature, so-called the Gay-Lussac law.
Later, this led to the introduction of the thermodynamic
temperature scale and to the formulation of the ideal gas
state equation. When only a 26-year-old chemist, he executed
many courageous high-altitude atmospheric measurements
by means of a balloon and with instruments he designed.
He studied terrestrial magnetism as well.
2.1.28 Geometric Coordinates X, Y, Z
X5x
L; Y 5
y
L; Z5
z
L
x, y, z (m) � dimensional coordinates; L (m) � characteristic length; S (m2) � surface
area; V (m3) � volume.
They express the ratio of the coordinate of a point in space to the characteristic
length of the system. In the dimensionless form, it characterizes the position of the
place M (X, Y, Z) in space. For a plate, the semi-thickness (symmetrical case) or
the thickness (unsymmetrical case) is usually chosen as the characteristic length.
In the case of a cylinder or a ball, the radius is chosen.
Sometimes for a general shape, it is convenient to choose the volume�surface
ratio of a body (module) L5V � S21. In the case of a plate, an unlimited cylinder
and a ball, the ratio 3:1.5:1 is obtained, which is also the ratio of mutually corre-
sponding process times according to the analytic heat transfer theory. However, the
relative length, determined in this manner, does not agree with the geometric coef-
ficient influence. Therefore, the generalized relative length L5KtV � S21 is used
sometimes where Kt is the relative shape coefficient. For a ball, an unlimited plate
and a cylinder, Kt5 1 holds, with Kt. 1 for other bodies.
Info: [A23].
2.1.29 Gravitational Coupling Constant αg
αg 5Gm2
e
hc5
me
mP
� �2
� 1; 7523 10245
G (N m2 kg22) � Newtonian constant of gravitation; me (kg) � electron mass; h� (J s) �Planck constant; c (m s21) � speed of light in vacuum; mP (kg)� Planck mass.
It represents a basic physical constant and characterizes the gravitational force
between typical elementary particles. It is related to gravitation as the fine structure
constant α (p. 13) is to electromagnetism and quantum electrodynamics.
Info: [C69],[C68].
15Physics and Physical Chemistry
2.1.30 Guldberg Number Gu
Gu5Tn
Tcrit
Tn (K) � thermodynamic temperature of saturation by pressure p5 105 Pa; Tcrit (K) �critical thermodynamic temperature.
This number expresses the relation between the thermodynamic saturation
temperature at an atmospheric isobar and the critical thermodynamic temperature.
GuAh0.37; 0.8i.Info: [A23].
Cato Maximilian Guldberg (p. 42).
2.1.31 Hadamard Number Hd
Hd53ηb 1 3ηf3ηb 1 2ηf
ηb, ηf (Pa s) � dynamic viscosity of fluid in bubble (b) and viscosity of ambient
fluid (f).
In the dimensionless form, it expresses the resulting dynamic viscosity of a
mixture of a fluid and bubbles contained therein.
Info: [A29].
Jacques Salomon Hadamard (8.12.1865�17.10.1963),
French mathematician.
He introduced the correct task concept in the partial differ-
ential equation theory. Hadamard’s matrices and the Hadamard
transformation are named for him, representing an example
of the generalized class of Fourier transformations. He edited
many publications, e.g. publications on geodesy and matrix
theory. After the WWII, his long and eventful life led to peace
activities and support for mathematicians all over the world.
2.1.32 Hamilton Numeric Number H
H5XNi.j
1
Rij
e2kRij 1XNi
R2i ;
where
Ri 5ri
rref; rref 5
ffiffiffiffiffiffiffiffiffiffiffi2q2
mεω2
3
r; k5
rref
lse
16 Dimensionless Physical Quantities in Science and Engineering
N (�) � number of the particles; k (�) � inverse dimensionless screening length;
Rij (�) � interparticle relative position of ith and jth particle; Ri (�) � relative
position of the ith particle; ri (m) � position of the ith particle; rref (m) � reference
position; q (C) � particle charge; m (kg) � mass of the particle; ε (F m2 1) �dielectric constant of the medium; ω (s21) � angular frequency of the particle;
lse (m) � screening length.
It characterizes the interaction of particles due to the Coulomb potential, depend-
ing on physical parameters and plasma background, and can have various forms.
Description by the Hamiltonian (H) starts from the isotropic potential between
particles and from the purely repelling and approximated potential. With k5 0,
the interaction between particles is purely the Coulomb potential. Mathematical
physics.
2.1.33 Kolmogorov Microscale λKol
λKol 5
ffiffiffiffiffiν3
ε4
rð1Þ; τ5
νε
ð2Þ
ν (m2 s21) � kinematic viscosity; ε (m2 s23) � speed of energy dissipation relative
to the mass unit; τ (s) � time.
It expresses the nominal length with which the viscous dissipation occurs in
three-dimensional turbulent flow. Together with the timescale (2), it characterizes
the turbulence beginning in a certain place in the flow. Physics. Fluid mechanics.
Info: [C75].
Andrey Nikolaevich Kolmogorov (25.4.1903�20.10.1987), Russian mathema-
tician (p. 104).
2.1.34 Lautrec Number for Fluid NL
NL 5L
δf; where δf 5
ffiffiffiffiffiffiffi2af
ω
r
L (m) � separated half-thickness of the plate; δf (m) � sound penetration in fluid
material; af (m2 s21) � thermal diffusivity of fluid; ω (s21) � angular frequency of
the acoustic oscillations.
It characterizes the thermoacoustic process originating in the thermal and acous-
tic energy transformation in a fluid. It expresses the influences in the acoustics, in
which the heat transfer and fluid entropy changes play an important role.
Thermoacoustics.
17Physics and Physical Chemistry
2.1.35 Lautrec Number for Solid NL
NL 5L
δf; where δs 5
ffiffiffiffiffiffiffi2af
ω
r
L (m) � plate half-thickness; δf (m) � sound penetration in solid material; as(m2 s21) � thermal diffusivity of material; ω (s21) � angular frequency of the
acoustic oscillations.
This number expresses the thermoacoustic process of acoustic wave penetration
into a material. Alternatively, it describes the thermal and acoustic energy transfor-
mation in a material. Thermoacoustics.
2.1.36 Lobachevsky Number Lo
Lo51
OA
ððOAÞ
dbmed
dblocdli ð1Þ; Lo5
1
SA
ððSAÞ
dbmed
dblocdSi ð2Þ
OA (m), SA (m2) � circumference and area of cross section; bmed, bloc (m) � mean and
local distance between isolines; dli (m) � elementary length; dSi (m2) � elementary
area.
In heat and mass transfer and in aero-hydrodynamics, it characterizes the isolines
shape. It represents the ratio, centred on the line of force, of the elementary distance
between the isolines to the local distance between two points of the same lines of
force. With the isolines and isoareas not changing their shape, the Lo number keeps
an equal value. In heat and mass transfer, it expresses the influence of a capillary
porous body on the heat and mass transfer.
Info: [A23],[A24].
Nikolay Ivanovich Lobachevsky (1.12.1792�24.2.1856),
Russian mathematician.
He founded non-Euclidean geometry, a geometry in which
Euclid’s fifth postulate is not true. He published a whole
range of books, starting with the Foundation of Geometry
(1835�1838) to Pangeometry (1855). Geometry with a con-
stant negative curvature is called Lobachevskian geometry.
2.1.37 Logarithmical Decrement Λ, ϑ
Λ5 lnsðtÞ
sðt1 TÞs(t) (m) � displacement of oscillation; t (s) � time; T (s) � time period of oscillation.
18 Dimensionless Physical Quantities in Science and Engineering
It is the logarithm of the ratio of the vibrating movement deviation s(t) to a
deviation in the time t1 T.
Info: [C28].
2.1.38 Lorentz Number Lo, L
Lo � ðΔUÞ2γΔT λ
ΔU (V) � voltage difference; γ (S m21) � specific electrical conductivity; ΔT (K) �temperature difference; λ (W m21 K21) � specific thermal conductivity.
It expresses the dependence between thermal and electrical conductivities
for metals. It starts with the Wiedemann�Franz law, according to which
both conductivities are proportional to the absolute temperature and depend
on the movement of free electrons. Physics. Thermomechanics. Electrical
engineering.
Info: [C79].
Hendrik Antoon Lorentz (p. 323).
2.1.39 Ludolph’s Number π
π5U
D
U (m) � circle circumference; D (m) � circle diameter.
It expresses the circumference-to-diameter ratio of a circle. Probably, it is
the oldest parametric criterion, known about the year 2000 BC, and approximated
by the Egyptians afterwards. Mathematics. Geometry.
Info: [C82].
Ludolph van Ceulen (28.1.1540�31.12.1610), Dutch math-
ematician and engineer.
In the year 1600, he was appointed as the first professor
of mathematics at Leyden University. He calculated the num-
ber π to 35 decimal places by applying the same methods as
Archimedes about 2000 years earlier. With this irrational
transcendental number, he confirmed that the classic circle
quadrature task is unsolvable. He wrote several works, of
which On the Circle (Van den Circkel) is the most popular.
The number is called Ludolph’s number after him.
19Physics and Physical Chemistry
2.1.40 Mass-to-Charge Ratio MZ
MZ5m
z
m (�) � atom (molecule) mass in atomic mass units; z (�) � atom (molecule)
electric charge.
This is an important quantity used in mass spectrometry, for example, in
depicting the ion signal dependence on this dimensionless parameter. In the case of
multi-degree mass spectrometry, this method can be applied to chemical structure
determination.
Info: [C86].
Joseph John Thomson (18.12.1856�30.8.1940), English
physicist. Nobel Prize in Physics, 1906.
He discovered the electron in cathode radiation and
presented the atom as having its own internal structure.
He clarified the properties of ions. He determined the mass-
to-charge ratio of the electron. He observed the first data
confirming the existence of isotopes. He received the Nobel
Prize for his research in electricity conduction in gas.
2.1.41 Mechanical Efficiency ηmech
ηmech 5P2
P1
5u F
P1
P2 (W) � work output; P1 (W) � mechanical advantage (work input); u (m s21) �speed; F (N) � force.
It expresses the effective input power ratio.
Info: [C88].
2.1.42 Mendeleev Number Me
Me5patm
pcrit
patm (Pa) � atmospheric pressure; pcrit (Pa) � critical pressure.
It expresses the atmospheric to critical pressure ratio on a static surface.
MeAh0.4303 1022; 0.4433 1022i.Info: [A23].
20 Dimensionless Physical Quantities in Science and Engineering
Dmitriy Ivanovich Mendeleev (7.2.1834�2.2.1907), Russian
chemist.
He assembled and published (1869) 63 elements,
unknown before, in the periodic table based on the atomic
number. Thus, he became the discoverer of the periodic law
of the elements, which was confirmed later by subsequent
discovery of other elements and by studying X-ray spectra
and quantum mechanics. He wrote about 400 published
works, concerning physics and technology, besides chemistry.
He wrote the textbook Foundations of Chemistry.
2.1.43 Moment of Inertia Ratio γ
γ5I
m r2
I (kg m2) � moment of inertia of the body; m (kg) � mass of the body; r (m) �radius.
The ratio γ represents the normalized dimensionless inertia moment.
Info: [C93].
2.1.44 Monin�Obukhov Length L
L52Rcpw3
gαK
R (kg m23) � air density; cp (J kg21 K21) � specific heat capacity; w (m s21) � air
velocity; g (m s22) � gravitational acceleration; α (K21) � coefficient of linear
thermal expansion; K (m kg K22) � Karman constant.
The length L corresponds to the measurement at which the Richardson
number Ri (p. 83) is Ri5 1 for the flow near a heated wall with free and forced
convections.
Info: [C95],[C94].
2.1.45 Neumann Numerical Number Neu
Neu5D1
ðΔxÞ2 11
ðΔyÞ2� �
Δτ#χ
D (m2 s21) � diffusion coefficient; Δx, Δy (m) � grid spacing in both coordinate
axes; Δτ (s) � time step; χ (�) � time stepping parameter.
21Physics and Physical Chemistry
This number expresses the stability criterion for a numerical solution, for
example, of two-dimensional diffusive flow (Neu# 1). Together with the Courant
numerical number Cou (p. 8), it characterizes the oscillating convection flow
through porous material. Numerical mathematics. Two-phase flow.
Info: [B44].
John von Neumann (28.12.1903�8.2.1957), American
mathematician of Hungarian origin.
He was one of the greatest mathematicians of the twentieth
century. He substantially influenced the development of the
structural and functional principles of computers. His scientific
sphere of interest was extraordinarily wide. He participated in
elaborating theoretical foundations for atomic energy utiliza-
tion. He founded the theory of sets, quantum theory and theory
of games, too, which represent important areas of mathematics
and economics.
2.1.46 Normalized Frequency, V number V
V 52πrλ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin21 2 n22
q
r (m) � core radius of optical fibre; λ (m) � wavelength in vacuum; n1 (�) � maxi-
mum refractive index of the core; n2 (�) � refractive index of the homogeneous
cladding.
It is important in telecommunication, to quantify the optical fibres especially,
to determine the actual to relative or nominal frequencies ratio. It is applied in
optoelectronics and telecommunications. For single-mode operations it is V, 2 and
for multi-mode ones V. 5. Optoelectronics, physics and telecommunications.
Info: [C97].
2.1.47 Optical Thickness τ, h
τ5 k L ð1Þ; Ix;λ 5 I0;λ exp 2x
τ
� �ð2Þ
k (m21) � monochromatic absorption coefficient; L (m) � characteristic length,
thickness; Ix,λ, I0,λ (�) � luminous intensity at the depth x and on the material
surface at the wavelength λ; x (m) � depth below the surface; λ (m) � wavelength.
It expresses the material depth at which the radiation intensity (of various wave-
lengths, e.g. of visible light) of a given frequency is reduced by a coefficient 1e.
In the optical thickness depth, about 23of the radiation is absorbed. Physics. Optics.
Atmospheric radiation.
Info: [C99].
22 Dimensionless Physical Quantities in Science and Engineering
2.1.48 Peclet Difference Number Penum
Penum 5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�wx
Δx
�2
1wy
Δy
� �2s
Dxx
ðΔxÞ2 1Dyy
ðΔyÞ2 2Dxy
Δx Δy
� �21
wx, wy (m s21) � flow velocity components in pores; Δx, Δy (m) � mesh increments
in x and y axes; Dxx, Dyy (m2 s21) � components of the dispersion coefficient;
Dxy (m2 s21) � transverse dispersion coefficient.
This number expresses the stability criterion of a numerical solution, for example,
of an underground water flow through dispersive surroundings. Physically, it is about
dynamic and dispersive forces acting. Numerical mathematics. Two-phase flow.
Info: [C98].
Jean Claude Eugene Peclet (p. 180).
2.1.49 Peclet Grid Number Pemesh
Pemesh 5w Δx
D, 2 ð1Þ;
Pemesh 5maxwx;maxΔx
D;wz;maxΔz
D
� �ð2Þ
w (m s21) � velocity; wx,max, wz,max (m s21) � maximum value of the velocity
components; Δx, Δz (m) � mesh increments in the x and z axes; D (m2 s21) � dif-
fusion coefficient.
Equation (1) is the stability criterion of a numerical process solution for one-
dimensional flow or transfer. Equation (2) is the limiting condition to choose a dif-
ference step for solution a two-dimensional diffusive process with a moving mem-
ber. Numerical fluid mechanics. Mathematics.
Info: [A47],[B77],[B44].
Jean Claude Eugene Peclet (p. 180).
2.1.50 Pitzer Number Pi
Pi � P0:7 5p
pref5 0:7 at Tred 5
T
Tref5 0:7
p, pref (Pa) � pressure, reference pressure; T, Tref (K) � temperature, reference
temperature; P0.7 (�) � reduced pressure; Tred (�) � reduced temperature.
It serves to determine the reduced steam pressure at reduced temperature.
Thermodynamics.
23Physics and Physical Chemistry
Kenneth Sanborn Pitzer (6.1.1914�26.12.1997), American
theoretical chemist.
Pitzer was the founder of modern theoretical chemistry.
He used quantum and statistical mechanics to explain the
thermodynamic and conformational properties of molecules,
and he pioneered quantum scattering theory for describing
chemical reactions at the most fundamental level. He also
made contributions to relativistic effects in chemical bonding
and the theory of fluids and electrolyte solutions.
2.1.51 Planck Number, Radiation Parameter Pl
Pl5λβ
4n2σ0T3ref
λ (W m21 K21) � intrinsic ambient thermal conductivity; β (m21) � parameters
vector of the absorption coefficient; n (�) � refractive index; σ0 (W m22 K24) �Stefan�Boltzmann constant; Tref (K) � reference temperature.
This number expresses radiation effects on a convective boundary layer with a
uniformly heated surface. An increasing Planck number leads to the reduction of
the thickness of the layer. On the contrary, an increasing temperature increases the
thickness of the layer. For gases, the Pl number depends on the Prandtl number Pr
(p. 197) and the temperature and is in the range of 100�150. For water steam with
the temperature 100�500�C and Pr5 1, it is PlAh30; 200i. It is analogous to the
radiation number (2.) N (p. 211).
Info: [B3].
Max Karl Ernst Ludwig Planck (23.4.1858�4.10.1947),
German physicist. Nobel Prize in Physics, 1918.
He was engaged in research related to the radiation of the
perfect black body and discovered the Planck radiation law,
which determines the dependence of the volume radiation
density of this body on the radiation frequency and body tem-
perature. This law is based on the hypothesis of discontinuous
radiation of electromagnetic energy in doses. For his discovery
of the electric quanta, he won the Nobel Prize. He was one of
the first who accepted and evolved Einstein’s relativity theory.
2.1.52 Porosity Φ, ϕ, p, n, e
Φ5Vf
Vt
Vf (m3) � volume of void space filled up by fluid; Vt (m
3) � total or bulk volume
of material including the solid and void components.
24 Dimensionless Physical Quantities in Science and Engineering
It characterizes the ability of a porous material to receive a fluid. The porosity
ΦAh0; 1i or ΦAh0; 100%i. Fluid mechanics. Geophysics. Sedimentation. Drying.
Info: [C105].
2.1.53 Reduced Boiling Temperature Nrbt
Nrbt 5Pi
4RePr
D
L
D (m) � circular pipe diameter; L (m) � characteristic length; Pi (�) � Pitzer number
(p. 23); Re (�) � Reynolds number (p. 81); Pr (�) � Prandtl number (p. 197).
It expresses the reduced normal boiling temperature. It is used for steam transfer
through a pipeline. Thermodynamics.15
2.1.54 Reduced Pressure pred
pred 5p
pcrit
p (Pa) � liquid pressure; pcrit (Pa) � critical pressure.
It serves to predict the properties (p, v, T) of gases from critical values.
Thermodynamics.
Info: [C110].
2.1.55 Reduced Volume vred
vred 5v
vcrit
v (m3 kg21) � gas or vapour specific volume; vcrit (m3 kg21) � gas or vapour spe-
cific volume at critical state.
It serves to compare and predict the properties (p, v, T) of gases from critical
values. Thermodynamics.
2.1.56 Relative Atomic (Molecular) Mass Ar, Mr
Ar 5mA
uð1Þ; Mr 5
mM
uð2Þ
mA (kg) � atomic mass; mM (kg) � molecular mass; u (kg) � atomic mass unit
(1.660573 10227 kg); mM (kg mol21) � molar mass.
25Physics and Physical Chemistry
The relative atomic mass of an atom is a dimensionless number indicating
how many times the atomic mass is greater than the atomic mass unit. Similarly,
the relative molecular mass of a molecule is a dimensionless number indicating
how many times the molecular mass is greater than the atomic mass unit. The
relative molecular mass can be calculated as a sum of all relative atomic masses of
all atoms of a molecule. The atomic mass unit is defined as 1/12 of the mass of the
carbon isotope nuclide 126C;
u5m 12
6C 12
:
The molar mass gives the 1-mole quantity substance mass. The pure substance
molar mass can be calculated as a quotient of its mass and its mass quantity or as a
product of the atom (molecular) mass and the number of atoms (molecules) in one
mole of the substance,
Mm 5mMNA5MruNA 5 0:000999995Mr
which is valid for the molar mass in the kg mol21. For the molar mass in g mol21
units, the relation Mm5Mr is valid. Therefore, the relative molecular (atomic)
mass corresponds numerically to the molar mass expressed in the g mol21 units.
For example, for the relative molecular mass of carbon dioxide CO2, one obtains
(12123 15.9994)5 43.9988. Hence, the CO2 molecule has the molecular mass of
43.9988u and the molar mass of 43.9988 g mol21.
The relative molecular (atomic) masses and the molar masses can be used espe-
cially in stochiometric calculations.
Info: [C91].
2.1.57 Relative Humidity ϕ
ϕ5φφv
φ (kg kg21) � specific absolute humidity; φv (kg kg21) � saturation-state-specific
absolute humidity.
It expresses the relation of the gas humidity to the saturation-state humidity, spe-
cifically under the same temperature and pressure. If saturated, the gas has the dew
point temperature.
Info: [A24].
2.1.58 Riedel Number Rie
Rie5T
p
dp
dT
� �crit
T (K) � temperature; p (Pa) � pressure.
26 Dimensionless Physical Quantities in Science and Engineering
It expresses the temperature�pressure relation of fluids under critical
conditions.
2.1.59 Shape Equivalence of System A
A5S
Sekv
S (m2) � heat transfer surface of warmed-up or cooled-down body; Sekv (m2) �surface area of equivalent body by the same volume; O, Oekv (m) � cross section
perimeters of the body; Vekv (m3) � sphere equivalent volume.
It characterizes the relation between the geometric shape of a body and its
thermal field. It is used to replace geometrically complicated bodies with an unlim-
ited finite thickness plate, a cylinder or a ball. For an infinitely long cylinder,
A5 S S21ekv5O O21
ekv, and for a ball
A5Sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
36πV2ekv
3p
For example, for a boundary condition, it is αekv 5α S S21ekv.
Info: [A23].
2.1.60 Size Factor x
x52πaλ
5 ka
a (m) � particle size; λ (m) � wavelength; k (m21) � wave number.
It expresses the relation between the size of a moving particle, the wavelength
and the wave number.
Info: [C121].
2.1.61 Snellen Number Sn
Sn5a
bð1Þ; Sn5
a
b5 2913 1026
a
hð2Þ
a (m) � distance at which an object can be recognized; b (m) � distance at which the
same object can be seen at a subtended angle of 1/60� to the eye; h (m) � object size.
It expresses the perfection of sight. For perfect sight it is Sn5 1 and for imperfect
sight it is Sn, 1. Optics. Ophthalmology.
Info: [A43].
27Physics and Physical Chemistry
Herman Snellen (19.2.1834�18.1.1908), Dutch ocularist
and physicist.
Above all, he was engaged in the physiology of the
eye. He created a very complex body of work in this special
sphere. Together with F.C. Donders he did great numbers
of experiments and participated in the development of oph-
thalmological instruments. He focused on corrective eye
glasses calculations. The visual Snellen chart, consisting of
variously sized letters, is still used to determine the quality
of sight all over the world.
2.1.62 Sound Emissivity NI
NI 5Ia
Ii
Ia, Ii (W m22) � intensity of absorbed (a) and incident (i) sound.
It expresses the ratio of the absorbed sound intensity to that of the incident
sound.
Info: [A24].
2.1.63 Sound Pressure Level L
L5 20 logp
p05 10 log
I
I0
p (Pa) � acoustic pressure; p0 (Pa) � reference pressure (p05 23 1025 Pa);
I (W m22) � acoustic intensity; I0 (W m22) � threshold acoustic intensity.
It expresses the logarithm of the acoustic-to-reference pressure ratio. Acoustics,
noise measurement.
2.1.64 Specific Heat Ratio k, γ
k5cp
cv
cp (J kg21 K21) � specific heat capacity at constant pressure; cv (J kg21 K21) �specific heat capacity at constant volume.
It expresses the ratio of the heat capacity at constant pressure to the heat capac-
ity at constant volume. Alternatively, it expresses the ratio of the enthalpy to the
internal energy. It characterizes thermodynamic relations in compressible fluid
flow. It is called also the adiabatic criterion, Poisson’s constant or the adiabatic
exponent. For air, it is γ5 1.4. Physics. Mechanics. Aerodynamics.
28 Dimensionless Physical Quantities in Science and Engineering
Info: [C108].
Simeon Denis Poisson (p. 143).
2.1.65 Strehl Ratio S
S5 expð22ð2πσÞ2Þ
σ (�) � root-mean-square deviation of the wavefront measured in wavelengths.
The Strehl ratio is expressed by the relation of the focus intensities peak in
deviated and ideal points of diffraction. Optics, light interference and diffraction.
Info: [C125].
2.1.66 Surface Elasticity Number NSE
NSE 52CS
DS
L
η@σ@CS
CS (K s21) � surface concentration of surfactant in undisturbed state; DS (m2 s21) �surface diffusivity; L (m) � film thickness; η (Pa s) � dynamic viscosity;
σ (N m21) � surface tension.
This number expresses the surface elasticity of solutions in diffusive mass
transfer.
Info: [A29].
2.1.67 Surface Tension Number NST
NST 5η2
hσR
η (Pa s) � dynamic viscosity; h (m) � ratio of surface area to perimeter; σ (N m21) �surface tension; R (kg m23) � density.
It characterizes the surface tension in mass transfer.
Info: [A31].
2.1.68 Surface Viscosity Number NSV
NSV 5ηSη h
ηS (N s m21) � surface viscosity; η (Pa s) � dynamic viscosity; h (m) � film depth
(thickness).
29Physics and Physical Chemistry
This number is used if there is a convective transfer zone in a fluid layer with
a surface-active substance.
Info: [A29].
2.1.69 Temkin Number Es
Es5
ffiffiffiS
pffiffiffiffiV3
p 5LS
LVð1Þ; Es5
OffiffiffiA
p ð2Þ
S (m2), V (m3) � surface and volume of thermal system; LS, LV (m) � characteristic
dimensions; O (m), A (m2) � perimeter and cross-sectional area.
It characterizes the body shape influence on its thermal field. However, in
comparing diverse bodies, the difference of physical parameters is not considered.
For limited bodies, expression (1) is valid and so is expression (2) for in-one-direction
non-limited bodies which have two finite shapes.
The lengths LS and LV depend on the prolongation and shape of the body. To
consider also the body size in the Es number, the auxiliary parameter Es0 5 L L21Vis used.
In general, the Es number is not a similarity criterion because the set of bodies
corresponds to its one value. However, from the equality of the criteria, no conclu-
sions can be made about their physical similarity even if their shapes are subject to
the isomorphism condition. The mutually unambiguous relation between the body
shape and the Es criterion can be determined if the body shape and size are the
functions of two physical dimensions only. For example, Es5 2.35 is valid for a
cylinder of a radius r and height 2r. For a ball of radius r, Es5 2.21 and Es5 2, 51
for a cone of the height of rffiffiffi3
pand a base of a radius r.
With the same values of the Biot number Bi (p. 173) and Fourier number Fo
(p. 175), the increasing of the number Es leads to faster cooling of the body, to
decreasing its surface temperature and to increasing the temperature non-uniformity.
The influence of the Es number on the temperature field of bodies, other than convex
ones, appears most expressively. It is also called the shape criterion or the geometric
cooling down criterion.
Info: [A23].
A.G. Temkin.
2.1.70 Timescale Number Nτ, Go
Nτ 5wτL
ð1Þ; Go5τDτR
ð2Þ
w (m s21) � speed; τ (s) � time scale; L (m) � length scale; τD (s) � dynamic time;
τR (s) � radiation time.
30 Dimensionless Physical Quantities in Science and Engineering
In expression (1), it is the dimensionless expression of time of a moving body
or fluid. The controlling timescale number for the atmosphere or the interior
of non-rotating planets (2) is called the Golitsyn number Go (p. 397). Dynamic pro-
cesses. Astronomy.
Info: [C134].
2.1.71 Tortuosity τ
τ5l
L
l (m) � mean length of the free path; L (m) � characteristic length.
It characterizes permeable substances and expresses the ratio of a true
mean length of free path, for example, of the flow, to the characteristic length
(thickness) of the substance. Besides the application in fluid mechanics in flow
through crystallic or porous materials, it is used in hydrogeology, in physical
technologies, in mathematics in functional minimizing by cubic splines and in
solving curvature problems. For example, in physical technologies, it expresses the
ratio of the molecule path, which must pass through the deposited layer tortuously,
to the thickness of the layer. Mathematics. Physics. Fluid mechanics. Geophysics.
Physical technology.
Info: [C135].
2.1.72 Tzou Number, Stability and Convergency Criterion Tz
Tz 5Δxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2aΔtð2τT 1ΔtÞ2τq 1Δt
s ð1Þ;
Tz 5Δx
vΔt
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11
Δt
2τq
s; v5
ffiffiffiffiffia
τq
sð2Þ
Δx (m) � mesh increment; a (m2 s21) � thermal diffusivity; Δt (s) � time step;
τT (s) � thermalization time; τq (s) � relaxation time; v (m s21) � thermal wave
speed.
For the finite differences method it determines the stability and convergence zone
of a solution if Tz$ 1. In the case of unbalanced heat propagation with the defined
relaxation time τq and the thermalization time τT, the criterion is in form (1). In the
case of wave-diffusive heat propagation (τT5 0), the criterion can be expressed in
form (2). In the case of balanced diffusive heat propagation (τT5 τq), the criterion
is reduced on the Crank�Nicolson parameter (p. 9).
Info: [A46],[B116].
31Physics and Physical Chemistry
2.1.73 Van der Waals Number (1.) Wa
Wa5rRcritTcrit
pcrit5
rTcrit
pcritvcrit
r (J kg21 K21) � specific gas constant; Rcrit (kg m23) � substance density at critical
state; Tcrit (K) � critical thermodynamic temperature; pcrit (Pa) � critical pressure;
vcrit (m3 kg21) � critical specific volume.
It is the fundamental criterion of thermodynamics. It expresses the state law of
fluids for a critical point and characterizes one substance or a whole group thereof in
accordance with their thermodynamical similarity. For various substances, its value
is various in general. However, in van der Waal’s equation its value is Wa5 8/3, but
it is WaAh3.3; 4.5i alternatively. It is called the critical coefficient as well.
Info: [A23].
Johannes Diderik van der Waals (23.11.1837�9.3.1923),
Dutch physicist. Nobel Prize in Physics, 1910.
He was engaged in research in gaseous and liquid states
of a mass. He started from the idea that the ideal gas state
equation can be deduced from the kinetic gas theory, pro-
vided the volume of molecules and the force between them
is neglected. With it, he succeeded in creating a new theory.
Subsequently, in the year 1881, he formulated a new state
equation named the van der Waals equation, the validity of
which he extended to all substances.
2.1.74 Van der Waals Number (2.) Wa
Wa5A
kT
A (J) � Hamaker constant; k (J K21) � Boltzmann constant; T (K) � absolute
temperature.
It characterizes the ratio of the van der Waals interaction energy to the heat
energy of particles. Thermodynamics of two-phase fluids. Filtration. Deposition.
Info: [A42].
Johannes Diderik van der Waals (see above).
2.1.75 Void Fraction ε
ε5Va
Vm
Va (m3) � volume of void space; Vm (m3) � total or bulk volume of material
including the solid and void components.
32 Dimensionless Physical Quantities in Science and Engineering
It expresses the ratio of the void space volume in the material to the total material
volume. Physics. Fluid mechanics. Flow through a porous material. Drying.
2.1.76 Wave Number K, k
K5L
λð1Þ; K5
2πLλ
5ωLa
ð2Þ; K5Krefn ð3Þ
L (m) � characteristic length; λ (m) � wavelength; ω (s21) � angular frequency;
a (m s21) � sound velocity; Kref (�) � wave number in the reference medium;
n (�) � medium’s index of refraction.19
In a dimensionless shape it expresses the wave length. The expressions (1) and
(2) are different forms only. For example, the expression (3) is used with electro-
magnetic waves. In acoustics, it is called the substance wave number (medium
wave number).
Info: [A2],[B7],[C139].
2.1.77 Zhukovsky Shape Number Ju
Ju5A
16Es2Es20 ð1Þ; Julam 5 32LoΨ ð2Þ;
Jurel 51
64Julam ð3Þ
Es, Es0 (�) � Temkin number and parameter (p. 30); A (�) � shape equivalence
of system (p. 27); Lo (�) � Lobachevsky number (p. 18).
It expresses the influence of clear area on the hydraulic resistance for fluid flow in
non-circular canals and pipelines. By using the Lobachevsky number Lo (p. 18) and
the non-uniformity criterion of the rate field Ψ5wmaxw21, the Zhukhovski laminar
number (2) is obtained. The expression (3) is the relative shape criterion. The corre-
spondence between the expressions (1) and (3) is given by various modifications of
Hagen�Poiseuille’s law for the laminar flow in a circular, cross-sectional tube.
Info: [A23],[A33].
Nikolay Jegorovich Zhukovsky (17.1.1847�17.3.1921),
Russian physicist and mechanicist.
He founded the Russian school of hydromechanics and
aerodynamics and is called the father of Russian aeronautics.
He was engaged in improving the wing profile (Zhukhovski’s
profile) and published two works (1906) in which he introduced
a mathematical expression for wing lift (Kutta�Zhukhovski’s
theorem). He devoted himself also to high-velocity aerodynam-
ics and to hydraulic shocks. In mathematics, he is known thanks
to his conformal mapping in a complex plane, which is known
as Zhukhovski’s transform.
33Physics and Physical Chemistry
2.2 Physical Chemistry
In physical chemistry, the dimensionless quantities mostly express the physical
and chemical conversions in homogeneous or heterogeneous processes or systems.
The focus is the non-isothermic or isothermic processes such as burning and
combustion, dynamics of flame propagation, baking, and heat and mass transfers
with chemical or phase reactions. Further, there are mixing, drying and corrosion
processes and physical and chemical processes in chemical reactors, mass diffusion
processes in catalyzers, granulation processes, dissociation of electrolytes in solu-
tions, separation of solid particles in solutions and others.
The Arrhenius, Damkohler, Frank-Kamenetskii, Guldberg�Waage, Knudsen
diffusion and Prater numbers are among the best known dimensionless numbers.
2.2.1 Arrhenius Number (1.) Energetic Ah1
Ah1 5E
RTð1Þ; Ah1 5U21 5
E
RTNð2Þ
E (J mol21) � molar activation energy; R (J mol21 K21) � molar gas constant;
T (K) � absolute temperature; TN (K) � ambient temperature.
This number expresses the activation-to-potential energy ratio of a fluid. It
characterizes the energetic conditions in a chemical�thermal reaction. For example,
it relates to the ability of a fuel to self-ignite. It is used, for example, to describe the
heat transfer in combustion chambers. Usually, the inverse value of equation (2) is
called the combustion parameter U (p. 35).
Info: [A29],[A33],[B17].
Svante August Arrhenius (19.2.1859�2.10.1927), Swedish
physical chemist. Nobel Prize in Chemistry, 1903.
He was engaged in chemical reaction description and
discovered the Arrhenius velocity law, which describes the
speed of chemical reactions. In addition, he elaborated on
the theory of electrolytes and made clear the electrical
conductivity of ion solutions. He received the Nobel Prize
for his electrolytic dissociation theory.
2.2.2 Arrhenius Number (2.) Velocity Ah2
Ah2 � Bd � Pem 5wL
Da
w (m s21) � speed; L (m) � characteristic length; Da (m2 s21) � axial diffusivity;
Bd (�) � Bodenstein number (p. 244); Pem (�) � Peclet mass number (p. 258).
34 Dimensionless Physical Quantities in Science and Engineering
It characterizes the mass diffusion process in chemical reactors. It is the measure
of the convective mass transfer related to the molecular transfer in flowing sur-
roundings. It expresses the degree to which an investigated system approaches an
idealized model of the reacting flow structure. With Ah2-N, it relates to a reactor
or system with total separation. With Ah2-0, it relates to a completely mixing sys-
tem. In intermediate models, N.Ah2. 0 is valid. It concerns diffusive, combined
or cellular models. Often, it is called the Bodenstein number Bd (p. 244). In
essence, it is the modified Peclet mass number Pem (p. 258).
Info: [A33].
Svante August Arrhenius (see above).
2.2.3 Arrhenius Number (3.) Time Ah3
Ah3 � Fo21ch 5uL2
RDa
5L2
Daτch
u (kg m23 s21) � chemical reaction rate; L (m) � characteristic length; R (kg m23) �density; Da (m2 s21) � axial diffusivity; τch (s) � chemical reaction time;
Foch (�) � Fourier chemical number (p. 39).
In the dimensionless shape, its reciprocal value characterizes the chemical reaction
time. Alternatively, it is also the Fourier chemical number Foch (p. 39). Sometimes it
is called the diffusion contact number Ko (p. 38), namely in a mononuclear reaction
in a stationary process with a simple conversion.
Info: [A33].
Svante August Arrhenius (see above).
2.2.4 Burnout Parameter γ
γRT2
N
E
cV RQ Ci
R (J mol21 K21) � molar gas constant; TN (K) � ambient temperature; E (J mol21) �molar activation energy; cV (J kg
21 K21)� specific heat capacity; R (kg m23)� density;
Q (J)� heat reaction effect; Ci (m23)� reagent numerical molecule concentration.
It characterizes the burnout in fuel combustion.
Info: [A23].
2.2.5 Combustion Parameter U
U � Ah211 5RTN
E
R (J mol21 K21) � molar gas constant; TN (K) � ambient temperature; E (J mol21) �molar activation energy; Ah1 (�) � Arrhenius number (1.) energetic (p. 34).
35Physics and Physical Chemistry
This parameter characterizes the temperature originating in a thermal system as
a chemical reaction result. See the Arrhenius number (1.) energetic Ah1 (p. 34).
Info: [A23].
2.2.6 Chemical Reaction Rate Γ
Γ 5 δγη expΘ
11UΘ
� �
δ (�) � Frank-Kamenetskii number (p. 35); γ (�) � burnout parameter (p. 40); η (�)
� critical reagent concentration; Θ (�) � reagent temperature (p. 47); U (�) �combustion parameter (p. 35).
It characterizes the reaction kinetics in a physical and chemical conversion
process.
Info: [A23].
2.2.7 Damkohler Number Da
Da5τDτR
τD (s) � characteristic diffusion timescale; τR (s) � characteristic reaction timescale.
It expresses the relation of diffusion and relaxation times in chemical processes.
It is used for turbulent combustion. With Da{1, the turbulence is much faster than
the chemical reaction.
Info: [A23],[A33].
Gerhard Friedrich Damkohler (1908�1944), German physical chemist.
2.2.8 Damkohler Number (1.) Hydrodynamic Da1
Da1 5uL
wRið1Þ; Da1 5
τcτr
ð2Þ
u (kg m23 s21) � chemical reaction rate; L (m) � characteristic length; w (m s21) �flow velocity; Ri (kg m
23) � mass concentration; τc, τr (s) � chemical conversion
and reaction time.
This number expresses the ratio of chemical conversion rate to that of the fluid
flow. Alternatively, it is the ratio of the number of moles entering the reaction to
that of moles induced by the fluid flow. It characterizes the hydrodynamic influence
in chemical reactions. Alternatively, together with other Damkohler numbers, it
characterizes the reaction time in general with transport phenomena.
Info: [A2],[A26],[A29],[B11].
Gerhard Friedrich Damkohler (see above).
36 Dimensionless Physical Quantities in Science and Engineering
2.2.9 Damkohler Number (2.) Diffusion Da2
Da2 5u L2
D Ri
u (kg m23 s21) � chemical reaction rate; L (m) � characteristic length; D (m2 s21) �diffusivity; Ri (kg m
23) � mass concentration.
It characterizes the diffusion relations in chemical reactions. It expresses the
ratio of the chemical conversion rate in a fluid flow to the molecular mass diffusion
rate; alternatively, it is the ratio of the number of reacting moles in a fluid flow to
the number of moles brought by the molecular diffusion.
Info: [A2],[A23],[A26],[A29],[B11].
Gerhard Friedrich Damkohler (see above).
2.2.10 Damkohler Number (3.) (1. Heat) Da3
Da3 5huL
cpRwT
h (J kg21) � specific enthalpy; u (kg m23 s21) � chemical reaction rate; L (m) �characteristic length; cp (J kg
21 K21) � specific heat capacity; R (kg m23) � density;
w (m s21) � flow velocity; T (K) � temperature.
This number expresses the ratio of the heat liberated in a chemical reaction in
the fluid flow to that transferred by convection. It characterizes the heat transfer by
convection in chemical reactions in fluids.
Info: [A2],[A23],[A26],[A29],[B11].
Gerhard Friedrich Damkohler (see above).
2.2.11 Damkohler Number (4.) (2. Heat) Da4
Da4 � Th2 5huL2
λTð1Þ;
Da4 � PoV � Os � Th2 � huL2
λTrefð2Þ
h (J kg21) � specific enthalpy; u (kg m23 s21) � chemical reaction rate; L (m) �characteristic length; λ (W m21 K21) � thermal conductivity; T (K) � temperature;
Tref (K) � reference temperature; Th (�) � Thiele modulus (1.) (p. 267); PoV (�) �Pomerantsev heat number (p. 181); Os (�) � Ostrogradsky number (p. 179).
It expresses the ratio of the heat liberated by a chemical reaction to that trans-
ferred by conduction. It characterizes the heat transfer process in chemical reactions.
Substantially, the matter is the Pomerantsev heat number PoV (p. 181), also
37Physics and Physical Chemistry
called the Ostrogradsky number Os (p. 179) in Russian literature. Its second root
corresponds to the Thiele modulus (1.) Th (p. 267).
Info: [A2],[A23],[A26],[A29],[B11].
Gerhard Friedrich Damkohler (see above).
2.2.12 Damkohler Number (5.) Hydrodynamic Da5
Da5 � Re5w L
vð1Þ;
Da2
Da1Da55 Sc ð2Þ; Da4
Da3Da55Pr ð3Þ
w (m s21) � flow velocity; L (m) � characteristic length; ν (m2 s21) � kinematic
viscosity; Re (�) � Reynolds number (p. 81); Sc (�) � Schmidt number (p. 263);
Pr (�) � Prandtl number (p. 197); Da1 (�) � Damkohler number (1.) hydrody-
namic (p. 36); Da2 (�) � Damkohler number (2.) diffusion (p. 37); Da3 (�) �Damkohler number (3.) (1. heat) (p. 37); Da4 (�) � Damkohler number (4.) (2.
heat) (p. 37).
For flowing fluid, it expresses the inertia-to-friction ratio. It characterizes the
hydrodynamic influence of the inertia and friction forces of the streaming fluid in
a chemical reaction. The relation of the Schmidt number (p. 263) and the Prandtl
number (p. 197) to the Damkohler numbers (p. 36, 38) is obvious from the expres-
sions (2) and (3).
Info: [A4],[A23],[A26],[A29],[B11].
Gerhard Friedrich Damkohler (see above).
2.2.13 Diffusion Contact Number Ko
See Arrhenius number (3.) time Ah3 (p. 35).
2.2.14 Djakonov Number (1.) Contact Dj, Ko
Dj5 k1ϕ1ðckÞτ5ττr
k1 ϕ1 (ck) (s21) � unit speed at a given concentration; k1 (�) � virtual speed constant
of direct reduction process; ck (�) � concentration; τ (s) � contact time; τr (s) �decay time.
Together with the thermodynamic equilibrium of chemical reduction Nreac
(p. 48), it expresses the general similarity condition for the physical and chemical
conversion process. It is the measure of the contact-to-decomposition time ratio
of exiting products of the chemical conversion. The contact time means the time
during which the corresponding mixture composition is in the reaction zone.
Info: [A13].
38 Dimensionless Physical Quantities in Science and Engineering
German Konstantinovich Djakonov (11.6.1907�1953),
Russian physical chemical engineer.
His work represents a new dimension in complex process
modelling. He was engaged especially in the theory and
methods of modelling physical and chemical processes and
conversions. He solved the problem of thermodynamic equi-
librium. From the analysis of all gained similarity conditions
and criteria, he determined that they cannot be fulfilled at
the same time. Therefore, he was engaged in approximate
modelling and created an original method of solution.
2.2.15 Djakonov Number (2.) Thermodynamic Dj, Po
Dj5k2ϕ2ðckÞk1ϕ1ðckÞ
k2 ϕ2 (ck) (s21) � unit rate of indirect change; k1 ϕ1(ck) (s
21) � unit rate of direct
change; k1, k2 (�) � virtual rate constant of direct and indirect reaction; ck (�) �critical concentration.
It expresses the thermodynamic equilibrium in the physical and chemical con-
version process. It represents the return-to-direct conversion rates ratio. Together
with the Djakonov number (1.) contact Dj (p. 38), it represents the condition for
the general physical and chemical conversion.
Info: [A13].
German Konstantinovich Djakonov (see above).
2.2.16 Djakonov Number (3.) Thermodynamic Di
Di5q
n
q (m3 s21 m23) � specific volume flux; n (s21) � rotational frequency.
It expresses the ratio of the substance, mixed in a unity reaction volume during
a time unit, to the amount of the substance conveyed into a reacting volume. It
characterizes the physical and chemical conversion process in the mixing of fluids.
Alternatively, it can also be expressed as the ratio of the substance conversion rate
to the rate of its feeding into the chemical fluid mixing facility.
Info: [A13].
German Konstantinovich Djakonov (see above).
2.2.17 Fourier Chemical Number Foch
See Arrhenius number (3.) time Ah3 (p. 35).
Jean Baptiste Joseph Fourier (p. 175).
39Physics and Physical Chemistry
2.2.18 Fraction Granulation Number Pfrac
Pfrac 5V
VV
V (m3) � particle volume; VV (m3) � vessel volume.
It represents the parameter expressing the ratio of the volume of particles to that
of the vessel in the granulation process. Physical chemistry. Granulation. Mixing.
Info: [C26].
2.2.19 Frank-Kamenetskii Number δ
δ5Q k CiL
2
λ TN
E
RTNexp 2
E
R TN
� �5
qv L2
λTNE
RTNexp 2
E
RTN
� �
5PoV
Uexp 2
1
U
� �5PoV Ah1 expð2Ah1Þ
Qk (J s21)� volume reaction heat flow rate; Ci (m23)� initial numerical concentration
of reagent molecules; L (m) � characteristic length; λ (W m21 K21) � thermal con-
ductivity of reacting mixture; TN (K)� ambient temperature; E (J mol21)� activation
energy per unit molar mass; R (J mol21 K21) � molar gas constant; qV (W m23) �volume heat flux at the beginning of chemical reaction; PoV (�) � Pomerantsev
heat number (p. 181); U (�) � combustion parameter (p. 35); Ah1 (�) � Arrhenius
number (1.) energetic (p. 34).
In the dimensionless shape, this number characterizes the reaction heat and the
heat transfer in reacting systems, for example, in burning.
Info: [A29],[A35].
David Albert Frank-Kamenetskii (born 1910), Russian engineer.
2.2.20 Frank-Kamenetskii Temperature Φ
Φ5EðT 2 T0Þ
RT2N
E (J mol21) � molar activation energy of reaction; T (K) � temperature after
reaction; T0 (K) � initial temperature of the reacting chemicals; R (J mol21 K21) �molar gas constant; TN (K) � reference temperature.
It characterizes the temperature or temperature stains (hot spots, higher tempera-
ture zones) originating in an exothermic reactive filling of a reservoir in which a
thermal explosion can occur if the heat generated in the reservoir is greater than
that let out of it. Physical chemistry. Ecology.
Info: [B106].
David Albert Frank-Kamenetskii (see above).
40 Dimensionless Physical Quantities in Science and Engineering
2.2.21 Froude Granulation Number Frgran
Frgran 5n2d
g
n (s21) � engine revolutions; d (m) � engine diameter; g (m s22) � acceleration
of gravity.
It expresses the centrifugal-to-gravitational energy ratio in mixing and granulating
processes. The criterion was deduced from empirical experience. Physical chemistry.
Granulation.
Info: [C67].
William Froude (p. 63).
2.2.22 Granulation Fluid Number Ngran
Ngran 5mτVR
m (kg s21) � quantity of granulating fluid added at time unit; τ (s) � time; V (m3) �volume of particles; R (kg m23) � density of particles.
This number expresses the specific quantity of granulating fluid in granulation.
Physical chemistry. Granulation.
Info: [C67].
2.2.23 Guldberg�Waage Number Gw
Gw5 yn11 yn22 . . . 2ðxm11 xm22 . . .Þ 1k
y1, y2, ... (�) � mole fractions of reaction products; x1, x2,... (�) � mole fractions
of reactants; n1, n2, ... (�) � stoichiometric coefficients of individual reaction
products; m1, m2, ... (�) � stoichiometric coefficients of individual reactants;
k (�) � dimensionless partial equilibrium constant.
It characterizes the similarity of chemical reactions in progress, for example,
in blast furnaces. It is expressed by the relation of volumes of reacting gases and
products of the reaction.
Info: [A23],[A29].
Cato Maximilian Guldberg (11.8.1836�14.1.1902), Norwegian
mathematician.
Together with his cousin Peter Waage, in the year 1864, he
formulated the law of mass behaviour in chemical reactions.
This law expresses in detail the influence of concentration,
mass and temperature on the speed of chemical reactions.
41Physics and Physical Chemistry
Peter Waage (29.6.1833�13.1.1900), Norwegian chemist.
In the year 1864, together with his cousin Cato
Maximilian Guldberg, he formulated the law of mass
behaviour in chemical reactions. He was also engaged in
mineralogy and crystallography.
2.2.24 Hatta Number Ha, Ht
Ha5γ
tanh γ; where γ5
ðknRn21i DÞ12β
ð1Þ;
Ha5k1D
β2ð2Þ
kn (m3 (n21) kg2 (n21) s21) � reaction rate constant for nth order chemical reaction;
Ri (kg m23) � mass concentration; D (m2 s21) � diffusivity; β (m s21) � mass
transfer coefficient.
Generally, in form (1), it characterizes the gas absorption with a chemical
reaction being under way. In form (2), it is about the first order of a chemical reac-
tion, for example, in cleaning (washing) the gas in fluid. Flow in geologic layers,
silts and deposits.
Info: [A39],[A50].
Shiroji Hatta (born 1895), Japanese chemist.
2.2.25 Helmholtz Pulsation Number, Helmholtz Resonator Group K
K5L3
V
� �12
M21
L (m) � characteristic length, diameter; V (m3) � volume; M (�) � Mach number
(p. 73).
It characterizes the processes in pulsating combustion.
Info: [A43].
Hermann Ludwig Ferdinand von Helmholtz
(31.8.1821�8.9.1894), German physiologist, physicist and
philosopher.
In the year 1847, he explained the energy conservation law
mathematically and was first to extend Joule’s results into a
general principle. In the year 1850, he measured the impulse
propagation velocity in a nerve tissue. He was engaged in the
physiology of eyesight and hearing. Furthermore, he devoted
himself to electromagnetism, acoustics and thermodynamics.
42 Dimensionless Physical Quantities in Science and Engineering
2.2.26 Henry Number He
He5R1R2
R1, R2 (kg m23) � mass concentration.
In a chemical reaction, it expresses the ratio of the mass concentration and the
phase interface.
Info: [A33].
William Henry (12.12.1775�2.9.1836), English chemist.
He was focused on research on the chemistry of gas, espe-
cially on gas absorption in water under diverse temperatures
and pressures. He formulated Henry’s law. At only 23 years
of age, he published his best works describing the experi-
ments he executed. In other works, gas analysis, humidity
in burning and other chemical problems are treated. The
book Elements of Experimental Chemistry (1799) was his
most important work.
2.2.27 Hess Number Hs
Hs5knL
2Rn21ini
am
kn (m3 (n21) kg2 (n21) s21) � reaction rate constant for nth order of chemical
reaction; L (m) � characteristic length; Rini (kg m23) � initial concentration of
solid material; am (m2 s21) � mass diffusivity of reaction products.
It characterizes the heat and mass transfer in processes with chemical and phase
reactions.
Info: [A35].
Germain Henri Hess (1802�1850), Russian chemist of Swiss origin.
2.2.28 Karlowitz Number (1.) Ka
Ka5dw
dy
L
w; where L5
λcpRgw
w (m s21) � gas velocity; y (m) � coordinate; L (m) � characteristic length;
λ (W m21 K21) � thermal conductivity; cp (J kg21 K21) � specific heat capacity;
Rg (kg m23) � gas density.
In the burning process, it characterizes the flame extending or increasing its
surface in flowing gas, the rate of which is strongly higher than the burning rate.
Due to the surface extending, the front of the flame front is curved. The criterion
43Physics and Physical Chemistry
expresses the critical degree of the flame divergence with which the equilibrium
between heat generation and thermal loss, in the reactive zone, can be destroyed
and the flame extinguished. With the burning wave extended on the path L being
longer than the critical path, the flame is extinguished.
Info: [A23],[A33].
2.2.29 Karlowitz Number (2.) Ka
Ka5τcharτKol
5w2Kol
w2lam
; where wKol 5ffiffiffiffiffivε4
p
τchar (s) � characteristic flame timescale; τKol (s) � smallest Kolmogorov turbulence
time; wKol (m s21) � Kolmogorov velocity; wlam (m s21) � laminar flame velocity;
ν (m2 s21) � kinematic viscosity; ε (m2 s23) � turbulence dissipation velocity.
It characterizes a thin reactive zone in the combustion area. It is used if the least
turbulent whirling in the flow is less than the flame width.
Info: [A103].
2.2.30 Karman Ratio N
N5λ
R cp
κ M um
R w25
at
ap
a2
w25 Le21p M22;
where
ap 5p
Mum; at 5
λcpR
; a5
ffiffiffiffiffikp
R
s
λ (W m21 K21) � gas thermal conductivity; Rg (kg m23) � gas density;
cp (J kg21 K21) � specific heat capacity; M (kg mol21) � molar density;
um (mol m23 s21) � molar rate of chemical reaction; w (m s21) � tributary velocity
of fuel mixture; at, ap (m2 s21) � temperature and pressure diffusivity; a (m s21) �
sound velocity; p (Pa) � pressure; k (�) � specific heat ratio (p. 28); Lep 5 ap a21t
(�) � Lewis pressure number; M (�) � Mach number (p. 73).
This ratio characterizes the steady-state burning of a gas mixture flow. It indicates
the relation between the flame propagation rate and the quantities that characterize
burning.
Info: [A23].
Theodore von Karman (p. 67).
44 Dimensionless Physical Quantities in Science and Engineering
2.2.31 Knudsen Number (2.) Diffusion Kn
K n � Sm21 5μ DAB
ζ DKA
μ (�) � porosity; DAB (m2 s21) � coefficient of binary volume diffusion of AB
system; ζ (�) � diffusion tortuosity; DKA (m2 s21) � Knudsen diffusion coefficient;
Sm (�) � Smoluchowski number (p. 86).
It expresses the ratio of the volume diffusion to that of Knudsen diffusion in
a granulated layer. It characterizes the diffusion in continuous granulated layers
of a catalyzer. It is especially applied in chemical technology. Its inverse value is
called the Smoluchowski number Sm (p. 86).
Info: [A29],[A35].
Martin Hans Christian Knudsen (p. 420).
2.2.32 Mass Fraction M
M5mAPmi
mA (kg) � mass of substance A; mi (kg) � mass of individual substances.
It characterizes the mass fraction of partial components, for example, in mixing.
Physical chemistry.
Info: [C85].
2.2.33 Mole Fraction xi
xi 5ni
n5
Ni
N
ni (�) � number of moles of ith substance; n (�) � total number of moles in a
mixture; Ni (�) � number of molecules of ith substance; N (�) � total number of
molecules in a mixture; NA (�) � Avogadro number (p. 4).
The mole fraction is one of the ways to express the concentration of various
chemical components in a mixture. It is about approximation to an ideal mixture,
but corrections (active coefficients) must be introduced in practice.
n5Xj
nj; Ni 5 niNA;Xi
xi 5 1
45Physics and Physical Chemistry
Physical chemistry. Thermodynamics.
Info: [C90].
2.2.34 Prater Number Pra, β
Pra52ΔH DeCA0
T λe
ΔH (kJ kmol21) � heat of reaction; De (m2 s21) � effective diffusivity; CA0
(kmol m23) � concentration of species A; T (K) � temperature; λ (W m21 K21) �effective thermal conductivity.
This number characterizes the non-isothermic chemical reaction with diffusion
and thermal transfer resistance in pores or the isothermic internal reaction with
the diffusion resistance in pores. It expresses the joining of physical and chemical
processes such as diffusive and convective heat and mass transfers. Above all, it is
important in designing, starting and controlling chemical reactors.
Info: [A49].
2.2.35 Reaction Enthalpy Number N
N5ΔhAΔcA
cpΔT
ΔhA (J kg21) � specific reaction enthalpy change of A product; ΔcA (�) � change
of mass fraction of A product; cp (J kg21 K21) � specific heat capacity; ΔT (K) �
temperature difference.
It expresses the ratio of the reaction energy change to the thermal energy
change. It characterizes the inter-phase transfer with chemical reactions.
Info: [A35].
2.2.36 Reaction Enthalpy Number (2.) Nre
Nre 5ðΔHÞAΔMA
CΔT
(ΔH)A (J kg21) � reaction enthalpy; ΔMA (kg) � mass of fraction A; C (J K21) �heat capacity; ΔT (K) � temperature difference.
It expresses the ratio of reactive energy change to that of thermal energy.
A phase transfer with a chemical reaction.
Info: [A29].
46 Dimensionless Physical Quantities in Science and Engineering
2.2.37 Reagent Concentration η, χ
η5C
Cini
ð1Þ; χ5R
E
Q
RC
cVð2Þ
C, Cini (m23) � numerical initial molecule concentration of reagent; R (J mol21 K21) �
molar gas constant; E (J mol21) � molar activation energy; Q (J) � thermal effect
of reaction; Rg (kg m23)� density; cV (J kg
21 K21) � specific heat capacity.
In burning, it characterizes the reagent concentration or the chemical enthalpy.
Info: [A48].
2.2.38 Reagent Temperature Θ
Θ5E
RT2N
ðT2 TNÞ
E (J mol21) � molar activation energy; R (J mol21 K21) � molar gas constant;
TN (K) � ambient temperature; T (K) � reagent temperature.
It characterizes the reagent temperature in fuel burning.
Info: [A23].
2.2.39 Rejection Coefficient R
R5cr 2 cp
cr; where RAh0; 1i
cr (kg m23) � separating and suspension with a solid concentration; cp (kg m23) �at the same time measure the solid concentration of the permeate.
It characterizes the solid substances separation process in suspensions.
Essentially, it expresses the diaphragm (of filters) efficiency. Physical chemistry.
Separation and filtration of materials.
Info: [C81],[C112].
2.2.40 Semenov Number N
N5βwch
β (m s21)� mass transfer coefficient; wch (m s21)� chemical reaction progress rate.
It characterizes the reaction kinetics in a physical and chemical conversion
process.
Info: [A23].
47Physics and Physical Chemistry
Nikolay Nikolajevich Semenov (3.4.1896�25.9.1986),
Russian physicist and chemist. Nobel Prize in Chemistry, 1956.
He was engaged in the mechanisms of chemical transforma-
tion involving the complete analysis of chain theory application
for diverse chemical reactions and combustion processes. He
proposed the degenerate branching theory, which explained the
phenomenon related to inducing periods in oxidation processes.
He wrote great numbers of works in the sphere of molecular
physics and investigated shock wave propagation and electron
phenomena.
2.2.41 Thermal Capacity of Combustion Chamber N
N5σ T4k
k u
σ (W m22 K24) � specific thermal absorption; T (K) � flue gas temperature;
k (m21) � coefficient of environment reduction in combustor; k (J kg21) � specific
combustion heat; u (kg m23 s21) � chemical reaction rate.
It expresses the ratio of the heat absorbed in a chamber filled with the mediums to
all heat released by fuel combustion. It characterizes the complicated heat transfer in
combustion chambers with physical and chemical processes of fuel burning.
Info: [A23].
2.2.42 Thermodynamic Equilibrium of Chemical Reduction Nreac
Nreac 5k2ϕ2ðRÞk1ϕ1ðRÞ
ð1Þ; Nreac 5k2R
Pα2i21
2i
k1RP
α1i21 1i
ð2Þ
k1, k2 (�) � constants of chemical reaction; ϕ1(R), ϕ2(R) (�) � mass concentration
function of chemical reaction products; R, R1i, R2i (kg m23) � densities of individual
components; α1i, α2i (�) � stoichiometric number of component change.
In the form (1), it expresses the ratio of outlet products of physical and chemical
reduction to inlet ones. It characterizes the internal process thereof. It represents
a local criterion of the thermodynamic equilibrium or the deviation degree from
the state of the thermodynamic equilibrium alternatively. For simple homogeneous
processes, (2) is valid. For quasi-static processes, Nreac5 1 holds and so does
Nreac, 1 for dynamic ones. For continuously changing states of the internal equi-
librium, it is Nreac-1. For the process rate approaching the change rate, Nreac-0
holds, whereas Nreac5 0 is valid for limiting irreversible processes.
Info: [A33].
48 Dimensionless Physical Quantities in Science and Engineering
2.2.43 Thermochemical Bond Ntch
Ntch 5lRmcpRT
l (J kg21) � latent heat; Rm (kg m23) � mass concentration; cp (J kg21 K21) � specific
heat capacity; R (kg m23) � density; T (K) � temperature.
It expresses the ratio of thermochemical heat reduction to that caused by a molar
movement of reacting surroundings. It characterizes the bond between the thermal
and concentrating fields. Heat transfer in physical and chemical processes.
Info: [A33].
2.2.44 Thiele Modulus (2.) Φ, Th
Φ5 L
ffiffiffiffiffik1
D
r5
ffiffiffiffiffiffiffiffiDa2
p
L (m) � effective particle diameter of porous catalyst (equals 6/S)26; k1 (s21) �reaction rate constant; D (m2 s21) � diffusivity; Da2 (�) � Damkohler number (2.)
diffusion (p. 37).
It expresses the isothermic reaction with the diffusion resistance in pores.
Diffusion in porous catalyzers.
Info: [A33],[A49].
Ernest William Thiele (p. 267).
2.2.45 Thring�Newby Number Tn
Tn 5Qmt 1QmP
QmP
r
L
Qmt (kg s21) � fluid mass flux in nozzle; QmP (kg s21) � ambient mass flux; r (m) �
equivalent nozzle radius; L (m) � characteristic length, wall half-thickness.
It characterizes the fuel combustion process in combustion spaces.
Info: [A23],[A35].
Meredith Wooldridge Thring (17.12.1915�15.9.2006), English engineer.
Maurice Purcell Newby (born 1917), English physicist.
2.2.46 Time of Chemical Reaction Kτ
Kτ 5τchτL
τch (s) � chemical reaction time necessary to reach the equilibrium state in flowing
fluid; τL (s) � time necessary to fluid flow through the characteristic length L.
49Physics and Physical Chemistry
It characterizes the time to reach the equilibrium state in chemical reactions
in flowing fluids. For a flow with homogeneous chemical reactions, with Kτ5 0,
the chemical equilibrium state occurs in the flow. It is the case in which the
chemical reaction rates are so high that the equilibrium state occurs in every
place of the surroundings with given mass densities. With Kτ-N, the chemical
reactions are so small that in every place of the surroundings, the mixture
composition is determined by mass flows of various components. It is the case
of frozen flow.
In flow with heterogeneous reactions with Kτ5 0, the chemically reacting
mixture, adhering to the surface, is in a chemical equilibrium state with the
temperature and pressure on the surface. This situation is called the ideally cata-
lytic surface. The non-catalytic surface case occurs for Kτ-N, in which the
catalytic reaction rate is so small that the mixture surface composition is deter-
mined by diffusion and convection processes not coupled with the reactions on
the wall.
Info: [A23].
2.2.47 Van ’t Hoff Factor i, Ni
i5 11 aðq21Þ
a (�) � degree of ionization; q (�) � number of ions formed for each dissociated
molecule.
It expresses the number of moles in a water solution which are created by a
mole of a solid admixture. The matter is the electrolyte dissociation into ions in the
solution. For example, NaCl-Na1 1Cl2 (i5 2), MgBr2-Mg21 1 2Br2 (i5 3).
For glucose it is i5 1 because one mole of it dissolved in water leads to the result
of one diluted mole.
Info: [C137].
Jacobus Henricus van ’t Hoff, Jr. (30.8.1852�1.3.1911),
Dutch physical chemist. Nobel Prize in Chemistry, 1911.
He was engaged in research on chemical kinetics, chemi-
cal equilibrium, osmotic pressure and crystallography. He
contributed to elaborating the branch of physical chemistry.
He was engaged also in organic chemistry. In the year 1874,
he described the optical activity phenomenon. He studied
three-dimensional chemical structures and other problems
of stereochemistry. In addition, he worked out Svante
Arrhenius’ theory of electrolytic dissociation.
50 Dimensionless Physical Quantities in Science and Engineering
2.2.48 Volume Fraction ϕ
ϕ5VAPVi
VA (kg) � volume of the A component; Vi (kg) � volumes of individual compo-
nents in a mixture.
It characterizes the volume fraction of partial components, for example, in mix-
ing. Physical chemistry.
Info: [C138].
2.2.49 Wagner Number Wa
Wa5γd
@U
@JA
γ (S m21) � electric conductivity of solid phase; U (V) � upper potential for reduc-
tion of O2; JA (A m22) � surface density of electric current; d (m) � thickness of
catalytic layer.
In catalytic layers it characterizes the ohmic drop, specifically by comparing
the charge transfer with the ohmic resistances. For Wa, 1, the ohmic resistance
predominates. It is used to design diaphragm�electrolyte systems, to solve cathodic
catalytic layers and gas diffusers and to fuel elements if need be. It can also be
applied to determine the current density distribution in electrochemical cells.
Info: [B42].
2.2.50 Weisz Modulus Wz, Ψ0
Wz5 ηkΦ2 5 ηkDa2; where ηk 5
uef
uðTS;CSÞ
uef (s21) � effective chemical reaction rate; u (s21) � chemical reaction rate; TS (K) �
outside surface temperature of catalyzer; ηk (�) � degree of required porosity; Φ (�) �Thiele modulus (2.) (p. 49); Da2 (�) � Damkohler number (2.) diffusion (p. 37);
CS (�)� gas constant for catalyzer surface.
It is applied to characterize the isothermal reaction with the diffusive resistance
of pores. Catalyzers. Diffusion in porous catalyzers.
Info: [A49].
51Physics and Physical Chemistry
3 Fluid Mechanics
The quantitative character of objects, one of the most general laws of the
existence.Leonhard Euler (1707�1783)
3.1 One-Phase Fluid Mechanics
In this field, the range of similarity criteria used is great. It involves the flow of
potential, compressible and incompressible fluids; viscous and non-viscous flows;
steady and unsteady flows; and laminar, turbulent and other flows. Some criteria
concern the related phenomena arising in flow, such as the boundary layer, friction,
acting forces, stroke waves, the sound barrier, surface tension and swirls. Among
the fundamental and most widespread similarity criteria are the Froude, Knudsen,
Mach, Prandtl, Richardson and Strouhal numbers.
3.1.1 Acceleration Number Ac
Ac5E3
R g2 η25Re2 Fr2 Ho23
E (Pa) � modulus of elasticity; R (kg m23) � density; g (m s22) � gravitational
acceleration; η (Pa s) � dynamic viscosity; Re (�) � Reynolds number (p. 81);
Fr (�) � Froude number (1.) (p. 62); Ho (�) � Hooke number (p. 138).
This number characterizes the acceleration flow. It depends on physical proper-
ties and gravitational acceleration only.
Info: [B20].
3.1.2 Archimedes Hydrodynamic Number Ar
Ar5g L3
v2Rs 2 R
R5Ga
Rs 2 RR
ð1Þ; Ar5g L3
v2εB
12 εBð2Þ
g (m s22) � gravitational acceleration; L� d (m) � particle diameter; ν (m2 s21) �fluid kinematic viscosity; Rs, R (kg m23) � particle and fluid density; εB (�) �volume fraction; Ga (�) � Galileo number (p. 123).
Dimensionless Physical Quantities in Science and Engineering. DOI: 10.1016/B978-0-12-416013-2.00003-8
© 2012 Elsevier Inc. All rights reserved.
It expresses the ratio of product of a floating body weight and the inertia force
to the square of the viscosity force. It characterizes the free flow of fluid caused by
the density difference of partial components. For example, it is about the floating of
solid particles, drops or bubbles (2) � or possibly the whole composition thereof �caused by uplift forces in the fluid volume. It is an unusual modification of the
Galileo number Ga (p. 123). It occurs in the heat transfer between the particles,
for example, in fluidization and in material bundles. The characteristic feature is
the heat transfer by convection and conduction in a wall-particle system.
Info: [A14],[A29],[B20].
Archimedes of Syracuse (287 BC�212 BC), Greek mathematician.
From our present point of view, he was a physicist and
engineer. He formulated the physical principle of upward
force and the weight of drawn fluid. This theorem became
known as the Archimedes law. He utilized Euclidean geome-
try to calculate the surfaces and volumes of bodies. He deter-
mined the approximate value of π.
3.1.3 Bairstow Number NBa
See Mach number M (p. 73).
Info: [A29].
Leonard Bairstow (1880�1963), English aerospace engineer.
3.1.4 Bejan Convective Number Be
Be5ΔpL2
ηa
Δp (Pa) � pressure drop; L (m) � characteristic length; η (Pa s) � dynamic viscos-
ity; a (m2 s21) � thermal diffusivity.
In fluid mechanics, it represents the dimensionless expression of the pressure
drop along an L-long canal. In convection, its role resembles the Rayleigh number
(2.) (heat instability) Ra2 (p. 187) in natural convection.
Info: [C6].
Adrian Bejan (p. 9).
3.1.5 Beranek Number Be
Be5w3sR
2l
ηlgðRs 2 RlÞ
54 Dimensionless Physical Quantities in Science and Engineering
ws (m s21) � fall velocity of solid particles; Rl, Rs (kg m23) � density of liquid and
solid particles; ηl (Pa s) � dynamic viscosity of liquid; g (m s22) � gravitational
acceleration.
It expresses the ratio of the inertia force square to the product of the viscosity
and the gravitational force. It characterizes the bypassing of free falling particles in
immobile unlimited fluid.
Info: [A29].
Josef Beranek, Czech chemist.
3.1.6 Blasius Shape of Boundary Layer δ=x
δx5
4; 9ffiffiffiffiffiffiffiRex
p ; where Rex 5wc
v
δ (m) � boundary layer thickness; x (m) � distance from entering edge; w (m s21) �flow velocity; ν (m2 s21) � kinematic viscosity; Rex (�) � local Reynolds number
relative to the x position.
It characterizes the viscous fluid flow in a laminar boundary layer above a float-
ing plate. The expression is used to monitor the boundary layer development and to
simplify the solution.
Info: [A21].
3.1.7 Boussinesq Number Bs
Bs5w2
2gL
w (m s21) � flow velocity; g (m s22) � gravitational acceleration; L (m) � charac-
teristic length, hydraulic radius.
It expresses the inertia-to-weight forces ratio. Weight influence on surface
waves in open canals.
Info: [A29], [B17], [B20].
Valentin Joseph Boussinesq (13.3.1842�19.2.1929), French
physicist and mathematician.
He was engaged in many areas of mathematical physics,
but especially in hydrodynamics, the undulation and flowing
of fluids, resistance of bodies in flowing fluid, the cooling
down effect of fluids and the problems of turbulence and
fluid elasticity.
55Fluid Mechanics
3.1.8 Boussinesq Approximation Parameter Bs
Bs5g0
g5
R1 2 R2Rref
g0 (m s22) � approximated gravitational acceleration; g (m s22) � gravitational
acceleration; R1, R2 (kg m23) � density of warm and cold liquid; Rref (kg m23) �reference density.
Compared to the often used parametric expression R1/R2, the flow tasks become
more precise by making use of this parameter. Fluid mechanics.
Info: [C109].
Valentin Joseph Boussinesq (see above).
3.1.9 Buoyancy Number NB
NB 5L2FβΔT
ηVw
L (m) � characteristic length; F (N) � buoyancy force; β (K21) � volume thermal
expansion coefficient; ΔT (K) � temperature difference; η (Pa s) � dynamic vis-
cosity; V (m3) � volume; w (m s21) � velocity.
It is the buoyancy-to-viscous forces ratio. It describes the influence of viscous
fluid natural convection on buoyancy.
3.1.10 Buoyancy Parameter NB
NB 5ΔT
T
gL
w25Gr1Re
22 5ΔT
TFr21
ΔT (K) � temperature difference; T (K) � environment temperature; g (m s22) �gravitational acceleration; L (m) � characteristic length; w (m s21) � flow velocity;
Gr1 (�) � Grashof heat number (p. 185); Re (�) � Reynolds number (p. 81);
Fr (�) � Froude number (1.) (p. 62).
It characterizes the buoyancy in free and forced convections. It is analogous to
the Archimedes thermodynamic number Ar2 (p. 184).
Info: [A35].
3.1.11 Capillary Number (2.) Ca
Ca5η2ERσ2
5ηwσ
� �25We21Ho
21Re22
56 Dimensionless Physical Quantities in Science and Engineering
η (Pa s) � dynamic viscosity; E (Pa) � bulk modulus of fluid; R (kg m23) � liquid
density; σ (N m21) � surface tension; w (m s21) � flow velocity; We1 (�) �Weber number (1.) (p. 91); Ho (�) � Hooke number (p. 138); Re (�) � Reynolds
number (p. 81).
It represents the influence of the surface tension acting on the flowing fluid and
its capillarity. It depends on physical properties of the fluid.
Info: [A29],[A35].
3.1.12 Centrifuge Number Ncf
Ncf 5Rr2hω2
σ
R (kg m23) � density; r (m) � radius; h (m) � fluid depth; ω (s21) � angular fre-
quency; σ (Pa) � surface tension.
It equals the centrifugal-to-capillary forces ratio. The flow in curved canals. The
centrifugal forces of stroke and surface waves. Hydromechanics.
Info: [B20].
3.1.13 Coefficient of Velocity ϕ
ϕ5w
wt
; whereas ϕ, 1
w (m s21) � real velocity; wt (m s21) � theoretic velocity.
This coefficient expresses the ratio of the actual fluid flow velocity, with friction
and without contraction, to the theoretical velocity without friction. It depends on
the bypassed wall roughness. Usually ϕAh0, 95; 1i. Together with the contraction
coefficient α (p. 57), it determines the outflow coefficient μ (p. 77). Aero-
hydrodynamics.
Info: [A39].
3.1.14 Contraction Coefficient α, ε
α5A
A0
A, A0 (m2) � cross section of nozzle spray and outlet area.
It is the ratio of the out-flowing beam diameter to the orifice area.
Hydromechanics.
Info: [A39].
57Fluid Mechanics
3.1.15 Correlation Coefficient R, .
R5wx1wx2ffiffiffiffiffiffiffiffiffiwx1
2p ffiffiffiffiffiffiffiffiffi
wx22
pwx1, wx2, wx1 ;wx2 (m s21) � pulsation velocities measured at the same time in posi-
tions 1 and 2 and their mean values.
It expresses the flow turbulence measure. Aerodynamics.
Info: [C22].
3.1.16 Crispation Number Ncr
Ncr 5ηaσL
η (Pa s) � dynamic viscosity; a (m2 s21) � thermal diffusivity; σ (Pa m) � undis-
turbed surface tension of the layer; L (m) � characteristic length, layer thickness.
This number characterizes the influence of the penetration of viscous fluid con-
vective flow in the layer thickness while considering the thermal diffusion and the
surface stress. One-phase fluid mechanics.
Info: [C25].
3.1.17 Crocco Number Cr
Cr5w
wmax
5 112
ðk21ÞM2
� �212
w (m s21) � local flow velocity; wmax (m s21) � velocity at adiabatic expansion;
κ (�) � specific heat ratio (p. 28); M (�) � Mach number (p. 73).
This number expresses the ratio of local flow rate to the highest possible fluid
rate in outlet into a vacuum (adiabatic expansion). It characterizes the compressible
fluid flow. It is also called the modified Mach number M.
Info: [A29], [B20].
3.1.18 Darcy Number (1.) Dc
Dc52ghd
w2L
g (m s22) � gravitational acceleration; h (m) � pressure loss; d (�) � pipe diame-
ter; w (m s21) � flow velocity; L (m) � characteristic pipe length of constant cross
section.
58 Dimensionless Physical Quantities in Science and Engineering
It expresses the pressure loss of fluid flowing through a pipeline. The Darcy
friction number fD (p. 119) or the Fanning friction number fF (p. 163) are modi-
fications of this. Fluid mechanics. Hydraulics. Tribology.
Info: [A26], [B20].
Henry Philibert Gaspard Darcy (p. 98).
3.1.19 Darcy Number (2.) Dc
Dc52ghrh
w2ð1Þ; Dc5
8ghrh
w2ð2Þ
g (m s22) � gravitational acceleration; h (m) � mean hydraulic depth; rh (�) �hydraulic gradient; w (m s21) � flow velocity.
It characterizes the fluid flows in open canals and over falls. It is analogous to
the Darcy friction number fD (p. 119). Hydraulics and hydromechanics.
Info: [A21],[B20].
Henry Philibert Gaspard Darcy (p. 98).
3.1.20 Dean Number Dn
Dn5wd
v
ffiffiffir
R
r5Re
ffiffiffir
R
r
w (m s21) � flow velocity; r5 d/2 (m) � canal radius; ν (m2 s21) � kinematics
viscosity; R (m) � radius of canal curvature; Re (�) � Reynolds number (p. 81).
It expresses the mutual relation between centrifugal, friction and inertia forces
and characterizes the flow in curved canals.
Info: [A14],[B17],[B20].
William Reginald Dean (1896�1973), British mathematician and physicist.
3.1.21 Degree of Turbulence E
E5w
wT
w (m s21) � mean square velocity of pulsation; wT (m s21) � mean velocity of tur-
bulence flow.
This degree expresses the ratio between the mean-quadratic pulsation velocity
and the mean turbulent flow velocity. Aero-hydrodynamics.
59Fluid Mechanics
3.1.22 Density Number N.
NR 5ΔRR0
ΔR (kg m23) � density difference of matter; R0 (kg m23) � initial density.
It expresses the ratio of the density difference, before and after loading, to the
initial density of solid, liquid or gaseous surroundings. For example, it is used in
fluid mechanics with the Boussinesq approximation when the original gravitational
acceleration g is transformed to g5 g(ΔR/R) in controlled buoyancy fluid flow.
Mechanics. Fluid mechanics. Buoyancy flow.
Info: [C10].
3.1.23 Derjaguin Number De
De5 L
ffiffiffiffiffiffiRg2σ
r
L (m) � characteristic length, film thickness; R (kg m23) � density; g (m s22) �gravitational acceleration; σ (N m21) � surface tension.
It expresses the ratio of the film thickness to the capillary length. It characterizes
the process of coatings and films generation of fluids on a body surface. Fluid
mechanics. Physical technology.
Info: [A35].
3.1.24 Drag Coefficient CD
CD 5FD
1
2Rw2A
ð1Þ; CD 5ðRS 2 RFÞLSg
RFw2ð2Þ
FD (N) � drag force on body; R, RS, RF (kg m23) � density, density of solid parti-
cles and fluid; w (m s21) � flow velocity; A (m2) � cross-sectional area of body
perpendicular to flow velocity; LS (m) � characteristic small dimension of parti-
cles; g (m s22) � gravitational acceleration; _m (kg s21) � mass flux.
In expression (1), this coefficient expresses the ratio of the drag force FD acting
perpendicularly to the dynamic force at a stagnation point on a transversal area A. It
characterizes the aerodynamic or hydrodynamic resistance in fluid flow. The expres-
sion (2) is valid in case of flow with small particles and steady distribution thereof.
From the relation (1), an equation expressing the drag force follows, which is
necessary to overpower the aerodynamic resistance and to set an object into
movement.
60 Dimensionless Physical Quantities in Science and Engineering
FD 5CD
1
2Rw2A ð3Þ
The expression (3) for the drag force is valid provided the flow is turbulent,
where CD is approximately constant. For laminar flow, it is
FD � _mw ð4Þand the force is, therefore, proportional to the velocity. With a turbulent flow, the
power is
P5FDw51
2Rw3A ð5Þ
The drag coefficient CD is important especially in the aircraft and car industries.
One tries to reach the least value of CD. For example, for automobiles it is CDAh0.35;0.45i and for sports cars CDAh0.25; 0.30i. For standard cars, it is CD�0.5, for a rac-
ing bicycle CD�0.9, and for a motorcycle CD�1.8. The coefficient CD is the physical
similarity criterion which enables � in the case of its equality between the model and
the object (CDM5CD), the provision of the behaviour similarity condition, for exam-
ple, with various geometric dimensions of the model and the object.
Info: [A21],[A35].
3.1.25 Euler Number (1.) Eu
Eu5Δp
Rw25
F
Rw2L2ð1Þ; Eu5
1
κM2ð2Þ
Δp (Pa) � pressure drop due to friction; R (kg m23) � density; w (m s21) � flow
velocity; F (N) � frictional force; L (m) � characteristic length; κ (�) � specific
heat ratio (p. 28); M (�) � Mach number (p. 73).
This number expresses the pressure gradient due to friction related to dynamic
pressure; alternatively, the ratio of the pressure surface force to the inertia force. It
characterizes the hydrodynamic pressure, pressure loss, hydraulic resistance and
fluid friction, and their influence on the proceeding process. With a compressible
environment flow, the expression (2) is used. For the pressure flow, it is Eu5Ne.
Info: [A29],[A35],[A43],[B20].
Leonhard Euler (15.4.1707�18.9.1783), Swiss mathemati-
cian and physicist.
He shifted the boundaries in modern analytical geometry
and trigonometry. He was characterized by his extraordinary
memory and, despite his initially partial and later total blind-
ness, he was most prolific mathematician of all time with his
more than 800 works. Great numbers of his works concern
optics, mechanics, electricity and magnetism.
61Fluid Mechanics
3.1.26 Euler Curved Number Eucurv
Eucurv 58Δpr3RDn2η2R
Δp (Pa) � pressure drop in the curved pipe; r (m) � pipe diameter; R (kg m23) �liquid density; η (Pa s) � dynamic viscosity; R (m) � curvature radius; Dn (�) �Dean number (p. 59).
It expresses the ratio of the pressure gradient due to friction in a curved tube to
the dynamic pressure. It characterizes the pressure loss, hydraulic resistance and
friction in curved canals. Fluid mechanics. Flow in curved canals.
Leonhard Euler (see above).
3.1.27 Force Coefficient CF
CF 5τw
1
2Rw2
5F
1
2Rw2A
5 2Ne
τw (Pa) � stress component of circumfluenced body; R (kg m23) � fluid density; w
(m s21) � flow velocity; F (N) � force; A (m2) � drag area; Ne (�) � Newton
number (p. 75).
It is important mainly in aerodynamics and expresses the resistance-to-inertia
forces ratio. As a vector, the force F has a drag component FD and that of uplift FL.
Info: [A20].
3.1.28 Fourier Hydrodynamic Number (Fourier Flow Number) Foh
Foh � Zh5vτL2
5ητRL2
ν (m2 s21) � kinematic viscosity; τ (s)� time; L (m) � characteristic time; η (Pa s)�dynamic viscosity; R (kg m23) � density; Zh (�) � Zhukovsky number (p. 93).
In the dimensionless shape, it characterizes the non-stationary flow time of a
viscous incompressible fluid. It is called the Zhukovsky number Zh (p. 93).
Info: [A29].
Jean Baptiste Joseph Fourier (p. 175).
3.1.29 Froude Number (1.) Fr
Fr5Ri21 5w2
gL5Ga21Re2 ð1Þ; Fr5
w2RgLΔR
ð2Þ
62 Dimensionless Physical Quantities in Science and Engineering
w (m s21) � flow velocity; g (m s22) � gravitational acceleration; L (m) � charac-
teristic length; R (kg m23) � liquid density; ΔR (kg m23) � density difference
between liquid and other phase; Ri (�) � Richardson number (p. 83); Ga (�) �Galileo number (p. 123); Re (�) � Reynolds number (p. 81).
This number expresses the ratio of the inertia force to the gravitational or uplift
force; alternatively, the ratio of the kinetic energy to the potential energy. For
example, it characterizes the surface phenomena in free-level fluid flow. It is
applied in hydraulics, wherever the weight influence must be considered. It can be
neglected in gases. With the bubble floating process in a fluid, the bubble radius
(L� r) is inserted for the characteristic length. The inverse value of the Froude
number (1.) is called the Richardson number Ri (p. 83) or the Reech number Ree
(p. 81) sometimes. The Boussinesq number (1.) Bs (p. 55) is its analogue, as well.
Info: [A7],[B20].
William Froude (28.11.1810�4.5.1879), English engineer
and naval architect.
He was engaged in hydrodynamics and hydraulics and
especially in surface phenomena in fluid flowing on free sur-
faces and in open canals, influenced by gravity. He also stud-
ied processes of the rising of bubbles in a fluid and other
phenomena connected to fluid flow and floating bodies.
3.1.30 Froude Number (2.) External Fr
Fr5wffiffiffiffiffiffigL
p 5ffiffiffiffiffiffiffiFr�
pð1Þ; Fr5
wffiffiffiffiffigh
p 5w
cð2Þ
w (m s21) � flow velocity; c (m s21) � velocity of the water level wave propaga-
tion at the position, where is the sea depth h; Fr� (�) � Froude number (1.) Fr
(p. 63); for other quantities see Froude number (1.) Fr (p. 62).
It expresses the relation of the inertia force to the gravitation force; alternatively,
the ratio of the flow rate in an open canal to the propagation rate of slight gravita-
tion waves. It characterizes the free-level fluid flow in open canals. It is also used
to determine the character of the flow. With Fr, 1, the flow is subcritical, with
Fr5 1, it is critical and with Fr. 1, it is supercritical or accelerated.
Info: [A43],[B20].
William Froude (see above).
3.1.31 Froude Internal Number (1.) Frint
Frint 5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRw2
gL2ðdR=dzÞ
s
63Fluid Mechanics
R (kg m23) � unperturbed density; w (m s21) � flow velocity; g (m s21) � gravita-
tional acceleration; L (m) � characteristic length; dR/dz (kg m24) � vertical density
gradient.
This number expresses the influence of the kinetic and potential energy and of
the vertical density gradient in sedimentation.
Info: [C73].
William Froude (see above).
3.1.32 Froude Internal Number (2.) Frint
Frint 5w
cg5
w
fBh
w (m s21) � flow velocity; cg (m s21) � velocity of gravitation wave propagation;
fB (s21) � buoyancy frequency; h (m) � depth.
It expresses the ratio of the current fluid acceleration to the wave-induced
acceleration.
Info: [C73].
William Froude (see above).
3.1.33 Froude Rotation Number Frrot
Frrot 5Ln2
g
L (m) � characteristic length, L� d � diameter of pipe, bubble, drop, shaft or
wheel; n (s21) � rotation speed; g (m s22) � gravitational acceleration.
It characterizes the forced fluid flow due to mixing, vibrating or other
movement.
Info: [A29].
William Froude (see above).
3.1.34 Gortler (Goertler) Number, Parameter Goe
Goe5Reδ
ffiffiffiδR
rð1Þ; whereas Reδ 5
wδv
ð2Þ
δ (m) � thickness of momentum boundary layer; R (m) � radius of wall edge; w
(m s21) � flow velocity; ν (m2 s21) � kinematic viscosity; Reδ (�) � Reynolds
boundary layer number (p. 82).
In the form (1), it characterizes the instability of laminar boundary layers on
curved concave walls. It expresses the origin of Karman swirls in the bypassing of
64 Dimensionless Physical Quantities in Science and Engineering
solid bodies. It is a modification of the Reynolds boundary layer number Reδ(p. 82) given by the expression (2). Therefore, it is also called the flow stabilization
criterion in a boundary layer.
Info: [A29].
Henry Gortler (born 1909), German mathematician.
3.1.35 Goucher Number Go
Go5r2Rg2σ
r (m) � radius of body, wire, etc.; R (kg m23) � density; g (m s22) � gravitation
acceleration; σ (N m21) � surface tension.
It is the gravitation-to-surface-tension forces ratio. The surface phenomena in
fluids. The tension. Making of films and coatings. See the Deryagin (p. 60), Bond
(p. 95) and Eotvos (p. 99) numbers.
Info: [A29],[B20].
Frederick Shand Goucher (30.9.1888�21.8.1973), Canadian physicist.
3.1.36 Hagen Number Hg
Hg52 Rdp
dx
L3
v2
rp52 dp/dx (Pa m21) � pressure gradient, for natural convection holds
rp5 gRβΔT; L (m) � characteristic length; ν (m2 s21) � kinematic viscosity;
g (m s22) � gravitational acceleration; R (kg m23) � liquid density; β (K21) � coef-
ficient of volume expansion at constant pressure; ΔT (K) � temperature difference.
It is used for forced flow. The Hg number approaches the Grashof heat number
Gr (p. 185) for a natural flow (Hg-Gr).
Info: [C70].
3.1.37 Hagen�Poiseuille Number Ha
Ha52dp
ds
L2
ηw
dp/ds (Pa m21) � pressure gradient; L (m) � characteristic length; η (Pa s) �dynamic viscosity; w (m s21) � flow velocity.
This number characterizes laminar viscous fluid flow. It is also called the
Poiseuille number.
Info: [A35].
Jean Louis Marie Poiseuille (p. 78).
65Fluid Mechanics
3.1.38 Hodgson Number Hs
Hs5VfΔp
QVpS
V (m3) � system volume; f (s21) � flow gas frequency; Δp (Pa) � pressure fall
due to friction loss, etc.; pS (Pa) � mean static pressure; QV (m3 s21) � mean vol-
ume flux.
It expresses the non-uniformity of the delivered amount in pulsating flow.
It states the ratio of a set time constant to the pulsation period time. Pulsating flow.
Info: [A29],[A33],[B20].
John Lawrence Hodgson (1881�1936), English engineer.
3.1.39 Hydraulic Resistance Group NRH
NRH 5Δp
RlgL5Eu Fr
Δp (Pa) � pressure drop across liquid on distillation tray; Rl (kg m23) � fluid den-
sity; g (m s22) � gravitational acceleration; L (m) � depth of liquid layer; Eu (�) �Euler number (1.) (p. 61); Fr (�) � Froude number (1.) (p. 62).
This number expresses the pressure drop, for example, in distillation columns.
Info: [A29].
3.1.40 Jeffrey Number Je
Je5RgL2
ηw5Re Fr21 5 Stk211
R (kg m23) � density; g (m s22) � gravitational acceleration; L (m) � characteristic
length; η (Pa s) � dynamic viscosity; w (m s21) � velocity; Re (�) � Reynolds
number (p. 81); Fr (�) � Froude number (1.) (p. 62); Stk1 (�) � Stokes number
(p. 130).
It expresses the slow fluid flow.
Info: [A26].
3.1.41 Karman Number (1.) Ka
Ka5 Rd3 2dp
dx
� �η22 5
U Rpd3
η2L5 2f
1=2F Re2
66 Dimensionless Physical Quantities in Science and Engineering
R (kg m23) � density; d (m) � pipe diameter; dp/dx (Pa m21) � pressure gradient;
η (Pa s) � dynamic viscosity; L (m) � characteristic pipe length; fF (�) � Fanning
friction number (p. 163); Re (�) � Reynolds number (p. 81).
It expresses the flow in a pipeline with friction.
Info: [A35],[B17].
Theodore von Karman (11.5.1881�6.5.1963), American
engineer of Hungarian origin.
His scientific work is characterized by the use of applied
mathematics in engineering. His work is very extensive and
includes fluid mechanics, turbulence theory, supersonic
flights, aircraft structures and wind erosion of soil. He was
engaged in the theory of missile movement in the atmosphere
and was a co-founder of magneto-hydrodynamics.
3.1.42 Karman Number (3.) Ka
Ka5h
v
ffiffiffiffiffiτsR
r
h (m) � mean height of surface roughness; ν (m2 s21) � kinematic viscosity;
τs (Pa) � shear stress near the wall surface; R (kg m23) � density.
It expresses the ratio of the shear stress and the viscous force in a fluid. It char-
acterizes the friction in floating fluids.
Info: [A29].
Theodore von Karman (see above).
3.1.43 Karman Turbulent Number Katur, Tu
Katur � Tu5
ffiffiffiffiffiffiffiffi2e
3w2r
s5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiw2x 1w2
y 1w2z
3w2r
s
e (m2 s22) � specific kinetic energy of turbulent flow; wr (m s21) � characteristic
flow velocity, for example, input flow; wx, wy, wz (m s21) � pulsation velocity com-
ponents in any points of turbulent flow; Tu (�) � turbulence number (p. 90).
It expresses the ratio of a mean-quadratic component of a local flow rate pulsa-
tion vector to the mean value of the relative flow rate. It characterizes the flow tur-
bulence and its influence on the heat transfer within forced convection. It is also
called the turbulence number Tu (p. 90).
Info: [A33].
Theodore von Karman (see above).
67Fluid Mechanics
3.1.44 Karman�Tsien Correction Factor Cp
Cp 5Cp0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
12M2N
p1
M2N
11ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12M2
N
p !
Cp0
2
Cp0 (�) � pressure coefficient for incompressible fluid, see pressure coefficient Cp
(p. 80); MN (�) � Mach number for high subsonic flow, see Mach number M
(p. 73).
This rule expresses the compressibility correction for the pressure distribution on
a bypassed body surface in high subsonic flow. Fluid mechanics. Aerodynamics.
Info: [A21].
Theodore von Karman (see above).
3.1.45 Keulegan Number Ke
Ke5wffiffiffiffiffi
gv3p
5ffiffiffiffiffiffiffiffiffiffiffiffiFr Re3
p
w (m s21) � flow velocity; g (m s22) � gravitational acceleration; ν (m2 s21) �kinematic viscosity; Fr (�) � Froude number (1.) (p. 62); Re (�) � Reynolds
number (p. 81).
This number expresses the dimensionless fluid flow rate, hydrodynamic stability
and structure and dynamics of the near-surface layer of the ocean. Hydraulics.
Oceanography.
Info: [A41].
3.1.46 Keulegan�Carpenter Number (1.) Kec, KC
Kec52πAL
A (m) � amplitude of sinusoidal flow; L (m) � characteristic length, height of the
circumfluenced body.
It characterizes the ratio of drag force to inertia for a bluff object in oscillatory
flow. It is analogous to the Strouhal number Sh (p. 87). A low value of KecAh0.4; 2ishows the inertial effect to be more important than the viscosity effect.
Hydrodynamics and aerodynamics. Oceanography. Flow oscillations in turbines and
pipelines.
Info: [A8],[A21].
68 Dimensionless Physical Quantities in Science and Engineering
3.1.47 Kirchhoff Number Kh
Kh5RpL2
η25Eu Re2
R (kg m23) � density; p (Pa) � pressure; L (m) � characteristic length; η (Pa s) �dynamic viscosity; Eu (�) � Euler number (1.) (p. 61); Re (�) � Reynolds
number (p. 81).
It expresses the pressure-to-molecular-friction forces ratio. It characterizes the
fluid flow process in flow-through canals.
Info: [A23],[A33].
Gustav Robert Kirchhoff (12.3.1824�17.10.1887), German
physicist.
He was engaged in mechanics and electricity. In the year
1859, he constructed a microscope by means of which he dis-
covered two new elements. In addition to applied spectral
analysis, he was engaged in electric circuits and in thermal
radiation. He formulated two Kirchhoff laws enabling the
calculation of the currents and voltages in electric circuits.
His work on black body radiation contributed to the develop-
ment of quantum theory.
3.1.48 Kirpitchev Hydrodynamic Number Kih
Kih 5
ffiffiffiffiffiffiffiffiffiRFR
η23
s5
ffiffiffiffiffiffiffiffiffiffiffiffiffiRe2CF
3p
R (kg m23) � density; FR (kg m s22) � hydrodynamic resistance; η (Pa s) � dynamic
viscosity; Re (�) � Reynolds number (p. 81); CF (�) � force coefficient (p. 62).
This number expresses the flow around obstacles and the resistance acting on a
body submerged in a viscous fluid flow.
Info: [A26],[A29].
Michail Viktorovich Kirpichev (1879�1955), Russian engineer and physicist
(p. 177).
3.1.49 Knudsen Number (1.) Kn
Kn5 Sm21 5lm
L5 1:28
ffiffiffiκ
pη
aRLð1Þ;
Kn5M Re21 ð2Þ; Kn5M Re212 ð3Þ;
69Fluid Mechanics
Kn5 1:28ffiffiffiκ
p M
Reð4Þ
lm (m) � mean free path; L (m) � characteristic length; η (Pa s) � dynamic viscosity;
a (m2 s21) � thermal diffusivity; R (kg m23) � density; Sm (�) � Smoluchowski
number (p. 86); κ (�) � specific heat ratio (p. 28); M (�) � Mach number (p. 73);
Re (�) � Reynolds number (p. 81).
It is also called the flow continuity criterion. It expresses the ratio of the average
free path of molecules in gas to the characteristic length dimension of a body.
It characterizes the aerodynamic gas dilution degree or, alternatively, its deflection
from the continuum state. It determines the conditions under which the phenomena
connected to gas dilution occur. With Kn. 10 the molecules flow freely, with
0.1,Kn, 10 transient flow occurs, and with 0.01,Kn, 0.1 slipping on a wall
sets in. With Kn, 0.01, a continual flow is achieved. The flow of artificial satel-
lites in an orbit without collisions above the earth is an example for Knc1. With
Re# 1 the expression (2) is applied, as is expression (3) for Rec1. The inverse
value of the Knudsen number is called the Smoluchowski number Sm (p. 86). In
kinetic theory, this number is used in the form (4).
Info: [A21],[A43],[B11],[B20].
Martin Hans Christian Knudsen (p. 420).
3.1.50 Lagrange Number (3.) Lg3
Lg3 5ΔpL
wη5Eu Re
Δp (Pa) � pressure difference; L (m) � characteristic length; w (m s21) �mean
velocity; η (Pa s) � dynamic viscosity; Eu (�) � Euler number (1.) (p. 61);
Re (�) � Reynolds number (p. 81).
This number expresses the ratio of the pressure force, under the action of the
hydraulic resistance, to the viscosity force. With fluid flowing in direct smooth and
even rough flow canals, it characterizes the relation between the pressure and
velocity fields. It is used in magneto-hydrodynamics especially.
Info: [A1],[A4],[A43],[B110].
Joseph-Louis Lagrange (25.1.1736�10.4.1813), French
mathematician and physicist of Italian origin.
He was the greatest mathematician of the nineteenth cen-
tury. He was engaged in the theory of numbers, algebra, the
theory of analytic functions, differential equations, variation
calculus, celestial mechanics and mathematical cartography.
The Lagrange equations represent the foundation of theoretical
mechanics. His Mecanique Analytique (Analytic Mechanics,
1788) was his principal work.
70 Dimensionless Physical Quantities in Science and Engineering
3.1.51 Laplace Number Lp
Lp5ΔpL
σ5Eu We1 ð1Þ; Lp5
σRLη2
5Ca Re ð2Þ;
Lp5
ffiffiffiffiffiffiffiffiffiffiσ
RgL2
r5Fr We
212
1 ð3Þ
Δp (Pa) � pressure difference; L (m) � characteristic length; σ (N m21) � surface
tension; R (kg m23) � density; η (Pa s) � dynamic viscosity; g (m s22) � gravita-
tional acceleration; Eu (�) � Euler number (1.) (p. 61); Ca (�) � capillary num-
ber (2.) (p. 96); Re (�) � Reynolds number (p. 81); Fr (�) � Froude number (2.)
external (p. 63); We1 (�) � Weber number (1.) (p. 91).
In expression (1), it expresses the ratio of the pressure force to the surface
stress. In expression (2), it expresses the ratio of the surface stress to the momen-
tum transfer in a fluid or to the gravitation force in a flowing mixture. In case (3),
it describes the gas�fluid mixture flow in a vertical pipeline or in distillation
plants.
Info: [A29],[A35],[C76].
Pierre-Simon, Marquis de Laplace (23.3.1749�5.3.1827),
French physicist, mathematician, astronomer and statesman.
He was engaged in mathematical analysis, probability
theory and celestial mechanics, in which he summarized
the results of his predecessors in the five-volume work
Mecanique Celeste (Celestial Mechanics). He formulated a
hypothesis on the origin of the solar system from a rotating
nebula. He was engaged in the integral transform known
today as the Laplace transform. He contributed to the calori-
metric theory and the determination of the thermal capaci-
ties of many materials. He discovered the gravitational
potential.
3.1.52 Laval Number La
La5w
acritð1Þ; La5w
2κκ1 1
rT
� �2 12
ð2Þ
w (m s21) � flow velocity; acrit (m s21) � critical sound velocity; r (J kg21 K21) �specific gas constant; T (K) � temperature; κ (�) � specific heat ratio (p. 28).
It expresses the ratio of the flow rate to the critical sound velocity. It charac-
terizes incompressible fluid flow. It is also called the critical Mach number Mcrit
and sometimes is denoted as M�. It is analogous to the Crocco number Cr (p. 58).
Info: [A26],[A29],[A35].
71Fluid Mechanics
Carl Gustaf Patrik de Laval (9.5.1845�2.2.1913), Swedish
engineer of French origin.
In 1887, he built small steam turbines to show how these
plants could be constructed on a large scale. Then, in the
year 1890, he developed the Laval nozzle enabling the
increase of the steam velocity entering a turbine. At present,
the principle of this nozzle is still utilized, for example, in
missile construction. He contributed significantly to the
development of centrifugal separators and other devices.
3.1.53 Lift Coefficient CL
CL 5FL
1
2Rw2A
FL (N) � lift force representing vertical aerodynamic drag; R (kg m23) � fluid den-
sity; w (m s21) � flow velocity; A (m2) � cross-sectional area of body.
Its main importance is in aerodynamics. It expresses the ratio of the vertical
component of the resistance force to the inertia force. It depends strongly on the
inclination angle. For the lift force perpendicular to the drag force, the following
equation for object lifting is valid:
FL 5CL
1
2Rw2A
Info: [A7],[A20],[A21].
3.1.54 Lift-to-Drag Ratio CLD
CLD 5FL
FD
5CL
CD
FL (N) � buoyancy force; FD (N) � drag force; CL (�) � lift coefficient (p. 72);
CD (�) � drag coefficient (p. 60).
It expresses the efficiency measure of an aerodynamic profile. The greater this
number is, the better the wing is.
Info: [A8],[A21].
3.1.55 Loss Factor ζ, ζc, K
ζ5ez
ekð1Þ; ζc 5 2
Xni5 1
ezi
w2ð2Þ; ζ5
Δp12Rw2
ð3Þ
ez (J kg21) � specific loss energy; ek (J kg
21) � specific kinetic energy; w (m s21) �flow velocity; Δp (Pa) � head loss; R (kg m23) � fluid density.
72 Dimensionless Physical Quantities in Science and Engineering
This factor expresses the ratio of the specific loss energy to the specific kinetic
energy (1). In expression (2), it is about the relation of the total energy loss to the mean
kinetic energy. In equation (3), it is the loss coefficient for pipelines. Fluid mechanics.
Info: [A21].
3.1.56 Lyashenko Number Ly
Ly5Re3Ar21
Re (�) � Reynolds number (p. 81); Ar (�) � Archimedes hydrodynamic number
(p. 53).
This number expresses the influence of the inertia force, gravitation force and
viscosity. Fluidization.
Info: [A4],[A35].
3.1.57 Mach Number M, Ma
M5w
að1Þ; M5
wffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðκ21ÞcpTp ð2Þ;
where a5ffiffiffiffiffiffiffiffiffiE=R
pfor any liquid, a5
ffiffiffiffiffiffiffiffiffikRT
pfor ideal gas; w (m s21) � local flow
velocity; a (m s21) � local sound velocity; κ (�) � specific heat ratio (p. 28); cp(J kg21 K21) � specific heat capacity; T (K) � gas temperature; E (Pa) � modulus
of elasticity; R (kg m23) � density.
In form (1), it expresses the ratio of the local flow rate to the sound velocity:
elastic deformation propagation or the mean velocity of the thermal molecule
movement in gases. It characterizes the fluid flow in a given place or, alternatively,
the enthalpy conversion into the kinetic energy in a flowing ideal gas. For M, 1 it
is about subsonic velocities, for M 5U 1 about sonic ones, for M. 1 about super-
sonic ones, for 0.8,M, 1.3 about transonic ones and for Mc1 about hypersonic
flow. With a hypersonic flow (M. 5) in the flow core, the fluid starts to behave as
a plasma due to high temperature. In the subsonic zone, it is a measure of the ideal
gas compressibility (2). The compressibility influence appears over M .~
0:3:In France, it is also called the Sarrau number Sa (p. 85) or the Bairstow number
NBa (p. 54) in England, though this is not yet used in general practice. Sometimes,
it is called the Maievski number Ma (p. 74).
Info: [A2],[B20].
Ernst Mach (18.2.1838�19.2.1916), Austrian physicist of
Moravian origin.
He influenced considerably the development of science in the
twentieth century. The idea that recognition of the laws of nature
is mediated by human senses forms the basis of his philosophy.
The Mach principle expresses inertia effects as a property of the
whole universe. Using this concept, Einstein began to formulate
the general relativity theory. He was engaged, too, in experimen-
tal physics and the construction of devices.
73Fluid Mechanics
3.1.58 Mach Angle μ
μ51
sinðM21Þ ; for M-1 it holds μ-90�
M (�) � Mach number (p. 73).
It expresses the Mach cone angle in supersonic fluid flow. High velocity
aerodynamics.
Info: [A21].
Ernst Mach (see above).
3.1.59 Mach Characteristic Number Machar, M0
Machar 5w
a0
w (m s21) � flow velocity; a0 (m s21) � sound velocity at Mchar5 1.
The Mchar is not the usual sonic Mach number M (p. 73), but the number related
to the sound velocity reached with the sonic Mach number. This Mach number form
is used to simplify the basic equations only; see the Prandtl relation M2 (p. 79).
Info: [A21].
Ernst Mach (see above).
3.1.60 Mach Number Across a Shock M
M5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðγ21ÞM�2 1 2
2γM�2 2 ðγ2 1Þ
s
γ (�) � specific heat ratio (p. 28); M� (�) � Mach number of inclination wave,
see Mach number M (p. 73).
For a normal shock wave, the Mach number across a shock expresses the
changes across the shock, which are a function of the drain Mach number only. For
the Mach number of inclination wave M� 5 1, the Mach number across a shock is
M5 2. For M�-N, it is M5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðγ21Þ=2γ
p: Fluid mechanics. Aerodynamics of
high velocities.
Info: [A21].
Ernst Mach (see above).
3.1.61 Maievski Number Ma
See Mach number M, Ma (p. 73).
Nikolaj Vladimirovic Majevskij (1823�1892), Russian physicist and
mechanist.
74 Dimensionless Physical Quantities in Science and Engineering
3.1.62 Mass Ratio Nm
Nm 5m
πRL3
m (kg) � mass; R (kg m23) � fluid density; L (m) � characteristic length.
It characterizes the ratio of a submerged body weight to that of the surrounding
fluid.
Info: [A29],[B20].
3.1.63 Moment Coefficient CM
CM 5M
12Rw2Ac
M (N m) � total moment acting on wing; R (kg m23) � fluid density; w (m s21) �flow velocity; A (m2) � drag area c (m) � mean chord of aerofoil section.
This moment is important in the aerodynamic stability of aircraft and other
objects. For the moment necessary for tilting, it is
M5CM
1
2Rw2Ac
Info: [A20].
3.1.64 Momentum Number NM
NM 5MNδRηΔw
MN (m2 s22) � specific energy of momentum flux; δ (m) � layer thickness; R (kg m23)
� density; η (Pa s)� dynamic viscosity;Δw (m s21)� velocity difference.
It expresses the ratio of the specific energy of a momentum flux to the momen-
tum change of viscous fluid.
Info: [B20].
3.1.65 Newton Number Ne
Ne5F
Rw2L25 0; 5CF
F (N) � hydrodynamic force; R (kg m23) � fluid density; w (m s21) � flow veloc-
ity; L (m) � characteristic length; CF (�) � force coefficient (p. 62).
75Fluid Mechanics
This number expresses the loading-to-inertia forces ratio. It characterizes the
friction in flowing fluids. In the case of viscous fluid flow, it expresses the turbuli-
zation process in flowing fluid. In a pressure flow, it is Ne5Eu; in a gravitation
flow, it is Ne5Fr2 1, and when considering the internal friction the Ne5Re2 1 is
valid. It also expresses dynamic force processes in vibration, mixing, material
deformation and other processes. The Fanning friction number fF (p. 163) and
the force coefficient CF (p. 62) are also analogues to the Newton number.
Hydrodynamics.
Info: [A4],[A29],[A35],[B17].
Isaac Newton (4.1.1643�31.3.1727), English physicist,
mathematician, astronomer and philosopher.
He laid the foundations of differential and integral calcu-
lus. His works on optics and gravitation made him one of
the greatest scientists of the world. In his book, Principia
(Principles), he formulated three movement laws: of inertia,
of action and reaction, and of acceleration proportional to
force. He was ingenious not only in mathematics but also in
experimentation, which enabled him to formulate new and
simpler laws of mechanics and hydromechanics.
3.1.66 Number of Velocity Heights Nvh
Nvh 5F
RL2
� �w2
2
� �21
F (N) � force; R (kg m23) � density; L (m) � characteristic length; w (m s21) �fluid velocity.
It expresses the ratio of the utilized pressure height to the velocity height.
Friction in pipelines and other water systems.
Info: [A35].
3.1.67 Nusselt Film Thickness Parameter Nuft
Nuft 5 L
ffiffiffiffiffig
v23
r� Ga
13
L (m) � characteristic length (layer thickness); g (m s2) � gravitational accelera-
tion; ν (m2 s21) � kinematic viscosity; Ga (�) � Galilei number (p. 123).
This parameter represents a special case of the Galilei number Ga (p. 123) for
thin film creation.
Info: [B20].
Ernst Kraft Wilhelm Nusselt (p. 196).
76 Dimensionless Physical Quantities in Science and Engineering
3.1.68 Oberbeck Number Ob
Ob5L3w2RQ
L (m) � characteristic length; w (m s21) � flow velocity; R (kg m23) � fluid den-
sity; Q (J) � heat.
It expresses the ratio of flowing fluid kinetic energy to heat transfer.
Info: [A6].
3.1.69 Outflow Coefficient μ
μ5αϕ; whereas μ, 1
α (�) � contraction coefficient (p. 57); ϕ (�) � coefficient of velocity (p. 57).
Using this, the velocities and contractions are considered in fluid outflow or pas-
sage. Aero-hydrodynamics.
3.1.70 Pipe Friction Coefficient Λ, Cf, cf
Λ5ez
ek
D
L
ez (J kg21) � friction loss energy; ek (J kg
21) � specific kinetic energy; D (m) �pipe diameter; L (m) � pipe length.
It expresses the friction loss in fluid flow in pipelines. Hydraulics.
Hydrodynamics.
3.1.71 Pipeline Parameter Np, .
Np 5wpwini
2ghs
wp (m s21) � pressure wave velocity; wini (m s21) � initial fluid velocity; g (m s22) �gravitational acceleration; hs (m) � static head.
It expresses the ratio of the highest pressure in a water stroke to the static pres-
sure. It characterizes the pressure originating in a pipeline during the water stroke.
Hydraulics.
Info: [B11],[B20].
77Fluid Mechanics
3.1.72 Pohlhausen Number Ph
Ph5δ2
vRwN
dp
dL
δ (m) � boundary layer thickness; ν (m2 s21) � kinematic viscosity; R (kg m23) �density; wN (m s21) � rise velocity; p (Pa) � pressure; L (m)� characteristic length.
In tasks on a boundary layer, it expresses the influence of the outer flow proper-
ties on the velocity distribution in a certain boundary layer cross section and on the
velocity profile change in the flow direction. It is the measure of the pressure-
to-viscous forces ratio in flowing fluid.
Info: [A19].
E. Pohlhausen.
3.1.73 Poiseuille Number (1.) Ps
Ps5D2Δp
ηwL
D (m) � pipe diameter; Δp (Pa m2 1) � loss of pressure along the pipe; η (Pa s) �dynamic viscosity; w (m s21) � flow velocity; L (m) � characteristic length.
It expresses the pressure-to-friction forces ratio in fluid laminar flow in a pipeline.
Info: [B20].
Jean Louis Marie Poiseuille (22.4.1797�26.12.1869),
French doctor and physiologist.
He was educated in physics and mathematics. He was
engaged in blood flowing in narrow tubes. In 1840 and 1846,
he formulated and published the Poiseuille law, which he
had deduced by experiment in 1838. This describes the
stationary flowing of incompressible viscous fluid through
cylindrical tubes. He applied the results to blood flowing
through capillaries, arteries, pulmonary alveolas and the like.
3.1.74 Prandtl Dimensionless Distance X
X5xffiffiffiffiffiffiffiffiRτw
pη
x (m) � distance from wall surface.; R (kg m23) � fluid density; τw (Pa) � surface
tension on the wall; η (Pa s) � dynamic viscosity.
It is used to study turbulent flow.
Info: [A29].
Ludwig Prandtl (p. 197).
78 Dimensionless Physical Quantities in Science and Engineering
3.1.75 Prandtl�Glauert rule Cp, CL, CD
Cp 5Cp0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12M2
N
p ð1Þ; CL 5CL0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12M2
N
p ð2Þ;
Cp 52ϑffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
M2N 2 1
p ð3Þ; CL 54αffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
M2N 2 1
p ð4Þ;
CD 54α2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2
N 2 1p ð5Þ;
where MN 5wN
að6Þ
Cp0 (�) � uncorrected pressure coefficient for incompressible fluid, see pressure
coefficient Cp (p. 80); CL0 (�) � uncorrected buoyancy coefficient for incompress-
ible fluid, see lift coefficient CL (p. 72); α (�) � incidence angle of thin aerofoil
section; ϑ (�) � fluid incidence angle; wN (m s21) � velocity of free-stream con-
ditions; a (m s21) � local sound velocity; MN (�) � Mach number of free-stream
conditions, see Mach number M (p. 73).
The dimensionless quantities Cp, CL and CD are the pressure, lift and drag coef-
ficients, respectively, corrected for a compressible fluid. See the pressure coeffi-
cient Cp (p. 80), the lift coefficient CL (p. 72) and the drag coefficient CD (p. 60).
The expressions (1) and (2) express the corrected relations for the fluid compress-
ibility in high subsonic flow which is characterized by the extent (0.3,MN, 0.8),
where MN is the Mach number of free-stream conditions, defined by the relation
(6). The expressions (3)�(5) are valid for supersonic flow.
Info: [A21].
Ludwig Prandtl (see above).
3.1.76 Prandtl Relation M2
M1;critM2;crit 5 1 ð1Þ;
M1;crit 5w1
acritð2Þ; M2;crit 5
w2
acritð3Þ
w1, w2 (m s21) � flow velocity before and after shock; acrit (m s21) � critical sound
velocity; M1,crit (�) � Mach number before shock considering the critical value
instead of local sound velocity, see Mach number M (p. 73); M2,crit (�) � Mach
number after shock considering the critical value instead of local sound velocity,
see Mach number M (p. 73).
It expresses the case of a fluid hit by a permanent shock wave. It simplifies both
Mach numbers (2) and (3) determination significantly because the sound velocity is
79Fluid Mechanics
equal on both sides of the shock, compared to the local velocity, and the velocities
can be measured very easily. In comparison to it, the pressure, temperature, density,
enthalpy and entropy increase. The expressions (1)�(3) are valid for a perfect gas,
and for supersonic and subsonic conditions. Fluid mechanics. High velocity aerody-
namics. Shocks.
Info: [A21].
Ludwig Prandtl (see above).
3.1.77 Prandtl Velocity Ratio Pr, u1
Pr5w
ffiffiffiRτ
r
w (m s21) � flow velocity; R (kg m23) � density; τ (Pa) � shear stress.
This is the ratio of the inertia force to the shear force arising on a wall by the
action of a flowing fluid. It characterizes the degree of fluid flow turbulence.
Hydromechanics.
Info: [A29],[B20].
Ludwig Prandtl (see above).
3.1.78 Pressure Coefficient Cp
Cp 5Δp
12RNw2
N
5p2 pN12RNw2
N
Δp (Pa) � pressure difference in fluid; p (Pa) � static pressure; pN (Pa), RN(kg m23), wN (m s21) � pressure, density and velocity of free flow.
It expresses the ratio of the overpressure in a fluid to the dynamic pressure in a
free flow. It is a special case of the Newton number Ne (p. 75).
Info: [A7],[A21],[A29].
3.1.79 Pressure Number (1.) Np
Np 5pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
gηðRl 2 RgÞp
p (Pa) � total pressure in system; g (m s22) � gravitational acceleration; η (Pa s) �dynamic viscosity; Rl, Rg (kg m23) � liquid and gas density.
It expresses the ratio of the total system pressure to the pressure gradient on a
fluid surface.
Info: [A29].
80 Dimensionless Physical Quantities in Science and Engineering
3.1.80 Pressure Recovery Coefficient Cp,rec
Cp;rec 5pout 2 pin
1
2R2in
pout (Pa) � output pressure; pin (Pa) � inlet flow pressure; R (kg m23) � liquid
density; win (m s21) � inlet flow velocity.
It expresses the ratio of the difference of static inlet and outlet pressures to the
inlet dynamic pressure. It characterizes the pressure rise in flow which is typical
for diffusers. Fluid mechanics. Aeromechanics.
Info: [A21].
3.1.81 Reech Number Ree
Ree5gL
w25Fr21
g (m s22) � gravitational acceleration; L (m) � characteristic length; w (m s21) �velocity; Fr (�) � Froude number (1.) (p. 62).
This number expresses the weight-to-inertia forces ratio. It can be neglected for
gases. It represents the movement of solid bodies in fluid, waves and surface phe-
nomena. Gravitation influence on the movement of surface ships.
Info: [A29],[A35].
Ferdinand Reech (1805�1880), French naval instructor.
3.1.82 Relative Amplitude of Fluid Displacement NA
NA 52wmax
ωL5
D
2L
Remax
Va
wmax (m s21) � maximal velocity (amplitude) at fluid flow oscillation; ω (s21) �angular frequency; L (m) � characteristic pipe length; D (m) � pipe diameter; Remax
(�) � Reynolds maximum number (p. 83); Va (�) � Valensi number (p. 90).
It characterizes the pulsation flow in a pipeline. With NA. 1, all initial fluid
passes outwards from a pipeline during one cycle. With NA5 1, all initial fluid is
moving in a pipeline. With NA, 1, a part of the initial fluid volume contained in a
pipeline does not leave the pipeline during one cycle.
Info: [A1],[A19].
3.1.83 Reynolds Number Re
Re5wLRη
5wL
vð1Þ; Rerot 5
ωL2
v5
RnL2
ηð2Þ
81Fluid Mechanics
w (m s21) � flow velocity; L (m) � characteristic length; R (kg m23) � density; η(Pa s) � dynamic viscosity; ν (m2 s21) � kinematic viscosity; ω (Hz) � angular
frequency; n (s21) � rotational frequency.
This number expresses the ratio of the fluid inertia force to that of molecular
friction (viscosity). It characterizes the hydrodynamic conditions for viscous fluid
flow. It determines the character of the flow (laminar, turbulent and transient flows).
For a laminar flow Re, 2000 is valid, for a transient flow 2000,Re, 4000, and
for a turbulent flow it is Re. 4000. With low values of the Re number, the viscous
friction muffles the originating dynamic influence of the flow relatively quickly and
intensively, due to which the streamlines and elementary fluid volumes cannot be
deformed substantially and the flow remains laminar. With large Re numbers, the
dynamic flow effect cannot be equalized by viscous friction and the flow stability is
lost, which is manifested by swirls and turbulence in the fluid. The expression (2) is
valid for flow in rotating canals and is often called the Reynolds rotary number.
Info: [A21],[A23],[A29],[A35],[B20].
Osborne Reynolds (23.8.1842�21.2.1912), English engineer
and physicist.
He opened the era of convective heat transfer (1838). He
showed that the flow of fluid through a tube depends on the
relation of the inertial force to the viscosity (Re number);
so he expressed the basic criterion for forced viscous fluid
convection and determined its critical value. In addition,
Reynolds’s studies relate to vapour condensation and fluid
boiling, heat transfer between solid materials and fluid,
lubrication theory (1886) and resistance laws in streaming
(1883).
3.1.84 Reynolds Boundary Layer Number Reδ
Reδ 5wδv
w (m s21) � velocity; δ (m) � layer thickness; ν (m2 s21) � kinematic viscosity.
It characterizes the viscous fluid flow in the boundary layer near a bypassed
body surface.
Osborne Reynolds (see above).
3.1.85 Reynolds Curved Number Re
Re5wfL
2v; where wf 5
ffiffiffiffiτRl
r
82 Dimensionless Physical Quantities in Science and Engineering
wf (m s21) � friction velocity; L (m) � characteristic length, curvature radius of layer;
ν (m2 s21) � kinematics viscosity; τ (Pa) � shear stress; Rl (kg m23) � fluid density.
It expresses the friction influence on the heat transfer in a curved fluid layer.
Osborne Reynolds (see above).
3.1.86 Reynolds Entry Number ReE
ReE 5L
DRe
L (m) � input length; D (m) � diameter; Re (�) � Reynolds number (p. 81).
It is applied to solve the flow in inlet flow-through parts.
Info: [B34].
Osborne Reynolds (see above).
3.1.87 Reynolds Maximum Number Remax
Remax 5wmaxL
v5
4Va
Sh
wmax (m s21) � maximal velocity amplitude at oscillation flow; L (m) � character-
istic length, pipe diameter; ν (m2 s21) � kinematic viscosity; Va (�) � Valensi
number (p. 90); Sh (�) � Strouhal number (p. 87).
It characterizes a pulsating flow. Fluid mechanics.
Info: [A1].
Osborne Reynolds (see above).
3.1.88 Reynolds Turbulence Number Returb
Returb 5e2
vε
e (J kg21) � turbulent specific kinetic energy; ν (m2 s21) � kinematic viscosity;
ε (s21) � dissipative velocity of turbulence.
For a turbulent flow, it expresses the kinetic-to-friction energy ratio. Fluid
mechanics.
Info: [A1].
Osborne Reynolds (see above).
3.1.89 Richardson Number Ri
Ri5gh
w25
1
Frð1Þ; Ri5
gredh
w2ð2Þ
83Fluid Mechanics
g (m s22) � gravitational acceleration; h (m) � characteristic length, vertical;
w (m s21) � characteristic flow velocity; gred (m s22) � reduced gravitational
acceleration; Fr (�) � Froude number (1.) (p. 62).
This number expresses the potential-to-kinetic energies ratio. More often, its
inverse value is used: the Froude number (1.) Fr (p. 62). With small flow density
changes (for example in the ocean or in the atmosphere), the Richardson number is
used in expression (2).
Info: [A21].
Lewis Fry Richardson (p. 403).
3.1.90 Roshko Number Ro
Ro5 Sh Re5fL2
v
f (s21) � whirl frequency; L (m) � characteristic length, for example, hydraulic
radius; ν (m2 s21) � kinematic viscosity; Sh (�) � Strouhal number (p. 87);
Re (�) � Reynolds number (p. 81).
It describes the mechanism of a swirl oscillating flow. Compared to the Strouhal
number Sh (p. 87), it describes the character and viscosity of the fluid flow. Its
validity has been verified by bypassing a cylinder in the range ReAh40; 10000i.Fluid mechanics. Turbulence.
Info: [A37],[C118].
Anatol Roshko (born 1911), American physical engineer of
Canadian origin.
He is an authority in aircraft and space engineering, espe-
cially in the spheres of turbulence, swirl flows, fluid mechan-
ics and gas dynamics. In turbulence research, he initiated
new trends primarily with his work on coherent structures. As
a co-author, he wrote the book Elements of Gasdynamics
(1956), which is well known all over the world. He worked
for 40 years as a professor at the California Institute of
Technology and gave remarkable lectures on fluid mechanics.
3.1.91 Russell Number Ru
Ru5w
fh; where f 5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2
g
RdRdz
s
w (m s21) � wind velocity; f (s21) � natural vertical frequency of the fluid part;
h (m) � height of obstruction; g (m s22) � gravitational acceleration; R (kg m23) �density; z (m) � vertical coordinate.
84 Dimensionless Physical Quantities in Science and Engineering
This number expresses the inertia-to-uplift forces ratio. It concerns buoyancy,
gravitation effects, two-phase flow, and shock and surface waves in hydromechan-
ics and waves in stratified flow.
Info: [B20].
3.1.92 Sarrau Number Sa
See Mach number M (p. 73).
Jacques Rose Ferdinand Emile Sarrau, French engineer.
3.1.93 Shear Stress Number NS
NS 5RwIF
R (kg m23) � fluid density; w (m s21) � flow velocity; IF (kg m22 s21) � force
impulse.
It characterizes the shear stress in fluid flow.
Info: [A49].
3.1.94 Schiller Number (1.) Sch
Sch5wL
ffiffiffiffiffiffiffiffiR2
2ηF3
s5
ffiffiffiffiffiffiRe
CD
3
r
w (m s21) � flow velocity; L (m) � characteristic length; R (kg m23) � gas or
liquid density; η (Pa s) � dynamic viscosity; F (N) � resistance force; Re (�) �Reynolds number (p. 81); CD (�) � drag coefficient (p. 60).
This number characterizes the flow around obstacles and expresses the ratio of
velocity to the force acting on a submerged body. It also characterizes the inertia
force of certain sized particles in relation to the resistance coefficient. It also
expresses the ratio of forces acting on a diving body.
Info: [A25],[A29],[A35].
L. Schiller.
3.1.95 Slosh Time NS
NS 5 τffiffiffiffiffiffiffiσgr3
3
r
τ (s) � time; σ (N m21) � surface tension; g (m s22) � gravitational acceleration;
r (m) � pipe radius.
85Fluid Mechanics
It expresses the dimensionless time of fluid strokes in pipelines and in surface
oscillation. Hydraulics.
Info: [B 20].
3.1.96 Smoluchowski Number Sm
It expresses the low pressure gas flow, see Knudsen number (1.) Kn (p. 69).
Info: [A29],[A35].
Marian Smoluchowski (28.5.1872�5.9.1917), Polish physicist.
He was engaged in the phenomena connected with the
thermal movement of molecules and atoms. Independently of
Einstein, he formulated the expression explaining the Brown
movement. He worked on the statistical expression of the
second law of thermodynamics. The whole range of his work
was an important contribution to science. It was not only
about physics, but also about physical chemistry, geophysics
and geology. He was the first to calculate the zeta potential
of electrokinetic movement.
3.1.97 Speed Number Pw, W
Pw 5w
wmax
w, wmax (m s21) � local and maximal velocity.
It expresses the local-to-maximum velocities ratio. Aero-hydrodynamics.
Info: [A24].
3.1.98 Stokes Number (6.) Stk
Stk5 1:042mFgR 12RRF
� �R3
η2
mF (kg) � float mass; g (m s22) � gravitational acceleration; R (kg m23) � fluid
density; RF (kg m23) � float density; R (�) � pipe and float radii ratio; η (Pa s) �dynamic viscosity.
It is used to calibrate the rotameters used in measuring fluid passage. Measuring
techniques.
Info: [A35].
George Gabriel Stokes (p. 131).
3.1.99 Stratification Parameter S
S5ω
2ωC
86 Dimensionless Physical Quantities in Science and Engineering
ω (s21) � angular frequency; ωC (s21) � Coriolis frequency.
It characterizes the fluid flow with rotation and action of the Coriolis force. For
S{1, stratified flow occurs (laminar flow, etc.). For Sc1, the rotation influence
prevails. Fluid mechanics. Geophysics.
Info: [C124].
3.1.100 Strouhal Number Sh
Sh � Th5wτL
ð1Þ; Sh5w
nLð2Þ;
Sh5fL
wð3Þ; Sh5
w
fLð4Þ
w (m s21) � velocity; τ (s) � time; L (m) � characteristic length; n (s21) � rota-
tional frequency; f (Hz) � frequency; Th (�) � Thomson number (p. 90).
This number describes the oscillating flow mechanism and expresses the vibration-
to-flow velocities ratio; alternatively, it relates the ratio of local force, caused by a
non-stationary process, to the inertia force. It represents the universal dynamic simi-
larity criterion of non-stationary processes in systems which are similar in terms of
geometry and kinematics. It determines the non-stationary to convective momentum
transfer in a system. It is the measure for movement non-stationarity. With Sh$ 1, the
local accelerations are commensurate with the convective ones and the movement is
non-stationary. On the contrary, in stationary movement, Sh 5_ 0 is valid.
It is known as the concept of reduced frequency. Sometimes, it is called
the Thomson number Th (p. 90), or the acceleration (accelerating) ratio in expres-
sion (2).
Info: [A29],[A43],[A95],[B20].
Vincenc Strouhal (10.4.1850�23.1.1922), Czech experi-
mental physicist.
He was engaged in hydrodynamic phenomena, acoustics
and electric and magnetic properties of steel. In the sphere of
gas and fluid dynamics, his experimental work led to the
determination of the dimensionless parameter which is called
the Strouhal number. He wrote several publications on exper-
imental physics: Mechanika (Mechanics, 1901), Akustika
(Acoustics, 1902), Thermika (Thermics, 1908) and Optika
(Optics, 1919).
3.1.101 Strouhal Turbulence Number Shturb
Shturb 5jðwrÞwj
@w
@τ
� wτcorLturb
~τcorτturb
87Fluid Mechanics
w (m s21) � flow velocity; τ (s) � running time; τcor (s) � correlation time; τturb(s) � fluctuation time; Lturb (m) � characteristic length of turbulence.
In the dimensionless form, it expresses the ratio of the correlation time to the
fluctuation time of turbulent flow. Fluid mechanics. Astrophysics.
Info: [B58].
Vincenc Strouhal (see above).
3.1.102 Swirl Number Sw
Sw5w
u
w (m s21) � tangential velocity component; u (m s21) � axial velocity component.
It expresses the swirl intensity in flow. It is given by the tangential-to-axial
momentum ratio. The tangential velocity is the mean tangential velocity component
in the outlet plane of a passage canal. The axial velocity is the longitudinal move-
ment velocity in the canal plane.
Info: [C127].
3.1.103 Szebehely Number Sz, Σ
Sz5j@w=@τjjðwUrÞwj
w (m s21) � flow velocity; τ (s) � time.
The Szebehely number characterizes the unsteadiness measure of a fluid flow. It
expresses the ratio of the fluid flow local acceleration to the convective accelera-
tion. Conforming to this criterion, the flow can be divided into the following
categories:
� steady flow with acceleration (Sz5 0),
� unsteady flow without acceleration (Sz5 1),
� steady flow without acceleration (Sz not determined),
� unsteady flow with acceleration (0, Sz#N, Sz 6¼ 1). Sz5N corresponds to unsteady
flow with zero convective acceleration.
The Szebehely number follows from two definitions for the velocity field:
� a steady flow is that for which @w=@τ5 0 is valid and
� a flow without acceleration is that for which Dw=Dt5 ð@w=@tÞ1ðwUrÞw5 0:
Therefore, an arbitrary velocity field can be assigned to one of the four catego-
ries together with the corresponding Szebehely number value.
Info: [B5].
88 Dimensionless Physical Quantities in Science and Engineering
Victor G. Szebehely (1921�13.9.1997), American space
engineer of Hungarian origin.
He was among the key persons in the US Apollo space
program, which resulted in the landing of a man on the
moon. He was engaged in applied mathematics, especially in
the dynamics of seaborne and space ships. In 1956, he used
the dimensionless number to express time-dependent nonsta-
tionary flowing, which is named the Szebehely number.
Similarly, the Szebehely equation, which determines the
gravitational potential of the earth, planets, satellites and
galaxies, is named after him.
3.1.104 Taylor Number Ta
Ta52wL2
v
� �2ð1Þ; Ta5
f 2L4
v25Re2rot ð2Þ;
Ta5ωr
121ðr2 2 r1Þ
32
vð3Þ; Ta5Re
12d
23v21 ð4Þ;
where Tacrit 5 41:3
ω (Hz) � angular frequency; L (m) � characteristic length (thickness of flow gap
or liquid); ν (m2 s21) � kinematic viscosity; f (s21) � frequency; r2, r1 (m) � outer
and inner gap radii; d (m) � gap width between the cylinders (d5 r22 r1); Rerot(�) � Reynolds rotation number (p. 383).
This number expresses the centrifugal and viscous forces; alternatively, the sec-
ond power of the Coriolis force and the viscous force. It characterizes the rotation
influence on free convection. It is used in expressions (1) or (2). In expression (3),
it expresses a flow with viscous fluid swirl instability originating in a ring canal
between two concentric circular cylinders of which the inner one rotates and the
outer one is stable. In this case, with Ta, 41.3 the flow is laminar, for
41.3# Ta, 400 the flow is laminar with steady origination of Taylor swirls, and
for Ta. 400 the flow is turbulent.
Info: [A21],[A35],[B11],[B20].
Geoffrey Ingram Taylor (7.3.1886�27.6.1975), English
mathematician, physicist, aerodynamicist and meteorologist.
Having started with the theoretical study of shock waves, he
continued with the work of J.J. Thompson. Taylor was engaged
in his proposals and verified quantum theory experimentally.
His other spheres of interest included dynamic meteorology �including his work on atmospheric turbulence � and
oceanography.
89Fluid Mechanics
3.1.105 Thien Parameter NT
NT 5Maϑ
ϑ (�) � flow angle; Ma (�) � local Mach number (p. 73).
It characterizes the hypersonic flow for cases of sin ϑ-0 (NT{1). In these cases,
it enables linearization of complicated non-linear equations of the supersonic flow.
Info: [A21].
3.1.106 Thoma Number (1.) Th
See Cavitation number Ncav (p. 366).
Dieter Thoma (1881�1942), German hydraulic engineer.
3.1.107 Thomson Number Th
Th � Sh5wτL
w (m s21) � flow velocity; τ (s) � time; L (m) � characteristic length; Sh (�) �Strouhal number (p. 87).
It describes oscillation flow. It is identical with the Strouhal number Sh (p. 87).
Info: [A29],[A35].
James Thomson (1882�1992), Irish engineer.
3.1.108 Transiency Group Nt
Nt 51
@p
@L
@@p
@L
� �@ðFoÞ ð1Þ; Nt 5
1
Re
@ðReÞ@ðFoÞ ð2Þ
@p=@L (Pa m21) � pressure gradient in the flow direction; Fo (�) � Fourier
number (p. 175); Re (�) � Reynolds number (p. 81).
It expresses transient changes of flowing fluid behaviour.
Info: [A35].
3.1.109 Turbulence Number Tu
See Karman turbulent number Ka, Tu (p. 67).
3.1.110 Valensi Number Va
Va5r2ωv
90 Dimensionless Physical Quantities in Science and Engineering
r (m) � inner pipe radius; ω (s21) � angular oscillation frequency; ν (m2 s21) �kinematic viscosity.
This number expresses the ratio of the inner tube radius to the viscous diffuse
length in fluid flowing through a tube with a periodic boundary condition. Fluid
dynamics.
Info: [A1],[A29],[A35].
Jacques Valensi (born 1903), French engineer.
3.1.111 Vedernikov Number Ve
Ve5ζξw
wA 2w
ζ (�) � exponent of hydraulic radius; ξ (�) � form factor of cross section;
w (m s21) � mean flow velocity; wA (m s21) � absolute velocity of wave
disturbance.
It describes the flow instability in an open canal. Hydraulics.
Info: [A33],[A35].
3.1.112 Wave Period NT
NT 5Twαh
T (s) � period of water wave; w (m s21) � mean velocity of steady flow; α (�) �canal slope; h (m) � steady water depth in canal.
It is applied to determine the opening time of irrigation canal gates. Hydraulics.
Info: [A47].
3.1.113 Weber Number (1.) and (2.) We
We1 5Rw2L
σð1Þ; We2 5 ðWe1Þ
12 5w
RLσ
� �12
ð2Þ
R (kg m23) � fluid density; w (m s21) � movement velocity; L (m) � characteristic
length; σ (N m21) � surface tension.
These numbers express the ratio of the inertia force to the surface stress force.
The Weber number is connected with surface stress waves, as the Froude number
(2.) external Fr (p. 63) is with gravitational waves. The We number characterizes
the originating process of bubbles or drops during fluid boiling or steam condensa-
tion. Sometimes, it is used in expression of the second root (2) and is called the
Weber number (2.).
Info: [A7],[A21],[A43],[B20].
91Fluid Mechanics
Ernst Heinrich Weber (24.6.1795�26.1.1878), German
psychophysicist, anatomist, physiologist.
He laid the foundation for psychophysics, which involves
psychological reactions to physical stimuli. Together with his
younger brother, Wilhelm Eduard Weber, he was engaged in
research on fluid flow. The well-known Weber criterion for
viscous liquid flow is named after them. Together, they
wrote the book Wave Theory and Fluidity.
Wilhelm Eduard Weber (24.10.1804�23.6.1891), German
physicist.
Already at only 20 years of age, he was engaged in fluid
flow research, with his elder brother Ernst Heinrich Weber.
He also devoted himself to the study of magnetism and mag-
netic tension measurement. He formulated a logical system
of units for electricity and tried to unify electricity and mag-
netism into a unique basic law of forces. He described elec-
tric current as a flow of electrons. The magnetic flux unit
was named after him.
3.1.114 Weber Rotation Number Werot
Werot 5n2L3Rσ
n (s21) � rotation frequency; L (m) � characteristic length, diameter; R (kg m23) �fluid density; σ (N m21) � surface tension.
It is the inertia-to-capillary forces ratio. It characterizes the process of forced
fluid mixing. Hydromechanics. Blade machines.
Info: [A35].
Ernst Heinrich Weber (see above).
Wilhelm Eduard Weber (see above).
3.1.115 Womersley Number Wo
Wo5 L
ffiffiffiffiωv
r5
ffiffiffiffiffiffiffiffiffiffiffiffiRe Sh
pð1Þ; Wo5 L
ffiffiffiffiffiffiffiffiffi2πfnv
rð2Þ
L (m) � characteristic length, half-thickness; ω (s21) � angular frequency (ω5 2πf);ν (m2 s21) � kinematic viscosity; f (s21) � pulsation frequency; n (�) � degree of
harmonic frequency; Re (�) � Reynolds number (p. 81); Sh (�) � Strouhal number
(p. 87).
92 Dimensionless Physical Quantities in Science and Engineering
It characterizes the non-stationary oscillating flow which manifests itself as the
response to pressure gradient oscillations. With composed polyharmonic pulsating
flow, the expression (2) is valid. The Wo number shows whether the flow is quasi-
stationary or not. It is also important in biomechanics in the flow of biologically
significant fluids such as air, water and blood. With Wo, 1, a quasi-stationary
flow can be presumed, with an oscillating pressure gradient and a parabolic veloc-
ity profile with the greatest amplitude. With Wo. 1 the flow is phase shifted in
time as compared with pressure gradient oscillations, and the fluid oscillation
amplitude can either grow or drop. Fluid mechanics. Aeroelasticity. Biomechanics.
Info: [B69].
3.1.116 Zhukovsky Number Zh
See Fourier hydrodynamic number Foh (p. 62).
Nikolay Yegorovich Zhukovsky (p. 33).
3.2 Multiphase Fluid Mechanics
Usually, this concerns the similarity criteria for two-phase flow in which mutual
chemical interference does not occur. It is about various combinations of two
phases, such as, for example, water�steam, water�air and solid particles (dust) in
the air or in fluid. Flow with more than two phases occurs, for example, with explo-
sions. The criteria concern such things as the solid particle movement in fluid, the
dynamics of increasing transfer and collapse of bubbles in fluids, flow accompanied
with boiling or condensation and granulation flow. The Bagnold, Camp, Darcy,
gravitational, capillary, Kolmogorov, Leibenzonov, Leverette, Morton, Ohnesorge
and Stokes numbers are among the wide range of similarity criteria applied.
3.2.1 Accelerating Frequency Nf
Nf 5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΔRω2L3
σ
r
ΔR (kg m23) � density difference; ω (s21) � angular velocity; L (m) � characteris-
tic length; σ (N m21) � surface tension.
This frequency characterizes the fluid flow acceleration in microgravitation
conditions. Essentially, it is an analogue to the Weber rotation number Werot(p. 92), which compares inertia and capillary forces. Two-phase flow.
3.2.2 Bagnold Number (1.) Bg
Bg53CDRgw
2
4dgRs
93Fluid Mechanics
CD (�) � drag coefficient (p. 60); Rg, Rs (kg m23) � density of fluid and solid
particles; w (m s21) � flow velocity; d (m) � particle diameter; g (m s22) � gravi-
tational acceleration.
It is the ratio of the front resistance of bodies (particles) to their weight. It char-
acterizes the carrying along of the solid particles by a fluid. Hydromechanics.
Ecology. Soil erosion.
Info: [A26],[B20].
Ralph Alger Bagnold (1896�1990), English geophysicist,
soldier and traveller.
His observations of sand movement while travelling in the
African desert led to his interest in the physical substance
and movement of sand dunes. His efforts resulted in the
monograph The Physics of Blown Sand and Desert Dunes
(1941). Later he also devoted himself to solid material water
stream transport. He was noted not only for his study of sand
dunes on earth, but also those on Mars.
3.2.3 Bagnold Number (2.) Bg
Bg5VS
12VS
RSL2 _τ
ηð1Þ; Bg5
V13S
V13max 2V
13S
0B@
1CA
12
RsL2 _τ
ηð2Þ
VS (m3) � volume of granular particles; RS (kg m23) � particle density; L (m) �characteristic length, grain diameter; _τ (s21) � unit speed of shear deformation;
η (Pa s) � fluid dynamic viscosity in pores; Vmax (m3) � maximal possible volume
grain concentration.
It expresses the relation between the number of collisions and the viscous tension
in steady and uniform shear flow. With Bg, 40, the macroviscous mode occurs, in
which the tensions are proportional to the shear tension velocity. With Bg. 200,
the collision mode prevails and the tension value is proportional to the second
power of the shear tension velocity. The expression (2) is the improvement of (1).
Info: [A7],[B13].
Ralph Alger Bagnold (see above).
3.2.4 Blake Number Bl
Bl5wRL
ηð12 eÞ
w (m s21) � flow velocity; R (kg m23) � density; L5V S2 1 (m) � characteristic
length of particle; V (m3) � volume of particle; S (m2) � surface of particle;
η (Pa s) � dynamic viscosity; e (�) � porosity (p. 24).
94 Dimensionless Physical Quantities in Science and Engineering
This number expresses the inertia-to-viscous forces ratio in momentum and mass
transfers through loose material layers. It characterizes the flow in loose material
transport. Essentially, it is a modification of the Reynolds number Re (p. 81).
Info: [A35],[B20].
J.R. Blake.
3.2.5 Bond Number Bo
Bo5Eo5ΔRgL2
σ5We1Fr
21
ΔR5 Rl2 Rg (kg m23) � difference between liquid and gas or vapour density; g
(m s22) � gravitational acceleration; L5 dB (m) � diameter of water drop or bub-
ble; σ (N m21) � surface tension; Eo (�) � Eotvos number (p. 99); We1 (�) �Weber number (1.) (p. 91); Fr (�) � Froude number (1.) (p. 62).
It expresses the ratio of the gravity force to the surface stress force. It charac-
terizes the movement of free bubbles or little drops in a stationary fluid, further the
vapour strokes arising due to energy release in fast mixing of heated and cold evap-
orating fluids. It applies to the processes in nuclear reactors and in combustion
engines, for example. Generally, it characterizes the fluid spraying process in a
two-phase environment flow. It is often called the Eotvos number Eo (p. 99).
See also Goucher number Go (p. 65). Fluids atomization. Capillary flow. Stroke
and surface waves. Bodies immersed in fluid. Spraying of fluids.
Info: [A35],[B20].
Wilfrid Noel Bond (1897�1937), English physicist.
3.2.6 Boussinesq Number (2.) Bob
Bob 5ηsηrb
ηs (Pa m s) � surface shear viscosity; η (Pa s) � liquid dynamic viscosity; rb (m) �bubble diameter.
It expresses the characteristic radius of bubbles in flowing foamed fluid. Two-
phase flow.
Info: [B109].
Valentin Joseph Boussinesq (p. 55).
3.2.7 Brownell�Katz Number BK
BK5Ca Bo5ηwRgL2
σ2
95Fluid Mechanics
η (Pa s) � dynamic viscosity; w (m s21) � flow velocity; R (kg m23) � density;
g (m s22) � gravitational acceleration; L (m) � characteristic length, diameter;
σ (N m21) � surface tension; Ca (�) � capillary number (1.) (p. 97); Bo (�) �Bond number (p. 95).
This number characterizes the flow through the porous material. It expresses the
combination of the capillary number (1.) Ca (p. 97) and the Bond number Bo (p. 95);
alternatively, it is the ratio of viscous and gravitational forces product to the second
power of the surface stress force. Two-phase flow. Filtration. Draining. Drying.
Info: [C38].
3.2.8 Camp Number Ca
Ca5
ffiffiffiffiffiffiffiffiPV
ηQ2
s
P (W) � energy dissipation in fluid volume V ; V (m3) � fluid volume; η (Pa s) �dynamic viscosity; Q (m3 s21) � volume flow.
It expresses the ratio of the elapsed time to the average shear off value in a
fluid. It is the criterion for free clustering of suspended particles.
Info: [A29].
Thomas Ringgold Camp (born 1895), American engineer.
3.2.9 Capillary Buoyancy Number CaB
CaB 5gη4
Rσ35We31Fr
21Re24
g (m s22) � gravitational acceleration; η (Pa s) � dynamic viscosity; R (kg m23) �density; σ (N m21) � surface tension; We1 (�) � Weber number (1.) (p. 91);
Fr (�) � Froude number (1.) (p. 62); Re (�) � Reynolds number (p. 81).
It characterizes the influences of the surface tension, viscosity and acceleration
in two-phase flow when fluid balls move in another fluid. It depends on physical
properties only. See the Morton number (p. 107). In the case of solid particles in
flowing fluid, the modified number (property number) K is used, which provides
better correlation with the experiment. Capillary flow.
Info: [A29],[B20].
3.2.10 Capillary Number (1.) Ca
Ca5σffiffiffik
p
ηwL
96 Dimensionless Physical Quantities in Science and Engineering
σ (N m21) � surface tension; k (m2) � permeability; η (Pa s) � dynamic viscosity;
w (m s21) � velocity; L (m) � characteristic length dimension.
It is the capillary-to-filtration forces ratio. Capillary flow and porous materials.
Info: [B20].
3.2.11 Capillary Number (1.) Ca
Ca5ηwσ
5We1Re21 ð1Þ; Ca5
ηwpcr
2cos ϑ ð2Þ;
where pc 52σ cos ϑ
r
η (Pa s) � dynamic viscosity; w (m s21) � characteristic velocity; σ (N m21) �surface or half-surface tension between two fluid phases; pc (Pa) � capillary pres-
sure; r (m) � diameter of pore throat; ϑ (�) � wetting angle, for hard
wettable material is valid cos ϑ5 1; We1 (�) � Weber number (1.) (p. 91);
Re (�) � Reynolds number (p. 81).
This number expresses the ratio of the viscosity force to the surface stress which
acts across the interface between a fluid surface and a gas or between two mixable
fluids. It characterizes the fluid spraying process in the flow of two-phase surround-
ings, in capillary tubes and porous material layers. With the pore structure and wet-
tability considered, expression (2) is valid. Capillary flow and porous materials.
Info: [A29],[B20].
3.2.12 Capillary Number (2.) Ca
Ca5L2Rgσ
L� d (m) � diameter (e.g. of pipe, particle, bubble, impeller or shaft); R (kg m23) �density; g (m s22) � gravitational acceleration; σ (N m21) � undisturbed surface
tension.
This number equals the ratio of the gravitational force to the surface tension
force. Two-phase medium flow.
Info: [B6].
3.2.13 Capillary Multiphase Number Ca
Ca5wDηpCR
; where pC 5 2σ cos ϑR21
wD (m s21) � characteristic velocity (Darcy velocity); η (Pa s) � dynamic viscos-
ity; pC (Pa) � capillary pressure; R (m) � mean pore radius; σ (N m21) � surface
tension; ϑ (�) � contact angle.
97Fluid Mechanics
It characterizes the capillary flow in porous material. For a fully wetted material,
it is cos ϑ5 1. With constant velocity, the capillary number is almost constant.
With the velocity changing in time, the number Ca becomes the dynamic parame-
ter. An example is petroleum driven out with gas.
Info: [A29].
3.2.14 Darcy Granulation Number (1.) Dc
Dc5η
VSRS _τk
η (Pa s) � fluid dynamic viscosity; VS (m3) � volume of grain particles;
RS (kg m23) � density of solid particles; _τ (s21) � unit speed of shear strain;
k (m21) � hydraulic permeability.
It expresses the ratio of tension caused by interaction to that due to particle inertia.
Info: [A29],[B13].
Henry Philibert Gaspard Darcy (10.6.1803�3.1.1858),
French engineer.
He executed research on flow and friction loss in a pipe-
line. This research resulted in the Darcy�Weisbach equation.
He improved the pitot tube structure and was the first to
express the boundary layer in flowing fluid. In the last years
of his life, he carried out many experiments leading to the
Darcy law for flow in sand. Since that time, this law has been
generalized and extended to unsaturated and multiphase flow.
3.2.15 Darcy Granulation Number (2.) Dc
Dc5wL
D0
w (m s21) � flow velocity; L (m) � characteristic length; D0 (m2 s21) � permeabil-
ity of granulated material.
It expresses the inertia-to-permeability forces ratio. It characterizes the flow in
porous material. It is an analogy of the Reynolds number Re (p. 81) and the Peclet
heat number Pe (p. 180).
Info: [A29].
Henry Philibert Gaspard Darcy (see above).
3.2.16 Driftage Elements Number ND
ND 5wvw
gL
98 Dimensionless Physical Quantities in Science and Engineering
wv (m s21) � free fall velocity of particle; w (m s21) � flow velocity; g (m s22) �gravitational acceleration; L (m) � characteristic length.
This number describes the drifting and depositing of fine particles in a fluid
flow. Creation of sediments in canals and pipelines. Pneumatic transport.15
3.2.17 Eotvos Number Eo
See the Bond number Bo (p. 95).
Info: [A29],[B7].
Lorand Baron von Eotvos (27.7.1848�8.4.1919),
Hungarian physicist.
Initially, he was engaged in capillary phenomena. Later
he devoted himself to gravitation. He used torsion balance to
prove that the ratio of the gravitational and inertia mass is
constant with the accuracy of 53 1029, which is fundamen-
tally important for general relativity theory because it con-
firms that both masses are equivalent.
3.2.18 Expansion Bubbling Number Ex
Ex5gd
w2
Rl 2 RvRl
5Fr21PR
g (m s22) � gravitational acceleration; d (m) � diameter of gas or vapour bubble;
w (m s21) � bubble velocity; Rl, Rv (kg m23) � liquid and gas density; Fr (�) �Froude number (1.) (p. 62); PR (�) � relative change of density, equivalent of
Boussinesq approximation number Bs (p. 56).
It expresses the lift-to-inertia forces ratio. It characterizes the process of gas or
vapour bubble propagation in a fluid.
Info: [A29].
3.2.19 Filtration Energy Gradient J
J52@H
@x5
w2
gLf ðReÞ
H (m) � energetic height; x (m) � axis x; w (m s21) � filtration velocity; g (m s22) �gravitational acceleration; L (m) � characteristic length of grains; Re (�) � Reynolds
filtration number, see Reynolds number (p. 81).
It expresses the energy gradient in the x-axis direction. Two-phase flow.
Hydraulics. Filtration.
Info: [A48].
99Fluid Mechanics
3.2.20 Fluidization Number Nfluid
Nfluid 5ζVsðRs 2 RfÞ
ηVf
ffiffiffig
L
r
ζ (m2) � permeability of porous material; Vs, Vf (m3) � volume of solid particles
and fluid; Rs, Rf (kg m23) � density of solid particles and fluid; η (Pa s) � dynamic
viscosity; g (m s22) � gravitational acceleration; L (m) � characteristic length, par-
ticle diameter.
It expresses the fluidization measure in relation to the velocity. For Nfluid{1, it
is used for the majority of flows. Two-phase fluid mechanics.
Info: [B54].
3.2.21 Frequency Number Nf
Nf 5ωLw
5 2πSh
ω (s21) � angular frequency; L (m) � characteristic length, thickness; w (m s21) �velocity of fluid infiltration; Sh (�) � Strouhal number (p. 87).
It characterizes the flow in condensed or suspension layers.
Info: [A29].
3.2.22 Friction Number Nf
Nf 5Vs
12Vs
nðRs 2 RfÞgL tan ϕ_τη
Vs (m3) � volume of solid particles; n (�) � grain number on the grain surface; Rs,
Rf (kg m23) � density of solid part and fluid; g (m s22) � gravitational accelera-
tion; L (m) � characteristic length (diameter); ϕ (�) � angle of inner friction; _τ(s21) � unit velocity of shear deformation; η (Pa s) � dynamic viscosity.
It expresses the ratio of friction tensions, originating with steady contact of
grains, to viscous shear tensions. It resembles the Bingham number Bm (p. 118),
but the characteristic tensions originate for solid and liquid parts differently.
Dominating viscous forces arise with Nf. 100.
Info: [B13].
3.2.23 Froude Multiphase Number Frmp
Frmp 5wgffiffiffiffiffiffigL
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiRg
Rl 2 Rg
s
100 Dimensionless Physical Quantities in Science and Engineering
wg (m s21) � velocity of moist gas flow; g (m s22) � gravitational acceleration;
L (m) � characteristic length; Rl, Rg (kg m23) � density of fluid and gas.
It expresses the inertia-to-gravitation forces ratio in two-phase fluid flow.
Multiphase fluid mechanics. Wet gas flow.
Info: [D3].
3.2.24 Gibbs Number Gb
Gb5E
kT
E (J) � local energy of perturbation; k (J K21) � Boltzmann constant; T (K) �absolute temperature.
This number expresses the kinetic theory of molecules. For example, it charac-
terizes cavitation processes and bubble dynamics in fluids. It is about the final
period in the homogeneous nucleon theory, in which the deposition energy E acts
and the bubbles originate. Two-phase flow. Boiling. Cavitation.
Info: [A7].
Josiah Willard Gibbs (11.2.1839�28.4.1903), American
mathematician, theoretical physicist and chemist.
He became an authority in theoretical thermodynamics. He
was a Professor in the Department of Mathematical Physics at
Yale University. He deduced the Gibbs phase rule. Free
enthalpy is denoted as the Gibbs thermodynamic potential.
Gibbs distribution, which is a basic law of statistical physics,
is connected with his name as well. His most important book
is Elementary Principles in Statistical Mechanics (1902).
3.2.25 Granulometric Number Gm
Gm5dmax
dekv
� �0:6
; where dekv 5Xni5 1
ðgid21i Þ
" #21
dmax (m)� maximal particle diameter; dekv (m)� equivalent particle diameter of layer
that polydisperses; gi (�), di (m) � weighted part and diameter of i-kind particles.
It expresses the fluid flow in dispersed layers.
Info: [A33].
3.2.26 Gravitational Number (1.) Ng
Ng 5kgΔRηw
5kgðRp 2 RfÞ
ηw
101Fluid Mechanics
k (m2) � permeability of environment; g (m s22) � gravitational acceleration;
ΔR (kg m23) � density difference between particles and fluid; Rp, Rf (kg m23) �density of particles and fluid; η (Pa s) � dynamic viscosity; w (m s21) � flow
velocity.
It expresses the gravitational-to-filtration forces ratio. Two-phase flow. Filtration.
Info: [B20],[B115].
3.2.27 Gravitational Number (2.) Ng
Ng 54πr4pðRp 2 RfÞg
3kT
rp (m) � particle radius; Rp, Rf (kg m23) � density of particle and fluid; g (m s22) �
gravitational acceleration; k (J K21) � Boltzmann constant; T (K) � fluid absolute
temperature.
It expresses the ratio of the gravitational potential of a particle to its thermal
energy. Physically chemical filtration in saturated porous materials.
Info: [B115].
3.2.28 Gravitational Number (3.) Ng
Ng 5ηw
gΔRL25
Ca
Bo
η (Pa s) � fluid dynamic viscosity; w (m s21) � flow velocity; g (m s22) � gravita-
tional acceleration; ΔR (kg m23) � density difference between two non-miscible
liquids; L (m) � characteristic length, thickness, diameter; Ca (�) � capillary
number (1.) (p. 97); Bo (�) � Bond number (p. 95).
It expresses gravitational force as the viscous-to-capillary forces ratio. It acts in
the penetration of two non-miscible fluids through a porous material. Two-phase
porous flow. Filtration.
3.2.29 Gravity Number NG
NG 5kgΔRηw
ð1Þ; NG 52
9
r2gðRp 2 RfÞηw
ð2Þ
k (m2) � permeability; g (m s22) � gravitational acceleration; ΔR (kg m23) � den-
sity difference of phases; η (Pa s) � dynamic viscosity; w (m s21) � flow velocity;
r (m) � particle radius; g (m s22) � gravitational acceleration; Rp, Rf (kg m23) �
density of particle and fluid.
102 Dimensionless Physical Quantities in Science and Engineering
This number denotes the gravity-to-filtration forces ratio or, alternatively, the
ratio of the Stokes rate of particle sedimentation to the fluid rate. It is expressed in
expression (2) as well. Flow through porous material. Two-phase flow. Filtration.
Info: [B20].
3.2.30 Haberman�Morton Number Hab, Hm
Hab5gη4lRlσ3
12mp
RlVp
� �5
We31Fr2Re4
g (m s22) � gravitational acceleration; ηl (Pa s) � fluid dynamic viscosity;
Rl (kg m23) � fluid density; σ (N m21) � surface tension; mp (kg) � particle mass;
Vp (m3) � particle volume; We1 (�) � Weber number (1.) (p. 91); Fr (�) �Froude number (1.) (p. 62); Re (�) � Reynolds number (p. 81).
This number expresses the deformation of moving particles, as drops, bubbles or
fine deforming (granular) materials. In the case of bubbles mp{Rl, mp=ðRlVpÞ-0
holds. Then the bubble dimension influence on the number Hab can be neglected.
For filtered water Hmi5 0:253 10210; for mineral oil Hmi5 1:453 1022 and for
syrup Hmi5 0:923 106:Info: [A2],[A7],[A8].
3.2.31 Inertia Parameter Ninert
Ninert 5d2RPw18ηL
d (m) � diameter (such as of pipe, particle, bubble or drop); RP (kg m23) � particle
density; w (m s21) � fluid velocity; η (Pa s) � dynamic viscosity; L (m) � charac-
teristic length (such as of channel width, distance from wall or height of liquid
layer).
It expresses the inertia to friction forces in particle flow in a fluid.
Info: [C72].
3.2.32 Keulegan�Carpenter Number (2.) KC
KC5wAτp2L
wA (m s21) � speed of body oscillating; τp (s) � time period; L (m) � characteristic
length, diameter.
It characterizes non-steady oscillation phenomena joined with the particle move-
ment in fluid. For KC, 5, the inertial phenomena prevail. Two-phase flow.
Info: [A4],[A7].
103Fluid Mechanics
3.2.33 Knudsen Number (3.) Diffusion Kn
Kn53
4
hDAB
ζKOAwA
h (m) � height of roughness; DAB (m2 s21) � binary bulk diffusion coefficient for
system AB; ζ (�) � diffusion tortuosity; KOA (m) � Knudsen coefficient of perme-
ability; wA (m s21) � equilibrium mean molecular speed of species A.
In contrast to the Knudsen number (2.) diffusion Kn (p. 45), it involves the influ-
ence of surface roughness and diffusive tortuousness. It expresses the gas diffusion
in bundles.
Info: [A29].
Martin Hans Christian Knudsen (p. 420).
3.2.34 Kolmogorov Number Kol
Kol5Rp 2 R
Rgwg
w2dyn
@w
@z
� �21
; where wdyn 5
ffiffiffiτR
r
Rp, R (kg m23) � density of particles and liquid; g (m s22) � gravitational accelera-
tion; wg (m s21) � free fall velocity of a particles in liquid; wdyn (m s21) � dynamic
velocity;@w
@z(s21) � vertical gradient of velocity flow; τ (m21 kg s22) � shear fric-
tion tension.
It expresses the relative loss of the turbulent energy expended to maintain the
equilibrium of particles in a fluid flow. It is a criterion of the dynamic balancing
activity, for example, the balanced particles’ influence on the flowing fluid dynam-
ics. It is analogous to the Kolmogorov parameter Kol (p. 399).
Info: [B6],[C75].
Andrey Nikolaevich Kolmogorov (25.4.1903�20.10.1987),
Russian mathematician.
He contributed substantially to the development of proba-
bility theory, topology and other mathematical branches.
However, he also worked in the sphere of classic mechanics,
especially on turbulence theory. He formulated the algorith-
mic complex theory. Beginning in the 1940s, he was engaged
in dynamic systems theory in relation to interplanetary flight
and other problems in space science. Besides mathematics,
he had an extraordinary interest in poetry, especially the
Russian poet Pushkin.
104 Dimensionless Physical Quantities in Science and Engineering
3.2.35 Kozeny Number Kz, k
Kz5Δp
ηLμ3S
ð12ϕ2Þw
Δp (Pa) � piezometric pressure drop in the thickness L of porous material; η (Pa s) �dynamic viscosity; L (m) � characteristic length, thickness; ϕ (�) � porosity (p. 24);
S (m2) � area; w (m s21) � mean velocity component in direction L.
It expresses the ratio of the pressure and volume deformation forces to the fluid
momentum. It characterizes the mass transfer through a porous material.
Info: [A29].
Josef Alexander Kozeny (born 1889), Austrian engineer of Czech origin.
3.2.36 Kubo Number of Percolation Kub
Kub5w
Lω
w (m s21) � characteristic velocity; L (m) � characteristic length, necking; ω (s21) �characteristic perturbation frequency.
It expresses the accidentality level in the percolation process of a moderately
compressible fluid under turbulent transfer conditions. Two-phase fluid mechanics.
Turbulent diffusion.
Info: [B4],[B53].
3.2.37 Leibenson Number (2.) Lb
Lb5RR3
HΔp
η2L
R (kg m23) � fluid density; RH (m) � hydraulic radius; Δp (Pa) � pressure drop;
η (Pa s) � dynamic viscosity; L (m) � characteristic length.
It characterizes the fluid filtration in a porous environment. It expresses the
porous environment layer resistance, which depends on its structure and the physi-
cal properties of the filtering environment. It is a measure of the ratio of pressure
forces under the action of hydraulic resistance to the viscosity forces.
Info: [A33].
Leonid Samuilovich Leibenzon (1879�1951), Russian physicist and
mechanist.
105Fluid Mechanics
3.2.38 Leverett Number Lt
Lt5
ffiffiffik
p
spc
σ
k (m2) � permeability; p (�) � porosity (p. 24); pc (Pa) � capillary pressure;
σ (Pa) � surface tension.
It expresses the ratio of the characteristic dimension of an inter-area curvature to
the characteristic dimension of pores. Two-phase flow in porous materials.
Info: [A29],[B20].
Miles Corrington Leverett (born 1910), American chemical engineer.
3.2.39 Lockhart�Martinelli Parameter χ
χ5ml
mg
ffiffiffiffiffiRgRl
r
ml, mg (kg s21) � liquid and gas mass flux; Rl, Rg � liquid and gas density.
This parameter is used for internal two-phase flow. It characterizes the influence
of the fluid fraction portion on the flow. It occurs mainly in pressure spraying and
heat transfer in boiling and condensation.
Info: [A12],[C9].
3.2.40 Lyashenko Number Lj
Lj5Re3Ar21
Re (�) � Reynolds number (p. 81); Ar (�) � Archimedes hydrodynamic number
(p. 53).
This number characterizes the hydraulic effect of the fluid flowing through a
granular material (fluidization).
Info: [A29].
3.2.41 Martinelli Parameter X
X2 5dp
dz
fr
� �liq
dp
dz
fr
� �21
g
dp
dz(N m23) � pressure gradient; subscripts: fr � friction; liq � liquid; g � gas or
vapour.
106 Dimensionless Physical Quantities in Science and Engineering
It expresses the ratio of the pressure caused by the friction of drops and the gas
flow. It is used for two-phase models to express the pressure of diversely shaped
drops.
Info: [A7].
3.2.42 Mass Number Nm
Nm 5VS
Vf
RSRf
VS, Vf (m3) � volume of solid and liquid components; RS, Rf (kg m23) � density of
solid and liquid components.
It expresses the volume to density fractions ratio of solid and liquid particles or,
alternatively, the inertia thereof in mixing. It quantifies the effect of the volume tol-
erance. For Nm. 1, the momentum transfer of the granules in the fluid prevails.
Two-phase fluid mechanics.
Info: [A4],[B53],[B54].
3.2.43 Mobility Number of Grains Nmov
Nmov 5Rw2
fVL
R (kg m23) � fluid density; w (m s21) � characteristic flow velocity; fV (N m23) �specific grain force relative to volume unit; L (m) � characteristic length of grains.
In a two-phase flow, it characterizes the ratio of the fluid flow dynamic force to
the dynamic force of flowing grains. Two-phase flow. Hydrodynamics. Filtration.
Info: [A48].
3.2.44 Morton Number Mo
Mo5gη4
Rσ35
We31Fr2Re4
ð1Þ; Mo5gη4ðR1 2 RgÞ
R2l σ3ð2Þ;
Mo5gη4ðRf 2 RsÞ
R2fσ3ð3Þ
g (m s2) � gravitational acceleration; η (Pa s) � dynamic viscosity; Rl, Rg, Rf, Rs(kg m23) � density of liquid, gas, fluid and solid particles; σ (N m21) � surface
tension; W e1 (�) � Weber number (1.) (p. 91); Fr (�) � Froude number (1.)
(p. 62); Re (�) � Reynolds number (p. 81).
107Fluid Mechanics
This number expresses the ratio of the gravitational acceleration to the molecular
acceleration of a fluid. It depends on fluid properties only. In expression (1), it is
called the Morton fluid number, which expresses the relations arising in the process
of immersing solid bodies into viscous fluid. It corresponds to the capillary buoy-
ancy number CaB (p. 96). Expression (2) describes the free bubble movement in
a steady state fluid. Expression (3) describes the solid particle movement in the
fluid. It is also called the property groups NP (p. 110) or the Haberman�Morton
number Hab (p. 103). For water at 20�C, its value is approximately 33 10211.
Info: [A26],[B20].
3.2.45 Non-dimensional Thermal Diffusivity Na
Na 5agf
c2g
ag (m2 s21) � gas thermal diffusivity; f (s21) � bubble frequency; cg (m s21) � gas
sound velocity; κ (�) � specific heat ratio (p. 28).
It expresses the internal thermal conductivity, including the thermal dumping
influence of bubbles, in two-phase surroundings. The effective polytropic exponent
k depends on internal thermal conductivity, and 1, k,κ holds. For high frequen-
cies, the mean free path in gas can be comparable with the bubble dimension and
the k can be outside the range mentioned as well.
3.2.46 Ohnesorge Number Oh, Z
Oh � Z5ηffiffiffiffiffiffiffiffiffiRLσ
p 5We121Re
21 ð1Þ; Oh5 Su12 ð2Þ
η (Pa s) � dynamic viscosity; R (kg m23) � density; L (m) � characteristic length;
σ (Pa) � surface tension; We1 (�) �Weber number (1.) (p. 91); Re (�) �Reynolds number (p. 81); Su (�) � Suratman number (p. 115).
It is the ratio of the viscous force to the surface stress force. It characterizes the
processes of fluid spraying (atomization) in a two-phase flow. Sometimes, the
Suratman number Su (p. 115) is used instead of it. Stroke and surface waves.
Spraying. Splashing.
Info: [A26],[B20],[B56].
Wolfgang von Ohnesorge (8.9.1901�26.5.1976), German hydrodynamics
specialist.
3.2.47 Pavlovsky Number Pa
Pa5d2Δp
wηδ
108 Dimensionless Physical Quantities in Science and Engineering
d (m) � particle diameter; Δp (Pa) � pressure drop; w (m s21) � characteristic
flow velocity; η (Pa s) � dynamic viscosity; δ (m) � layer thickness.
It characterizes the fluid flow through a granular material layer. It expresses the
pressure force, with consideration of the hydraulic resistances, to the viscosity force.
Info: [A33].
3.2.48 Poiseuille number (2.) Ps
Ps5wv
ðRs 2 RfÞgD2
w (m s21) � flow velocity; ν (m2 s21) � kinematic viscosity; Rs, Rf (kg m23) � den-
sity of solid particles and liquid; g (m s22) � gravitational acceleration; D (m) �diameter of solid particle.
It expresses the fraction and gravitation forces in a fluid flow with mass parti-
cles. Two-phase flow.
Info: [A29].
Jean Louis Marie Poiseuille (p. 78).
3.2.49 Porous Flow Number NP
NP 5wηLffiffiffik
pσ cos ϑ
w (m s21) � flow velocity; η (Pa s) � dynamic viscosity; L (m) � characteristic
length; k (m2) � permeability; σ (N m21) � surface tension; ϑ (�) � angle of cap-
illary depression.
It is the viscous-to-capillary pressures ratio. It is the inverse value of the capil-
lary number (1.) Ca (p. 56). Filtration with a porous material.
Info: [B20].
3.2.50 Porous Pressure Number NP
NP 5D
h2
ffiffiffiL
g
s
D (m2 s21) � porous pressure diffusivity; h (m) � height; L (m) � characteristic
length; g (m s22) � gravitational acceleration.
This number expresses the ratio of the gravity flow time to the time of the pore
pressure diffusion which is perpendicular to the flow direction. The small value
Npor{1 shows that the porous pressure and the diffusion act at different times.
Two-phase fluid mechanics.
109Fluid Mechanics
3.2.51 Property Groups NP
NP 5gη4ðRf 2 RsÞ
R2fσ3
g (m s22) � gravitational acceleration; η (Pa s) � dynamic viscosity; Rf, Rs (kg m23) �
density of fluid, solid particles or bubbles; σ (N m21) � surface tension.
It is a widely used and specifying modification of the capillary buoyancy
number CaB (p. 96).
Info: [A29].
3.2.52 Rabin Number Rab
Rab5We1ffiffiffiffiffiffiRe
p ð1Þ; Rab5
ffiffiffiffiffiffiffiffiffiWe31Lp
4
sð2Þ
d (m) � drop diameter; R (kg m23) � drop density; σ (N m21) � surface tension;
η (Pa s) � dynamic viscosity; We1 (�) � Weber number (1.) (p. 91); Re (�) �Reynolds number (p. 81); Lp (�) � Laplace number (p. 71).
In expression (1), it serves to determine various drop decomposition. It can be rewrit-
ten into expression (2), which is essentially analogous to the Weber number (1.) We1(p. 91), with certain corrections for the drop dimension expressed by the Laplace
number Lp (p. 71). Two-phase flow. Condensation. Getting frostbitten. Spraying.
Info: [A15].
3.2.53 Radial Frequency Parameter (1.) Pfr
Pfr 5ωrD
w2
ωr (s21) � angular frequency; D (m2 s21) � mass diffusion or dispersion;
w (m s21) � fluid velocity.
It expresses the cyclic transfer of molecular mass diffusion with dynamic fluid
acting on a rotating system. It serves to monitor the resonating frequency and is
related to control. With its value increasing, the radial characteristics strengthen
repeatedly. Fluidized coating.
Info: [A35].
3.2.54 Radial Frequency Parameter (2.) Pfr
Pfr 5ωrL
2
a
110 Dimensionless Physical Quantities in Science and Engineering
ωr (s21) � angular frequency; a (m2 s21) � thermal diffusivity; w (m s21) � fluid
velocity.
It expresses the heat diffusion cyclic transfer with dynamic fluid acting on rotat-
ing systems. It is also called the wave thermal parameter. It serves to monitor the
resonating frequency. An increasing value leads to repeated strengthening.
Fluidized coating.
Info: [A35].
3.2.55 Radial Frequency Parameter (3.) Pfr
Pfr 5ω2rDL
w35 ðP�
frÞ2Pe21m
ωr (s21) � angular frequency; D (m2 s21) � mass diffusivity or dispersion; L (m) �
characteristic length; w (m s21) � fluid velocity; P�fr (�) � radial frequency param-
eter (1.) (p. 110); Pem (�) � Peclet mass number (p. 258).
It characterizes the cyclic transfer of molecular mass diffusion with dynamic
fluid acting on a rotating system. Fluidized coating.
Info: [A35].
3.2.56 Radial Frequency Parameter (4.) Pfr
Pfr 5 L
ffiffiffiffiffiffiωr
2D
r
L (m) � characteristic length; D (m2 s21) � diffusivity or dispersion; ωr (s21) �
angular frequency.
It characterizes the cyclic transfer of molecular mass diffusion. Fluidized
coating.
Info: [A35].
3.2.57 Rayleigh Pore Flow Number Rapor
Rapor 5ϕgβL3ΔT
av
ϕ (�) � porosity (p. 24); g (m s22) � gravitational acceleration; β (K21) � volume
thermal expansion coefficient; L (m) � characteristic length, thickness; ΔT (K) �temperature difference; a (m2 s21) � thermal diffusivity; ν (m2 s21) � kinematic
viscosity.
111Fluid Mechanics
This number expresses the viscous fluid infiltration through a porous layer
caused by the thermal gradient, gravitation force and diffusion heat propagation. It
is analogous to the Rayleigh modified number Ramod (p. 382).
Info: [A35].
Lord Rayleigh (p. 187).
3.2.58 Relative Filtration Velocity Pw
Pw 5c
w
c (m s21) � velocity propagation of surface wave; w (m s21) � filtration velocity.
It expresses the influence of the water wave propagation velocity on the filtra-
tion velocity. Two-phase flow. Filtration. Geophysics.
Info: [A48].
3.2.59 Reynolds Quasi Number Req
Req 5Rh
ffiffiffiffiffiffigL
p
Vfη
R (kg m23) � density; h (m) � layer thickness; L (m) � characteristic length;
g (m s22) � gravitational acceleration; Vf (m3) � fluid volume; η (Pa s) � dynamic
viscosity.
In infiltration, it represents the dynamic coefficient analogous to the Reynolds
number Re (p. 81) in Newtonian flow. Usually, it reaches the value Req. 106.
Two-phase fluid mechanics.
Info: [B12],[B13],[B505].
Osborne Reynolds (p. 82).
3.2.60 Savage Number Sav
Sav5_γRsL
ðRs 2 RfÞghð1Þ; Sav5
_γRsLnðRs 2 RfÞgh tan ϕ
ð2Þ
_γ (s21) � unit speed of shear deformation; Rs, Rf (kg m23) � density of particles
and fluid; L (m) � characteristic length, grain diameter; g (m s22) � gravitational
acceleration; h (m) � layer thickness; n (�) � number of surface grains; ϕ (�) �angle of inner friction.
This number expresses the ratio of tensions caused by grain collision to gravita-
tion tensions arising due to the contact friction of grains. With Sav. 0.1 in the
usual layer depth, the tensions originating due to collision have great influence.
Info: [B13].
112 Dimensionless Physical Quantities in Science and Engineering
3.2.61 Schiller Number (2.) Sch
Sch5w
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3R
4gvðRs 2 RÞ3
s5
ffiffiffiffiffiffiffiffiReδ
W
r
where W 5wδ
w; Reδ 5
wδv
w (m s21) � fluid flow velocity; R, Rs (kg m23) � density of fluid and solid parti-
cles; g (m s22) � gravitational acceleration; ν (m2 s21) � kinematic viscosity; wδ
(m s21) � boundary layer velocity; Reδ (�) � Reynolds boundary layer number,
see Reynolds number Re (p. 81); W (�) � relative velocity.
It expresses the equilibrium rate of ball-shaped particles floating in flowing
fluid. Fluidization.
Info: [A25],[A35].
L. Schiller.
3.2.62 Spray Number NA
NA 5gðR2 RfÞL2
σ
g (m s22) � gravitational acceleration; R, Rf (kg m23) � density of bubble and
ambient fluid; L (m) � characteristic length; σ (N m21) � surface tension.
It expresses the ratio of the weight force to the surface stress force. It corre-
sponds to the Bond number Bo (p. 95). Atomization of fluids.
Info: [B7].
3.2.63 Stokes Foam Drainage Number St
St5ηws
Rgr2b5 fðε;ReÞ
η (Pa s)� dynamic fluid viscosity; ws (m s21) � liquid surface velocity; R (kg m23) �density; g (m s22) � gravitational acceleration; rb (m) � bubble diameter; ε (�) �volume liquid fraction in foam; Re (�) � Reynolds number (p. 81).
It expresses the fluid foaming in steady state flowing through sewers or other
canals, and the fact that the flow is influenced by the relative volume fluid fraction
and the Reynolds number Re (p. 81). Two-phase fluid mechanics.
Info: [B109].
George Gabriel Stokes (p. 131).
113Fluid Mechanics
3.2.64 Stokes Particle Number St
St5τrxτk
5R
18
d
η
� �2
where τrx 5Rd2
18v; τk 5
ffiffiffiv
ε
r; η5 v
34ε
14
τrx (s) � relaxation time of particle; τk (s) � Kolmogorov time; R (�) � density
ratio of particle and fluid; d (m) � particle diameter; η (m) � Kolmogorov length;
ν (m2 s21) � kinematic viscosity; ε (m2 s21) � dissipation rate of turbulent energy.
This number characterizes the particles moving by inertia in a turbulent flow
and clustering in a low swirl zone of the fluid due to the inertia equilibrium among
denser particles and the lighter surrounding fluid. The clustering peaks occur with
St�1. Particle clustering in turbulent aerosols. Two-phase flow.
George Gabriel Stokes (see above).
3.2.65 Stokes Time Number Stk
Stk5τpτf
5Rd2wCc
18ηdc
τp (s)� time response; τf (s)� characteristic flow time; R (kg m23)� density; d (m)�particle diameter; w (m s21) � flow velocity; Cc (�) � Cunningham�Stokes correction
factor; η (Pa s) � dynamic viscosity; dc (m)� cylinder diameter.
It expresses the ability of aerosol particles to collide. It represents the ratio of
the response time to the characteristic slow time. Sometime, it is called the inertia
parameter.
George Gabriel Stokes (see above).
3.2.66 Strain Rate of Grain N, B
N5 _ε
ffiffiffiffiffiffiffiffiffiffiRsD3
k
r
_ε (s21) � speed of relative elongation; Rs (kg m23) � particle density; D (m) � par-
ticle diameter; k (kg s22) � linear stiffness of particle.
It characterizes the deformation rate of grains in the two-phase slide flow of
granulated fluid where the deformation can be understood as plastic or viscous.
Usually, real granulated flow is between these two extremes. Two-phase flow.
Info: [B4].
114 Dimensionless Physical Quantities in Science and Engineering
3.2.67 Suratman Number Su
Su5Oh22 5RlLση2l
5Re2We211
Rl (kg m23)� fluid density; ηl (Pa s)� liquid dynamic viscosity; L (m)� characteristic
length; σ (N m21) � liquid surface tension; Oh (�) � Ohnesorge number (p. 108);
Re (�)� Reynolds number (p. 81);We1 (�) � Weber number (1.) (p. 91).
It expresses the ratio of the product of the inertia force and the surface stress to
the square of the viscous force. Alternatively, it expresses the relation between the
surface stress and the friction of plunged particles. It characterizes the fluid spray-
ing process. Sometimes, the Ohnesorge number Oh (p. 108) is used instead of it.
Hydromechanics. Dynamics of particles.
Info: [A29],[B20].
3.2.68 Thermal Diffusivity of Gas Ag
Ag 5agωc2g
ag (m2 s21) � gas thermal diffusivity; ω (s21) � bubble frequency; cg (m s21) �gas sound velocity.
This diffusivity expresses the influence of the frequency of the bubbles of
diverse gases in a fluid on the change of thermal diffusivity and the polytropic gas
exponent k. At low frequencies, the time is sufficient for thermal isothermal diffu-
sion (k5 1). With higher frequencies, the time for the thermal diffusion is insuffi-
cient and the behaviour causes an isentropic tendency. With a constant high
frequency, the mean free path in the gas is comparable to the bubble size and
the polytropic exponent can reach a value outside the range kAh1; κi, where κ is
the specific heat ratio (p. 28). Thermodynamics. Multiphase fluids mechanics.
Cavitation.
Info: [A2].
3.2.69 Two-Phase Flow Number NP2
NP2 5ηdwσL
η (Pa s) � dynamic viscosity; d (m) � bubble or particles diameter; w (m s21) �flow velocity; σ (N m21) � surface tension; L (m) � characteristic length.
It is the ratio of the viscous (friction) force to that of surface stress. Capillary
flow. Dispersion materials.
Info: [B20].
115Fluid Mechanics
3.2.70 Two-Phase Porous Flow Number NPP
NPP 5wηffiffiffiffiffiffiffiffiffi
k1k2p
gΔR
w (m s21) � flow velocity; η (Pa s) � dynamic viscosity; k1, k2 (m2) � longitudinal
and transversal permeability; g (m s22) � gravitational acceleration; ΔR (kg m23) �density difference of both phases.
It is the viscous-to-gravitational pressures ratio. Capillary flow in porous
materials.
Info: [B20].
3.2.71 Valensi Two-Phase Number Va
Va5ωL2
v
ω (s21) � angular oscillation frequency of object in fluid with zero dynamic viscos-
ity; L (m) � characteristic length; ν (m2 s21) � kinematic viscosity.
Physically, it characterizes the ratio of the viscous diffusion time L2/(4ν) to the
oscillating frequency ω2 1. It expresses the oscillation of objects, drops and bubbles
in a fluid especially.
Info: [A35].
Jacques Valensi (born 1903), French engineer.
3.2.72 Viscous Inverse Number Nvis
Nvis 5
ffiffiffiffiffiffiffiffigD3
t
pv
5
ffiffiffiffiffiffiffiffiEo3
Mo
4
r
g (m s22) � gravitational acceleration; Dt (m) � characteristic pipe diameter;
ν (m2 s21) � kinematic viscosity; Eo (�) � Eotvos number (p. 99); Mo (�) �Morton number (p. 101).
This number characterizes the influence of the viscosity and other properties of
a two-phase fluid with the generation of individual Taylor bubbles and formation
of wake in a vertical pipeline or canal. With high gas velocities (Nvis. 1500), the
liquid part can be pushed to the pipeline wall and the gas flows through the core
continuously. With 500,Nvis, 1500, the end of bubbles is nearly in plane and
the wake part is closed immovably. It tends to lose symmetry with respect to the
tube axis and provides periodic waves, the frequency of which grows with increas-
ing Nvis. With Nvis, 500, the wake part is divided into two closed ring swirls
which act mutually on each other.
Info: [B112].
116 Dimensionless Physical Quantities in Science and Engineering
3.2.73 Weber Spray Number Wespray
Wespray 5Rq2VLσ
R (kg m23) � fluid density; qV (m s21) � volume flux density; L (m) � characteristic
length, diameter; σ (N m21) � surface tension.
It characterizes the shower sprinkler spraying of a fluid. Among other things, it
applies to cooling in electrical engineering, diverse industrial, apartment or office
rooms, gardening and agriculture.
Info: [B7].
Ernst Heinrich Weber (p. 92).
Wilhelm Eduard Weber.
3.3 Rheology
In rheology, the similarity criteria express the deformation and flow of non-
Newtonian fluids and plastic materials with various resistances thereof. It concerns
the mechanical properties of these substances, such as relations between stress and
deformation and deformation rate. The applications of rheology involve a wide
range of fields, from engineering to geophysics to physiology. These include, for
example, diverse polymer and biopolymer systems, terrestrial lava movement and
blood flow. The Bingham number, characterizing the flow of Bingham plastics, is a
fundamental similarity criterion. As for other criteria, these include the Deborah,
elastic, Elvis, Galilei, Hedstrom and Weissenberg numbers.
3.3.1 API Gravity Degree API
API5RrefR
2131:5
Rref (kg m23) � reference density (Rref5 1.4153 105 kg m23); R (kg m23) � liquid
density at temperature 15.55�C.The API gravity degree characterizes the density of liquid petroleum products.
Info: [A43].
3.3.2 Bingham Compression Number Bn
Bn5σminH
2
Fð1Þ; Bn5
σminHn
Kð2Þ
σmin (Pa) � minimal yield strength; H (m) � initial height of material sample;
F (N) � constant loading force; n (�) � constant of time independent deformation
speed; K (N) � dynamic loading force.
117Fluid Mechanics
It expresses the state of a semi-solid Bingham material if compressed. In
expression (1), it is about a constant load; in expression (2), it is about the condi-
tion with a constant deformation rate. Rheology. Bingham fluid.
Info: [B2],[C7].
Eugene Cook Bingham (1878�1945), American physicist.
3.3.3 Bingham Growth Exponent M
M5m
ffiffiffiffiffiffiffiffiffiffiF
H2K
n
rð1Þ; M5
mu
Hð2Þ
m (s) � exponent of stress raising; n (�) � power index of non-linear fluid behav-
iour; F (N) � loading force; H (m) � initial sample height; K (m21 kg s21) � con-
sistent index corresponding to dynamic viscosity; u (m s21) � compression speed.
This exponent characterizes the rheological behaviour dynamics of semi-solid
pulps. The expression (1) holds for a constant load; the expression (2) is valid for a
constant deformation rate. The index n has the value of n, 1 for a thinner fluid
layer and n. 1 for a thicker one. For the Bingham model, n5 1 is valid.
Info: [B2],[C7].
Eugene Cook Bingham (see above).
3.3.4 Bingham Number Bm
Bm � P5σkL
ηw
σk (Pa) � yield strength; L (m) � channel width; η (Pa s) � dynamic viscosity of
plastic; w (m s21) � plastic flow rate; P (�) � plasticity number (p. 129).
This number expresses the ratio of the yield point stress to the viscous tension.
It characterizes the creep of Bingham plastics and the rheological phenomena in
viscous material flow, and material creep and forming. It is connected with the
Hedstrom number (1.) Hd1 (p. 124). Sometimes, it is called the plasticity number
P (p. 129). Its inverse value is the plastic deformation K (p. 128).
Info: [A13],[B20],[C7].
Eugene Cook Bingham (see above).
3.3.5 Brinkman Rheological Number Br
Br5w2ηλΔT
ð1Þ; Br � Gn5ηw2
λΔTprocð2Þ
118 Dimensionless Physical Quantities in Science and Engineering
w (m s21) � flux velocity; η (Pa s) � dynamic viscosity; λ (W m21 K21) � thermal
conductivity; ΔT (K) � reference temperature difference; ΔTproc (K) � tempera-
ture difference between fluid and boundary; Gn (�) � heat build-up number
(p. 124); Pr (�) � Prandtl number (p. 197).
It characterizes the mutual relation between the dissipation and heat conduction
in non-isothermal rheological systems, such as molten polymers. With the relative
temperature gradient represented by the difference in a thermal process
(ΔT5ΔTproc), it is called the Heat build-up number Gn (p. 124) or the Eckert
rheological number Ecrh (p. 121), Ecrh5GnPr2 1. Large values of the numbers
Gn or Ecrh mean that, in the flowing polymer, the heat development is determined
by the viscous dissipation, but very little by the conduction.
Info: [A23].
Henri Coenraad Brinkman, German physicist.
3.3.6 Darcy Friction Number fD
fD 52dp
dx
DH
R w2ð1Þ; fD 5 4 fF 5
8 Δp
R w2
DH
Lð2Þ;
fD 58τwR w2
Nð3Þ; where DH 5
4 A
U
dp
dx(Pa m21) � pressure gradient; DH (m) � hydraulic diameter; R (kg m23) �
liquid density; w (m s21) � flow velocity; Δp (Pa) � pressure difference; L (m) �characteristic length; A (m2) � cross-section area; U (m) � wetted perimeter of
cross-section area; τw (Pa) � shear stress on the wall; wN (m s21) � free flow
velocity; fF (�) � Fanning friction number (p. 163).
In expression (1) or (2), it describes the pressure loss caused by the friction in
the flow in a constant cross-section pipeline. In expression (3), it is analogous to
the Darcy�Weisbach friction coefficient. See Fanning friction number fF (p. 163).
Info: [A2],[A35].
Henry Philibert Gaspard Darcy (p. 98).
3.3.7 Deborah Generalized Number De
De5 τffiffiffiffiffiffiffiffiffiffiffiffiffiIe 2 Iv
pτ (s) � natural time of viscoelastic material; Ie (s
22) � invariant of rate of strain
tensor; Iv (s22) � invariant of vorticity tensor.
In the dimensionless form, it characterizes the viscoelastic process time. See
also the Weissenberg generalized number Ws (p. 132).
Info: [A29],[A35].
119Fluid Mechanics
3.3.8 Deborah Mixing Number Demix
Demix 5τmix
τpol
τmix (s) � mixing time; τpol (s) � polymerization time.
It expresses the ratio of the characteristic mixing time to that of a polymerizing
process. It is analogous to the Deborah number (1.) De (p. 120). Rheology. Mixing
of melted polymers.
Info: [A29].
3.3.9 Deborah Number (1.) De
De5τrxΔτ
ð1Þ; De5τmΔτ
ð2Þ
De{1 ðliquid-likeÞ ð3Þ;
De-0 ðviscoelasticÞ ð4Þ;
Dec1 ðsolid-likeÞ ð5Þ;
τrx (s) � relaxation time; Δτ (s) � natural time process interval (experimental or
observation time); τm (s) � molecular time.
This number expresses the ratio of the relaxation time to the process observation
(experimental) time interval. It characterizes the elastic behaviour � depending on
the relaxation time length � of the polymeric fluid. A long relaxation time is con-
nected with further stress preservation, whereas a short relaxation time means purely
viscous behaviour of the fluid. A Newtonian fluid is the limit case (De5 0). With a
great value of the number De, the material behaves as elastically solid; with extremely
small De numbers, it behaves as a purely viscous fluid. In expression (2), for example,
it holds for the flow of nematic liquid crystals. Rheology of viscoelastic fluids.
Info: [A29].
3.3.10 Deborah Pressurization Number Dep
Dep 5τrxvh
ð1Þ; Dep;crit 5 τrxωcrit ð2Þ
τrx (s) � relaxation time; v (m s21) � moving plate velocity; h (m) � gap width
between two plates; ωcrit (s21) � critical angular velocity.
In expression (1), it characterizes the dynamics of viscoelastic fluid pressuriza-
tion in a gap between two parallel plates. The critical value is given by expression
(2). Rheology.
Info: [A35],[B98].
120 Dimensionless Physical Quantities in Science and Engineering
3.3.11 Eckert Rheological Number Ecrh
See Heat build-up number Gn (p. 124).
Ernst Rudolf Georg Eckert (p. 192).
3.3.12 Elasticity Number (1.) El
El5τrxvL2
5De Re21 � Forx
τrx (s) � relaxation time; ν (m2 s21) � kinematic viscosity; L (m) � characteristic
length; De (�) � Deborah number (1.) (p. 120); Re (�) � Reynolds number
(p. 81); Forx (�) � Fourier viscoelastic relaxation number (p. 121).
This number expresses the elastic-to-inertia forces ratio. Alternatively, it is the
ratio of the relaxation (polymerization) time to the diffusion time. It characterizes
the non-isotropic flow of polymer fluids. Elc1. Rheology.
Info: [A26],[B20].
3.3.13 Elasticity Number (2.) Nelast, El
Nelast 5cpRβK
5Gc Ho Du21
cp (J kg21 K21) � specific heat capacity; R (kg m23) � density; β (K21) � volume
thermal expansion coefficient; K (Pa) � bulk modulus of elasticity; Gc (�) � Gay-
Lussac number (p. 14); Ho (�) � Hooke number (p. 138); Du (�) � Dulong num-
ber (p. 191).
It expresses the elasticity effect in the viscoelastic fluid flow process. It depends
only on the physical properties of the fluid.
Info: [A26],[B20].
3.3.14 Elasticity Number (3.) Nelast
Nelast 5cp
βa2
cp (J kg21 K21) � specific heat capacity; β (K21) � volume thermal expansion
coefficient; a (m s21) � sound velocity.
It expresses the elastic-to-inertia forces ratio or, alternatively, the elasticity
effect in fluid flow processes.
Info: [A152],[B20].
3.3.15 Elasto-capillary Number Nec, Γ0
Nec 5τrxσηL
5De Ca21
121Fluid Mechanics
τrx (s) � relaxation time; σ (N m21) � surface tension; η (Pa s) � dynamic viscos-
ity of viscoelastic fluid; De (�) � Deborah number (1.) (p. 120); Ca (�) � capil-
lary number (1.) (p. 97).
It characterizes the elastic and capillary properties of adhesive and other non-
Newtonian fluids in the production of microfibres. Rheology. Microbiology.
Info: [C136].
3.3.16 Ellis Number El
El5ηiniwτd
ηini (Pa s) � initial limiting value of viscosity as shear nears to zero; w (m s21) �flow velocity; τ (Pa) � shear stress by condition η5 ηini/2; d (m) � pipe diameter.
It characterizes the non-Newtonian fluid flowing through a pipeline. Rheology.
Info: [A35],[B20].
Samuel Benjamin Ellis (born 1904), American chemist.
3.3.17 Ericksen Number Er
Er58wLτMUR2
58
U
L
R
� �2
De
w (m s21) � velocity of movement of crystal wall; L�Δs (m) � dislocation; τM(s) � molecular time; U (�) � dimensionless molecular concentration parameter;
R (m) � space position; De (�) � Deborah number (1.) (p. 120).
This number expresses the ratio of the viscous moment of rheological fluid flow
to the moment of elastic flow and, alternatively, it quantifies the hydrodynamic and
elastic couplings. For example, it describes the Poiseuille flow of nematic liquid
crystals with diverse degrees of space orientation. Here, it expresses the intensity
of the narrow extent of nematic potential, this intensity being related to the broad
extent of the elastic potential.
Info: [C51].
3.3.18 Fourier Rheology Number Forh
Forh 5στηL
σ (N m21) � surface tension; τ (s) � natural process time; η (Pa s) � liquid
dynamic viscosity; L (m) � characteristic length.
It expresses the dimensionless rheological process time expressed by means of
the elastic and capillary properties of fluid. Rheology. Microrheology.
Info: [C136].
Jean Baptiste Joseph Fourier (p. 175).
122 Dimensionless Physical Quantities in Science and Engineering
3.3.19 Fourier Viscoelastic Relaxation Number Forx
Forx � Zhrx 5vτrxL2
ν (m2 s21) � kinematic viscosity; τrx (s) � relaxation time; L (m) � characteristic
length; Zhrx (�) � Zhukovsky viscoelastic number (p. 133).
It expresses the elastic-to-inertia forces ratio and, alternatively, the relaxation-to-
diffusion time ratio. In the dimensionless form, it characterizes the stroke flow of vis-
coelastic fluids. In addition, it characterizes the stroke wave propagation time and
viscoelastic vibration of mechanical stress. Sometimes, it is called the Zhukovsky vis-
coelastic number Zhrx (p. 133) or the viscoelastic number (1.) Nη (p. 131).
Info: [A23].
Jean Baptiste Joseph Fourier (see above).
3.3.20 Galilei Number Ga
Ga1 5gL3
v25Re2Fr21 ð1Þ;
Ga2 5Ga1
βΔTð2Þ; Ga3 5
RΔR
Ar ð3Þ
g (m s22) � gravitational acceleration; L (m) � characteristic length; ν (m2 s21) �kinematic viscosity; β (K21) � volume thermal expansion coefficient;ΔT (K) � tem-
perature difference; R, ΔR (kg m23) � liquid density and their difference in various
places of non-isothermic flow; Re (�) � Reynolds number (p. 81); Fr (�) � Froude
number (1.) (p. 62); Ar (�) � Archimedes hydrodynamic number (p. 53).
This number characterizes the molecular friction and gravity forces in flowing
fluid. In expression (2), it expresses the mutual action of the molecular friction
force and the buoyancy force in a non-isothermal process. In expression (3), it is
about the mutual action of the molecular friction force and the Archimedes force
determined by different densities of the fluid in a system. Rheology.
Info: [B17],[B20].
Galileo Galilei (15.2.1564�8.1.1642), Italian mathematician,
astronomer and physicist, originally doctor.
In addition to extensive astronomical observations with a
telescope he had improved, he was also engaged in the
dynamics of bodies. For example, he discovered inertia, the
law of superposition of velocities, the free fall law and
the pendulum swing time dependence on its length. With his
telescope, he was the first to discover the craters on the
Moon, solar spots, the four moons of Jupiter and the Milky
Way as an aggregation of remote stars.
123Fluid Mechanics
3.3.21 Graetz Rheological Number Gzr
Gzr 5RcpwL2nλL
R (kg m23) � density; cp (J kg21 K21) � specific heat capacity; w (m s21) � refer-
ence velocity; L, Ln (m) � length of heat transfer in front and vertical flow;
λ (W m21 K21) � thermal conductivity.
It expresses the ratio of heat transferred by convection to that transferred by con-
duction. It characterizes the mutual ratio of the heat transfer by convection to the
transfer by conduction in rheological systems such as molten polymers.
Info: [A29],[B17].
Leo Graetz (p. 193).
3.3.22 Heat Build-up Number Gn
See Brinkman rheological number Br (p. 118).
3.3.23 Hedstrom Number (1.) Hd1
Hd1 5σkL
2
Rv25Re Bm
σk (Pa) � yield limit; L (m) � characteristic length; R (kg m23) � fluid density;
ν (m2 s21) � kinematic viscosity of plastic; Re (�) � Reynolds number (p. 81);
Bm (�) � Bingham number (p. 118).
It characterizes non-Newtonian fluid flow (the creep of Bingham plastics). See
the Bingham number Bm (p. 118) and the plastic deformation K (p. 128).
Info: [A26],[A29],[B17].
Bengt Olaf-Arvid Hedstrom (born 1926), Swedish chemist.
3.3.24 Hedstrom Number (2.) Hd2
Hd2 5σEL
ηw
σE (Pa) � elastic limit stress; L (m) � characteristic length; η (Pa s) � dynamic
viscosity in the plastic state; w (m s21) � mean velocity.
It relates to Bingham plastics flow and corresponds to the Bingham number Bm
(p. 118).
Info: [A26].
Bengt Olaf-Arvid Hedstrom (see above).
124 Dimensionless Physical Quantities in Science and Engineering
3.3.25 Ilyushin Number Il
Il54
3
Lτηw
5Re4
3
τRw2
� �
L (m) � characteristic length, channel diameter; τ (Pa) � maximum dynamic slip
stress; η (Pa s) � dynamic viscosity; w (m s21) � mean flow velocity; R (kg m23) �density; Re (�) � Reynolds number (p. 81).
It characterizes the viscoplastic fluid flow under the action of a pressure gradient
in a ring canal consisting of two immovable cylinders. With Il-0, a Newtonian
fluid flows.
Info: [A29].
Alexey Antonovich Ilyushin (born 1911), Russian engineer.
3.3.26 Mackley�Sherman Number MS
MS5βw
5 tgϑ
β (m s21) � mass transfer coefficient of particle; w (m s21) � velocity of incident
particle; ϑ (�) � particle impact angle.
It expresses the ratio of the particle mass transfer velocity to its velocity of
impact on a diaphragm surface which equals the tangent of the particle impact
angle. The Stanton mass number Stm (p. 265) is another analogue of the number
MS for a cluster of particles.
Malcolm Robert Mackley (born 1947), English chemical
engineer.
He is engaged in the flow of rheological fluids and its
industrial applications in the polymerization process in the
food industry. Especially, his research concerns the oscillation
mixing of liquids and the cool processing thereof in production
lines. It includes the drawing-through of molten polyethylene
and other industrial and nutritional materials. In his work, he
uses both direct experiment and numerical modelling.
3.3.27 Magnetic and Thermal Energy Ratio λ
λ5Emg
Et
5Emg
kT5
μ0μm2
16πd3kT; where m5
4
3πd3H
Emg (J) � magnetic energy; Et (J) � thermal energy; k (J K21) � Boltzmann con-
stant; T (K) � absolute temperature; μ0 (H m21) � vacuum magnetic permeability;
125Fluid Mechanics
μ (H m21) � dissolvent magnetic permeability; d (m) � particle diameter; m
(A m2) � magnetic moment; H (A m21) � magnetization.
It describes the microstructure development in magnetorheological suspensions,
which represent artificial suspensions of magnetized particles immersed in non-
magnetic fluid. Together with the Mason number Mas (p. 126), it represents the
basic criterion in magnetorheology. Rheology.
Info: [B74].
3.3.28 Mach Viscoelastic Number Ma
Ma5wffiffiffiffiffiffiffiffiffiη
Rτrx
r
w (m s21) � flow velocity; η (Pa s) � dynamic viscosity; R (kg m23) � fluid den-
sity; τrx (s) � time of stress relaxation.
It expresses the viscoelastic fluid flow rate at the inlet of a pipeline or canal. It
is important in the extruding or injecting of plastics, particularly with geometrically
simple input parts when complex singularities arise in the fluid flow. Fluid mechan-
ics. Rheology.
Info: [B34].
Ernst Mach (p. 73).
3.3.29 Mason Number Mas
Mas5122ηωμμ0H
2
η (Pa s) � solvent dynamic viscosity; ω (s21) � rotating frequency; μ (H m21) �solvent magnetic permeability; μ0 (H m21) � vacuum magnetic permeability;
H (A m21) � particle magnetization.
This number expresses the viscous-to-magnetic forces ratio. It characterizes the
ability of particles suspended in a non-magnetic fluid to be magnetized (the diame-
ter 1�10 μm approximately) with respect to a rotating magnetic field under the
influence of modulation according to the Mason number. Magnetorheological pro-
cesses of sedimentation.
Info: [B74].
3.3.30 Nahme Number Na
Na5w2η
λΔTrh; where ΔTrh 52
@ ln Ah1
@T
� �21
Tini
126 Dimensionless Physical Quantities in Science and Engineering
w (m s21) � characteristic velocity; η (Pa s) � dynamic viscosity; λ (W m21 K21) �thermal conductivity; ΔTrh (K) � temperature difference in the rheological process;
T (K) � temperature; Ah1 (�) � Arrhenius number (1.) energetic (p. 34).
It characterizes the mutual relation between the heat transfer by dissipation and
conduction in non-isothermal rheological systems such as molten polymers. It is a
special case of the Heat build-up number Gn (p. 124) pro ΔT5ΔTrh.
Info: [A23].
3.3.31 Nahme Nonisothermal Number Na
Na5ηβw2
λT; where w5 L _γ; β5
T
ηdηdT
η (Pa s) � dynamic viscosity in the limit of zero shear rate; β (�) � thermal sensi-
tivity of the fluid viscosity; w (m s21) � characteristic velocity; λ (W m21 K21) �thermal conductivity; T (K) � absolute temperature; L (m) � characteristic length
(L5 h); _γ (s21) � shear rate.
It characterizes the non-isothermal instability in the torsion flow of polymeric
fluids and expresses the influence of viscous fluid heating. NaAh0; 1i.Info: [B98].
3.3.32 Nahme�Griffith Number Na
Na5ΔTdis
ΔTvis5Br Ps
ΔTdis (K) � temperature difference developed by viscous dissipation of fluid;
ΔTvis (K) � temperature difference developed by temperature-dependent viscosity;
Br (�) � Brinkman number (p. 174); Ps (�) � Pearson number (p. 127).
This number expresses the viscous heating influence in the action of both vis-
cous dissipation and fluid viscosity depending on temperature. Nahme�Griffith
number is usually in the range 0,Na, 200, for 0.1,Na, 0.5 viscosity is unaf-
fected by temperature. Rheology.
Info: [A9].
3.3.33 Pearson Number Ps
Ps5ΔTop
ΔTvis5αðTS 2 TNÞ
ΔTop (K) � temperature difference imposed by operating conditions; ΔTvis (K) �temperature difference required to change the viscosity; TS (K) � surface wall
127Fluid Mechanics
temperature; TN (K) � temperature of thermal undisturbed fluid area; α (K21) �thermal expansion coefficient; η0, η (Pa s) � basic and real dynamic viscosities.
This number expresses the ratio of the operating temperature range to the tem-
perature gradient necessary to change the viscosity. It characterizes the viscosity
temperature dependence in cases when the temperature gradient must be considered
in boundary conditions. For the viscosity exponential dependence on the tempera-
ture, it is
η5 η0 expð2αΔTopÞ
Rheology.
Info: [A23].
3.3.34 Peclet Rheological Number Perh
Perh 5L2 _γa
L (m) � characteristic length, gap width; _γ (s21) � sound velocity; a (m2 s21) �thermal diffusivity.
It characterizes the diffusion heat transfer in non-isothermal polymer fluid flow.
PerhBh0; 1i.Info: [B25].
Jean Claude Eugene Peclet (p. 180).
3.3.35 Plastic Deformation K
See Bingham number Bm (p. 118).
3.3.36 Plastic Viscosity Ratio Pη, k
Pη 5ηp
ηs 1 ηp
ηs, ηp (Pa s) � dynamic viscosities of dissolvent and plastic.
It expresses the total viscosity as the ratio of the Newtonian fluid (solvent) vis-
cosity to the polymeric viscosity. It is used, for example, in drawing or injecting
plastics. Fluid mechanics. Rheology.
Info: [B34].
128 Dimensionless Physical Quantities in Science and Engineering
3.3.37 Plasticity Number Npl
Npl 5τELηw
τE (Pa) � yield limit; L (m) � characteristic length; η (Pa s) � dynamic viscosity;
w (m s21) � velocity.
It is used in rheology and the creep of plastics. It is an equivalent of the
Bingham number Bm (p. 118) and analogous to the Hedstrom number (2.) Bm
(p. 124).
Info: [A35].
3.3.38 Prandtl Generalized Number Prgener
Prgener 5wL
aðw22 nLnkR21Þ2 1
11 n
w (m s21) � flow velocity; L (m) � characteristic length; a (m2 s21) � thermal dif-
fusivity; n (�) � exponent of non-Newtonian fluid; κ (kg m21 s21) � consistency
coefficient of pseudoplastic and dilating fluids; R (kg m23) � density.21 It expresses the relation between the velocity and temperature processes in
flowing non-Newtonian fluids.
Info: [A33].
Ludwig Prandtl (p. 197).
3.3.39 Processability Parameter Np, P
Np 5βησ
5 Sh CaðRe ScÞ21 5 StmCa
β (m s21) � mass transfer coefficient; η (Pa s) � dynamic viscosity; σ (N m21) �surface tension; Sh (�) � Sherwood number (p. 264); Ca (�) � capillary number
(1.) (p. 97); Re (�) � Reynolds number (p. 81); Sc (�) � Schmidt number
(p. 263); Stm (�) � Stanton mass number (p. 265).
In microfibre rheometry, it is applied to predict the fibre shape and to support
the process using adhesive and other non-Newtonian fluids. This parameter serves
to determine the critical time within which the fibre is broken. Above all, this time
depends on the rheological properties of the fluid and the mass transfer as
expressed by this parameter. Rheology. Microrheology.
Info: [C136].
129Fluid Mechanics
3.3.40 Reynolds Generalized Number for Non-Newtonian Fluids Renon
Renon 58Rw2
τcharð1Þ; Renon 5
Rw2
τcharð2Þ
R (kg m23) � fluid density; w (m s21) � fluid velocity; τchar (Pa) � characteristic
shear stress depending on Boltzmann constant and absolute temperature.
It is used in form (1) for the flow of non-Newtonian fluids in tubes of circular
cross section. In form (2) to liquid crystals flow, it expresses the inertia-to-
characteristic tensions ratio which originates in liquid crystal flow.
Info: [A29].
Osborne Reynolds (p. 82).
3.3.41 Reynolds Rheological Number (1.) Rerh
Rerh 5L2 _γv0
L (m) � characteristic length, gap width; _γ (s21) � shear velocity; ν0 (m2 s21) �kinematic viscosity at zero shear velocity.
It characterizes the convective transfer in the non-isothermic flow of polymer
fluids. Rerh{1. Rheology.
Info: [B98].
Osborne Reynolds (see above).
3.3.42 Stokes Number Stk
Stk1 5vw
gL25Ca Bo21 ð1Þ; Stk2 5
vτL2
5 Sh21 Re21 ð2Þ;
Stk3 5ωLv
ð3Þ; Stk4 5LΔp
ηwð4Þ
ν (m2 s21) � kinematic viscosity; w (m s21) � flow velocity; g (m s22) � gravita-
tional acceleration; L (m) � particle characteristic dimension; τ (s) � particle
vibration time in fluid flow; ω (s21) � angular frequency of particle; Δp (Pa) �pressure difference; η (Pa s) � dynamic viscosity; Ca (�) � capillary number (1.)
(p. 97); Bo (�) � Bond number (p. 95); Sh (�) � Strouhal number (p. 87);
Re (�) � Reynolds number (p. 81).
In expression (1), it expresses the viscosity-to-gravity forces ratio. In cases (2)�(4),
it expresses the inertia-to-friction forces ratio. Hydromechanics. Dynamics of particles.
Rheology.
Info: [A7],[A29],[A35],[B20].
130 Dimensionless Physical Quantities in Science and Engineering
George Gabriel Stokes (13.8.1819�1.2.1903), Irish mathe-
matician and physicist.
He was engaged in the mechanics of continua, especially
in hydrodynamics, waves in elastic bodies, acoustics and dif-
fraction. The fundamental continuum mechanics equation is
called the Navier�Stokes equation after him. In addition to
continuum mechanics, he measured gravitational field
changes on the surface of the earth. He was engaged in vec-
tor analysis, and in chemistry and botany as well.
3.3.43 Thermoelastic Rheological Number Nelast
Nelast 5 τrxL
ffiffiffiffiffiffiffiffiη0βλT
r
τrx (s) � relaxation time of viscoelastic stresses; L (m) � characteristic length; η0(Pa s) � dynamic viscosity at zero shear rate; β (�) � temperature sensibility of fluid
viscosity; λ (W m21 K21) � thermal conductivity; T (K)� absolute temperature.
It characterizes the non-isentropic flow of polymer fluids. It expresses the ratio
of the heat propagation time to the polymerization time. NelastAh0; 1i.Info: [B98].
3.3.44 Truncation Number, Shearing Failure Parameter NT
NT 5ηωp
η (Pa s) � dynamic viscosity; ω (s21) � angular speed of shear deformation; p (Pa) �pressure.
It is the ratio of the shear stress to the normal pressure. It relates to viscous non-
Newtonian fluid flow. Viscoelasticity. See the Hersey number He (p. 165).
Info: [A35].
3.3.45 Viscoelastic Number (1.) Nη, E
Nη 5τrxηRL2
ð1Þ; Nη 5M2Re22 ð2Þ
τrx (s) � stress relaxation time; η (Pa s) � dynamic viscosity; R (kg m23) � den-
sity; L (m) � characteristic length; M (�) � Mach number (p. 73); Re (�) �Reynolds number (p. 81).
In expression (1) or (2), it expresses the elastic properties of viscoelastic sur-
roundings and their flow. In the case of viscometry, L denotes the capillary tube
131Fluid Mechanics
diameter. It is also called the Fourier viscoelastic relaxation number Forx (p. 123)
or the Zhukovsky viscoelastic number Zhrx (p. 133).
Info: [A23].
3.3.46 Viscoelastic Number (2.) Nη
Nη 5G
ηω
G (Pa) � shear modulus; η (Pa s) � dynamic viscosity; ω (s21) � angular speed.
It is the elastic-to-viscous forces ratio. It describes the flow (outflow) of viscous
fluids. Viscometry.
Info: [A24],[B20],[B59].
3.3.47 Viscoelastic Parameter Pη
Pη 5ηSηP
5ηPwLτchar
ηS (Pa s) � dynamic viscosity of dissolvent matter; ηP (Pa s) � characteristic
dynamic viscosity of the polymer; w (m s21) � motion velocity; L (m) � wall dis-
placement; τchar (Pa) � characteristic stress.
It expresses the ratio of the diluting material viscosity to the characteristic vis-
cosity of the polymer or, alternatively, the ratio of the shear rate to the characteris-
tic tension. It appears in the flow of polymer liquid crystals.22
3.3.48 Weissenberg Generalized Number Wi
Wi5 τffiffiffi_γ
pτ (s) � time scale for viscoelastic stress relaxation; _γ (s22) � shear rate.
It expresses the characteristic material time (relaxation time) and the shear
velocity. It characterizes the velocity and time relations in rheological processes in
viscoelastic shear flow. See also the Deborah generalized number De (p. 119).
Info: [A23],[A35].
Karl Weissenberg (born 1893), German physicist.
3.3.49 Weissenberg Number (1.) Ws
Ws5Rw2
τ5Re El ð1Þ; Ws5
wτrxL
5Re Forx ð2Þ;
132 Dimensionless Physical Quantities in Science and Engineering
Ws5ðt1 2 t2Þw
Lð3Þ; where τ1 t1 _τ5 η0ðε1 t2 _εÞ;
Ws5Ma2Re21 ð4Þ
R (kg m23) � fluid density; w (m s21) � flow velocity; τ (Pa) � shear stress; _τ(Pa s21) � shear stress rate; τrx (s) � relaxation time; L (m) � characteristic length,
diameter; t1, t2 (s) � time constants; η0 (Pa s) � dynamic viscosity; ε (�) � rela-
tive elongation; _ε (s21) � relative elongation rate; Re (�) � Reynolds number
(p. 81); El (�) � Ellis number (p. 122); Forx (�) � Fourier viscoelastic relaxation
number (p. 123); Ma (�) � Mach viscoelastic number (p. 126); Re (�) �Reynolds number (p. 81).
It characterizes the influence of the viscoelastic fluid properties on proceeding
processes. In expression (1), it expresses the ratio of the dynamic viscoelastic force
to the viscous force. In expression (2), it expresses the time change of the visco-
elastic fluid flow close to a bypassed wall. The expression (3) is analogous to (2),
and expresses more exactly the ratio of viscoelastic force � considering the relaxa-
tion modulus � to the elastic force only. In expression (4), it expresses that the
shear wave propagation rate is analogous to the sound propagation velocity in com-
pressible fluid flow. The Ws is a special case of the Strouhal number Sh. Fluid
mechanics. Rheology.
Info: [A29],[A33],[B20],[B34].
Karl Weissenberg (see above).
3.3.50 Zhukovsky Viscoelastic Number Zhrx
See the Fourier viscoelastic relaxation number Forx (p. 123), or the viscoelastic
number (1.) Nη (p. 131).
Nikolay Yegorovich Zhukovsky (p. 33).
133Fluid Mechanics
4 Solid Mechanics
I hold a single experiment higher than thousands of opinions originated by
imagination only.Mikhail Vasilyevich Lomonosov (1711�1765)
4.1 Linear and Non-Linear Solid Mechanics
In solid mechanics, the dimensionless quantities involve the elastic, viscoelastic
and plastic fields. For the elastic, the linear tension-dependent deformation is valid
and for the viscoelastic�plastic sphere, the non-linear behaviour is valid. The rela-
tive prolongation, Hooke and Johnson numbers are among the most widespread
similarity criteria.
4.1.1 Angular Displacement Number ND
ND 5EJkαqL3
E (Pa) � modulus of elasticity; Jk (m4) � shear modulus; α (rad m21) � angular
displacement; q (N m21) � length loading; L (m) � characteristic length.
It expresses the angular displacement of an elastically laid beam.
Info: [A24].
4.1.2 Beam Loading N
N5q
ELð1Þ;
N5qA
Eð2Þ
q (N m21) � length loading; qA (N m22) � surface loading; E (Pa) � modulus of
elasticity; L (m) � characteristic length.
It expresses longitudinal (1) and planar (2) beam loading. Elasticity, statics.
Info: [A24].
Dimensionless Physical Quantities in Science and Engineering. DOI: 10.1016/B978-0-12-416013-2.00004-X
© 2012 Elsevier Inc. All rights reserved.
4.1.3 Crater Scaling Relationship Ncra
Ncra 5hmax
r5αFr2β 5α
gr
w2im
� �β
hmax (m) � maximum crater depth; r (m) � radius of the projectile; α, β (�) �empirically or numerically determined constants; g (m s22) � gravitational acceler-
ation; wim (m s21) � impact velocity; Fr (�) � Froude number (1.).
It expresses the dynamics of projectile penetration into a purely hydrodynamic
target. The data of the constants (α, β), determined by experiment and numerically,
are presented in the literature.
Info: [B122].
4.1.4 Criterion of Stringed Musical Instruments Nsmi
Nsmi 5RL2f 2
σ
R (kg m23) � string density; L (m) � characteristic length of string; f (s21) � string
frequency; σ (Pa) � mechanical stress of string.
It expresses the dynamic similarity criterion for string instruments provided the
string tension and density are equal for all instruments (violin, viola, violoncello
and double bass). When building these instruments, it has the same value, for
example, Nsmi5 0.2 usually. The criterion has very deep musical and physical
meaning. Its value is deduced from the instrument prototype and involves all previ-
ous experiences. Diverse relative length of the string instruments is 0.9 for the vio-
lin, 1 for the viola, 2 for the violoncello and 2.8 for the double bass. With this,
corresponding frequencies are obtained for certain musical tones.
Info: [A24].
4.1.5 Decibel Scales SPL, Lv, La, LF, LP, LI , LE , Lw
SPL5 20 logp
p0
� �; where p0 5 20 μPa ð1Þ;
Lv 5 20 logv
v0
� �; where v0 5 1 nm s21 ð2Þ;
La 5 20 loga
a0
� �; where a0 5 1 μm s22 ð3Þ;
LF 5 20 logF
F0
� �; where F0 5 1 μN ð4Þ;
136 Dimensionless Physical Quantities in Science and Engineering
LP 5 20 logP
P0
� �; where P0 5 1 pW ð5Þ;
LI 5 20 logI
I0
� �; where I0 5 1 pW m22 ð6Þ;
LE 5 20 logE
E0
� �; where E0 5 1 pJ ð7Þ;
Lw 5 20 logw
w0
� �; where w0 5 1 pJ m23 ð8Þ;
P0 (Pa) � reference pressure value; v0 (m s21) � reference speed value; a0 (m s22) �reference acceleration value; F0 (N) � reference force value; P0 (W) � reference
power value; I0 (W m22) � reference intensity value; E0 (J) � reference energy
value; w0 (J m23) � reference energy density value.
In the area of dynamic quantities measurement, the decibel scales express the loga-
rithmic dimensionless useful-to-threshold signals ratio denoted hereafter by a zero
index. Usually the quantities are sound, noise, velocity, acceleration, force, power,
intensity, energy and density thereof. Then, the sound (SPL), velocity (Lv), accelera-
tion (La), force (LF), power (LW), intensity (LI), energy (LE) and energy density (Lw)
levels are the corresponding logarithmic scales. The logarithmic scale introduction is
based on the fact that the human ear perceives logarithmically due to which a loga-
rithmic unit can be applied advantageously (the bel or the decibel as the tenth part of
it). Most frequently, the decibel scale is used in acoustics and in measuring instru-
mental techniques, electronics and other ranges to express the signal-to-noise ratio.
Info: [C34].
4.1.6 Explosion Number Ex
Ex5 rpE
R
� �215
τ225
rp (m) � blast wave radius; E (J) � explosive energy; R (kg m23) � density; τ (s) �time.
It characterizes the pressure stroke waves in detonations. Special techniques.
Info: [A24], [B20].
4.1.7 Force Numbers NF
NF 5Fτ2
mLð1Þ;
NF 5F
qLð2Þ;
137Solid Mechanics
NF 5FL2
GJð3Þ;
F (N) � force; τ (s) � time; m (kg) � density; L (m) � characteristic length;
q (N m21) � length loading; G (Pa) � shear modulus; J (m4) � square moment of
cross section.
In equation (1), they express the dynamic force in damped mass movement. In
equation (2), they express the static force with longitudinal beam load and � in
equation (3), they express the static force with static torsion stress.
Info: [A24].
4.1.8 Fourier Thermoelastic Number Fote
Fote 5wteτL
; where wte 5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiλ1 2μ
R
s
wte (m s21) � velocity propagation of the thermoelastic deformation; τ (s) � time;
L (m) � characteristic length; λ, μ (Pa) � Lame functions; R (kg m23) � density.
It expresses the ratio of the thermoelastic wave penetration depth to the charac-
teristic system length. In the dimensionless form, it characterizes the thermoelastic
process propagation time in a system.
Info: [A23].
Jean Baptiste Joseph Fourier (p. 175).
4.1.9 Fourier Wave Number Fow,mech
Fow;mech 5c2τ2
L2; where c5
ffiffiffiffiE
R
s
c (m s21) � velocity propagation of the mechanical wave; τ (s) � time; L (m) �characteristic length; E (Pa) � modulus of elasticity; R (kg m23) � density of
material.
This number expresses the dimensionless time of the mechanical wave propaga-
tion in the material under strokes, for example. Solid phase mechanics. Stroke
phenomena.
Info: [A24].
Jean Baptiste Joseph Fourier (see above).
4.1.10 Hooke Number Ho
Ho � Cau5RσσF
ð1Þ;
138 Dimensionless Physical Quantities in Science and Engineering
Ho5Rw2
Eð2Þ
R (kg m23) � density; σ (Pa) � stress; σF (Pa) � stress from the loading force;
w (m s21) � velocity; E (Pa) � modulus of elasticity; Cau (�) � Cauchy number
(aeroelasticity parameter) (p. 155).
It is the ratio of the yield point or the breaking strength σ to the tension due to a
loading force (1), which can be the dynamic pressure of the fluid (2). Elasticity
hydrodynamics.
Info: [A23],[A29],[B17].
Robert Hooke (18.7.1635�3.3.1703), English scientist.
The Hooke’s law is his important discovery, according to
which the body deformation is proportional to the acting
force. This is valid for small deformations to the extent of
the elasticity limit. In the year 1678, he tried to prove that
the Earth’s orbit around the Sun is an ellipse. In the same
year, he discovered that the gravity force drops with the sec-
ond root of the distance. He informed Newton about this in
writing, but subsequently an authorship conflict arose
between them which led to extreme hostility.
4.1.11 Impact Factor Nimp, I
Nimp 5Mw2
ini
σcd3
M (kg) � mass of a projectile; wini (m s21) � impact velocity of a projectile;
σc (Pa) � unconfined compressive strength of a concrete target; d (m) � diameter
of a projectile.
This factor expresses the ratio of the projectile energy to the inverse deforming
pressure energy in a target. Impact mechanics. Ballistics.
Info: [B66].
4.1.12 Johnson’s Damage Number Dn
Dn5Rwini
σk
R (kg m23) � material density; wini (m s21) � initial impulse velocity; σk (Pa) �yield stress of the material.
This number is the measure of the inertia loading force related to the material
resistance against loading. It represents the basic parameter in impact mechanics to
solve the material plastic response especially with dynamic loading. It is the
139Solid Mechanics
deformation state measure in the zone where plastic deformation arises abruptly.
The Cauchy number (aeroelasticity parameter) Cau (p. 155) is a similar criterion
for the dynamic elastic response. Non-linear mechanics.
Info: [B127],[B47].
4.1.13 Johnson’s Damage Number Jo
Jo5w2ini Rtσc
5Nimp
NA
wini (m s21) � impact velocity of a projectile; Rt (kg m23) � density of a concrete
target; σc (Pa) � unconfined compressive strength of a concrete target; Nimp (�) �impact factor (p. 139); NA (�) � slenderness factor (p. 145).
It expresses the degree of the relative damage in the projectile impact on a tar-
get. Usually, it is applied to classify the projectile impact. Stroke mechanics.
Ballistics.
Info: [B66].
4.1.14 Loss Coefficient η
η5Δu
2πuð1Þ;
where u5
ðσmax
0
dεσ2max
2Eð2Þ;
Δu5
Iσ dε ð3Þ
Δu (J m23) � change of specific volume energy; u (J m23) � specific volume
energy; σmax (Pa) � maximal stress; σ (Pa) � stress by loading; E (Pa) � modulus
of elasticity; ε (�) � relative elongation (p. 144).
It expresses the material dynamic memory function and, alternatively, the degree
to which the material dissipates the energy of vibration. With the elastic load up to
the highest stress σmax, expression (2) is valid for the elastic energy related to the
volume unit. Expression (3) concerns the loading and following unloading in which
the energy dispersion corresponds to the area of the hysteresis stress deformation
loop. Usually, the loss coefficient is time and cycling frequency dependent.
Info: [C65].
4.1.15 Mach Thermoelastic Number Mte
Mte 5wt
wte
; where wt 5a
L; wte 5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiλ1 2μ
R
s
140 Dimensionless Physical Quantities in Science and Engineering
wt, wte (m s21) � velocity propagation of thermal and thermoelastic wave in the
system; a (m2.s21) � thermal diffusivity; L (m) � characteristic length; λ, μ (Pa) �Lame functions; R (kg m23) � density.
This number expresses the ratio of the heat propagation rate in elastic surround-
ings to the propagation rate of mechanical waves (sound rate). It characterizes the
mutual relation between the heat propagation rate and the thermal stress in a ther-
mal system.
Info: [A23].
Ernst Mach (p. 73).
4.1.16 Mass of the Projectile Nm
Nm 5M
Rd3
M (kg) � mass of a projectile; R (kg m23) � material density of a target; d (m) �projectile diameter.
In the dimensionless shape, it expresses the ratio of the projectile mass to the
target density. Solid mechanics. Ballistics.
Info: [B22].
4.1.17 Noise Number Noi
Noi 5NiT12refS
212
Ni (m) � mutually uncorrelated stochastic processes with zero mean value;
Tref (s) � dimensionless reference time; S (m2 s) � power density spectrum of the
stochastic process.
It characterizes a random process. It appears in a dimensionless mathematical
model to solve the stochastic equations set by means of the continuous hybrid
Monte Carlo method.
Info: [A24],[B36].
4.1.18 Nose Factor Pn
Pn 51
3Ψ2
1
24Ψ2; where Ψ5
R
dð1Þ; ðogive noseÞ
Pn 51
11 4Ψ2; where Ψ5
h
dð2Þ; ðconical noseÞ
Pn 5 121
8Ψ2; where Ψ5
r
dð3Þ; ðblunt=spherical noseÞ
141Solid Mechanics
Ψ (�) � calibrated radius head (CRH); R (m) � calibrated radius or diameter of
the ogive; d (m) � projectile diameter; h (m) � projectile nose length; r (m) �projectile nose radius.
This factor expresses the influence of the projectile nose shape on penetration.
The expression (1) is valid for the calibrated head radius or a broken Gothic arch,
the expression (2) holds for a conical peak, whereas the expression (3) holds for a
spherical top. The projectile nose factor becomes values in the range 0,Pn# 1.
For a flat top of a projectile, Pn5 1 is valid. For a semispherical projectile point,
Pn5 0.5 is valid. Stroke mechanics. Ballistics.
Info: [B66].
4.1.19 Oscillation Amplitude NA
NA 5EJy
qL4
E (Pa) � modulus of elasticity; J (m4) � square moment of cross section; y (m) �displacement of oscillation; q (N m21) � length loading; L (m) � characteristic
length.
This number expresses the amplitude of an oscillating beam. Elasticity.
Dynamics.
Info: [A24].
4.1.20 Oscillation Frequency Nf
Nf 5RL4f 2
EJð1Þ;
Nf 5ω2L
gð2Þ;
Nf 5ω2I
GL3ð3Þ;
R (kg m23) � density; L (m) � characteristic length; f (s21) � frequency; E (Pa) �modulus of elasticity; J (m4) � square moment of cross section; ω (s21) � angular
frequency; g (m s22) � gravitational acceleration; I (kg m2) � moment of inertia;
G (Pa) � shear modulus.
In form (1), it expresses the oscillating beam frequency. In form (2), it expresses
the mathematical pendulum oscillation frequency and in form (3), it means the
eigenfrequency of shaft torsion oscillations. Dynamics. Elasticity. Physics.
Info: [A24].
142 Dimensionless Physical Quantities in Science and Engineering
4.1.21 Peclet Thermoelastic Number Pete
See Vernotte thermoelastic number Vete (p. 283).
Jean Claude Eugene Peclet (p. 180).
4.1.22 Penetration Depth Hp
Hp 5h
d5 f ðNimp; NA; PnÞ
h (m) � penetration depth; d (m) � diameter of a projectile; Nimp (�) � impact
factor (p. 139); Na (�) � slenderness factor (p. 145); Pn (�) � nose factor.
It expresses the projectile penetration depth of a non-deformable projectile strik-
ing a target. It is the function of the impact factor Nimp (p. 139), of the slenderness
factor NA (p. 149) and of the nose factor Pn (p. 141). Stroke mechanics. Ballistics.
Info: [B66].
4.1.23 Poisson’s Ratio ν
ν52ε1ε2
5E
2G21
ε1 (�) � relative deformation in the cross section (contraction) by uniaxial stress;
ε2 (�) � relative deformation in axial direction (elongation) by uniaxial stress;
E (Pa) � modulus of elasticity; G (Pa) � shear modulus.
It characterizes the relative deformation in single axis stress. For perfect non-
compressible materials, it would be ν5 0.5 in theory; for most technical materials
it is νA 0; 0:5h i; for cork it is ν-0; for most steels it is ν � 0.3; for rubber it is
almost 0.5. For the Poisson constant, it is μ5 ν21. Mechanics of solid bodies.
Elasticity.
Info: [A23],[A29].
Simeon Denis Poisson (21.6.1781�25.4.1840), French math-
ematician and physicist.
He was one of the founders of mathematical physics and
one of Laplace’s students. Especially, he was engaged in
mathematical analysis, variation calculus, probability theory
and hydromechanics. By extending and generalizing the
Laplace equation with source members, he formed the
Poisson equation describing steady-state fields with internal
sources and sinks.
143Solid Mechanics
4.1.24 Ratio of the Plug Mass to the Projectile Mass NΔm
NΔm 5πd2RH4M
d (m) � projectile diameter; R (kg m23) � material density of a plate; H (m) pene-
tration depth of a projectile; M (kg) � projectile mass.
It expresses the ratio of the material mass loss of a thin plate, after a projectile
impact, to the projectile mass. Solid mechanics. Ballistics.
Info: [B22].
4.1.25 Relative Elongation ε
ε5σE5
ΔL
L5
F
EA
σ (Pa) � stress; E (Pa) � modulus of elasticity; ΔL (m) � absolute elongation;
L (m) � characteristic length; F (N) � force; A (m2) � cross-sectional area.
It expresses the relative elongation of a body.
Info: [A24].
4.1.26 Relative Stiffness Ns
Ns 5ksy
F
ks (kg s22) � stiffness of elastic element, spring constant; y (m) � compression of
elastic component; F (N) � force.
In the dynamics of mechanical sets, it expresses the ratio of elastic element com-
pression to the acting force.
4.1.27 Relative Volume Change PV
PV 5ΔV
V0
ΔV (m3) � volume change; V0 (m3) � initial volume.
It expresses the ratio of the volume change to the original volume before defor-
mation. Elasticity, mechanics.
Info: [A24].
144 Dimensionless Physical Quantities in Science and Engineering
4.1.28 Sachs Number Sa
Sa5 r
ffiffiffiffiffiPa
E
3
r
r (m) � blast wave radius (distance from explosion place to reference one); pa (Pa) �atmospheric pressure; E (J) � explosive energy.
It expresses the superficial explosion propagation. See the analogous explosion
number Ex (p. 137).
Info: [A24].
4.1.29 Slenderness Factor NA
NA 5M
Rchard3
M (kg) � mass of a projectile; Rchar(kg m23) � characteristic density of environ-
ment between projectile and target; d (m) � projectile diameter.
It expresses the ratio of the pressure cross section of a projectile to the charac-
teristic density zone for an unlimited target. It represents the projectile slenderness
for a solid characteristic density of the environment between a projectile and a tar-
get. Stroke mechanics. Ballistics.
Info: [B66].
4.1.30 Statical Moment Number NMS
NMS 5M
EL3ð1Þ;
NMS 5M
GL3ð2Þ;
NMS 5M
qL2ð3Þ
M (N m) � moment of force; E (Pa) � modulus of elasticity; G (Pa) � shear mod-
ulus; L (m) � characteristic length; q (N m21) � length loading.
In form (1), it expresses the moment for the static tensile or pressure stress; in
form (2), it expresses the torsion stress. The bending moment of a beam, laid on a
common elastic base and loaded longitudinally, is given by form (3).
Info: [A24].
4.1.31 Structural Property Number NS
NS 5σLE
145Solid Mechanics
σ (N m23) � volume loading; L (m) � characteristic length; E (Pa) � modulus of
elasticity.
This number expresses the mass-to-stiffness ratio of a structure. In a complex
mechanical system, it expresses the resulting value of its partial subsystems.
4.1.32 Thermoelastic Coupling Number (1.) N1
N1 5ð3λ1 2μÞα
cR5
Eαð12 2νÞcR 5
3KαcR
λ, μ (Pa) � Lame functions; α (K21) � linear thermal expansion coefficient;
cR (J m23 K21) � specific heat capacity; E (Pa) � modulus of elasticity; K (Pa) �volume modulus of elasticity; ν (�) � Poisson’s ratio (p. 143).
This number expresses the ratio of the specific thermal stress to the specific vol-
ume thermal capacity. It characterizes the relation between the thermoelastic stress
and the thermal capacity in a compound thermal and thermal stress process. In a
non-compound process, α5 0.
Info: [A23].
4.1.33 Thermoelastic Coupling Number (2.) (Gay-Lussac ThermoelasticNumber) N2, Gcte
N2 � Gcte 5ð3λ1 2μÞðλ1 2μÞ αTref 5βTref
λ, μ (Pa) � Lame functions; α (K21) � linear thermal expansion coefficient;
Tref (K) � reference temperature; β (K21) � volume thermal expansion coefficient.
In an elastic environment, it expresses the ratio of the thermal deformation prop-
agation velocity to the electromagnetic waves propagation velocity. It characterizes
the velocity relation between thermal and mechanical deformation energy propaga-
tion. With α5 0, the thermal and deformation processes do not depend on each
other. It enables the solution of temperature rise connected to the body deformation
in tasks on thermal energy dissipation.
Info: [A23].
Joseph Louis Gay-Lussac (p. 15).
4.1.34 Thermoelastic Coupling Number Resultant Nυ
Nυ 5N1N2 5ð3λ1 2μÞ2α1α2Tref
ðλ1 2μÞcR 5Eα1βTrefð12 2νÞcR 5
3Kα1βTrefcR
146 Dimensionless Physical Quantities in Science and Engineering
λ, μ (Pa) � Lame functions; α1, α2 (K21) � linear thermal expansion coefficient
in system 1 and system 2; Tref (K) � reference temperature; cR (J m23 K21) � spe-
cific volume heat; E (Pa) � modulus of elasticity; β (K21) � volume thermal
expansion coefficient; K (Pa) � volume modulus of elasticity; ν (�) � Poisson’s
ratio (p. 143); N1 (�) � thermoelastic coupling number (1.) (p. 146); N2 (�) �thermoelastic coupling number (2.) (p. 146).
It characterizes a compound thermal and thermal stress process in linear sys-
tems. The independence of both processes is expressed by the value of Kυ, because
α15 0 or the reference temperature Tref5 0.
Info: [A23].
4.1.35 Thermoelastic Deformation Nε
Nε 5ε
αTrefð1Þ;
Nε 5ε
ð11 νÞαTrefð2Þ;
Nε 5ð12 νÞε
ð11 νÞαTrefð3Þ
α (K21) � linear thermal expansion coefficient; Tref (K) � reference temperature;
ν (�) � Poisson’s ratio (p. 143); ε (�) � relative elongation (p. 144);
This quantity expresses the relative thermoplastic deformation with single axis (1),
planar (2) and space (3) thermal stresses.
Info: [A23].
4.1.36 Thermoelastic Displacement Nu
Nu 5u
αTrefLð1Þ;
Nu 5u
αLTrefð11 νÞ ð2Þ;
Nu 5uð12 νÞ
ð11 νÞαTrefLð3Þ
u (m) � displacement; α (K21) � linear thermal expansion coefficient; Tref (K) � ref-
erence temperature; L (m) � characteristic length; ν (�) � Poisson’s ratio (p. 143).
It characterizes the dimensionless thermoelastic displacement with single axis (1),
planar (2) and space (3) thermal stresses.
Info: [A23].
147Solid Mechanics
4.1.37 Thermoelastic Displacement Potential NΦu
NΦu 5Φ
βTrefL25
ð12 νÞΦð11 νÞαTrefL2
5ðλ1 2μÞΦ
ð3λ1 2μÞαTrefL2
Φ (m2) � thermoelastic potential of the displacement; β (K21) � volume thermal
expansion coefficient; Tref (K) � reference temperature; L (m) � characteristic
length; α (K21) � linear thermal expansion coefficient; λ, μ (Pa) � Lame func-
tions; ν (�) � Poisson’s ratio (p. 143).
It characterizes the dimensionless thermoelastic displacement potential.
Info: [A23].
4.1.38 Thermoelastic Potential NΦ
NΦu 5ΦuL
ð1Þ;
NΦε 5ΦεL2
ð2Þ;
NΦσ 5GΦσL2
ð3Þ
Φ (m2) � thermoelastic potential of the displacement; u (m) � displacement; L (m)�characteristic length; G (Pa) � shear modulus; σ (Pa) � stress; ε (�) � relative elon-
gation (p. 144).
It characterizes the dimensionless thermoelastic potential related to displacement (1),
deformation (2) or stress (3).
Info: [A23].
4.1.39 Thermoelastic Stress Function NF
NF 5ð12 νÞFEαTrefL2
F (N) � thermoelastic stress function; E (Pa) � modulus of elasticity; α (K21) lin-
ear thermal expansion coefficient; Tref (K) � reference temperature; L (m) charac-
teristic length; ν (�) � Poisson’s ratio (p. 143).
In the dimensionless form, it expresses the thermoelastic stress function.
Info: [A23].
148 Dimensionless Physical Quantities in Science and Engineering
4.1.40 Thermoelastic Stress Number Nσ
Nσ 5σ
αETrefð1Þ;
Nσ 5σð12 νÞαETref
ð2Þ;
Nσ 5σk
αEΔTlsð3Þ
σ (Pa) � stress; α (K21) � linear thermal expansion coefficient; E (Pa) � modulus
of elasticity; Tref (K) � reference temperature; σk (Pa) � yield limit; ΔTls (K) �temperature difference between liquidus and solidus; ν (�) � Poisson’s ratio
(p. 143).
In the dimensionless form, this number characterizes the thermoelastic stress for
planar (1) and space (2) linear stresses. In expression (3), it is about the criterion of
internal residual thermoelastic stress.
Info: [A23].
4.1.41 Viscous Damping Coefficient Nd, .
Nd 5kdffiffiffiffiffiffiffiffiksm
p
kd (kg s21) � coefficient of viscous damping; ks (kg s22) � stiffness of elastic com-
ponent, spring constant; m (kg) � mass.
In the dynamics of mechanical sets, it expresses the ratio of the viscous damping
force to the elastic element stiffness.
4.2 Fracture Mechanics and Micromechanics
In fracture mechanics, the physical similarity criteria are focused to prevent struc-
ture defects from originating, especially those which could cause cracks and mate-
rial fractures. They characterize the material resistance. This relates to the
elastoplastic and/or fully plastic material loading conditions. Well-known dimen-
sionless numbers include the Fick number, resistance against crack generation,
fracture strength and the crack propagation number.
149Solid Mechanics
4.2.1 Crack Filling Number N
N5pmτνR
5EuFoPrRe2
pm (Pa) � metallostatic pressure; τ (s) � local solidification time; ν (m2 s21) �kinematic viscosity; R (kg m23) � density; Eu (�) � Euler number (1.) (p. 61); Fo
(�) � Fourier number (p. 175); Pr (�) � Prandtl number (p. 197); Re (�)
Reynolds number (p. 81).
It characterizes the possibility to fill the crack with residual interdendrite molten
material or alternatively to liquefy the existing crack. It is the fracture mechanics
criterion.
Info: [A23].
4.2.2 Crack Propagation Number, Ice Number NCR
NCR 5Rw2
E
� �2ffiffiffiffiffiffiEL
R
r; where R5
πσ2a
E
R (kg m23) � density; w (m s21) � crack propagation velocity; E (Pa) � modulus
of elasticity; L (m) � characteristic length; R (kg s21) � mechanical resistance;
σ (Pa) � stress; a (m) � half-length of a crack.
It is the extension of the Cauchy number (aeroelasticity parameter) (p. 155) in
the fracture mechanics. It enables one to follow the evolution of crack propagation
and the influence of resistance on crack freezing.
Anthony G. Atkins, American engineer.
Robert M. Caddell, American engineer.
4.2.3 Energy Release Number Nσ,res
Nσ;res 5NF
qEf
σ2resh
; where Ef 5E
ð12 ν2Þ
NF (�) � acting force on the notch forehead; q (N m21) � length loading; Ef (Pa) �plane elasticity modulus of layer and substrate; σres (Pa) � residual stress; h (m)
distance in the layer thickness; E (Pa) � modulus of elasticity; v (�) � Poisson’s
ratio (p. 143).
It expresses the inner-to-outer stress ratio created due to loosening in the loca-
tion of a longitudinal notch, crack or cutting in a surface layer.
Info: [B50].
150 Dimensionless Physical Quantities in Science and Engineering
4.2.4 Fick Number Fi
Fi � Fo21D 5
L2
Dτ
L (m) � characteristic length; D (m2 s21) � diffusivity; τ (s) � time; FoD (�) �Fourier mass number (p. 249).
It characterizes the diffusion intensity of the elements in the solid phase in rela-
tion to the distance of primary or secondary dendritic axes and to the local solidifi-
cation time. With increasing Fi, the mixing off of the elements during
solidification rises, the solidification interval increases and the trend to cracks
between dendrites increases (e.g. in steel especially). Of the metallurgic factors, the
steel temperature magnifies the value of the Fi. By refining the dendrite structure,
for example, by modifying the steel at the end of the reduction time in a furnace or
in a pan in tapping, the unfavourable influence of the number Fi can be limited
substantially. It is a criterion of fracture mechanics.
Info: [A23].
Adolf Eugen Fick (3.9.1829�21.8.1901), German
physiologist.
After an initial study of mathematics, he devoted himself
primarily to physiology. He formulated the diffusion laws
and an equation for neutral particles (1855). With this he laid
an important base for subsequent research of diffusion pro-
cesses, which led later to contact lens development, among
others. In addition, he was engaged in studying the activity
of the sense organs and in examining electric signals under
the excitation of nerves.
4.2.5 Fracture Strength Nf
Nf 5σfrac
σth
5 60
ffiffiffiffiffiffiσEh
r
σfrac (Pa) � fraction stress of material; σth (Pa) � theoretical stress of material
(σth5E/30); σ (N m21) � surface stress; E (Pa) � modulus of elasticity; h (m) �thickness of material.
This number characterizes fraction stress in nanomaterials. Fracture mechanics
and nanomaterials.
Info: [B38].
4.2.6 Interval of Solidification N
N5ΔTls
Tc
ΔTls (K) � temperature interval of solidification; Tc (K) � casting temperature.
151Solid Mechanics
It expresses the relation of the thermal solidification interval to the pouring tem-
perature. It characterizes the material trend to cracks between the dendrites during
solidification. With optimal pouring temperature, the tendency increases with the
solidification interval. Then, the crack tendency increases with the increasing crite-
rion value. It is a fracture mechanics criterion.
Info: [A23].
4.2.7 Metallostatic Pressure N
N5pm
gRL5EuFr
pm (Pa) � metallostatic pressure; g (m s22) � gravitational acceleration;
R (kg m23) � density; L (m) � characteristic length; Eu (�) � Euler number (1.)
(p. 61); Fr (�) � Froude number (1.) (p. 62).
It characterizes the ratio of the metallostatic pressure in a critical place of a cast-
ing to its characteristic wall thickness. The criterion application is joined with the
building of risers on castings. The greater its value, the less the probability of a
crack originating. For certain characteristic casting thicknesses, the tendency to cre-
ate interdendrite cracks can be suppressed by increasing the metallostatic height
(e.g. riser heightening, riser filling or overpressure riser application). It is a fracture
mechanics criterion.
Info: [A23].
4.2.8 Microcrack Initial Number Ncrini
Ncrini 5FiniEf
σ2A
Fini (N) � acting dynamic force; Ef (Pa) � elasticity modulus of thin layer; σ (Pa) �tension stress in thin layer; A (m2) � opening surface in the crack size of surface
layer.
This number characterizes the crack evolution in a thin surface layer or on an
elastic substrate or a viscous sublayer. It expresses the relation between an outer
initializing force acting on the layer and an internal stress force in the layer. For
example, it relates to electronic parts such as solar cells and flat panel display tech-
nology. Microelectronics in macroelectronic systems. Fracture mechanics.
Info: [B110].
4.2.9 Microcrack Steady Number Ncrst
Ncrst 5FstEf
σ2A
152 Dimensionless Physical Quantities in Science and Engineering
Fst (N) � steady dynamic force; Ef (Pa) � elasticity modulus of thin layer; σ (Pa) �tension stress in thin layer; A (m2) � opening surface in the crack size of sur-
face layer.
It characterizes a steady crack in a thin surface layer or on an elastic substrate
or a viscous sublayer. It expresses the relation between an outer steady-state force
acting on a layer and an internal stress force in a layer. Microelectronics in macro-
electronic systems. Fracture mechanics.
Info: [B110].
4.2.10 Relative Fracture Strain εfrac
εfrac 5 2
ffiffiffiffiffiffiEs
Eh
r
Es (J m22) � density of surface energy; E (Pa) � modulus of elasticity; h (m) �
depth of the crack opening.
It characterizes the fracture deformation in nanomaterials. Fracture mechanics.
Nanotechnology.
Info: [B38].
4.2.11 Relative Strain Index Pε
Pε 5εyieldεfrac
εyield (�) � yield strain; εfrac (�) � fracture strain.
This index expresses the relation of a relative yield point deformation to its
material destruction value. Fracture mechanics.
4.2.12 Resistivity Against Crack Generation (1.) N
N5pmL
νRgτ5EuFrReSh21
pm (Pa) � metallostatic pressure; L (m) � critical defect size; ν (m2 s21) � kinematic
viscosity; R (kg m23) � density; g (m s22) � gravitational acceleration; τ (s) �solidification time; Eu (�) � Euler number (1.) (p. 61); Fr (�) � Froude number
(1.) (p. 62); Re (�) � Reynolds number (p. 81); Sh (�) � Strouhal number (p. 87).
It characterizes the resistivity against crack generation in a solidifying casting. It
is a fracture thermomechanics criterion.
Info: [A23].
153Solid Mechanics
4.2.13 Resistivity Against Crack Generation (2.) N
N5σ
νRgτ5 LeFrðFoWeScÞ21
σ (Pa) � surface stress; ν (m2 s21) � kinematic viscosity; R (kg m23) � density;
g (m s22) � gravitational acceleration; τ (s) � solidification time; Le (�) � Lewis
number (p. 254); Fr (�) � Froude number (1.) (p. 62); Fo (�) � Fourier number
(p. 175); We (�) � Weber number (p. 389); Sc (�) � Schmidt number (p. 263).
It characterizes the resistivity against crack generation as a function of the
molten stuff surface energy, viscosity and solidification time. It is a fracture ther-
momechanics criterion.
Info: [A23].
4.2.14 Strain at Fracture εfrac
εfrac 53fr sin α
L2
f (m) � deflection of the lateral twig at breakage; r (m) � radius of the lateral
twig, distance from the neutral plane; α (�) � angle just before breakage; L (m) �distance from the twig base to the point where the force acts.
It expresses the deformation at the point where the fracture arises. Fracture
mechanics.
4.2.15 Stress Index Pσ
Pσ 5σyield
σfrac
σyield (Pa) � yield stress; σfrac (Pa) � fraction stress.
It expresses the ratio of the yield point stress to the material destruction stress.
Fracture mechanics.
4.3 Aeroelasticity
In aeroelasticity, the dimensionless quantities express the influence of the interac-
tion between inertia, elasticity and aerodynamic forces. In static aeroelasticity, the
dimensionless quantity relates to the problems of divergence and regressive control
of a bypassed surface. In dynamic aeroelasticity, the dimensionless quantity relates
to the problems of fluttering and the dynamic response of mechanical parts which
are bypassed. The aeroelasticity parameter, Cauchy’s, Connor’s, Frueh’s, Regier’s
154 Dimensionless Physical Quantities in Science and Engineering
and other numbers are among the well-known dimensionless quantities in
aeroelasticity.
4.3.1 Aeroelasticity Parameter Ae
See the Cauchy number (aeroelasticity parameter) Cau (p. 155).
Info: [A29].
4.3.2 Aeroelasticity Stiffness NAE
NAE 52E
Rw2
E (Pa) � modulus of elasticity; R (kg m23) � density; w (m s21) � flow velocity.
It expresses the ratio of the structure stiffness to the aerodynamic force. It is the
inverse value of the aeroelasticity parameter AE (p. 155). Aeroelasticity.
4.3.3 Cauchy Number (Aeroelasticity Parameter) Cau
Cau � M2 � Ho5R w2
Kð1Þ;
Cau5R w2
Eð2Þ
R (kg m23) � density; w (m s21) � velocity; K (Pa) � volume modulus of elastic-
ity; E (Pa) � modulus of elasticity; M (�) � Mach number (p. 73); Ho (�) �Hooke number (p. 138).
This parameter expresses the ratio of the inertia force to the material compress-
ibility force. It characterizes the compressible fluid flow (1) and the dynamic mate-
rial strain (2) by inertia forces. It is often called the aeroelasticity parameter as
well. Aerodynamics, aeroelasticity, dynamics.
Info: [A23],[A43],[B127],[C37].
Augustin Louis Cauchy (21.8.1789�23.5.1857), French
mathematician.
He was known for his precision and consistency in mathe-
matics. He introduced many concepts such as the determi-
nant, limit, continuity and convergence. He founded complex
analysis and deduced the Cauchy�Riemann conditions with
Riemann. He was very prolific, publishing nearly 800 works.
155Solid Mechanics
4.3.4 Connors Number Con
Con5w
fL5K
ffiffiffiffiffiffiffiffimδRL2
s
w (m s21) � mean velocity of two-phase flow; f (s21) � oscillation frequency;
L (m) � characteristic length, pipe diameter; K (�) � instability factor;
m (kg m21) � length density of pipe material; δ (�) � damping decrement;
R (kg m23) � fluid density.
It relates to the fluid elastic vibration characteristics in the two-phase flow of
fluids. For example, it relates to tubes exposed to a dynamic two-phase flow of air
and water in heat exchangers. The number Con serves to determine the critical
flow rate and is based on the mean flow rate, mean fluid density and damping in a
two-phase flow. Aeroelasticity, two-phase flow, heat exchangers.
Info: [G28].
H. J. Connors, American scientific researcher.
Together with P.M. Moretti, Connors is among the significant personalities in
the aeroelasticity field, in dynamic induced oscillation and fluttering.
Peter M. Moretti, American scientist.
His scientific work is very wide-ranging and involves the
dynamics of mechanical systems, especially those of induced
vibrations excited by fluid streaming. Primarily, this relates
to the fluid stream dynamic acting on the nest of tubes in
heat exchangers, further about the instability solution in
transversal streaming through these nests and the eigenfre-
quency determination and damping of nests. He was also
engaged in research related to thermal stratification in lakes,
in supersonic streaming and heat transfer problems.
4.3.5 Flutter Number F
F5weq
wR
5M
Rg
weq (m s21) � equivalent air velocity near the sea level; wR (m s21) � Regier sur-
face velocity index (flutter parameter); M (�) � Mach number (p. 73); Rg (�) �Regier number (p. 157).
High velocities aerodynamics. Aeroelasticity. Flutter.
Info: [B94].
4.3.6 Frequency Parameter Pf
Pf 5ωLw
5 2πSh
156 Dimensionless Physical Quantities in Science and Engineering
ω (s21) � angular velocity; L (m) � characteristic length (e.g. channel width or dis-
tance from a wall); w (m s21) � flow velocity; Sh (�) � Strouhal number (p. 87).
It characterizes the unsteady flow in bundles, fluidization and other dynamic
processes acting on mechanical systems, as an example.
Info: [A29],[A35].
4.3.7 Frueh Number Fh
Fh5Lω1
a
ffiffiffiffiffiffiNm
cL
r
L (m) � characteristic length, (e.g. half-length of wing chord); ω1 (s21) � first har-
monic frequency of torsional vibration; a (m s21) � sound velocity; cL (�) � uplift
curve inclination; m (kg m21) � mass to the wing length; R (kg m23) � air density;
Nm (�) � mass ratio (p. 75).
Aeroelasticity, transonic flutter of wings.
Info: [A24].
Frank J. Frueh, American aerodynamic engineer.
4.3.8 Mass Ratio Nm
Nm 5m
πRL3
m (kg) � mass of body; R (kg m23) � fluid density; L (m) � characteristic length
of a body.
Mechanical systems dynamics. Aircraft fluttering and stability. Aeroelasticity.
Info: [A24].
4.3.9 Regier Number Rg
Rg5Lωa
ffiffiffiffiffiffiffiffiffiffiRl
πRL2
s
L (m)� characteristic length; ω (s21)� angular velocity; a (m s21) � sound velocity;
Rl (kg m23) � fluid density; R (kg m23) � material density.
It expresses aeroelastic relations, transonic flutter of wings and blades of flow
machines. It is a modification of the Strouhal number Sh (p. 87). Aeroelasticity.
Info: [A24].
157Solid Mechanics
4.3.10 Regier Surface Number Rgs
Rgs 5wR
a
wR (m s21) � Regier surface velocity index (flutter parameter); a (m s21) � sound
velocity.
It is a simpler expression of the Regier number Rg (p. 157) and represents the
elastic-to-aerodynamic forces ratio at sea level. High velocity aerodynamics.
Aeroelasticity. Flutter.
Info: [A24],[B94].
4.4 Tribology
The tribologic dimensionless quantities involve especially the subjects of friction,
lubrication and wearing. Usually, they express the interaction of surfaces in their
relative movement. For example, they relate to the friction and lubrication of bear-
ings and the friction and wear of tools in machining. The dimensionless quantities
especially apply to new micro- and nanotechnologies, in which the tribological pro-
cesses represent a basic problem. Among the well-known tribologic dimensionless
quantities are Gumbel’s, Hersey’s and Sommerfeld’s numbers, as well as wear
intensity and friction factor.
4.4.1 Adhesion Parameter Nadh,Θ
Nadh 5Eh
32
r12aΔσ
; where Δσ5 σ1 1σ2 2σ1;2
E (Pa) � modulus of elasticity; h (m) � standard deflection of peak heights; ra (m) �curve radius of surface roughness; Δσ (N m21) � Dupre adhesion, adhesion surface
stress; σ1, σ2, σ1,2 (N m21) � surface stress of two spheres and their boundary line.
It expresses the ratio of the elastic energy to the adhesion work, provided a con-
tact has occurred. With Nadhc1; there is a partial contact in which the elastic mate-
rial has contact only on the highest roughness peaks. A full contact occurs for
Nadh{1: The parameter is based on the Gauss transversal roughness distribution on
the elastic material surface. Micro and nanotechnology. Tribology.
Info: [B126].
4.4.2 Coefficient of Bearing Friction f
f1 5 2π2 ηnp
r
cð1Þ
158 Dimensionless Physical Quantities in Science and Engineering
f2 5η n
pð2Þ
η (Pa s) � dynamic viscosity; n (s21) � rotational speed; p (Pa) � pressure induced
by weight or loading force on bearing surface (2rl ); r (m) � radius of shaft journal;
l (m) � length of shaft journal; c (m) � angular clearance.
Expression (1) serves to estimate the bearing friction or the energy loss in a
bearing approximately. For stable lubrication (boundary, hydrodynamic or mixed),
relation (2) is valid. For fully hydrodynamic lubrication in a plain bearing, it is
ηnp
$ 1:73 1026
The course of the friction coefficient dependence on the Sommerfeld number Sm
(p. 385), expressed with the Stribeck curve, can be replaced by three zones �boundary, mixed and hydrodynamic lubrication � in each of which the effective
coefficient value can be determined.
Sm{1 fef 5 fs ðboundary lubricationÞ
Sm � 1 fef 5 fsð12 SmÞ1 SmhL
ðmixed lubricationÞ
Smc1 fef 5 SmhL
ðhydrodynamic lubricationÞ
Tribology, plain bearings.
4.4.3 Coefficient of Kinetic Friction fk,μk
fk 5F
Fn
ð1Þ;
fk # fs ð2Þ
F (N) � friction force; Fn (N) � normal force pressing friction surfaces together;
fs (�) � coefficient of static friction (p. 160).
It expresses the friction in movement. In contrast to the coefficient of static fric-
tion (p. 160), its size depends on the surface roughness and conditions thereof. In
general, expression (2) holds for the rate range of cm s21 to m s21, and it is sup-
posed to approximate a constant. For comparison, for an iron casting, for example,
the coefficient is fs5 1.1, fk5 0.15; for steel/bronze, it is fs5 0.51, fk5 0.44; for
steel/steel, it is fs5 0.74, fk5 0.57. Mechanics. Tribology.
159Solid Mechanics
4.4.4 Coefficient of Static Friction fs,μs
fs 5Ff
Fn
Ff (N) � friction force; Fn (N) � normal force pressing standing friction surfaces
together.
This coefficient expresses the friction of solid bodies at a standstill. It depends
on the surface roughness and is greater than the coefficient of kinetic friction fk(p. 157), generally ( fk , fs ). The rolling coefficient fr belongs here, which occurs
in rolling operations, such as wheel contact with the rolling surface of vehicles.
It expresses the friction of solid bodies at a standstill. Its size does not depend
on the contact surface area. The static friction force is a reaction the size of which
depends on the size of some outer forces tending to shift the surfaces mutually.
The static friction force rises to the maximum value (1), with which the object
starts to move. The static friction coefficient is equal to or greater than the kinetic
friction coefficient (2). For example, for a few materials and dry friction the values
are as follows: for wood/wood fs A h0.25; 0.5i, for steel/steel fs A h0.5; 0.8i, foraluminium/steel fs A h0.6; 0.8i, for glass/glass fs A h0.9; 1.0i. The oxidation sur-
face cover strongly influences this coefficient, for example, for clean surfaces of
steel/steel fs5 0.78; for an oxidized surface fs5 0.27; and for a sulfidated surface
fs5 0.39. A survey of the static and sliding friction coefficient values is given in
[C103] for various combinations of materials. Mechanics. Tribology.
Info: [C57].
4.4.5 Critical Frictional Temperature Θf
Θf 5η2wsut
L2σ2f
ð1Þ;
Θf 5η2wsut
L2σ2m
ð2Þ
η (Pa s) � dynamic viscosity at the oil temperature in input to the sliding part;
ws (m s21) � sliding velocity; ut (m s21) � peripheral velocity; L (m) � character-
istic length (disc radius); σf (Pa) � contact stress considering the surface rough-
ness; σm (Pa) � maximum contact stress according to Hertz definition at arbitrary
loading and without seizing.
It characterizes the heating and seizing of discs in rolling movement and lubrica-
tion. The expressions (1) and (2) are linearly dependent, approximately.
Info: [B16].
160 Dimensionless Physical Quantities in Science and Engineering
4.4.6 Dimensionless Film Pressure NP; p
NP 5p
pHZ5
πbLp2F
; where pHZ5
ffiffiffiffiffiffiffiffiffiffiffiFE
2πLR
r5
2F
πbL
p (Pa) � oil-film pressure; pHZ (Pa) � maximal dry (Hertzian) friction in contact;
b (m) � half width of dry contact; L (m) � characteristic length (bearing width);
F (N) � force loading of bearing; E (Pa) � reduced modulus of elasticity; R (m) �reduced curve radius.
It expresses the dimensionless oil-film pressure.
Info: [B117].
4.4.7 Dimensionless Film Thickness h
h5πLEh4F
; where E5 212 ν21E1
112 ν22E2
� �
L (m) � characteristic length (bearing width); E (Pa) � reduced modulus of elastic-
ity; h (m) � film thickness; F (N) � force loading of bearing; E1, E2 (Pa) � elas-
ticity modulus of bearing parts; ν1, ν2 (�) � Poisson’s ratio (p. 143) of bearing
parts.
The dimensionless oil-film thickness. Tribology, plain bearings.
Info: [B117].
4.4.8 Drag Friction Factor fD
fD 5Ff 1Fd
12ApRw2
N
Ff (N) � friction force; Fd (N) � drag force; Ap (m2) � cross-section area;
R (kg m23) � fluid density; wN (m s21) � free flow velocity.
This factor characterizes friction under the action of dynamic drag and friction
forces. Fluid mechanics. Tribology.13
4.4.9 Duty Cycle Group NC
NC 524η0TE
π2
� �32 ELR
F
� �32
η0 (Pa s) � dynamic viscosity at normal conditions; T (s) � period of load cycle;
E (Pa) � reduced modulus of elasticity; L (m) � characteristic length (bearing
width); R (m) � reduced curve radius; F (N) � force bearing load.
161Solid Mechanics
It expresses the influence of the bearing duty cycle on the thickness and pressure
of an oil film in a bearing. The elastohydrodynamic lubrication is considered.
Tribology, lubrication, plain bearings.
Info: [B117].
4.4.10 Dynamic Force NFd
NFd 5Rw2E
σ2P
R (kg m23) � density; w (m s21) � sliding velocity; E (Pa) � modulus of elasticity;
σP (Pa) � tensile strength.
It expresses the ratio of the dynamic loading force to the material strength force.
Together with the static force NFs (p. 170), it represents two basic parameters influ-
encing the coefficient size of the friction between a rotating wheel and a support
(e.g. road or rail).14
4.4.11 Energetic Wear Intensity NE
NE 5ΔV
FLλ
ΔV (m3) � volume of wearout layer; F (N) � friction force; L (m) � length of
sliding; λ (J m23) � volume density of deformation energy.
It expresses the ratio of the energy consumed for wearing to the friction energy.
Tribology, material wearing.
4.4.12 Euler Number (2.) Eu2
Eu2 52dp
dl
d
Rw25 2fF
dp
dl(Pa m21) friction pressure gradient; d (m) � pipe diameter; R (kg m23) � fluid
density; w (m s21) � flow velocity; fF (�) � Fanning friction number (p. 163).
This number expresses the fluid friction if flowing through a pipeline.
Info: [A35].
Leonhard Euler (p. 61).
4.4.13 Fanning Number Fa
Fa52 τR w2
τ (Pa) � shear stress; R (kg m23) � fluid density; w (m s21) � flow velocity.
162 Dimensionless Physical Quantities in Science and Engineering
It expresses the ratio of the shear stress to the fluid dynamic pressure. It charac-
terizes the friction loss for flow around bodies or in a pipeline. See also the
Fanning friction number fF (p. 163).
Info: [B20].
John Thomas Fanning (1837�1911), American engineer.
4.4.14 Fanning Friction Number fF, cF
fF 52Δp
Rw2
DH
L5
fD
4
Δp (Pa) � pressure difference; R (kg m23) � fluid density; w (m s21) � flow veloc-
ity; DH (m) � hydraulic diameter; L (m) � characteristic pipe length; fD (�) �Darcy friction number (p. 119).
This number characterizes the friction in turbulent isothermic flow in a bare
pipeline. In the range of the Reynolds number (p. 81) Re A h53 103; 2003 103i,the value fF is given by the relation fF5 0.046 Re20.2.
Info: [A2],[B17].
John Thomas Fanning (see above).
4.4.15 Flexible Number NF, F
F51
Ered
R
α4ηus
� �13
Ered (Pa) � reduced modulus of elasticity; R (m) � effective radius curve;
α (m2 N21) � piezoviscosity coefficient; η (Pa s) � input dynamic viscosity;
us (m s21) � sum of tangential surface velocities.
It influences the minimum thickness of a bearing film. Elastohydrodynamic
lubrication of a bearing.
4.4.16 Friction Factor f
f 52dp
dxDH
1
2Rw2
� �21
2dp
dx(Pa m21) � pressure drop per unit length; DH5 4 SO21 (m) � hydraulic diam-
eter; S (m2) � surface area; O (m) � perimeter; R (kg m23) � fluid density;
w (m s21) � fluid density.
163Solid Mechanics
It represents the general definition of the friction factor. The Fanning friction
number fF (p. 163) and the Darcy friction number fD (p. 119) are practical expres-
sions thereof.
Info: [C29].
4.4.17 Gumbel Number (1.) Gu
Gu5Fh2
2ηuR2
F (N m21) � force on length of bearing; h (m) � angular clearance; η (Pa s) �dynamic viscosity; u (m s21) � peripheral velocity; R (m) � shaft radius.
It expresses the ratio of the bearing loading force to the dynamic friction force.
Tribology, lubrication, bearings.
Info: [A29].
Ludwig Karl Friedrich Gumbel (1874�1923), German naval engineer.
4.4.18 Gumbel Number (2.) Gu
Gu5ηωD2
F
η (Pa s) � dynamic viscosity; ω (s21) � angular velocity; D (m) � diameter of
bearing journal; F (N) � loading force.
It expresses the ratio of the dynamic friction force to the loading force of a plain
bearing. Tribology, lubrication, bearings.
Info: [A29].
Ludwig Karl Friedrich Gumbel (see above).
4.4.19 Harrison Number Hr
Hr56ηuLpaδ2
η (Pa s) � dynamic viscosity; u (m s21) � peripheral velocity; L (m) � length of
bearing pad in direction of motion; pa (Pa) � ambient of inflow pressure; δ (m) �bearing film thickness by output.
It expresses the ratio of the dynamic friction force to the static pressure force in
loading a bearing. Tribology. Lubrication. Bearings.
Info: [A29].
William John Harrison (1884�1969), English mathematician.
164 Dimensionless Physical Quantities in Science and Engineering
4.4.20 Hersey Number He
He5FL
ηað1Þ;
He5F
ηuLð2Þ;
Fl (N m21) � length force loading (bearing); η (Pa s) � dynamic viscosity;
a (m s21) � sound space; F (N) � load force; u (m s21) � peripheral velocity;
L (m) � bearing length.
It expresses the loading-to-viscosity forces ratio and characterizes the force rela-
tion in tribology (e.g. for bearing loading and lubrication).
Info: [A29],[B20].
Mayo Dyer Hersey (born 1886), American engineer.
4.4.21 Hydrodynamic Load Group NH
NH 5 3π2 η0uER
ELR
F
� �2
η0 (Pa s) � dynamic viscosity at normal conditions; u (m s21) � peripheral velocity
E (Pa) � reduced modulus of elasticity; R (m) � reduced radius curve; L (m) �characteristic length (bearing width); F (N) � bearing load.
It expresses the hydrodynamic bearing load influence on the thickness and pres-
sure of the oil film in a bearing. Elastohydrodynamic lubrication. Tribology, lubri-
cation, plain bearings.
Info: [B117].
4.4.22 Mass Wear Intensity Nm
Nm 5Δm
LAR
Δm (kg) � mass of wearout layer; L (m) � length of sliding; A (m2) � nominal
area; R (kg m23) � material density.
It represents the mass wear of friction surfaces. Tribology, material wearing.
4.4.23 Material Abradability NA
NA 5ΔV
fΔmL5
kAQA
f
165Solid Mechanics
ΔV (m3) � volume wearout of material by abrasion; f (�) � coefficient of sliding
friction; Δm (kg) � volume wearout of material by sliding; L (m) � length of
sliding; kA (m2 kg�1) � abrasivity factor; QA (kg m22) � surface loading limit.
It expresses the ratio of the volume decrease due to abrasion, to the mass loss
due to sliding friction. Tribology, material wear.
4.4.24 Material Wear Resistance NR
NR 5N 21A 5
f
kAQA
f (�) � coefficient of sliding friction; kA (m2.kg21) � abrasivity factor;
QA (kg m22) � surface loading limit.; NA (�) � material abradability (p. 165).
It expresses the material wear resistance. It is the inverse value of the material
abradability NA (p. 165). Tribology, material wear.
Info: [B92].
4.4.25 Mechanical Severity of Contact MS
MS5ð11 10 fsÞσmax
ffiffiffid
p
K
σmax (Pa) � maximum occurring Hertzian stress18; d (m) � crack length;
K (m21/2 kg s22) � fracture toughness of the material; fs (�) � coefficient of static
friction (p. 160).
It expresses the contact mechanical resistivity under force loading. Tribology.
Info: [B75].
4.4.26 Minimal Film Thickness NH,H
HH 5 hmin
1
α2η2u2sR
hmin (m) � minimal film thickness; α (m2 N21) � piezoviscosity coefficient;
η (Pa s) � input dynamic viscosity; us (m s21) � the sum of tangential surfaces
velocities; R (m) � effective curve radius.
It expresses the minimum thickness of a lubricating film. In general, it increases
with increasing flexibility, though this is not valid in some applications.
Elastohydrodynamic lubrication, bearings.
Info: [B31].
166 Dimensionless Physical Quantities in Science and Engineering
4.4.27 Moody Friction Factor fM
fM 5 fD 5 4fF
FD (�) � Darcy friction number (p. 119); fF (�) � Fanning friction number
(p. 169).
It corresponds to the Darcy friction number and fourfold of the Fanning friction
number (p. 169).
4.4.28 Piezoviscous Number NL , L
NL 5αηusE3
red
R
� �14
α (m2 N21) � piezoviscosity coefficient; η (Pa s) � dynamic viscosity; us (m s21) �the sum of tangential surface velocities; Ered (Pa) � reduced modulus of elasticity;
R (m) � effective curve radius.
This number expresses the influence of the lubricating film physical properties,
such as viscosity, velocity, shape geometry and elasticity in the sliding lubrication
process. Electrodynamic lubrication, plain bearings.
Info: [B77].
4.4.29 Reduced Frequency Nf,red
Nf ;red 5ωLc
ω (s21) � angular frequency; L (m) � characteristic length; c (m s21) � undisturbed
sound velocity.
It is the measure for the ratio of the gap width to the wavelength of sound propa-
gation. Tribology, bearings.
4.4.30 Relative Pipe Roughness εrel
εrel 5εD
ε (m) � mean height, roughness height of inner pipe surface; D (m) � inner pipe
diameter.
It is analogous to the roughness factor fr (p. 168). It serves to determine the fric-
tion factor. Usually, this is determined from the Moody diagram, which represents
167Solid Mechanics
its dependence on the Reynolds number Re (p. 81) and the relative pipeline rough-
ness. For example, for Re5 33 104 and εrel5 1022, the pipeline fracture coeffi-
cient is f5 0.02.
4.4.31 Relative Roughness εrel
εrel 5Ag
Arough
; where εrelAh0; 1i
Ag (m2) � geometric sample surface; Arough (m2) � surface determined by
roughness of the sample.
It expresses the relative planar roughness of a specimen. Tribology.
4.4.32 Reynolds Friction Number Rer,δ1
Reτ 5wτδν
; where wτ 5w0
ffiffiffifficf
2
r5
ffiffiffiffiffiffiτwR
r
wτ (m s21) � friction velocity; δ (m) � boundary layer thickness; ν (m2 s21) �kinematic viscosity; w0 (m s21) � reference velocity of the undisturbed flow in
pipes and channels; cf (�) � friction coefficient; τw (Pa) � shear stress on circum-
fluenced wall.
It characterizes the viscous fluid flow in a boundary layer close to walls, in a
pipeline and canals. Tribology.
Osborne Reynolds (p. 82).
4.4.33 Roughness Factor, Roughness Ratio fr
fr 5εL
ð1Þ;
fr 5Ar
Ag
ð2Þ
ε (m) � height of roughness; L � d (m) � characteristic length, pipe diameter;
Ar (m2) � real surface (interface) area ; Ag (m
2) � geometric surface (interface) area.
It expresses the ratio of the real interface height to the geometric one (1) or
the ratio of the real surface area (of the interface) to the geometric surface (2).
In the case of a circular cross-section pipeline, the diameter is the characteristic
dimension; the equivalent diameter is that for the non-circular one. It characterizes
the roughness influence of a bypassed surface on the friction of fluids.
Info: [A29].
168 Dimensionless Physical Quantities in Science and Engineering
4.4.34 Shear Wave Number Nws
Nws 5 L
ffiffiffiffiων
r
L (m) � characteristic length (thickness); ω (s21) � angular frequency; ν (m2 s21) �kinematic viscosity.
It expresses the inertia-to-friction forces ratio. For low values of the number, the
viscosity influence predominates. On the contrary, for high values of the number,
the inertia influence dominates. From the physical point of view, it represents the
ratio between the gap width and the boundary layer thickness. Tribology, bearings.
4.4.35 Skin Friction Coefficient, Wall Shear Stress Coefficient Cf
Cf 5τw
12Rw2
N
ð1Þ; where τw 5 ηdw
dyy50:��
Cf 50:664ffiffiffiffiffiffiffiRex
p ð2Þ
Cf 5 0:0576ffiffiffiffiffiffiffiRex
5p ð3Þ
τw (Pa) � shear stress on the wall (y5 0); R (kg m21) � fluid density; wN (m s21) �free fluid flow; η (Pa s) � dynamic viscosity; y (m) � coordinate perpendicular to
the wall surface; Rex (�) � Reynolds number.
It expresses the dynamic friction resistance originating in viscous fluid flow
around a fixed wall. The expression (2) is valid for laminar flow along a flat plate
(Blasius’ boundary layer) and the expression (3) holds for turbulent flow.
Tribology. Aerohydrodynamics.
Info: [A21].
4.4.36 Squeeze Number Sq
Sq512ηωpa
L
δ0
� �2
η (Pa s) � dynamic viscosity; ω (s21) � oscillation frequency; pa (Pa) � ambient
pressure; L (m) � characteristic length, radius; δ0 (m) � initial gap thickness.
It characterizes the damping of a thin compressible film squeezed between a
fixed and movable oscillating plate. It expresses the friction and damping condi-
tions in bearings and lubrication. Tribology.
Info: [A26],[B20].
169Solid Mechanics
4.4.37 Stanton Pannell Friction Factor fSP
fSP 51
2Fa
Fa (�) � Fanning number (p. 162).
It corresponds to half of the Fanning number (p. 162).
Info: [C49].
Thomas Edward Stanton (p. 201).
4.4.38 Static Force NFs
NFs 5FE
σ2P
F (N) � static force; E (Pa) � modulus of elasticity; σP (Pa) � tensile strength.
It expresses the ratio of the static loading force to the material strength force.
Together with the dynamic force NFd (p. 162), it represents two basic parameters
acting on the coefficient size of the friction between a rotating wheel and a support
(road, rail, etc.).
4.4.39 Thermal Severity of Contacts TS
TS5fsFu
ΔTsbλef
F (N) � contact load; u (m s21) � velocity;ΔTs (K) � temperature difference; b (m) �half width of the elliptic contact in the direction of sliding; λef (W m21 K21) � effective
thermal conductivity; fs (�)� coefficient of static friction (p. 160).
It characterizes the thermal resistivity of a shock loaded contact. Tribology.
Info: [B75].
4.4.40 Torque Frictional Coefficient CM
CM 52Mf
r3ω2AR
Mf (N m) � friction moment; r (m) � characteristic radius; ω (s21) � angular
velocity; A (m2) � surface; R (kg m23) � fluid density.
It expresses the ratio of a friction moment in rotation to the moment due to
dynamic force. Hydromechanics.
Info: [A8].
170 Dimensionless Physical Quantities in Science and Engineering
4.4.41 Transfer Number (4.) Friction Nf
Nf 5τwLwNη
τw (Pa) � shear stress on circumfluenced wall; L (m) � characteristic length
(e.g. radius of pipe, wall and curvature); wN (m s21) � free flow fluid velocity;
η (Pa s) � dynamic viscosity.
This number expresses the friction-to-viscous forces ratio or dimensionless sur-
face friction alternatively. Laminar flow in a boundary layer.
4.4.42 Wear Intensity PW
Pw5h
Lð1Þ;
Pw5ΔV
LAð2Þ
h (m) � thickness of wearout layer; L (m) � length of sliding; ΔV (m3) � volume
of wear layer; A (m2) � nominal area.
In expression (1), it expresses linear wear intensity. In expression (2), it
expresses volume intensity. Tribology, material wearing.
171Solid Mechanics
5 Thermomechanics
The theory becomes the more attractive the simpler are its premises and the
expressive is the heterogeneity of phenomena it involves and the wider is the area
of applicability thereof.Albert Einstein (1879�1955)
5.1 Heat Conduction
In a thermal system, the heat conduction manifests itself as a thermal field. The
thermal field can be steady or unsteady, linear or non-linear, with inner sources
or without them, with movable or stationary borders, with basic, composed or
combined boundary conditions, or with a phase conversion. Conjugate processes
of heat and mass transfers, heat and thermally induced stress, as well as non-
equilibrium heat propagation processes belong to this area. Dimensionless quanti-
ties also correspond to this classification. Among them are the following numbers:
the Biot, Brinkmann, Fourier, Kirpichev, Peclet, Pomerantsev and Stefan two-phase
conduction number.
5.1.1 Biot Number Bi
Bi5αLλ
ð1Þ;
Bi5αð2πfλcRÞ21=2 ð2Þ
α (W m22 K21) � heat transfer coefficient; L (m) � characteristic length; λ(W m21 K21) � thermal conductivity; f (Hz) � frequency; c (J kg21 K21) � spe-
cific heat capacity; R (kg m23) � density; V (m3), S (m2) � body volume and sur-
face; TS, TP (K) � temperature of the body and environment.
This number expresses the ratio of the heat flow transferred by convection on a
body surface to the heat flow transferred by conduction in a body. It characterizes
the third-type boundary condition. The equation Bi5 L (λα21)21 expresses the ratio
of a characteristic body length to an equivalent environmental thickness adhering
to the surface. With Bi$ 100 in the heat transfer on the surface, the thermal resis-
tance is slight in comparison to the heat transfer by conduction. The temperature
Dimensionless Physical Quantities in Science and Engineering. DOI: 10.1016/B978-0-12-416013-2.00005-1
© 2012 Elsevier Inc. All rights reserved.
field is characterized by considerable non-uniformity. On the contrary, with
Bi# 0.1, difference (TP2 TS) is great on the body surface and slight inside the
body and the field is uniform. With the general shape of body, the ratio L5VS21 is
inserted for the characteristic length. With periodically changing heat transfer, unit
amplitude and frequency, the modified Biot number is used in expression (2).
Info: [A23],[A29],[A31],[A35],[B20].
Jean-Baptiste Biot (21.4.1774�3.2.1862), French physicist.
He was engaged in a wide range of physics � especially
in optics � specifically in polarized light, changes in polari-
zation and optical gas refractions. He also studied heat, con-
duction analysis and the external convection influence on
conduction. He was noted for requiring precision in mathe-
matical formulations and experiments. With Gay-Lussac, he
collaborated to design the first balloon for scientific purposes
(1804) and discovered the Biot�Savart law (1870) with
Felix Savart.
5.1.2 Brinkmann Number Br
Br5ηw2
λΔT5
ηw2
λðTS 2 TNÞ
η (Pa s) � dynamic viscosity; w (m s21) � flow velocity; λ (W m21 K21) � thermal
conductivity; ΔT (K) � temperature difference; TS (K) � surface temperature;
TN (K) � temperature in the thermal undisturbed fluid area.
It expresses the ratio of the heat arising due to viscous friction of a fluid to the
heat transferred by molecular conduction. It characterizes the heat conduction in
viscous fluid flow. For high fluid viscosity values and low thermal conductivity
values (e.g. molten polymers), the value is Brc1.
Info: [A23],[A29],[A31],[B20].
Henri Coenraad Brinkmann, German physicist.
5.1.3 Clausius Number Cl
Cl5w3LRλΔT
w (m s21) � velocity of fluid or solid body relative to undisturbed fluid; L (m) �characteristic length; R (kg m23) � density; λ (W m21 K21) � thermal conductivity;
ΔT (K) � temperature difference.
174 Dimensionless Physical Quantities in Science and Engineering
It is the ratio of the kinetic energy propagation rate to the heat propagation rate
by conduction. It characterizes the heat conduction in forced fluid flow or in a
moving environment generally. The characteristic length can be the canal width,
distance from the wall or fluid layer thickness.
Info: [A23],[A29],[A31],[A35],[B20].
Rudolf Julius Emanuel Clausius (2.1.1822�24.10.1888),
German mathematician and physicist.
He worked in the field of thermodynamics and formulated
the first and second thermodynamic theorems. The first is
about the equivalence of heat and work. By introducing the
entropy concept, he discovered that entropy can never drop
during physical processes, but can only stay constant in
reversal processes, as expressed by the second thermody-
namic theorem. Together with Maxwell, he formulated the
kinetic theory of gases.
5.1.4 Fourier Number Fo
Fo5aτL2
ð1Þ;
Fo5λLτcRL3
ð2Þ
a (m2 s21) � thermal diffusivity; τ (s) � time; L (m) � characteristic length;
λ (W m21 K21) � thermal conductivity; c (J kg21 K21) � specific heat capacity;
R (kg m23) � density.
It expresses the ratio of the time of a proceeding thermal process to that of the
molecular diffusion of the heat. In this time, expressed in a dimensionless way, the
relations between the thermal field change rate, physical parameters and thermal
system dimensions appear. In form (2), it expresses the ratio of the heat transferred
by conduction in a system to the heat accumulated in the system.
Info: [A23],[A31],[A19],[A35],[A43],[B20].
Jean Baptiste Joseph Fourier (21.3.1768�16.5.1830),
French mathematician and physicist.
With his work Theorie Analytique de la Chaleur
(Analytic Theory of Heat) of 1822, he created the founda-
tions for the mathematical heat conduction theory. First, he
formulated the important property of mathematical physics
equations: their dimensional homogeneity. His basic contri-
bution was very important for the theory of trigonometric
arrays and that of real variable functions.
175Thermomechanics
5.1.5 Heat Capacities Relation K
K5cR
ðcRÞref
cR, (cR)ref (J m23 K21) � specific volume heat capacity and its reference value.
It is the parametric criterion expressing the dimensionless material thermal
capacity of a body. It can be used in non-linear heat conduction. Usually cR 5c(T)R(T) is valid.
Info: [A23].
5.1.6 Heat Diffusivities Relation K
K5a
aref
a, aref (m2 s21) � thermal diffusivity and its reference value.
It is the parametric criterion expressing the dimensionless material thermal dif-
fusivity of a body. It can be used in non-linear heat conduction. Usually a5 a(T) is
valid.
Info: [A23].
5.1.7 Irregularity of Temperature Field Ψ
Ψ5Tp 2 Ts
Tp 2 Tmed
Tp (K) � temperature of external environment; Ts (K) � wall surface temperature;
Tmed (K) � mean integral volume body temperature.
It expresses the ratio of the environment and surface temperature difference of a
body to the difference between the environment temperature and the mean integral
volume temperature. It characterizes the temperature field non-uniformity in a
body.
Info: [A23].
5.1.8 Kirpichev Heat Number Ki
Ki � PoA 5qAL
λΔTð1Þ;
Ki5BiTS 2 TP
ΔTð2Þ
176 Dimensionless Physical Quantities in Science and Engineering
qA (W m22) � surface heat flux density; L (m) � characteristic length;
λ (W m21 K21) � thermal conductivity; ΔT (K) � reference temperature differ-
ence or temperature; TS, TP (K) � body surface and environment temperatures;
PoA (�) � Pomerantsev heat number (p. 181); Bi (�) � Biot number (p. 173).
This number expresses the ratio of the thermal heat flow surface density to the
density of the thermal heat flow conducted in a body. It characterizes the second-
kind boundary condition or, alternatively, a planar heat source.
Info: [A4],[A23],[A26],[A29],[A31],[A20],[B20].
Mikhail Viktorovich Kirpichev (1879�1955), Russian
engineer and physicist.
Using thermal similarity theory, he laid the foundations of
systematic modelling research for heat transfer, especially in
liquid metals, and elaborated the regular mode theory in ther-
mal systems with internal heat sources. He was engaged in
research of the heat transfer in boiling and condensation.
Essentially, he interpreted similarity theory as theory of an
experiment. His book Teorija podobija i rnodelirovanije
(The Theory of Similarity and Modeling, 1956) has substan-
tial significance.
5.1.9 Kondratiev Number Kd
Kd5mL2
að1Þ;
Kd5BiΨ5αλLΨ ð2Þ
m (s21) � unit speed of cooling; L5VS2 1 (m) � characteristic length dimension;
a (m2 s21) � thermal diffusivity; α (W m22 K21) � heat transfer coefficient;
λ (W m21 K21) � thermal conductivity; Bi (�) � Biot number (p. 173); Ψ (�) �irregularity of temperature field (p. 176).
It characterizes the regular state of the heat transfer between a fluid and a body
in which the temperature space distribution in the body preserves the thermal simi-
larity in time. In this state, the temperature is delayed regularly at an arbitrary point
of the body against the external environment temperature. It expresses the intensity
of mutual action of the body surface with the surrounding environment and, further,
the non-uniformity of the body temperature field in the regular state. With uniform
temperature distribution (Bi-0, for Bi, 0.1, essentially), Ψ5 1 and Kd5Bi are
valid. On the contrary, with the greatest thermal field non-uniformity, Ψ5 0 holds
(Bi-N, for Bi. 100, essentially), and then Kd is constant. The number Kd
involves the zone from zero to a value given by the body shape. It is often called
the thermal inertia criterion as well.
Info: [A23],[A29],[A35].
Georgy Mikhailovich Kondratiev, Russian engineer.
177Thermomechanics
5.1.10 Local Temperature Θ
Θ5T 2 Tmin
Tmax 2 Tmin
T, Tmin, Tmax (K) � local, minimal and maximal temperatures.
It expresses the ratio of the difference between the local and minimal tempera-
ture to that between the maximal and minimal temperatures. It is the parametric cri-
terion expressing the dimensionless temperature in a system. Usually, it represents
the dependent variable in the solution of thermal tasks. ΘAh0; 1i.Info: [A23].
5.1.11 Maxwell Number Ml
Ml5ληcV
5 kPr21
λ (W m21 K21) � thermal conductivity; η (Pa s) � dynamic viscosity;
cV (J kg21 K21) � specific heat capacity; κ (�) � specific heat ratio (p. 28);
Pr (�) � Prandtl number (p. 197).
It expresses the relation between gas thermal conductivity and gas viscosity.
It is a modification of the Prandtl number Pr (p. 197).
Info: [A23].
James Clerk Maxwell (p. 197).
5.1.12 Mikheyev Number Mi
Mi5ατcRL
5BiFo ð1Þ;
Mi5α2τλcR
5Bi2Fo ð2Þ
α (W m22 K21) � heat transfer coefficient; τ (s) � time; c (J kg21 K21) � specific
heat capacity; R (kg m23) � density; L (m) � characteristic length; λ (W m21 K21) �thermal conductivity; Bi (�) � Biot number (p. 173); Fo (�) � Fourier number
(p. 175).
This number characterizes the dependence of the mean integral quantities in
space and time on a body shape. It expresses the ratio of the heat passed by the
body to the volume heat. For finite volume bodies L5ffiffiffiS
pis valid, and for longitu-
dinally unlimited bodies it is L � 0: In equation (2), it expresses the ratio of the
heat transferred by convection to the body thermal activity ε5ffiffiffiffiffiffiffiffiλcR
p: It is applied
in cases in which L5λα2 1 is a characteristic length dimension.
178 Dimensionless Physical Quantities in Science and Engineering
Info: [A23].
Mikhail Alexandrovich Mikheyev (1902�1970), Russian engineer.
5.1.13 Number of Phase Changes Nph, K
Nph 51
cpΔT5 Sf 21 ð1Þ;
Nph 5rRwq
5NuPe21 ð2Þ;
Nph 52rRνqL
5NuPe21 ð3Þ
l (J kg21) � specific latent heat of phase change; cp (J kg21 K21) � specific heat
capacity; ΔT (K) � temperature difference; r (J kg21) � reaction heat of chemical
change; R (kg m23) � fluid density; w (m s21) � flow velocity; q (W m22) � sur-
face heat flux; ν (m2 s21) � kinematic viscosity; Sf (�) � Stefan two-phase con-
duction number (p. 183); Nu (�) � Nusselt number (p. 196); Pe (�) � Peclet heat
number (p. 180).
It expresses phase conversions. It is the inverse value of the Stefan two-phase
conduction number Sf (p. 183). Expression (1) is the ratio of the thermal flow nec-
essary to the phase conversion in a material, to the overheating or subcooling heat
of one of the phases. Expressions (2) and (3) represent the number Nph with phase
transformations in flowing fluid. Expression (2) represents the ratio of the phase
conversion rate to the flow rate. Expression (3) is the ratio of an inertia force in
flowing fluid, originating due to the phase conversion, to the inner friction force.
Therefore, it represents a specific form of the Reynolds number Re (p. 81). It is
analogous to the Jakob evaporation number Jak (p. 220).
5.1.14 Ostrogradsky Number Os
See Pomerantsev heat number PoV (p. 181) and Damkohler number (4.) (2. heat)
Da4 (p. 37).
Mikhail Vasilyevich Ostrogradsky (24.9.1801�1.1.1862),
Russian mathematician and physicist.
He was the first mathematician to publish the proof of
the divergence theorem. In the year 1822 he went to Paris,
where he worked on heat theory and on hydrodynamic pro-
blems. He applied the divergence theorem as a tool to con-
vert the volume integral to the planar one. He wrote many
articles on partial differential equations, algebra, elasticity,
hydromechanics, heat and electricity. He is considered the
founder of the Russian school of theoretical mechanics.
179Thermomechanics
5.1.15 Peclet Heat Number Pe
Pe5wL
a5RePr
w (m s21) � motion velocity; L (m) � characteristic length; a (m2 s21) � thermal
diffusivity; Re (�) � Reynolds number (p. 81); Pr (�) � Prandtl number (p. 197).
This number is the criterion for the mutual action of the convective and molecu-
lar heat transfers in flowing fluid. It expresses the ratio of the convective thermal
flow transferred by the fluid to that transferred by conduction. It characterizes the
thermal process with consideration of the movement of the environment and physi-
cal properties. It can be understood as the ratio of the enthalpy changes of the fluid
flowing in the axial direction to the heat penetrating into the flow by conduction in
the normal direction. With the Pe number increasing, the heat conduction portion
decreases and the convective heat portion grows. Thermal motion processes.
Info: [A4],[A23],[A35],[A43],[B20].
Jean Claude Eugene Peclet (10.2.1793�6.12.1857),
French physicist.
He was engaged in research on heat transfer by flow and
conduction especially, and in radiation and mass transfer as
well. His publications are well known for his clear and dis-
tinct opinions and well-executed experiments. His principal
work is the Traite de la Chaleur Consideree dans ses
Applications (Essay on Heat and its Applications in Crafts
and Manufacture, 1829).
5.1.16 Peclet Non-Linear Number of Solidification Pe
Pe5ωLλðTÞ
dh
dΘ1
TS 2 TC; where Θ5
T 2 TC
TS 2 TC
w (m s21) � interface motion velocity; L (m) � characteristic length; λ(W m21 K21) � thermal conductivity; h (J m23) � specific volume enthalpy of
solid; TS, TC, T (K) � solidifying interface, cooler and local temperatures.
It characterizes the motion of the isothermic solid�liquid interface, for example,
in continuous casting and in solidification of castings.
Info: [A23].
Jean Claude Eugene Peclet (see above).
5.1.17 Phase Change of Enthalpy N
N5h2 hs
lslð1Þ;
180 Dimensionless Physical Quantities in Science and Engineering
N5cðTs 2 TslÞ
lslð2Þ;
N5cðTi 2 TslÞ
lslð3Þ
h (J kg21) � specific enthalpy; hs (J kg21) � specific enthalpy of solid phase; lsl
(J kg21) � specific latent heat of melting; c (J kg21 K21) � specific heat capacity;
Ts (K) � solid phase temperature; Tsl (K) � temperature of solidification or melt-
ing; Ti (K) � initial (casting) temperature of overheated melting.
In the dimensionless form, it characterizes phase conversion enthalpy. It is anal-
ogous to the Stefan two-phase conduction number Sf (p. 183).
Info: [A23].
5.1.18 Pomerantsev Heat Number, Heat Generation Term Po, G
PoV � Os5qVL
2
λΔTrð1Þ;
PoA 5Ki5qAL
λΔTrð2Þ;
PoL 5qL
λΔTrð3Þ;
PoB 5qB
λLΔTrð4Þ
qV (W m23), qA (W m22), qL (W m21), qB (W) � volume, area, linear and point
density of heat flux (source); L (m) � characteristic length; λ (W m21 K21) �thermal conductivity; ΔTr (K) � characteristic temperature difference; Os (�) �Ostrogradsky number (p. 179).
It characterizes volume (1), area (2), linear (3) or point (4) heat sources in a sys-
tem. Sometimes in expression (1), it is called the Ostrogradsky number Os (p. 179),
whereas it is called the Kirpichev number Ki (p. 176) in expression (2).
Info: [A4],[A23],[A26],[A35],[A42],[B20].
Alexey Alexandrovich Pomerantsev, Russian engineer.
5.1.19 Predvoditelev Number Pd
Pd5dΘP
dFo
� �max
ð1Þ;
181Thermomechanics
Pd5bL2
aTP;ið2Þ;
Pd5b2L
2
að3Þ;
Pd5 2πfL2a21 ð4Þ
ΘP (�) � dimensionless environment temperature; b (K s21) � constant; L (m) �characteristic length; a (m2 s21) � thermal diffusivity; TP,i (K) � initial tempera-
ture; b2 (s21) � constant; f (Hz) � oscillation frequency of external temperature;
Fo (�) � Fourier number (p. 175).
This number characterizes the temperature time change of the outer environment
or of internal heat sources. For a linear change, TP5 TP,i1 bτ holds for expression
(2), and for T(τ)5 T(N)2 (T(N)2 Ti) exp (2b2τ), expression (3) is valid. For
the thermal flow density changing exponentially from an internal source, q(τ)5 qiexp (2b2τ) holds, when the constant b2 expresses the maximum change rate of the
thermal source. Expression (4) holds for the environmental temperature T(τ)�Ti5 Tm cos (2πfτ), which changes in a harmonic way.
Info: [A4],[A23],[A26],[B20].
Alexander Savviich Predvoditelev (1891�1973), Russian physicist.
5.1.20 Stefan�Fourier Number N
N5cðTls 2 TsÞ
lls
aτL2
5 Sf Fo ð1Þ;
N5cðTs 2 TslÞ
lsl
aτL2
5 Sf Fo ð2Þ
c (J kg21 K21) � specific heat capacity at solidification or melting temperature;
Tls, Tsl (K) � temperature of solidification or melting; Ts (K) � surface temperature
of solidification body (ingot); lls, lsl (J kg21) � specific latent heat of solidification
or melting; a (m2 s21) � thermal diffusivity; τ (s) � time; L (m) � characteristic
length; Sf (�) � Stefan two-phase conduction number (p. 183); Fo (�) � Fourier
number (p. 175).
In the dimensionless form, it characterizes the time of the thermal two-phase
process of solidification (1) or melting (2).
Info: [A123].
Josef Stefan (see below).
Jean Baptiste Joseph Fourier (p. 175).
182 Dimensionless Physical Quantities in Science and Engineering
5.1.21 Stefan Two-Phase Conduction Number Sf
Sf 5cðTls 2 TsÞ
llsð1Þ;
Sf 5cðTs 2 TslÞ
lslð2Þ
c (J kg21 K21) � specific heat capacity at solidification or melting temperature; Tls,
Tsl (K) � temperature of solidification or melting; Ts (K) � surface temperature of
solidification body (ingot); lls, lsl (J kg21) � specific latent heat of solidification or
melting.
It expresses the ratio of the specific enthalpy of an ingot to the specific latent
heat of solidification. It characterizes the sensitivity to the thermal process latent
energy of solidification (1) or melting (2) in a solidus�liquidus system under the
condition of linear heat conduction, neglecting the difference of densities of both
phases and considering the phase conversion at an equal temperature.
Info: [A23].
Josef Stefan (p. 214).
5.1.22 Thermal Conductivity Relation Λ
Λ5λλref
λ, λref (W m21 K21) � thermal conductivity and its reference value.
It is the parametric criterion expressing the dimensionless thermal conductivity
of a material. It can be used in non-linear heat conduction.
Info: [A23].
5.1.23 Two-Phase Heat Conduction K
K5λlðT1 2 TsÞλsðTs 2 T2Þ
λl, λs (W m21 K21) � thermal conductivity of liquid and solid phases; T1, T2, Ts(K) � input liquid casting, output and solid phase temperatures.
It expresses the thermal flows ratio in liquid and solid phases. It characterizes
the heat conduction process in a two-phase environment during solidification or
melting.
Info: [A23].
183Thermomechanics
5.2 Free Convection
In free (natural) convection, the dimensionless quantities express the spontaneous
heat flow in fluids or gases due to the thermal difference caused by the difference
of fluid densities. The force fields setting the fluid in motion originate directly in
the fluid. Free convection occurs in the atmosphere, in oceans and near the surface
of heated or cooled bodies, for example. For free convection, the Rayleigh,
Grashof, Marangoni, Archimedes, Grigull and Schwarzschild numbers are
characteristic.
5.2.1 Archimedes Thermodynamic Number Ar
Ar5ΔT
T0
gL
w25ΔΘFr21 ð1Þ;
Ar2 5gL
w2βΔT 5
L3gβΔT
ν2ν2
w2L25GrRe22 ð2Þ
ΔT (K) � temperature difference; T0 (K) � steady-state temperature; g (m s22) �gravitational acceleration; L (m) � characteristic length; w (m s21) � flow velocity;
β (K21) � thermal volume expansion coefficient; ν (m2 s21) � kinematic viscosity;
Θ (�) � relative temperature; Fr (�) � Froude number (1.) (p. 62); Gr (�) �Grashof heat number (p. 185); Re (�) � Reynolds number.
This number characterizes the free and forced convection caused by the temper-
ature gradient and the fluid flow rate. In equation (2), it is often called the buoy-
ancy parameter NB (p. 56).
Info: [A4],[A12],[A29],[B20].
Archimedes of Syracuse (p. 54).
5.2.2 Bejan Pressure Number Be
Be5ΔpL2
ηa
Δp (Pa) � pressure difference in channel; L (m) � characteristic length of circum-
fluenced channel; η (Pa s) � dynamic viscosity; a (m2 s21) � thermal diffusivity.
It expresses the pressure gradient in a flow-through canal with consideration of
the heat diffusion. It has an analogous role as the Rayleigh number (2.) (heat insta-
bility) Ra (p. 187) in natural convection. Natural convection.
Info: [C23].
Adrian Bejan (p. 5).
184 Dimensionless Physical Quantities in Science and Engineering
5.2.3 Crispation Number Cr
Cr5ηaσL
η (Pa s) � dynamic viscosity; a (m2 s21) � thermal diffusivity; σ (N m21) �surface
tension; L (m) � characteristic length, layer thickness.
This number expresses the ratio of a viscous force and diffusive heat action to a
surface tension force. It characterizes the influence of convective flows in heat and
momentum transfers.
Info: [A26],[A29].
5.2.4 Grashof Heat Number Gr
Gr5L3gΔRRν2
5L3gβΔT
ν25 βΔTGa ð1Þ;
Gr5L3gΔT
ν2Tð2Þ;
Grrot 5Gr�ω2L
gð3Þ
L (m) � characteristic length dimension; g (m s22) � gravitational acceleration;
ΔR (kg m23) � change of fluid density due to temperature change ΔT; R (kg m23) �density; ν (m2 s21) � kinematic viscosity; β (K21) � volume thermal expansion coef-
ficient; T (K) � temperature; ω (Hz) � angular frequency; Ga (�) � Galilei number
(p. 123); Gr� (�) � Grashof heat number defined in expression (1).
It expresses the buoyancy-to-viscous forces ratio and its action on a fluid. It
characterizes the free non-isothermal convection of the fluid due to the density dif-
ference caused by the temperature gradient in the fluid. It is often used in expres-
sion (2) for an ideal gas. In form (3), it is used for convective transfer in rotating
canals of which the axis is parallel with the rotation axis. With this modified Gr
number, the centrifugal force influence on the heat transfer is considered. Usually,
the Coriolis force influence is negligible.
Info: [A23],[A26],[A43],[B20].
Franz Grashof (11.7.1826�26.10.1893), German engineer.
Within the framework of applied mechanics and general
machinery, he was engaged in research on material stiffness,
hydraulics and heat theory. As for heat transfer, he was
engaged in the problem of free convection caused by the
density difference due to the temperature gradient in a fluid.
185Thermomechanics
5.2.5 Grashof Modified Number Grmod
Grmod 5gR2βL4
λη2
g (m s22) � gravitational acceleration; R (kg m23) � liquid density; β (K21) � vol-
ume thermal expansion coefficient; L (m) � characteristic length; λ (W m21 K21) �thermal conductivity; η (Pa s) � dynamic viscosity.
It is a modification of the Grashof heat number Gr (p. 185). It expresses the natural
convective transfer from a heated or cooled wall into the surrounding viscous fluid.
Info: [B60].
Franz Grashof (see above).
5.2.6 Grigull Number Gg
Gg5βgqAτ2
ηcp
β (K21) � volume thermal expansion coefficient; g (m s22) � gravitational acceler-
ation; qA (W m22) � heat flux density; τ (s) � time; η (Pa s) � dynamic viscosity;
cp (J kg21 K21) � specific heat capacity.
It characterizes the conditions for the origin of the development of free convec-
tion in a horizontal layer of a fluid or gas which is heated from below.
Info: [A23],[A33].
Ulrich Grigull (12.3.1912�20.10.2003), German engineer.
He was engaged in research on thermodynamics and heat
transfer, and in research on turbulent film condensation. He
published the book Grundgesetze der Warmeubertragung
(Basic Laws of Heat Transfer). His research included the
investigation of water and water steam properties, thermal
physical properties � such as viscosity under high pressure
and temperature � and determination of critical water steam
properties.
5.2.7 Marangoni Number Mr
Mr5dσdT
ΔTL
ηa5
ΔσΔT
ΔT
ΔL
L2
ηa
σ (N m21) � surface tension; ΔT (K) � temperature difference between upper surface
layer and lower one; L (m) � characteristic length (layer thickness); η (Pa s) �dynamic viscosity; a (m2 s21) � thermal diffusivity.
It is the criterion for mutual influence of capillary forces and molecular friction
forces. It characterizes the thermal capillary phenomena on a fluid-free surface and
186 Dimensionless Physical Quantities in Science and Engineering
the surface tension influence on the origin of the free convection in a thin fluid
layer. The Mrcrit5 80 is the critical value leading to instability.
Info: [A4],[A29],[A35],[B20],[C7].
Carlo Guiseppe Matteo Marangoni (1840�1925), Italian physicist.
5.2.8 Rayleigh Number (2.) (Heat Instability) Ra2
Ra2 5gL3ΔTβ
νa5
gL3ΔRηa
5GrPr
g (m s22) � gravitational acceleration; L (m) � characteristic length; ΔT (K) �temperature difference; β (K21) � volume thermal expansion coefficient;
ν (m2 s21) � kinematic viscosity; a (m2 s21) � thermal diffusivity; ΔR (kg m23) �density difference; η (Pa s) � dynamic viscosity; Gr (�) � Grashof heat number
(p. 185); Pr (�) � Prandtl number (p. 197).
It characterizes the free convection heat transfer along a heat-exchanging sur-
face. It expresses the buoyancy-to-diffusion ratio or, alternatively, the free convec-
tion thermal instability in fluids. In a closed space with Ra, 103, the heat is
transferred between warmer and cooler walls by conduction only.
Info: [A4],[A23],[A29],[A43],[B20].
Lord Rayleigh, John William Strutt (12.11.1842�30.6.1919), British physicist. Nobel Prize in Physics, 1904.
He was engaged in research on optics, acoustics, electro-
magnetism and molecular light dispersion. He discovered
one of the radiation laws for perfect black body radiation in
the long-wave zone. He intensively studied Maxwell’s elec-
tromagnetism theory, both theoretically and experimentally.
He researched the density of rare gases and discovered
argon.
5.2.9 Rayleigh Number (3.) Ra3
Ra3 5qAL
5gβνλax
5GrNuPrX21
qA (W m22) � heat flux density; L (m) � characteristic length, pipe diameter;
g (m s22) � gravitational acceleration; β (K21) � volume thermal expansion coef-
ficient; ν (m2 s21) � kinematic viscosity; λ (W m21 K21) � thermal conductivity;
a (m2 s21) � thermal diffusivity; x (m) � distance from inlet to vertical tube;
Gr (�) � Grashof heat number (p. 185); Nu (�) � Nusselt number (p. 196);
Pr (�) � Prandtl number (p. 197); X (�) � geometrical coordinates (p. 15).
187Thermomechanics
It characterizes the combined free and forced convections in vertical tubes.
Info: [A4],[A23].
Lord Rayleigh (see above).
5.2.10 Rayleigh Number (4.) Ra4
Ra4 5qVL
5gβvλa
5GrPrPoV
qV (W m23) � volume density of heat flux; L (m) � characteristic length;
g (m s22) � gravitational acceleration; β (K21) � volume thermal expansion coef-
ficient; v (m2 s21) � kinematic viscosity; λ (W m21 K21) � thermal conductivity;
a (m2 s21) � thermal diffusivity; Gr (�) � Grashof heat number (p. 185); Pr (�) �Prandtl number (p. 197); PoV (�) � Pomerantsev heat number (p. 181).
It characterizes the free convection in a horizontal fluid layer with a constant
internal volume heat source.
Info: [A4],[A23],[A29],[A35].
Lord Rayleigh (see above).
5.2.11 Schwarzschild Number Sch
Sch5gT
cpv
@v
@T
� �p
dT
dl
� �21
g (m s22) � gravitational acceleration; T (K) � temperature; cp (J kg21 K21) � specific
heat capacity; v (m3 kg21) � specific volume; l (m) � length.
This number characterizes the stability conditions for a fluid mechanical balance
with its longitudinally changing temperature or the condition under which no ther-
mal convection occurs in the fluid. In a fluid, over the horizontal surface of a wall,
of which the temperature changes perpendicularly to the surface up to the free fluid
level, the following condition is valid to cause convection
dT
dl
��������. gT
cpv
dv
dT
� �p
Info : [A23],[A33].
188 Dimensionless Physical Quantities in Science and Engineering
Karl Schwarzschild (9.10.1873�11.5.1916), German
astronomer, physicist and mathematician.
He was focused on electrodynamics, optics and the radi-
ation of stars related to stellar atmosphere research. He
applied photography to measure the instability of stars and
investigated internal star structures and star gas equilibrium.
He devoted himself also to relativity theory, and the
Schwarzschild radius, which represents the critical gravita-
tional solid-body radius at which a body becomes a black
hole according to general relativity theory, is named after
him.
5.2.12 Thompson Number Th
See Marangoni number Mr (p. 186).
Info: [A29].
5.3 Forced Convection
The dimensionless quantities expressing forced convection heat transfer are among
the most widely used. Forced convection is characterized by a flow with mutual
action of inertia and viscous forces. The predominance of the viscous flow charac-
terizes a laminar flow. On the contrary, turbulent flow arises under the action of
external influences and its development depends on the viscous-to-inertia forces
ratio exclusively. Among the numerous dimensionless quantities, the Reynolds,
Prandtl, Colburn, Dulong, Eckert, Nusselt, Fliegner, Graetz, Stanton and other
numbers are especially important.
5.3.1 Bansen Heat Number Ba
Ba5T1 5 T2
T 2 Ts
T1, T2 (K) � input and output temperatures in channel; T, Ts (K) � fluid flow and
channel wall temperatures.
This number expresses the ratio of the temperature gradient between a channel
inlet and outlet to the temperature gradient between the fluid and the channel wall.
It characterizes the convection heat transfer intensity for fluid flowing in channels.
Info: [A14],[A23],[A29].
R.V. Bansen.
189Thermomechanics
5.3.2 Boundary Layer Number N
N5τsLwNη
τs (N m21) � shear surface stress; L (m) � characteristic length; wN (m s21) �flow velocity far from boundary layer; η (Pa s21) � dynamic viscosity.
It characterizes the surface friction in flow in a laminar boundary layer. It is
analogous to the Bingham number Bm (p. 118).
Info: [A23].
5.3.3 Colburn Number JQ, Jh
JQ � Jh 5α
Rcpwcpηλ
� �23
5 StPr23 5NuRe21Pr2
13
α (W m22 K21) � heat transfer coefficient; R (kg m23) � density; cp (J kg21 K21) �
specific heat capacity; w (m s21) � fluid flow velocity; η (Pa s) � dynamic viscosity;
λ (W m21 K21) � thermal conductivity; St (�) � Stanton number (p. 201);
Pr (�) � Prandtl number (p. 197); Nu (�) � Nusselt number (p. 196); Re (�) �Reynolds number (p. 81); Jh (�) � J-heat transfer factor (p. 195).
It characterizes the heat transfer in forced and free flows of a viscous fluid. The
Colburn number is identical with the J-heat transfer factor (p. 195).
Info: [A23],[A29].
Allan Philip Colburn (8.6.1904�1955), American chemi-
cal engineer.
He was engaged in water steam condensation from a sat-
urated air flow, which was his main interest during his
entire life. It was due to him that the common foundations
of momentum, heat and mass transfer, following from ther-
modynamic principles, were formulated in the US for the
first time.
5.3.4 Crispation Group Ncr
Ncr 5ηaσL
η (Pa s) � dynamic viscosity; a (m2 s21) � thermal diffusivity; σ (kg s22) � undis-
turbed surface tension; L (m) � characteristic length (thin layer thickness).
It expresses the possibility of fluid crispation (tearing) originating in convective
flow and heat transfer.
Info: [A23],[A35].
190 Dimensionless Physical Quantities in Science and Engineering
5.3.5 Dulong Number Du
Du � r21h 5w2
cpΔT
w (m s21) � flow velocity; cp (J kg21 K21) � specific heat capacity; ΔT (K) �temperature difference.
This number expresses the ratio of the flowing fluid kinetic energy to the
enthalpy change in adiabatic compression. It characterizes the convection heat
transfer between a wall and fast flowing compressible gas with viscous energy dis-
sipation and adiabatic compression. It represents the volume density of the heat
flow that arises and the heating of the bypassed surface. It corresponds to the
Eckert number Ec (p. 191). Its inverse value corresponds to the restitution enthalpy
coefficient rh (p. 198).
Info: [A23],[A29],[B20].
Pierre Louis Dulong (12.2.1785�19.7.1838), French phys-
icist and chemist.
Above all, he is well known for the Dulong�Petit law
(1819), which joins the specific thermal capacity of metals
with their molar mass. It is a classic relation that expresses
the specific thermal crystal capacity dependence on the grid
oscillating frequency. Despite its simplicity, this relation pro-
vides a very good result for solid materials with relatively
simple crystal structure in the high-temperature range.
5.3.6 Eckert Number Ec
Ec5w2N
cpΔT5 2Rf ð1Þ;
Ec5 ðκ21ÞM2N
TN
ΔT5 2
ΔTad
ΔT5 2
Tad;N 2 TN
TN 2 Tð2Þ
wN (m s21) � fluid flow velocity far from body; cp (J kg21 K21) � specific heat
capacity of fluid; ΔT (K) � temperature difference; T, Tad, TN (K) � static, adia-
batic and far from body temperatures; k (�) � specific heat ratio (p. 28); Rf (�) �temperature recovery factor (p. 202).
It expresses the ratio of kinetic energy to a thermal energy change. Ec1 is a
special case of the Dulong number Du (p. 191) for w5wN. Ec2 is valid for
a compressible fluid.
Info: [A23],[A26],[A29],[A35].
191Thermomechanics
Ernst Rudolf Georg Eckert (13.9.1904�8.7.2004),
American engineer of Czech-German origin.
He researched a wide range of heat transfer problems,
from cryogenic temperatures to plasma. His research
involves basic studies on humidity migration in porous mate-
rials and on related problems of gas turbine cooling. His
principal works are Introduction to the Transfer of Heat and
Mass (1959) and Analysis of Heat and Mass Transfer (1972).
5.3.7 Fliegner Number Fl
Fl5Rwpst
ffiffiffiffiffiffiffiffiffifficpTst
pð1Þ;
Fl5Rwp
ffiffiffiffiffiffiffifficpT
pð2Þ;
Fl5Rw
ðps 1 Rw2ÞffiffiffiffiffiffiffifficpT
pð3Þ;
Fl5Qm
ffiffiffiffiffiffiffifficpT
pAðps 1 Rw2Þ 5
kMffiffiffiffiffiffiffiffiffiffik21
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11
ðk21ÞM2
2
rð4Þ
R (kg m23) � density; w (m s21) � flow velocity; pst (Pa) � stagnation absolute
pressure of gas flow; cp (J kg21 K21) � specific heat capacity; Tst (K) � stagnation
temperature; p (Pa) � static absolute pressure; T (K) � absolute temperature;
ps (Pa) � static pressure; Qm (kg s21) � mass flux; A (m2) � clear area; k (�) �specific heat ratio (p. 28); M (�) � Mach number (p. 73).
It characterizes the coupling between the fluid flow and the heat transfer. It
expresses the ratio of the heat related to the velocity unit to the fluid momentum
flow. In expression (1), it is the stagnation number; in expression (2), it is the static
number; and in expression (3), it is the impulse number. Generally, the Fl5 f(M)
expressed in form (4) holds for the compressible fluid flow.
Info: [A23],[A29],[A35],[B20].
Albert Friedrich Fliegner (1842�1928), Swiss engineer.
5.3.8 Frossling Heat Number Fs
Fs5Nu22
Re12Pr
13
ð1Þ;
192 Dimensionless Physical Quantities in Science and Engineering
Fs5Nu
Re12
ð2Þ
Nu (�) � Nusselt number (p. 196); Re (�) � Reynolds number (p. 81); Pr (�) �Prandtl number (p. 197).
It expresses the influence of the fluid molecular properties and character of the
flow in a boundary layer on the average value of the specific thermal transference.
Expression (1) holds for turbulent flow around a cylinder or a ball. Expression (2)
is valid for an isothermic plate with laminar flow in a boundary layer.
Info: [A23],[A29],[A35].
Karl Gustav Frossling (born 1913), Swedish engineer.
5.3.9 Graetz Number Gz
Gz5Qmcp
λL5
cpRwAλL
ð1Þ;
Gz5wD2
aL5RePr
D
Lð2Þ
Qm (kg s21) � mass flux; cp (J kg21 K21) � specific heat capacity; λ (W m21 K21) �
thermal conductivity; L (m) � characteristic length; ρ (kg m23) � density; w (m s21)
� velocity; A (m2) � area; D (m) � pipe diameter; a (m2 s21) � thermal diffusivity;
Re (�) � Reynolds number (p. 81); Pr (�) � Prandtl number (p. 197).
This number expresses the ratio of the fluid thermal capacity to its thermal con-
ductivity. It characterizes the heat transfer in the laminar flow of fluids. The rela-
tion (2) is valid for a circular pipeline. A large Gz number represents a
predominating convection influence. A small Gz number indicates to a predominat-
ing heat conduction influence. In the last case, the temperature does not change
along the path and the thermal field is fully developed. The Gz number is used
mainly in the American science texts, in which it often replaces the Peclet heat
number Pe (p. 180).
Info: [A23],[A29],[A35],[B20].
Leo Graetz (26.9.1856�12.11.1941), German physicist.
His scientific work was focused on heat transfer by con-
duction and radiation, and on the problems of friction and
elasticity. For a long time, he was devoted to problems of
electromagnetic waves and cathode radiation. His publica-
tions Electricity and Magnetism and Electricity and its
Applications are especially significant.
193Thermomechanics
5.3.10 Heat Non-Stationarity Number (1.) NT
NT 5@TS@τ
L2
ðTS 2 TmedÞaTS (K) � wall temperature; τ (s) � time; L (m) � characteristic length; Tmed (K) �mean fluid temperature; a (m2 s21) � thermal diffusivity.
This number characterizes the influence of the time-dependent temperature of a
wall on the non-stationary heat transfer of single-phase fluids in passageways.
Info: [A23].
5.3.11 Heat Non-Stationarity Number (2.) NZ
NZ 5@TS@z
L
TS 2 Tmed
Ts (K) � wall temperature; z (m) � axis along the pipeline; L (m) � characteristic
length; Tmed (K) � mean fluid temperature.
It characterizes the influence of the temperature gradient change in a wall on the
heat transfer of single-phase fluids in passageways.
Info: [A23].
5.3.12 Heat Source Effect Number F
F5DqVL
2RcpνΔT
D (m) � tube diameter; qV (W m23) � volume density of heat flux; L (m) � char-
acteristic length; R (kg m23) � density; cp (J kg21 K21) � specific heat capacity;
ν (m2 s21) � kinematic viscosity; ΔT (K) � temperature difference.
It characterizes the heat source influence on the heat transfer by convection in
wholly developed laminar fluid flow from below upwards in inclined tubes with
uniformly distributed inner sources and uniform thermal flow on the walls. With
increasing parameter F, the Nusselt number Nu (p. 196) drops. With F 5 const, the
heat transfer intensity depends on the Rayleigh number (2.) (heat instability) Ra2.
With large Ra2 (p. 187) numbers and arbitrary F, the tube orientation influences
the heat transfer intensity.
Info: [A23].
5.3.13 Heat Transfer Efficiency N
N5αLRw2
λΔp5BiEu21
194 Dimensionless Physical Quantities in Science and Engineering
α (W m22 K21) � heat transfer coefficient; L (m) � characteristic length;
R (kg m23) � density; w (m s21) � velocity; λ (W m21 K21) � thermal conductiv-
ity; Δp (Pa) � pressure difference; Bi (�) � Biot number (p. 173); Eu (�) �Euler number (1.) (p. 61).
It characterizes the external heat transfer intensity related to system hydraulic
resistance.
Info: [A23],[A33].
5.3.14 Hydrodynamic Non-Stationarity Number NG
NG 5@Qm
@τL2
νQm
Qm (kg s21) � mass flux; τ (s) � time; L (m) � characteristic length; ν (m2 s21) �kinematic viscosity.
It expresses time change in the passage of a mass which influences the non-
stationary heat transfer of single-phase fluid.
Info: [A23].
5.3.15 J-Heat Transfer Factor Jh
Jh � JQ 5α
cpqm
cpηλ
� �23
α (W m22 K21) � heat transfer coefficient; cp (J kg21 K21) � specific heat capac-
ity; qm (kg m22 s21) � mass flux density; η (Pa s) � dynamic viscosity;
λ (W m21 K21) � thermal conductivity; JQ (�) � Colburn number (p. 190).
Generally, it characterizes the heat transfer in free and forced flows. It is identi-
cal with the Colburn number JQ (p. 190).
Info: [A35],[B20].
5.3.16 Leibenzon Number (1.) Lb
Lb5NuPr1 2
Pr
Nu (�) � Nusselt number (p. 196); Pr (�) � Prandtl number (p. 197).
It is an extended modification of the Nusselt number Nu (p. 196), with which
convective and diffuse heat transfers are considered.
Info: [A23],[A33].
Leonid Samuilovich Leibenzon (1879�1951), Russian physicist and
mechanist.
195Thermomechanics
5.3.17 Margoulis Number Mg
Mg5 St21 5cpRwα
5RePr
Nu
cp (J kg21 K21) � specific heat capacity; R (kg m23) � fluid density; w (m s21) �fluid flow velocity; α (W m22 K21) � heat transfer coefficient; St (�) � Stanton
number (p. 201); Re (�) � Reynolds number (p. 81); Pr (�) � Prandtl number (p.
197); Nu (�) � Nusselt number (p. 196).
It expresses forced convection. It is also called the thermal resistance number.
Info: [A23],[A35].
Wladimir Margoulis (born 1886).
5.3.18 Nusselt Number Nu
Nu5αLλ
5ReStPr5PeSt
α (W m22 K21) � heat transfer coefficient; L (m) � characteristic length;
λ (W m21 K21) � thermal conductivity; Re (�) � Reynolds number (p. 81); St (�)
� Stanton number (p. 201); Pr (�) � Prandtl number (p. 197); Pe (�) � Peclet
heat number (p. 180).
It expresses the ratio of the total heat transfer in a system to the heat transfer by
conduction. In characterizes the heat transfer by convection between a fluid and the
environment close to it or, alternatively, the connection between the heat transfer
intensity and the temperature field in a flow boundary layer. It expresses the
dimensionless thermal transference. The physical significance is based on the idea
of a fluid boundary layer in which the heat is transferred by conduction. If it is not
so, the criterion loses its significance. In the expression α (λ/L)21, it expresses the
ratio of the heat transfer intensity to heat conduction intensity in a boundary layer.
In the expression L (λ/α)21, it represents the ratio of the characteristic length to the
boundary layer thickness. In the expression L/λ (1/α)21, it expresses the ratio of
thermal resistances by conduction to those by convection in a boundary layer.
Info: [A4],[A29],[A35],[A43],[B20].
Ernst Kraft Wilhelm Nusselt (25.11.1882�1.9.1957),
German engineer.
In 1915, he published an important work on basic heat
transfer laws. In this work, he first suggested the dimension-
less thermal similarity criteria. He was engaged in research
on the analogy between the transfer of heat and mass, and
convective heat transition. He formulated the basic theory of
heat regenerators. He was also engaged in work related to
thermal processes in membrane steam condensation and in
powder fuel combustion.
196 Dimensionless Physical Quantities in Science and Engineering
5.3.19 Prandtl Number Pr
Pr5ηcpλ
5νa5PeRe21
η (Pa s) � dynamic viscosity; cp (J kg21 K21) � specific heat capacity;
λ (W m21 K21) � thermal conductivity; ν (m2 s21) � kinematic viscosity; a (m2 s21) �thermal diffusivity; δ, δt (m) � thicknesses of thermodynamic and thermal boundary
layer; Pe (�) � Peclet heat number (p. 180); Re (�) � Reynolds number (p. 81).
This number expresses the ratio of the momentum diffusivity (viscosity) to the
thermal diffusivity. It characterizes the physical properties of a fluid with convec-
tive and diffusive heat transfers. It describes, for example, the phenomena con-
nected with the energy transfer in a boundary layer. It expresses the degree of
similarity between velocity and diffusive thermal fields or, alternatively, between
hydrodynamic and thermal boundary layers. With Pr5 1 and grad p5 0, the ther-
mal and hydrodynamic fields are similar. For example, if diverse molten materials
have equal Prandtl numbers, they have similar velocity and temperature fields in
crystallization. The following are valid:
Pr{1 δ{δt;Pr61 δ5 δt;Prc1 δcδt:
With small Pr numbers (Pr, 1), the molecular heat transfer by conduction
predominates over that by convection. With Pr. 1, it is the opposite case.
For example, for air and many other gases, the Pr value is about 0.7, for
coolants PrAh2; 10i, for machine oils PrAh102; 43 104i, and for mercury it is
approximately 153 1023.
Info: [A23],[A29],[A43],[B20].
Ludwig Prandtl (4.2.1875�15.8.1953), German physicist.
His principal works relate to the hydrodynamic and ther-
mal boundary layers, turbulent boundary layer analysis and,
especially, heat transfer. He published (1910) his elabora-
tion of Reynolds’ original concept (1874) of the analogy
between turbulent momentum and heat transfer. He applied
modelling methods extensively, above all the hydraulic
analogy to visualize flow.
5.3.20 Prandtl Resultant (Effective, Total) Number Pref
Pref 5εM 1 νεT 1 a
197Thermomechanics
εM (m2 s21) � vortex momentum diffusivity; εT (m2 s21) � vortex thermal diffu-
sivity; ν (m2 s21) � kinematic viscosity; a (m2 s21) � thermal diffusivity.
This number expresses the ratio of the resulting momentum diffusivity to the
resulting thermal diffusivity in mixed laminar and turbulent flow. It characterizes
the influence of the physical properties of a fluid on convective heat transfer and
momentum transfer under turbulent flow conditions.
Info: [A23],[A29],[A33].
Ludwig Prandtl (see above).
5.3.21 Prandtl Turbulent Number Prtur
Prtur � Scv 5εMεT
5L
LT
εM (m2 s21) � vortex momentum diffusivity; εT (m2 s21) � vortex thermal diffusiv-
ity; L, LT (m) � characteristic lengths for momentum and heat transfer by mixing;
Scv (�) � Schmidt turbulent number (p. 264).
It expresses the ratio of turbulent momentum diffusivity to turbulent thermal dif-
fusivity. It characterizes the influence of turbulent flow and turbulence in heat
transfer.
Info: [A23],[A29],[A33].
Ludwig Prandtl (see above).
5.3.22 Rayleigh Number (1.) Ra1
Ra1 5w
ffiffiffiffiffiffiRLσ
r
w (m s21) � flow velocity; R (kg m23) � liquid density; L (m) � characteristic
length; σ (N m21) � surface tension.
It characterizes the condition of fluid jet decomposition. It is equivalent to the
Weber number (2.) We2 (p. 91) and it is often called the Rayleigh parameter.
Info: [A23].
Lord Rayleigh (p. 187).
5.3.23 Restitution Enthalpy Coefficient rh
rh 5ha 2 hl
hst 2 hlð1Þ;
198 Dimensionless Physical Quantities in Science and Engineering
rh 5cpðTaS 2 TlÞ
w2ð2Þ
ha (J) � specific adiabatic enthalpy of high-speed fluid flow on the wall; hl (J) � spe-
cific enthalpy of fluid flow; hst (J) � specific enthalpy of retarded flow;
cp (J kg21 K21) � specific heat capacity; TaS (K) � adiabatic temperature of circum-
fluenced surface; Tl (K) � static temperature of fluid flow; w (m s21) � flow velocity.
This coefficient expresses the ratio of the fluid flow adiabatic enthalpy on a wall
to the sum of the total flow enthalpy and the flow kinetic energy or, alternatively,
the ratio of the real conversion temperature to the theoretical conversion tempera-
ture. It characterizes the change of the specific fluid flow enthalpy when a high
velocity flow bypasses a wall; see the Eckert number Ec (p. 191) and the Dulong
number Du (p. 191). It expresses the ratio of the real flow heating due to conver-
sion to the ideal gas heating. It is used for the convective heat transfer in compress-
ible fluid flow.
Info: [A23].
5.3.24 Restitution Temperature Coefficient r
r5Taw 2 Tl
Tst 2 Tl
Taw (K) � adiabatic wall temperature; Tst (K) � stagnation temperature of fluid
flow; Tl (K) � static fluid temperature.
It expresses the ratio of the heating of a real wall due to flowing fluid kinetic
energy to the highest theoretical heating of the wall. It characterizes the heating
degree of a wall surface due to flowing fluid kinetic energy in relation to the high-
est theoretical heating.
Info: [A23].
5.3.25 Shukhov Number Su
Su5πkdLQcpR
k (W m22 K21) � heat passage coefficient; d (m) � pipe diameter; L (m) � charac-
teristic length; Q (m3 s21) � volume flow; cp (J kg21 K21) � specific heat capacity;
R (kg m23) � density.
It characterizes the influence of the radial and axial temperature gradients, in
a heated or cooled fluid flowing through a pipeline, on the hydraulic resistance
(pressure loss) in a considered pipeline section.
Info: [A23],[A33].
Vladimir Grigoryevich Shukhov (1853�1939), Russian engineer.
199Thermomechanics
5.3.26 Spalding Function Sp
Sp52@Θ@u1
� �u150
where Θ5T 2 TN
Tw 2 TN; u15w
ffiffiffiRτ
r
T (K) � local temperature; TN (K) � free stream temperature; Tw (K) � wall tem-
perature; w (m s21) � velocity; τ (Pa) � shear stress; R (kg m23) � density;
u1 (�) � Prandtl velocity ratio (p. 80).
This function characterizes the dimensionless temperature gradient in a wall
with forced convection.
Info: [A23].
Dudley Brian Spalding (see above).
5.3.27 Spalding Number (1.) Sp
Sp 5αν
wλffiffiffiffiffiffiffiffiffiffi0:5fF
p 5ανλ
τR
0@1A
21
2
5 StPrNe21
2
5NuRe21Ne21
25NuPe21PrNe
21
2
α (W m22 K21) � heat transfer coefficient; ν (m2 s21) � kinematic viscosity;
w (m s21) � velocity; λ (W m21 K21) � thermal conductivity; fF (�) � Fanning
friction number (p. 163); St (�) � Stanton number (p. 201); Pr (�) � Prandtl num-
ber (p. 197); Ne (�) � Newton number (p. 75); Nu (�) � Nusselt number (p. 196);
Re (�) � Reynolds number (p. 81); Pe (�) � Peclet heat number (p. 180).
It characterizes the convective heat transfer intensity in a fully developed boundary
layer.
Info: [A23].
Brian Spalding (born 1923), English engineer.
Initially, he devoted himself to the liquid fluid combus-
tion problem and to thermodynamics. He published a whole
range of original scientific works concerning this sphere.
He has also worked on the transfer of heat and mass,
boundary layer theory and the fluid dynamics simulations,
making use of the rapid evolution of calculation techniques
in recent years.
200 Dimensionless Physical Quantities in Science and Engineering
5.3.28 Stanton Number St
St � Mg21 5α
Rcpw5
Nu
RePr5NuFo ð1Þ;
St5αS
qmVVcpð2Þ;
St5qA
RwcpðTr 2 TsÞð3Þ
α (W m22 K21) � heat transfer coefficient; R (kg m23) � density; cp (J kg21 K21) �
specific heat capacity; w (m s21) � fluid velocity; S (m2) � surface area;
qmV (kg m23) � volume density of mass flux; V (m3) � volume; qA (W m22) �surface heat flux density; Tr (K) � reference temperature; TS (K) � wall tempera-
ture; Nu (�) � Nusselt number (p. 196); Re (�) � Reynolds number (p. 81);
Pr (�) � Prandtl number (p. 197); Fo (�) � Fourier number (p. 175).
It expresses the ratio of the heat transferred in a system by convection to the
thermal capacity of heat-bearing surroundings. It characterizes the forced convec-
tion heat transfer. In form (2), it is used to express the convective heat transfer in
burning chambers and in steam boiler spaces. In form (3), it expresses the heat
flow through a laminar boundary layer when solid bodies are passed by with fluid.
In form (1), it is also called the Margoulis number Mg (p. 196).
Info: [A4],[A21],[A43],[B11],[B20].
Thomas Edward Stanton (12.12.1865�1931), English
engineer.
His principal interest was the flowing of fluids and the
friction problem, with its related heat transfer. From 1902
to 1907, he was engaged in wide research on the influence
of wine on structures such as bridges and roofs. After the
first flight of the Wright brothers in an aircraft in 1908, he
devoted himself enthusiastically to airplane and airship
design, and above all to the heat transfer in air-cooled
engines.
5.3.29 Temperature Factor ΘS
ΘS 5TS
Trð1Þ;
ΘS 5TS
TaSð2Þ
201Thermomechanics
TS (K) � surface temperatures of body wall; Tr (K) � characteristic temperature of
fluid flow; TaS (K) � adiabatic temperature of body surface.
It expresses the ratio of the absolute surface temperature of a body wall to the
characteristic absolute temperature of a flowing fluid. It characterizes the influence
of the flowing fluid temperature on a body. Instead of the relative temperature Tr,
sometimes the adiabatic temperature of a solid, ideally insulated and non-radiating
wall is considered, which is passed by a fluid with heat development caused by
energy dissipation. With outer bypassing of a body, usually the outer flow tempera-
ture is chosen as the characteristic temperature Tr. The mean flow temperature is
chosen as the characteristic temperature in flow in tubes and canals.
Info: [A23].
5.3.30 Temperature Recovery Factor Rf
Rf 52cpΔT
w25 2Ec21
cp (J kg21 K21) � specific heat, capacity of fluid flow; ΔT (K) � difference
between static temperature of flowing gas and adiabatic temperature of circum-
fluenced wall; w (m s21) � fluid velocity; Ec (�) � Eckert number (p. 191).
For an ideal gas, this factor expresses the ratio of the real temperature of a
bypassed wall to the theoretical temperature.
Info: [A15].
5.3.31 Thermal Coupling Number K, χ
K5 0:5λl
λs
Re12; KA 0;Nh Þ
λl, λs (W m21 K21) � specific thermal conductivity of fluid and solid; Re (�)
Reynolds number (p. 81).
This number characterizes the influence of thermally physical properties on the
convective heat transfer in compound tasks. It occurs in compressible fluid flow
when considering the mechanical energy dissipation and heat propagation in a solid
body. Usually, its value is approximately K5 1. The case of K-N (λs-0) occurs
when a perfect thermal insulator is inserted into the fluid flow.
Info: [A23].
5.3.32 Heat Transfer Number Nq
Nq 5q
Rw3L2
202 Dimensionless Physical Quantities in Science and Engineering
q (W) � heat flux; R (kg m23) � density; w (m s21) � flow velocity; L (m) � char-
acteristic length.
It expresses the ratio of the heat transferred in a system to the kinetic energy of
the flowing environment. Heat transfer in flowing fluid.
Info: [A23],[A29],[A33].
5.3.33 Unsteady Heat Transfer Number N
N5Nu
Nu0
Nu (�) � Nusselt unsteady number, see Nusselt number (p. 196); Nu0 (�) �Nusselt quasi-stationary number, see Nusselt number (p. 196).
It expresses the ratio of unsteady heat transfer to quasi-stationary heat transfer.
It characterizes the dimensionless transfer in unsteady flow.
Info: [A23].
5.4 Radiation
Here radiation denotes thermal radiation, of which the main aspect is electromag-
netic radiation. Besides conduction and convection, it describes the third kind of
heat transfer, which is of use with high surface temperatures especially. Among the
basic dimensionless quantities are the Boltzmann and Stefan numbers. The former
expresses the relation of the heat transferred by convection and radiation, whereas
the latter expresses that transferred by radiation and conduction. The Bansen,
Bouguer, Nusselt radiation, Peclet radiation and Schuster numbers are also among
the dimensionless values, and so are the radiation pressure and radiation viscosity
numbers.
5.4.1 Bansen Radiation Number Ba
Ba5αRS
Qmcp
αR (W m22 K21) � radiation heat transfer coefficient; S (m2) � surface area of
channel wall; Qm (kg s21) � mass flux; cp (J kg21 K21) � specific heat capacity.
It expresses the ratio of the heat transferred by radiation to the fluid thermal
capacity.
Info: [A4],[A35],[B20].
R.V. Bansen.
203Thermomechanics
5.4.2 Biot Radiation Number BiR
See the Stefan number Sf (p. 213).
Jean-Baptiste Biot (p. 174).
5.4.3 Boltzmann Number Bo
Bo � Tg5cpRwεσ0T3
ð1Þ;
Bo5 4NRePrτ21 5 4NPeτ21 ð2Þ;
Bo53
16τBo1 ð3Þ;
Bo5Bo1
2τð4Þ;
Bo5Rwðh1 2 h2Þ
εσ0T4ð5Þ;
Bo5εσ0SRT
3
Qmcpð6Þ
cp (J kg2 1 K2 1) � specific heat capacity; R (kg m2 3) � flue gases density;
w (m s21) � flue gases velocity; σ0 (W m22 K24) � Stefan�Boltzmann constant;
T (K) � flue gas thermodynamic temperature; τ (�) � optical thickness, ratio of
characteristic length to mean free path; h1, h2 (J kg21) � specific enthalpy of flue
gases; SR (m2) � radiation surface; Qm (kg s21) � mass flux; ε (�) � emissivity
(p. 206); Tg (�) � Thring radiation number (p. 214); N (�) � radiation number
(2.) (p. 211); Re (�) � Reynolds number (p. 81); Pr (�) � Prandtl number
(p. 197); Pe (�) � Peclet heat number (p. 180).
This number expresses the ratio of the heat transferred by forced convection to
that transferred by radiation. It characterizes the relation between the convective
energy transfer (in the flow direction) and the heat radiation transfer. It expresses
also the ratio of the absorbing surroundings’ enthalpy to the thermal flow radiated
by a surface which limits the surroundings. In heat transfer by radiation, it has sim-
ilar significance to the Peclet heat number Pe (p. 180) in heat conduction.
Characteristic optical thickness is important for determining the relative ratio of the
heat radiation transfer to the convective transfer for optically thick (τc1) or
optically thin (τ{1) layers. This ratio is expressed by the modified Bo number (3)
for an optically thick layer or by (4) for a thin one. Sometimes, the modified
enthalpy expression (5) is used. For radiation in combustion spaces, expression (6)
is applied.
Info: [A23],[B20].
204 Dimensionless Physical Quantities in Science and Engineering
Ludwig Boltzmann (20.2.1844�5.10.1906), Austrian
physicist.
He founded gas kinetic theory and statistical physics. He
was engaged in entropy and probability. He deduced one of
the radiation laws now known as the Boltzmann law. His
works on electromagnetism and thermodynamics are espe-
cially important.
5.4.4 Bouguer Number (1.) Bu1
Bu1 53RilR2Rmd
Ri (kg m23) � mass concentration of particles in reference gas volume; lR (m) �
mean length of radiation path; Rm (kg m23) � density of mass particles; d (m) �diameter of mass particles.
It relates to fluid flow with mass particles. It characterizes the heat transfer by
radiation into mass particles in flowing gas.
Info: [A23],[A29],[B20].
Pierre Bouguer (16.2.1698�15.8.1758), French mathema-
tician, astronomer and physicist.
He was engaged in research on earth density, photome-
try, astronomical photometry and general light propagation
problems. He is often considered the founder of photome-
try. The Bouguer law expresses the relation between radia-
tion energy absorption and the absorbing surroundings.
5.4.5 Bouguer Number (2.) Bu2
Bu2 5κL
κ (m21) � attenuation coefficient of environment including absorption and disper-
sion coefficients; L (m) � reference length.
It expresses the ratio of the characteristic relative length of a ray path to the free
ray path caused by absorption and radiation scatter. It characterizes the degree of
attenuation (absorbance) of the surroundings in heat radiation transfer in a gas. It is
analogous to the Knudsen number (1.) Kn (p. 69) in gas dynamics.
Info: [A23],[A29].
Pierre Bouguer (see above).
205Thermomechanics
5.4.6 Emissivity ε
ε5εgbεbb
5 12 r; where εAh0; 1i
εgb (�) � grey body absorptivity; εbb (�) � ideal black body absorptivity; r (�) �surface reflectance; Q (W m22) � heat flux transferred by radiation;
σ0 (W m22 K24) � Stefan�Boltzmann constant; T1, T2 (K) � temperature of radi-
ating and absorbing surfaces.
It expresses the relative radiativity (absorbance) of the surface of grey bodies. It
is a non-linear function of the surface temperature and depends on the radiation
wave length. For the amount of heat transferred by radiation, the
Stefan�Boltzmann law is valid.
Q5 εσ0ðT41 2 T4
2 Þ
Thermomechanics. Radiation. Contactless measurement of surface temperatures.
Info: [C4].
5.4.7 Fourier Radiation Number FoR
FoR 5 aκ2τ
a (m2 s21) � thermal diffusivity; κ (m21) � attenuation coefficient; τ (s) � time.
It expresses the ratio of the radiation heat transferred in a system of molecular
diffusion to the radiation heat absorbed in the system. In the dimensionless form, it
characterizes the time of non-stationary heat transfer by radiation. It determines the
relation between the velocity due to temperature field radiation and the physical
parameters of the system.
Info: [A23].
Jean Baptiste Joseph Fourier (p. 175).
5.4.8 Hottel Number Hot
Hot5α
εσ0T3
α (W m22 K21) � convective heat transfer coefficient; σ0 (W m22 K24) �Stefan�Boltzmann constant; T (K) � absolute temperature; ε (�) � emissivity
(p. 206).
It expresses the ratio of heat flows by convection to those by radiation. Heat
transfer by radiation. Thermomechanics. High-temperature devices. Industrial
furnaces.
Info: [A2].
H.C. Hottel.
206 Dimensionless Physical Quantities in Science and Engineering
5.4.9 Kirpichev Radiation Number KiR
KiR � N21 5σT3
λκ5 SfBu212
σ (W m22 K24) � absorptance (absorption capacity, radiating capacity); T (K) �temperature; λ (W m21 K21) � thermal conductivity; κ (m21) � attenuation coeffi-
cient; N (�) � radiation number (2.) (p. 211); Sf (�) � Stefan number (p. 213);
Bu2 (�) � Bouguer number (2.) (p. 205).
It expresses the ratio of the heat transferred by radiation to that transferred by
molecular conduction in a turbulent flow. It characterizes the heat transfer by radia-
tion in an environment with attenuation effect. Its inverse value is called the radia-
tion number (2.) N (p. 211).
Info: [A23],[A33].
Mikhail Viktorovich Kirpichev (p. 177).
5.4.10 Local Radiation Heat Transfer ξ
ξ5εσ0T
3κxRcpwr
5κxBo
5KReEc21 5KηEi2Re
Bo2Ec
σ0 (W m22 K24) � Stefan�Boltzmann constant; T (K) � temperature; κ (m21) �attenuation coefficient; x (m) � coordinate; R (kg m23) � density; cp (J kg
21 K21) �specific heat capacity; wr (m s21) � reference velocity; ε (�) � emissivity (p. 206);
Bo (�) � Boltzmann number (p. 204); K (�) � radiation high-temperature number
(p. 210); Re (�) � local Reynolds number, see Reynolds number (p. 81); Ec (�) �Eckert number (p. 191); Kη (�) � radiation viscosity (p. 212); Ei (�) � Einstein
number (p. 317).
This transfer expresses the ratio of the heat transferred by radiation in absorbing
surroundings to that transferred by convection. It characterizes the influence of the
radiation and convective heat transfer on the local heat transfer in high-temperature
processes, where the hydrodynamic and radiation gas viscosities and gas flow rate
are considered. The thermal radiation dissipation by gas molecules is presumed to
be negligible.
Info: [A23],[A33].
5.4.11 Nusselt Radiation Number NuR
NuR 5qAκL
ðT 2 TSÞσT3
qA (W m22) � heat flux density; κ (m21) � attenuation coefficient; L (m) � char-
acteristic length; T, TS (K) � radiating environment and wall surface temperature;
σ (W m22 K24) � absorptance (absorption capacity, radiating capacity).
207Thermomechanics
It characterizes the radiation heat transfer intensity of the radiating turbulent
flow into a canal wall. It expresses the size of the thermal impact on the interface
between the radiating gas and a wall.
Info: [A23],[A33].
Ernst Kraft Wilhelm Nusselt (p. 196).
5.4.12 Optical Thickness of Heat Boundary Layer τ
τ5 kδ ð1Þ;
τ5 5κffiffiffiffiffiνxwr
r5 5
ffiffiffiffiffiffiffiffiffiffiffiPrNξ
pð2Þ
κ (m21) � attenuation coefficient; δ (m) � thickness of thermal boundary layer;
ν (m2 s21) � kinematic viscosity; x (m) � coordinate; wr (m s21) � reference
velocity; Pr (�) � Prandtl number (p. 197); N (�) � radiation number (2.)
(p. 211); ξ (�) � local radiation heat transfer (p. 207).
It expresses the ratio of the thickness of the thinned thermal boundary layer to
that of the unthinned thermal boundary layer. In the dimensionless form, it charac-
terizes the optical thickness of the thermal boundary layer. With the expressionffiffiffiffiffiffiNξ
pc1; it is about an optically thick boundary layer, but with
ffiffiffiffiffiffiNξ
p {1; it isabout an optically thin one. The size of the τ depends strongly on pressure. It drops
with increasing pressure and vice versa.
Info: [A23].
5.4.13 Peclet Radiation Number PeR
PeR 5wRcpεσ0T3
κL
w (m s21) � velocity; R (kg m23) � density; cp (J kg21 K21) � specific heat capacity;
σ0 (W m22 K�4) � Stefan�Boltzmann constant; T (K) � temperature; κ (m21) �attenuation coefficient; L (m) � characteristic length; ε (�) � emissivity (p. 206).
It characterizes the relation between the convective and radiation heat transfers
in the turbulent flow of a radiating media.
Info: [A23],[A33].
Jean Claude Eugene Peclet (p. 180).
5.4.14 Pomerantsev Radiation Number PoR
PoR 5qRV
κσT4ð1Þ;
208 Dimensionless Physical Quantities in Science and Engineering
PoR 5qVL
RwcpTð2Þ
qRV (W m23) � volume density of radiation heat flux; κ (m21) � attenuation coef-
ficient; σ (W m22 K24) � absorptance (absorption capacity, radiating capacity);
T (K) � temperature; qV (W m23) � volume density of heat flux; L (m) � charac-
teristic length; R (kg m23) � density; w (m s21) � velocity; cp (J kg21 K21) � spe-
cific heat capacity.
This number expresses the ratio of the internal volume heat source to the heat
transferred by radiation. It characterizes the internal volume heat source in radia-
tion heat transfer.
Info: [A23].
Alexey Alexandrovich Pomerantsev, Russian engineer.
5.4.15 Radiation Heat Flow (1.) QR1
QR1 5qRA
σT45
qRA
pR
qRA (W m22) � radiation heat flux density; σ (W m22 K24) � absorptance (absorp-
tion capacity, radiating capacity); T (K) � temperature of outer environment;
pR (W m22) � radiation pressure.
It expresses the ratio of the heat flow transferred by radiation to that absorbed
by a surface. It characterizes the resulting radiation heat flow in the heat transfer in
a system.
Info: [A23].
5.4.16 Radiation Heat Flow (2.) QR2
QR2 5DRσT3
λw5
DRaRT3
λ
DR (m2 s21) � radiation diffusion coefficient; σ (W m22 K24) � fluid absorptance
(absorption capacity, radiating capacity); T (K) � temperature; λ (W m21 K21) �thermal conductivity; w (m s21) � flow velocity; aR (J m23 K24) � radiation
constant.
It expresses the ratio of the heat transferred by flowing fluid radiation to that
transferred by conduction. It characterizes the heat transfer in various high-temper-
ature heat-exchanging devices.
Info: [A23].
209Thermomechanics
5.4.17 Radiation Heating Up (1.) N1
N1 5σT3τcRL
σ (W m22 K24) � absorptance (absorption capacity, radiating capacity); T (K) �temperature; τ (s) � time; cR (J m23 K21) � specific volume heat; L5 h (m) �depth of warming through.
It expresses the ratio of the heat, carried to a body by radiation, to the thermal
content of the body. It characterizes the radiation heat transfer on a body surface.
Info: [A23].
5.4.18 Radiation Heating Up (2.) N2
N2 5ðσT3Þ2τλcR
σ (W m22 K24) � absorptance (absorption capacity, radiating capacity); T (K) �flue gases temperature; τ (s) � time; λ (W m21 K21) � thermal conductivity;
cR (J m23 K21) � specific volume heat.
It characterizes the velocity of the radiation heating of bodies, where not the
geometrical body size but other thermal parameters act strongly. These parameters
are expressed by the body thermal conductivity and the boundary condition of sur-
face heat transfer.
Info: [A23].
5.4.19 Radiation High-Temperature Number K
K5σT3ηκR2c2p
5ηc2κσT4
σT3
Rcpc
0@
1A
2
5ηηR
σT3
Rcpc
0@
1A
2
5ηηR
w
c
0@1A
2
σT3
Rcpw
0@
1A
2
5KηEi
Bo
0@
1A
2
σ (W m22 K24) � absorptance (absorption capacity, radiating capacity); T (K) �temperature; η, ηR (Pa s) � dynamic (hydrodynamic) viscosity, radiation dynamic
viscosity; κ (m21) � attenuation coefficient; R (kg m23) � density; cp (J kg21 K21) �
specific heat capacity; c (m s21) � light velocity; w (m s21) � flow velocity;
Kη (�) � radiation viscosity (p. 212); Ei (�) � Einstein number (p. 317); Bo (�) �Boltzmann number (p. 204).
210 Dimensionless Physical Quantities in Science and Engineering
This criterion expresses the ratio of the energy transferred by hydrodynamic and
radiation viscosities to the total energy transferred by radiation and convection. It
characterizes the influence of the hydrodynamic and radiation viscosities and the
flow rate on the resulting energy transfer by radiation and convection in high-
temperature processes.
Info: [A23].
5.4.20 Radiation Number (1.) K
K5λE
εσ0σT35WeHo21Sf 21
λ (W m21 K21) � thermal conductivity; E (Pa) � modulus of elasticity;
σ0 (W m22 K24) � Stefan�Boltzmann constant; σ (N m21) � surface tension;
T (K) � temperature; ε (�) � emissivity (p. 206); We (�) � Weber number
(p. 91); Ho (�) � Hooke number (p. 138); Sf (�) � Stefan number (p. 213).
It characterizes the radiation heat transfer (Sf ) under conditions of inertia force
and surface tension (We) and elasticity (Ho) acting on the flowing fluid.
Info: [A23].
5.4.21 Radiation Number (2.) N
N5λκσT3
λ (W m21 K21) � thermal conductivity; k (m21) � attenuation coefficient;
σ (W m22 K24) � absorptance (absorption capacity, radiating capacity); T (K) �temperature.
It expresses the ratio of heat transferred by conduction to that transferred by
radiation. It characterizes the radiation heat transfer in attenuating the surroundings.
It is analogous to the Planck number (p. 24).
Info: [A23],[A29].
5.4.22 Radiation Parameter Φ
Φ5εσ0T
3Lh
λð1Þ;
Φ5f εσ0T
3Lh
λð2Þ
σ0 (W m22 K24) � Stefan�Boltzmann constant; T (K) � gas temperature;
Lh5AO21(m) � hydraulic diameter; A (m2) � channel cross-section surface area;
211Thermomechanics
O (m) � channel perimeter with fluid contact; λ (W m21 K21) � thermal conduc-
tivity; ε (�) � emissivity of inner channel wall (p. 206); f (�) � function of mean
surface wall temperature.
This parameter expresses the ratio of the heat transferred by radiation in a pas-
sageway to that transferred by conduction in a channel wall. It characterizes the
relation between the radiation and conduction heat transfers in passageways.
Alternatively, it expresses the influence of the radiation on the convective transfer.
Info: [A23],[A29].
5.4.23 Radiation Pressure Rp
Rp 5σT4
cpð1Þ;
Rp 5ηRckp
ð2Þ
σ (W m22 K24) � absorptance (absorption capacity, radiating capacity); T (K) �temperature; c (m s21) � light velocity; p (Pa) � gas pressure; ηR (Pa s) �radiation dynamic viscosity; k (m21) � attenuation coefficient.
It expresses the ratio of the radiation pressure to the static pressure of a gas. It
characterizes the relation between the radiation heat transfer energy and the gas
pressure energy. The value of the number Rp is very small and so is the signifi-
cance of the radiation pressure if the temperature T is not greater than approxi-
mately 105 K or if the pressure p is not too low. In most technical applications, the
Rp number can be neglected. However, it is very important, for example, in high-
temperature plasma physics.
Info: [A23],[B20].
5.4.24 Radiation Viscosity Kη
Kη 5ηηR
5ηc2κσT4
η, ηR (Pa s) � dynamic (hydrodynamic) viscosity, radiation dynamic viscosity;
c (m s21) � light velocity; κ (m21) � attenuation coefficient; σ (W m22 K24) �absorptance (absorption capacity, radiating capacity); T (K) � temperature.
It characterizes the relation between hydrodynamic and radiation dynamic
viscosities.
Info: [A23].
212 Dimensionless Physical Quantities in Science and Engineering
5.4.25 Schuster Number Sch
Sch5κp
κc
κp (m21) � attenuation coefficient by absorption of radiation; κc (m
21) � attenua-
tion coefficient by absorption and dispersion on particles in gas flow.
This number expresses the ratio of the radiation attenuation by dissipation to the
total radiation attenuation by dissipation and absorption. It characterizes the rela-
tion between absorbing and dissipating properties of particles floating in a radiating
gas flow. It states the dispersion portion of the total attenuation of radiation due to
surroundings.
Info: [A23],[A33].
Sir Franz Arthur Friedrich Schuster (1851�1934),
British physicist of German origin.
He is well known for his work in spectroscopy, in which
he is a renowned authority. In addition, he was engaged in
research on electricity behaviour in gases, terrestrial magne-
tism, radiometry, calorimetry and the mathematical period-
icity theory. He wrote the book An Introduction to the
Theory of Optics (1904).
5.4.26 Stark Number Sk
See the Stefan number Sf (p. 213).
5.4.27 Stefan Number Sf
Sf � Sk5εσ0T
3L
λ
σ0 (W m22 K24) � Stefan�Boltzmann constant; T (K) � temperature; L (m) �characteristic length; λ (W m21 K21) � thermal conductivity; Sk (�) � Stark num-
ber (p. 213); ε (�) � emissivity of inner channel wall (p. 206).
It expresses the ratio of the heat transferred by radiation to that transferred by
conduction. It characterizes the coupling between the thermal field in a solid body
and the heat transfer by radiation on the body surface. It is analogous to the Biot
number Bi (p. 173) and therefore is called the Biot radiation number BiR (p. 204)
sometimes. It is often called the Stark number Sk (p. 213).
Info: [A4],[A23],[B20].
213Thermomechanics
Josef Stefan (24.3.1835�7.1.1893), Slovenian�Austrian
mathematician and physicist.
He was engaged in research on the kinetic theory of
gases, thermodynamics and heat transfer by radiation. In
the year 1891, he published his famous work on iceberg
melting in polar seas, in which he laid the foundation for
the non-linear heat conduction theory with phase change:
the eponymous Stefan problem. Contrary to Kirchhoff,
Stefan showed that radiated energy exists in all wavelengths
and is governed by the 4th power of the absolute
temperature.
5.4.28 Thring Radiation Number Tg
See the Boltzmann number Bo (p. 204).
Meredith Wooldridge Thring (17.12.1915�15.9.2006), English engineer.
5.5 Boiling
Boiling represents a phase conversion with fast fluid evaporation within the entire
fluid mass. It passes through three stages involving the initial creation of vapour
kernels, a transient zone and a film boiling. The complicated phase conversion
process is described by a lot of dimensionless quantities. Among the basic ones are
the following: the Bond, Archimedes, Jakob, Kutateladze, Lockhart�Martinelli,
Stanton, Sterman and other numbers, including classic criteria modified for boiling.
5.5.1 Archimedes Steam Number Ar
Ar5R12l 2σ
32
η2l g12
RvRl
� �21
Rl, Rv (kg m23) � liquid and vapour density; σ (N m21) � surface tension; ηl (Pa s) �
liquid dynamic viscosity; g (m s22) � gravitational acceleration.
It expresses the ratio of the buoyancy force (the Archimedes force) of the vapour
phase in a non-isotropic fluid to the molecular friction force in the fluid. It charac-
terizes the mutual action of capillary, viscous and gravity forces in a vapour�fluid
mixture. It influences the origin and development of gravity capillary waves in a
viscous fluid. It is a special modification of the Galilei steam number (p. 217).
Heat transfer in boiling.
Archimedes of Syracuse (p. 54).
214 Dimensionless Physical Quantities in Science and Engineering
5.5.2 Beginning of Boiling Instability N
N5ðTS 2 TvÞllvRlL
Tvσ
TS (K) � wall temperature before rising of vapour bubbles (initial maximal super-
heating of liquid); Tv (K) � vapour temperature; llv (J kg21) � specific latent heat
of vaporization; Rl (kg m23) � density of liquid; L (m) � characteristic length;
σ (N m21) � surface tension.
It characterizes the origin of the instability of a fluid as it begins boiling.
Info: [A23].
5.5.3 Boiling Number Boi
Boi5qA
llvRlwð1Þ;
Boi5L2qA
Qm llv5
qA
qm llvð2Þ
qA (W m22) � area heat flux density; llv (J kg21) � specific latent heat of vaporiza-
tion; Rl (kg m23) � liquid density; w (m s21) � flow velocity; L (m) � characteristic
length; Qm (kg s21) � mass flux; qm (kg s21 m22) � area mass flow density.
It expresses the ratio of the specific vapour heat flow perpendicular to a
bypassed wall to the total heat flow parallel with the wall. It characterizes the pro-
cess of creating vapour kernels in fluid boiling and the process of heat transfer
intensification. For flow through a pipeline, with the internal diameter d, L5 d
holds.
Info: [A23],[B56].
5.5.4 Boiling Stability K
K5qcrit
llvR12v½σgðRl 2 RvÞ�
14
qcrit (W m22) � surface density of critical heat flux; llv (J kg21) � specific latent
heat of vaporization; Rv, Rl (kg m23) � density of vapour and liquid; σ (N m21) �surface tension; g (m s22) � gravitational acceleration.
It expresses the ratio of the evaporation rate on a heated surface to the mean
velocity of vapour bubble growth. It characterizes the degree of the dynamic action
of the evaporation process on the vapour�fluid system stability in boiling.
Sometimes, it is called the criterion of the heat transfer hydrodynamic crisis in
boiling.
Info: [A23].
215Thermomechanics
5.5.5 Convection Number Ncon
Ncon 512 x
x
� �0:9 RgRl
� �0:5
x (�) � mass ratio of vapour fraction; Rg, Rl (kg m23) � density of gas (vapour)
and liquid.
Essentially, it represents the modified Martinelli parameter X (p. 106) used for
vapour bubble flow in boiling fluid.
Info: [A23],[B56].
5.5.6 Frequency Evolution of Steam Bubbles N
N5RlRv
clσdllvpβ
RlRl 2 Rv
Rl, Rv (kg m23) � liquid and vapour; cl (J kg21 K21) � liquid and vapour density;
σ (N m21) � surface tension; d (m) � characteristic dimension of vapour bubble;
llv (J kg21) � specific latent heat of vaporization; p (Pa) � pressure; β (K21) �thermal expansivity of vapour.
It expresses the ratio of the surface stress force to the thermogravitational force.
It characterizes the evolution frequency of vapour bubbles in boiling fluid.
Info: [A23].
5.5.7 Froude Boiling Number Frb
Frb 52
R3g
ðTS 2 TvÞclRlllvRv
� �4ðπalÞ2 ð1Þ;
Frb2 5w2
gR12
RvRl
� �21
ð2Þ
R (m) � radius of vapour bubble or drop; g (m s22) � gravitational acceleration;
TS, Tv (K) � temperature of heated wall and vapour; cl (J kg21 K21) � specific
heat capacity of liquid; Rl, Rv (kg m23) � density of liquid and vapour; llv (J kg
21) �specific latent heat of vaporization; al (m2 s21) � thermal diffusivity of liquid;
w (m s21) � motion velocity.
This number expresses the inertia-to-buoyancy forces ratio. It characterizes the
influence of force relations on the origin and evolution of the vapour bubbles in
boiling. In form (1), it expresses the condition for bubbles originating; in form (2),
it expresses the condition for vapour bubble motion. The vapour bubble radius can
216 Dimensionless Physical Quantities in Science and Engineering
be determined in a linear scale R5V1/3, where V denotes the bubble volume, or an
effective radius R5 (3V/4πn)1/3 can be used.
Info: [A23].
William Froude. (p. 63)
5.5.8 Froude Heat Number Frt
Frt 5gL3
a2
g (m s22) � gravitational acceleration; L (m) � characteristic length; a (m2 s21) �thermal diffusivity.
It expresses the gravitation-to-thermodiffusion forces ratio. It characterizes the
mean intensity of the heat transfer under forced convection and fluid boiling in a
tube.
Info: [A23].
William Froude (see above).
5.5.9 Galilei Steam Number Ga
Ga5g
ν2l
σgðRl 2 RvÞ
� �32
g (m s22) � gravitational acceleration; νl (m2 s21) � kinematic viscosity of liquid;
σ (N m21) � surface tension; Rl, Rv (kg m23) � density of liquid and vapour.
It characterizes the heat transfer in closed two-phase thermosiphon systems
(a fluid boiling in an evaporator, vapour motion in a transfer part, condensation,
condensate return to the evaporator), all under the influence of gravity and inertia
forces.
Info: [A23].
Galileo Galilei (p. 123).
5.5.10 Heat Activity K
K5ðRcλÞlðRcλÞw
R (kg m23) � density; c (J kg21 K21) � specific heat capacity; λ (W m22 K21) �thermal conductivity; subscripts: l � liquid; w � channel wall.
This activity expresses the ratio of the fluid thermal activity to the canal wall
thermal activity. It characterizes the influence of material thermal properties on the
non-stationary heat transfer in fluid boiling. Together with the off liquid heating Ψ,
217Thermomechanics
(p. 222) it expresses the temperature of the film boiling crisis. In bubble boiling, it
is of use similarly as the characteristic roughness dimension.
Info: [A23].
5.5.11 Impulse Heating NP
NP 5qA
Rlllv
RvRl
Rb� �2
1k
qA (W m22) � heat flux density; Rl, Rv (kg m23) � density of liquid and vapour;
R (kg m23) � density of generated vapour nuclei; llv (J kg21) � specific latent heat
of vaporization; b (�) � mean value of coefficient depending on liquid superheat-
ing; k (�) � boiling constant.
This quantity expresses the conditions to bring a fluid to boiling with impulse
heating when the vapour origin is caused spontaneously by originating random-
character kernels. For k. 1, the temperature of intensive kernel nuclei origin is
reached. For kc1, the vapour phase nuclei are created in a fluctuating way even
for small masses of the fluid, and the created vapour conversion centres do not
play any role. With k5 3 in an experiment, the growth of bubbles is delayed by
inertia forces (Rayleigh’s case). With k5 0.5, the growth rate is limited by the heat
supply. Heat transfer in boiling.
Info: [A33].
5.5.12 Jakob Evaporation Speed Number Ja
Ja � To5qA
llvRvdf
qA (W m22) � heat flux density; llv (J kg21) � specific latent heat of vaporization;
Rv (kg m23) � density of saturated vapour; d (m) � diameter of vapour bubble;
f (Hz) � generation frequency of vapour bubbles; To (�) � Tolubinsky number
(p. 226).
It expresses the ratio of the vapour phase generation (bubbles) rate, on a heated
surface, to the mean growth rate of vapour bubbles. It characterizes the heat trans-
fer intensity in boiling. It is the equivalent of the modified Peclet boiling speed
number Pewb (p. 223). It expresses also the ratio of the maximum vapour bubble
diameter to the overheated fluid film thickness. Sometimes, it is called the
Tolubinsky number To (p. 226).
Info: [A23].
Max Jakob (see below).
218 Dimensionless Physical Quantities in Science and Engineering
5.5.13 Jakob Number Ja1
Ja1 5c1ΔT
llv
RlRv
cl (J kg21 K21) � specific heat capacity of liquid; ΔT5 Tl�Tn (K) � liquid super
heating; Tl, Tn (K) � temperature of liquid and saturated vapour; llv (J kg21) � spe-
cific latent heat of vaporization; Rl, Rv (kg m23) � liquid and vapour density.
This number expresses the ratio of the heat supplied to overheat the unit volume
of a fluid to the specific volume latent heat of evaporation. It characterizes the heat
transfer intensity in fluid boiling. It depends on the fluid pressure and overheating.
It drops with the pressure increasing because the vapour density increases, and vice
versa. With the pressure greater than the atmospheric pressure (Ja1# 20), the
vapour bubbles grow in size due to the heat supplied from a heated surface over
the adhering fluid layer. With low pressures (Ja1$ 20), the heat that led to a
vapour bubble is transferred from the overheated fluid on the inter-phase surface.
To determine the vapour bubble diameter, the formula is in the first case
R5ffiffiffiffiffiffiffiffiffiffiffiffi2βJa1
p ffiffiffiffiffiaτ
p
and in the second case
R5 2γJa1ffiffiffiffiffiaτ
p
and generally
R5 γJa1 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiγ2Ja21 1 2βJa1
q� � ffiffiffiffiffiaτ
p
The last expression is valid for Ja15 0.1�1000, which corresponds to pressures
of 0.01�10 MPa. The factors β and γ depend on geometric parameters. For exam-
ple, for the wettability angle ϑ5 40�90�, it is γ5 0.1�0.49, β66.
Info: [A23],[B20],[B56].
Max Jakob (20.7.1879�1955), American physicist of
German origin.
After the year 1910, he was engaged in research on ther-
modynamics and heat transfer and carried out a great deal
of work in this area. His work concerned, especially, steam
and air properties under high pressure, devices to measure
thermal conductivity, explanations of the boiling mecha-
nism and condensation, and flow in pipelines and channels.
In this sphere, he published approximately 500 works.
219Thermomechanics
5.5.14 Kutateladze Evaporation Number Ku
Ku5llv
clΔTð1Þ;
Ku5llvRlwqA
5Ku�PeNu21 ð2Þ;
Ku5llvRlνlqAL
5Ku�PrNu21 ð3Þ;
Ku4 5 Ja211
RlRv
ð4Þ
llv (J kg21) � specific latent heat of vaporization; cl (J kg21 K21) � specific heat
capacity of liquid; ΔT5 Tv2 Ts (K) � difference of vapour and wall temperatures;
Rl, Rv (kg m23) � density of liquid and vapour; w (m s21) � liquid velocity;
qA (W m22) � heat flux density; vl (m2 s21) � kinematic viscosity of liquid; L (m) �
characteristic length; Ku� (�) � Kutateladze evaporation number defined by the rela-
tion (1); Pe (�) � Peclet heat number (p. 180); Nu (�) � Nusselt number (p. 196);
Pr (�) � Prandtl number (p. 197); Ja1 (�) � Jakob number (p. 219).
It expresses the ratio of the specific evaporation heat to the heat content increase
in the overheating of one of the phases. In form (2), it expresses the ratio of the
phase conversion rate to the phase flow rate. In form (3), it expresses the ratio of
flow inertia forces originating due to the phase conversion process to viscous
forces. It is a special expression of the Reynolds number Re (p. 81). In form (1), it
is also called the Trawton number Tr (p. 232), and in form (4) it is the Jakob
vapour number.
Info: [A23].
Samson Semenovich Kutateladze (p. 237).
5.5.15 Local Boiling Stability K
K5Rvw
2vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
σgðRl 2 RvÞp ð1Þ;
K5Rlw
2lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
σgðRl 2 RvÞp ð2Þ
Rv, Rl (kg m23) � density of vapour and liquid; wv, wl (m s21) � velocity of vapour
and liquid; σ (N m21) � surface tension; g (m s22) � gravitational acceleration.
It characterizes the local vapour�fluid set stability under dynamic and capillary�gravitational mutual action. Equation (1) expresses the approximate transition of the
non-viscous fluid bubble boiling to the film boiling, and equation (2) expresses the
220 Dimensionless Physical Quantities in Science and Engineering
effect of non-viscous fluid being driven out from the microporous surface in the
barbotage.
Info: [A23].
5.5.16 Lockhart�Martinelli Number (1.) LM1, Lp
LM1 5RvσLη2v
Rv (kg m23) � vapour density; σ (N m21) � surface tension; L (m) � characteristic
length; ηv (Pa s) � dynamic viscosity of vapour.
It expresses the ratio of the fluid surface stress force to the vapour viscosity
force. It characterizes the influence of hydrodynamic forces on the heat transfer in
film boiling, for example, of cryogen fluids in a pipeline.
Info: [A23].
5.5.17 Lockhart�Martinelli Number (2.) LM2, Xtt
LM2 512 x
x
� �0:9 RvRl
� �0:5 ηlηv
� �0:1
x (�) � relative mass flux of vapour fraction; Rv, Rl (kg m23) � density of vapour
and liquid; ηv, ηl (Pa s) � dynamic viscosity of vapour and liquid.
It characterizes the convective fluid boiling in vertical tubes and constant cross-
section canals. Together with the boiling number Boi (p. 215), it determines the
heat transfer. The formula
ααl
5C1 Boi1C2
1
Xtt
� �C3
( )C4
is valid, where C15 6.73 103; C25 3.53 1024; C35 0.67; C45 1; αl is the heat
transfer coefficient on the fluid side of a channel; α5αk1αv, where αk is the con-
vection part of heat transfer and αv is the part corresponding to the formation of
vapour nuclei.
Info: [A23].
5.5.18 Number of Steam Nucleus N
N5dp
dT
� �Tn
d2ðRl 2 RvÞqAλlσRl
221Thermomechanics
p (Pa) � pressure; T, Tn (K) � temperature and saturation temperature; d (m) �characteristic dimension of vapour bubble; Rl, Rv (kg m23) � density of liquid and
vapour; qA (W m22) � area heat flux density; λl (W m21 K21) � thermal conduc-
tivity of liquid; σ (N m21) � surface tension.
It characterizes the number of vapour nuclei originating in the vapour creation
in fluid.
Info: [A23].
5.5.19 Nusselt Boiling Number Nub
Nub 5αdλl
5αλl
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσ
gðRl 2 RvÞr
α (W m22 K21) � heat transfer coefficient; d (m) � characteristic dimension of
vapour bubble; λl (W m21 K21) � thermal conductivity of liquid; σ (N m21) �surface tension; g (m s22) � gravitational acceleration; Rl, Rv (kg m23) � density
of liquid and vapour.
This number characterizes the convective heat transfer in the heating and boiling
of fluids.
Info: [A23].
Ernst Kraft Wilhelm Nusselt (p. 196).
5.5.20 Off Liquid Heating Ψ
Ψ5clðTn 2 TlÞ
llv
cl (J kg21 K21) � specific heat capacity of liquid; Tn, Tl (K) � temperature of satu-
ration and liquid; llv (J kg21) � specific latent heat of vaporization.
It expresses the ratio of the specific heat required to finish the fluid heating to
the boiling point to the specific latent heat of evaporation. It characterizes the
degree of insufficient fluid heating in boiling. With increasing Ψ, the effect of heat
flow intensification is reduced.
Info: [A23].
5.5.21 Peclet Boiling Number Peb
Peb 5qAcl
llvλl
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσ
gðRl 2 RvÞr
qA (W m22) � area heat flux density; cl (J kg21 K21) � specific heat capacity of
liquid; λl (W m21 K21) � thermal conductivity of liquid; llv (J kg21) � specific
222 Dimensionless Physical Quantities in Science and Engineering
latent heat of vaporization; σ (N m21) � surface tension; g (m s22) � gravitational
acceleration; Rl, Rv (kg m23) � density of liquid and vapour.
It characterizes the heat transfer by convection and conduction in a fluid bound-
ary layer in boiling.
Info: [A23].
Jean Claude Eugene Peclet (p. 180).
5.5.22 Peclet Boiling Speed Number Pewb
Pewb �fLRvcld
λl
5wRvclλl
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσ
gðRl 2 RvÞr
ð1Þ;
Pewb � Sh21 5df
wt
ð2Þ
f (Hz) � frequency of rising bubbles; L (m) � characteristic length; Rv, Rl (kg m23) �
density of vapour and liquid; cl (J kg21 K21) � specific heat capacity of liquid;
λl (W m21 K21) � thermal conductivity of liquid; d (m) � characteristic dimension of
vapour bubble; w5 qA (llvRv)21 (m s21) � boiling velocity; qA (W m22) � heat flux
density; llv (J kg21) � specific latent heat of vaporization; σ (N m21) � surface ten-
sion; g (m s22) � gravitational acceleration; wt (m s21) � heat diffusion velocity;
Sh (�) � Strouhal number defined by the relation (3) (p. 87).
It expresses the ratio of the velocity of vapour bubbles originating in boiling to
the heat diffusion rate in the fluid. It characterizes the rate at which vapour bubbles
originate in boiling. In equation (2), it represents the inverse value of the Strouhal
number Sh defined by equation (3) (p. 87).
Info: [A23].
Jean Claude Eugene Peclet (see above).
5.5.23 Rayleigh Steam Number Ra
Ra5GrPr5gL3βlΔT
νlalð1Þ;
Ra5gL3critνvav
Rl 2 RvRv
ð2Þ
where Lcrit 5 2πσ
gðRl 2 RvÞ
� �12
g (m s22) � gravitational acceleration; L (m) � characteristic length; βl (K21) �
volume thermal expansion coefficient of liquid; ΔT (K) � temperature difference;
νl, νv (m2 s21) � kinematic viscosity of liquid and vapour; al, av (m
2 s21) � thermal
223Thermomechanics
diffusivity of liquid and vapour; Rl, Rv (kg m23) � density of liquid and vapour; σ(N m21) � surface tension; Lcrit (m) � critical wavelength; Gr (�) � Grashof heat
number (p. 185); Pr (�) � Prandtl number (p. 197).
It characterizes the heat transfer by free convection in boiling and thermal insta-
bility in fluid. In the basic form (1), it is valid for fluid heating, and in the modified
form (2), it holds for the boiling and resulting instability of a fluid. In the latter
case, the critical wavelength is taken as the characteristic length.
Info: [A23],[A29].
Lord Rayleigh, John William Strutt (p. 187).
5.5.24 Reynolds Boiling Number Reb
Reb 5qA
llvRvνl
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσ
gðRl 2 RvÞr
5NubPr21Ku21 Rl
Rvð1Þ;
Reb 5qAd
llvRvνlð2Þ
qA (W m22) � heat flux density; llv (J kg21) � specific latent heat of vaporization;
Rv, Rl (kg m23) � density of vapour and liquid; νl (m2 s21) � kinematic
viscosity of liquid; σ (N m21) � surface tension; g (m s22) � gravitational acceler-
ation; d (m) � characteristic tube dimension; Nub (�) � Nusselt boiling number
(p. 222); Pr (�) � Prandtl number (p. 197); Ku (�) � Kutateladze evaporation
number (p. 220).
It expresses the ratio of the inertia force arising in flowing fluid with phase con-
version to the internal friction force. It characterizes the heating and boiling of a
fluid. It is the hydrodynamic boiling criterion. In equation (2), it expresses the ratio
of the pipe surface heat loading to the specific latent evaporating heat. It is the
measure for pipe wall thermal loading in fluid boiling.
Info: [A33].
Osborne Reynolds (p. 82).
5.5.25 Reynolds Boiling Speed Number Rewb
Rewb 5d
νlRl
π6d3Rv fn
�
d (m) � characteristic dimension of vapour bubble; νl (m2 s21) � kinematic viscos-
ity of liquid; Rl, Rv (kg m23) � density of liquid and vapour; f (Hz) � generation
frequency of vapour bubbles; n (�) � number of vapour nuclei passing through
surface unit.
It characterizes the fluid boiling rate.
224 Dimensionless Physical Quantities in Science and Engineering
Info: [A23].
Osborne Reynolds (see above).
5.5.26 Reynolds Bubbling Evaporation Number Rebb
Rebb 5Rlwdηl
5wd
νl
Rl (kg m23) � liquid density; νl (m
2 s21) � kinematic viscosity of liquid; ηl (Pa s) �dynamic viscosity of liquid; w (m s21) � propagation velocity of vapour bubble;
d (m) � characteristic dimension of vapour bubble.
It expresses the ratio of the vapour bubble inertia force to the fluid friction force.
It characterizes the hydrodynamic relations in fluid bubble boiling.
Info: [A23].
Osborne Reynolds (see above).
5.5.27 Stanton Boiling Number Stb
Stb 5αllvqAcl
ð1Þ;
Stb 5qA
RvclwlðTS 2 TnÞð2Þ
α (W m22 K21) � heat transfer coefficient; llv (J kg21) � specific latent heat of
vaporization; qA (W m22) � surface density of heat flux; cl (J kg21 K21) � specific
heat capacity of liquid; Rv (kg m23) � vapour density; wl (m s21) � liquid motion
velocity; TS, Tn (K) � wall and saturation temperatures.
This number characterizes the convective heat transfer in fluid boiling. In equa-
tion (2), it is valid for the shooting process4 in film boiling.
Info: [A23],[A29].
Thomas Edward Stanton (p. 201).
5.5.28 Steam Pressure Np
Np 5p
σ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσ
gðRl 2 RvÞr
p (Pa) � pressure; σ (N m21) � surface tension; g (m s22) � gravitational accelera-
tion; Rl, Rv (kg m23) � density of liquid and vapour.
It characterizes the pressure in heat transfer, for example, in closed two-phase
thermosiphon systems in which the boiling and condensation processes proceed
under the action of gravitation and inertia forces.
Info: [A23],[A29].
225Thermomechanics
5.5.29 Steam Superheating Number Ψ
Ψ5cpvðTv 2 TnÞ
llv
cpv (J kg21 K21) � specific heat capacity at constant pressure; Tv (K) � vapour
temperature; Tn (K) � saturation temperature; llv (J kg21) � specific latent heat of
vaporization.
It expresses the superheated vapour specific heat to the specific latent heat of
evaporation. With Ψ-0, the heat flow from a wall causes vapour generation and
superheating.
Info: [A23].
5.5.30 Sterman Number Sn
Sn5llv
cpvTn
llv (J kg21) � specific latent heat of vaporization; cpv (J kg21 K21) � specific heat
capacity of saturated vapour; Tn (K) � temperature of saturated vapour.
It expresses the ratio of the specific evaporation heat to the specific heat of satu-
rated steam. It characterizes the developed bubble boiling.
Info: [A23].
5.5.31 Tolubinsky Number To
See the Jakob evaporation speed number Ja (p. 218).
5.5.32 Wallis Number Ws
Ws51
η½d3gðRl 2 RvÞRl�
12
η (Pa s) � dynamic viscosity of liquid; d (m) � tube diameter; g (m s22) � gravita-
tional acceleration; Rl, Rv (kg m23) � density of liquid and vapour or gas.
It characterizes the fluid viscosity influence on stroke flow in boiling.
Info: [A23].
5.5.33 Weber Boiling Number Web
Web 5Rvðw2
v 2w2l ÞL
σ
226 Dimensionless Physical Quantities in Science and Engineering
Rv (kg m23) � vapour density; wv, wl (m s21) � flow velocity of vapour and liquid;
L (m) � characteristic length dimension (wall thickness, boundary layer thickness,
drop or bubble diameter); σ (N m21) � surface tension.
It expresses the inertia-to-capillary forces ratio. It characterizes the hydrody-
namic forces influence on the process of bubbles originating in fluid boiling.
Info: [A23].
Ernst Heinrich Weber (p. 92).
Wilhelm Eduard Weber.
5.6 Evaporation
Evaporation is the phase conversion in which a fluid is converted to vapour on its
surface. The evaporation occurs under an arbitrary fluid temperature. The higher
the temperature is, the faster the evaporation. Further, the evaporation is influenced
by the fluid’s properties, by gas pressure and motion over the fluid and by the
evaporating area size. In addition to several evaporation numbers, some of the fun-
damental dimensionless quantities are the Borishanskyi, Bulygin, Gukhman, Jakob
evaporation, Lomonosov, Richman, Spalding and Trawton numbers.
5.6.1 Borishansky Number Bs
Bs5pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
σðRl 2 RvÞgp
p (Pa) � pressure; σ (N m21) � surface tension; Rl, Rv (kg m23) � density of liquid
and vapour; g (m s22) � gravitational acceleration.
This number expresses the ratio of the absolute pressure in the system to the
pressure gradient on the interface of phases. It characterizes the pressure conditions
in fluid evaporation.
Info: [A23].
Veniamin Mironovich Borishansky (born 1917), Russian engineer.
5.6.2 Bulygin Number Bu
Bu5llvcmpp
clðT 2 TiÞ
llv (J kg21) � specific latent heat of evaporation; cmp (Pa
21) � pressure mass capac-
ity of vapour; p (Pa) � pressure; cl (J kg21 K21) � specific heat capacity of liquid;
T, Ti (K) � material and initial temperatures.
It expresses the ratio of the evaporation heat to the fluid heat (the heat necessary
to bring the fluid to boiling). It characterizes the evaporation heat transfer and the
227Thermomechanics
body accumulating ability in high-intensity heat transfer. The influence of the heat
which is lost during evaporation in molecular transfer is considered.
Info: [A23],[A29].
N.P. Bulygin.
5.6.3 Drag Evaporation Coefficient CD,ev
CD;ev 5FD
12Rcw2
FD (N) � drag force acting on particle; Rc (kg m23) � density of cold liquid;
w (m s21) � velocity difference between particle and cold liquid.
It expresses the resistance of the vapour layer arising on the surface of a hot par-
ticle during its penetration into cold fluid. It is analogous to the Dalton number Dal
(p. 394). Evaporation.
Info: [B15].
5.6.4 Euler Modified Number Eumod
Eumod 5HLRlgw2GRg
HL (m) � pressure height of liquid; Rl (kg m23) � liquid density; Rg (kg m23) �vapour density; g (m s22) � gravitational acceleration; WG (m s21) � vaporization
velocity from free surface.
It expresses the vapour flow across a mass transfer.
Info: [A29].
Leonhard Euler (p. 61).
5.6.5 Evaporation Flow Number Ev
Ev5wevDp
η
wev (kg m22 s21) � vaporization velocity; Dp (m) � particle diameter; η (Pa s) �dynamic viscosity.
It expresses the dynamic-to-viscous forces ratio. It characterizes the vapour flow
inside a vapour film, with the vapour being induced during cold fluid evaporation
at the interface between the cold fluid and the vapour. Evaporation.
Info: [A23].
228 Dimensionless Physical Quantities in Science and Engineering
5.6.6 Evaporation Number (1.) Nev
Nev 5w2
llv
w (m s21) � flow velocity; llv (J kg21) � specific latent heat of vaporization.
It characterizes the evaporation process. It expresses the ratio of the flow kinetic
energy to the fluid latent heat.
Info: [A23].
5.6.7 Evaporation Number (2.) Nev
Nev 5cl
βllv5N�
evGcDu21
cl (J kg21 K21) � specific heat capacity of liquid; β (K21) � volume thermal
expansion coefficient; llv (J kg21) � specific latent heat of evaporation; N�
ev (�) �evaporation number (1.) (p. 229); Gc (�) � Gay-Lussac number (p. 14); Du (�) �Dulong number (p. 191).
It characterizes the evaporation process. It expresses the ratio of the heat neces-
sary to heat the fluid to the evaporation temperature to the specific latent heat of
evaporation.
Info: [A23].
5.6.8 Evaporation Number (3.) Elasticity Nev
Nev 5K
llvRl5
a2
llv5N�
evCau
K (Pa) � bulk modulus; llv (J kg21) � specific latent heat of vaporization;
Rl (kg m23) � liquid density; a (m s21) � sound velocity; N�ev (�) � evaporation
number (1.) (p. 229); Cau (�) � Cauchy number (aeroelasticity parameter)
(p. 155).
It characterizes the fluid evaporation process. It expresses the ratio of the energy
of forces acting with fluid evaporation to the specific latent evaporation heat. It is
called the elasticity evaporation number as well.
Info: [A23].
5.6.9 Gravitational Evaporation Number Ngv
Ngv 5D3
pR2cg
η2c
Dp (m) � particle diameter; Rc (kg m23) � density of cooling agent; g (m s22) �gravitational acceleration; ηc (Pa s) � dynamic viscosity of cooling agent.
229Thermomechanics
It describes the particles in a cooling system under the influence of gravitation.
It expresses the gravity-to-viscosity forces ratio. Together with the evaporation
flow number Ev (p. 228) and the Reynolds number Re (p. 81), it is among the fun-
damental criteria influencing the force acting on a cooled particle under laminar
flow conditions during film evaporation.6
5.6.10 Gukhman Evaporation Number Gu
Gu5tg 2 ts
Tg
tg, ts (K) � hot gas temperature, moist material surface (liquid level) temperature;
Tg (K) � thermodynamic gas temperature.
With evaporation from a free fluid surface, it expresses the ratio of the outer
mass transfer intensity (moisture), expressed by the volume fluid evaporation inten-
sity (moisture) in the air boundary layer, to the outer heat transfer intensity. It is
the thermodynamic criterion of evaporation in isobaric, adiabatic conditions. It is
used to describe the evaporation from a fluid-free surface or moist material drying.
With non-adiabatic evaporation, an additional criterion Θ5 T/Tl must be intro-
duced to express the thermal conditions’ influence if the fluid surface temperature
Tl differs from the adiabatic air saturation temperature close to the moist body sur-
face. In moist material drying, it is called the Gukhman drying number Gu (p. 251).
Info: [A4],[A23].
Alexander Adolfovich Gukhman, Russian chemical engineer.
5.6.11 Jakob Evaporation Number Jak
Jak5llv
cpΔT
llv (J kg21) � specific latent heat of vaporization; cp (J kg21 K21) � specific heat
capacity; ΔT (K) � temperature difference.
It expresses the ratio of the specific latent heat of evaporation to the fluid heat
capacity. It is analogous to the number of phase changes N (p. 179).
Info: [B20].
Max Jakob (p. 219).
5.6.12 Leidenfrost Number Lei
Lei5Bi
Pe2
Bi (�) � Biot number (p. 173); Pe (�) � Peclet heat number (p. 180).
230 Dimensionless Physical Quantities in Science and Engineering
It characterizes the heat transfer by evaporation with hot wall cooling by a thin
fluid film flowing over its surface. The hardening of bodies in fluids and other
technological processes are examples. The Leidenfrost phenomenon.
Info: [B30].
Johann Gottlob Leidenfrost (27.11.1715�2.12.1794),
German physicist and chemist.
Above all, his basic contribution consisted in the discov-
ery of the Leidenfrost phenomenon. With this, in the vicin-
ity of a hot body being much hotter than the boiling
temperature, which is the so-called Leidenfrost point, the
fluid forms a thin lagging steam layer. This steam layer
forms a thermal barrier which prevents the fluid from get-
ting into fast boiling. Therefore, it takes much longer for
the fluid to evaporate. With the temperature under the
Leidenfrost point, this thermal barrier does not exist, due to
which the fluid evaporates swiftly.
5.6.13 Lomonosov Number Lo
Lo5gΔRLRw2
5ArRe22
g (m s22) � gravitational acceleration; ΔR (kg m23) � vapour and air density dif-
ference; L (m) � characteristic length; R (kg m23) � air density; w (m s21) � air-
flow velocity; Ar (�) � Archimedes hydrodynamic number (p. 53); Re (�) �Reynolds number (p. 81).
It expresses the ratio of the buoyancy force (Archimedes force) to the inertia
force. It acts in processes of non-isothermal water evaporation from surfaces of
vessels and tanks into flowing air.
Info: [A23],[A33].
Mikhail Vasilyevich Lomonosov (19.11.1711�15.4.1765),
Russian scientist, encyclopedist, writer and poet.
He was a founder of both Russian science and culture.
He studied physics, chemistry, metallurgy, history and arts,
and he was interested in other fields as well. By creating
the mechanical heat theory, expressing and proving the law
of mass conservation, he laid the first foundations for gen-
eral heat theory. He showed that heat is a form of move-
ment of particles. He wrote a whole range of works,
including the first history of Russia.
231Thermomechanics
5.6.14 Richman Number Ri
Ri5qmL
Rν5
wL
ν
qm (kg s21 m22) � mass flow density; L (m) � characteristic length; R (kg m23)
liquid density; ν (m2 s21) � kinematic viscosity; w (m s21) � flow velocity.
It expresses the ratio of inertia forces to viscous forces which act in condensate
evaporation. It characterizes the influence of hydrodynamic conditions on the evap-
oration process. Sometimes, it is called the transversal flow criterion for parallel
running heat and mass transfers during evaporation or vapour condensation in a
vapour�gas mixture. It can be called the Reynolds mass number Rem (p. 262).
Info: [A33].
Georg Vilhelm Richman (1711�1753), Russian physicist.
5.6.15 Spalding Number (2.) Sp, B’
Sp5 cpΔT llv 2ΦQm
� �21
cp (J kg21 K21) � specific heat capacity of surroundings; ΔT (K) � surroundings
and liquid level temperature difference; llv (J kg21) � specific latent heat of vapori-
zation; Φ (W) � radiation heat flux; Qm (kg s21) � mass flux.
This number expresses the ratio of the received heat to the latent heat of an
evaporating material. It characterizes the heat transfer in the evaporation of fluid
drops.
Info: [A23].
Brian Spalding (born 1923), English engineer (p. 200).
5.6.16 Trawton Number Tr
Tr � Ku5llv
ΔTcp
llv (J kg21) � specific latent heat of vaporization; ΔT (K) � temperature differ-
ence; cp (J kg21 K21) � specific heat capacity; Ku (�) � Kutateladze evaporation
number defined by equation (1) (p. 220).
It expresses the ratio of the specific latent evaporation heat to that delivered to
heat a fluid volume unit. It characterizes the fluid evaporation process with
the temperature corresponding to normal conditions. It is a special case of the
Kutateladze evaporation number Ku defined by relation (1) (p. 220).
Info: [A23].
Frederic Thomas Trawton (1863�1922), Irish physicist.
232 Dimensionless Physical Quantities in Science and Engineering
5.7 Condensation
For condensation, several dimensionless quantities are used, to represent the gas
or vapour transformation of the fluid. For example, there is a set of quantitative
condensation criteria expressing energetic, force and other influences which
are connected with condensation. Other dimensionless quantities are the following:
the Archimedes, Galilei, Kapica, Grigul, Kirbride, Kutateladze, Laplace,
McAdams, Nusselt, Prandtl, Reynolds, Sherwood, Stanton and Weber numbers. In
addition, some dimensionless quantities are used for such things as various binary
mixtures and thermocapillary processes.
5.7.1 Archimedes Condensation Number Ark
Ark 5gL3
ν2l12
RvRl
� �5Gak 12
RvRl
� �
g (m s22) � gravitational acceleration; L (m) � characteristic length; νl (m2 s21) �
kinematic viscosity of liquid; Rv, Rl (kg m21) � density of vapour and liquid; Gak(�) � Galilei condensation number (p. 235).
This number expresses the ratio of the product of the static buoyancy, arising
from the difference of densities, and the inertia force to the square of the friction
force. It characterizes the influence of the free two-phase vapour�condensate envi-
ronment on the heat transfer during condensation.
Info: [A23].
Archimedes of Syracuse (p. 54).
5.7.2 Condensation Number (2.) K
K5L3R2glvlληΔT
L (m) � characteristic length; R (kg m23) � condensation liquid density; g (m s22) �gravitational acceleration; lvl (J kg21) � specific latent heat of condensation;
λ (W m21 K21) � thermal conductivity of condensation liquid; η (Pa s) � dynamic
viscosity of condensation liquid; ΔT (K) � temperature difference during
condensation.
It characterizes the condensation process on a vertical wall.
Info: [A23].
5.7.3 Condensation Number of Binary Steam Mixtures Kb
Kb 5 2σ@σ@cm
ðlvlη2l Þ21
233Thermomechanics
σ (N m21) � surface tension; cm (�) � mass fraction; lvl (J kg21) � specific latent
heat of condensation; ηl (Pa s) � dynamic viscosity of liquid.
It characterizes the influence of a vapour binary mixture on drop condensation.
The surface stress is presumed to be determined by the condensation liquid first.
Info: [A23].
5.7.4 Condensation Number of Flowing Steam Kw
Kw 52w2
v
ðgνlÞ23
RvRl
wv (m s21) � vapour velocity; g (m s22) � gravitational acceleration; νl (m2 s21) �
kinematic viscosity of liquid; Rl, Rv (kg m23) � density of liquid and vapour.
It characterizes the influence of clean vapour velocity on the heat transfer during
drop condensation on a vertical surface. With it, the mutual vapour friction force
on the condensate film surface, the gravity force and the molecular film friction are
considered.
Info: [A23].
5.7.5 Condensation Number of Thermocapillary Movement Kk
Kk 5βσΔTcritσRcrit
Rlν2l5βσΔTcritLak
where βσ 51
σ@σ@T
; Rcrit 52σTn
RllvlΔTcrit
βσ (K21) � temperature coefficient of surface tension; ΔTcrit (K) � critical temper-
ature difference; σ (N m21) � liquid surface tension; Rcrit (m) � critical radius
of condensation nucleus; Rl (kg m23) � liquid density; νl (m2 s21) � kinematic
viscosity of liquid; T (K) � temperature; Tn (K) � saturation temperature; lvl(J kg21) � specific latent heat of condensation; Lak (�) � Laplace condensation
number (p. 238).
It expresses the thermocapillary-to-viscosity forces ratio. It characterizes the
influence of thermocapillary motion in immovable, clean vapour condensation.
Info: [A23].
5.7.6 Condensation Number of Thermocapillary Separation Kδ
Kδ 5βpΔTE
Rlν2l Rcrit
; where Rcrit 52σTn
RllvlΔTcrit
234 Dimensionless Physical Quantities in Science and Engineering
βp (K21) � thermal coefficient of displaced pressure; ΔT (K) � temperature differ-
ence between saturation temperature and surface wall temperature; E (J) � pressure
energy of separation; Rl (kg m23) � liquid density; νl (m2 s21) � kinematic viscos-
ity of liquid; σ (N m21) � surface tension; Tn (K) � saturation temperature;
lvl (J kg21) � specific latent heat of condensation; ΔTcrit (K) � critical temperature
difference.
It expresses the ratio of mutually acting forces between condensate molecules
and those of molecularly active surface material on a condensation surface. It char-
acterizes the influence of the wall surface material on the drop condensation of
immobile clean vapour.
Info: [A23].
5.7.7 Degree of Vapour Condensation ε
ε5Qm1 2Qm2
Qm1
Qm1, Qm2 (kg s21) � vapour mass fluxes on input and output of vapour condenser.
It expresses the ratio of the vapour mass flow on a condenser inlet to that on its
outlet. It characterizes the condensation efficiency of the condenser.
Info: [A23].
5.7.8 Galilei Condensation Number Gak
Gak 5gL3
ν2lð1Þ;
Gak 5gx3
ν2lð2Þ
g (m s22) � gravitational acceleration; L (m) � characteristic length (wall height);
νl (m2 s21) � kinematic viscosity of liquid; x (m) � distance in direction of con-
densation liquid layer generation.
It expresses the ratio of the molecular friction force during condensation to the
gravitation force. It characterizes the influence of friction and gravitation on the
heat transfer in condensation. Expression (1) holds for developed condensation, and
expression (2) is valid for a developing condensation layer.
Info: [A23].
Galileo Galilei (p. 123).
235Thermomechanics
5.7.9 Grigull Condensation Number Ggk, Z
Ggk 5g
ν2l
� �13 λlΔTL
lvlηl5Ga
13k
λlΔT
lvlηl
g (m s22) � gravitational acceleration; νl (m2 s21) � kinematic viscosity of conden-
sate liquid; λl (W m21 K21) � thermal conductivity of condensate liquid
ΔT5 Tn�TS (K) � temperature difference during condensation; Tn (K) � saturation
temperature; TS (K) � condensation wall surface temperature; L (m) � characteristic
length; lvl (J kg21) � specific latent heat of condensation; ηl (Pa s) � dynamic
viscosity of condensation liquid; Gak (�) � Galilei condensation number (p. 235).
It characterizes the condensate film flow. It represents the relative effective
length of the condensate flow-down surface. The condensate film flow is laminar
in horizontal tubes with Ggk, 3900 and in vertical ones with Ggk, 2300. A turbu-
lent flow occurs with Ggk. 3900 in horizontal tubes and with Ggk. 2300 in the
case of vertical tubes.
Info: [A23].
Ulrich Grigull (p. 186).
5.7.10 Kapica Number Ka
Ka5σ3
gR3l ν4l
σ (N m21) � surface tension; g (m s22) � gravitational acceleration; Rl (kg m23) �liquid density; νl (m
2 s21) � kinematic viscosity of liquid.
This number expresses the ratio of the surface stress force to the product of the
volume force and the viscosity force. It characterizes fluid film thermal conductiv-
ity increase due to wave periodic flow on a vertical wall.
Info: [A23].
Peter Leonidovich Kapica (8.7.1894�8.4.1984), Russian
physicist. Nobel Prize in Physics, 1978.
He was engaged in research on magnetism under low
temperatures and helium superfluidity problems. He
designed several unique devices during his stay in England,
where he collaborated with Ernst Rutherford, and in the for-
mer Soviet Union as well. His research on microwave gen-
erators led him to the study of controlled nuclear reaction
starting in the 1960s.
236 Dimensionless Physical Quantities in Science and Engineering
5.7.11 Kirbride Condensation Number Kb
Kb5αλl
ν2lg
� �13
5NuGa213
k
α (W m22 K21) � heat transfer coefficient; λl (W m21 K21) � thermal conductivity
of condensation liquid; νl (m2 s21) � kinematic viscosity of condensation liquid;
g (m s22) � gravitational acceleration; Nu (�) � Nusselt number (p. 196); Gak (�)
� Galilei condensation number (p. 235).
It characterizes the heat transfer in membrane clean vapour condensation and
turbulent condensate flow. It is often called the condensation number (1.).
Info: [A23].
5.7.12 Kutateladze Condensation Number Ku
Kup 5lvl
cpvðTv 2 TlÞð1Þ;
Kuk 5lvl
cplðTl 2 TSÞð2Þ
lvl (J kg21) � specific latent heat of condensation; cpv, cpl (J kg
21 K21) � specific
heat capacity of vapour and condensation liquid; Tv (K) � vapour temperature;
Tl (K) � mean temperature of condensation liquid; TS (K) � external surface wall
temperature.
This number expresses the ratio of the condensation specific latent heat to the
specific heat cooling of one phase. In equation (1), it characterizes the vapour con-
densation process, and in equation (2), it characterizes the condensate subcooling.
Therefore, it is called the vapour (1) or condensate (2) Kutateladze number.
Info: [A23].
Samson Semenovich Kutateladze (18.7.1914�20.3.1986),
Russian engineer and physicist.
In the year 1936, he formulated the basic similarity condi-
tions for heat-exchange processes with the change of state
phase of material. His experimental work with the heat trans-
fer in liquid metals in nuclear engineering is extensive.
He formulated the basic ideas of the hydrodynamic theory of
the boiling crisis. His books Osnovy teorii teploobmena
(Fundamentals of Heat Exchange Theory, 1979) and Analiz
podobija v teplofizike (Similarity Analysis in Thermophysics,
1982) are significant in the evolution of the field.
237Thermomechanics
5.7.13 Laplace Condensation Number Lak
Lak 5ΔpL
σð1Þ;
Lak 5RlσLη2l
ð2Þ
Δp (Pa) � pressure difference across condensation liquid layer; L (m) � thickness
of condensation layer; σ (N m21) � surface tension; Rl (kg m23) � density of
condensation liquid; ηl (Pa s) � dynamic viscosity of condensation liquid.
It expresses the ratio of the pressure force to the surface stress force. It charac-
terizes the force relations in the membrane or drop condensation of vapour.
Info: [A23].
Pierre-Simon Laplace (p. 71).
5.7.14 McAdams Number Mc
Mc5α4LηlΔT
λ3l R
2l glvl
α (W m22 K21) � heat transfer coefficient; L (m) � characteristic length; ηl (Pa s) �dynamic viscosity of condensation liquid layer; λl (W m21 K21) � thermal conduc-
tivity of condensation liquid layer; Rl (kg m23) � density of condensation liquid;
ΔT (K) � mean temperature difference between vapour and wall g (m s22) �gravitational acceleration; lvl (J kg
21) � specific latent heat of condensation.
It characterizes the heat transfer in fluid membrane condensation. It is constant
for a specific surface orientation.
Info: [A23].
William Henry McAdams (15.3.1892�2.5.1975), American
chemical engineer.
He was engaged in the heat transfer, distillation and flow-
ing of viscous fluids. His publication Heat Transmission is
widely known. In addition, he edited many of technical
papers. His work on the Manhattan project and his original
work on nuclear submarines and jet aircraft are well known.
5.7.15 Middle Mass Temperature Θ
Θ5T 2 T1
Tn 2 T1
238 Dimensionless Physical Quantities in Science and Engineering
T (K) � mean temperature in condenser; T1 (K) � temperature in input in con-
denser; Tn (K) � temperature of saturated vapour.
It characterizes the mean mass temperature of the dispersed water flow in a con-
denser (water and water vapour).
Info: [A23].
5.7.16 Nusselt Condensation Number (1.) Nukl
Nukl 5L3R2l glvlλlηlΔT
5PrKuGak ð1Þ;
Nukl 5qAL
λlΔTð2Þ
L (m) � characteristic length; Rl (kg m23) � condensation liquid density;
λl (W m21 K21) � thermal conductivity of condensation liquid; ηl (Pa s) �dynamic viscosity of condensation liquid; g (m s22) � gravitation acceleration;
lvl (J kg21) � specific latent heat of condensation; ΔT (K) � temperature difference
during condensation; qA (W m22) � surface heat flux density; Pr (�) � Prandtl
number (p. 197); Ku (�) � Kutateladze evaporation number (p. 220); Gak (�) �Galilei condensation number (p. 235).
It characterizes the influence of molecular friction and gravity acting on the heat
transfer in membrane condensation.
Info: [A23].
Ernst Kraft Wilhelm Nusselt (p. 196).
5.7.17 Nusselt Condensation Number (2.) Nuk2
Nuk2 5λl
αR2l gη2l
� �13
5R2l gη2l
� �13
5Ga13k
λl (W m21 K21) � thermal conductivity of condensation liquid; Rl (kg m23) � den-
sity of condensation liquid; ηl (Pa s) � dynamic viscosity of condensation liquid;
g (m s22) � gravitational acceleration; α (W m22 K21) � heat transfer coefficient;
δ (m) � thickness of condensation liquid layer; Gak (�) � Galilei condensation
number (p. 235).
It characterizes the heat transfer in membrane condensation on a vertical wall
under the mutual influence of friction force, gravity and molecular friction in
flowing-down condensate film.
Info: [A23].
Ernst Kraft Wilhelm Nusselt (see above).
239Thermomechanics
5.7.18 Phase Transformation Number N
N5lvl
clðTn 2 T1Þlvl (J kg
21) � specific latent heat of condensation; cl (J kg21 K21) � specific heat
capacity of condensation liquid; Tn (K) � saturation temperature of vapour; T1 (K) �steam condenser input temperature.
It expresses the ratio of the specific latent condensation heat to that withdrawn
from the saturated vapour by a unit quantity of the cooling water. It is among the
set of quantitative condensation criteria F, G, J, K, L, M, R (p. 240).
Info: [A23].
5.7.19 Prandtl Condensation Number Prk
Prk 5νlal
5PeRe21
νl (m2 s21) � kinematic viscosity of condensation liquid; al (m
2 s21) � thermal dif-
fusivity of condensation liquid; Pe (�) � Peclet heat number (p. 180); Re (�) �Reynolds number (p. 81).
It characterizes the relation between the temperature and velocity fields in con-
densation or, alternatively, between the thickness of the temperature and hydrody-
namic boundary layers in gradientless bypassing of solid bodies. In liquid metals, it
is often Prk{1, which can be of use both in drop condensation (mercury) and
membrane condensation (sodium, potassium).
Info: [A23].
Ludwig Prandtl (p. 197).
5.7.20 Proportion of Vapour Content x
x5Qmv
Qmv 1Qml
Qmv, Qml (kg s21) � vapour and condensation liquid mass fluxes.
It expresses the ratio of the vapour mass flow to the mass flow of a
vapour�condensate mixture. It characterizes the relative vapour content in the mix-
ture, for example, at the inlet and outlet of a condenser tube.
Info: [A23].
5.7.21 Quantitative Condensation Criteria F, G, J, K, L, M, R
F5gdηllvl
w2NλlΔT
5Fr21PrN ð1Þ;
240 Dimensionless Physical Quantities in Science and Engineering
G5ΔTλl
ηllvl
ffiffiffiffiffiffiffiffiffiRlηlRvηv
r5Pr21N 21M5RM ð2Þ;
J5glvl
qA
ffiffiffiffiffiffiffiffiffiffiRlηldw3N
s5Re
12ðFrLÞ21 ð3Þ;
K5qA
lvl
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid
wNRvηv
s5MLRe2
12 ð4Þ;
L5qAd
ηllvlð5Þ;
M5
ffiffiffiffiffiffiffiffiffiRlηlRvηv
rð6Þ;
R5Pr21N 21 ð7Þ
g (m s22) � gravitation acceleration; d (m) � outer tube diameter; ηl, ηv (Pa s) �dynamic viscosity of condensation liquid and vapour; lvl (J kg
21) � specific latent
heat of condensation; wN (m s21) � velocity of vapour free flow; λl, λv(W m21 K21) � thermal conductivity of condensation liquid and vapour; ΔT (K) �temperature difference; Rl, Rv (kg m23) � density of condensation liquid and
vapour; qA (W m22) � surface density of heat flux; Fr (�) � Froude number (1.)
(p. 62); Pr (�) � Prandtl number (p. 197); N (�) � phase change (p. 180); Re (�)
� Reynolds number (p. 81).
These criteria characterize the energetic, force and other quantitative relations
under certain conditions, which are, for example, constant density of the through-
wall heat flow (criteria J, K, L), constant wall temperature (criteria F, G) and physi-
cal parameters of phases (M). The phase change N (p. 180) is also among the quan-
titative criteria.
Info: [A23].
5.7.22 Reynolds Number of Film Condensation Rebk
Rebk 5αΔTL
lvlηl11ϕ
clΔT
lvl
� �21ð1Þ;
Rebk 5Qm
2πrηlð2Þ
α (W m22 K21) � heat transfer coefficient; ΔT (K) � temperature difference dur-
ing condensation; L (m) � characteristic length; lvl (J kg21) � specific latent
heat of condensation; ηl (Pa s) � dynamic viscosity of condensation liquid;
241Thermomechanics
cl (J kg21 K21) � specific heat capacity of condensation liquid; ϕ (�) � relative
subcooling of condensation liquid; Qm (kg s21) � mass flux in condensation layer;
r (m) � inner tube radius.
It expresses the ratio of the condensate film inertia force to the molecular fric-
tion in a condensate film. It characterizes the hydrodynamic force relations in a
thin condensate film. In equation (2), it is used with partial condensation in tubes.
Info: [A23].
Osborne Reynolds (p. 82).
5.7.23 Reynolds Number of Two-Phase Condensation Rek
Rek 5wNL
νv5
λlðTn 2 TSÞlvlηl
ð1Þ;
Rep 5wvx
νvð2Þ
wN (m s21) � vapour free flow velocity; L (m) � characteristic length; νv (m2 s21) �
kinematic viscosity of vapour; λl (W m21 K21) � thermal conductivity of condensa-
tion liquid; ηl (Pa s) � dynamic viscosity of condensation liquid; Tn (K) � temperature
of saturated vapour; TS (K) � temperature of wall surface; lvl (J kg21) � specific latent
heat of condensation; wv (m s21) � local vapour velocity; x (m) � distance in direction
of condensation liquid layer generation.
It expresses the ratio of the flowing fluid inertia force to the molecular flowing
fluid friction force. It characterizes the hydrodynamic force relations in vapour con-
densation. In expression (1), it represents a resultant criterion, and in expression
(2), it represents a local criterion.
Info: [A23].
Osborne Reynolds (see above).
5.7.24 Sherwood Condensation Number Shk
Shk 5L3gRllvl4νlλlΔT
� �14
L (m) � characteristic length; g (m s22) � gravitational acceleration; Rl (kg m23) �density of condensation liquid; νl (m
2 s21) � kinematic viscosity of condensation
liquid; λl (W m21 K21) � thermal conductivity of condensation liquid; lvl (J kg21) �
specific latent heat of condensation; ΔT (K) � temperature difference between
vapour and wall.
It characterizes the influence of molecular friction and gravity on the heat trans-
fer in vapour membrane condensation on a vertical wall.
Info: [A23].
Thomas Kilgore Sherwood (p. 265).
242 Dimensionless Physical Quantities in Science and Engineering
5.7.25 Stanton Condensation Number Stk
Stk 5α
Rlclw5NuPe21 ð1Þ;
Stk 5αlvlqAcl
ð2Þ
α (W m22 K21) � heat transfer coefficient; Rl (kg m23) � density of condensation
layer; cl (J kg21 K21) � specific heat capacity of condensation layer; w (m s21) �
velocity growth of condensation film; lvl (J kg21) � specific latent heat of condensa-
tion; qA (W m22) � surface heat flux density; Nu (�) � Nusselt number (p. 196); Pe
(�) � Peclet heat number (p. 180).
It expresses the ratio of the condensate temperature change along a wall to the
temperature drop between the wall and the condensate. It characterizes the convec-
tive heat transfer process in drop and membrane condensations. Its significance is
similar to that of the Nusselt number Nu (p. 196).
Info: [A23].
Thomas Edward Stanton (p. 201).
5.7.26 Weber Condensation Number Wek
Wek 5ðRl 2 RvÞgL2
σð1Þ;
Wek 5RlgL
2
σð2Þ
Rl, Rv (kg m23) � density of liquid and vapour; g (m s22) � gravitational accelera-
tion; L (m) � characteristic length; σ (N m21) � surface tension.
It expresses the ratio of the gravity force to the surface stress force. It charac-
terizes the force relations in water condensation which appear, for example, in the
formation of the arising condensate drop. Sometimes, it is used in form (2), in the
description of the surface distillation process.
Info: [A23].
Ernst Heinrich Weber (p. 92).
Wilhelm Eduard Weber.
5.8 Heat and Mass Transfer
Compound heat and mass transfer is characterized by a mathematical model in the
expression of second-order partial equations set with composed boundary condi-
tions. A great number of dimensional quantities are used to describe heat and mass
243Thermomechanics
transfer. Among the most important are the Bulygin, Fedorov, Fourier, Kirpichev,
Kossovitch, Lykov and Posnov numbers, and also many mostly modified, dimen-
sionless quantities for heat or mass transfer, such as the Sherwood, Schmidt and
Stanton numbers.
5.8.1 Absorption Number Ab
Ab5 βffiffiffiffiffiffiffix
Dw
rð1Þ;
Ab5 βffiffiffiffiffiffiffiffiffixL
Dw1
rð2Þ
β (m s21) � absorption coefficient of liquid transfer; x (m) � surface length with
liquid layer; D (m2 s21) � gas diffusivity in liquid layer; w (m s21) � mean liquid
velocity in the layer; L (m) � layer thickness; w1 (m2 s21) � volume velocity.
In the dimensionless form, it expresses the mass transfer coefficient, for exam-
ple, in gas absorption on wet walls of chambers. In expression (2), the fluid film
thickness is considered.
Info: [A23].
5.8.2 Biot Mass Number Bim
Bim 5αmL
λm
5βLam
αm (kg m22 s21) � specific mass transfer; L (m) � characteristic length;
λm (kg m21 K21) � specific mass transfer; β (m s21) � mass transfer coefficient;
am (m2 s21) � mass diffusivity.
This number expresses the ratio of the mass transfer on a fluid�body interface
to the mass transfer inside the L-thick wall of the body. It characterizes the relation
between the mass transfer on the outer surface of the body by convection and that
by conduction inside the body. The third-type boundary condition is used in solving
heat transfer tasks, for example, in the drying or wetting of various materials.
Info: [A23].
Jean-Baptiste Biot (p. 174).
5.8.3 Bodenstein number Bd
Bd � Pem 5w L
D
w (m s21); L (m) � characteristic axial length; D (m2 s21) � effective axial diffu-
sivity; Pem (�) � Peclet mass number (p. 258).
244 Dimensionless Physical Quantities in Science and Engineering
It expresses the ratio of the convective heat transfer rate to the diffusion rate. It
characterizes the diffusion process in chemical reactors. In substance, it is about
expressing the degree of the approximation of a system being examined to ideal-
ized models of reacting flows. With Bd-N, it is about total displacement of a
system and with Bd-0, it is about total mixing. For real diffusion and combined
models, N,Bd. 0 is valid. It is a special case of the Peclet mass number Pem(p. 258) for mass transfer. It describes the diffusion in granulated material layers in
forced convection.
Info: [A23],[A33],[B20].
Max Ernst August Bodenstein (15.7.1871�3.9.1942),
German physical chemist.
In the year 1916, he explained the velocity of certain
chemical reactions with the chain reaction concept. He was
convinced that in those cases the process starts by random
collision of two molecules of different chemicals. This col-
lision generates heat, causing another couple of molecules
to react and generate further heat, and so on. In later years,
this chain reaction concept was applied to designing the
atomic reactor and the atomic bomb.
5.8.4 Bulygin Number Bu
Bu2llvδpΔp
clΔT5RmKo
llv (J kg21) � specific latent heat of evaporation; δp (Pa21) � Soret pressure coeffi-
cient; p (Pa) � pressure difference; cl (J kg21 K21) � specific heat capacity of liquid;
ΔT (K) � temperature difference; Rm (�) � Ramzin number (p. 261);
Ko (�) � Kossovitch number (p. 253).
It expresses the ratio of the heat consumed to evaporate a portion of the mois-
ture in a molecular vapour�gas mixture transfer to that consumed to heat the wet
material to the boiling point. It characterizes the phase conversion in high-tempera-
ture mass transfer especially. It is about a filtering vapour transfer in wet material
and the wet material’s ability to accumulate heat, connected with this transfer.
With it, the heat which is lost to change the moisture to vapour participating in the
molecular transfer is considered.8
Info: [A23],[A33].
N.P. Bulygin.
5.8.5 Colburn Mass Number Jm, Com
Jm � Sc � Prm 5νD
5Sh
ReffiffiffiffiffiSc3
p
245Thermomechanics
ν (m2 s21) kinematic viscosity; D (m2 s21) diffusion coefficient; Sc (�) � Schmidt
number (p. 263); Prm (�) � Prandtl mass number (p. 260); Sh (�) � Sherwood
number (p. 264); Re (�) � Reynolds number (p. 81).
This number characterizes the mass transfer in forced viscous fluid flow. It is
also called the Schmidt number Sc (p. 263) or the Prandtl mass number Prm(p. 260). Heat and mass transfers.
Info: [A20].
Allan Philip Colburn (p. 190).
5.8.6 Cooling of Porous Bodies Np
Np 5qmcpL
λð12μÞ
qm (kg s21 m22) � mass flux density; cp (J kg21 K21) � specific heat capacity;
L (m) � characteristic length; λ (W m21 K21) � thermal conductivity of porous
wall; μ (�) � porosity (p. 24).
It expresses the ratio of heat transferred by convection inside a porous wall to
that transferred by conduction with an equal temperature gradient.
Info: [A23].
5.8.7 Coupling Energy Number E
E5l
llv
l (J kg21) � specific bond heat; llv (J kg21) � specific latent heat of evaporation.
It expresses the ratio of the heat consumed to overcome the moisture-material
coupling to the specific evaporation heat. It characterizes the energy consumed to
release the moisture in hygroscopic material drying.
Info: [A23],[A33].
5.8.8 Drew Number Dr
Dr5ðMA 2MBÞxA 1MB
xA 2 xASðMB 2MAÞlnMv
MS
MA, MB (kg mol21) � molar mass (gram molecule) of components A, B; xA (�)
molar fraction of A in diffusion flow; xAS (�) � molar fraction of component A in
the wall; Mv, MS (kg mol21) � molar mass (gram molecule) of the mixture in
vapour and wall.
246 Dimensionless Physical Quantities in Science and Engineering
It characterizes the mass transfer intensity in a boundary layer with said inten-
sity causing the change in the velocity profile in binary systems. For these systems,
it also represents the drag coefficient CD (p. 60).
Info: [A33].
5.8.9 Dufour Number Du
Du5r
cpLe
ðN1N21Þ0
ðN1N 21ÞðN2N 21Þ 5r
cpLe
ðp1p21Þ0ðp1p21Þðp2p21Þ
ð1Þ;
Du5qD
cpTð2Þ
r (J kg21 K21) � specific gas constant; cp (J kg21 K21) � specific heat capacity;
N1, N2 (�) � molecule number of components 1 and 2 in volume of binary gas
mixture, N5N11N2; p1, p2 (Pa) � partial pressures, p5 p11 p2; qD (J kg21) �heat transferred by diffusion in gas mixture; T (K) � gas mixture temperature;
Le (�) � Lewis number (p. 254).
It characterizes the ratio of the diffuse heat and mass transfers, in a binary mix-
ture of gases under isotropic conditions, to the enthalpy of the unit mixture mass,
provided the linear diffusion rate equals that of conduction. Thermodiffusion.
Info: [A23],[A33].
Louis Dufour (1832�1892), Swiss physicist.
5.8.10 Fedorov Number (1.) Fe
Fe5εδtllvcp
5 εKoPn
δt (K21) � Soret thermal coefficient; llv (J kg
21) � specific latent heat of evapora-
tion; cp (J kg21 K21) � specific heat capacity of damp material; ε (�) � liquid
phase conversion in vapour (p. 255); Ko (�) � Kossovitch number (p. 253);
Pn (�) � Posnov diffusion number (p. 259).
This number expresses the ratio of the heat necessary to evaporate the moisture
in a material internally to the average specific thermal capacity of the wet material.
It characterizes the heat and mass transfer process, especially with respect to the
drying of wet material, and defines the transition between quiet and boiling fluid
layers. The criterion does not depend on the choice of heat and mass transfer poten-
tials, but only on ε, δt coefficients and the thermodynamic characteristics. It is
applied as a generalized variable in the analytical solution of material drying
problems.9
Info: [A23],[A33].
Igor Mikhailovich Fedorov, Russian physicist.
247Thermomechanics
5.8.11 Fedorov Number (2.) Fe
Fe5 δtβ� 5NtrPn
δt (K21) � Soret thermal coefficient; β� (K) Dufour coefficient; Ntr (�) � transfer
number (p. 268); Pn (�) � Posnov diffusion number (p. 259).
It characterizes the transfer processes. It is analogous to the Posnov diffusion
number Pn (p. 259) for mass transfer.
Info: [A23].
Igor Mikhailovich Fedorov (see above).
5.8.12 Fedorov Number (3.) Fe
Fe5 dekv
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4gR2g3η2
RsRg
21
!3
vuut 5
ffiffiffiffiffiffiffiffiffi4
3Ar
3
r
dekv (m) � equivalent particle diameter; g (m s22) � gravitational acceleration;
Rg, Rs (kg m23) � density of gas and particles; η (Pa s) � dynamic viscosity;
Ar (�) � Archimedes hydrodynamic number (p. 53).
It characterizes the transition from the immobile layer to the uplifting one in
fluidizing processes. The criterion is analogous to the Archimedes hydrodynamic
number Ar (p. 53).
Info: [A23],[A33].
Igor Mikhailovich Fedorov (see above).
5.8.13 Fourier Filtration Number Fop
Fop 5apτL2
5FoLuf ; where ap 5kh
μ
ap (m2 s21) � pressure filtration velocity; τ (s) � time; L (m) � characteristic
length; k (m s21) � filtration coefficient; h (m) � mean static height; μ (�) �porosity (p. 24); Fo (�) � Fourier number (p. 175); Luf (�) � Lykov filtration
number (p. 256).
It characterizes the time of non-stationary filtration of moisture in the porous
material.
Info: [A23].
Jean Baptiste Joseph Fourier (p. 175).
248 Dimensionless Physical Quantities in Science and Engineering
5.8.14 Fourier Mass Number FoD, Fom
FoD 5DτL2
ð1Þ;
Fom 5amτL2
5FoLu ð2Þ
D (m2 s21) � diffusivity; τ (s) � time; L (m) � characteristic length; am (m2 s21) �mass diffusivity; Fo (�) � Fourier number (p. 175); Lu (�) � Lykov number
(p. 256).
This number expresses the ratio of the intensity of the volume mass diffusion
change to that of the time change of moisture content in a material. In a dimension-
less expression, it characterizes the non-stationary mass (moisture) transfer. It
determines the relation between the change rate of the mass transfer potential field
and the physical properties and size of the system in molecular material transfer in
a non-stationary state.
Info: [A23],[A33].
Jean Baptiste Joseph Fourier (see above).
5.8.15 Frossling Mass Number Fsm
Fsm 5Sh22
Re12Sc
13
Sh (�) � Sherwood number (p. 264); Re (�) � Reynolds number (p. 81);
Sc (�) � Schmidt number (p. 263).
It characterizes the convective mass transfer for a ball located in flowing fluid.
Info: [A23].
Karl Gustav Frossling (born 1913), Swedish engineer.
5.8.16 Geometric Number Ge
Ge5h
H
h (m) � bundle perimeter; H (m) � bundle height.
It expresses a dimensionless bundle height. It is used to determine the mass
transfer in bundles, for example, in drying wood.
249Thermomechanics
5.8.17 Graetz Mass Number Gzm
Gzm 5RemPrm 5PemL
dH
Rem (�) � Reynolds mass number (p. 262); Prm (�) � Prandtl mass number (p.
260); Pem (�) � Peclet mass number (p. 258); L (m) � characteristic length; dH(m) � hydraulic diameter.
It expresses the ratio of the convective mass transfer to the diffusion heating or
cooling in fluid flow inside closed pipelines and other systems.
Info: [A23].
Leo Graetz (p. 193).
5.8.18 Granulation Number Au
Au5lls
llvΔu
lls (J kg21) � specific heat of crystallization; llv (J kg21) � specific latent heat of
evaporation; Δu (�) � moisture difference.
It expresses the ratio of the heat arising in crystallization and drying to the total
heat necessary to evaporate moisture from a dried solution. With it, the influence
of the phase conversion (crystallization) of the solution (molten material) layer on
the moisture evaporation intensity on the granule surface is considered. It deter-
mines the heat flow direction among the granules (grains) and bypassing gas. With
the granulation in a fluidization layer, it determines the boundary between the gran-
ulation zones in drying (Au, 1) and cooling of molten material (Au. 1).
Info: [A23],[A33].
5.8.19 Grashof Mass Number Grm
Grm 5gL3βc
ν2lðRs 2 RÞ5 gL3
ν2l
MNTS
MTP21
� �ð1Þ;
Grm 5gL3
ν2l
ΔRR
ð2Þ
g (m s22) � gravitational acceleration; L (m) � characteristic length; βc (m3 kg21) �
volume expansion coefficient inducing solution; ν1 (m2 s21) � kinematic viscosity of
fluid; Rs (kg m23) � mass concentration of dissolved matter at surface; R (kg m23) �
mass concentration of dissolved matter; M, MN (kg mol21) � molar mass of
steam�gas mixture at liquid surface and in sufficient distance from surface; TS, TP(K) � liquid surface temperature and external environment temperature.
250 Dimensionless Physical Quantities in Science and Engineering
It expresses the ratio of the product of inertia and buoyancy forces to the square
of a viscous force. It characterizes the mass transfer by natural convection in a
non-isotropic vapour�gas mixture, with the transfer being caused by the tempera-
ture difference and vapour concentration in the air of the vapour�gas mixture.
Info: [A23].
Franz Grashof (p. 185).
5.8.20 Gukhman Drying Number Gu
See the Gukhman evaporation number Gu (p. 230).
Alexander Adolfovich Gukhman, Russian chemical engineer.
5.8.21 Heating Up of Moist Particles Nt, Nm
Nt 5
ffiffiffiffiffiffiffiffiffiffibw
aL
rð1Þ;
Nm 5
ffiffiffiffiffiffiffikL2
a
rð2Þ
w (m s21) � motion velocity of material in dryer; a (m2 s21) � thermal diffusivity;
L5Vd S21 (m) � ratio of absolute dry body volume to its surface; Vd (m
3) � vol-
ume of absolute dry body; S (m2) � surface of absolute dry body; k (s21) � dry
coefficient; b (�) � logarithmical decrement (p. 18).
It characterizes the heating intensity of dispersive spherical particles in gas flow
drying with convective and radiation heat supply of constant value and temperature
changing T5 To exp (28wτ). It also characterizes material motion velocity in a
drying stove w(Nt) and evaporation rate Nm.
Info: [A33].
5.8.22 Internal Heat and Mass Transfer N
N5Bi
εKo
Bi (�) � Biot number (p. 173); ε (�) � liquid phase conversion in vapour
(p. 255); Ko (�) � Kossovitch number (p. 253).
It expresses the ratio of the heat supplied to a body material to the heat con-
sumed in internal mixture evaporation in the material.
Info: [A23],[A33].
251Thermomechanics
5.8.23 J-Mass Factor Jm
Jm 5βRqm
νD
� 235
βw
νD
� 235NumRe
21Pr213
m 5βwSc
23
β (m s21) � mass transfer coefficient; R (kg m23) � density; qm (J kg21 m22) �mass flux density; ν (m2 s21) � kinematic viscosity; D (m2 s21) � diffusivity;
w (m s21) � flow velocity; Num (�) � Nusselt mass number (p. 258); Re (�) �Reynolds number (p. 81); Prm (�) � Prandtl mass number (p. 260); Sc (�)
Schmidt number (p. 263).
It characterizes the convective transfer of mass, heat and momentum.
Info: [A23].
5.8.24 Jermakov Number Er
Er5kqS
Rl
k (m s21) � energy absorbing coefficient by freeze-drying; qS (W m22) � surface
heat flux density; R (kg m23) � density of liquid or vapour; l (J kg21) � specific
latent heat.
This number characterizes the sublimation drying process in electromagnetic
heating with a constant drying rate. It is the measure of the ratio of the heat amount
delivered to the drying material in a certain time to that consumed to evaporate
unit volume of drying material in the same amount of time.
Info: [A33].
E.A. Jermakova.
5.8.25 Kirpichev Mass Number Kim
Kim 5qmL
λmtΔϑ5
qmL
amtRΔuΔϑð1Þ;
Kim 5BimϑS 2ϑP
Δϑð2Þ
qm (kg s21 m22) � mass flux density; L (m) � characteristic length; λmt(kg m21 s21 K21) � thermal mass conductivity; Δϑ (K) � potential difference of
mass transfer; amt (m2 s21 K21) � thermal mass diffusivity; R (kg m23) � density;
Δu (�) � moisture content difference; ϑS, ϑP (K) � mass transfer potential of wall
and environment; Bim (�) � Biot mass number (p. 244).
It characterizes the relation between outer and inner mass flow intensities. In
mass transfer tasks, it represents the second-type boundary condition.
252 Dimensionless Physical Quantities in Science and Engineering
Info: [A23].
Mikhail Viktorovich Kirpichev (p. 177).
5.8.26 Kossovitch Number Ko
Ko5llvcm
cpΔT5
llvΔu
cpΔT
llv (J kg21) � specific latent heat of evaporation; cm (�) � mass fraction;
cp (J kg21 K21) � specific heat capacity of damp material; ΔT (K) � temperature
difference; Δu (�) � dampness difference.
It is the phase conversion criterion. It expresses the ratio of the evaporation heat
for moisture evaporation to the heat for wet body heating. It characterizes the con-
vective heat transfer in evaporation. It represents the ratio of the specific heat con-
sumed to evaporate the whole amount of moisture to that consumed by wet
material when heating to maximum temperature.
Info: [A23].
5.8.27 Lagrange Number (1.) Lg1
Lg1 5D1 εD
D
D (m2 s21) � molecular diffusivity; εD (m2 s21) � vortex mass diffusivity.
It expresses the ratio of the sum of molecular and swirl mass transfer intensities
to the molecular heat transfer intensity. It characterizes the mass transfer in a turbu-
lent flow.
Info: [A23].
Joseph-Louis Lagrange (p. 70).
5.8.28 Lebedev Number Lb
Lb5μbðTP 2 TÞRscmpp
ð1Þ;
Lb5TR
TPð2Þ;
Lb5umed
ucritð3Þ
μ (�) � porosity (p. 24); b5 @ðmvV21v Þ=@Tðkg m23 K21Þ � temperature gradient
of vapour concentration; mv (kg) � vapour mass; Vv (m3) � vapour volume;
253Thermomechanics
T, TP (K) � initial temperature of material and environment; Rs (kg m2 3) �density of solid phase; cmp (kg kg21 Pa2 1) � pressure mass capacity of vapour
(vapour mass per unit mass of dry gas and per unit of pressure difference); p (Pa) �pressure; TR (K) � temperature of radiating source; umed (�) � mean moisture
content in a body; ucrit (�) � critical moisture content.
This number expresses the ratio of the vapour flow by molecular expansion to
the total macroscopic vapour flow from porous material. It characterizes thermal
processes in drying intensification. In expression (2), it represents the volume evap-
oration of dispersed fluid in a boundary layer. It determines the heat transfer
increase due to boundary layer thickness reduction with the temperature increase of
a radiating source. In expression (3), it represents the ratio of a material’s mean
moisture with drying rate reduction to a material’s mean critical moisture. With it,
the heat transfer coefficient reduction due to material moisture reduction is consid-
ered. Sometimes expression (2) is called the Gukhman modified criterion.
Info: [A23].
Peter Nikolaevich Lebedev (8.3.1866�14.3.1912),
Russian physicist.
He founded the Russian school of physicists. He was the
first to discover and investigate mm-range electromagnetic
waves (1895) and addressed himself to the nature of the ter-
restrial magnetic field. He was engaged in the action of
light on solid substances (1900) and gases (1908). He con-
firmed the electromagnetic theory of light from the quanti-
tative point of view. In his last years, he devoted himself to
the problem of ultrasound.
5.8.29 Lewis Number Le
Le5λ
RcpD5
a
D5 ScPr21 5PrmPr
21 ð1Þ;
Le5D
að2Þ
λ (W m21 K21) � thermal conductivity; R (kg m23) � density; cp (J kg21 K21) �specific heat capacity; D (m2 s21) � molecular diffusivity; a (m2 s21) � thermal
diffusivity; Sc (�) � Schmidt number (p. 263); Prm (�) � Prandtl mass number
(p. 260); Pr (�) � Prandtl number (p. 197).
It expresses the thermal-to-molecular diffusivities ratio. It characterizes the
mutual relation of heat and mass transfers in various materials. It is the ratio of
chemical potential field change to a thermal field in the flow of gas or fluid mix-
tures. For gases, it is LeAh0.8; 1.2i, for fluids LeAh70; 100i. In expression (2), it is
usually called the Lewis�Semenov number, or sometimes only the Lewis number.
Info: [A23].
254 Dimensionless Physical Quantities in Science and Engineering
Warren Kendall Lewis (21.8.1882�9.3.1975), American
chemical engineer.
He is called the father of modern chemical engineering.
He was engaged in researching distillation and evaporation,
petroleum flow modelling, predicting the lifetime of petro-
leum deposits and developing methods for exploiting them.
His work concerning powder fluidization and the control of
its movement in combustion chambers, and in heat and cat-
alytic cracking, was very significant.
Nikolay Nikolayevich Semyonov (p. 48).
5.8.30 Lewis Turbulence Number Letur
Letur 5R cp εD
λT
5lD
lT5
εDεT
R (kg m23) � density; cp (J kg21 K21) � specific heat capacity; εD, εT (m2 s21) �vortex mass and thermal diffusivity; λT (W m21 K21) � vortex thermal diffusivity;
lD, lT (m) � length of mixing for mass and heat.
This number expresses the ratio of the molecular mass transfer intensity of a
given component in a mixture to the turbulent heat transfer intensity. It charac-
terizes the kinematic properties in heat and mass transfers in gas mixture flow. It
expresses the relation between thermal and concentration fields.
Info: [A23].
Warren Kendall Lewis (see above).
5.8.31 Liquid Phase Conversion in Vapour ε
ε5duf
duð1Þ;
ε5qmv
qmv 1 qmlð2Þ
0# ε# 1
uf (�) � body moisture content with phase change; u (�) � body moisture content;
qmv, qml (kg s21 m22) � mass flux density of vapour and liquid.
It expresses the ratio of the relative moisture content change by evaporation to
the total moisture content change in a given place of a body under conditions of
non-stationary moisture transfer in drying. It characterizes the vapour amount, dif-
fusing in capillary porous material, with respect to fluid and vapour flow without
255Thermomechanics
considering the convective transfer. For the moisture content change in an arbitrary
place, the following limits are valid:
ε5 0 � the moisture is changed by fluid transfer only,
ε5 1 � the moisture is changed only by evaporation.
In hygroscopic materials drying, it characterizes, in equation (2), the ratio of the
absolute value of the resulting vapour mass flow density to the sum of densities of
the vapour and fluid mass flows in non-stationary moisture transfer.
Info : [A23],[A33].
5.8.32 Lukomsky Number Lk
Lk � Lu21 5a
am
a (m2 s21) � thermal diffusivity; am (m2 s21) � mass diffusivity; Lu (�) � Lykov
number (p. 256).
It characterizes combined heat and mass transfer. It is analogous to the Lykov
number Lu (p. 256).
Info: [A29].
5.8.33 Lykov Filtration Number Luf
Luf 5ap
a
ap, a (m2 s21) � pressure filtration and thermal diffusivity.
It expresses the pressure to temperature fields propagation intensities ratio. It
characterizes the pressure to thermal diffusion ratio in heat and mass transfers.
Usually, it is Lufc1, but it is 100�1000 most frequently.
Info: [A23].
Alexey Vasilievich Lykov (see below).
5.8.34 Lykov Number Lu
Lu5am
a
am, a (m2 s21) � mass and thermal diffusivity.
This number expresses the ratio of the mass to temperature field propagation
intensities. It characterizes the relation between the temperature field propagation
and the mass (moisture) field. It expresses the mutual heat-to-mass transfer ratio in
a capillary porous material and in dispersive materials. It is the degree of the poten-
tial molecular field relaxation of the materials. The difference between the Lykov
256 Dimensionless Physical Quantities in Science and Engineering
and Lewis number Le (p. 254) is analogous to that between the Biot Bi (p. 174) and
Nusselt number Nu (p. 196), for example, as the intensities of corresponding phe-
nomena ratio.
Info: [A23].
Alexey Vasilievich Lykov (20.9.1910�28.6.1974), Russian
physicist.
In the year 1935, he discovered the thermodiffusion of
the humidity in capillary porous material. It was called the
Lykov effect. He was engaged in solving complicated pro-
blems of heat and mass transfer in capillary porous material
and dispersal environments under acting of phase and
chemical conversions. Of his many monographs, the
Teorija suski (Theory of Drying, 1951) and Teorija teplo-
provodnosti (Theory of Thermal Conductivity, 1969) are his
most important works.
5.8.35 Margoulis Mass Number Mgm
See the Stanton mass number Stm (p. 256).
Wladimir Margoulis (born 1886).
5.8.36 Merkel Number Me
Me5qmS
Qm
qm (kg s21 m22) � mass flux density; S (m2) � total surface of water level that is
in contact with air; Qm (kg s21) � mass flux.
It expresses the ratio of water mass, transferred in cooling, to dry gas mass. It
characterizes the mass transfer in basin coolers and cooling towers.
Info: [A23],[B11].
Friedrich Merkel (born 1892), German engineer.
5.8.37 Mikhailov Number Mi
Mi5 δpdp
du5BuRb
δp (Pa21) � Soret pressure coefficient; p (Pa) � pressure; u (�) � moisture con-
tent; Bu (�) � Bulygin number (p. 227); Rb (�) � Rebinder number (p. 262).
It characterizes the filtering mass transfer kinetics in intense heat and mass
transfer in drying. It expresses the degree to which pores are filled by a fluid at a
257Thermomechanics
given temperature or, alternatively, the total pressure change of the vapour�gas
mixture or a fluid inside a material.
5.8.38 Miniovich Number Mn
Mn5Sr
Vp
S (m2) � particle surface, V (m3) � particle volume, r (m) � pore radius; p(�) �porosity (p. 24).
In heat and mass transfer processes in a very small area of a layer of material, it
expresses the layer geometry influence on temperature and moisture fields in the
layer. Drying of porous materials.
Info: [A33].
J.M. Miniovich.
5.8.39 Nusselt Mass Number Num
See the Sherwood number Sh (p. 264).
Ernst Kraft Wilhelm Nusselt (p. 196).
5.8.40 Peclet Mass and Vortex Number Pemu
Pemu 5wL
εD
w (m s21) � velocity; L (m) � characteristic length; εD (m2 s21) � vortex mass
diffusivity.
It expresses the ratio of the convective mass propagation to the propagation rate
caused by molecular vortex diffusion. It characterizes the convective-vortex diffu-
sive mass transfer in flowing fluid.
Info: [A23].
Jean Claude Eugene Peclet (p. 180)
5.8.41 Peclet Mass Number Pem
Pem 5wL
D5RePrm 5ReSc ð1Þ; Pem 5
J
kBDð2Þ
w (m s21) � velocity; L (m) � characteristic length; D (m2 s21) � diffusivity;
J (kg s21) � convective mass flux through the membrane; kBD (kg s21) � total
258 Dimensionless Physical Quantities in Science and Engineering
coefficient of back diffusion; Re (�) � Reynolds number (p. 81); Prm (�) �Prandtl mass number (p. 260); Sc (�) � Schmidt number (p. 263).
In expression (1), it expresses the ratio of the volume mass transfer rate,
with forced convection, to the diffusive mass transfer rate. It characterizes the
convective�diffusive mass transfer in flowing fluid. It is also called the Reynolds
mass number Rem (p. 262).
For diaphragm systems, the dimensionless mass transfer parameter (2) is used,
which expresses the ratio of convective flow through a diaphragm to a phenomenon
which causes return diffusion expressed by the total coefficient kBD.
Info: [A23].
Jean Claude Eugene Peclet (see above).
5.8.42 Pomerantsev Mass Number Pom
Pom 5qmVL
2
amRΔu
qmV (kg s21 m23) � volume density of mass flux; L (m) � characteristic length;
am (m2 s21) � mass diffusivity; R (kg m23) � material density; Δu (�) � differ-
ence of moisture content.
This number expresses the ratio of unit mass development by an internal source
to the maximum possible unit mass amount transferred by conduction. It charac-
terizes an internal mass source in a dimensionless way.
Info: [A23].
Alexey Alexandrovich Pomerantsev, Russian engineer.
5.8.43 Posnov Diffusion Number Pn
Pn5δtΔT
Δu5
amtRΔT
amtRΔu5
amtRΔT
λmtΔϑð1Þ;
Pn5cmt
@ϑ@T
�uΔT
ðcmtÞmedΔϑ5
@ ϑΔϑ
�@ T
ΔT
�" #
u
ð2Þ
δt (K21) � Soret thermal coefficient; ΔT (K) � temperature difference; Δu (�) �
difference of moisture content; amt (m2 s21 K21)� thermal mass diffusivity; R (kg m23)�
density of material; am (m2 s21) � mass diffusivity; λmt (kg m21 s21 K21) � thermal
mass conductivity; Δϑ (K) � difference of mass transfer potential; cmt (K21) � thermal
mass capacity.
It expresses the ratio of the thermodiffusive moisture transfer intensity to the dif-
fusive moisture transfer intensity. It characterizes the internal moisture transfer by
conduction under non-isentropic conditions. The relative non-uniformity of a mois-
ture field in a material is caused by heat and mass transfer in a steady state. With
259Thermomechanics
small Δu or linear dependence of u(ϑ), the Pn number becomes a dynamic transfer
parameter � a relative temperature product of the mass transfer potential expressed
by expression (2). In this form, it expresses the ratio of the potential change of
mass (moisture content) to the relative temperature change with a given mass steady-
state content in a body. It is analogous to the Fedorov number (2.) Fe (p. 248).
Info: [A23].
5.8.44 Posnov Filtration Number Pnp
Pnp 5δpΔp
Δu5
λmpΔp
RamΔu
δp (Pa21) � Soret pressure coefficient; Δp (Pa) � pressure difference; Δu (�) �difference of moisture content; λmp (kg m21 s21 Pa21) � pressure mass conductiv-
ity; R (kg m23) � density; am (m2 s21) � mass diffusivity.
It expresses the ratio of the filtration to diffusive moisture transfer intensities. It
characterizes the internal moisture transfer in material during filtration. The relative
moisture field non-uniformity in the material is caused by a pressure gradient
(filtration potential difference).
Info: [A23].
5.8.45 Prandtl Mass Number Prm
See the Schmidt number Sc (p. 263).
Ludwig Prandtl (p. 197)
5.8.46 Predvoditelev Mass Number Pdm
Pdm 52dðqmq21
mi ÞdFo
� �max
ð1Þ;
Pdm 5kL2
amð2Þ
qm, qmi (kg s21 m23, or kg s21 m22) � volume and surface mass flow intensity and
its initial value; k (s21) � drying coefficient; L (m) � characteristic length;
am (m2 s21) � mass diffusivity; Fo (�) � Fourier number (p. 175).
It expresses the ratio of the material concentration change rate in the surround-
ings to that in a body. It characterizes the mass flow density time change, for exam-
ple, that of an internal source (qm� qmV) or of the second-type boundary condition
(qm� qmA). In tasks involving wet material heating in constant temperature sur-
roundings, when the surface moisture evaporation occurs in the second drying
260 Dimensionless Physical Quantities in Science and Engineering
period according to the law qm5 qmi exp (2 kτ), expression (2) is valid, and the
first period of drying Pdm5 0 is valid.
Info: [A23].
Alexander Savviich Predvoditelev (1891�1973), Russian physicist.
5.8.47 Psychrometric Relation K
K5αcmqmc
α (W m22 K21) � heat transfer coefficient; cm (�) � mass ratio; qm (kg s21 m22) �mass flux density; c (J kg21 K21) � specific heat capacity.
It expresses the ratio of the heat transferred by convection to that transferred by
the mass transfer. It characterizes wet air as a heat transfer means and a drying
environment simultaneously. Wet and dry thermometer well.
Info: [A23].
5.8.48 Ramzin Number Rm
Rm5δpΔp
Δu5
δpΔp
δtΔϑ5BuKo21
δp (Pa21) � Soret pressure coefficient; Δp (Pa) � pressure difference; Δu (�) �difference of moisture content; δt (K
21) � Soret thermal coefficient; Δϑ (K) � dif-
ference of moisture transfer potential; Bu (�) � Bulygin number (p. 227); Ko (�) �Kossovitch number (p. 253).
In molecular transfer, it characterizes the vapour content described as a mass
content of the material which takes part in the molecular transfer. See the Posnov
filtration number Pnp (p. 260).
Info: [A23].
5.8.49 Rayleigh Mass Number Ram
Ram 5XgL3Δc
νD5
gL3
νDΔRR
5GrSh
X (�) � dimensionless diffusion number; g (m s22) � gravitational acceleration;
L (m) � characteristic length; Δc (�) � difference of initial and final mass ratio of
chemical reaction products; ν (m2 s21) � kinematic viscosity; D (m2 s21) � diffu-
sivity; ΔR (kg m23) � density difference; R (kg m23) � density; Gr (�) � Grashof
heat number (p. 185); Sh (�) � Sherwood number (p. 264).
It characterizes the natural convection influence on the mass transfer in fluids,
for example, the concentration and temperature propagation influence on the
261Thermomechanics
thermodiffusion separation of liquid mixtures in a fluid thermodiffusion separating
column.
Info: [A23].
Lord Rayleigh (p. 187).
5.8.50 Rebinder Number Rb
Rb5cb
llv5
dTmed
dumed
� �c
llv5BKo21
c (J kg21 K21) � specific heat capacity; b5 dTmed/dumed (K) � change of mean
integral temperature to change of mean integral moisture content ratio; llv (J kg21) �
specific latent heat of vaporization; B5ΔubΔT21 (�) � dimensionless drying
coefficient; Ko (�) � Kossovitch number (p. 253).
This number expresses the ratio of the heat consumed in material heating to that
consumed in moisture evaporating during an infinitely short time interval. It relates
to the kinematic characteristics of integral properties of heat and mass transfer. It
expresses the characteristics of local changes of temperature and moisture content.
Info: [A23].
Peter Aleksandrovich Rebinder (21.9.1898�12.7.1972),
Russian physical chemist.
He was engaged in the problems of the origin, stability,
surface phenomena and structure formation in disperse sys-
tems. In addition, he explained the idea of the molecular
mechanism of surface active substances and their applica-
tions in diverse technological processes. In the year 1928,
he clarified the influence of absorption in reducing the
rigidity of solid bodies.
5.8.51 Reynolds Mass Number Rem
See the Peclet mass number Pem (p. 258), Richman number Ri (p. 232).
Osborne Reynolds (p. 82).
5.8.52 Romankov Number Ro
Ro5Tg 2 Ts
Tgð1Þ;
Ro5Tg
Tsð2Þ
262 Dimensionless Physical Quantities in Science and Engineering
Tg (K) � hot gas temperature used for drying; Ts (K) � temperature of dried-up
material.
This number expresses the ratio of the temperature gradient between gas and
material in drying to the heated gas temperature at the drying stove outlet. In dry-
ing, it characterizes the potential wet gas possibilities to heat the dried material.
Sometimes, it is used in the form of equation (2).
Info: [A23].
Peter Grigorievich Romankov (17.1.1904�1.10.1990),
Russian physical chemist.
He was engaged in the analysis of kinetic laws and the
generalization of the methods of similarity theory and
dimensional analysis. He applied the results on heat and
mass transfer processes and hydromechanical ones in
chemical technology. His principal work is the monograph
Gidravliceskije processy v chimiceskoj technologii
(Hydraulic Processes in Chemical Engineering). He was
also engaged in fluid mixing, filtration, centrifugal separa-
tion, cleaning of gas, drying, absorption, extraction and
distillation.
5.8.53 Schmidt Effective Number Scef
Scef 5εM 1 νεD 1D
εM, εD (m2 s21) � vortex momentum diffusivity and vortex mass diffusivity; ν(m2 s21) � kinematic viscosity; D (m2 s21) � diffusivity.
It expresses the ratio of the total momentum diffusivity to the total mass diffu-
sivity in the mass transfer in combined laminar and turbulent flow. It characterizes
the mass transfer in combined turbulent and laminar flow.
Info: [A23].
Ernst Schmidt (see below).
5.8.54 Schmidt Number Sc
Sc � JQ � Prm 5νD
ν (m2 s21) � kinematic viscosity; D (m2 s21) � diffusivity; JQ (�) � Colburn
number (p. 190); Prm (�) � Prandtl mass number (p. 260).
This number expresses the ratio of the kinematic viscosity, or momentum trans-
fer by internal friction, to the molecular diffusivity. It characterizes the relation
between the material and momentum transfers in mass transfer. It provides the
263Thermomechanics
similarity of velocity and concentration fields in mass transfer. For example, mol-
ten materials with an equal Schmidt number have similar velocity and concentration
fields. Higher Sc number values characterize slower mass exchange and higher
values of dividing coefficients. This leads to higher mixing and a tendency to crack
in a solidified casting. The criterion was first introduced by Schmidt in 1929. It
is also called the Colburn number Jq (p. 190) or the Prandtl mass number Prm(p. 260).
Info: [A23],[A43].
Ernst Schmidt (11.2.1892�22.1.1975), German engineer.
He was a pioneer in engineering thermodynamics, espe-
cially in heat and mass transfer. Initially, he was engaged in
measuring the radiation properties of solid materials, which
was the reason to design an effective thermal shield of alu-
minium foil. In addition, he was engaged in solving non-
stationary thermal fields by a graphic difference method, in
determining the local heat transfer coefficient and in anal-
ogy between heat and mass transfer.
5.8.55 Schmidt Turbulent Number Scv
Scv 5εMεD
εM, εD (m2 s21) � vortex momentum and mass diffusivity.
It expresses the ratio of the vortex momentum diffusivity to the vortex mass dif-
fusivity. It characterizes the mass transfer in a turbulent flow.
Info: [A23].
Ernst Schmidt (see above).
5.8.56 Sherwood Number Sh
Sh � Num � Tam 5βLD
β (m s21) � mass transfer coefficient; L (m) � characteristic length; D (m2 s21) �molecular diffusivity; Num (�) � Nusselt mass number (p. 258); Tam (�) � Taylor
mass number (p. 266).
It expresses the ratio of the heat transfer to the molecular diffusion. It charac-
terizes the mass transfer intensity at the interface of phases. It is also called the
Nusselt mass number Num (p. 258) or the Taylor mass number Tam (p. 266).
Info: [A23].
264 Dimensionless Physical Quantities in Science and Engineering
Thomas Kilgore Sherwood (25.7.1903�14.1.1976),
American chemical engineer.
He was engaged primarily in research on mass transfer
and its interactions with flow, chemical reactions and indus-
trial applications in which these processes play a principal
role. After having published Absorption and Extraction
(1937), which was the first, sui generis in this field, he
became a world famous personality.
5.8.57 Soret Thermodiffusion Number So
So5 LeðN2N
21ÞiniðN1N 21ÞðN2N 21Þ
N1, N2 (�) � molecule number of components 1 and 2 in binary gas mixture,
N5N11N2; Le (�) � Lewis number (p. 254).
It characterizes the thermodiffusion effect in mass transfer. In binary gas mix-
tures with equal initial content of components, it equals half of the thermodiffusion
constant. It expresses the coupling between heat and mass transfers. It represents
the dimensionless thermodiffusion coefficient.
Info: [A23].
Charles Soret (1854�1904), Swiss physicist.
5.8.58 Spalding Transfer Number Spp
Spp 5 cpΔT llv 2ΦR
Qm
� �21
cp (J kg21 K21) � specific heat, capacity; ΔT (K) � temperature difference;
llv (J kg21) � specific latent heat of vaporization; ΦR (W) � radiation heat flux;
Qm (kg s21) � mass flux.
It expresses the ratio of the thermal energy change to the latent evaporation heat
of material. It characterizes heat and mass transfers under conditions of thermal
radiation and phase conversion.
Info: [A23].
Brian Spalding (p. 200).
5.8.59 Stanton Mass Number Stm
Stm � Mg5βw
5 ShRe21Sc21 ð1Þ;
265Thermomechanics
Stm 5qm
Rw5
1
αmpwð2Þ
β (m s21) � mass transfer coefficient; w (m s21) � flow velocity; qm (kg s21 m22) �mass flux density; R (kg m23) � fluid density; αmp (kg m22 s21 Pa21) � pressure
specific mass transfer; Mg (�) � Margoulis number (p. 196); Sh (�) � Sherwood
number (p. 264); Re (�) � Reynolds number (p. 81); Sc (�) � Schmidt number
(p. 263).
It expresses the ratio of the mass transfer perpendicular to a solid phase surface
to the mass transfer by flowing in parallel with the solid phase surface. It charac-
terizes the mass transfer at the interface between the solid and fluid phases. It is
called the Margoulis number Mg (p. 196) as well.
Info: [A23].
Thomas Edward Stanton (p. 201).
5.8.60 Stefan Parameter Sf
Sf 5p
ppS 2 ppPlnp2 ppP
p2 ppS
p (Pa) � local absolute static pressure; ppS (Pa) � partial vapour pressure above
the surface of evaporation or condensation; ppP (Pa) � partial vapour pressure in
environment.
It expresses the influence of the transversal material flow on the size of the
mass transfer coefficient in evaporation or vapour condensation from a vapour�gas
mixture.
Info: [A23].
Josef Stefan (p. 214).
5.8.61 Taylor Mass Number Tam
See the Sherwood number Sh (p. 264).
Geoffrey Ingram Taylor (p. 89).
5.8.62 Temkin Heat and Mass Transfer Number Di
Di5qmllvL
λΔT5KoLuKim
qm (kg s21 m22) � density of mass flux; llv (J kg21) � specific latent heat of evapo-
ration; L (m) � characteristic length; λ (W m21 K21) � thermal conductivity of
material; ΔT (K) � initial and final temperature difference; Ko (�) � Kossovitch
266 Dimensionless Physical Quantities in Science and Engineering
number (p. 253); Lu (�) � Lykov number (p. 256); Kim (�) � Kirpichev mass
number (p. 252).
This number expresses the ratio of the external heat transition intensity, caused
by the external mass transition, to the internal heat transfer intensity by conduction
in wet material. It characterizes the mean thermal gradient value in wet material.
Info: [A23].
A.G. Temkin.
5.8.63 Temperature Criterion N
N5Ta 2 Ts
Ta 2 Tad
Ta (K) � temperature of moist air; Ts (K) � material temperature after drying-up;
Tad � air temperature in a state of adiabatic saturation (temperature of a wet
thermometer).
It characterizes the potential possibilities of wet air as a heating material and
drying agent as well.
Info: [A23].
5.8.64 Thiele Modulus (1.) Th, mT
Th � Da4 5euL2
λT
e (m2 s22) � specific energy; u (kg m23 s21) � chemical reaction rate; L (m) �characteristic length; λ (W m21 K21) � thermal conductivity; T (K) � temperature;
Da4 (�) � Damkohler number (4.) (p. 37).
It expresses the ratio of the surface reaction rate to the thermal diffusion rate. It
characterizes the non-isothermal heat and mass transfer in physical, chemical and
biotechnical processes. Catalysis. Diffusion in porous catalysers.
Info: [A29].
Ernest William Thiele (8.12.1895�29.11.1993), American
chemical engineer.
Initially, he was engaged in work related to carbon reac-
tions and evolved an idea leading to the McCabe�Thiele
method of fractional column design. During the war, he
participated in the atomic energy development program and
was engaged in heavy water extraction. After the war
(1948), he worked on a project to exploit the nuclear drive
in aeronautics. Then he was engaged in refining processes
and the distillation of hydrocarbon mixtures. In connection
with this, he solved the heat and mass transfer problems.
267Thermomechanics
5.8.65 Transfer Number Ntr
Ntr 5β�Δn
ΔTð1Þ;
Ntr 5β�Δu
ΔTð2Þ
β�(K) � Dufour coefficient; Δn (�) � difference of mass concentration; ΔT (K) �
temperature difference; Δu (�) � difference of relative moisture content.
In equation (1), it expresses the general transfer phenomena which represent the
mass concentration change due to a temperature change, and in equation (2) it
expresses a moisture change. Thermomechanics. Heat and mass transfer.
Info: [A23].
5.8.66 Transfer Number (1.) Thermodynamic Ntr , ε
Ntr 5βc
r
β (K21) � volume thermal expansion coefficient; c (J kg21 K21) � specific heat
capacity; r (J kg21) � specific heat of evaporation.
It characterizes the thermodynamic heat and mass transfer processes. It is analogous
to the Posnov diffusion number Pn (p. 259) and the Fedorov number (2.) Fe (p. 248).
Info: [A23].
5.8.67 Transfer Number (2.) Concentration Ntr
Ntr 5 δtΔcm
ΔT
δt (K21) � Soret thermal coefficient; Δcm (�) � difference of relative substance
concentration; ΔT (K) � temperature difference.
It characterizes the mass concentration process in heat and mass transfer.
Info: [A23].
5.8.68 Transfer Number (3.) Diffusion Ntr , ε
Ntr 5Dx
Qδ5Ab2Sh22
268 Dimensionless Physical Quantities in Science and Engineering
D (m2 s21) � molecular diffusivity; x (m) � wetted length; Q (m3 s21) � volume
flow; δ (�) � relative film thickness; Ab (�) � absorption number (p. 244);
Sh (�) � Sherwood number (p. 264).
It characterizes dimensionless diffusivity. It expresses the gas absorption on wet
chamber walls. Mass transfer.
Info: [A23].
5.9 Non-Equilibrium Thermomechanics
The dimensionless quantities for non-equilibrium thermomechanics involve the
processes of wave heat propagation and thermal stress in material and parallel
heat and mass propagation. The wave propagation is closely related to thermal
strokes and the action of intensive heat sources on the material. A survey of the
wave propagation theory is given in [A46]. Parallel heat and mass propagation is
applied, for example, in the transfer in porous and capillary materials and is related
to new technologies and materials especially. Most of the dimensionless quantities
are more complicated modifications of basic criteria and represent the influence of
the heat and mass propagation velocities, relaxation action, thermalization and
internal heat transfer. The following numbers are important: the Fourier, Biot,
Nusselt, Peclet, Vernotte, Sparrow and others. In this section, the more detailed
explanation of some dimensionless quantities relates, above all, to their specific
significance in non-equilibrium thermomechanics and in the actual complexity of
wave and wave-diffusion processes.
5.9.1 Biot Volumetric Number of Internal Heat Transfer BiV
BiV 5αVL
2
λð1Þ;
BiV 5τ2α2
V
c1c25Fo2ðBi�V Þ2Pc ð2Þ
αV (W m23 K21) � volume coefficient of internal heat transfer; L (m) � character-
istic length; λ (W m21 K21) � thermal conductivity; τ (s) � time; c1, c2(J m23 K21) specific volume heat capacity of material components; Fo (�) �Fourier number (p. 175); Bi�V (�) � Biot volumetric number of internal heat trans-
fer defined by the relation (1); Pc (�) � relative parallel heat capacity (p. 279).
The Biot volumetric number of internal heat transfer expresses the ratio of the
internal inter-component conductivity (heat transfer between the material compo-
nents) and the standard volume heat conductivity.
269Thermomechanics
With great internal heat transfer, the heat propagates as in homogeneous mate-
rial and only one diffusive heat conduction equation can be used for the
description.
With the internal heat transfer comparable to the material thermal conductivity,
the heat propagates in parallel within the framework of material components and
heat exchange occurs among them. To describe it mathematically, one equation of
parallel heat propagation must be used or, alternatively, two diffusive equations
with mutual heat transfer coupling.
With less intensive internal heat transfer, the heat propagates in the material in
parallel within the framework of the components, between which thermal interac-
tion scarcely occurs. Then, the mathematical description of the heat propagation
consists of two independent equations for the heat conduction in material compo-
nents. If the thermal conductivities of the material components do not differ too
much, only one single equation can be used for heat conduction, as in a homoge-
neous material.
In contrast to the standard expression (1), with the characteristic dimension
being replaced by the product of the thermal process time and the equivalent paral-
lel heat propagation rate, the Biot volumetric number of internal heat transfer can
be defined as in expression (2).
Info: [A17],[B45].
Jean-Baptiste Biot (p. 174).
5.9.2 Cattaneo Number Cat
Cat5τλw2
ν2Rcp
τ (s) � cas; λ (W m21 K21) � thermal conductivity of fluid flow; w (m s21) �flow velocity; ν (m2 s21) � kinematic viscosity; R (kg m23) � fluid density;
cp (J kg21 K21) � specific heat capacity.
It expresses the existence and propagation of a discontinuity in the velocity gra-
dient and temperature of a flowing fluid along a wall, heated with an impulse
method or randomly by an intensive heat source. Non-equilibrium
thermomechanics.
Info: [B93].
5.9.3 Coupling Factor Ncoup
Ncoup 5GL2
λref
G (W m23 K211) � electron�phonon coupling factor; L (m) � characteristic
length; λref (W m21 K21) � thermal conductivity.
270 Dimensionless Physical Quantities in Science and Engineering
It represents the ratio of the square of the convective electron heat transfer to
the heat transferred by conduction. It expresses an energetic coupling between elec-
tron gas and a grid. Wave heat propagation. Non-equilibrium thermomechanics.
Physical technology.
Info: [A70],[B63].
5.9.4 Dominance Number K
K5a
c2@2T
@t2@T
@t
� �21
a (m2 s) � thermal diffusivity; c (m s21) � velocity propagation of thermal wave;
T (K) � temperature; t (s) � time; τ (s) � relaxation time ðτ5 a=c2Þ:The wave character of heat transfer predominates for Kc1; i.e. (1) if the critical
frequency c2/a is small (it is the inverse value of the relaxation time τ), (2) if theprocess time t is very short or (3) if the heating or cooling rate (caused by the heat
flow) is high. The first condition involves the solid material properties only,
whereas the second and third depend also on the combined influence of the geo-
metric arrangement and system thermal load. This criterion is more general than
the usual wave propagation one.
Info: [A46].
Robert Tzou (born 1955), American physical engineer.
He is engaged in unbalanced macro- and microscopic
heat propagation, in shock heat processes and their model-
ling. Furthermore, he was engaged in the diffusion anomaly
in amorphous surroundings, control of fast shock thermal
processes, thermomechanical properties of composites, ther-
momechanical interactions in fractures and high-velocity
penetration through surroundings. His monographs Macro-
to Microscale Heat Transfer (1997) and Ultrafast Heating
and Thermomechanical Coupling Induced by Femtosecond
Lasers (2007) are especially significant.
5.9.5 Fourier Mass Relaxation Number Form
Form 5amτrxL2
am (m2 s21) � mass diffusivity; τrx (s) � relaxation time of material; L (m) � char-
acteristic length.
It characterizes the relaxation time of a non-stationary mass transfer.
Info: [A23].
Jean Baptiste Joseph Fourier (p. 175).
271Thermomechanics
5.9.6 Fourier Porous Number (1.) Fop
Fop 5λekvτtcL2
5cf
cSp21
where τt �Rhcf
α; c5 εcf 1ð12 εÞcs; ε5
Vf
V
λekv (W m21 K21) � equivalent thermal conductivity; τt (s) � time delay of tem-
perature; c, cf, cs (J m23 K21) � specific volume heat capacity: final, fluid and
solid material; L (m) � thickness of porous layer; Rh (m) � hydraulic diameter;
α (W m22 K21) � internal heat transfer; ε (�) � volume ratio; Vf, V (m3) � fluid
volume, total volume; Sp (�) � Sparrow number (p. 281).
This number characterizes the dimensionless time in heat transfer by fluid flow-
ing through porous material. It expresses the influence of the non-equilibrium heat
propagation.
Info: [B76].
Jean Baptiste Joseph Fourier (see above).
5.9.7 Fourier Porous Number (2.) Fop
Fop 5c2fw
2τcλekv
5λekvτcL2
cfwL
λekv
5Fo�pPe2p
cf (J m23 K21) � fluid specific volume heat capacity; w (m s21) � constant mean
velocity; τ (s) � time; c (J m23 K21) � final specific volume heat capacity; λekv(W m21 K21) � equivalent thermal conductivity; L (m) � porous layer thickness;
τq, τt (s) � time delay of heat flux and temperature; Fo�p (�) � Fourier porous
number (1.) (p. 272); Pep (�) � Peclet porous number (p. 279).
It characterizes the dimensionless time in forced flow and heat transfer in a
porous material. The equilibrium state, constant velocity and times τq5 τt are
considered.
Info: [B79].
Jean Baptiste Joseph Fourier (see above).
5.9.8 Fourier Relaxation Heat Number Forx
Forx 5aτrxL2
5 ShPe21
a (m2 s21) � thermal diffusivity; τrx (s) � relaxation time; L (m) � characteristic
length; Sh (�) � Strouhal number (p. 87); Pe (�) � Peclet heat number (p. 180).
272 Dimensionless Physical Quantities in Science and Engineering
It characterizes the inertia of temperature equalizing in a thermal system. It
represents the part of the Fourier number Fo (p. 175) corresponding to the thermal
field relaxation time in a system. The relaxation time is that during which all para-
meters of the considered thermal system are equalized.
The influence of the Fourier relaxation heat number can be expressed, analo-
gously to the molecular heat and mass transfers, in three bands: wave conduction
for Forx. 102; wave-diffusive conduction for 1024,Forx, 102; and diffusive con-
duction for Forx, 1024. Non-equilibrium thermomechanics. Physical technology.
Info: [A23],[B45].
Jean Baptiste Joseph Fourier (see above).
5.9.9 Fourier Relaxation Parallel Number Foq
Foq 5aτqL2
5Bi21V P1c ð1Þ;
Foq 51
L2λ1 1λ2
c1 1 c2
� �1
αV
1
c11
1
c2
� �21ð2Þ
a (m2 s21) � thermal diffusivity; τq (s) � relaxation time of parallel propagation;
L (m) � characteristic length; λ1, λ2 (W m21 K21) � thermal conductivity of mate-
rial components; c1, c2 (J m23 K21) � specific volume heat capacity of material
components; αV (W m23 K21) � volume coefficient of internal heat transfer;
BiV (�) � Biot volumetric number of internal heat transfer (p. 269); Pc (�) �relative parallel heat capacity (p. 279).
It determines the timescale for material transition from the non-equilibrium
state, which is caused by different temperatures of its individual components in one
place. The velocity, with which the material reaches equilibrium by the action of
internal heat transfer, depends on the relaxation time value of parallel heat propaga-
tion, where the equilibrium is expressed by equal temperatures of all transition
components. The relaxation time value is defined as the product of the material
thermal capacity and the internal transfer flow of unit energy. The relaxation time
represents the effective time during which the whole material thermal content could
be transported among its components. The Fourier relaxation parallel number
expresses the ratio of the series and parallel material thermal capacities and the
inner heat transfer (1). In the case of two-component material, it can be expressed
as in expression (2).
Info: [A17].
Jean Baptiste Joseph Fourier (see above).
5.9.10 Fourier Relaxation Wave Number Foq
Foq 5aτqL2
5l
L
� �2
273Thermomechanics
a (m2 s21) � thermal diffusivity; τq (s) � relaxation time of wave heat propaga-
tion; L (m) � characteristic length; l (m) � mean free path of heat carrier.
The relaxation time of heat wave propagation describes the effect of inelastic
collisions with which the thermal wave energy converts to the system internal
energy and is damped in passing through the material. The Fourier relaxation
wave number equals the second power of the ratio of the mean free path and the
characteristic system dimension.
By making use of the analogy with low pressure gas flow, the influence of Foqon the propagation character can be determined. With Foq. 102, almost no colli-
sions occur in the considered material volume, there is no diffusion, and the energy
carriers transfer the heat freely directly inside the material. With 1024,Foq, 102,
a transient state exists. With the Foq value decreasing, the heat transfer wave char-
acter weakens steadily because of the effect of collisions, damping the wave propa-
gation, increases, and, therefore, the propagation character changes to diffusive
propagation. For Foq, 1024, the collisions break the propagation of heat and dis-
tribute it in all directions, resulting in the known character of diffusive propagation.
Info: [A17],[B45].
Jean Baptiste Joseph Fourier (see above).
5.9.11 Fourier Thermalization parallel Number (1.) FoPT
FoPT 5aPτTL2
5Bi21V P21λ ð1Þ;
FoPT 51
L2λ1λ2
λ1c2 1λ2c1
� �1
αV
λ1c2 1λ2c1
λ1 1λ2
� �ð2Þ
aP (m2 s21) � parallel thermal diffusivity; τT (s) � thermalization time of parallel
heat propagation; L (m) � characteristic length; λ1, λ2 (W m21 K21) � thermal
conductivity of material components; c1, c2 (J m23 K21) � specific volume heat
capacity of material components; αV (W m23 K21) � volume coefficient of internal
heat transfer; BiV (�) � Biot volumetric number of internal heat transfer (p. 269);
Pλ (�) � relative parallel thermal conductivity (p. 280).
It expresses the ratio of series and parallel thermal material conductivities and
internal heat transfer (1). In the case of two-component material, it can be
expressed as in equation (2).
Info: [A17],[B45].
Jean Baptiste Joseph Fourier (see above).
5.9.12 Fourier Thermalization Parallel Number (2.) FoT
FoT1 5aτTL2
5Bi21V P21λ P21
1 5FoPTP21a ð1Þ;
274 Dimensionless Physical Quantities in Science and Engineering
FoT 51
L2λ1 1λ2
c2 1 c1
� �1
αV
λ1c2 1λ2c1
λ1 1λ2
� �ð2Þ
a (m2 s21) � thermal diffusivity; τT (s) � thermalization time of parallel heat prop-
agation; L (m) � characteristic length; λ1, λ2 (W m21 K21) � thermal conductivity
of material components; c1, c2 (J m23 K21) � specific volume heat capacity of
material components; αV (W m23 K21) � volume coefficient of internal heat transfer;
BiV (�) � Biot volumetric number of internal heat transfer (p. 269); Pλ (�) � rela-
tive parallel thermal conductivity (p. 280); Pa (�) � relative parallel thermal diffu-
sivity (p. 280); FoPT (�) � Fourier thermalization parallel number (1.) (p. 274).
It expresses the relation between internal heat transfer, series and parallel ther-
mal material conductivities and diffusivity (1). It is a modification of the Fourier
thermalization parallel number (1.) (p. 274) by the mutual ratio of the series and
parallel material thermal diffusivities. In the case of two-component material, it
can be expressed as in equation (2).
Info: [A17],[B45].
Jean Baptiste Joseph Fourier (see above).
5.9.13 Fourier Thermalization Wave Number FoT
FoT 5vτqL
vτTL
5aτTL2
v (m s21) � heat carrier velocity propagation; τq (s) � relaxation time of wave
heat propagation; τT (s) � thermalization time of wave heat propagation; L (m) �characteristic length; a (m2 s21) � thermal diffusivity.
The thermalization time of the wave heat propagation describes the effect of
elastic collisions with which the thermal wave momentum is not lost, but the veloc-
ity distribution of heat carriers is formed. The Fourier thermalization wave number
expresses the product of the ratio of the path covered by a heat carrier during the
relaxation time to the characteristic length of the observed material zone and
the ratio of the path covered by a heat carrier during the thermalization time to the
characteristic length of the observed material zone.
Info: [A17],[B45].
Jean Baptiste Joseph Fourier (see above).
5.9.14 Fourier Wave Number Fo
Fo5vτL
vτqL
v (m s21) � heat carrier velocity propagation; τ (s) � time; τq (s) � relaxation
time of wave heat propagation; L (m) � characteristic length of a system.
275Thermomechanics
It characterizes the dimensionless time of the wave heat propagation process. It
is the product of the ratio of the path covered by a heat carrier during the time τ to
the characteristic system length and the ratio of the path covered by a heat carrier
during the relaxation time to the characteristic system length.
Info: [A17],[B45].
Jean Baptiste Joseph Fourier (see above).
5.9.15 Heat Transfer in Micropores Kτ
Kτ 5τekvτt
5ð12 εÞcs
c
where τt �Rhcf
α; τekv 5
ð12 εÞcsτtc
; ε5Vf
V
τekv (s) � equivalent time; τt (s) � time delay, thermalization time of non-equilib-
rium heat propagation; ε (�) � volume ratio; Vf, V (m3) � fluid volume and total
volume; c, cs, cf (J m23 K21) � specific volume heat capacity: total, solid and fluid;
Rh (m) � hydraulic radius of nano or microchannels; α (W m22 K21) � heat trans-
fer coefficient.
It limits the character of non-equilibrium heat propagation in fluid flow through
micropores.
5.9.16 Knudsen Phonon Number Knph
Knph 5
ffiffiffiffiffiffiffiffiffiffi3aτd
p
h5
ffiffiffiffiffi3τ
p; where τd 5
h2
a
a (m2 s21) � thermal diffusivity; τd (s) � diffusion time; h (m) � thickness; τ (s) �time constant of Cattaneo�Vernotte.
It characterizes heat wave propagation and determines when the classic
Cattaneo�Vernotte mathematical model can be used. This model cannot be used in
cases in which it exceeds the limits of KnphAh0.35; 0.5i.Info: [B35].
Martin Hans Christian Knudsen (p. 420).
5.9.17 Mikheyev Volumetric Number of Internal Heat Transfer MiV
See the non-equilibrium parallel heat transfer number Knep (p. 277).
Mikhail Alexandrovich Mikheyev (1902�1970), Russian engineer.
276 Dimensionless Physical Quantities in Science and Engineering
5.9.18 Number of Non-Equilibrium Heat Transfer Kτ
Kτ 5τTτq
5FoT
Foq
τT (s) � thermalization time of non-equilibrium (wave or parallel) heat propaga-
tion; τq (s) � relaxation time of non-equilibrium (wave or parallel) heat propaga-
tion; FoT (�) � Fourier thermalization wave number (p. 275) or Fourier
thermalization parallel number (2.) (p. 274); Foq (�) � Fourier relaxation wave
number (p. 273) or Fourier relaxation parallel number (p. 273).
This number defines the physically allowed zone of wave and parallel non-equi-
librium heat propagation. The character of non-equilibrium heat propagation is
determined by the thermalization to relaxation time ratio of material. The value
Kτ5 1 represents equilibrium diffusion propagation. With Kτ, 1, it expresses the
non-equilibrium heat wave propagation which changes to diffusion with increasing
time. For Kτ. 1, it expresses the non-equilibrium parallel heat propagation.
Info: [A17],[B45].
5.9.19 Number of Non-Equilibrium Parallel Heat Transfer Knep
Knep 5Fo
Foq5
ταV
cS5FoBiVPc
τ (s) � time; αV (W m23 K21) � volume coefficient of internal heat transfer;
cS (J m23 K21) � series specific volume heat capacity; Fo (�) � Fourier number
(p. 175); Foq (�) � Fourier relaxation parallel number (p. 273); BiV (�) � Biot
volumetric number of internal heat transfer (p. 269); Pc (�) � relative parallel heat
capacity (p. 279).
This criterion is defined as the ratio of the Fourier number (p. 175) to the
Fourier relaxation parallel number (p. 273). It determines the time zone of
the action of parallel non-equilibrium heat propagation. It expresses the ratio of the
heat transferred inside a material between its components to the total thermal con-
tent of a body. The Biot volumetric number of internal heat transfer BiV (p. 269)
and the relative parallel heat capacity Pc (p. 279) decide whether in the time Fo,
since the origin of the non-equilibrium state, the material is already in an equilib-
rium state or still in a non-equilibrium state. The criterion is also called the
Mikheyev volumetric number of internal heat transfer MiV (p. 276).
With Knep, 1022, the internal energy exchange between material components
with unequal temperatures is not able to manifest itself in the short time, since the
origin of the non-equilibrium state with the action of a heat source. In this phase,
the thermal process can be described by a set of independent diffusive heat conduc-
tion equations for individual material components. If 1022,Knep, 101, the energy
transfer appears between the components inside the material. With increasing time,
the temperatures of individual components come near to the equilibrium values.
277Thermomechanics
The heat propagation process must be expressed by the parallel heat propagation
equation or by the set of diffusive equations with mutual coupling. For Knep. 101,
sufficient time has elapsed already since the origin of the non-equilibrium state that
the material may get, due to internal redistribution of energy, into the equilibrium
state with equal temperature of all of its components. Then, the heat propagation
can be expressed as a rule by one diffusion heat propagation equation for the mate-
rial as a whole.
Info: [A17],[B45].
5.9.20 Number of Non-Equilibrium State of Matter Knsm
Knsm 5Fo
Forx
Fo (�) � Fourier number Fo (p. 175); Forx (�) � Fourier relaxation heat number
(p. 272).
It expresses the time zone of non-equilibrium wave heat conduction. It is the cri-
terion which enables the division of the wave-diffusive heat conduction process
into three time zones: the non-equilibrium heat conduction with Knsm, 1021; the
transient zone with 1021,Knsm, 101; and the equilibrium diffusive heat conduc-
tion with Knsm. 101. Non-equilibrium thermomechanics. Physical technology.
Info: [A17],[B45].
5.9.21 Number of Non-Equilibrium Wave Heat Transfer Knev
Knev 5Fo
Foq
Fo (�) � Fourier wave number (p. 275); Foq (�) � Fourier relaxation wave
number (p. 273).
This criterion is defined as the ratio of the Fourier wave number (p. 275) to the
Fourier relaxation wave number (p. 273). It determines the time of acting of the
wave non-equilibrium heat propagation.
With Knev, 1022, the heat carriers forming the thermal wave propagate through
a material without collisions. In the short time since the origin of non-equilibrium
state due to a heat source, no collision process creating material thermal resistance
occurred yet. In this phase, the heat propagation process can be described by a
wave propagation equation. If 1022,Knev, 101, the wave propagation damping
appears with increasing time and the material gets into the equilibrium state gradu-
ally. The heat propagation process is expressed by a wave-diffusion equation. With
Knev. 101, sufficient time has elapsed since the origin of the non-equilibrium state
that the wave propagation character may be damped. Then, the heat propagation
can be expressed by a heat conduction diffusion equation.
Info: [A17], [B45].
278 Dimensionless Physical Quantities in Science and Engineering
5.9.22 Nusselt Porous Number Nup
Nup 5αRp
λf
α (W m22 K21) � heat transfer coefficient; Rp (m) � hydraulic radius expressed as
pore volumes ratio to their surface (ΔVp/ΔAp); λf (W m21 K21) � thermal conduc-
tivity of fluid.
This number characterizes the convection heat transfer in porous material. It is
applied especially if the heat conduction predominates, due to which its size is con-
stant. It occurs in cases where the Rayleigh pore flow number Rapor (p. 111) is
small and the natural convection in pores predominates in the heat conduction.
Equilibrium and non-equilibrium heat transfers. Two-phase flow.
Info: [B55].
Ernst Kraft Wilhelm Nusselt (p. 196).
5.9.23 Peclet Porous Number Pep
Pep 5cfwL
λekv
cf (J m23 K21) � specific volume heat capacity of fluid; w (m s21) � mean flow
velocity; L (m) � characteristic length, porous layer thickness; λekv (W m21 K21) �equivalent thermal conductivity; Sp (�) � Sparrow number (p. 281); Pe (�) �Peclet heat number (p. 180).
It characterizes the velocity of fluid flowing through porous material. The vari-
able velocity frequency has a strong influence on the heat transfer. The ratio
SpPe21 shows approximately whether there is a local equilibrium state in the
porous material.
Info: [B76].
Jean Claude Eugene Peclet (p. 180).
5.9.24 Relative Parallel Heat Capacity Pc
Pc 5cp
cs5X
ciX 1
ci
cp (J m23 K21) � parallel heat capacity ðcp 5
PciÞ; csðJ m23 K21Þ � series specific
volume heat capacity cs 5P
1ci
� 21� �; ciðJ m23 K21Þ � specific volume heat
capacity of ith material component.
It expresses the parallel-to-series thermal capacities ratio of a material, with said
capacities being determined by the heat capacities of individual material components.
Info: [A17],[B45].
279Thermomechanics
5.9.25 Relative Parallel Heat Transfer Rate Pv
Pv 51
wt
ffiffiffiffiffia
τq
rð1Þ;
Pv 51
wt
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiαV
λ1 1λ2
c1c2
rð2Þ
wt (m s21) � heat propagation velocity; a (m2 s21) � thermal diffusivity; τq (s) �relaxation time of parallel heat propagation; αV (W m23 K21) � volume coefficient
of internal heat transfer; λ1, λ2 (W m21 K21) � thermal conductivity of material
components; c1, c2 (J m23 K21) � specific volume heat capacity of material
components.
It expresses the ratio of the apparent parallel heat transfer rate to the heat propa-
gation rate in a material (1). In the case of a two-component material, it can be
expressed as in equation (2).
Info: [A17],[B45].
5.9.26 Relative Parallel Thermal Conductivity Pλ
Pλ 5λp
λs
5X
λi
X 1
λi
λp (W m21 K21) � parallel thermal conductivity ðλp 5P
λiÞ;λs (W m21 K21) � series thermal conductivity ðλs 5 ðP 1
λiÞ21Þ;
λi (W m21 K21) � thermal conductivity of ith material component.
It expresses the ratio of the parallel thermal material conductivity to the series
conductivity, with said conductivities being determined by the thermal conductivi-
ties of individual material components.
Info: [A17],[B45].
5.9.27 Relative Parallel Thermal Diffusivity Pa
Pa 5ap
a
a (m2 s21) � thermal diffusivity; ap (m2 s21) � parallel thermal diffusivity.
It expresses the character of non-equilibrium heat propagation by means of the
parallel-to-standard heat diffusivities ratio.
The relative parallel diffusivity has two limits. In the case Pa-0, i.e. ap-0, the
heat propagates in material mostly by mediation of one kind of heat carrier. With
Pa-1, i.e. ap-a, the heat propagates nearly exclusively by a parallel mechanism.
280 Dimensionless Physical Quantities in Science and Engineering
In materials with two kinds of heat carriers, the thermal capacity of one of the
carriers is often many times greater than that of the second carrier and, on the con-
trary, the thermal conductivity of the first carrier is substantially less than that of
the second one.
Info: [A17],[B45].
5.9.28 Relaxation Time Nτ,rx
Nτ;rx 5τrxw2
ν5CatPr
τrx (s) � relaxation time; w (m s21) � fluid flow velocity; ν (m2 s21) � kinematic
viscosity; Cat (�) � Cattaneo number (p. 270); Pr (�) � Prandtl number (p. 197).
In non-equilibrium thermodynamics, this number expresses the relaxation time �in dimensionless form � in flowing fluid along a plate wall, heated by impulses or
strokes. It is used, for example, in observing the discontinuities of velocity gradients
and temperatures.
Info: [B77].
5.9.29 Sound Speed Number K
K5vτqL
� 2v (m s21) � adiabatic sound velocity in liquid; τq (s) � relaxation time; L (m) �characteristic length.
It expresses the ratio of the product of the relaxation time characterizing the
hyperbolic fluid behaviour and the in-fluid propagation rate in the fluid to the char-
acteristic dimension of the system.
Info: [B43].
5.9.30 Sparrow Numbers Sp
Sp5αL2
λekvRh
5Nuλf
λekv
� �L
Rh
� �2
α (W m22 K21) � equivalent heat transfer coefficient; L (m) � characteristic
length, porous layer thickness; λekv (W m21 K21) � effective thermal conductivity;
λf (W m21 K21) � fluid thermal conductivity; Rh 5ΔVp=ΔApðmÞ � hydraulic
radius; ΔVp (m3) � pore volume; ΔAp (m2) � pore surface; Nu (�) � Nusselt
number (p. 196); Pe (�) � Peclet heat number (p. 180).
281Thermomechanics
It characterizes the non-equilibrium heat transfer in porous material. Essentially,
it expresses the extension of the Nusselt number Nu (p. 196) for equilibrium heat
propagation to non-equilibrium heat propagation, which is of use in the flow in
porous materials. Usually λf,λekv holds. The number Sp enables the estimation of
the time delay due to the process non-equilibrium. The fraction SpPe21 enables
estimating local equilibrium in fluid flow through porous material.
Info: [B76].
5.9.31 Thermal Wave Speed Number K
K5Lv
að1Þ;
K5Ll
aτq5
1ffiffiffiffiffiffiffiffiFoq
p ð2Þ
L (m) � characteristic length; v (m s21) � velocity of the heat wave propagation;
a (m2 s21) � thermal diffusivity; l (m) � mean free path; τq (s) � relaxation time;
Foq (�) � Fourier relaxation wave number (p. 273).
It expresses the ratio of the thermal wave propagation rate to the characteristic
system dimension and the thermal diffusivity of the material or surroundings. With
the propagation rate expressed by the mean free path and the wave propagation
relaxation time v5 1=τq; the criterion is simplified to equation (2). Then, it
expresses the inverse root of the Fourier relaxation wave number Foq (p. 273).
Info: [B24].
5.9.32 Vernotte Heat Number Ve
Ve5Pe21 5a
wtLð1Þ;
Ve5a
wtrLð2Þ;
where wtr 5
ffiffiffiffia
τr
r
a (m2 s21) � thermal diffusivity; wt (m s21) � velocity of heat propagation;
wtr (m s21) � relaxation velocity of heat propagation; L (m) � characteristic
length; τr (s) � relaxation time; Pe (�) � Peclet heat number (p. 180).
It characterizes the influence of the finite heat propagation rate in a body. Wave
heat transfer in stroke phenomena. Non-equilibrium thermomechanics. The action
of intensive heat sources on material. New technologies.
Info: [A23].
Pierre Vernotte (born 18.5.1898), French physicist.
282 Dimensionless Physical Quantities in Science and Engineering
5.9.33 Vernotte Thermoelastic Number Vete
Vete 5Pe21te 5
a
wteL; where wte 5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiλ1 2μ
R
s
a (m2 s21) � thermal diffusivity; L (m) � characteristic length; wte (m s21) �velocity of thermoelastic deformation propagation; λ, μ (Pa) � Lame functions;
R (kg m23) � density; Pete (�) � Peclet thermoelastic number (p. 143).
It characterizes the wave propagation rate of the thermoelastic stress or deforma-
tion in heat systems with stroke phenomena. Its inverse value is called the Peclet
thermoelastic number Pete (p. 143). Thermomechanics of stroke thermal processes.
New technologies.
Info: [A23].
Pierre Vernotte (see above).
5.9.34 Wave Penetration Depth (1.) H
H5FoffiffiffiffiffiffiffiffiffiForx
p ð1Þ;
ΘðFoÞ5Θð0Þexp 2Fo
2Forx
� �ð2Þ
Fo (�) � Fourier number Fo (p. 175); Forx (�) � Fourier relaxation heat number
(p. 272).
It characterizes a material zone in which the heat wave propagation influence
predominates. In the time Fo, it expresses the heat wave penetration depth
into material with wave-diffusive heat propagation and impulse action of surface
thermal flow. The front of the arising thermal wave is damped gradually (2).
Non-equilibrium thermomechanics. Physical technology.
Info: [A17],[B63].
5.9.35 Wave Penetration Depth (2.) Effective Hef
Hef 5 2ffiffiffiffiffiffiffiffiffiForx
p
Forx (�) � Fourier relaxation heat number (p. 272).
It expresses the heat wave penetration depth into a material with which the
wave temperature drops e times in passing through the material. It also confirms
that the penetration depth is proportional to the mean free path of the heat carriers.
Non-equilibrium thermomechanics. Physical technology.
Info: [A17],[B63].
283Thermomechanics
6 Electromagnetism
Finally, some observations of the momentum transfer from the Sun to the Planets
have been carried out and have fundamental significance for the theory. The signif-
icance of magneto-hydrodynamic waves is remarkable from this aspect.Hannes Olof Gosta Alfven (1908�1955)
Electromagnetism represents the fundamental interaction between electricity and
magnetism. The electric dimensionless quantities involve electric fields and cir-
cuits, electronics, illumination, electrical energy and a wide range of applications
in various fields. The magnetic dimensionless quantities express, above all, basic
magnetic forces caused by electric charge movement, propagation of electromag-
netic energy and waves, and other transfer phenomena, such as plasma physics and
magneto-hydrodynamics. The magneto-hydrodynamic dimensionless quantities
express the propagation of magneto-hydrodynamic waves, dynamic pressure, force
relations and energy conversion.
6.1 Electricity
A relatively large number of dimensionless quantities involve both electric circuits
and fields, and are as diverse as electrothermal, ionization and corona diffusion
processes; natural and forced electrothermal convection; heat transfer intensifica-
tion with dielectric fluid condensation; condensation in an electric field; condensate
film stabilization; mass transfer in electrochemical processes; the motion of
charged particles in flowing gas; and many other applications. Among them, the
dimensionless criteria for diverse boundary conditions and internal sources express
the inhomogeneity and non-linearity of the environment, charge density, electric
field intensity, corona phenomenon, earthing and puncture. Of the many dimension-
less numbers, the following are especially important: the electromechanical and
electronic numbers, the Fourier electrical number, the electric field intensity num-
ber, the Joule electrothermal number, the corona number, the Kronig number, the
Ohm number and many equivalents of basic quantities.
Dimensionless Physical Quantities in Science and Engineering. DOI: 10.1016/B978-0-12-416013-2.00006-3
© 2012 Elsevier Inc. All rights reserved.
6.1.1 Biot Electrical Number Bie
Bie 5L
RαγS
L(m) � characteristic length; RαðΩÞ � electrical resistance equivalent to convective
thermal resistance; γ (S m21) � specific electrical conductance; S (m2) � surface
area in accordance with resistance Rα.
It characterizes the third-type boundary condition of an electric system. It is
analogous to the thermal Biot number Bi (p. 173).
Info: [A23].
Jean-Baptiste Biot (p. 174).
6.1.2 Breakdown Coefficient N
N5Iρ
L2Ep
5U
LEp
I (A) � electric current; ρ (Ω m)� specific electrical resistance; L (m) � characteristic
length; Ep (V m21) � earth breakdown gradient; U (V) � spark zone surface voltage.
This coefficient expresses the dimensionless value of the earth breakdown gradi-
ent. Earthing.
Info: [A24].
6.1.3 Characteristic Electrical Number Nche
Nche 5 ρdεdT
� �L2ΔTE2
η2
ρ (kg m23) � density; ε (F m21) � permittivity; T (K) � temperature; L (m) �characteristic length; ΔT (K) � temperature difference; E (V m21) � electric field
intensity; η (Pa s) � dynamic viscosity.
It characterizes the effect of the temperature change of environmental dielectric
parameters with the electrostatic field in relation to the heat transfer from a heated
surface. Sometimes, it is incorrectly called the Kronig number Kr (p. 299).
Info: [A23].
6.1.4 Charge Density Number NQ
NQ 5wLρeH
w (m s21) � velocity; L (m) � characteristic length; ρe (C m23) � volume charge
density; H (A m21) � magnetic field intensity.
286 Dimensionless Physical Quantities in Science and Engineering
It expresses the ratio of the electric charge flow intensity to the magnetic field
intensity.
Info: [C173].
6.1.5 Clausius�Mossotti relation K
KðωÞ5 ε�1 2 ε�2ε�1 1 2ε�2
; where ε� 5 ε2jγω
ε�1; ε�2 (F m21) � permittivity of two different environments; γ (S m21) � specific
electrical conductance; ω (s21) � angular frequency of electrical field; j (2) �imaginary unit.
This relation is used in electrokinetics. The real part determines the coefficient
of the electromagnetic force acting on a particle. The imaginary part expresses the
electrorotating moment. The particle size and the electric field are other influences.
Electricity. Electrokinetics.
Info: [C20].
Rudolf Julius Emanuel Clausius (p. 175).
Ottaviano-Fabrizio Mossotti
6.1.6 Communication Coefficient N
N5ω ε0 β AR
ω (s21) � angular frequency; ε0 (F m21) � vacuum permittivity; β (m21) � wave
number (β5 2πλ21); A (m2) � cross section area of electron flux; R (Ω) �resistance.
It expresses the relative efficiency of the electron flow and the electric field
wave. Communications engineering.
Info: [A24].
6.1.7 Condensate Film Stability (1.) N
N5εE2L2
ρlν2l
ε (F m21) � permittivity; E (V m21) � electric field intensity; L (m) � characteristic
length; ρl (kg m23) � condensation layer density; νl (m2 s21) � condensation layer
kinematic viscosity.
It expresses the Coulomb-to-viscosity forces ratio. It characterizes the degree of
the stability damage of a condensate film and its deformation in an electric field.
This appears as formation of Coulombs and abruptly formed condensate drops on
287Electromagnetism
the surface due to longitudinally nonuniformly distributed pressure (εE2/2). With
sufficiently great E, the film is damaged and the condensate drops are sprayed,
with the electric force overcoming the viscosity force.
Info: [A23].
6.1.8 Condensate Film Stability (2.) N
N5εE2L
σ
ε (F m21) � permittivity; E (V m21) � electric field intensity; L (m) � characteris-
tic length; σ (N m21) � surface tension.
It expresses the ratio of the Coulomb force to the surface stress force. It charac-
terizes the stability degree of a condensate film and its deformation in an electric
field in relation to the surface stress force.
Info: [A23].
6.1.9 Condensation Electrostatic Number N
N5ðεgrad E2Þlδl
gρLð1Þ; N5
ðεE2ÞυgρL
ð2Þ
ε (F m21) � permittivity; E (V m21) � electric field intensity; δl (m) � condensate
film thickness; g (m s22) � gravitational acceleration; ρ (kg m23) � condensate
density; L (m) � characteristic length; index y: l � condensate, υ � vapour.
This number expresses the ratio of the gravity force to the ponderomotive force
of an electric field. It characterizes the heat transfer intensity with dielectric fluid
condensation on an outer tube surface (L5D/2) in an inhomogeneous electrostatic
field. In expression (1), it represents the volume ponderomotive force; in expres-
sion (2), it represents the surface force. The surface electrostatic force, acting on
the interface of phases, has an especially strong influence on heat transfer
intensification.
Info: [A23].
6.1.10 Corona Number NK
NK 5r0
ffiffiffiffiffiffiffiffiε0ω
pffiffiffiffiffiτk
p
r0 (m) � wire radius; ε0 (F m21) � vacuum permittivity; ω (s21) � angular fre-
quency; τ (A s m21) � linear density of electric charge; k (A s2 kg21) � ion
mobility.
It expresses the corona phenomenon.
Info: [A24].
288 Dimensionless Physical Quantities in Science and Engineering
6.1.11 Current Source Number NJ
NJ 5J0 R Δ x
U
J0 (A m21) � linear density of electric current; R (Ω) � resistance; Δx (m) �length of element; U (V) � voltage.
It concerns the current source related to 1 m of the conductor length. It deter-
mines, for example, the electric loss in conductors. Electric machines and devices.
Info: [A24].
6.1.12 Dielectric Constant εr
εr 5εsε0
ð1Þ; ε5 ε0εr ð2Þ
εs (F m21) � static material permittivity; ε0 (F m21) � vacuum permittivity
(ε05 8,8541873 10212 F m21); ε (F m21) � absolute permittivity.
In the case of a static field, it expresses the insulation properties of a dielectric
material. In the case of an alternating field or electromagnetic oscillation, it
expresses the relation between the vectors of electric induction and electric field
and depends on the frequency. Expression (2) represents the absolute permittivity.
The values of εr are: for air ðεr61Þ, for glass ðεr67:6Þ and for water ðεr680Þ.Electromagnetism. Electric fields. Physical properties.
Info: [C113].
6.1.13 Diffusion Corona Number N
N5D
LkE
D (m2 s21) � mass diffusion coefficient in electric field; L (m) � characteristic
length; k (A s2 kg21) � ion mobility; E (V m21) � electric field intensity.
It expresses the ratio of the diffusion mass transfer to rectified ion movement. It
characterizes the mass diffusion process in a corona discharge. The diffusion begins
to appear with N$ 1, approximately.
Info: [A23].
6.1.14 Dynamics of Inner Heat Source Number Dy
Dy5w2tδT 2w1t
w1t
; where w1t 5w1qV
cρ; w2t 5
w2qV
cρ
289Electromagnetism
δT (2) � thermogradient coefficient; w1, w2 (K s21) � warming velocity;
qV (W m23) � volume density of heat flux; c (J kg21 K21) � specific heat capacity;
ρ (kg m23) � density.
It characterizes the influence of an inner heat source on heat and mass transfers,
when power changes are affected by material temperature changes. When Dy, 0,
the power increase; when Dy. 0, it decreases. In dielectric heating, the internal
volume source qV is known. Induction and dielectric heating. Electric heat.
Info: [A33].
6.1.15 Electric Field Intensity N
N5E
wμHð1Þ; N5
EL
w
ffiffiffiγη
rð2Þ
E (V m21) � electric field intensity; w (m s21) � velocity; μ (H m21) � permeability;
H (A m21) � magnetic field intensity; L (m) � characteristic length; γ (S m21) �specific electrical conductance; η (Pa s)� dynamic viscosity.
It characterizes the ratio between the fed and induced electric field intensities in
moving electrically conductive fluids. In examining the outer electric field influ-
ence independently from the induced electric field � e.g. with a zero value for the
outer magnetic field � it is more suitable to use expression (2). Sometimes it is
called the criterion of loading characteristics MHD canal.
Info: [A23],[A33].
6.1.16 Electrical Circuit Number N
N5UτLI
5R2C
Lð1Þ; N5
τRC
ð2Þ
U (V) � voltage; τ (s) � time; L (H) � inductance; I (A) � electric current; R (Ω) �resistance; C (F) � capacitance.
Expression (1) describes the transient phenomena in an electric RLC circuit after
a sudden alternating voltage connection. The ratio of time to the electric circuit
time constant is expressed by (2).
Info: [A24].
6.1.17 Electrical Convection Number N
N5αη2
grad E2ρ
α (A2 s4 kg21) � molecule polarization coefficient; η (Pa s) � dynamic viscosity;
E (V m21) � electric field intensity; ρ (kg m23) � density.
290 Dimensionless Physical Quantities in Science and Engineering
It expresses the ratio of the molecular friction force to the electroconvective
one. It characterizes the dielectric fluids flow intensity influenced by the
electric field inhomogeneity. In natural convection, it appears as the Grashof heat
number. (p. 185)
Info: [A23],[A33].
6.1.18 Electrical Field Number NE
NE 5QNL4
τkU2
Q (C) � electric charge; N (m23 s21) � number of ion pairs; L (m) � length;
τ (C m21) � linear density of an electric charge; k (A s2 kg21) � anion mobility;
U (V) � voltage.
This number represents an electric field in ionized gas.
Info: [A12].
6.1.19 Electric Instability N2
N2 5βεdT
dxL
βε (K21) � thermal coefficient of permittivity; dT/dx (K m21) � temperature gradi-
ent; L (m) � characteristic length.
It characterizes the inhomogeneity grade of the electric environment permittivity
in solving problems of equilibrium fluid stability in a horizontal planar condenser.
Info: [A23].
6.1.20 Electric Relaxation Number Nrelax
Nrelax 5νεEJAL2
ν (m2 s21) � kinematic viscosity; ε (F m21) � permittivity; E (V m21) � electric
field intensity; JA (A m22) � surface density of electric current; L (m) � character-
istic length.
It expresses the ratio of electric phenomena relaxation in surroundings to the
mechanical relaxation time. It characterizes the relaxation effects related to electric
wind. When the ratio is small, electric relaxation occurs independently of mechani-
cal relaxation. Heat transfer in electrothermal systems.
291Electromagnetism
6.1.21 Electrocalorimetric Effect N
N5cpρΔT
εE2
cp (J kg21 K21) � specific heat capacity; ρ (kg m23) � density; ΔT (K) � tempera-
ture difference; ε (F m21) � permittivity; E (V m21) � electric field intensity.
This effect expresses the ratio of the heat obtained by transfer in the unit volume
to the energy density in an electric field. It characterizes the thermal effect of elec-
tric current passage through fluids without considering the fluid heating by friction,
polarization energy and energy of electrostatic charges. (K{1).
Info: [A12].
6.1.22 Electromechanical Number Es
Es5JAρL3
k1 η2
JA (A m22) � surface density of electric current at corona discharge in gas;
ρ (kg m23) � gas density; L (m) � characteristic length; k1 (A s2 kg21) � positive
charge carrier mobility; η (Pa s) � dynamic viscosity.
It characterizes the influence of the gas boundary layer damage, by corona elec-
tric discharge, on the intensity of the convective heat transfer between an electrode
and gas under conditions of natural or forced convection near the coronizing elec-
trode surface.
Info: [A23],[A33].
6.1.23 Electronic Number N
N5JAε0BLρ2
ð1Þ; N5Uε0L2ρ
ð2Þ;
N5JAL
Dρð3Þ; N5
Dε0BL2ρ
ð4Þ;
JA (A m22) � surface density of electric current; ε0 (F m21) � vacuum permittivity;
B (T) � magnetic induction; L (m) � characteristic length; ρ (C m23) � volume
density of electric charge; U (V) � voltage; D (m2 s21) � diffusion coefficient.
In expression (1), it expresses the current, limited by the volume charge, in real
dielectrics and semiconductors. In expression (2), it represents a voltage. In expres-
sion (3), it describes the current, limited by the volume charge, when considering
the diffusion of charge carriers in semiconductors and dielectrics with the diffusion
292 Dimensionless Physical Quantities in Science and Engineering
predominating over drift. In expression (4), the equivalent drift and diffusion influ-
ence is considered.
Info: [A24].
6.1.24 Electrothermal Convection Number Net
Net 5εβτE
2L3grad T
ρνa
ε (F m21) � permittivity; βτ 5 τ21r dτr=dTðK 21Þ � temperature coefficient of
relaxation time of electrical effects; E (V m21) � electric field intensity; T (K) �temperature; L (m) � characteristic length; ρ (kg m23) � density; ν (m2 s21) �kinematic viscosity; a (m2 s21) � thermal diffusivity; τr5 εγ21 (s) � relaxation
time of electrical effects.
It characterizes the electrothermal convection in fluids under the action of a
homogeneous or inhomogeneous field. It expresses also the forces of the electro-
thermal substance acting with the electrothermal convection. With no electric field
acting, it is Net5 0. The number Net expresses the electrothermal convection suit-
ably, especially with a homogenous field.
Info: [A23].
6.1.25 Electrothermal Convection Parameter N, r
N5εβτE
2
ρβgL5NetRa
212 5
εβτΔV2
ρβgL3
ε (F m21) � permittivity; βτ 5 τ21r dτr=dTðK 21Þ � temperature coefficient of elec-
trical effects on relaxation time; E (V m2 1) � electric field intensity; ρ (kg m23) �density; β (K21) � volume thermal expansion coefficient; g (m s22) � gravitational
acceleration; L (m) � characteristic length; V ðVÞ � electric potential; Net (2) �electrothermal convection number (p. 293); Ra2 (2) � Rayleigh number (2.)
(p. 187); τr (s) � relaxation time of electrical effects.
This parameter expresses the electrothermal to buoyancy forces ratio. It charac-
terizes the relation of electrothermal convection to free convection.
Info: [A23].
6.1.26 Electrothermal Conversion (1.) N
N5ρwhJAE L
293Electromagnetism
ρ (kg m23) � density; w (m s21) � flow velocity; h (J kg21) � specific enthalpy;
JA (A m22) � surface density of electric current; E (V m21) � electric field inten-
sity; L (m) � characteristic length.
It characterizes the process of electric energy conversion to thermal energy,
which proceeds in a gas arc. This criterion is among the fundamental ones of use,
for example, in work related to plasmatrones with heated gas.
Info: [A23].
6.1.27 Electrothermal Conversion (2.) N
N5ρw3
JAE L
ρ (kg m23) � density; w (m s21) � velocity; JA (A m22) � surface density of electric
current; E (V m21) � electric field intensity; L (m) � characteristic length.
It characterizes the process of electric energy conversion to kinetic energy of a
rectified flow. The sudden temperature increase of the heated gas and correspond-
ing density drop, with a given passage and cross section of a canal, are connected
to a remarkable velocity increase of the gradual gas flow. The criterion is of use,
for example, in a gas heater solution with supersonic flow in aerodynamic tunnels
if an electric arc is used in an accelerating nozzle.
Info: [A23].
6.1.28 Electrothermal Coupling Number NET
NET 5J2AL
2
γλΔT
JA (A m22) � surface density of electric current; L (m) � characteristic length;
γ (S m21) � specific electric conductance; λ (W m21 K21) � thermal conductivity;
ΔT (K) � temperature difference.
It expresses the coupling between electric current and heat conductions. With
this relation as defined by the number, further criteria expressing Joule heat action
can be obtained, for example, in flowing surroundings with physical and chemical
conversions.
Info: [A28].
6.1.29 Electrothermal Power N
N5γ2U4
gρ2c2T2d3βΔT5
γ2U4
ηλcT2
ηλgρ2cd3βΔT
5K1Gr21Pr21
294 Dimensionless Physical Quantities in Science and Engineering
γ (S m21) � specific electrical conductance; U (V) � voltage; g (m s22) � gravita-
tional acceleration; ρ (kg m23) � density; c (J kg21 K21) � specific heat capacity; T
(K) � local temperature in tank furnace; d (m) � electrode diameter; β (K21) � vol-
ume thermal expansion coefficient; ΔT (K) � temperature difference; η (Pa s) �dynamic viscosity; λ (W m21 K21) � thermal conductivity; K1 (2) � criterion K1;
Gr (2) � Grashof heat number (p. 185); Pr (2) � Prandtl number (p. 197).
It expresses the ratio of electric forces in molten material to mechanical forces
in free moving molten material with thermoelectric convection. It characterizes the
mutual relation among fundamental physical processes in an electric induction fur-
nace and their influence on heat transfer. The influence of the product (Gr Pr)
characterizes mechanical forces distributed in a tank that depends on slag para-
meters, temperature distribution and geometric dimensions. When Gr Pr. 23 107,
the process becomes automatic modelling. The criterion K1 depends on the charac-
ter of the electric field and inner source distribution.
Info: [A23].
6.1.30 Erlang Unit Eb, P
EbðN;AÞ5 AEbðN21;AÞN1AEbðN21;AÞ ; Ebð0;AÞ51 ð1Þ;
Pð. 0Þ5AN
N!N
N2APN21x5 0
Ax
x!AN
N!N
N2A
ð2Þ
N (2) � number of sources, for example, phones, servers or circuits; A (2) �operation size in dimensionless Erlang units.
The Erlang unit expresses the statistical degree of telecommunications opera-
tions used in telephony or internet connections, for example. One Erlang expresses,
for example, one source in uninterrupted operation or two sources at half of their
capacities. The Erlang unit is used to determine whether a telecommunications sys-
tem is overloaded or, on the contrary, underutilized � in other words, whether it
has many or very few utilized sources. It is applied to calculate the grade of service
(GoS) and the quality of service (QoS) as well. There are several diverse Erlang
equations, among which the Erlang B equation (1) and Erlang C equation (2) are
especially important.
Erlang relation B (1) expresses the probability Eb that a system is blocked. This
means that with a request not being realized immediately, it is blocked and can-
celled immediately. This relation is used for telephone systems which do not use
the so-called queues.
The Erlang relation C (2) expresses the probability P that a request remains in
the system queue for a certain length of time. In such a way, the quantity P(. 0)
expresses the probability that the response delay to the request is greater than zero,
meaning that the request is not realized immediately. This relation is used for tele-
phone ‘call centre’ systems, in which blocked requests remain in the queue as long
295Electromagnetism
as they can be realized. This system is utilized also in networks transmitting data
sets, for example, on the internet.
In addition to the Erlang unit, the Engset relation is applied to express the statis-
tical degree of a telecommunications operation where there is a narrow relation
between these quantities.
Info: [C52].
Agner Krarup Erlang (1.1.1878�3.2.1929), Danish math-
ematician, statistician and engineer.
He was the first to systematically deal with the problems
of telephone networks. In examining the activities of
municipal phone systems, he elaborated a dimensionless
expression which is known as the Erlang unit nowadays. In
particular, it represents the telecommunications criterion.
Even though the Erlang model is simple, the mathematical
foundation of present complex telecommunications net-
works is substantially based on this principle.
6.1.31 Fourier Electrical Number Foe
Foe 5τΔx2
RCL25
τRCn2
τ (s) � time; Δx (m) � geometric step; R (Ω) � resistance; C (F) � capacitance;
L (m) � characteristic length; n (2) � number of cells.
This is the dimensionless description related to time of electric current propaga-
tion through a conductor, e.g. through a geometrically discretized RC model.
Info: [A23].
Jean Baptiste Joseph Fourier (p. 175).
6.1.32 Fourier Relaxation Number of Electrical Field Foe
Foe 5νeτeL2
5νeεL2γ
; where τe 5εγ
νe (m2 s21) � electric viscosity; τe (s) � relaxation time of electric field propaga-
tion; L (m) � characteristic length; ε (F m21) � permittivity; γ (S m21) � specific
electrical conductance.
This number expresses the charge relaxation time which influences the electric
and electromagnetic field distribution. Together with the Fourier magnetic number
Fom (p. 308), it expresses the Fourier electromagnetic number Foem (p. 308).
Electric fields.
Jean Baptiste Joseph Fourier (see above).
296 Dimensionless Physical Quantities in Science and Engineering
6.1.33 Gain
Gain5 10 logP2
P1
ðdBÞ ð1Þ ; Gain5 20 logV2
V1
ðdBÞ ð2Þ
P1, P2 (W) � input and output power signal; V1, V2 (V) � input and output voltage
signal.
It is used in electronics especially and expresses the output-to-input power (1)
or voltage (2) signals ratio. It is used widely in amplifiers. Electronics,
Telecommunications. Automatic control.
Info: [C62].
6.1.34 Grashof Electrical Number Gre
Gre1 5F
ρν25
εE2L2
ρν2ð1Þ; Gre2 � Kr1 ð2Þ
F (N) � Coulomb force; ρ (kg m23) � density; ν (m2 s21) � kinematic viscosity;
ε (F m21) � permittivity; E (V m21) � electric field intensity; L (m) � characteristic
length; Kr1 (2) � Kronig number (p. 299).
It expresses the electric-to-viscous forces under thermoelectric convection con-
ditions. It characterizes the Coulomb forces influencing the origin of the electric
convection and wind which occurs due to gradual environmental neutrality loss. In
gases, it is about a secondary phenomenon arising with corona discharge. The force
F expresses the force close to the coronizing electrode. However, the number Gredoes not express the field inhomogeneity as a necessary condition for electric con-
vection generation. The Grashof electrical number Gre2 is known as the Kronig
number Kr1 (p. 299) as well.
Info: [A23].
Franz Grashof (p. 185).
6.1.35 Grashof Electrical Relative Number Grrel
Grrel 5εE2
LgΔρ5Gre1Gr
21
ε (F m21) � permittivity; E (V m21) � electric field intensity; L (m) � characteristic
length; g (m s22) � gravitational acceleration; Δρ � density difference; Gre1 (2) �Grashof electrical number (p. 297); Gr (2) � Grashof heat number (p. 185).
It expresses the ratio of a free electrothermic convection force to a free thermal
convection force. It characterizes the ponderomotive force influence on the heat
transfer under the action of electric convection.
Info: [A23].
Franz Grashof (see above).
297Electromagnetism
6.1.36 Ground Parameter N
N5UL
Iρð1Þ; N5
ρλΔT
U2ð2Þ
U (V) � voltage; L (m) � characteristic length; I (A) � electric current; ρ (Ω m) �specific electrical resistance; λ (W m21 K21) � thermal conductivity; ΔT (K) �ground warming around earthed system.
Expression (1) describes the voltage-to-grounding ratio recalculation, and
expression (2) represents the relation between the voltage and the heating of the
earth.
Info: [A24].
6.1.37 Joule Electrothermal Number Jo
Jo5λΔT
L2JAEð1Þ; Jo5
λTγU2
5λT
γE2L2ð2Þ
λ (W m21 K21) � thermal conductivity; ΔT (K) � temperature difference; L (m) �characteristic length; JA (A m22) � surface density of electric current; E (V m21) �electric field intensity; T (K) � temperature; γ (S m21) � specific electric
conductance; U (V) � voltage.
This number expresses the ratio of Joule heat to electric energy. Equation (1)
can be expressive for devices with low efficiency and low gradual motion velocity,
for example, the plasma in an electric arc. Equation (2) is called the Joule effect
criterion, which expresses the thermal to electric flows ratio in a fluid. With higher
values, the fluid Joule heating influence can be neglected in comparison to the
heating by simple heat conduction in the fluid.
Info: [A23].
James Prescott Joule (1818�1889), English physicist.
Above all, he is well known due to his experimental
determination of mechanical heat theory. This was con-
nected with his initial effort to determine the efficiency of
electric motors. He was one of the authors of the energy
conservation law, discoverer of the Joule heat unit and the
co-discoverer of the Joule�Thomson phenomenon. He
determined the mechanical equivalent of heat created by
friction.
298 Dimensionless Physical Quantities in Science and Engineering
6.1.38 Kirpichev Electric Number Kie
Kie 5JAL
γU
JA (A m22) � surface density of electric current; L (m) � characteristic length;
γ (S m21) � specific electrical conductance; U (V) � voltage.
It relates to the planar current density, planar source, surface loss and second-
type boundary condition. It is also called the Pomerantsev electrical number PoeA(p. 302).
Info: [A23].
Mikhail Viktorovich Kirpichev (p. 177).
6.1.39 Kronig Number Kr
Kr1 � Gre2 5L2βΔTE2Nk
Mν2α1
2p2
3kTN
� �ð1Þ;
Kr2 5εð3β 1 2βγÞE2L2ΔT
ρν25NetPr
21 ð2Þ
L (m) � characteristic length; β (K21) � volume thermal expansion coefficient;
ΔT5 TS2 TN (K) � temperature difference small against TN; TS, TN (K) � wall
and undisturbed fluid flow temperatures; E (V m21) � electric field intensity;
Nk (mol21) � Avogadro constant; M (kg mol21) � molar mass; ν (m2 s21) �kinematic viscosity; α (A2 s4 kg21) � gas molecule polarization coefficient;
p (C m) � electrical molecular dipole moment; k5RN 21k ðJ K21Þ � Boltzmann
constant; R (J mol21 K21) � molar gas constant; ε (F m21) � permittivity;
βε (K21) � thermal coefficient of permittivity; βγ (K21) � thermal coefficient of
specific electrical conductance; ρ (kg m23) � density; Gre2 (2) � Grashof electrical
number (p. 297); Net (2) � electrothermal convection number (p. 293); Pr (2) �Prandtl number (p. 197).
This number expresses the ratio of the product of the inertia and electrostatic
forces to the square of the viscous force. It characterizes the electrostatic field
effect on the heat transfer from a surface heated by convection. It expresses the
force relations caused by the inhomogeneity of electrophysical properties and vis-
cous friction in a fluid. Essentially, it is analogous to the Archimedes thermody-
namic number Ar (p. 184). In form (2), it is a modified Kronig electrothermal
convection number. The Kronig number Kr1 is also called the Grashof electrical
number Gre2 (p. 297).
Info: [A23].
Ralph De Laer Kronig (born 1904), German-American physicist.
299Electromagnetism
6.1.40 Kutateladze Electrothermal Number Kue
Kue 5JAEL
ρwΔh
JA (A m22) � surface density of electric current; E (V m21) � electric field inten-
sity; L (m) � characteristic length; ρ (kg m23) � density; w (m s21) � fluid velocity;
Δh (J kg21) � increment of specific enthalpy.
It characterizes the electrothermal process with an electric discharge in a flow-
ing fluid.
Info: [A23].
Samson Semenovich Kutateladze (18.7.1914�20.3.1986), Russian engineer
and physicist (p. 327).
6.1.41 Length Current Density K
K5γE2d2
cηT
γ (S m21) � specific electrical conductance; E (V m21) � electric field intensity;
d (m) � electrode diameter; c (J kg21 K21) � specific heat capacity; η (Pa s21) �dynamic viscosity; T (K) � temperature.
It expresses the ratio of longitudinal densities of electric and thermal flows. It
characterizes the electrothermal process in diverse electrothermal systems, for
example, in electric furnaces.
Info: [A23].
6.1.42 Motulevitch Number Mo
Mo5 ρL2E2
η2Δε; where Δε5
dεdT
� �ΔT
ρ (kg m23) � density; L (m) � characteristic length; E (V m21) � electric field
intensity; η (Pa s) � dynamic viscosity; ε (F m21) � permittivity; T (K) �temperature.
It expresses the ratio of the electric force caused by dielectric susceptibility
dependence on the environmental density to the viscosity force. It characterizes the
influence of the inhomogeneity, and the thermal system non-uniformity connected
with it, on the electric convection in dielectric fluids. It expresses the influence of
an electric field on the transfer phenomenon.
Info: [A23].
300 Dimensionless Physical Quantities in Science and Engineering
6.1.43 Nusselt Electrical Number Nue
Nue 5wL
D; where D5
1
2ðD11D2Þ
w (m s21) � flow velocity; L (m) � characteristic length; D (m2 s21) � final ion
diffusivity.
It expresses the ratio of the convective flow to the diffusion electric flow. It
characterizes the electrothermal convection in electrochemical processes.
Sometimes, it is called the Reynolds electrical number (2.) NuEL2 (p. 304).
Info: [A23].
Ernst Kraft Wilhelm Nusselt (p. 196).
6.1.44 Ohm Number Oh
Oh5γEJA
γ (S m21) � specific electrical conductance; E (V m21) � electric field intensity;
JA (A m22) � surface density of electric current.
In the dimensionless state, it characterizes the electric conductance of a system.
Info: [A23].
Georg Simon Ohm (16.3.1789�6.7.1854), German
physicist.
By experiment, he deduced the Ohm law and introduced
the electrical resistance concept. He was engaged in acous-
tics as well. He was influenced by French mathematicians
and physicists. On the base of the Fourier heat conduction
law, he modelled the electric current conduction in an elec-
tric network and described it mathematically.
6.1.45 Peclet Electrical Number Pee
Pee 5wLRC
Δx2
w (m s21) � velocity; L (m) � characteristic length; RC (s) � time constant of
electrical RC network; R (Ω) � resistance; C (F) � capacitance; Δx (m) � element
length dimension.
It expresses the relationship among motion velocity of a source, zone boundaries
and conduction in a geometrically discrete RC model.
Info: [A24].
Jean Claude Eugene Peclet (p. 180).
301Electromagnetism
6.1.46 Pomerantsev Electrical Number Poe
PoeV 5JVL
2
γUr
ð1Þ; PoeA � Kie 5JAL
γUr
ð2Þ;
PoeL 5JL
γUr
ð3Þ; PoeB 5I
γUrLð4Þ
JV, JA, JL � volume density of electric current (A m23), surface density (A m22)
and linear density (A m21); L (m) � characteristic length; γ (S m21) � specific
electrical conductance; Ur (V) � reference voltage; I (A) � electric current; Kie(2) � Kirpichev electrical number (p. 299).
It expresses the ratio of the fed electric flow to that transferred by conduction in
a conductor. It characterizes volume (1), planar (2), longitudinal (3) or spot (4)
inner current sources. It represents thermal loss caused by electric current passage
through conductors.
Info: [A23].
Alexey Alexandrovich Pomerantsev, Russian engineer.
6.1.47 Rayleigh Electrothermal Condensation Number Rael
Rael 5FehρlvlηλΔT
Feh (N) � equivalent excitation electrodynamic force acting on condensing film;
ρ (kg m23) � density; lvl (J kg21) � specific latent heat of condensation; η (Pa s) �
dynamic viscosity; λ (W m21 K21) � thermal conductivity; ΔT (K) � temperature
difference.
It expresses the ratio of the product of the buoyancy electrohydrodynamic force
and the inertia force to the square of the viscosity force. It characterizes the acting
forces influence on the condensation process in an electric field. Condensation in
the electric field.
Info: [A23].
Lord Rayleigh (p. 187).
6.1.48 Rayleigh Electrothermal Instability Number Rae1
Rae1 5β2εD
2ðrTÞ2L4ερνa
βε (K21) � thermal coefficient of permittivity for τr=τD{1; τr (s) � relaxation
time of temperature field; τD (s) � characteristic time of electric induction change;
D (C m21) � electric induction; T (K) � temperature; L (m) � characteristic
302 Dimensionless Physical Quantities in Science and Engineering
length; ε (F m21) � permittivity; ρ (kgτr m23) � density; ν (m2 s21) � kinematic
viscosity; a (m2 s21) � thermal diffusivity.
It expresses the ratio of the product of the electric buoyancy force and the inertia
force to the square of the viscosity force. It characterizes the electrothermal con-
vection in ideal liquid dielectrics. It occurs in the stability description of a balanced
planar horizontal layer in a vertical electric field.
Info: [A23].
Lord Rayleigh (see above).
6.1.49 Rayleigh Electrothermal Number Rae
Rae 5 nεβεL
3E3rTρνa
n5 0, 1, 2; ε (Fτr m21) � permittivity; βε (K
21) � thermal coefficient of permittivity;
L (m) � characteristic length; E (Vτr m21) � electric field intensity; T (K) �
temperature; ρ (kg τ m23) � density; ν (m2 s21) � kinematic viscosity; a (m2 s21) �thermal diffusivity.
This number characterizes the electrothermal convection originating in a fluid
due to field inhomogeneities which are influenced by the geometric shape of
electrodes.
Info: [A23].
Lord Rayleigh (see above).
6.1.50 Reynolds Electrical Number (1.) ReEL1
ReEL1 5wL
νe5wL
γL2
ε
� �21
5wεγL
ð1Þ;
ReEL1 5wερkL
5w
kEð2Þ
w (m s21) � flow velocity; L (m) � characteristic length; νe (m2 s21) � electrical
kinematic viscosity; γ (S m21) � specific electrical conductance; ε (F m21) �permittivity; k (A s2 kg21) � charge carrier mobility; E5 ρL /ε (V m21) � electric
field intensity; ρ (C m23) � volume density of electric charge.
This number expresses the ratio of the fluid inertia force to the friction electric one
(of atoms, electrons or ions). It characterizes the influence of the inertia force and
that of the electric viscosity with forced electrothermal convection. In equation (2), it
expresses the volume density of charges. Magneto-hydrodynamics.
Info: [A23].
Osborne Reynolds (p. 82).
303Electromagnetism
6.1.51 Reynolds Electrical Number (2.) ReEL2
See the Nusselt electrical number Nue (p. 301).
Osborne Reynolds (see above).
6.1.52 Schmidt Electrodiffusion Number Sc3
Sc3 5νεγL2
ν (m2 s21) � kinematic viscosity; ε (F m21) � permittivity; γ (S m21) � specific
electrical conductance; L (m) � characteristic length.
It expresses the ratio of the swirl diffusivity of a system to the mass diffusivity
of ions. It characterizes the mass transfer in electrochemical processes.
Info: [A23].
Ernst Schmidt (p. 264).
6.1.53 Senftleben Number Se
Se5NkE
2
LMgα1
2
3
p2
kTN
� �5KrGr21
Nk (mol21) � Avogadro constant; E (V m21) � electric field intensity; L (m) �characteristic length; M (kg mol21) � molar mass; g (m s22) � gravitational accel-
eration; α (A2 s4 kg21) � gas molecule polarization coefficient; p (C m) � electric
molecular dipole moment; k (J K21) � Boltzmann constant; TN (K) � temperature
of undisturbed fluid flow; Kr (2) � Kronig number (p. 299); Gr (2) � Grashof
heat number (p. 185).
This number characterizes the electric field influence on the heat transfer of
paraelectric fluids (with constant dipole moment). Essentially, it is another expres-
sion of the Kronig number Kr (p. 299).
Info: [A23].
Hermann Senftleben (born 1890), German physicist.
6.1.54 Source of Electrical Heat N
N5J2AL
γρwΔh5
u
Δh
JA (A m22) � surface density of electric current; L (m) � characteristic length; γ(S m21) � specific electrical conductance; ρ (kg m23) � density; w (m s21) �motion velocity; Δh (J kg21) � specific enthalpy change; u (J kg21) � specific
inner energy.
304 Dimensionless Physical Quantities in Science and Engineering
It expresses the ratio of the fed specific inner energy to the specific enthalpy
change of moving surroundings. It characterizes the thermal source caused by elec-
tric current passage through moving surroundings.
Info: [A23].
6.1.55 Stefan Electrical Number Sfe
Sf e 5σγU3L
σ (A m22 V24) � electric equivalent of surface immisivity or emissivity; γ (S m21) �specific electrical conductance; U (V) � voltage; L (m) � characteristic length.
This number expresses the ratio of the electric energy transmitted by a surface
under a non-linear boundary condition to that transmitted by conduction in a sys-
tem. It characterizes a non-linear boundary condition, expressed by a power func-
tion, in an electric system which is analogous to a thermal one. It is of use, for
example, in the electrothermal modelling of heat transfer by radiation. It is the
physical analogue to the Stefan number Sf (p. 213).
Info: [A23].
Josef Stefan (p. 214).
6.1.56 Temperature Non-Uniformity N
N5εβ2
εL2E2ðrTÞ2ρνa
ε (F m21) � permittivity; βε (K21) � thermal coefficient of permittivity; L (m) �
characteristic length; E (V m21) � electric field intensity; T (K) � temperature;
ρ (kg m23) � density; ν (m2 s21) � kinematic viscosity; a (m2 s21) � thermal
diffusivity.
It characterizes the influence of environmental thermal non-uniformity on an
inhomogeneous electric field and the phenomena it causes. Especially with high
field intensities, it has influence but does not have any substantial influence on
thermal process intensification except in cases of thin fluid layers.
Info: [A23].
6.1.57 Viscous Electrical Number E
E5
ffiffiffiffiffiffiffiffiffiffiρ
2πε0
rρL2Qηm
305Electromagnetism
ρ (kg m23) � density; ε0 (F m21) � permittivity; L (m) � characteristic length;
Q (C) � electric charge; η (Pa s) � dynamic viscosity; m (A m2) � magnetic
moment.
This number expresses the relation between electrostatic and hydrodynamic
quantities or, alternatively, the motion of charged particles in flowing gas.
Info: [A24].
6.2 Magnetism
In magnetism, the dimensionless quantities are related to magnetic processes and
forces caused by the electric charge motion. Some of them express steady-state
magnetic fields or unsteady electromagnetic ones, propagation of electromagnetic
energy and magnetic waves, thermomagnetic convection and transfer phenomena
in plasma and magnetic turbulence. Among the important numbers are the Ampere,
magnetic field, Joule magnetic, Kubo, Lorentz, Maxwell and Stuart, and other num-
bers. A great number of magnetic dimensionless quantities are related to magneto-
hydrodynamics.
6.2.1 Ampere Number Am
Am5JAL
H
JA (A m22) � surface density of electric current; L (m) � characteristic length;
H (A m21) � magnetic field intensity.
This number expresses the ratio of flow density to magnetic field intensity. It
characterizes the mutual action of the current and the magnetic field in magneto-
hydrodynamic processes.
Info: [A23].
Andre-Marie Ampere (20.1.1775�10.6.1836), French
mathematician and physicist.
He contributed markedly to the electricity and magne-
tism theory which became the base of the nineteenth cen-
tury scientific development. He executed many experiments
and, in addition to other things, he demonstrated the origin
of a magnetic field around a conductor through which elec-
tric current was flowing. The unit of electric current was
named in his honour.
6.2.2 Electromagnetic Coupling Constant α
α5e2
hcB
1
137:03599 . . .
306 Dimensionless Physical Quantities in Science and Engineering
e (C) � elementary charge; h (J s) � Dirac constant; c (m s21) � speed of light in
vacuum.
This constant characterizes an electromagnetic field. It can be utilized to demon-
strate the probability that an electron emits or absorbs a photon. It is a very impor-
tant phenomenon in modern physics and in quantum electrodynamics especially. It
is a non-linear expression. With extremely high energy levels, the effective value
of the electric charge changes the coupling constant value. Electromagnetic fields.
Quantum physics. Astrophysics.
Info: [C23].
6.2.3 Electromagnetic Effect K
K5μH2
εE2
μ (H m21) � permeability; H (A m21) � magnetic field intensity; ε (F m21) �permittivity; E (V m21) � electric field intensity.
This effect expresses the ratio of the magnetic field energy density to that of the
electric field. It characterizes the degree of the hydrodynamic action of electric and
magnetic fields in fluids. It measures the degree of magnetic effect on electric
convection.
Info: [A23].
6.2.4 Electrothermal Time Nτ
Nτ 5c
wA
� �2
5τrmτ
5 γμc2τrm;
where
τrm 5ρ
γμ2H2; τ5
1
γμc2
c (m s21) � speed of light; wA (m s21) � speed of Alfven waves; τrm (s) � relaxa-
tion time of magnetic field; τ (s) � conversion time of electromagnetic field energy
into Joule heat; γ (S m21) � specific electrical conductance; μ (H m21) � perme-
ability; ρ (kg m23) � fluid density; H (A m21) � magnetic field intensity.
It expresses the ratio of the lines-of-force relaxation time in an electrically con-
ductive fluid to the conversion time of the electromagnetic field energy to the heat
in Joules. It characterizes time relations involved in the conversion of the electro-
magnetic field to heat in Joules.
Info: [A23].
307Electromagnetism
6.2.5 Fourier Electromagnetic Number Foem
Foem 5νmτemL2
; where τem 5ffiffiffiffiffiffiffiffiffiffiτeτm
p; τe 5
εγ; τm 5μγL2
νm (m2 s21) � magnetic viscosity; τem (s) � propagation time of electromagnetic
wave; L (m) � characteristic length; τe (s) � relaxation time of electrical field
propagation; τm (s) � time of magnetic diffusion; ε (F m21) � permittivity;
γ (S m21) � specific electrical conductivity; μ (H m21) � permeability.
In the dimensionless expression, it characterizes the magnetic field propagation
time, for example, in an electrically conductive fluid to the depth L.
Info: [A23],[A33].
Jean Baptiste Joseph Fourier (p. 175).
6.2.6 Fourier Magnetic Number Fom
Fom 5νmτmL2
5 νmμγ; where τm 5μγL22
νm (m2 s21) � magnetic viscosity; τm (s) � time of magnetic diffusion; L (m) �characteristic length; γ (H m21) � magnetic permeability; γ (S m21) � specific
electrical conductivity.
It expresses magnetic diffusion time. It points to the electromagnetic effects
influence on electromagnetic wave propagation. Together with the Fourier relaxa-
tion number of electrical field Foe (p. 296), it influences the characteristics of an
electromagnetic field. Magnetism.
Info: [A24].
6.2.7 Grashof Magnetic Number Grm
Grm 5μ0
@M
@T
rHL3βρν2
ð1Þ;
Grm 5μ0βMMðTS 2 TNÞL3
ρν2dH
dxð2Þ;
Grm 5 4πγμνGr ð3Þ
μ0 (H m21) � vacuum permeability; M (A m21) � magnetization; T (K) �temperature; H (A m21) � magnetic field intensity; L (m) � characteristic length;
β (K21) � volume thermal expansion coefficient of fluid; ρ (kg m23) � density;
ν (m2 s21) � kinematic viscosity; βM (K21) � volume thermal expansion of
magnetization; TS, TN (K) � wall and undisturbed fluid flow temperature;
308 Dimensionless Physical Quantities in Science and Engineering
x (m) � length, coordinate; γ (S m21) � specific electrical conductance;
μ (H m21) � permeability; Gr (2) � Grashof heat number (p. 185).
This number characterizes the heat transfer by free thermomagnetic convection
in paramagnetic fluids with a gravity acceleration which depends on the space
coordinates. It is of use under non-isothermic conditions with the action of thermo-
magnetophoresis transfer of particles.
Info: [A23],[A29].
Franz Grashof (p. 185).
6.2.8 Interaction Parameter Ninter
See the Stuart number (1.) (magnetic force) St (p. 329).
6.2.9 Joule Magnetic Number Jom
Jom 52ρcpΔT
μH2
ρ (kg m23) � density; cp (J kg21 K21) � specific heat capacity; T (K) � tempera-
ture; μ (H m21) � permeability; H (A m21) � magnetic field intensity.
It expresses the ratio of the heat in Joules to the total magnetic field energy. It
characterizes the heat transfer in Joules when acted upon by a magnetic field, for
example, in magneto-hydrodynamics and inductive heating.
Info: [A23].
James Prescott Joule (p. 298).
6.2.10 Kubo Number Kub, R
Kub5δBB0
Lpar
Lper
δB (T) � mean fluctuation of magnetic field; B0 (T) � mean value of magnetic
induction; Lpar, Lper (m) � parallel and perpendicular correlation lengths in regard
to B0.
The Kubo number is used to classify the transfer phenomena in turbulent mag-
netic and other systems. With a small Kubo number, the quasi-linear diffuse coeffi-
cient is of use, as is the filtration diffusion coefficient with a large Kubo number.
The Kubo number expresses the level of a magnetic flux lines chaos. For Kub{1,
a weak chaos is valid and for Kub. 0.3, the extent of the chaos damages the sur-
faces which had been closed before. With Kubc1, a complete stochastic behaviour
occurs in which the filtration diffusion coefficient is of use. Transient phenomena
in plasma. Magnetic turbulence. Diffusion.
Info: [B130].
309Electromagnetism
6.2.11 Lorentz�Lorenz Number n
n �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11
3Ap
R T
r
A (m3 mol21) � molar index of refraction; p (Pa) � gas pressure; R (J mol21 K21) �molar gas constant; T (K) � absolute temperature.
This number represents the refraction index of rarefied gas. It expresses its
dependence on absolute temperature, pressure and the refraction index.
Electrodynamics. Magnetism. Electromagnetic radiation.
Info: [C109].
Hendrik Antoon Lorentz (p. 323).
Ludvig Valentine Lorenz.
6.2.12 Magnetic Dimensionless Frequency Nω
Nω 5τrxwref
L
τrx (s) � magnetic relaxation time of particles; wref (m s21) � reference velocity of
particles; L (m) � characteristic length.
It expresses the relation between the relaxation time of particles and the convec-
tion time. Usually, Nω{1 holds, which means that the influence of the angular
velocity of fluid particles is not as effective as the convective influence. Magneto-
hydrodynamics. Magnetic fields. Technology of magnetic separation, emulsifi-
cation, connection, etc.
Info: [B27].
6.2.13 Magnetic Field Number N
N5JAL
H5
I
LHð1Þ; N5 L2f 2με ð2Þ;
N5 L2fμγ5L2μγτ
ð3Þ; N5U
I
ffiffiffiεμ
rð4Þ
JA (A m22) � surface density of electric current; L (m) � characteristic length;
H (A m21) � magnetic field intensity; I (A) � electric current; f (s21) � frequency;
μ (H m21) � permeability; ε (F m21) � permittivity; γ (S m21) � specific electri-
cal conductance; τ (s) � time; U (V) � voltage.
In equation (1), it expresses a steady-state magnetic field. In expression (4), it
relates to an unsteady electromagnetic field. Further, expression (2) represents
310 Dimensionless Physical Quantities in Science and Engineering
electromagnetic energy propagation and expression (3) represents it electromag-
netic waves propagation.
Info: [A24].
6.2.14 Magnetic Force Parameter NmgF
See Stuart number (1.) (magnetic force) St (p. 329).
6.2.15 Maxwell Number (1.) Fundamental Mx
Mx5ε E2
μ H
ε (F m21) � permittivity; E (V m21) � electric field intensity; μ (H m21) �magnetic permeability; H (A m21) � magnetic field intensity.
It expresses the electric-to-magnetic field intensity ratio. It is the basic criterion
in the set of dimensionless Maxwell equations. Electromagnetic fields.
Info: [C87].
James Clerk Maxwell (13.6.1831�5.11.1879), Scottish
theoretical physicist.
He made revolutionary works in electrical engineering,
magnetism and the kinetic theory of gases. He predicted the
existence of electromagnetic waves, which propagate at
the same velocity as light. He created the electromagnetic
theory describing the phenomena of electrodynamics, optics
and thermal radiation. The substance of the Maxwell theory
is expressed by four Maxwell equations describing the time
and space dependence of the electromagnetic field.
6.2.16 Maxwell Number (2.) Electrodynamical Mx
Mx5τ2
εμL25Fom
τ (s) � time; ε (F m21) � permittivity; μ (H m21) � magnetic permeability; L (m) �characteristic length; Fom (2) Fourier magnetic number (p. 308).
This number expresses the dynamics of the electromagnetic field. It occurs in
the system of dimensionless Maxwell equations. Electromagnetic fields.
Info: [C87].
James Clerk Maxwell (see above).
311Electromagnetism
6.2.17 Maxwell Number (3.) Time Mx
Mx5ρτμL2
5Foem
ρ (Ω m) � specific electrical resistance; τ (s) � time; μ (H m21) � magnetic per-
meability; L (m) � characteristic length; Foem (2) � Fourier electromagnetic num-
ber (p. 308).
Essentially, it expresses the dimensionless time of an electromagnetic process. It
is also called the Fourier electromagnetic number (p. 308). Electromagnetic
processes.
Info: [A23],[C87].
James Clerk Maxwell (see above).
6.2.18 Ohmic Heating Oht
Oht 5λ γ ΔT
J2A L2
λ (W m21 K21) � thermal conductivity; γ (S m21) � specific electrical conduc-
tance; ΔT5 T2 TS (K) � temperature difference, difference of mean plasma tem-
perature and temperature of channel wall surface; JA (A m22) � surface density of
electric current; L (m) � characteristic length.
It expresses the ratio of the heat transferred by conduction to the electric energy
transferred by plasma. It characterizes the influence of the heat led away by con-
duction on the electric arc motion in a transversal electric field. It expresses the
heat transfer intensity from a plasma into a canal wall with laminar equilibrium
and fully developed flow in a canal spot heated with an arc plasma generator.
Info: [A23].
6.2.19 Peclet Magnetic Rotational Number Pemg
Pemg 5τBτrx
5L2
Dmgτrx
τB (s) � Brown relaxation time; τrx (s) � magnetic relaxation time of particles;
L (m) � characteristic length; Dmg (m2 s21) � magnetic diffusivity.
It expresses the relation between the Brown time and the magnetizing relaxation
time and, alternatively, it defines how fast the magnetic particles can be oriented
according to the local magnetic field. Magnetic field. Magnetic emulsification.
Magnetic separation. Magnetic spraying. Destructive methods.
Info: [B27].
Jean Claude Eugene Peclet (p. 180).
312 Dimensionless Physical Quantities in Science and Engineering
6.2.20 Pomerantsev Electromagnetic Number Pomg
Pomg 5Lεqmg
λT
L (m) � characteristic length of solid particle; qmg (W m22) � surface density of
electromagnetic flux; λ (W m21 K21) � thermal conductivity; T (K) � source tem-
perature; ε (2) � emissivity (absorptivity) (p. 206).
This number characterizes the heating of solid particles in an electromagnetic
field. It is applied, for example, to discover the thermal properties of microscopic
particles by means of the impulse laser heating method. It is analogous to the
Pomerantsev heat number Po (p. 181).
Alexey Alexandrovich Pomerantsev, Russian engineer.
6.2.21 Prandtl Magnetic Number Prm
Prm5El215νμγ5ννm
5RemRe21
ν (m2 s21) � kinematic viscosity; μ (H m21) � permeability; γ (S m21) � specific
electrical conductance; νm (m2 s21) � magnetic viscosity; El (2) � Elsasser num-
ber (p. 318); Rem (2) � Reynolds magnetic number (p. 328); Re (2) � Reynolds
number (p. 81).
It expresses the ratio of viscosity to magnetic diffusion. It characterizes the simi-
larity between the magnetic and the velocity fields of an electrically conductive
fluid flowing in the magnetic field. It is used to express the exciting effect on the
induced magnetic field current.
Info: [A23],[B20].
Ludwig Prandtl (p. 197).
6.2.22 Rayleigh Magnetic Number Ram
Ram 5 γ2μ2L3gβΔT 5GrPr2m
γ (S m21) � specific electrical conductance; μ (H m21) � permeability; L (m) �characteristic length; g (m s22) � gravitational acceleration; β (K21) � volume
thermal expansion coefficient; ΔT (K) � temperature difference; Gr (2) �Grashof heat number (p. 185); Prm (2) � Prandtl magnetic number (p. 313); Ra2(2) � Rayleigh number (2.) (p. 187).
It characterizes the magnetic convection influence on heat transfer intensifica-
tion. The influence of the Ram number appears from Ram. 150 Ra2 approximately.
With RamcRa2, only the thermomagnetic convention is of use.
Info: [A23].
Lord Rayleigh (p. 187).
313Electromagnetism
6.3 Magneto-hydrodynamics
Magneto-hydrodynamics is focused on the hydrodynamics of electrically conduc-
tive fluids on plasma and liquid metals, especially. In this field, the range of dimen-
sionless quantities used is extensive, which corresponds to the significance of
dimensionless quantities of this field and to the wide range of applications in
applied physics, engineering branches, geophysics, astrophysics and other spheres.
Individual dimensionless quantities are focused on dynamic velocity, pressure and
force conditions in MHD, on the continuity of flow of electrically conductive fluid
or plasma and on the level of non-stationarity. The Alfven, Cowling, Einstein,
Elsasser, Hartman, Karman, Mach magnetic, Reynolds and Strouhal numbers are
among the basic dimensionless quantities used within this area.
6.3.1 Alfven Number Al
Al5Co12 5M21
m 5wA
w5
B
wffiffiffiffiffiffiμρ
p ð1Þ; where wA5H
ffiffiffiμρ
r;
Al5Eu12m 5
H
w
ffiffiffiμρ
rð2Þ
wA (m s21) � Alfven’s wave velocity; w (m s21) � velocity; B (T) � magnetic
induction; μ (H m21) � permeability; ρ (kg m23) � density; H (A m21) �magnetic field intensity; Co (2) � Cowling number (p. 316); Mm (2) � Mach
magnetic number (p. 324); Eum (2) � Euler magnetic number (p. 319).
This number expresses the magnetic-to-inertia forces ratio or the ratio of the
Alfven wave propagation rate to the fluid flow propagation rate. It characterizes
the propagation of magneto-hydrodynamic waves (Alfven waves) along the mag-
netic field flux lines in a flowing electrically conductive fluid flow (the plasma). It
corresponds to the Euler magnetic number Eum (p. 319) and sometimes is used in
equation (2) to express the magnetic-to-kinetic energy ratio in unit volume. Its
inverse value is called the Mach magnetic number Mm (p. 324) or the Karman num-
ber (2.) Ka (p. 321). Magneto-hydrodynamics.
Info: [A23],[B20].
Hannes Olof Gosta Alfven (30.5.1908�2.4.1995),
Swedish astronomer. Nobel Prize in Physics, 1970.
He was engaged in research in electrical engineering,
electronics and astrophysics, and is one of the founders of
modern plasma physics. He published the theory of mag-
netic storms and of polar glares and laid the foundation of
the Earth magnetosphere theory. His principal works con-
cern theoretical research of magneto-hydrodynamics, for
which he received the Nobel Prize.
314 Dimensionless Physical Quantities in Science and Engineering
6.3.2 Batchelor Number Bt
Bt5wγLc2ε
w (m s21) � velocity; γ (S m21) � specific electrical conductance; L (m) � charac-
teristic length; c (m s21) � speed of light; ε (F m21) � permittivity.
It characterizes the dynamic relations in magneto-hydrodynamics.
Info: [A23],[A35]
George Keith Batchelor (8.3.1920�30.3.2000), Australian
mathematician and physicist.
He worked in hydrodynamics and magneto-hydrodynam-
ics. The focus of his work was primarily in turbulence
research. Of his monographs, Homogenous Turbulence
(1953), Turbulent Diffusion (1956) and An Introduction to
Fluid Dynamics (1957) are the most important.
6.3.3 Brinkmann Modified Number Brmod
Brmod 5BrLffiffiffiζ
p 5EcPrLffiffiffiζ
p
L (m) � characteristic length (width) of channel; ζ (m2) � permeability of porous
material; Br (2) � Brinkmann number (p. 174); Ec (2) � Eckert number (p. 191);
Pr (2) � Prandtl number (p. 197).
It expresses the ratio of the heat caused by viscous friction of the fluid in micro-
scopic canals of a porous material to the heat transferred by molecular conduction.
Two-phase fluid mechanics. Magneto-hydrodynamics.
Info: [B70]
Henri Coenraad Brinkmann, German physicist.
6.3.4 Chandrasekhar Number Ch
Ch5B2L2
ρνρe
B (T) � magnetic induction; L (m) � characteristic length; ρ (kg m23) � fluid density;
ν (m2 s21) � kinematic viscosity; ρe (Ω m) � specific electrical resistance.
It expresses the magnetic-to-dispersion forces ratio. Magneto-hydrodynamics.
315Electromagnetism
Subrahmanyan Chandrasekhar (19.10.1910�21.8.1995),
Indian astrophysicist. Nobel Prize in Physics, 1983.
He was engaged in research in various areas of physics,
in astrophysics especially. Besides the theory of the structure
and dynamic stability of stars, he devoted himself to radia-
tion, the mathematical theory of black holes and general rel-
ativity. His name has been given to the Chandrasekhar limit.
According to this, a cooling star, with 1.5 times the Sun’s
mass, cannot resist its own gravitation. He received the
Nobel Prize for his theoretical work about the gravitational
collapse of stars.
6.3.5 Cowling Number, Magnetic Force Number Co
Co5Al2 5wA
w
� �2
5B2
μρw25
BH
ρw2
wA (m s21) � Alfven’s wave velocity; w (m s21) � velocity; B (T) � magnetic
induction; μ (H m21) � magnetic permeability; ρ (kg m23) � density; H (A m21) �magnetic field intensity; Al (2) � Alfven number defined by equation (1) (p. 314).
This number expresses the magnetic-to-inertia forces ratio. It is also called the
magnetic force number. It expresses the second power of the ratio of the Alfven
wave velocity to the fluid velocity. Magneto-hydrodynamics.
Info: [A29],[A43],[B17]
Thomas George Cowling (17.6.1996�16.6.1990), English
mathematician and physicist.
He contributed significantly to modern research on stel-
lar energy, especially that of the sun. He played an impor-
tant role in finding the convective nucleus in stars. Due to
this convection, the sun can behave as a giant dynamo, the
rotation, internal circulation and flow of which generate
electric currents and magnetic fields connected with solar
spots.
6.3.6 Eckert Magnetic Number Ecm
Ecm 5B2
μρcpΔT5
w2A
cpðT 2 TSÞ
B (T) � magnetic induction; μ (H m21) � permeability; ρ (kg m23) � density;
cp (J kg21 K21) � specific heat capacity; ΔT5 T2 TS (K) � fluid and wall tem-
peratures difference; wA (m s21) � Alfven’s wave velocity.
316 Dimensionless Physical Quantities in Science and Engineering
It characterizes heat transfer under the conditions of a laminar magneto-hydro-
dynamic boundary layer caused by a magnetic field of induced currents. The value
of Ecm. 0 corresponds to outer heating and the Ecm, 0 represents fluid flow
cooling.
Info: [A23].
Ernst Rudolf Georg Eckert (p. 192).
6.3.7 Einstein Number Ei
Ei5w
cð1Þ; Ei � R5
w2
c25
w
c2γμLð2Þ
w (m s21) � flow velocity; c (m s21) � speed of light; γ (S m21) � specific electri-
cal conductance; μ (H m21) � permeability; L (m) � characteristic length; R (2) �relativistic parameter (p. 327).
This number expresses the ratio of the local flow rate of electrically conductive
fluid to the conversion rate of electromagnetic field energy to heat in Joules. In the
form of equation (2), it is also called the velocity criterion or the relativistic param-
eter R (p. 327).
Info: [A23].
Albert Einstein (14.3.1879�18.4.1955), American physi-
cist of German origin. Nobel Prize in Physics, 1921.
He is the ingenious physicist who brought to physics the
modern vision of physical reality, particularly with his spe-
cial and general relativity theories. He also explained
the concept of the photoelectric phenomenon. He received
the Nobel Prize for his photoelectric phenomenon explana-
tion, but not for the theory of general relativity, which was
his main contribution. He spent the last years of his life try-
ing to create a unified theory which could explain all
known forces in nature.
6.3.8 Electric Field Number RE
RE 5E
wμH
E (V m21) � electric field intensity; w (m s21) � flow velocity; μ (H m21) �permeability; H (A m21) � magnetic field intensity.
It characterizes the mutual action of hydrodynamic and electromagnetic fields in
magneto-hydrodynamics.
Info: [A23],[A33].
317Electromagnetism
6.3.9 Electric Field Parameter NE
NE5E
wμH5
E
Bw
E (V m21) � electric field intensity; w (m s21) � velocity; μ (H m21) � permeability;
H (A m21) � magnetic field intensity; B (T) � magnetic induction.
It expresses the ratio of the active electrodynamic force to the induced electro-
dynamic force. Magneto-hydrodynamics.
Info: [A23].
6.3.10 Elsasser Number El
El5ReRe21m 5Pr21m 5
ρηγμ
ρ (kg m23) � density; η (Pa s) � dynamic viscosity; γ (S m21) � specific electrical
conductance; μ (H m21) � permeability; Re (2) � Reynolds number (p. 81);
Rem (2) � Reynolds magnetic number (p. 328); Prm (2) � Prandtl magnetic num-
ber (p. 313).
Under magneto-hydrodynamic conditions, it expresses the hydrodynamic-to-
magnetic field propagation ratio. It characterizes the relation between hydrody-
namic and magnetic quantities in the magneto-hydrodynamics field.
Info: [A23],[B20].
Walter Maurice Elsasser (20.3.1904�14.10.1991),
American theoretical physicist of German origin.
He occupied himself with fundamental problems of
atomic physics, geophysics, radiation heat transfer in the
atmosphere, terrestrial magnetism and � in the last years of
his life � the theory of organisms. In 1939, he proposed
that the Earth’s rotation generates turbulent currents in both
its liquid nucleus and in its terrestrial magnetism.
6.3.11 Entropy Generation Number NS
NS 5 SgL2 T0
λ ðΔTÞ2
Sg (W m23 K21) � entropy generation rate; λ (W m21 K21) � thermal conductiv-
ity; ΔT (K) � temperature difference; L (m) � characteristic length (width) of
channel; T0 (K) � initial temperature.
This number expresses the ratio of the generated entropy rate change to the
characteristic entropy transfer. In magneto-hydrodynamics, it is used to express the
318 Dimensionless Physical Quantities in Science and Engineering
entropy changes when acted upon by mixed convection and radiation in vertical
porous canals. Magneto-hydrodynamics.
Info: [B70].
6.3.12 Euler Magnetic Number Eum
Eum 5μH2
ρw25
B2
μρw2
μ (H m21) � permeability; H (A m21) � magnetic field intensity; ρ (kg m23) �density; w (m s21) � velocity; B (T) � magnetic induction.
In a fluid moving in a magnetic field, it expresses the magnetic-to-dynamic pres-
sures ratio. It characterizes the voltage acting along the magnetic lines of force in
an electrically conductive fluid which moves in a magnetic field. Essentially, it is
the degree of the magnetic-to-kinetic energies ratio for a unit fluid volume.
Sometimes, it is also called the magnetic pressure N (p. 325).
Info: [A23].
Leonhard Euler (p. 61).
6.3.13 Fourier Magnetic Relaxation Number Fomr
Fomr 5τrmνmL2
5ReRe21m Ha22 5Eu21
m Re22m 5 St21Re21
m ;
where
τrm 5ρ
γμ2H2; νm 5
1
γμ
τrm (s) � magnetic relaxation time; νm (m2 s21) � magnetic viscosity; L (m) �characteristic length; ρ (kg m23) � density; γ (S m21) � specific electrical
conductance; μ (H m21) � permeability; H (A m21) � magnetic field intensity;
Re (2) � Reynolds number (p. 81); Rem (2) � Reynolds magnetic number
(p. 328); Ha (2) � Hartmann number (p. 320); Eum (2) � Euler magnetic number
(p. 319); St (2) � Stuart number (1.) (magnetic force) (p. 329).
In an electrically conducting fluid, moving at the characteristic velocity
we (γμL)21, it expresses the ratio of lines of force relaxation (attenuation) time to
the lines of force passage time through a magnetic field and the conversion of its
energy to electric heat.
Info: [A23].
Jean Baptiste Joseph Fourier (p. 175).
319Electromagnetism
6.3.14 Hall Coefficient CH
CH 5λrL
λ (m) � mean free path; rL (m) � Larmor radius. This coefficient expresses the
ratio of the gyro-frequency to the frequency of collisions. Magneto-hydrodynamics.
Info: [A29].
Edwin Herbert Hall (see below).
6.3.15 Hall Parameter, Hall Coefficient H, β
H5ωτ5eBτm
5KnLr21
ω (s21) � resonant frequency of electron; τ (s) � mean time of electron relaxation;
e (C) � electron charge; B (T) � magnetic induction; m (kg) � electron mass;
Kn (2) � Knudsen number (1.) (p. 69); Lr (2) � Larmor number (p. 322).
It represents the ratio of the gyro-frequency to the frequency of collisions. In the
magneto-hydrodynamics, it expresses the ratio of the mean time between collisions
to the time of their circulation around a magnetic flux line.
Info: [B20].
Edwin Herbert Hall (7.11.1855�20.11.1938), American
physicist.
In the year 1879, he discovered the Hall effect, the sub-
stance of which is the transversal electric voltage in a cur-
rent loaded conductor or in a semiconductor located in an
outer magnetic field. In two-dimensional electron systems,
the quantum Hall effect appears in strong magnetic fields
and under temperatures close to those of liquid helium.
Together with superconductivity, this effect is one of the
most significant manifestations of quantum theory.
6.3.16 Hartmann Number Ha
Ha5μHLffiffiffiγη
r5BL
ffiffiffiγη
r5
ffiffiffiffiffiffiffiffiffiStRe
p5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiReEumRem
p
μ (H m21) � permeability; H (A m21) � magnetic field intensity; L (m) � charac-
teristic length; γ (S m21) � specific electrical conductance; η (Pa s) � dynamic vis-
cosity; B (T) � magnetic induction; St (2) � Stuart number (1.) (magnetic force)
(p. 329); Re (2) � Reynolds number (p. 81); Eum (2) � Euler magnetic number
(p. 319); Rem (2) � Reynolds magnetic number (p. 328).
320 Dimensionless Physical Quantities in Science and Engineering
It is an important criterion of magneto-hydrodynamics. It expresses the ratio of
the induced electrodynamic (magnetic) force to the hydrodynamic force of the vis-
cosity or, alternatively, the ratio of the ponderomotive force (the electromagnetic
volume force by means of which the magnetic field acts on a conductor through
which electric current flows, which causes magnetic pressure) to the molecular fric-
tion force. It characterizes the magnetic field influence on the flow of viscous, elec-
trically conducting fluid. With small Ha values, the motion proceeds as if no
magnetic field were acting. With great Ha values, the viscosity forces act only on a
thin layer of the electrically conducting fluid (ionized gas) which adheres closely
to a by-passed wall surface. In other cases, the motion resistance does depend on
the viscosity and is determined completely by electromagnetic volume forces which
are acting on the fluid. With high velocities and turbulent flow, it is more
suitable to use the Stuart number (2.) St (p. 330), expressing the mutual magneto-
hydrodynamic action, instead of the Ha number.
Info: [A23],[A43],[B20].
Julius Frederik Georg Poul Hartmann (1881�1951), Danish physicist.
6.3.17 Hydromagnetic Velocity Number Nhm
Nhm 5w
wmg
w (m s21) � fluid flow velocity; wmg (m s21) � velocity of magnetic field
propagation.
It expresses the ratio of the fluid flow rate to the magnetic field propagation
rate. It corresponds to the Reynolds magnetic number Rem (p. 328). Magneto-
hydrodynamics.
6.3.18 Karman Number (2.) Ka
Ka5w
wA
5Al21 5w
ffiffiffiffiffiffiμρ
pB
; where wA 5H
ffiffiffiμρ
r
w (m s21) � local fluid velocity; wA (m s21) � velocity propagation of Alfven’s
magnetic wave; μ (H m21) � permeability; ρ (kg m23) � fluid density; B (T) �magnetic induction; H (A m21) � magnetic field intensity; Al (2) � Alfven num-
ber (p. 314).
It characterizes the influence of the mutual action of hydrodynamic and mag-
netic fields in magneto-hydrodynamics. It expresses the ratio of the fluid flow rate
to that of the Alfven waves. It expresses also the kinetic-to-magnetic energies ratio
in a unit volume. Magneto-hydrodynamics.
Info: [A35].
Theodore von Karman (p. 67).
321Electromagnetism
6.3.19 Larmor Number Lr
Lr5rL
L
rL (m) � Larmor radius; L (m) � characteristic length.
In the magnetic field, it expresses the ratio of the Larmor rotation radius of a
charged mass particle to a characteristic longitudinal dimension. It characterizes
the continuity of the electrically conductive flow. With the Lr negligibly small, the
fluid or plasma flow can be described as a continuous environment exposed to
magnetic field influence. For a large Lr value and the Knudsen number (1.) (p. 69)
Knc1, the fluid flow must be described as the flow of mutually acting discrete
charged particles, exposed to magnetic field influence. Magneto-hydrodynamics.
Info: [A23].
Joseph Larmor (11.7.1857�19.5.1942), Irish physicist.
He was engaged in researching electricity, dynamics and
thermodynamics. He suggested the physical theory of light,
according to which ether can represent a homogenous fluid
which would be both perfectly incompressible and perfectly
elastic. He developed a theory according to which the spec-
tra could be explained by oscillations of electrons. He
determined the energy released with radiation of an acceler-
ated charge and explained the spectral lines splitting in a
magnetic field. The Larmor radius represents that of the
charged particle movement in a magnetic field.
6.3.20 Lorentz Factor, Lorentz Term γ, Lof
γ5dτmdτ
51ffiffiffiffiffiffiffiffiffiffiffiffiffi
12 β2p ð1Þ; γ5
cffiffiffiffiffiffiffiffiffiffiffiffiffiffic2 2 u2
p ð2Þ;
where
β5u
c
τm (s) � measured time; τ (s) � supposed time; u (m s21) � velocity at reference
point where time τm is measured; c (m s21) � speed of light.
It is used in the special relativity theory to simplify the recording of equations
and, especially, to express the related time dilation, length contraction and rest-mass
conversion to the relativistic mass. Physics. Electromagnetism. Electrodynamics.
Magneto-hydrodynamics.
Info: [C79].
322 Dimensionless Physical Quantities in Science and Engineering
Hendrik Antoon Lorentz (18.7.1853�4.2.1928), Dutch
physicist. Nobel Prize in Physics, 1902.
He was engaged in optics, electrodynamics and atomis-
tics. He contributed significantly to the general relativity
theory, elaborated by Einstein, specifically by experimen-
tally verifying the special relativity theory (1919). He is
known for his work on the electric and magnetic field influ-
ence on electromagnetic radiation. He laid the foundation
for the electron theory of the electric conductivity of sub-
stances. He received the Nobel Prize for his work on elec-
tromagnetic radiation.
6.3.21 Lundquist Number Ld
Ld5γHμ
32L
ρ12
5HaRe12mRe
212
γ (S m21) � specific electrical conductance; H (A m21) � magnetic field intensity;
μ (H m21) � permeability; L (m) � characteristic length (thickness of conductive
fluid); ρ (kg m23) � density; Ha (2) � Hartmann number (p. 320); Rem (2) �Reynolds magnetic number (p. 328); Re (2) � Reynolds number (p. 81).
This number characterizes the size of the attached magnetic field influence on
the distribution of the induced magnetic field in the flow of an electrically conduc-
tive fluid. It has been used in magneto-hydrodynamics to express unidirectional
Alfven waves or, alternatively, the waves caused by the magnetic field in the flow
of the electrically conductive fluid.
Info: [A23].
6.3.22 Lykoudis Number Ly
Ly5γμ2H2L
12
ðgβΔTÞ12ρ5Ha2Gr2
12
γ (S m21) � specific electrical conductance; μ (H m21) � permeability; H (A m21) �magnetic field intensity; L (m) � characteristic length; g (m s22) � gravitational
acceleration; β (K21) � volume thermal expansion coefficient; T (K) � temperature;
ρ (kg m23) � density; Ha (2) � Hartmann number (p. 320); Gr (2) � Grashof heat
number (p. 185).
It expresses the ratio of the ponderomotive force to the product of buoyancy and
inertia forces. It characterizes the magnetic field influence on the convection of an
electrically conductive fluid in magneto-hydrodynamics.
Info: [A23].
Paul S. Lykoudis (born 1926), American engineer of Greek origin.
323Electromagnetism
6.3.23 Mach Electromagnetic Number Mae
Mae 5E
Bw5ReEL1Rem 5
we
w
E (V m21) � electric field intensity; B (T) � magnetic induction; w (m s21) � fluid
flow velocity; we (m s21) � drift velocity; ReEL1 (2) � Reynolds electrical
number (1.) (p. 303); Rem (2) � Reynolds magnetic number (p. 328).
It expresses the drift-to-flow rates ratio. Magneto-hydrodynamics.
Info: [A24].
Ernst Mach (p. 73).
6.3.24 Mach Magnetic Number Mm
Mm 5Al21 5w
wA
5w
ffiffiffiffiffiffiρμ
pB
ð1Þ; Mm 5w
ða2 1w2AÞ
12
ð2Þ;
Mm 5w
a1wA
ð3Þ
w (m s21) � velocity; wA (m s21) � Alfven wave velocity; ρ (kg m23) � density;
μ (H m21) � permeability; B (T) � magnetic induction; a (m s21) � sound speed;
Al (2) � Alfven number (p. 314).
This number expresses the ratio of the local flow rate of an electrically conduc-
tive fluid to the propagation rate of the Alfven (magneto-hydrodynamic) wave.
Like the Alfven number (p. 314) or the Euler magnetic number Eum (p. 319), it
characterizes the flow rate in magneto-hydrodynamics. Sometimes, expression
(2) or (3) is used. It is also called the Alfven�Mach number.
Info: [A23].
Ernst Mach (p. 73).
6.3.25 Magnetic Fluidization Number Nmgf
Nmgf 5μsH
2x
ρsu2f
μs (H m21) � powder permeability; H (A m21) � magnetic field intensity; x (2) �volume fraction of magnetic powder; ρs (kg m23) � mean density of fluidized bed;
uf (m s21) � surface fluid velocity.
It characterizes the magnetic fluidization of powders in a flowing fluid. It
expresses the magnetic-to-kinetic energy ratio for particles. It is a modification of
the Euler magnetic number Eum (p. 319). Magneto-hydrodynamics.
Info: [B129].
324 Dimensionless Physical Quantities in Science and Engineering
6.3.26 Magnetic Force Parameter Nmg,F
Nmg;F 5μHγLρw
μ (H m21) � permeability; H (A m21) � magnetic field intensity; γ (S m21) �specific electrical conductance; L (m) � characteristic length; ρ (kg m23) � density
of environment; w (m s21) � flow velocity.
It expresses the magnetic-to-dynamic forces ratio. It is analogous to the mag-
netic number Nmg (p. 325). Magneto-hydrodynamics.
Info: [A29],[B17].
6.3.27 Magnetic Interaction Nmg
Nmg 5μ0H
2r
2σ
μ0 (H m21) � vacuum permeability; H (A m21) � magnetic field intensity; r (m) �radius; σ (kg s21) � liquid surface tension.
It relates to ferrous fluids dynamics. Magneto-hydrodynamics.
Info: [A24],[B20].
6.3.28 Magnetic Number Nmg
Nmg 5B
ffiffiffiffiffiffiγLρw
s5N
12mgF
B (T) � magnetic induction; γ (S m21) � specific electrical conductance; L (m) �characteristic length; ρ (kg m23) � density; w (m s21) � fluid flow velocity;
NmgF (2) � magnetic force parameter (p. 311).
It expresses the magnetic-to-inertia forces ratio. It is analogous to the magnetic
force parameter NmgF (p. 311). Magneto-hydrodynamics.
Info: [C84].
6.3.29 Magnetic Pressure N, S
N5p
pm5 nkT
2μB2
ð1Þ; S5μH2
ρw2ð2Þ
p (Pa) � gas pressure; pm (Pa) � magnetic pressure; n (m23) � numerical
density of particles; k (J K21) � Boltzmann constant; T (K) � temperature;
325Electromagnetism
μ (H m21) � permeability; B (T) � magnetic induction; H (A m21) � magnetic
field intensity; ρ (kg m23) � density; w (m s21) � velocity.
It expresses the ratio of the magnetic pressure to that of the gas. It characterizes
the pressure in magneto-hydrodynamics. pm5B2/2μ holds for magnetic pressure
and p5 nkT for gas pressure. It is also called the Euler magnetic number Eum(p. 319).
Info: [A23],[B20].
6.3.30 Magnetic�Dynamic Number Nmgd
Nmgd 5γwB2L
ρw2
γ (S m21) � specific electrical conductivity; w (m s21) � fluid flow velocity;
B (T) � magnetic induction; L (m) � characteristic length; ρ (kg m23) � fluid
density.
It expresses the magnetic-to-dynamic pressures ratio of the fluid. Magneto-
hydrodynamics.
Info: [A24],[B20].
6.3.31 Magnetization Parameter Ω
Ω5ωef
νef
ωef (s21) � angular frequency; νef (s
21) � effective electron collision.
It represents the local electron magnetization parameter. Magneto-hydrodynamics.
Flow in accelerator canals.
6.3.32 Morozov-Hall Parameter Mor
Mor5ΩRem
Ω (2) � magnetization parameter (p. 329); Rem (2) � Reynolds magnetic number
(p. 328).
It characterizes the magneto-hydrodynamic processes for flow in accelerator
canals. Magneto-hydrodynamics.
Info: [C96].
V.A. Morozov, Russian physicist.
Edwin Herbert Hall (p. 320).
326 Dimensionless Physical Quantities in Science and Engineering
6.3.33 Naze Number Na
Na5wA
a5MAl
wA (m s21) � Alfven’s wave velocity; a (m s21) � sound speed; M (2) � Mach
number (p. 73); Al (2) � Alfven number (p. 314).
In an electrically conductive fluid, it expresses the ratio of the propagation rate
of magneto-hydrodynamic waves (the Alfven waves) to the sound propagation rate
in the fluid. It characterizes the velocity relations in electro-magneto-
hydrodynamics.
Info: [A23].
Jacqueline Naze (born 1935), French mathematician.
6.3.34 Oseen Magnetic Number Os
Os51
2ð12Al2ÞNmg
Al (2) � Alfven number (p. 314); Nmg (2) � magnetic number (p. 325), or the
Stuart number (1.) (magnetic force) (p. 329).
It expresses the magnetic-to-inertia forces ratio. It characterizes the magneto-
hydrodynamic relations in electrically conductive fluid flow in the magnetic field.
Info: [A23].
6.3.35 Porous Media Inertia Coefficient Γ, Npmi
Γ 5Ckε2Lffiffiffi
ζp
Ck (2) � Forchheimer coefficient; ε (2) � porosity (p. 24); ζ (m2) � permeability
of porous material; L (m) � characteristic length (width of channel).
It characterizes the porous environment behaviour with convective and mixed
convective radiation flow and interaction in vertical porous canals. It occurs, for
example, in magneto-hydrodynamics and is accompanied by entropic changes.
Two-phase flow. Magneto-hydrodynamics.
Info: [B70].
6.3.36 Relativistic Parameter R
See the Einstein number Ei defined by relation (2) (p. 317).
327Electromagnetism
6.3.37 Reynolds Magnetic Number Rem
Rem 5RePrm 5wL
νm5 γμwL5
w
wm
ð1Þ;
Rem 5γμELB
5Re�mM21m ð2Þ
w (m s21) � velocity; L (m) � characteristic length; νm5 (μmσ)21 (m2 s21) �
magnetic viscosity or magnetic diffusivity; γ (S m21) � specific electrical conduc-
tance; μ (H m21) � permeability; wm5 (γμL)21 (m s21) � characteristic velocity;
E (V m21) � electric field intensity; B (T) � magnetic induction; Re (2) � Reynolds
number (p. 81); Prm (2) � Prandtl magnetic number (p. 313); Re�m(2) � Reynolds
magnetic number defined by the equation (1); Mm (2) � Mach magnetic number
(p. 324).
This number expresses the ratio of a movable induced magnetic field to an intro-
duced outer magnetic field. It characterizes the influence of the electrically conduc-
tive fluid flow rate on the magnetic field distribution in the fluid. Sometimes, it is
also called the velocity number Rw.
It can be expressed as the ratio of the fluid flow rate to the characteristic velocity
wm5 (γμL)21 of a magnetic field moving over a conductor, or as the ratio of the
flow dimension L to the characteristic length of a magnetic field Lm5 (γμwm)21
which moves through a conductor. It is the degree of outer magnetic field excita-
tion. For Rem5 0, the fluid flow does not influence the magnetic flux lines of the
field in which the fluid is moving. The magnetic field of induced currents does not
originate in the fluid. For Rem{1, or w{wm, L{Lm alternatively, the magnetic
field influence can be ignored and it can be assumed that the flow is due to outer
magnetic field influence only, for example, without fluid motion influence on the
magnetic field. The plasma moving in an electromagnetic field or liquid metal in a
pipeline, in canals of magnetic pumps, or in MHD generators, are examples. For
Rem�1, the induced magnetic field can influence the outer magnetic field action
considerably. It causes some entraining of a resulting magnetic field by the fluid,
this entraining growing greater with the increasing Rem. For Remc1, the fluid flow
becomes a good electric conductor and acts on a magnetic field strongly. It is as if
the magnetic flux lines are frozen in the fluid flow and entrained by it. These ques-
tions are the subject of magneto-hydrodynamics research in astrophysics.
Info: [A23],[A29],[B20].
Osborne Reynolds (p. 82).
6.3.38 Roberts Number Rob
Rob5a
νm5RemPe
21
a (m2 s21) � thermal diffusivity; νm (m2 s21) � magnetic viscosity; Rem (2) �Reynolds magnetic number (p. 328); Pe (2) � Peclet heat number (p. 180).
328 Dimensionless Physical Quantities in Science and Engineering
It characterizes the magnetic convection of an electrically conducting fluid in
fast rotating systems under action of a toroidal magnetic field. It expresses the ratio
of induced and outer magnetic fields to convective and molecular heat transfers in
the flowing fluid. It is a very important criterion in examination of the convection
around planets. Magneto-hydrodynamics. Astrophysics.
Info: [B125].
6.3.39 Strouhal Electromagnetic Number Shm
Shm 5Eτ
BL5
weT
Lð1Þ; Shm 5
wB
Eð2Þ;
Shm 5τ
γμL2ð3Þ
E (V m21) � electric field intensity; τ (s) � time; B (T) � magnetic induction;
L (m) � characteristic length; we5E/B (m s21) � drift velocity; w (m s21) �velocity; γ (S m21) � specific electrical conductance; μ (H m21) � permeability.
It expresses the ratio of the local inertia force to the convective electromagnetic
force. It characterizes the dynamic relations in non-stationary electromagnetic pro-
cesses. It expresses the degree of the movement non-stationarity in a system.
Info: [A23].
Vincenc Strouhal (p. 87).
6.3.40 Stuart Number (1.) (Magnetic Force) St
St5γμ2H2L
ρw5
B2Lγρw
5Ha2Re21 5EumRem
γ (S m21) � specific electrical conductance; μ (H m21) � permeability; H (A m21) �magnetic field intensity; L (m) � characteristic length; ρ (kg m23) � density;
w (m s21) � velocity; B (T) � magnetic induction; Ha (2) � Hartmann number
(p. 320); Re (2) � Reynolds number (p. 81); Eum (2) � Euler magnetic number
(p. 319); Rem (2) � Reynolds magnetic number (p. 328).
This number expresses the ratio of the ponderomotive force caused by induced
currents to the fluid inertia force. It characterizes the electrically conductive fluid
flow in a magnetic field. This criterion can also be held as the ratio of the persis-
tence time of the fluid in a magnetic field to the relaxation time of magnetic field
flux lines. Sometimes, it is called the magnetic parameter or the parameter of mag-
neto-hydrodynamic mutual interaction (interaction parameter). For St. 1 with
Rem{1, the magnetic forces strongly influence the flow. Magneto-hydrodynamics.
Info: [A23],[A33].
John Trevor Stuart (born 1929), English mathematician.
329Electromagnetism
6.3.41 Stuart Number (2.) (Charge Volume Density) St
St5εE2
ρw25
ρeELρw2
ε (F m21) � permittivity; E (V m21) � electric field intensity; ρ (kg m23) � fluid
density; w (m s21) � flow velocity; ρe (C m23) � volume density of an electric
charge; L (m) � characteristic length.
It expresses the electrostatic-to-dynamic pressures ratio for a fluid. It is also
called the electrodynamic number Nmgd. It characterizes the electrodynamic process
in magneto-hydrodynamics.
Info: [A23].
John Trevor Stuart (see above).
330 Dimensionless Physical Quantities in Science and Engineering
7 Physical Technology
There is a Plenty of Room at the BottomRichard Phillips Feynman (1918�1988)
7.1 Micro- and Nanotechnology
In micro- and nanotechnology, the dimensionless quantities differ substantially
from those in macroscopically oriented physics and mechanics. Here, specific phys-
ical problems of these technologies appear as adhesion, tension, deformation, plas-
ticity, peeling and processes of flow in microchannels and nanopores. The
microfluidic phenomena � the flow of micro- and nanoscopic particles, such as
carbon soot, dust and tissue cells � are connected with these technologies. The
Knudsen, Mason, microfluidic, Tabor and Zhang-Zhan numbers are among the
most popular dimensionless quantities in micro- and nanotechnology.
Richard Phillips Feynman (11.5.1918�15.2.1988),
American theoretical physicist. Nobel Prize in Physics, 1965.
The main area of his research was in the field of quan-
tum physics, in which he participated in the origin of quan-
tum electrodynamics. He contributed to defining the
mechanisms of interactions between elementary particles
and elaborated on the technique to describe them, namely
the eponymous Feynman diagrams. He predicted the exis-
tence of the internal structures of the proton and the neu-
tron. His works contributed significantly to describing the
microworld we now know. He played an important role in
creating a new sphere: nanotechnology.
7.1.1 Adhesion Parameter Nadh, Θ
Nadh 5Eh3=2
r1=2a Δσ
5Eh
Wadh
f 1=2r ; where Δσ5σ1 1σ2 2 σ1;2
E (Pa) � modulus of elasticity; h (m) � standard deviation of the peak heights of
roughness; ra (m) � radius of curvature of an asperity; Δσ (N m21) � Dupre
Dimensionless Physical Quantities in Science and Engineering. DOI: 10.1016/B978-0-12-416013-2.00007-5
© 2012 Elsevier Inc. All rights reserved.
adhesion, adhesion surface tension; σ1, σ2, σ1,2 (N m21) � surface tension between
two spheres; Wadh (J) � adhesion work; fr (2) � roughness factor.
It expresses the ratio of the elastic energy to the adhesion work, provided there
is a contact. For Nadhc1, there is only a partial contact, where the elastic material
has contact only on peaks of the highest roughness. Full contact occurs with
Nadh{1 only. The parameter is based on the Gauss transversal roughness distribu-
tion on an elastic material surface. Micro- and nanotribology.
Info: [B128].
7.1.2 Benard�Marangoni Instability Number Mrcrit
Mrcrit 5σ0ΔTh
ηa5 80; where σ0 5
dσdT
σ0 (N m21 K21) � rate of change of surface tension with temperature; ΔAT (K) �temperature difference between lower level and free liquid level; h (m) � depth of
fluid layer; η (Pa s) � dynamic viscosity; a (m2 s21) � thermal diffusivity;
σ (N m21) � surface tension; T (K) � temperature.
This number is expressed by the value of the critical Marangoni number Mr
(p. 186), with which instability occurs in a fluid surface layer. This instability man-
ifests itself by loss of contact with the bottom wall, of which the temperature must
be higher than the air temperature above the fluid layer surface. The
Benard�Marangoni set of hexagonal cells builds a system in which the fluid flows
upwards through the centre of the cells, and downwards near the walls.
Benard�Marangoni instability arises from the temperature nonuniformity in the
fluid and is a certain modification of the classic Rayleigh�Benard instability num-
ber, which dates from the beginning of the 20th century. It relates to hydrothermal
flow and surface waves under the action of an intensive heat source (i.e. laser or
electron beam) on the molten material surface. Thermocapillary phenomena.
Microgravitational applications. Microtechnology.
Info: [C7].
Carlo Guiseppe Matteo Marangoni (1840�1925), Italian physicist.
7.1.3 Darcy Number (3.) Porous Da
Da5h2
ξ
h (m) � thickness of porous layer; ξ (m2) � permeability of porous material.
It characterizes the permeability in porous material and in microchannels. It is
analogous to the Darcy granulation number (2.) Dc (p. 98). Two-phase flow.
Micro- and nanotechnologies. Thin layers.
Info: [B95].
Henry Philibert Gaspard Darcy (p. 98).
332 Dimensionless Physical Quantities in Science and Engineering
7.1.4 Deborah Number (2.) De
De5 τrxγ52τrxwmed
kð1Þ;
De5ReNelast ð2Þ
τrx (s) � fluid relaxation time; γ (s21) � shear rate; wmed (m s21) � mean fluid
flow velocity; k (m) � contraction width of the channel; Re (2) � Reynolds num-
ber (p. 81); Nelast (2) � elasticity number (4.) (p. 333).
It characterizes the low viscosity elastic fluid flow in industrially made micro-
channels with various degrees of narrowing when the fluid contracts and expands.
It is a modification of Deborah number (1.) De (p. 120). It relates to rheologic fluid
flow in technological microchannels with diverse geometric contraction.
Info: [B97].
7.1.5 Dispersion Number Ndis
Ndis 52kd
σwR2h
kd (N) � dispersion coefficient; σw (N m21) � fluid surface tension; Rh (m) �hydraulic diameter.
It represents the quantity of the dispersion force in the thin surface layer of a
microchannel. Two-phase flow in microchannels. Micro- and nanotechnology. Thin
layers.
Info: [B95].
7.1.6 Elasticity Number (4.) Nelast , El
Nelast 5τrxηRbDh
5DeRe21; where Dh 52bh
b1 h
τrx (s) � fluid relaxation time; η (Pa s) � dynamic viscosity; R (kg m23) � fluid
density; b (m) � width of contraction; Dh (m) � hydraulic diameter; h (m) �height of channel; De (2) � Deborah number (1.) (p. 120); Re (2) � Reynolds
number (p. 81).
It characterizes the non-Newtonian flow of the low viscosity elastic fluid
flow in industrially made microchannels in which the concentration and expansion
processes of the fluid flow occur. It is a modification of elasticity number (1.) El
(p. 121). It relates to the flow of polyethylene and similar rheologic fluids through
technological channels with diverse geometric contraction.
Info: [B97].
333Physical Technology
7.1.7 Evaporation Momentum Force (1.) Nv
Nv 5q
qmlv
� �2 RlRv
q (W m22) � heat flux; qm (kg m22 s21) � mass flux density; lv (J kg21) � latent
heat of vapourization; Rl, Rv (kg m23) � density of liquid and vapour, respectively.
It expresses the ratio of the momentum force, during fluid evaporation, to the
inertia force in microchannels. Primarily, it involves the boiling number Boi
(p. 215) and fluid to vapour densities ratio. Boiling. Heat transfer in microchannels.
Microtechnology.
7.1.8 Evaporation Momentum Force (2.) Nv
Nv 5q
lv
� �2L
Rvσ
q (W m22) � heat flux; lv (J kg21) � latent heat of vapourization; L (m) � charac-
teristic length (bubble diameter); Rv (kg m23) � vapour density; σ (N m21) �surface tension.
It expresses the ratio of the momentum force, in fluid evaporation, to the surface
strain force with fluid passage and evaporation in microchannels. Heat transfer.
Boiling. Microtechnology.
7.1.9 Force Flow Boiling Number in Microchannels (1.) NF1
NF1 5qA
qmlv
� �2 RlRv
qA (W m22) � surface heat flux; qm (kg m22 s21) � mass flux density; lv (J kg21) �
specific latent heat of vapourization; Rl, Rv (kg m23) � density of liquid and vapour.
It expresses the ratio of the evaporating momentum force to the inertia force in
fluid flow and evaporation in microchannels. For these relations, low values of the
hydraulic diameter and Reynolds numbers ReAh(100; 1000i (p. 81) are characteris-
tic. Two-phase flow. Boiling. Nonequilibrium processes. Micro- and
nanotechnologies.
Info: [B57].
334 Dimensionless Physical Quantities in Science and Engineering
7.1.10 Force Flow Boiling Number in Microchannels (2.) NF2
NF2 5qA
lv
� �2L
Rvσ
qA (W m22) � surface heat flux; lv (J kg21) � specific latent heat of vapourization;
L (m) � characteristic length, bubble diameter; Rv (kg m23) � vapour density;
σ (N m21) � surface tension.
This number expresses the ratio of the evaporating momentum force to the sur-
face strain force in fluid flow and evaporation in microchannels. The expression
does not involve the influence of the contact angle. Two-phase flow. Boiling.
Nonequilibrium processes. Micro- and nanotechnologies.
Info: [B57].
7.1.11 Heat Removal Nrem
Nrem 5λlðTsat 2 TwÞ
Rlνlllv
λl (W m21 K21) � fluid thermal conductivity; Tsat (K) � saturation temperature;
Tw (K) � wall temperature; Rl (kg m23) � liquid density; νl (m2 s21) � kinematic
viscosity; llv (J kg21) � latent heat of evaporation.
It expresses the ratio of the heat transfer rate, in terms of the channel length unit,
to the transfer of fluid properties. Two-phase flow. Micro- and nanotechnology.
Info: [B95].
7.1.12 Kandlikar Number (1.) Kan
Kan5q
Gr
� �2 RlRg
q (W m22) � heat flux; G (kg m22 s21) � mass flux density; r (J kg21) � specific
heat of evaporation; Rl, Rg (kg m23) � fluid and vapour density.
This number expresses the ratio of the evaporation momentum force to the iner-
tia force. Comparing to the boiling number Boi (p. 215) � which does not, itself,
represent the real evaporation momentum effect � the Kandlikar number involves
the ratio of fluid and vapour densities, too. Among other things, the number Kan
describes the heat transfer mechanism of a boiling fluid flowing through micro-
channels. Two-phase flow.
Info: [B56].
335Physical Technology
7.1.13 Kandlikar Number (2.) Kan
Kan5q
r
� �2 L
Rgσ
q (W m22) � heat flux; r (J kg21) � specific heat of evaporation; L�D (m) �characteristic number, hydraulic diameter D; Rg (kg m23) � vapour density;
σ (N m21) � surface tension.
It expresses the ratio of the evaporation momentum force to the surface stress
force. It describes the heat transfer mechanism of a boiling fluid flowing through
microchannels. Two-phase flow.
Info: [B56].
7.1.14 Knudsen Micro- and Nanometric Number Kn
Kn5
ffiffiffiffiffiπk2
rM
Re5
ffiffiffiffiffiffiffiffiffiπ
2RT
rηRL
R (J mol21 K21) � molar gas constant; T (K) � absolute fluid temperature; η (Pa s) �dynamic viscosity; R (kg m23) � fluid density; L (m) � characteristic length
(microchannels and nanopores dimension); K (2) � specific heat ratio (p. 28);
M (2) � Mach number (p. 73); Re (2) � Reynolds number (p. 81).
It characterizes the flow in microchannels and nanopores. For KnAh0; 1023i, itdescribes continuous flow; for KnAh1023; 1021i it describes shear flow; for
KnAh1021; 101i it describes transient flow; and for KnAh101; Ni it describes freemolecular flow. Micro- and nanomechanics.
Info: [B99].
Martin Hans Christian Knudsen (p. 420).
7.1.15 Mason Dielectrophoretic Number Masdf
Masdf 5Mf1
Mfdf15
8ηwε0εfβ2dE2
η (Pa s) � fluid dynamic viscosity; w (m s21) � characteristic velocity; ε0,εf (F m21) � vacuum and fluid permittivity, respectively (ε05 8.8542310212 F m21); β (2) � Clausius�Mossotti factor; d (m) � particle diameter;
E (V m21) � electric field intensity; Mf1 (2) � microfluidic number (p. 338);
Mfdf1 (2) � microfluidic dielectrophoretic number (1.) (p. 337).
It expresses the ratio of the viscosity force to the electrostatic interaction force
between particles. It characterizes the microfluidic phenomena in the flow of
micro- and nanoscopic particles (i.e. carbon black, glass dust or tissue cells).
Info: [B55].
336 Dimensionless Physical Quantities in Science and Engineering
7.1.16 Mass Flux Number Nqm
Nqm 522 c
2c
λv
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRðTw 2 TlÞ3
qσwllv
ffiffiffi2
pπ
c (2) � constant, usually c5 1; λv (W m21 K21) � vapour thermal conductivity;
R (J kg21 K21) � gas constant; Tw (K) � wall temperature; Tl (K) � fluid tempera-
ture; σw (N m21) � fluid surface tension at the wall; llv (J kg21) � latent heat of
vapourization.
It characterizes the mass flow on the interface between phases. It is implied
from the kinetic theory of fluid flow in thin layers and microchannels under phase
conversion conditions.
Info: [B95].
7.1.17 Microchannel Flow Boiling Number Boim, Co
Boim 5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσgðRl 2 RgÞD2
h
r
σ (N m21) � surface tension; g (m s22) � gravitational acceleration; Rl, Rg(kg m23) � density of liquid, gas or vapour; Dh (m) � hydraulic diameter.
When boiling fluid passes through microchannels, it expresses the ratio of the
surface strain force to the buoyancy force acting on a vapour bubble.
Thermomechanics. Boiling. Micro- and nanotechnologies.
Info: [B56].
7.1.18 Microfluidic Dielectrophoretic Number (1.) Mfdf1
Mfdf1 53πε0β2d3E2
4mw2
ε0 (F m21) � vacuum permittivity (ε05 8.85423 10212 F m21); β (2) � positive
or negative direction of dielectrophoretic force; d (m) � particle diameter;
E (V m21) � electric field intensity; m (kg) � particle mass; w (m s21) � charac-
teristic velocity of particle.
It expresses the ratio of the electrostatic interaction force, between particles, to
the inertia force. It relates to microfluidic phenomena in the flow of micro- and
nanoscopic particles in an electrical field.
Info: [B55].
337Physical Technology
7.1.19 Microfluidic Dielectrophoretic Number (2.) Mfdf2
Mfdf2 54πε0εfβd4E2
mw2L; where β5
εp 2 εfεp 1 2εf
ε0 (F m21) � vacuum permittivity (ε05 8.85423 10212 F m21); εf (F m21) � fluid
permittivity; β (2) � Clausius�Mossotti factor; d (m) � particle diameter;
E (V m21) � electric field intensity; m (kg) � particle mass; w (m s21) � particle
flow velocity; L (m) � characteristic length (electrodes distance).
It expresses the dielectrophoresis and inertia forces ratio in the flow of particles
in an electric field. Microfluidic phenomena.
Info: [B55].
7.1.20 Microfluidic Dielectrophoretic Number (3.) Mfdf3
Mfdf3 5Mfdf1
Mfdf25
3βL16d
β (2) � Clausius�Mossotti factor; L (m) � characteristic length (electrodes dis-
tance); d (m) � particle diameter; Mfdf1 (2) � microfluidic dielectrophoretic num-
ber (1.) (p. 337); Mfdf2 (2) � microfluidic dielectrophoretic number (2.) (p. 338).
When the particles flow in an electric field, it characterizes the force relations.
It always applies in cases of important influence of mutual interaction of particles
and the dielectrophoresis force. Microfluidic phenomena.
Info: [B55].
7.1.21 Microfluidic Number Mf1
Mf1 56πηd2
mw
η (Pa s) � dynamic viscosity of fluid; d (m) � particle diameter; m (kg) � particle
mass; w (m s21) � characteristic velocity.
It characterizes the behaviour of micro- and nanoscopic particles in a flowing
fluid, for example, that of live cells which are suspended in the fluid. It expresses
the viscous to inertia forces ratio.
Info: [B55].
7.1.22 Mismatch Parameter δ
δ5 100sff 2 sss
sss; where sfs 5
sff 1 sss
2
338 Dimensionless Physical Quantities in Science and Engineering
sff (m) � atom geometric parameter in thin layer; sss (m) � atom geometric param-
eter in substrate; sfs (m) � atom geometric parameter on the boundary line of thin
layer and substrate.
It characterizes the growth of thin layers, with deposition processes, and the
structure originating on the interface between a thin layer and the substrate mate-
rial. It is applied, for example, in the simulation of deposition processes to create
interatomic potentials. Micro- and nanotechnology. Thin layers.
Info: [B84].
7.1.23 Peel Number Np
Np53Es3h2
2L4Wadh
E (Pa) � modulus of elasticity; s (m) � layer thickness; h (m) � distance of
unstuck layer from substrate; L (m) � characteristic length of crack; Wadh (J) �adhesion work.
With the deposited layer deviated from the substrate, it characterizes the rela-
tions of adhesion. It expresses the ratio of the elastic deformation energy, stored in
the deformed microstructure memory, to the adhesion work between the micro-
structure and the substrate. When Np. 1, the elastic deformation stored in the
memory is greater than the work of adhesion and the microstructure will not adhere
to the substrate. When Np# 1, the deformed layer does not have enough energy to
overpower the adhesion.
Info: [B128]
7.1.24 Plastic Adhesion Index Npl.ad.
Npl:ad: 5π2H4raσ8E2
ekvW2a
H (N) � material hardness; ra (m) � curvature radius of the roughness peaks,
in point as a sphere; σ (m) � standard deviation of the roughness height distribu-
tion; Eekv (Pa) � equivalent modulus of elasticity (see Tabor number Tab)
(p. 343); Wa (J) � Dupre adhesion work.
It characterizes the quantitative point of view of the relation between adhesion,
plastic deformation and other corresponding phenomena. Micro- and nanotechnology.
Info: [B100].
7.1.25 Plasticity Index Nplast
Nplast 5Eekv
H
ffiffiffiffiσra
r5
Eekv
H
ffiffiffif
p
339Physical Technology
Eekv (Pa) � equivalent modulus of elasticity (see Tabor number Tab) (p. 343);
H (N) � material hardness; σ (m) � standard deviation of the roughness height dis-
tribution; ra (m) � curvature radius of the roughness peaks, in point as a sphere;
f (2) � relative surface roughness.
It is the plastic deformation criterion of a microscopic contact. For elastic rough-
ness it is Nplast, 0.6; for plastic roughness it is Nplast. 1; and for elastoplastic
roughness it is 0.6,Nplast, 1. Micro- and nanotechnology.
Info: [B128].
7.1.26 Prediction Number of Adhesion Npred
Npred 5 24
3πEσ2
WadhL
� �2=3
E (Pa) � equivalent modulus of elasticity; σ2 (m2) � quadratic deviation of the
roughness height distribution; Wa (J) � adhesion work; L (m) � characteristic cor-
relation length.
It characterizes the quantitative predictions of adhesion between deformable
fractal surfaces. It expresses the ratio between the standard deviation of the rough-
ness heights and the maximum height of the roughness peaks before transforming.
Micro- and nanotechnologies.
Info: [B128].
7.1.27 Pulse Electromagnetic Heating of Microparticles Nheat , β
Nhead 5dεP2λTref
d (m) � microparticle diameter; P (W m22) � power density of the electromag-
netic field; λ (W m21 K21) � thermal conductivity of microparticles; Tref (K) �reference temperature; ε (2) � emissivity (p. 206).
It characterizes the pulse heating of microscopic particles with a laser, as an
example. Among other things, it is used to determine the thermal, physical proper-
ties of the particles. Physical technology. Microtechnology. Thermomechanics.
Electromagnetism.
7.1.28 Pumping Intensity Npi
Npi 5ðpl 2 pvÞL2
ν2l Rl
340 Dimensionless Physical Quantities in Science and Engineering
pl (Pa) � fluid pressure; pv (Pa) � vapour pressure; L (m) � characteristic pressure
length; νl (m2 s21) � kinematic viscosity of liquid; Rl (kg m23) � liquid density.
It represents the ratio of the maximum power � which can be obtained with the
pressure difference between a fluid and a vapour � to the transfer parameters.
Porous flow through microchannels. Two-phase flow. Micro- and nanotechnology.
Thin layers.
Info: [B95].
7.1.29 Pumping Resistance Npr
Npr 5pRh
4σ
p (Pa) � fluid pressure; Rh (m) � hydraulic diameter; σ (N m21) � fluid surface
tension on the channel wall.
It characterizes the resistance of the two-phase miniature microchannel flow in
porous layers which serve, for example, to cool down the electronic circuits or as
miniature heat exchangers or pumping microcircuits. Two-phase flow. Micro- and
nanotechnology. Thin layers.
Info: [B95].
7.1.30 Response Number Rn
Rn5Rw2
ini
τL
H
� �25Dn
L
H
� �2ð1Þ;
Rn5I2
RτH2
L
H
� �2ð2Þ
R (kg m23) � material density; wini (m s21) � initial velocity of particles; τ (Pa) �shear stress of material; L (m) � characteristic length; H (m) � thickness of beam
or slab; I (Pa s21) � pressure impulse velocity of rectangular shape; Dn (2) �Johnson’s damage number.
This number characterizes the dynamic plastic response in a material exposed to
impulse or stroke pressure loading. It especially relates to diverse beams or plates.
However, for the dynamic load of generally shaped bodies, expression (2) is valid.
Info: [B127].
7.1.31 Reynolds Rheological Number (2.) Rerh
Rerh 5wmedDh
ν05
2Q
ðb1 hÞν0;
where wmed 5Qbh; Dh 5
2bhb1 h
341Physical Technology
wmed (m s21) � average fluid velocity; Dh (m) � hydraulic diameter; ν0 (m2 s21) �
kinematic viscosity at zero shear speed; Q (m3 s21) � volume flow rate; b (m) �width of contraction; h (m) � height of the channel.
It characterizes the forced flow of a low viscosity elastic fluid through micro-
channels with a range of technological cross section narrowing when the fluid is
contracted and expanded. Rheology. Micro- and nanotechnologies.
Osborne Reynolds (p. 82).
7.1.32 Strain Peel Number Npε
Npε 5128Es3h2
8L4Wadh
114σrL
2
21Es21
256
2205
h
s
� �2" #ð1Þ;
Npε 5186Es3h2
ð12 ν2Þb4Wadh
1127ð12 ν2Þσrb
2
310Es21
12
31
h
s
� �2" #ð2Þ;
Npε 540Es3h2
3ð12 ν2ÞR4Wadh
1151ð12 ν2ÞσrR
2
160Es2
� �ð3Þ
E (Pa) � modulus of elasticity; s (m) � layer thickness; h (m) � distance of
unstuck layer from substrate; L (m) � length of double-sided, close-set beam;
Wadh (J) � adhesion work; σr (Pa) � residual stress; ν (2) � Poisson’s ratio
(p. 143); b (m) � width of beam; R (m) � radius of circular plate.
In a dimensionless shape, the number Npε expresses the residual strain in a
beam fixed at both ends (1), in a square plate (2) or in a round plate (3). From these
equations, the maximum dimensions in the microstructure can be determined
(the beam length or fixation, the width or the plate radius), with which the layer
will not be connected with the substrate. It is valid provided the condition Np5 1
holds, with the inner energy of the elastic deformation corresponding to the
adhesion.
Info: [B127].
7.1.33 Subcooling Number (2.) Nsc
Nsc 52ηvλvðTw 2 TlÞRhRvπllvσw
ηv (Pa) � vapour dynamic viscosity; λv (W m21 K21) � vapour thermal conductivity;
Tw (K) � wall temperature; Tl (K) � fluid temperature; Rh (m) � hydraulic diameter;
Rv (kg m23) � vapour density; llv (J kg
21) � latent heat of vapourization; σw (N m21) �fluid surface tension at the wall.
342 Dimensionless Physical Quantities in Science and Engineering
In porous material microchannels, this number expresses the condensation
process and its influence on the heat dissipation with transfer zone reduction and
very high dissipation capability. Heat transfer in microchannels. Condensation and
controlled cooling. Microtechnology.
Info: [B95].
7.1.34 Tabor Number Tab, μ
Tab5RW2
a
E2ekvε2
� �1=3; h5
RW2a
E2ekv
� �1=3
where E21ekv 5
12 ν21
E11
12 ν22
E2; R (m) � equivalent radius of curvature; Wa (J) �
Dupre adhesion work; Eekv (Pa) � equivalent modulus of elasticity; E1, E2 (Pa) �modulus of elasticity of two hemispheres; ν1, ν2 (2) � Poisson’s constant of two
hemispheres; ε (m) intermolecular distance; h (m) � depth of penetration.
The Tabor number is the degree of elastic deformation, which depends on the
surface forces’ size. It characterizes the nonequilibrium adhesion mechanism of a
contact in micromechanical systems. Alternatively, it expresses the ratio of the
impression height to the intermolecular distance. Bodies in contact characterized
by a small number, Tab , 0.1, have slight influence on the elastic deformation
and adhere one to the other more easily. For Tab. 5, it is the opposite. With both
materials and quantities Wadh, Eekv and ε being equal, TabBR1=3 holds. Micro- and
nanotribology.
Info: [B128].
7.1.35 Transport Heat Number Nheat
Nheat 5Rlν2lσL
Rl (kg m23) � liquid density; νl (m
2 s21) � liquid kinematic viscosity; σ (N m21) �surface tension; L (m) � characteristic length of channel.
It characterizes the condition for the maximum heat transfer on the microchan-
nel length unit. Two-phase flow. Micro- and nanotechnologies.
Info: [B95].
7.1.36 Zhang-Zhao Number (1.) Zh1
Zh1 5Wa
σH
Wa (J) � Dupre adhesion work; σ (m) � standard deviation of the roughness height
distribution; H (N) � hardness of material.
343Physical Technology
It expresses the ratio of the adhesion work to the product of the standard deviation
of the roughness peak heights and the material hardness. The greater this number is,
the more easily the plastic deformation occurs. Micro- and nanotechnologies.
Info: [B90].
7.1.37 Zhang-Zhao Number (2.) Zh2
Zh2 5d
σ
d (m) � difference between mean surface and glare surface during loading; σ (m) �standard deviation of the roughness heights.
It expresses the ratio of the difference between the central surface height and the
smooth surface during loading. It affects the plastic deformation adhesion.
7.2 Plasma Physics and Technology
The dissimilarity of plasma � in the form of ionized gas � from the solid, liquid
or gaseous phases manifests itself in dimensionless quantities as well. The quanti-
ties relate to gas particle behaviour; wavelength; frequency and thermal equilibrium
of the plasma; magnetic field influence on the plasma; and plasma pressure obser-
vation with regard to magnetic pressure, to magnetohydrodynamic load and to other
quantities. Plasma is associated with various technologies utilizing intensive
sources such as the laser, an electric arc in arc welding, etching of dielectric layers
in electronics with integrated circuits manufacture, thermal loading of the front sur-
face of space bodies in descent into the atmosphere, nuclear fusion research and
other areas of physics and technology. The de Broglie and Larmor numbers, Hall’s
parameter and many other numbers are well known dimensionless quantities in
plasma physics. Systematic using of the dimensionless quantities for plasma tech-
nologies has been lacking up to now.
7.2.1 Anisothermal Plasma Number Nta,τ
Nta 5Te
Ti
Te, Ti (K) � kinetic temperature of electrons and ions.
This number expresses a dimensionless parameter characterizing thermal plasma
anisotropy. It appears in plasma density determination and discovering out the ion
collision influence in measurement with the Langmuir cylindrical probe. The
344 Dimensionless Physical Quantities in Science and Engineering
anisothermal plasma number and the Knudsen number for electrons (ions) Kne,i(p. 349) represent two parameters which presume the Maxwell distribution of elec-
tron energy. Plasma physics and technology.
Info: [B88].
Irving Langmuir (31.1.1881�16.8.1957), American chem-
ist and physicist. Nobel Prize in Chemistry, 1932.
He devoted himself to studying light and contributed sig-
nificantly to vacuum engineering improvement, which led
to the idea of a high vacuum tube. He was engaged in
research related to glowing filament behaviour in a vacuum
and in the thermal emission problem. He was one of the
first scientists who worked with plasma, which he called
ionized gas. He introduced the concept of electron tempera-
ture and the diagnostic method to measure vacuum temper-
ature and density with the Langmuir probe. He devoted also
to atmospheric sciences and meteorology.
7.2.2 Attraction Number Na
Na 5A
12πηr2pw
A (J) � Hamaker constant; η (Pa s) � dynamic viscosity; rp (m) � particle radius;
w (m s21) � approximate fluid velocity.
It represents the combined influence of the van der Waals attraction forces and
the fluid rate on the deposition velocity with particle capture. Deposition processes.
Physical and chemical filtration in saturated porous materials.
Info: [B48].
7.2.3 de Broglie Thermal Wavelength Λ
Λ5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffih2
2πmkT
rð1Þ;
V
nΛ3# 1 ð2Þ;
V
nΛ3c1 ð3Þ;
Λ5ch
2kT3ffiffiffiπ
p ð4Þ
345Physical Technology
h (J s) � Planck constant; m (kg) � mass of gas particles; k (J K21) � Boltzmann
constant; T (K) � absolute temperature; c (m s21) � speed of light; V (m3) � vol-
ume; η (m23) � number of particles in volume V.
This number represents roughly the average of the de Broglie wavelength of ideal
gas particles with specific density. It can be held as a mean distance between the par-
ticles in the gas, approximatelyffiffiffiffiffiffiffiffiffiffiffiVn213
p: When the thermal wavelength is much less
than the distance between the particles, the gas behaves as a classic
Maxwell�Boltzmann gas. However, when the thermal de Broglie wavelength is
greater � by one or more orders � than the distance between the particles, the quan-
tum effects become dominant and the gas must be considered a Fermi gas or a Bose
gas, depending on the substance of gas particles. In this type of situation, expression
(2) is valid, and for the preceding situation, the Maxwell�Boltzmann expression (3)
holds. For a massless particle, the thermal wavelength can be expressed by expres-
sion (4). Plasma physics.
Info: [C133].
Louis-Victor Pierre Raymond, duc de Broglie(15.8.1892�19.3.1987), French mathematician and physi-
cist. Nobel Prize in Physics, 1929.
He studied quantum theory and the wave theory of elec-
trons based on Einstein’s and Planck’s works. This led to
the proposal of the wave particle duality theory. According
to it, a substance has properties both of particles and of
waves. The results of de Broglie’s theory of electromag-
netic waves later served Schrodinger, Dirac and other scien-
tists in the creation of wave mechanics.
7.2.4 de Broglie Wavelength λ
λ5h
pð1Þ;
λ5h
mvð2Þ
h (J s) � Planck constant; p (kg m s21) � momentum of particle; m (kg) � mass of
particle; v (m s21) � speed of particle.
It represents a wavelength as the ratio of the kinetic energy of a particle to its
momentum. In physics, de Broglie’s hypothesis primarily expresses the fact that all
materials have properties of waves. Plasma physics.
Info: [C31].
Louis-Victor Pierre Raymond, duc de Broglie (see previous entry).
346 Dimensionless Physical Quantities in Science and Engineering
7.2.5 de Broglie Wave Number k
k52πph
5p
h
p (kg m s21) � momentum; h (kg m s21) � Planck constant; h (kg m s21) � Planck
constant (h-bar).
This number expresses the ratio of a particle’s momentum to its kinetic energy.
Plasma physics.
Info: [C32].
Louis-Victor Pierre Raymond, duc de Broglie (see previous entry).
7.2.6 Debye Number ND
ND 54
3πneλ2
D ð1Þ;
λD 5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiε0kq22
e
neT 21e 1
Pij j
2nijT21i
sð2Þ;
λD 5
ffiffiffiffiffiffiffiffiffiffiffiε0kTeneq2e
sð3Þ
ne (m23) � electron density; λd (m) � Debye length ε0 (F m21) � vacuum permittiv-
ity; k (J K21) � Boltzmann constant; qe (C) � electron charge; Te, Ti (K) � electron
and ion temperatures, respectively; nij (m23) � density of atom particles i with posi-
tive ion charge jqe.
This number represents the relation between the Debye length and the probe
radius. It expresses the state of the thermal mode and the density during a plasma’s
existence. For ND5 1, the state occurs which is denoted as the ratio of the collec-
tive behaviour to the independent behaviour of particles. When NDc1, it repre-
sents the collective behaviour mode with which the dynamic properties of the
plasma are changing. When ND, 1, it is called the mode of independent particles,
in which the collective phenomenon has slight influence in the plasma. Expressions
(2) and (3) represent the Debye length, which expresses the shielding diameter in
the plasma. Expression (3) is valid only if the ions are much colder than the elec-
trons. The size of λD changes substantially from 10211, for the plasma in the Sun’s
nucleus (ne5 1032), over 1024 for a gas discharge (ne5 1016), to values of 105
(ne5 1) for the intergalactic environment. Plasma physics and technology.
Tokamak. Astrophysics.
Info: [A24], [C33].
347Physical Technology
Peter Josephus Wilhelmus Debye (24.3.1884�2.11.1966),
Dutch-American electrical engineer, physicist, chemist.
Nobel Prize in Chemistry, 1936.
He formulated the specific heat theory (1911) and
formed the theory of electrolytic solutions. He engaged in
work related to dipole moments of molecules. He had
remarkable knowledge of the arrangement of atoms and
molecules and of the distances between atoms. Together
with P. Scherrer, he developed (1916) the method to deter-
mine the atomic structure of crystals by means of X-rays.
This is called the Debye�Scherrer method.
7.2.7 Hall Parameter, Hall Coefficient H, β
H5ωτ5eBτm
5KnLr1
ω (s21) � resonant frequency of electron; τ (s) � average relaxation time of elec-
tron; e (C) � electron charge; B (T) � magnetic induction; m (kg) � electron
mass; Kn (�) � Knudsen number (1.) (p. 69); Lr (�) � Larmor number (p. 322).
In magnetohydrodynamics, it expresses the ratio of the mean time between colli-
sions of particles to the time of their circulation around a magnetic flux line. It is
an important criterion for examining three plasma acceleration modes in a magnetic
field. There is no dispersion for βc1 , but it occurs for β5 1 and is aperiodic for
β{1 . Plasma physics.
Info: [A29].
Edwin Herbert Hall (p. 320).
7.2.8 Ionization Plasma Number Ni
Ni 5ni
nn� 33 1027
T32
niexp 2
Ui
kT
� �
ni (m23) � number of ionized atoms per unit volume; nn (m
23) � number of neu-
tral atoms per unit volume; T (K) � absolute temperature; Ui (J) � ionizing energy;
k (J K21) � Boltzmann constant.
It expresses the general approximation of the thermal plasma equilibrium. It pre-
dicts which ionization degree can be expected in a gas with thermal equilibrium.
Plasma physics.
Info: [C103].
348 Dimensionless Physical Quantities in Science and Engineering
7.2.9 Knudsen Number for Electrons (Ions) Kne,i
Kne;i 5λe;i
rp
λe,i (m) � electron (ion) mean free path; rp (m) � probe radius.
It is the characteristic number for the theory of collisions, as it describes the
mean free path of electrons or ions in relation to the dimension of the applied
probe. For collisionless conditions, Kne,i - N. In other words, with a probe, the
collisions act on the charged particles more intensely as the probe radius increases,
especially under experimental conditions. Plasma physics and technology.
Info: [B88].
Martin Hans Christian Knudsen (p. 420).
7.2.10 Larmor Plasma Number Lrpl
Lrpl 5rB
rt5
v
rtωB
; where ωB 5 gB
rB (m) � Larmor radius; rt (m) � toroid radius; v (m s21) � particle velocity per-
pendicular to magnetic field; ωB (s21) � Larmor frequency; g (s21 T21) � gyro-
magnetic radius; B (T) � magnetic induction.
It expresses the ratio of the Larmor radius to that of a toroid. Alternatively, it is
the ratio of the particle velocity to the magnetic field intensity. Plasma and atomic
physics.
Info: [B86],[C131],[C104].
Joseph Larmor (p. 322).
7.2.11 Magnetization Parameter δ
δ5RL
ð1Þ
R (m) � Larmor radius, gyroradius; L (m) � characteristic length.
It characterizes the magnetic field influence on a plasma. The plasma process or
system is magnetized if its characteristic length is much greater than the gyrora-
dius. When R{L, the charged particles follow the magnetic field lines of trajecto-
ries directly. In the opposite case, when RcL; the charged particles strike against a
chamber wall before they can be accelerated due to the magnetic field influence.
Plasma physics.
Info: [C131].
349Physical Technology
7.2.12 Controlled Thermonuclear Fusion number
β52μ0p
B2
μ0 (H m21) � vacuum permeability; p (Pa) � plasma pressure; B (T) � magnetic
induction.
It plays a key role in controlled thermonuclear fusion. It expresses the plasma to
magnetic pressures ratio. To maintain the plasma, the value must be β, 1 and the
magnetic flow must be higher than the plasma flow. Nevertheless, in reality it is
often β# 0.2, which limits, for example, the pressure p# 107 Pa with BB10 T.
Info: [C131].
7.2.13 Plasma Collision Frequency Nfpl ,γ
Nfpl 5Nπd2w
ω5Nπd2
ffiffiffiffiffiffiffiffiffi8πTmπ
r1
ω
N (m23) � number of particles per unit volume m3; d (m) � atom diameter;
w (m s21) � average atom velocity; ω (s21) � characteristic frequency of plasma;
k (J K21) � Boltzmann constant; T (K) � absolute temperature; m (kg) � atom
mass.
This expression represents an estimate � based on the gas kinetic theory � of
the plasma collision frequency. When Nfpl{ω; then Nfpl5 0.01. Plasma physics.
Info: [B86],[C104].
7.2.14 Plasma Parameter Λ
Λ5 4πnλ3D
n (m23) � number of particles per unit volume; λD (m) � Debye length.
It is among the basic plasma parameters. Usually, it expresses the number of
particles contained in a Debye sphere. When Λ{1, the Debye zone is rarely occu-
pied, which corresponds to a weakly coupled plasma. When Λc1; in the Debye
sphere there is a great concentration of particles to which strongly coupled plasma
corresponds. For strongly coupled plasma, lower temperatures and great densities
of particles are typical, whereas weakly coupled plasma is diffusive with low con-
centrations of particles and high temperatures. Plasma with a laser ablation and
plasma with an arc discharge are examples of strongly coupled plasma. Some
examples of typical values of a weakly coupled plasma are Λ5 53 108 for nuclear
fusion and Λ5 43 104 for the interstellar environment. Λ5 23 103 is valid for a
chromosphere and Λ5 33 102 for a glow discharge. Plasma physics.
Info: [C131].
350 Dimensionless Physical Quantities in Science and Engineering
7.2.15 Pressure Plasma Number Nβ , β
Nβ 52μ0p
B2; where p5 nkT
μ0 (H m21) � vacuum permeability; p (Pa) � plasma pressure; B (T) � magnetic
induction; n (m23) � number of particles per unit volume; k (J K21) � Boltzmann
constant; T (K) � absolute temperature.
It expresses the ratio of plasma to magnetic pressures or, alternatively, that of
thermal energy density to magnetic energy density. It characterizes the conditions
under which certain limitations are valid to maintain the plasma for a magnetic
field. The value must be less than unity (Nβ, 1). As a matter of fact, the most
suitable value is NβB0.2. The pressure plasma number plays a key role, for exam-
ple, in controlled thermonuclear fusion. Plasma physics and technology.
Info: [B86],[C131],[C104].
7.2.16 Thrust Coefficient Ct
Ct 54πμ0
FT
I2
μ0 (H m21) � vacuum permeability; FT (N) � thrust force; I (A) � total current
between electrodes.
It characterizes the magnetoplasmadynamic load by the total current.
Magnetodynamics. Electromagnetic plasma accelerators.
Info: [B23].
351Physical Technology
8 Technology and MechanicalEngineering
The intuition is the only valuable thing indeed.Albert Einstein (1879�1955)
8.1 Technology and Material
In this section, quantities are presented which are applied only in specific areas
of a technology or materials processing. Others are not considered here because
their main application is in fundamental fields, and they are presented, therefore,
in other chapters of this book. In this section, technologies such as heating in
diverse combustion devices, welding, melting, surface treatment and solidification
processes, and other technologies, are discussed. The following numbers are
among the known quantities: the Boltzmann and Craya�Curtet numbers for fire-
places and furnaces, the Christensen, Rosenthal and Rykalin numbers for welding,
the Tikhonov number for thermochemical materials processing, and the Chvorinov
number for solidification time.
8.1.1 Agglomerate Number Ag
Ag5 Rw2
ffiffiffiffiffiffiffiffiffiffiffiD5E2
Γ 5
3
s
R (kg m23) � particle density; w (m s21) � particle velocity; D (m) � particle
diameter; E (Pa) � modulus of elasticity; Γ (J m22) � surface energy.
It expresses the degree of contact damage during agglomeration due to the
impact rate, the adhesion energy, and the properties and shape of particles. It is
based on the assumption that the energy used to erode the material is proportional
to the incidental kinetic energy of the agglomerate. The number is a modification
of the Weber agglomerate number Weagl (p. 363).
Dimensionless Physical Quantities in Science and Engineering. DOI: 10.1016/B978-0-12-416013-2.00008-7
© 2012 Elsevier Inc. All rights reserved.
8.1.2 Boltzmann Number of Combustion Chamber Bo
Bo5k1Qmcp
σk2ST3sp
k1 (2) � heat hold coefficient in combustion chamber, comprising heat losses to
ambient; Qm (kg s21) � fuel mass flux calculated; cp (J kg21 K21) � specific heat
capacity; σ (W m22 K24) � absorptance (absorption capacity, radiating capacity);
k2 (2) � contamination coefficient of wall surface; S (m2) � surface area of
absorption walls; Tsp (K) � theoretical combustion temperature.
It expresses the ratio of the heat transferred to a fireplace surface to the maxi-
mum heat which can be transferred by radiation. It characterizes the heat transfer
in boiler fireplaces. Their inner space is considered as an absolutely black body.
Info: [A23].
Ludwig Boltzmann (p. 205).
8.1.3 Christensen Number Chr
Chr5w2A
a2
w (m s21) � motion velocity; A (m2) � surface; L (m) � characteristic length,
weld width; a (m2 s21) � thermal diffusivity.
It is often called the dimensionless weld width. It expresses the coupling
between the weld dimension and the dynamic thermal process in electric arc weld-
ing. It is valid for both 3D and 2D solutions.
Info: [B37].
N. Christensen.
8.1.4 Chvorinov Cast Mould System Number Ch
Ch5τL2
RclΔTcfRf
Vc
Vf
λf
λc
� �12
5FoPVP12λ
τ (s) � time; L (m) � characteristic length L5Vc
S
� �; Rc, Rf (kg m23) � density of
casting and mould; l (J kg21) � specific latent heat of solidification; ΔT (K) �superheating of melting; cf (J kg21 K21) � specific heat capacity of a mould;
Vc (m3) � casting volume; Vf (m
3) � mould volume; S (m2) � surface area of cast-
ing; λf, λc (W m21 K21) � thermal conductivity of mould and casting, respectively;
Fo (2) � Fourier number (p. 175); PV (2) � volume parameter; Pλ (2) �thermal conductivity parameter.
354 Dimensionless Physical Quantities in Science and Engineering
In the case of a contact task, it expresses the mutual dynamic coupling between
thermomechanical quantities, during the process of casting solidification in a mould
and the cooling time of the casting. Foundry technology.
Nikolai Chvorinov (15.11.1903�7.11.1987), Czech engi-
neer of Russian origin.
He was an engineer, but a talented painter too, enabling
him to introduce abstraction into his foundry work, which
was only empirical before. With extensive experiments
researching the solidification and crystallization of steel
and diversely formed castings, he deduced that solidifica-
tion time is proportional to the second power of the vol-
ume-to-surface ratio. His work formed the basis for
subsequent theoretical research on foundry processes.
8.1.5 Chvorinov Shrink Formation Number Chsh
Chsh 5grad Tffiffiffiffiffiffi
wTp grad Tffiffiffiffiffiffi
wTp
� �21ref
grad T (K m21) � temperature gradient in solidifying casting;WT (K s21) � solidifi-
cation rate of casting; subscript ref � limiting value for shrink hole formation.
This criterion is based on Chvorinov’s idea about the influence of the simulta-
neous action of the geometric and time temperature gradients on a shrink hole in a
solidifying casting. The solution process of the casting solidification dynamics
starts by modeling the temperature field and � from it � the module and argument
fields of the geometric and time temperature gradients. From these, the distribution
field of the criterion for the shrink hole arising in a casting is determined. In the
casting, the zones having the value of Chsh, 1 point to the place of possible shrink
hole origin. For Chsh$ 1, no shrink hole appears. It is analogous to the Niyam
shrink formation number Φ (p. 358).
Nikolai Chvorinov (see above).
8.1.6 Chvorinov Time Number Ch, Chf
Ch5a1τL21
5λ1ΔTl
llR1
τL21
ð1Þ;
where l1 5 lls 1ΔTlc1 5lls
ΔTl1 c1
� �ΔTl 5 cekvΔTl;
355Technology and Mechanical Engineering
Chf 5a2τL22
5λ2
ðcRÞ2τL22
ð2Þ
a1, a2 (m2 s21) � thermal diffusivity; τ (s) � time; L1, L2 (m) � characteristic
length L1 5V1
S
� �; L2 5
V2
S
� �; V1, V2 (m
3) � volume; S (m2) � surface area of cast-
ing; λ1, λ2 (W m21 K21) � thermal conductivity; ΔTl (K) � superheating of melt-
ing; l1, lls (J kg21) � specific latent heat of solidification; R1, R2 (kg m23) �density; c1, c2 (J kg
21 K21) � specific heat capacity; cekv (J kg21 K21) � equivalent
specific heat capacity; subscripts: 1 � casting, 2 � mould, ls � liquidus�solidus.
This number expresses the dimensionless solidification time of a casting (1) and
a mould (2). It is a modification of the Fourier number Fo (p. 175).
Nikolai Chvorinov
8.1.7 Craya�Curtet Number Ct
Ct5wkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
w2d 2
12w2k
q ;
where wk 5Qm1 1Qm2
RA; wd 5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiH1 1
12H2
RA
s
wk (m s21) � mean kinematic velocity; wd (m s21) � mean dynamic velocity; Qm1,
Qm2 (kg s21) � mass flux relative to burner and to induced air stream; R (kg m23) �gas density; A (m2) � cross section of furnace; H1, H2 (kg m s21) � momentum rela-
tive to burner and to induced air stream.
It characterizes the radiation flow and the heat transfer in industrial furnaces by
means of a passing mass and the momentum of basic and induced currents. For
Ct-0, the axial flow has great momentum and recirculation. When Ct. 0.74, the
return flow disappears.
Info: [A23].
Antoine Joseph Edouard Craya (1911�1976), French engineer.
Roger Michel Curtet (born 1923), French engineer.
8.1.8 Heat Diffusivity B
B5a
aref; b5
ffiffiffiffiffiffiffiffiRcλ
p
a (m2 s21) � thermal diffusivity; aref (m2 s21) � reference value of thermal diffu-
sivity; R (kg m23) � material density; c (J kg21 K21) � specific heat capacity;
λ (W m21 K21) � thermal conductivity.
356 Dimensionless Physical Quantities in Science and Engineering
Most frequently, it characterizes the diffusivity in a cast-mould system with the
relative value corresponding to the casting diffusivity and the value of b corre-
sponding to the mould. Foundry work technology. Solidification and cooling of
castings.
Info: [B81].
8.1.9 Latent Heat NL
NL 5lsl
cðTl 2 TiniÞð1Þ;
NL 5lls
cðTl 2 TsÞð2Þ
lsl (J kg21) � specific latent heat of melting; lls (J kg
21) � specific latent heat of
solidification; c (J kg21 K21) � specific heat capacity; Tl, Ts, Tini (K) � tempera-
ture of liquidus and solidus, initial mould temperature.
It expresses the latent heat of melting (1) and solidification (2) of a casting in a
mould. Foundry work technology.
Info: [B81].
8.1.10 Loss Dissipation Coefficient η, Ndis
η5ΔU
2πU; where U5
ðσmax
0
dε1
2
σ2
E
U (J m23) � elastic energy in material per volume unit; ΔU (J m23) � difference
of dissipation energy between loading and unloading; σmax (Pa) � maximal stress
at elastic loading; E (Pa) � modulus of elasticity; ε (2) � relative elongation
(p. 144).
It expresses the degree which is reached by a material due to vibration energy
dissipation. The cycle can be expressed in various ways, either as fast or slow.
Usually, the coefficient value depends on a timescale or a cycling frequency.
Info: [C65].
8.1.11 Melting Efficiency ηW
ηW 5wAQV
Q
w (m s21) � motion velocity; A (m2) � weld surface;QV (J m23) � heat per unit
volume; Q (W) � heat flux.
357Technology and Mechanical Engineering
It expresses the melting efficiency as a ratio of the power necessary to melt the
welding zone to the net power absorbed in a part. It represents an important aspect
of the welding process. The theoretical maximum value is 0.48 for the 2D heat
propagation process. For the 3D process, it is 0.37.
Info: [B37].
8.1.12 Niyam Shrink Formation Number Φ
Φ5rΘ N21η
rΘ (2) � temperature gradient (p. 361); Nη (2) � solidification rate constant (p. 360).
It is an approximate criterion of the shrink hole originating in a casting. It is
based on the assumption that geometric and time gradients influence shrink hole
formation in a solidifying casting. It is analogous to the Chvorinov shrink formation
number Chsh (p. 355). Foundry work technology. Solidification of castings.
Info: [B81].
8.1.13 Part of Liquid Phase β
β5T 2 Ts
Tl 2 Tsfor Ts , T, Tl ð1Þ;
β5 0 for T , Ts ð2Þ;
where h5 href
ðTTref
cp dT ; ΔH5βL ð3Þ;
H5 h1ΔH ð4Þ;
where h5 href
ðTTref
cp dT ; ΔH5βL
T (K) � temperature of melting; Ts (K) � solidus temperature; Tl (K) � liquidus
temperature; H (J) � enthalpy of material; h (J) � enthalpy of heating up; href (J) �reference enthalpy; Tref (K) � reference temperature; cp (J kg2 1 K2 1) �specific heat capacity; ΔH (J) � addition of enthalpy; L (J) � latent heat of
melting.
In equation (1), it expresses the liquid phase portion of material melting. The
enthalpy increment can be determined from the latent heat of the material and is in
the range of h0; Li for the solidus and liquidus. For the enthalpy of the material,
expression (4) holds.
358 Dimensionless Physical Quantities in Science and Engineering
8.1.14 Power of Smeltery N
N5P
cηTdð1Þ;
N5Pz
cηTdð2Þ
P (W) � furnace power per electrode; c (J kg2 1 K2 1) � specific heat capacity;
η (Pa s) � dynamic viscosity; T (K) � temperature; d (m) � electrode diameter;
Pz (W) � power dissipation per electrode.
It expresses the electrical-to-thermal-power ratio. In the dimensionless state, it
characterizes the electrical power (1) or the power dissipation (2) of an electric
induction furnace.
Info: [A23].
8.1.15 Pressure in Smeltery N
N5p
g R β ΔT d
p (Pa) � pressure in a molten material in smelting furnace; g (m s22) � gravita-
tional acceleration; R (kg m22) � density; β (K21) � volume thermal expansion
coefficient; ΔT (K) � temperature difference; d (m) � electrode diameter.
It expresses the relationship between the pressure in a molten material and the
pressure caused by the temperature non-uniformity in the molten material. It char-
acterizes the pressure relations in a melting pan of an electric smelting furnace
with free convection of the molten material.
Info: [A23].
8.1.16 Rockwell Scale HR
HR A fHRA; HRB; HRC;?g
It expresses the hardness of a material based on the depth of a body impression
in a material sample compared to that in a reference material. The HR hardness is a
dimensionless quantity which is characterized by its simplicity and wide applica-
tion in mechanical engineering and � most especially � in metallurgy. Several
hardness scales are used, mostly the B and C scales, in which the B scale applies to
soft material (aluminium, bronze and mild steel). In the B scale, hard steel balls
and 100 kg loads are used in the measurement. The measured value is denoted as
HRB. With the C scale, a diamond cone and 150 kg loads are used. For very hard
steel, HRC A h55; 62i. For the steel for spades, axes and chisels, for example,
359Technology and Mechanical Engineering
HRC A h40; 45i. The values below HRC5 20 are considered unreliable, as are
those over HRC5 100. Many other scales, including the HRA, have specific
applications.
Info: [C117].
8.1.17 Rosenthal Number Ro
Ro5Q
LλΔT
Q (W) � heat flux; L (m) � characteristic length, width of weld; λ (W m21 K21) �thermal conductivity; ΔT (K) � temperature difference.
Together with the Rykalin number Ry (p. 360), it is used for the 3D and 2D ther-
mal process analyses of electric arc welding. It expresses the influence of a charac-
teristic weld dimension in welding. It is analogous to the Pomerantsev heat number
Po (p. 181). Welding.
Info: [B37].
D. Rosenthal.
8.1.18 Rykalin Number Ry
Ry5Qw
a2QV
Q (W) � heat flux; w (m s21) � motion velocity; a (m2 s21) � thermal diffusivity;
QV (J m23) � heat per unit volume.
This number expresses the dynamic thermal process in electric arc welding. It is
used to analyse the heat conduction in melting down. It can be used for 3D and 2D
thermal processes.
Info: [B37].
N. N. Rykalin, Russian engineer.
8.1.19 Solidification Rate Constant Nη
Nη 5η
2ffiffiffia
p
η (m s21/2) � solidification rate; a (m2 s21) � thermal diffusivity.
It expresses the solidification rate of a casting in a mould. Together with the
temperature gradient rΘ (p. 361), it influences the shrink hole formation in a cast-
ing. Foundry work technology. Solidification of castings.
Info: [B81].
360 Dimensionless Physical Quantities in Science and Engineering
8.1.20 Solidification Shrinkage β, Nsh
β5vl 2 vs
vs
vl, vs (m3 kg21) � specific volume of liquidus and solidus.
It expresses the volume changes with different molten materials and casting
shrinkage during solidification. Foundry work technology. Solidification of
castings.
Info: [B81].
8.1.21 Specific Heat of Melting Nmelt
Nmelt 5ΔTc
lsl5
ðT 2 TminÞclsl
T (K) � temperature; c (J kg21 K21) � specific heat capacity; lsl (J kg21) �specific latent heat of melting.
It characterizes the specific heat which is necessary to melt the material in a
furnace.
Info: [A23].
8.1.22 Temperature Gradient rΘ
rΘ5
ffiffiffiffiffiaτ
p rTrefTl 2 Tini
a (m2 s21) � thermal diffusivity; τ (s) � time; rTref (K m21) � reference tempera-
ture gradient; Tl (K) � liquidus temperature; Tini (K) � initial mould temperature.
It expresses the temperature gradient in a solidifying casting during time
τ, related to the temperature difference between the molten material and a
mould. Together with the solidification rate constant Nη (p. 360), it is the main
quantity acting during shrink hole formation. Foundry work technology.
Solidification of castings.
Info: [B81].
8.1.23 Temperature Range of Solidification Θsl
Θsl 5Tl 2 Ts
Tl 2 Tini
Tl, Ts, Tini (K) � temperature of solidus and liquidus, initial mould temperature.
361Technology and Mechanical Engineering
It expresses the temperature range during casting solidification in a mould.
Foundry work technology. Solidification of castings.
Info: [B64].
8.1.24 Tikhonov Number (1.) Ti
Ti5w
ffiffiffiffiffiτnD
r
w (m s21) � reaction rate; τn (s) � surface saturation time; D (m2 s21) �diffusivity.
This number expresses the ratio of the velocity of transferring an element from
the surroundings on the surface of an object to that of the element propagation in a
body surface layer. It characterizes the process of diffusion saturation of a rigid
body surface by various atoms from the gas or vapour phase, for example, in
cementation, oxidation or chrome plating of steel or cast iron parts. It is the crite-
rion of the thermochemical treatment of materials.
Info: [A33].
Andrey Nikolayevich Tikhonov (30.10.1906�8.11.1993),
Russian mathematician and physicist.
He was engaged in research in topology, functional anal-
ysis and differential equations. He focused especially on
mathematical problems in geophysics, the physics of
plasma, gas dynamics and electrodynamics, and solved
many mathematical and physical problems in the area of
thermomechanics, for example, in heating, cooling and
solidification. He formulated and proved the Tikhonov the-
orem in the field of topology.
8.1.25 Tikhonov Number (2.) Ti
Ti5αchar
b
ffiffiffiτ
p ð1Þ;
Ti5αchar
b
ffiffiffiffiffiτ1
p αmed
αchar
ð2Þ;
where αmed 5
ðτ10
α dτ
α (W m22 K21) � heat transfer coefficient at the melt�mould interface; αchar
(W m22 K21) � characteristic heat transfer coefficient; αmed (W m22 K21) � mean
heat transfer coefficient from the initial time of mould heating-up to the time
τ1; b ðkg s232 K21Þ � mould capability of heat accumulation; τ (s) � heating time
of monitored mould section; τ1 (s) � heating time.
362 Dimensionless Physical Quantities in Science and Engineering
It characterizes the mould heating time by considering the heat transfer on the
boundary between molten material and a mould, and the thermal accumulating
capabilities of the mould, whereas the Chvorinov cast mould system number Ch
(p. 354) solves the heat transfer between a casting and a mould as a contact prob-
lem (a fourth-type boundary condition).
Info: [B33].
Andrey Nikolayevich Tikhonov (see above).
8.1.26 Weber Agglomerate Number Weagl
Weagl 5RDw2
Γð1Þ;
Weagl 5RDðw2w0Þ2
Γð2Þ
R (kg m23) � particle density; D (m) � particle diameter; w (m s21) � particle
velocity; w0 (m s21) � limiting velocity at which no agglomeration of particles
occurs; Γ (J m22) � surface energy.
It expresses the ratio of the dynamic agglomeration energy to the surface energy.
Expression (2) is a modification of expression (1), a modification taking into con-
sideration the limiting velocity at which no agglomeration occurs.
Ernst Heinrich Weber (p. 92).Wilhelm Eduard Weber
8.1.27 Weld Size Parameter PW
PW 5wA
La
w (m s21) � motion velocity; A (m2) � weld surface; L (m) � characteristic weld
length; a (m2 s21) � thermal diffusivity.
It is often called the dimensionless weld width. It characterizes the weld width in
considering 3D and 2D thermal processes.
Info: [B37].
8.1.28 Weld Width Nww
Nww 5wA
La
363Technology and Mechanical Engineering
w (m s21) � motion velocity; A (m2) � weld surface; L (m) � characteristic length,
weld width; a (m2 s21) � thermal diffusivity.
It expresses the weld width in 2D analysis of an electric arc welding process.
Together with the Rykalin number Ry (p. 360) and the Rosenthal number Ro
(p. 360), it is among the fundamental welding criteria. Welding technology.
Info: [B37].
8.2 Mechanical Engineering
In mechanical engineering, many dimensionless quantities are applied in all aspects
of the design and construction of machines or parts thereof. This includes, for
example, the design and construction of water pumps, compressors, water and
steam turbines, fans and mixers [A29].
These quantities relate to rotating parts dynamics in moving blades, airscrews,
propellers and worm conveyers. Other dimensionless quantities relate to the design
of bearings, their loading and lubrication, dimensionless cryogenic cooling of
machines and devices, the observation of water drops in steam, including the flow
through part of the last stages of steam turbines, and the dimensioning of combus-
tion chambers, rocket engines and other machines and devices. Of the applied
dimensionless quantities, the Harrison and Addison numbers are used, for example,
to design a water turbine or a rotary pump. Similar to the Harrison and Ocvirk
numbers for bearings, the Leroux and Thoma numbers can be used for the cavita-
tion phenomena in water machines, and the Tomson number can be used for fuel
consumption depending on environmental resistance in aircraft and water
machines. Many of the numbers are modifications of applied dimensionless
quantities.
8.2.1 Addison Shape Number Ad
Ad5 103Kn; where Kn is specific velocity
Kn 5nP
12
R12ðghÞ54
for water turbines;
Kn 5nQ
12
ðghÞ34for rotary pumps
n (2) � revolutions per time interval; P (W) � power; R (kg m23) � density;
g (m s22) � gravitational acceleration; h (m) � hydrostatic head of machine;
Q (m3 s21) � volume flow.
364 Dimensionless Physical Quantities in Science and Engineering
This number is a multiple of the specific velocity. It is used with water
machines. It is an empirically obtained dimensionless number.
Info: [A29].
Herbert Addison (born 1889), English engineer.
8.2.2 Advance Ratio J
J5w
nD
w (m s21) � motion velocity; n (s21) � revolutions per second; D (m) � outer
diameter of screw or propeller, impeller.
It characterizes the relation between the velocity and the geometric parameters
of an airscrew or a ship propeller. With the ultrafiltration of mixing chambers,
it serves to predict the permeability and clogging degree thereof. It enables
dimensioning of a drive system.
Info: [A29].
8.2.3 Bearing Damping Coefficient Ndam, D0
Ndam 5DωRFηðTÞ
D (m kg2 s22) � general damping coefficient; ω (s21) � angular frequency; R (m) �radius of bearing journal; F (N) � bearing loading; η (Pa s) � dynamic viscosity
T (K) � temperature.
This coefficient expresses the plain bearing damping capabilities. Mechanical
engineering. Tribology. Bearings.
8.2.4 Bearing Modulus Nmod
Nmod 5ηnp
η (Pa s) � dynamic viscosity; n (s21) � revolutions per second; p (Pa) � bearing
pressure.
It is an important bearing characteristic, according to which three bands of the
bearing friction coefficients can be distinguished (thin film friction, transient
boundary and hydrodynamic friction). Mechanical engineering. Tribology. Plain
bearings.
Info: [C77].
365Technology and Mechanical Engineering
8.2.5 Bearing Number Nbea, S0
Nbea 5F
ηðTÞRωL
F (N) � loading force; η (Pa s) � dynamic viscosity; T (K) � temperature; R (m) �radius of bearing journal; ω (s21) � angular frequency; L (m) � characteristic length
of bearing.
It expresses the plain bearing load size. It is a modification of the Sommerfeld
number Sm (p. 385).
8.2.6 Bearing Stiffness Nst
Nst 5kst
EðTÞtp
kst (N m21) � stiffness; E(T ) (Pa) � modulus of elasticity; tp (m) � thickness of
lubricating film.
It expresses the stiffness of various kinds of fluid and gas plain bearings.
Mechanical engineering. Plain bearings.
8.2.7 Betz Number Be
Be5ðωrÞ2Lw2r
ω (s21) � angular frequency; r (m) � radius; L (m) � characteristic length, chord
of profile; ω (m s21) � flow velocity.
It expresses the flow in rotating channels of flow machines.
Info: [A24].
8.2.8 Cavitation Number Ncav, σ
Ncav 5p2 pv
Rw2ð1Þ;
Ncav � Th5ha 2 hs 2 hv
hð2Þ
366 Dimensionless Physical Quantities in Science and Engineering
p (Pa) � total static pressure; pv (Pa) � partial vapour pressure; R (kg m23) � den-
sity; w (m s21) � mean flow velocity; h, ha, hs, hv (m) � total, atmospheric, suck-
ing and water vapour pressure heads; Th (2) � Thoma number (1.) (p. 90).
It characterizes the formation of cavitation in a flowing fluid. Above all, it
depends on the flow rate, the kind and purity of the fluid, and the inlet flow turbu-
lence. In equation (1), it expresses the ratio of the static pressure drop, between the
fluid and vapour, to the fluid dynamic pressure. It characterizes the thermohydrody-
namic process of the cavitation origin in flow through parts of water machines. It is
a modification of the Euler number (1.) Eu (p. 61). In equation (2), it is called the
Thoma number (1.) Th (p. 90), and it expresses the cavity relations in water pumps.
See the Leroux number Lx (p. 374). Hydrodynamics. Hydraulics. Water machines.
Info: [A7],[A26].
8.2.9 Critical Number of Brake Disc Ndisc, crit
Ndisc; crit 5η20wsu
R2σ2max
ð1Þ;
Pe5uΔx
a. 2 ð2Þ
η0 (Pa s) � dynamic viscosity of oil at the temperature at input to contact area;
ws (m s21) � shear speed; u (m s21) � surface speed; R (m) � disc radius; σmax (Pa) �maximum Hertz stress under different loading, not leading to seizing; Δx (m) � grid
space; a (m2 s21) � thermal diffusivity.
The number size expresses the brake disc critical temperature and the possibility
of it subsequently seizing up. Usually, a high disc velocity and arising superficial
thermal instabilities cause brake disc oscillations. The Peclet heat number Pe
(p. 180), expressing heat transfer by convection and conduction, is the characteris-
tic dimensionless number for formation of motion thermal field. In the case of disc
brakes, the value of this number is Pe� 10.5 With the numerical solution using the
finite elements method (FEM) application, expression (2) is valid for the selection
of the grid space.
Info: [B16].
8.2.10 Cryogenic Cooling of Superconductor Nsupra
Nsupra 5λqV1OαqV1A
5Bi21PoisoPo21supra
λ (W m21 K21) � thermal conductivity of isolation; qV1, qV2 (W m23) � volume
density of heat flux in isolation and superconductor; O (m) � helium cooled
367Technology and Mechanical Engineering
perimeter; α (W m22 K21) � effective heat transfer coefficient; A (m2) � cross sec-
tion area of supraconductor; Bi (2) � Biot number (p. 173); Poiso (2) �Pomerantsev number for isolation; Posupra (2) � Pomerantsev number for super-
conductor, see Pomerantsev heat number Po (p. 181).
It characterizes the affect of insulation on superconductor cooling. Cryogenic
cooling.
Info: [B118].
8.2.11 Damping Coefficient Ndb
Ndb 5hωkdpabL
h (m) � bearing clearance; ω (s21) � angular frequency of bearing journal;
kd (kg m21 s21) � damping coefficient of bearing; pa (Pa) � ambient pressure;
b (m) � bearing width; L (m) � bearing length.
It characterizes the plain bearing damping. It depends on the geometric proper-
ties, revolving speed and damping properties of the bearing. Mechanical engineer-
ing. Aerostatic bearings.
8.2.12 Delivery Number ND
ND 5Q
Au
Q (m3 s21) � volume flow; A (m2) � impeller surface A5 14πd2
� �; u (m s21) �
peripheral speed.
It characterizes the volume throughflow in rotary machines. It is used in
throughflow blade machines.
Info: [A35].
8.2.13 Diameter Group ND
ND 51
4πD2
� �12
ð2ghÞ14w12
D (m) � diameter of screw propeller, impeller; g (m s22) � gravitational accelera-
tion; h (m) � hydrostatic head; w (m s21) � flow velocity.
It serves to dimension water and turbojet machines.
368 Dimensionless Physical Quantities in Science and Engineering
8.2.14 Discharge Number NΦ
See the flow coefficient φ (p. 370).
8.2.15 Droplets Formation in Jet N
N5ηlrmin
RlL2ffiffiffiffiffiffiffirT1
p ð1Þ;
N5λrmin
clRlL2ffiffiffiffiffiffiffirT1
p ð2Þ;
N5σRlLη2l
ð3Þ;
N5L
rmin
ð4Þ;
N5 5cl
rð5Þ;
N5rT1Rlp1
ð6Þ;
N5LRl
ffiffiffiffiffiffiffirT1
p
ηlð7Þ
ηl (Pa s) � dynamic viscosity; rmin (m) � radius of minimal jet cross section;
Rl (kg m23) � drop density; L (m) � characteristic length, initial drop diameter;
r (J kg21 K21) � specific gas constant; T1 (K) � temperature in input cross section; p1(Pa) � pressure in input cross section; λ (W m21 K21) � thermal conductivity; cl(J kg21 K21) � drop specific heat capacity; σ (N m21) � surface tension.
The set of criteria (1)�(7) describes gas flow influence on the laws of droplets
crumbling in nozzles without considering evaporation and condensation. The frac-
tional composition of droplets is presumed according to the law of normal logarith-
mic distribution. Temperature dependence of physical properties is neglected.
Info: [B119]
8.2.16 Eccentricity of the Bearing ε
ε5e
c
e (m) � eccentricity (distance between centre of journal and bearing); c (m) �interspace between journal and bearing.
369Technology and Mechanical Engineering
It expresses the bearing journal deviation which occurs in the hydrodynamic
mode. Sliding bearings, lubrication.
Info: [B31].
8.2.17 Ekman Number (2.) Ek
Ek5w
ωL2
w (m s21) � motion velocity; ω (s21) � angular frequency; L (m) � characteristic
length.
It characterizes, for example, the inertia to centrifugal forces ratio during
rotation of a vessel with the fluid in motion. It appears in cases of heat and mass
transfer solutions in rotary systems.
Info: [A5].
8.2.18 Elastic Component of Indentation Energy ηIT
ηIT 5Welast
Wcelk
100; where Wcelk 5Welast 1Wplast
Welast, Wplast, Wcelk (J) � elastic, plastic and total indentation energy, respectively.
It characterizes the hardness measurement. It expresses the portion of the elastic
energy in measurement of macro-, micro-, and nanohardness by means of diverse
kinds of indenters. The elastic energy component is given with the test conditions,
of which the first is the loading force in newtons (N) and the second is the loading
time in seconds (s), for example, ηIT (0, 5/10)5 36, 5%.
Info: [B111].
8.2.19 Fineness Coefficient, Waterplane Coefficient Ψ
Ψ5 LW13
L (m) � characteristic length; W (m3) � volume draft.
It is applied in ship modelling and design.
8.2.20 Flow Coefficient φ
φ5Q
nD3ð1Þ;
370 Dimensionless Physical Quantities in Science and Engineering
φ5Q
12πD2u2
ð2Þ;
φ5Q
14πð12D2
r ÞD2u2ð3Þ;
φ5v
u2ð4Þ
Q (m3 s21) � volume flow; n (s21) � rotational frequency; D (m) � diameter of
flow part; u2 (m s21) � peripheral flow velocity; Dr (2) � ratio of inner to outer
diameter; v (m s21) � mean flow velocity.
It expresses the throughflow through turbojet rotary machines. In equation (1), it
is often called the outlet number. Together with modifications (2) through (4), it is
used in the hydrodynamics and aerodynamics of rotary machines, fans and blowers.
Info: [B20].
8.2.21 Froude Number (3.) Rotation Fr3
Fr3 5Dn2
g
D (m) � diameter of impeller; n (s21) � revolutions; g (m s22) � gravitational
acceleration.
It expresses the influence of rotation on the flow through a pump running wheel.
Hydrodynamics. Water machines. Mixing.
Info: [A29].
William Froude (p. 63)
8.2.22 Geometric Parameter of Water Jet A
A5Rzrd
Nr21
Rz (m) � curvature radius of liquid flow in sprayer chamber; rd (m) � radius of
jet; N (2) � number of canals; r1 (m) � radius of input canals.
It characterizes the geometric parameter of a fluid sprayer. It is applied in the
heat transfer with condensation of the sprayed fluid.
Info: [B39].
8.2.23 Gravitational to Centrifugal Acceleration Ratio Ngc
Ngc 5q
ω2R
371Technology and Mechanical Engineering
g (m s22) � gravitational acceleration; ω (s21) � angular frequency; R (m) �radius.
It characterizes the dynamic force relations, for example, in rotary systems.
Info: [A5].
8.2.24 Head Coefficient CH, Ψ
CH 5H
n2d2
H (J kg21) � specific pressure energy; n (s21) � rotational frequency; d (m) �impeller diameter of pump, ventilator, etc.
It characterizes the flow in flow machines. It serves to dimension rotary pumps
and ventilators.
Info: [A4],[A29].
8.2.25 Head Coefficient, Loading Coefficient, Energy TransferCoefficient Ψ
Ψ5ΔP
Rn2D2ð1Þ;
Ψ5ΔP
Rω3D5ð2Þ;
Ψ5gh
n2D2ð3Þ;
Ψ5gh
u2ð4Þ
ΔP (Pa) � pressure difference; R (kg m23) � density; n (s21) � rotational fre-
quency; D (m) � diameter of flow part; ω (s21) � angular frequency; g (m s22) �gravitational acceleration; h (m) � hydrostatic head; u (m s21) � peripheral speed.
It is used in rotary machine dynamics. Expressions (1) and (2) are used to design
fans and rotary blowers. Expressions (3) and (4) are applied in the designing of
water pumps and turbines.
Info: [A29].
8.2.26 Homochronicity Number Ho
Ho5 nτ
n (s21) � revolutions; τ (s) � mixing time.
372 Dimensionless Physical Quantities in Science and Engineering
It is used for time comparison of mixing, whisking and other processes in time.
Info: [A29].
8.2.27 Homochronous Number Ho
Ho5wτL
w (m s21) � motion velocity; τ (s) � transfer time to the distance L; L (m) �distance.
It serves to choose the timescale for similar processes.
Info: [A29].
8.2.28 Kennedy Number Ω
Ω5τdapfRb
Tda (s) � deaerating time; pf (m3 kg21 s21) � constant of material permeability
(specific volume per second); Rb (kg m23) � material density.
It expresses the ratio of the deaerating time to the delivery time of the unit mate-
rial amount in pneumatic transport. Mechanical engineering. Pneumatic transport.
8.2.29 Lagrange Number (2.) Lg2
Lg2 5P
ηL3n25RerotNp; where Np 5
P
L5Rn3
P (W) � power input of impeller; η (Pa s) � dynamic viscosity; L (m) � character-
istic length; n (Hz) � revolutions per second; R (kg m23) � density; Rerot (2) �Reynolds number (p. 81); Np (2) � power input criterion.
It characterizes the energy loss from mixing viscous fluids in vessels, depending
on the physical properties of the fluid and the dimensions and speed of the mixer.
Info: [A14],[A24].
Joseph-Louis Lagrange (p. 70).
8.2.30 Lautrec Number Lau
Lau5L
δsð1Þ;
373Technology and Mechanical Engineering
Lau5R
δkð2Þ
L (m) � characteristic length of the wall; R (m) � characteristic length (gap thick-
ness) between two walls; δs (m) � penetration depth in solid wall; δk (m) � pene-
tration depth in liquid.
It characterizes the thermoacoustic phenomenon of the thermal and acoustic
energy transformations in a wall (1) or in a gap between two walls (2) in thermoa-
coustic devices such as thermal pumps and cooling devices. It appears together
with the Mach number M (p. 73) and the Prandtl number Pr (p. 197), especially in
observing the interaction between porous surroundings and a sound field in pipes
and pipelines.
Info: [C128].
8.2.31 Leroux Number Lx, σc
Lx5p2 pv12Rlw2
p (Pa) � local static pressure; pv (Pa) � vapour pressure; Rl (kg m23) � liquid den-
sity; w (m s21) � velocity.
This number expresses the ratio of the difference between the local static pres-
sure head and the vapour head to the velocity head. In water machines it charac-
terizes the cavitation phenomenon with water flow through. See the cavitation
number Ncav (p. 366).
Info: [A35].
8.2.32 Load Parameter of Journal Bearing NB
NB 5ηur2
FL2
η (Pa s) � dynamic viscosity; u (m s21) � peripheral speed; r (m) � radius; F (N) �force; L (m) � characteristic length.
It is another variant of the Sommerfeld number Sm (p. 385). Mechanical engi-
neering. Tribology.
8.2.33 Loading Number of Bearing NM, M
NM 5 qα2
ηusR2
� �13
374 Dimensionless Physical Quantities in Science and Engineering
q (N m21) � length loading; α (m2 N21) � coefficient of piezoviscosity; η (Pa s) �input dynamic viscosity; us (m s21) � sum of tangential surface velocities: R (m) �effective radius curvature.
It expresses the ratio of the plain bearing load to surface friction forces.
Elastohydrodynamic lubrication. Bearings.
Info: [B77].
8.2.34 Loading Number of Sliding bearing NB
NB 5ηuR2
Fh25 2Gu
η (Pa s) � dynamic viscosity; u (m s21) � peripheral speed; R (m) � radius of
bearing journal; F (N m21) � length force loading of bearing; h (m) � thickness of
round gap; Gu (2) � Gumbel number (1.) (p. 164).
It expresses the ratio of the dynamic friction force to the loading force of a bear-
ing. It is another variant of the Sommerfeld number Sm (p. 385). Tribology.
Lubrication. Bearings.
8.2.35 Lock Number (1.) Lk
Lk5RR4la
I
R (kg m23) � liquid density; R (m) � rotor radius; l (m) � blade chord;
a (m2 kg21) � steepness of lifetime curve of rotor; I (m4) � moment of inertia of
blade at hangings.
It expresses the dynamics of rotor blades. It is an empirical dimensionless
number.
Info: [A35].
Christopher Noel Hunter Lock (1894�1949), English mathematician.
8.2.36 Lock Number (2.) Lk
Lk5dCL
dαRlR4
I
dCL
dα(2) � angular coefficient of rotor lifting curve; CL, (2) � lifting coefficient
of rotor; α (rad) � angle of pitch of copter rotor; R (kg m23) � liquid density;
l (m) � blade chord; R (m) � rotor radius; I (m2 kg) � inertia moment of rotary
rotor blades.
375Technology and Mechanical Engineering
It expresses the buoyancy to inertia forces ratio in helicopters.
Info: [29].
Christopher Noel Hunter Lock (see above).
8.2.37 Mach Blowing Number Mb
Mb 5wb
a
Wb (m s21) � inject velocity at surface of porous boundary; a (m s21) � mean stag-
nation sound speed.
It is the basic criterion of combustion chambers. The effect of fuel viscous dissi-
pation is important with a low Mb value occurring with small fuel injection, high
frequency and kinematic viscosity, which is possible in hot gases. With a very
small Mach number (p. 73) corresponding to slight fuel injection, the solution
approaches the injectionless Stokes solution. Jet engines. Combustion chambers.
Info: [B119].
Ernst Mach (p. 73).
8.2.38 Mach Propeller Blade Tip Number Ma
Ma5πnDa
n (s21) � revolutions per second; D (m) � diameter; a (m s21) � sound speed.
It characterizes the velocity of a propeller blade tip or of a pump propeller run-
ning wheel, as a couple of examples.
Ernst Mach (see above).
8.2.39 Mechanical Load Group NM
NM 5ασHZ 5αE
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiF
2πLER
r
α (m2 N21) � pressure coefficient of viscosity; σHZ (Pa) � maximal dry (Hertzian)
stress; E (Pa) � reduced modulus of elasticity; F (N) � force loading of bearing;
L (m) � characteristic length (width of bearing); R (m) � reduced radius of
curvature.
It expresses the mechanical contact load influence on the oil film thickness and
pressure in a bearing. Electrohydrodynamic lubrication. Tribology. Lubrication.
Plain bearings.
Info: [B117].
376 Dimensionless Physical Quantities in Science and Engineering
8.2.40 Mishken Number Mi
Mi5DefL
q
Def (m2 s21) � effective diffusivity; L (m) � characteristic length (endless screw
diameter); q (m3 s21) � volume flow.
It describes the mixing process in worm conveyors from the kinematic point of
view.
8.2.41 Modified Power Number NPM
NPM 5Np
D
L
� �Δs0:5
ðnbnsÞ0:67
D (m) � effective diameter of blade wheel; L (m) � effective wheel length;
Δs (2) � wall closeness factor; nb, ns (2) � blade number of impeller or stator;
Np (2) power coefficient, power number (p. 379).
It is used to recalculate the power output of rotary mixers and pumps after
changing the rotating wheel parameters.
8.2.42 Muramtsev Number Mo
Mo5Raλbccp
ηd
R (kg m23) � fluid density; λ (W m21 K21) � thermal conductivity; cp(J kg21 K21) � specific heat capacity; η (Pa s) � dynamic viscosity; a, b, c, d (2) �dimensionless constants.
It is the criterion used to choose the heat transferring material with the heat
transfer properties in a limited range of working temperatures, specifically from the
point of view of heat transfer intensity under the equal velocity condition. It can be
used to design energy generating plants. It is an empirical dimensionless number.
Info: [A33].
8.2.43 Nusselt Modified Number Numod
Numod 5αλcrit
λv
; where λcrit 5 2πffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
σgðRl 2 RvÞ
s
α (W m22 K21) � heat transfer coefficient; λcrit (m) � critical wavelength;
λv (W m21 K21) � vapour thermal conductivity; σ (N m21) � surface tension;
377Technology and Mechanical Engineering
g (m s22) � gravitational acceleration; Rl, Rv (kg m23) � density of liquid and
vapour, respectively.
It expresses the ratio of the load due to convection heat transition on a corre-
sponding transfer surface to the nucleon film boiling, to the thermal load by con-
duction in vapour. Cryogenic cooling of machines and plants.
Ernst Kraft Wilhelm Nusselt (p. 196).
8.2.44 Ocvirk Number Oc
Oc5q
ηwΔs
r
2r
L
� �2
q (kg s22) � length loading; η (Pa s) � dynamic viscosity; w (m s21) � surface
speed of bearing journal; Δs (m) � bearing clearance; r (m) � bearing radius;
L (m) � bearing length.
This number expresses the ratio of the bearing load to the friction force.
Tribology. See also Hersey number He (p. 165) and Sommerfeld number Sm
(p. 385).
Info: [A29].
Frederick A. Ocvirk (1913�1967), American engineer.
8.2.45 Orifice Design Parameter Nλ, λ
Nλ 5
ffiffiffi23
pcdD
20ηLffiffiffiffiffiffi
Rpp
e3D
cd (2) � orifice outflow coefficient; D0 (m) � nozzle diameter; η (Pa s) �dynamic viscosity; L (m) � characteristic bearing length; R (kg m23) � lubricant
density p (Pa) � pressure; e (m) � eccentricity clearing; D (m) � bearing journal
diameter.
Mechanical engineering. Tribology. Bearings.
8.2.46 Peclet Brake Number Pebrake
Pebrake 5uL
a. 2; where u5ωr
u (m s21) � sliding speed; L (m) � characteristic length; a (m2 s21) � thermal dif-
fusivity; ω (s21) � angular speed; r (m) � radius of disc.
It characterizes the convectively diffusive heat transfer in disc brakes from the
point of view of the finite elements method solution. Incorrect size of the brake
378 Dimensionless Physical Quantities in Science and Engineering
disc causes often inadmissible oscillations and instability. This is one o- the main
problems of brakes, in which the typical size of the number Pebrake is approxi-
mately 105.
Info: [B123].
Jean Claude Eugene Peclet (p. 180).
8.2.47 Penetration Number Npen, Sp
Npen 5w3z
ω2νL5M3
bN22ω Re21
wz (m s21) � inject velocity at surface of porous boundary; ω (s21) � angular fre-
quency; v (m2 s21) � kinematic viscosity; L (m) � characteristic length, radius
Mb (2) � Mach blowing number (p. 376); Nω (2) � wave number, angular speed
(p. 389); Re (2) � Reynolds number based on sound speed (p. 383).
This number governs the depth and structure of the unsteady boundary layer,
e.g. in jet engines, particularly the shape and dimension of the outer casing Jet
engines. Combustion chambers.
Info: [B71].
8.2.48 Petrov Number Pt
Pt5Δp1 1Δp2
Δp3
Δp1 (Pa) � pressure difference on water part of boiler; Δp2 (Pa) � pressure differ-
ence on economizer; Δp3 (Pa) � pressure difference on vapour�liquid part and
superheating part.
It characterizes the stability of the flow parameters of a heat transferring
medium which moves in a steam boiler heating system. For steam boilers, Pt . 1
is valid. Generally, it is Pt A h0.1; 10i.Info: [A33].
P. A. Petrov.
8.2.49 Power Coefficient, Power Number NP
NP 5M
Rn2D5ð1Þ;
NP 5Mω
Rn3D5ð2Þ;
379Technology and Mechanical Engineering
NP 5P
Rn3D5ð3Þ;
where P5 2πnM
M (m2 kg s21) � force moment; R (kg m23) � fluid density; n (s21) � rotational
frequency; D (m) � rotor diameter; ω (rad s21) � angular speed; P (W) � shaft
power.
This coefficient relates to the dynamic criteria for rotary machines. For example,
it represents the hydrodynamic resistance on the mixer blade against the inertia
force. It expresses the energy loss in mixer vessels. In the shapes (1) and (2), these
numbers express the ratio of the shaft moment to the inertia force. In the shape (3),
it is the ratio of the thrust force (resistance) to the inertia force. It characterizes the
power output of some rotary machines, such as pumps, turbines, fans, mixers and
other devices. It is among the widely used dimensionless numbers in mechanical
engineering. Mechanical engineering.
Info: [A29],[A20],[A35],[B20].
8.2.50 Power Pressure Number NP
NP 5Δp
RD2Sn
2ð1Þ;
NP 5P
12RAw3
ð2Þ;
Δp (Pa) � pressure difference; R (kg m23) � fluid density; Ds (m) � impeller
diameter; n (s21) � rotation of mixer, etc.; P (W) � impeller power; A (m2) �effective surface; w (m s21) � wind speed.
This number expresses the ratio of the pressure gradient to the dynamic pres-
sure. It serves to design and dimension rotary mixers (1) and wind energy
plants (2).
Info: [B19].
8.2.51 Pressure Number (2.) NP2
NP2 5 e1
2u2
� �21
e (m2 s22) � specific energy; u (m s21) � peripheral speed.
It describes the flow through turbines, pumps and other flow machines.
Info: [17].
380 Dimensionless Physical Quantities in Science and Engineering
8.2.52 Propeller Torque Coefficient Mt
Mt 5MF
Rn2D5
MF (N m) � torque of propeller or impeller; R (kg m23) � density; n (s21) � revo-
lutions per second; D (m) � diameter of propeller or impeller.
It expresses the torque caused by the force acting on a rotating propeller or a
rotating wheel in rotary machines. It is not derived strictly from similarity theory
but from empirical experience.
Info: [A29].
8.2.53 Pulsation Number Pu
Pu5fdRqm
f (s21) � frequency of pulsation; d (m) � equivalent pipe or channel diameter;
R (kg m23) � fluid density; qm (kg m22 s21) � mass flux density.
This number characterizes the pulsation transport of fluids.
8.2.54 Rayleigh Centrifugal Number Rac
Rac 5ω2TβΔCR3
wD5Re2ScRo
ω (s21) � angular frequency; T (K) � temperature; β (K21) � volume thermal coef-
ficient of expansion; ΔC (2) � difference of ion concentration; R (m) � character-
istic length, vessel diameter with fluid; w (m s21) � travel speed; D (m2 s21) �diffusivity; Re (2) � Reynolds number (p. 81); Sc (2) � Schmidt number (p. 263);
Ro (2) � Rossby number (p. 406).
It expresses the influence of temperature, mass diffusion and motion in rotating
systems, such as volume conveyors and containers, as some examples. Mechanical
engineering.
Lord Rayleigh (p. 187).
8.2.55 Rayleigh Modified Number Ramod
Ramod 5gλ3
cr
νvavRl 2 RvRv
ð1Þ;
381Technology and Mechanical Engineering
Ramod 5PrGrmod ð2Þ;
where λcrit 5 2πffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
σgðRl 2 RvÞ
r
g (m s22) � gravitational acceleration; λcrit (m) � critical wavelength; νv (m2 s21) �
vapour kinematic viscosity; av (m2 s21) � vapour thermal diffusivity; Rl, Rv (kg m
23) �liquid and vapour density; σ (N m21) � surface tension; Pr (2) � Prandtl number
(p. 197); Grmod (2) � Grashof modified number (p. 186).
In expression (1), it is a modification of the Rayleigh number (2.) (heat instability)
(p. 187). It is used in thermal calculations of cryogenic cooling systems such as cryo-
genic turboalternators. Expression (2) is a general equation expressing the natural
convective heat transfer from a heated or cooled wall.
Info: [B60].
Lord Rayleigh (see above).
8.2.56 Reynolds Acoustic Number Rea
Rea 5ωL2
ν5NωRe
ω (s21) � angular frequency; L (m) � characteristic length, radius; ν (m2 s21) �kinematic viscosity; Nω (2) � wave number, angular speed (p. 389); Re (2) �Reynolds number based on sound speed (p. 383).
In comparison to the Reynolds number based on sound speed (p. 383), it
involves the influence of the angular velocity of combustion products. Rocket
machines. Combustion chambers.
Info: [B71].
Osborne Reynolds (p. 82).
8.2.57 Reynolds Mixing Number Remix
Remix 5D2nRmηm
D (m) � propeller diameter; n (s21) � propeller rotational frequency; Rm (kg m23) �mixture density; ηm (Pa s)� mixture dynamic viscosity.
It characterizes the ratio of dynamic to viscous forces which act on an airscrew
with mixing. It is used to design and dimension rotary mixers.
Info: [C3].
Osborne Reynolds (see above).
382 Dimensionless Physical Quantities in Science and Engineering
8.2.58 Reynolds Number Based on Sound Speed Re
Re5aL
ν
a (m s21) � mean stagnation sound speed; L (m) � characteristic length, radius;
ν (m2 s21) � kinematic viscosity.
It expresses the ratio of the mean dynamic force of acting combustion products
to the viscous force. Rocket engines. Combustion chambers.
Info: [B71].
Osborne Reynolds (see above).
8.2.59 Reynolds Oscillatory Number Re
Re52πDfA
ν
D (m) � column diameter of oscillatory reactor; f (s21) � oscillatory frequency;
A (m) � oscillatory amplitude; ν (m2 s21) � kinematic viscosity.
It characterizes the intensity of mixing and flow through a circulation reactor.
The fluid (e.g. denatured protein) is delivered into the mixing chamber where it is
vibrated with an oscillating piston. Mixing of fluids. Physical chemistry. Chemical
mechanical engineering.
Info: [C130].
Osborne Reynolds (see above).
8.2.60 Reynolds Rotation Number Rerot
Rerot 5L2nRη
ð1Þ;
Rerot 5ωr2
νð2Þ
L�D (m) � characteristic length (diameter); n (s21) � rotational frequency;
R (kg m23) � fluid density; η (Pa s) � dynamic viscosity; ω (s21) � angular fre-
quency; r (m) � characteristic radius; ν (m2 s21) � kinematic viscosity.
This number characterizes the rotating viscous fluid flow. It expresses the
dynamic similarity of turbojet rotary machines (e.g. fluid mixers).
Info: [A29],[B17].
Osborne Reynolds (see above).
383Technology and Mechanical Engineering
8.2.61 Richardson Gradient Number of Wind Turbine Rig
Rig 5g
Tmed
ΔT
Δz
Δw
Δz
� �22
g (m s22) � gravitational acceleration; Tmed (K) � mean air temperature; w (m s21) �wind speed; z (m) � height above ground.
This number expresses the ratio of the generated turbulence, with vertical tem-
perature (density) difference or buoyancy, to the wind velocity change according to
height. A negative value of Rig corresponds to an unstable condition of the flow in
a layer between the surface and the turbine rotor height. On the contrary, its posi-
tive value corresponds to a stable condition. The zero value of Rig represents a
neutral condition under which the buoyancy does not have any influence and the
turbulence is generated only by wind action. Mechanical engineering. Wind tur-
bines. Aerodynamics.
Lewis Fry Richardson (p. 403).
8.2.62 Size Number NSN
NSN 5DðghÞ14Q
12
D (m) � diameter of reactor or propeller; g (m s22) � gravitational acceleration;
H (m) � pressure head difference across turbine; Q (m3 s21) � volume flow.
It is also called the specific diameter. It serves to dimension water turbines and
pumps. It is an empirical dimensionless number.
Info: [A29].
8.2.63 Slosh Time Nτ
Nτ 5
ffiffiffiffiffiffiffiffiσRR3
rτ
σ (N m21) � surface tension; R (kg m23) � density; R (m) � tank radius; τ (s) �time.
It expresses the time of non-linear free oscillations of the interface of immiscible
fluids in containers under zero friction and zero gravity conditions. Tanking of fluid
in a low gravitation field. Astronautics. Space mechanical engineering.
Info: [B8].
384 Dimensionless Physical Quantities in Science and Engineering
8.2.64 Sommerfeld Number Sm
Sm5ηnp
D
h
� �2
ð1Þ;
Sm5ηnp
R
h
� �2
ð2Þ;
Sm5F
ηuh
R
� �2
ð3Þ;
Sm5F
2ηnh
R
� �2
ð4Þ
η (Pa s) � dynamic viscosity; n (s21) � rotational frequency; p (Pa) � mean bear-
ing pressure; D (m) � shaft diameter; h (m) � gap thickness; R (m) � shaft radius;
F (N m21) � length loading of bearing; u (m s21) � peripheral speed of shaft.
In equations (1) and (2), this number expresses the viscosity to loading forces
ratio. In equations (3) and (4), it represents the inverse value. It is analogous to the
Gumbel number (1.) Gu (p. 164), and in equation (4), it is identical to it. It charac-
terizes the relation of forces in a bearing with lubrication.
For small Sommerfeld numbers ðSm{1Þ; the lubrication pressure represents a
small part of the increase in the loading capability of a plain bearing. Two surface
sides are contacted by means of the roughness. For larger numbers Smc1; thelubrication pressure wholly supports the bearing load and the friction strain corre-
sponds to the viscous tension in the wholly developed lubrication mode. For
Sm�1, a mixed mode exists, with which the friction is influenced both by the vis-
cous resistance and by the roughness of contacts.
Info: [A29],[B20],[C77].
Arnold Johannes Wilhelm Sommerfeld (5.12.1868�26.4.1951), German physicist.
He was engaged in research in atomic physics, electro-
dynamics, mechanics and partial differential equations. He
modified the Niels Bohr atomic theory by considering
the elliptical path of electrons. Similarly, he also modified
the special relativity theory by formulating the Sommerfeld
model for energy in which the energetic levels are distrib-
uted into numerous parts. He was the first to come up with
an exact solution to the diffraction problem by making use
of the Maxwell equations.
385Technology and Mechanical Engineering
8.2.65 Speed of the Bearing Ns
Ns 56ηω
paR2e22
η (Pa s) � dynamic viscosity; ω (s21) � angular frequency; pa (Pa) � ambient pres-
sure; R (m) � radius of bearing journal; e (m) � bearing gap.
It characterizes the lubrication rate of a plain bearing. Tribology. Bearings.
8.2.66 Speed Specific Number N3
N3 5nQ
12
ðghSÞ34
5N12DN
234
P2
n (s21) � rotational frequency; Q (m3 s21) � volume flow; g (m s22) � gravita-
tional acceleration; hS (m) � hydrostatic head; ND (2) � delivery number
(p. 368); NP2 (2) � pressure number (2.) (p. 380).
It expresses the dimensionless velocity in rotary flow machines, such as pumps,
water turbines and compressors.
Info: [A29],[A35],[B17].
8.2.67 Squeeze Number NS
NS 512ηL2ωpah2
η (Pa s) � dynamic viscosity; L (m) � characteristic length (length of plate);
ω (s21) � oscillation frequency; pa (Pa) � ambient pressure; h (m) � gap height.
It characterizes the squeezing of a thin isothermic film between two flat plates
of which the bottom one is steady and the top one is oscillating. Among other
things, it is used to damp dynamic systems and microsystems.
Info: [B20].
8.2.68 Stability of Cryogenic Cooling Nstab
Nstab 5qmax
LffiffiffiffiffiRv
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσgðRl 2 RvÞ4
pqmax (W m22) � maximal surface heat flux; L (J kg21) � specific latent heat of
evaporation; Rv, Rl (kg m23) � vapour and liquid density, respectively; σ (N m21) �surface tension; g (m s22) � gravitational acceleration.
386 Dimensionless Physical Quantities in Science and Engineering
It expresses the ratio of the maximum planar thermal surface load, correspond-
ing to the transfer with nucleon film boiling, to the load caused by the surface
strain force and the force of gravity. It characterizes the stability of the cryogenic
cooling process in machines and plants.
8.2.69 Storage Energy Number Nen
Nen 5Ef
p
E (J) � accessible energy; P (W) � nominal actual output; f (s21) � synchroniza-
tion frequency.
It expresses the number of basic cycles which should be chosen for accessible
energy to reach the power corresponding to the nominal energy. When
P5 100 MW, f5 50 Hz, E5 20 MW, the number of cycles is Nen5 10.
Info: [C47].
8.2.70 Strouhal Boundary Layer Number St
St5ωLwb
5NωM21b
ω (s21) � angular frequency; L (m) � characteristic length, radius, ratio of volume
to half porous part; wb (m s21) � inject velocity at surface of porous boundary;
Nω (2) � wave number, angular speed (p. 389); Mb (2) � Mach blowing number
(p. 376).
It is a factor in the wavelength control of the unsteady swirl velocity of the
oscillatory flow in a cylindrical combustion space with side injection. It provides
precise observation of the amplitude and phase between non-stationary velocity
and pressure. Rocket engines.
Info: [B71].
Vincenc Strouhal (p. 87).
8.2.71 Structural Utilization Number N
N5gRLE
g (m s22) � gravitational acceleration; R (kg m23) � density; L (m) � characteris-
tic length; E (Pa) � modulus of elasticity.
It expresses the ratio of the weight of a structural part to its rigidity.
387Technology and Mechanical Engineering
8.2.72 Subcooling Number (1.) Nsubc
Nsubc 5cpðTs 2 TlÞ
l
cp (J kg21 K21) � specific heat capacity of liquid; TS (K) � temperature of satura-
tion; Tl (K) � temperature of liquid; l (J K21) � specific latent heat of evaporation.
It expresses the heat transfer in subcooled liquid cryogenic materials such as
helium, hydrogen and nitrogen. Cryogenic cooling of machines and plants.
8.2.73 Thoma Number (2.) Tm
Tm5pH 2 pV
p2 2 p1
pH (Pa) � local total hydrostatic pressure; pV (Pa) � vapour pressure under given
temperature; p2 (Pa) � absolute input pressure; p1 (Pa) � absolute output pressure.
It expresses the ratio of the pressure gradient in cavitation to the pressure
increase in a pump. It characterizes the cavitation in water pumps.
Info: [A29].
Dieter Thoma (1881�1942), German hydraulic engineer.
8.2.74 Thrust Coefficient CDP
CDP 5FT
Rn2D4
FT (N) � drag force of propeller or impeller; R (kg m23) � fluid density; n (s21) �rotational frequency; D (m) � propeller diameter.
It characterizes the ratio of thrust to dynamic forces which act on a propeller.
Info: [A29].
8.2.75 Tomson Number To
To5gQ
Rw3L
g (m s22) � gravitational acceleration; Q (kg s21) � fuel usage; R (kg m23) � envi-
ronment density; w (m s21) � motion velocity; L (m) � characteristic length.
It expresses the ratio of fuel time unit consumption to environmental resistance.
Flying or floating machines. Aeronautics. Ships.
Info: [B20].
388 Dimensionless Physical Quantities in Science and Engineering
8.2.76 Torque Coefficient N
N5Mk
Rv2D3ð1Þ;
N5Mk
Rn2D5ð2Þ
Mk (N m) � torsion moment; R (kg m23) � density; v (m s21) � peripheral speed;
D (m) � impeller diameter; n (s21) � revolutions per second.
It is used to design propellers and running wheels. It is an empirically deter-
mined dimensionless number.
Info: [A29],[A35].
8.2.77 Wave Number, Angular Speed Nω
Nω 5ωLam
ω (s21) � angular frequency; L (m) � characteristic length, radius; am (m s21) �mean stagnation sound speed.
It expresses the ratio of the dynamic force of the combustion products flow to
the dynamic force corresponding to the stagnation sound velocity. Rocket engines.
Combustion chambers.
Info: [B71].
8.2.78 Weaver Flame Speed Number Wea
Wea5w
wH
100
w (m s21) � laminar combustion rate of gas; wH (m s21) � laminar combustion
rate of hydrogen.
It expresses the ratio of the laminar combustion rate of the considered gas to that
of hydrogen gas (Wea5 100). It characterizes the tendency of the gas to react. A
lower value of this number represents a lower flame rate. The value of a high com-
bustion rate of the gas is Wea A h32; 45i; of a medium rate, it is Wea A h25; 32i;and of a low rate, it is Wea A h13; 25i.
Info: [C37].
8.2.79 Weber Number of Damp Steam Wev
Wev 5Rvðwv 2w2ÞL
σ
389Technology and Mechanical Engineering
Rv (kg m23) � vapour density; wv (m s21) � absolute vapour speed; w (m s21) �absolute speed of water drop; L (m) � characteristic length, maximal length of sta-
bilized water drop dimension; σ (N m21) � surface tension.
It characterizes the process of the erosive influence of water droplets in damp
steam on the throughflow parts, for example, of the last steam turbine stages.
Wev A h21; 24i.Info: [C37].
Ernst Heinrich Weber (p. 92).
Wilhelm Eduard Weber.
8.2.80 Wobbe Number, Wobbe Index Wob
Wob5λV1
ffiffiffiffiffiR2
pλV2
ffiffiffiffiffiR1
p ð1Þ;
Wob5λV
pð2Þ
λV1, λV2 (J m23) � volume density of energy in gas 1 and 2; R1, R2 (kg m23) den-
sity of gas 1 and 2; λV (J m23) � gas total volume energy density; p (Pa) � gas
pressure.
In equation (1), this number enables a comparison of the energies of changing
gases, provided the burner is the same (having an equal cross section and pressure
drop). In equation (2), it expresses the ratio of the energy density and pressure in
the same gas. Wob A h17.8; 35.8i is valid for a coke oven gas, Wob A h35.8; 71.5iholds for natural gas, and Wob A h71.5; 87.2i is for liquefied gases (LPG).
Info: [C63].
390 Dimensionless Physical Quantities in Science and Engineering
9 Geophysics and Ecology
To withdraw from Nature would mean to languish physically and a total divorce
would be death.Jan Neruda (1834�1891)
9.1 Geophysics, Meteorology and Astrophysics
In recent years, in geophysics, meteorology and astrophysics, the number of dimen-
sionless quantities give evidence of the increasing importance of these scientific
disciplines. Geophysics is about geodynamic, geothermic, geomagnetic, gravimetric
and seismologic processes, to which natural catastrophes are closely related, for
example, eruptions of volcanoes, tsunamis and earthquakes. Meteorology, as atmo-
spheric physics, deals especially with climatological conditions and accompanying
phenomena. In geophysics, the Ekman, Hatta, Rossby, Argand and Love numbers,
as well as the tsunami velocities and the Shida number, are very important. In
meteorology, the Richardson number and the Burger, Davies and Rossby atmo-
spheric numbers are among the best known. In astrophysics, the redshift number
and the Eddington, Golitsyn, Hubble, and Wesson numbers are among the most
significant.
9.1.1 Albedo Number A
A5 124σ0T
4
εC
σ0 (W m22 K24) � Stefan-Boltzmann constant; T (K) � absolute temperature;
ε (2) � emissivity (p. 206) of the atmosphere, around the height of about 5.5 km
where long-wave radiation is emitted; C (W m22) � solar constant.
It expresses the reflectance of surfaces and bodies. It characterizes the climatic
conditions on the Earth which are produced by the solar radiation reflected from
the Earth. The global value of the Albedo number can be determined by measuring
the amount of solar radiation, reflected from the Earth, and back to the Earth during
rotation from the dark surface of the Moon. Meteorology. Climatology. Astronomy.
Info: [B73],[C42].
Dimensionless Physical Quantities in Science and Engineering. DOI: 10.1016/B978-0-12-416013-2.00009-9
© 2012 Elsevier Inc. All rights reserved.
9.1.2 Argand Number Arg
Arg5gLRcðRm 2 RcÞ
BRm
ffiffiffiffiffiL
wc
n
r
g (m s22) � gravitational acceleration; L (m) � characteristic length, thickness of
lithosphere; Rc (kg m23) � crustal density; Rm (kg m23) � mantle density;
B (N s m22) � constant characterizing vertical average of compositional and ther-
mal influences on the lithosphere rheology; wc (m s21) � imposed collision veloc-
ity; n (2) � stress�strain exponent.
It expresses the deformation in the terrestrial crust as the ratio of the buoyancy
force to the boundary force acting on the terrestrial plate. Geophysics.
Info: [B73],[C42].
9.1.3 Argand Number (1.) Arg
Arg5ΔRghη_ε
ΔR (kg m23) � density difference of the oceanic and continental lithospheres;
g (m s22) � gravitational acceleration; h (m) � thickness; η (Pa s) � lithosphere
dynamic viscosity; _ε (s21) � strain rate.
This number expresses the ratio of the tension, induced by the density difference
on the bottom part of the lithosphere, to the ductile deformation of the lithosphere.
With a certain simplification (the Newtonian fluid is considered instead of the
material creep law), it describes the lithosphere rheology. The criterion Arg charac-
terizes the interface area between the ocean and the mainland. For Arg. 1, the lith-
osphere deformation force cannot lead to lower energetic potential so that the
ocean lithosphere will act on the mainland lithosphere and the interface angle
decreases. Geophysics. Terrestrial crust geology. Rheology.
Info: [B73].
9.1.4 Argand Number (2.) Arg
Arg5Rcghc
C1μcRchc
Rc (kg m23) � density of continental lithosphere; g (m s22) � gravitational acceler-
ation; hc (m) � thickness of the continental brittle lithosphere, which is unsup-
ported by a juxtaposed oceanic lithosphere; C (kg m21 s22) � cohesion;
μc (kg m21 s22) � constant dependent on friction coefficient (usually 0.5).
It expresses the ratio of the tension, on the bottom part of the non-hardened
mainland brittle lithosphere, generated with various increases of the mainland and
392 Dimensionless Physical Quantities in Science and Engineering
oceanic lithospheres, to the brittle tension of the lithosphere. The Arg reproduces
the development of the destruction in the upper lithosphere along the interface of
the mainland plates. Geophysics. Terrestrial crust geology. Rheology.
Info: [B73].
9.1.5 Argand Number (3.) Arg
Arg5τ
μσn
τ (Pa) � shear stress; μ (2) � coefficient of friction between the lithospheres;
σn (Pa) � normal stress.
It expresses the shear strain-to-resistance ratio along the interface between the
ocean and the mainland. Therefore, with Arg. 1, the slip of the lithosphere and
its shift down into the asthenosphere can be expected. The shear resistance can be
used to represent the main impediment force in the subduction process (subshifting).
Geophysics. Terrestrial crust geology. Rheology.
Info: [B73].
9.1.6 Beaufort Number B
B5w
0:836
� �23
for w½ �Bm s21 ð1Þ;
B5w
1:87
� �23
for w½ �Bmile hour21 ð2Þ
w (m s21) � wind speed.
It characterizes the wind velocity on the sea. The value is in the range
BAh0; 15i.Info: [A43].
Francis Beaufort (7.5.1774�17.12.1857), Irish oceanographer.
He formulated the Beaufort scale indicating wind force.
His digital scale extended from zero (calm) to 12 (hurricane)
and determined the wind force and possible thunderstorm
activity. Later, the scale was extended to 17 (tropical
cyclones) and supplemented by a formula to determine the
Beaufort number conversion to wind velocity. He was a
significant contributor to the development of the fields of
geography, astronomy, oceanography and meteorology.
393Geophysics and Ecology
9.1.7 Buoyancy Wave Number NB
NB 5
ffiffiffiffif 3
ε
r
f (s21) � buoyancy frequency; ε (s21) � dissipation of turbulent kinetic energy.
This number characterizes the buoyancy vertical oscillations originating in a
stable environment as considered in meteorology. This is different to gravitational
waves arising in a gravitation field. On the turbulent kinetic energy spectrum, the
wave number separates the buoyancy band from the inertia band. The measure-
ments show a relatively small portion of this number on the spectrum.
Info: [C13].
9.1.8 Businger�Dyer Relationship NBD
NBD 5wmed
wf
51
klnz2 d
z01Ψ
z2 d
L
� �
wmed (m s21) � mean wind velocity; wf (m s21) � friction velocity; k (2) �Karman constant; z (m) � height above the ground; d (m) � distance of transloca-
tion; z0 (m) � roughness length; Ψ (2) � empirical correction; L (m) �Monin�Obukhov length.
It characterizes the similarity relation between the surface flow of a variable
quantity and the mean profile value of this variable. Micrometeorology of surface
layer. Boundary conditions.
Info: [C15].
9.1.9 Centripetal Acceleration Ac
Ac 5ac
g; where ac 5ω2R
ac (m s22) � centrifugal acceleration; g (m s22) � gravitational acceleration;
ω (s21) � angular frequency of body (planet) rotation; R (m) � body radius.
It expresses the ratio of the centrifugal acceleration, with uniform circular
motion of a body, to the gravitational acceleration. It is used for bodies in approxi-
mate hydrostatic equilibrium. Geophysics. Astrophysics.
Info: [C18].
9.1.10 Dalton Number Dal
Dal5wv
wðks 2 kvÞ5CD;evSc
223
394 Dimensionless Physical Quantities in Science and Engineering
wv (m s21) � resultant evaporation velocity; w (m s21) � wind speed above water
level; ks (2) � saturation vapour concentration; kv (2) � vapour concentration;
CD,ev (2) � drag evaporation coefficient (p. 228); Sc (2) � Schmidt number
(p. 263).
It expresses the local evaporation rate of a substance from the free surface into
the atmosphere. It characterizes the atmospheric diffusion in a thin subsurface layer
between the surface and the surrounding atmosphere. Meteorology. Geophysics.
Info: [B48].
9.1.11 Davies Number Dav
Dav5CDRe2
CD (2) � drag coefficient (p. 60); Re (2) � Reynolds number (p. 81).
It is used in meteorology to calculate the final velocity of water droplets falling
from the atmosphere to the earth. It is also called the Best number. Meteorology.
Info: [C30].
9.1.12 Dissipation Number Dn
Dn5αgLcp
α (K21) � linear thermal expansion coefficient; g (m s22) � gravitational accelera-
tion; L (m) � characteristic length (depth); cp (J kg21 K21) � specific heat
capacity.
It expresses viscous dissipation with rapidly rotating thermal convection. It fol-
lows the dynamics of the terrestrial crust, but without the influence of the magnetic
field. Geology. Geophysics.
9.1.13 Ekman Number (1.) Ek
Ek5
ffiffiffiffiffiffiffiffiv
ωL2
r5
ffiffiffiffiffiffiRo
Re
rð1Þ;
Ek5
ffiffiffiffiffiffiffiffiffiffiffiν
2ωL2
rð2Þ;
Ek5ν
2L2ω sin ϕð3Þ
ν (m2 s21) � kinematic viscosity; ω (s21) � angular frequency of planetary rota-
tion; L (m) � characteristic length (usually vertical); ϕ (2) � latitude;
395Geophysics and Ecology
2w sin ϕ (s21) � Coriolis frequency; Ro (2) � Rossby number (p. 406); Re (2) �Reynolds number (p. 81).
This number expresses the ratio of the viscosity force to the Coriolis force. It
characterizes the flow in rotating channels. In expression (2) Ek, as defined by rela-
tion (1), it is called the velocity pressure criterion. Sometimes, equation (2) is
applied. In magnetohydrodynamic processes, it expresses the viscous fluid friction
influence and the effect of a secondary carrying motion, for example, the rotation of
the Earth. In equation (3), it describes geophysical phenomena in oceans and in the
atmosphere. With a small Ek, the arising disturbances can propagate due to a weak-
ening friction effect. Magnetohydrodynamics. Geophysics of viscous friction forces.
Info: [C45].
Vagn Walfrid Ekman (3.5.1874�9.3.1954), Swedish
oceanographer and physicist.
He is among the most famous oceanographers of his
time. He observed the movement of icebergs and found that
they did not move in the wind direction, but had a deviation
of 20�40�. He published his theory of the Ekman spiral, in
which this phenomenon was explained as the equilibrium
between friction effects and the fictive forces due to the
Earth’s rotation. He examined how the steady-state winds
formed a thin layer on the ocean surface, the eponymous
Ekman layer, and defined its velocity and thickness.
9.1.14 Froude Biomechanical Number Fr
Fr5w2
gLl
w (m s21) � forward speed of animal; g (m s22) � gravitational acceleration;
Ll (m) � leg length, usually measured from coxa.
It expresses the classic Froude number (1.) Fr (p. 62), but with parameters which
correspond to the motion of a person or animal. It is an important similarity criterion
in modelling the motion of a person under conditions of reduced gravitation, for
example, of the astronauts on other planets. Biomechanics. Astrophysics.
Astronautics.
Info: [B29].
William Froude (p. 63).
9.1.15 Geometry Erosion Parameter Pg
Pg 5
ffiffiffih
L
r
h (m) � depth of surface water; L (m) � slope surface size.
396 Dimensionless Physical Quantities in Science and Engineering
It characterizes the geometric parameters of the hydrodynamic and kinematic
processes of soil erosion. The parameter value is very small, among other things,
with respect to physical conditions connected to a channel acting with surface
water flow.
Info: [B65].
9.1.16 Golitsyn Energy Balance Number Goenergy, E
Goenergy 5qabs 1 qint
qabs5
Tef
Tekv
� �4
qabs (W m22) � absolute planet radiation; qint (W m22) � internal planet radiation;
Tef (K) � effective temperature; Tekv (K) � reference temperature.
This number expresses the thermal equilibrium between an internal heat source
and planet radiation. When Goenergy. 1.27, the internal heat source can be
expected to play an important role in controlling atmospheric dynamics.
Geophysics and astrophysics. Dynamics of the atmosphere.
Info: [B111].
Georgy Sergeyevich Golitsyn (see below).
9.1.17 Golitsyn Number Go
Go5τdynτrad
; where τdyn 5Rp
a; τrad 5
cppef
σ0gT3ef
τdyn (h) � dynamic time constant; τrad (day) � radiative time constant; Rp (m) �planetary radius; a (m s21) � sound speed; cp (J kg
21 K21) � specific heat capacity;
pef (Pa) � effective pressure characterizing the condition when hydrogen
optical depth is 1; σ0 (W m22 K24) � Stefan�Boltzmann constant; g (m s22) �gravitational acceleration; Tef (K) � effective temperature.
It characterizes the thermal inertia of the atmosphere or its structure, which is
important from an atmospheric radiative heating or cooling point of view, as com-
pared to the dynamics thereof. With Go{1, the atmosphere would be in equilib-
rium from a radiative point of view. In this case, the thermal mode plays a
controlling role over atmospheric dynamics and leads to low temperature gradients.
For Go$ 1, the global circulation requires longer time periods for the transference
of heat. The atmosphere is closed for local radiation equilibrium originating and
the dynamics is set on this thermal mode. The Golitsyn number Go is the time mea-
suring number Nτ (p. 30) for the atmosphere. Geophysics and astrophysics.
Dynamics of the atmosphere.
Info: [B111].
Georgy Sergeyevich Golitsyn, Russian astrophysicist.
397Geophysics and Ecology
9.1.18 Grouch Number Gro
Gro5wvωg
5Fr Sh21
wv (m s21) � vertical landing velocity; ω (s21) � natural frequency of mass
dynamical system during impact (incidence); g (m s21) � gravitational accelera-
tion; Fr (2) � Froude number (1.) (p. 62); Sh (2) � Strouhal number (p. 87).
It expresses the ratio of the vertical Froude number (1.) Fr (p. 62) and the verti-
cal Strouhal number Sh (p. 87). In dimensionless form, it represents the landing
velocity of a person or animal under various gravitational conditions. Astrophysics.
Astronautics.
Info: [B29].
9.1.19 Hubble Number Hu, z
Hu5δλλ
5H
cr
δλ (m) � wavelength shift of spectral lines; λ (m) � wavelength of spectral line;
H (s21) � Hubble constant; c (m s21) � speed of light; r (m) � distance of each
nebula from the Earth.
Essentially, it expresses Hubble’s law. According to it, the electromagnetic radi-
ation propagating from the Universe to the Earth has shifted towards longer
wavelengths (the so-called redshift in the visible light band) and is proportional to
the distance the radiation travelled. Together with the Doppler law, it confirms the
expansion of the Universe.
Info: [C44],[C71].
Edwin Powell Hubble (20.11.1889�28.9.1953), American
astronomer.
He was a prestigious astronomer because of his discovery
of galaxies behind the Milky Way and his cosmological sub-
stantiation of the wavelength redshift. He was the first to
argue that this shift was due to the Doppler effect, induced
by the expansion of the Universe. He was one of the leading
astronomers of his time who laid the foundation for physical
cosmology. Shortly before his death, the Hale telescope was
finished and Hubble was the first to use it.
9.1.20 Jeffreys Phase Φ
Φ52π
ðFo=2Þ13Ai
X2Fo
ðFo=2Þ13
0@
1A; where Fo5 τ
ffiffiffig
h
r; X5
x
h
398 Dimensionless Physical Quantities in Science and Engineering
τ (s) � time; g (m s22) � gravitational acceleration; h (m) � depth of ocean; x (m) �distance; Fo (2) � Fourier number characterizing tsunami propagation; X (2) �geometrical coordinates (p. 15).
It characterizes the phase shift of tsunamis. Geophysics. Oceanography.
9.1.21 Kibel Number Ki
See the Rossby number Ro (p. 406).
9.1.22 Kirpichev Number of Solar Radiation KiS
KiS 5εSLλT
ε (2) � relative absorption of solar radiation; S (W m22) � solar constant, density
of solar radiation; L (m) � characteristic length; λ (W m21 K21) � thermal conduc-
tivity of absorption environment; T (K) � temperature.
It expresses the ratio of the specific solar heat flow, transferred by the solar
radiation to the system, to the specific heat flow transferred by conduction in the
system. It characterizes the solar radiation heat flow onto the surface of a body. It
is the modified Kirpichev heat number Ki. (p. 176)
Info: [A23].
Mikhail Viktorovich Kirpichev (p. 177).
9.1.23 Kolmogorov Parameter Kol
Kol5Rgwffc
w2m
@w
@z
� �21
where R5Rw 2 R
R; wm 5
ffiffiffiτR
r; 0#Kol, 1
R (2) � relative density; g (m s22) � gravitational acceleration; wff (m s21) � free
fall velocity of droplets in air; c (2) � droplet concentration in thin layer;
wm (m s21) � shear stress velocity; @w@z ðs21Þ � gradient of flow velocity in the
z axis; w (m s21) � mean flow velocity at water level; Rw (kg m23) � water den-
sity; R (kg m23) � air density; τ (Pa) � shear stress.
For the suspension of droplets, it expresses the ratio of the energy dissipation
rate to the turbulent energy generation rate. The acceleration rate and the creation
of water spray connected with it on the surface of oceans is an example. The
Coriolis effect and the effect of cooling act with this, causing water droplets to be
399Geophysics and Ecology
formed and to evaporate. It is a specific analogy of the Kolmogorov number Kol
(p. 104). Geophysics. The physics of water spray generation in ocean surf.
Info: [C75].
Andrey Nikolayevich Kolmogorov (p. 104).
9.1.24 Langmuir Turbulent Number Lat
Lat 5
ffiffiffiffiffiuf
uS
r
uf (m s21) � water friction velocity; uS (m s21) � Stoke’s drift velocity.
It expresses the ratio of the friction force to the Stokes drift velocity. It charac-
terizes the turbulent process in the ocean surface layer with this process causing
mixing in a layer which becomes deeper due to Langmuir circulation. The
Langmuir turbulence has various characteristics, depending on the wind acting at
the surface. Geophysics.
Info: [C35].
Irving Langmuir (p. 345).
9.1.25 Lochtin Number Lo
Lo5Q
d2ffiffiffiffiffiffiffiffigdL
p
Q (m3 s21) 2 volume flow; d (m) � size of particles; g (m s22) � gravitational
acceleration; L (m) � characteristic length.
It characterizes the steady-state natural river flow which depends on the size of
particles of sediments, forming the river bed, and on the bottom inclination and
flow. It determines the width and depth. It is analogous to the Froude river inflow
number Frin (p. 417).
Info: [B52].
V.M. Lochtin.
9.1.26 Love Number Lov, k
Lov5Vred
Vtide
r
R
� �3
ð1Þ; Lov5ΔVred;2
Vtide
r
R
� �3
ð2Þ;
Lov53
211
19w2s
2gr
� �21ð3Þ; Lov � 3
19
gr
w2s
ð4Þ;
400 Dimensionless Physical Quantities in Science and Engineering
Vred (m2 s22) � deformation potential due to influence of tidal mass redistribution;
Vtide (m2 s22)2 tidal potential; ΔVred,2 (m2 s22) � increase of deformation poten-
tial at excitation8; r (m) � local radius of planet; R (m) � outer radius of planet;
ws (m s21) � shear speed dependent up to tidal excitation; g (m s22) � gravitational
acceleration.
The Love numbers characterize the influence of the Moon’s rotation around the
Earth and its periodic action on oceans (ebb and flow) and on the atmosphere and
rigid surface of the Earth. Due to the tide, this surface is deformed alternatively
and changes the gravitational field. As a result, the tide potential is created, which
differs on opposite sides of the Earth and causes two different shifts. The Love
number expresses the ratio of the vertical shift, leading to surface heightening and
to the tide potential. The expressions (1)�(4) are valid for homogeneous and
incompressible planets. For the Earth, it is Lov, 10 and for the Mars Lov. 10.
The expression (4) gives an approximate value. Geophysics. Mineralogy.
Info: [C81].
Augustus Edward Hough Love (17.4.1863�5.6.1940),
English mathematician.
He became famous by his work on the mathematical
elasticity theory. He was also engaged in the wave propaga-
tion and elaborated the mathematical model of terrestrial
surface waves which are known as the Love waves. The
book Some Problems in Geodynamics is among his most
popular works. Also important is his two-volume work, A
Treatise of the Mathematical Theory of Elasticity.
9.1.27 Moment Magnitude Scale Mw, Me, Mn
Mw 52
3log10
M0
ðN mÞ 29:1
� �ð1Þ;
Me 52
3log10
Es
ðN mÞ 22:9 ð2Þ;
where Es 5 1024:8M0 5 1:63 1025M0;
Mn 52
3log10
mtnt
ðkgÞ ð3Þ pro mtnt½ �5 kg;
Mn 52
3log10
mtnt
ðktÞ ð4Þ pro mtnt½ �5 kt;
Mn 52
3log10
mtnt
ðMtÞ ð5Þ pro mtnt½ �5Mt;
401Geophysics and Ecology
M0 (N m) � seismic moment of the total amount of energy that is transformed dur-
ing an earthquake; Me (N m) � small fraction of the moment M0 converted into
radiation seismic energy; Mn (N m) � comparative nuclear detonation moment;
Es (J) � radiation seismic energy; mtnt (kg, kt, Mt) � mass of explosive TNT serv-
ing for comparison.
The expression (1) represents the successor of the well-known Richter’s scale. It
serves to compare the energy loosened during the earthquake. At this logarithmic
scale, the rise by 1 degree corresponds to the loosened energy increase multiplied
by 101.55 31.65. The degree 2 corresponds to 1035 1000 times greater energy.
The expression (2) shows a part of the energy, released during the earthquake, in
the form of the radiation seismic energy. Expressions (3) and (4) present the com-
parison with nuclear detonations of the trinitrotoluene in kg, kilotons and
megatons.
Info: [C92].
9.1.28 Pratt Number Pra
Pra5Ls
Ll5
τwLl
Ls (m) � step length; Ll (m) � leg length, usually measured from the coxa; τ (s) �time duration of step; w (m s21) � forward speed of animal motion.
Together with the Froude biomechanical number Fr (p. 396), it expresses the
fundamental similarity criteria in modelling human motion under reduced gravity
conditions, such as in the motion of astronauts on other planets. Biomechanics.
Astrophysics. Astronautics.
9.1.29 Redshift Number Z
See the Hubble Number Hu (p. 398).
9.1.30 Richardson Bulk Number Rib
Rib 5gT 21
V ΔTVΔz
ðΔw1Þ2 1 ðΔw2Þ2ð1Þ; Rib 5
gΔRh
R0ðΔwÞ2 ð2Þ
g (m s22) � gravitational acceleration; TV (K) � absolute virtual temperature;
ΔTV (K) � virtual potential temperature difference across a layer of thickness Δz;
h�Δz (m) � layer thickness; Δw1, Δw2 (m s21) � changes in horizontal wind
components across the same layer; ΔR (kg m23) � change of density across a
layer; R0 (kg m23) � initial density; Δw (m s21) � change of resulting horizontal
wind component across a layer.
402 Dimensionless Physical Quantities in Science and Engineering
In meteorology, it represents the relation between the vertical flow stability and
the vertical slip. A high value indicates the instability and a low one shows stabil-
ity. The value RibAh50; 100i represents advantageous development. It is the differ-
ence analogy of the Richardson gradient number Rig (p. 384) and expresses the
approximation of local gradients by finite differences across a layer. Both numbers
characterize a slip instability model which expresses mixing. It is used to simulate
daily fluctuations in the depth of an instable layer and to determine the flow pro-
files, for example from subtropical regions. A complete mathematical model also
involves the influence of thermal flows and wind tension on layer instability. The
number Rib itself expresses the approximation of local gradients by finite differ-
ences across a layer. It is used in expression (1) or (2). The mathematical difference
model works, for example with the so-called surface box, and with the slip being
greater than the stratification density necessary to support it (Rib. 0.55), the prop-
erties of both adhering boxes are averaged because both have been stirred. This
process drops continuously up to Rib. 0.65. Meteorology.
Info: [C12].
Lewis Fry Richardson (11.10.1881�30.9.1953), English
mathematician, physicist, meteorologist, psychologist and
pacifist.
His interest was concentrated on meteorology and, above
all, on the design of a mathematical model, using differen-
tial equations (1922), to predict the weather, which is still
used today. He was also engaged in atmospheric turbulence
and executed many terrestrial experiments. The dimension-
less parameters of turbulence theory are named after him.
9.1.31 Richardson Flux Number Rif
Rif 5gT 21
VP wTV
uwðrUÞ1 vwðrVÞ
g (m s22) � gravitational acceleration; TVP (K) � virtual potential
temperature (theoretic temperature of dry air having the same density as wet air);
TV (K) � virtual temperature (coming from state equation of dry air for wet air
except substitution of T for TV); U, V (m s21) � horizontal Cartesian wind compo-
nents from the west and from the east; rU, rV (s21) � vertical wind velocity gra-
dients @U=@z; @V=@z; u, v, w (m s21) � axial components of wind velocity.
It expresses the ratio of the buoyancy force limit of turbulent kinetic energy to
the negative slip limit. It is the measure of the dynamic instability describing the
flow capability to stay turbulent in case of a slip wind and certain static stability.
In the expression, the denominator is usually negative. Therefore, with a positive
numerator, the Rif is negative for statically instable flow. For Rif, 1, the flow is
dynamically instable and turbulent. For Rif. 1, the flow is dynamically stable, the
403Geophysics and Ecology
turbulence tends to decrease and, according to the number Rif, the flow cannot be
defined. Meteorology.
Info: [C12].
Lewis Fry Richardson (see above).
9.1.32 Richardson Gradient Number Rig
Rig 5gT 21
A rTPðrw1Þ2 1 ðrw2Þ2
ð1Þ; Rig 5 g@R@z
R210
@w
@z
� �22
ð2Þ
g (m s22) � gravitational acceleration; TP (K) � virtual potential temperature;
TA (K) � absolute virtual temperature; w1, w2 (m s21) � wind components towards
the east and north; R (kg m23) � air density; R0 (kg m23) � initial density; z (m) �height of the layer.
This number is used in expression (1) or (2). It expresses the capability of gener-
ation or dissipation of the turbulence, divided according to direction, in the slip
development thereof. It is used to determine the dynamic stability and formation of
turbulence. For Rig less than the Richardson critical number (approximately
Rig# 0.25), the air becomes dynamically instable and turbulent. The origin of hys-
teresis can be considered too, with which the laminar flow must sink under the
above-mentioned critical value in order to pass to a turbulent flow. However, turbu-
lent flow can exist up to Rig5 1.0, provided it was primarily laminar before.
Meteorology.
Info: [C12].
Lewis Fry Richardson (see above).
9.1.33 Richardson Meteorology Number (1.) Rim
Rim 5gαrTðrwÞ2
g (m s22) � gravitational acceleration; α (K21) � linear thermal expansion coeffi-
cient; rT 5 @T=@z (K m21) � temperature gradient expressing vertical stability;
rw5 @w=@z (m s22) � velocity gradient expressing characteristic vertical wind
shear.
It expresses the buoyancy-to-inertia forces ratio or, alternatively, the capability
to suppress turbulence in the formation of the turbulence slip. It is used as a stabil-
ity measure to determine whether turbulence arises. Meteorology.
Info: [C12].
Lewis Fry Richardson (see above).
404 Dimensionless Physical Quantities in Science and Engineering
9.1.34 Richardson Meteorology Number (2.) Ri
Ri5gLΔRRw2
5Gr Re22 ð1Þ; Ri5gαΔTL
w2ð2Þ
g (m s22) � gravitational acceleration; L (m) � characteristic length; ΔR (kg m23)
� density difference; R (kg m23) � density; w (m s21) � flow velocity; α (K21) �linear thermal expansion coefficient; ΔT (K) � temperature difference; Gr (�) �Grashof heat number (p. 185); Re (�) � Reynolds number (p. 81).
It expresses the ratio of buoyancy effects to vertical slip effects. With Ri{1, the
buoyancy influences the turbulence with forced convection. For natural convection,
it is Ric1. At Ri�1, it expresses mixed convection. The value of the number
Ri5 1 is known as the Monin�Obukhov length L (p. 21). Mechanics of fluids.
Meteorology.
Info: [C12].
Lewis Fry Richardson (see above).
9.1.35 Richardson Overall Number Ri
Ri5g0Lw2
g0 (m s22) � reduced gravitational acceleration; L (m) � characteristic length;
w (m s21) � velocity imposed by the boundary condition of the problem.
In the above-mentioned expression, this number presents a basic view of its
physical meaning only. A more detailed description is in the following Richardson
numbers.
Lewis Fry Richardson (see above).
9.1.36 Rossby Atmospheric Number (1.) Roatm
Roatm 5τrotτdyn
Ba
Rpω
τrot (h) � time period of rotation; τdyn (h) � dynamic time constant; a (m s21) �sound speed; Rp (m) � planet radius; ω (s21) � angular frequency of planet
rotation.
It expresses the importance of planet rotation velocity (the Coriolis force) to cre-
ate atmospheric motion. Alternatively, it expresses the ratio of the rotation period
to the dynamic time scale. With Roatm, 1, the Coriolis forces play an important
role in atmospheric motion control.
Carl Gustaf Arvid Rossby (see below).
405Geophysics and Ecology
9.1.37 Rossby Atmospheric Number (2.) Ro
Ro5τrotτconv
5w
ωH; where τconv 5
H
w; τrot 5
1
ω
τrot (h) � time period of rotation; τconv (h) � time period of convection; w (m s21) �vertical velocity in the convective layer; ω (s21) � angular frequency of planet
rotation; H (m) � atmospheric height at static equilibrium.
This number characterizes the rotation acting on the vertical motion which
causes buoyancy. Therefore, it is also called the Rossby buoyancy number. With
Ro, 1, the planetary rotation causes the deviation of moving objects.
Carl Gustaf Arvid Rossby (see below).
9.1.38 Rossby Concentration Number RoC
RoC 5 βCΔC
βC (M21) � volume ion concentration coefficient11; ΔC (M) � ion concentration
difference between two electrodes12.
It expresses the change of the ion concentration in the atmosphere or other sys-
tems. Geophysics. Meteorology.
Carl Gustaf Arvid Rossby (see below).
9.1.39 Rossby Heat Number RoT
RoT 5βΔT
β (K21) � volume thermal expansion coefficient; ΔT (K) � temperature difference
between surfaces of warm and cold systems.
It expresses the thermal dilatability of solid, liquid or gaseous materials.
Mechanics. Geophysics. Meteorology.
Carl Gustaf Arvid Rossby (see below).
9.1.40 Rossby Number Ro
Ro5w
2ωL sin αð1Þ; Ro5
w
ωLð2Þ
w (m s21) � fluid flow velocity; ω (s21) � angular frequency; L (m) � characteris-
tic length; α (�) � angle between the axis of rotation and the direction of fluid
motion.
406 Dimensionless Physical Quantities in Science and Engineering
It expresses the ratio of the fluid inertia force to the Coriolis force in the atmo-
sphere or in an ocean, based on the planetary rotation. It characterizes the influence
of the rotation of the Earth on the flow in channels and pipelines. It describes the
geophysical phenomena in oceans and in the atmosphere. With a large number Ro,
usually Ro. 1, for example in the tropics or lower geographical latitudes, the plan-
etary rotational influence is insubstantial and can be neglected. If the Ro number is
small (Ro, 1), this influence cannot be neglected. It is also used in expression (2).
It is also known as the Kibel number (p. 399).
Info: [A21],[B20].
Carl Gustaf Arvid Rossby (28.12.1898�19.8.1957),
Swedish-American meteorologist and oceanographer.
He was the first to clarify the wide movements of the ter-
restrial atmosphere by using fluid mechanics. He introduced
a mathematical description (model) for these movements.
He was engaged in atmospheric thermodynamics, in mixing
and turbulence, and in the ocean�atmosphere interaction.
He identified and characterized nozzle flowing and Rossby
waves in the atmosphere. After the Second World War, he
modified his previous mathematical model of atmospheric
dynamics for the computer and weather predictions.
9.1.41 Rossby Temporal Number Roτ
Roτ 52πτω
τ (s) � time, time of the one rotation period; ω (s21) � angular frequency of the
Earth’s rotation.
It compares the local velocity changes in the atmosphere to the Coriolis force.
Geophysics. Meteorology.
Info: [C48].
Carl Gustaf Arvid Rossby (see above).
9.1.42 Seismic Efficiency η
η5Erad
Etot
5Δσ2σ
; where σ5σ1 1Δσ2
Erad (J) � radiated seismic energy; Etot (J) � total loading energy; Δσ (Pa) �difference of stresses before and after dislocation; σ (Pa) � average stress during
dislocation; σ1 (Pa) � stress after dislocation.
It expresses the ratio of the radiated seismic wave energy during earthquakes to
the total energy. Alternatively, it also expresses the ratio of the tension increment
to the mean tension value. Geophysics. Earthquakes.
Info: [C41].
407Geophysics and Ecology
9.1.43 Shida Number Shi, l
Shi5uλg sin λR gradλV
ð1Þ; Shi5uϑg
R gradϑVð2Þ
uλ, uϑ (m) � components of horizontal shift; g (m s22) � gravitational accelera-
tion; λ, ϑ (�) � shift angles; R (m) � outer Earth radius; V (m2 s22) � tide
potential.
Like the Love number, it characterizes the influence of the moon’s rotation
around the Earth, which causes the action of the tide on the surface deformation
thereof. The number expresses the ratio of the angles of horizontal shifts with two
components of the tide potential gradient.
9.1.44 Smith Erosion Number Sm
Sm5εwL
ε (�) � scale factor, usually ε{1; w (m s21) � mean effective velocity of rain,
incident to Earth surface; τ (s) � time; L (m) � characteristic length of a surface,
vertical or horizontal.
It characterizes the erosion time during which measurable changes occur on the
soil surface. Especially, it is used with physical modelling and analysis of land-
scape terrain changes. Geophysics. Morphology. Ecology.
Info: [B65].
9.1.45 Strouhal Biomechanical Number St
St5wvL
21l
wn
ð1Þ; St5whL
21l
ws
ð2Þ
wv (m s21) � vertical velocity of landing; Ll (m) � length of leg, usually measured
from coxa; ωn (s21) � natural frequency; wh (m s21) � horizontal speed at jumping
motion; ωs (s21) � jumping frequency at horizontal motion.
It is a modification of the basic Strouhal number Sh (p. 87) for the dynamics of
vertical motion with landing (1) or with horizontal spring motion (2). Astrophysics.
Astronautics.
Info: [B29].
Vincenc Strouhal (p. 87)
408 Dimensionless Physical Quantities in Science and Engineering
9.1.46 Tidal Horizontal Displacement Number Nhor
Nhor 5uϑ
R5
Shi
g
@V
@ϑð1Þ;
Nhor 5uλ
R5
Shi
g sin λ@V
@λð2Þ;
uϑ, uλ (�) � components of horizontal displacement; R (m) � outer Earth radius;
g (m s22) � gravitational acceleration; V (m2 s22) � tide potential; ϑ, λ (�) �angles of deviation; Shi (�) � Shida number (p. 408).
It expresses the horizontal displacement components caused by the tidal redistri-
bution of the terrestrial mass.
Info: [C81].
9.1.47 Tidal Radial Displacement Number Nrad
Nrad 5ur
R5
Lov2Vtide
g
ur (�) � radial displacement; R (m) � outer Earth radius; Vtide (m2 s22) � tide poten-
tial; g (m s22) � gravitational acceleration; Lov2 (�) � Love number.
It expresses the radial displacement caused by the tidal redistribution of the
mass of the Earth.
Info: [C81].
9.1.48 Tide Height Ratio H, h
H5 Lov5 Shi5 0 ð1Þ; H5 Shi5 1 ð2Þ;Hn 5 11 Lovn ð3Þ
Lov (�) � Love number (p. 400); Shi (�) � Shida number (p. 408); Lovn (�) �Love number (p. 400) at excitation potential of nth degree (here n5 2).
It expresses the ratio of the main tide height to the static height of the sea tide.
The expression (1) holds for a fully rigid body, the expression (2) for a liquid body
and the expression (3) for real body. If they have equal density, then Lov2 5 3=2and H2 5 5=4 are valid. Geophysics.
Info: [C81].
9.1.49 Tsunami Frequency Number Ω
Ω5ω
ffiffiffih
g
sð1Þ; Ω2 5K tanh K ð2Þ; where K5 kh
409Geophysics and Ecology
ω (s21) � angular frequency; g (m s22) � gravitational acceleration; h (m) � char-
acteristic length, depth of flat ocean; k (m21) � dimension wave number; K (�) �wave number (p. 33).
In expression (1), it represents the tsunami propagation frequency on the ocean
surface. In expression (2), it represents the dispersion relation for created gravita-
tional waves. Geophysics. Oceanography.
Info: [B18].
9.1.50 Tsunami Velocities C, W
C5cffiffiffiffiffigh
p 5 12K2
6ð1Þ;
W 5wffiffiffiffiffigh
p 5 12K2
2ð2Þ
c, w (m s21) � velocity propagation of water wave; g (m s22) � gravitational accel-
eration; h (m) � sea depth; K (�) � wave number (p. 33).
It expresses the approximate velocity, provided the sea depth is constant and the
velocity and phase approximation is quadratic. In addition, it is valid for long-wave
tsunamis (K{1). Geophysics. Oceanography.
9.1.51 Wesson Coupling Constant β
β5G
pcB33 1023Bα
G (m3 kg21 s22) � Newtonian constant of gravitation; p (kg21 m2 s21) � Wesson’s
empirical parameter expressing the ability of planets, stars and galaxies to super-cluster-
ing; c (m s2 1) � speed of light; α (�) � electromagnetic coupling constant (p. 306).
It is an analogy to the gyromagnetic ratio, but is more suitable from the general
relativity point of view. Astronomy. Astrophysics.
9.2 Ecology and Biology
Above all, ecology, as a multidisciplinary field, is based on biology and other
branches such as physics, chemistry, meteorology, geology and other technical as
well as non-technical fields. The increasing significance of ecology is demonstrated
in a lot of dimensionless quantities. They involve general ecology as well as
410 Dimensionless Physical Quantities in Science and Engineering
industrial ecology, ecology of seas, landscapes and forests, and global ecology.
Among the dimensionless quantities, the Brown number for atmospheric pollution,
the Burger number for the influence of the earth’s rotation on the water movement
in water systems, and the Imberger number to express the tendency of a surface
water layer to be mixed are presented. Several dimensionless quantities relate to
exergy and industrial ecology. As for biology, the following numbers are presented:
the sperm number for viscous and elastic tensions in microbiologic fibres, the radi-
ation weighting factor and the breakup wavenumber expressing the influence of
biological heterogeneity on intercellular relations in wave propagation in cardiac
tissue. A survey of fundamental dimensionless quantities for biological systems,
with more detailed comments on their formulation is in [B95]. In addition to the
dimensionless quantities in biology, microbiology, and agrobiology, these quanti-
ties are presented for pulmonology and botany as well.
9.2.1 Absorbed Agrobiological Dose Ndose
Ndose 5RedE2γτ
where edBE2γτR
for ω{ωcr; edBE2εR
for ωcωcr
R (kg m23) � density of soil; ed (J kg21) � specific energy; E (V m21) � electric
field intensity; γ (S m21) � specific electrical conductance; τ (s) � time; ω (s21) �frequency of supply; ωcr (s
21) � critical frequency; ε (F m21) � permittivity.
Together with other dimensionless quantities, it characterizes both the capability
of soil to absorb electric field energy and the efficiency of the delivered energy. It
is used in agricultural soil refining technology, based on the electric field method,
enabling inactivation of various harmful microorganisms in the soil. Agrobiology.
Info: [B113].
9.2.2 Association Constant kA
kA 5konC0
koff
kon (m3 mol21 s21) � kinetic speed constant of direct reaction; C0 (mol m23) �volume concentration of the target DNA in solution; koff (s21) � kinetic speed
constant of reversed reaction.
It expresses the ratio of the maximum direct biochemical reaction rate to the
maximum reversed reaction rate. It characterizes the biochemical processes, for
example, in DNA microorganisms in the hybridizing process and diffusion and
411Geophysics and Ecology
convective motions. In kA{1; it corresponds to the equilibrium relative molar sur-
face concentration of the formation of hybrid pairs. Microbiology. Microorganisms.
Info: [B86].
9.2.3 Bond Natural Number Bo
Bo5RghL2σ
R (kg m23) � sap density; g (m s22) � gravitational acceleration; h (m) � height in
capillary; L (m) � characteristic length, capillary radius; σ (N m21) � surface
tension.
It expresses the ratio of the gravitational force to that of the surface tension
force in the sap movement in plants. It characterizes this movement originating
from water loss as a result of evaporation from leaves. In the trunk or stem of a
plant, the radius of capillary tubes equals several tens of micrometres, and the
radius of leaves equals several tenths of micrometres. Biomechanics. Botany.
Info: [A49].
Wilfrid Noel Bond (1897�1937), English physicist.
9.2.4 Breakup Wave Number NS
NS 5 τr
ffiffiffiffiffiffiffiffiSRD
τ
r
τr (s) � enhanced time of output phase; SR (m22) � density downtrend per unit
area; D (m2 s21) � diffusion coefficient; τ (s) � transfer time rising between
inhomogeneities.
This number expresses the heterogeneity of a biological system and the proper-
ties of a medium capable of being excited. The isolated heterogeneity causes trans-
fer wave decomposition. Microbiology. The influence of heterogeneity and
intercellular coupling on wave propagation in the heart tissue.
Info: [B104],[B108].
9.2.5 Brown Number Bro
Bro5
ffiffiffiffiffiffiffiffiwp
2
wf2
s5
wp0
wx0
�������� ð1Þ;
where wf0�� ��5
ffiffiffiffiffiffiffiffiffiffi8kT
πmf
;
r
412 Dimensionless Physical Quantities in Science and Engineering
w51
2cfλ ð2Þ;
where λ51ffiffiffi
2p
πnd2m5
kTffiffiffi2
pπd2mp
wp ;wf (m s21) � mean velocity of particles (p) and fluid (f); wp0;wf
0 (m s21) � mean
temperature fluctuation of the velocity of particles (p) and fluid (f); k (J K21) �Boltzmann constant; T (K) � absolute temperature; mf (kg) � mass of fluid;
w (m s21) � velocity of fluid; cf (m s21) � sound velocity in fluid; λ (m) � mean free
path of molecules in gas; n (cm23) � numerical density of molecules in gas; dm (m) �diameter of gas molecule; p (Pa)� pressure in fluid.
Most frequently, it characterizes aerosols as suspensions of solid or liquid parti-
cles in a fluid and in the air. Dust, smoke, smog and mist are common aerosols. It
describes a colloid system in which the dispersed phase is represented by sprayed
solid or liquid particles. Usually, the radius of aerosol particles is in the range of
0.01�10 μm. The Brown number represents an important ecological criterion
expressing atmospheric pollution conditions, among other things. Ecology.
Meteorology.
Info: [A0].
Robert Brown (21.12.1773�10.6.1858), English botanist.
Above all, he dedicated himself to botany. In Australia,
his research over many years led him to discover roughly
2000 hitherto unknown plants. He edited a whole range of
works about the results of his Australian flora research. He
became most famous for his microscopic observations of
the random movement of dust pollen flecks and for formu-
lating the law of this movement. To honour him, this move-
ment has been named Brown movement. He also
discovered and described vegetal cells’ nuclei.
9.2.6 Burger Number, Si Number Bur
Bur5fBLV
ωLh5
Ro
Fr
� �2
5Rr
L
� �2
fB (s21) � buoyancy frequency; LV (m) � vertical depth of flow; ω (s21) � inertial
angular frequency of the Earth; Lh (m) � characteristic horizontal length of flow;
Rr (m) � Rossby deformation radius; L (m) � characteristic length; Ro (2) �Rossby number (p. 406); Fr (2) � Froude number (1.) (p. 62).
It is also called the Si number. It expresses the influence of the earth’s rotation
on the motion in a lake or other large water basin. With its value less than the criti-
cal one (Bur 5 1), the waves have the character of Kelvin or topographic waves.
413Geophysics and Ecology
For Bur , 1, the earth’s rotation influences the dynamics and the waves have an
oscillating character with most of their energy in the form of kinetic energy. The
Burger number characterizes the atmospheric or oceanographic flow expressed by
the ratio of the vertical stratification density to the horizontal terrestrial rotation.
For Bur-0, the rotational flow predominates, whereas the vertical stratification
does for Burc0: Geophysics. Ecology. Meteorology.
Info: [B52],[C14].
Alewyn Burger, mathematician and meteorologist.
9.2.7 Crop Coefficient KC
KC 5ETC
ET0
ETC (mm day21) � crop evapotranspiration; ET0 (mm day21) � reference crop
evapotranspiration.
It expresses the ratio of the real evaporation rate from plants and the soil under
them to the relative rate, generally determined by calculation according to the
Penman�Monteith equation for grass. The coefficient value is used to determine
the amount of water in soil irrigation. The evaporation coefficient value is usually
within the range KCAh0.1; 1.5i. Ecology. Botany. Agriculture.Info: [C26].
9.2.8 Damkohler Biological Number Dabiol
Dabiol 5konHmaxh
Dmol
kon (m3 mol21 s21) � constant of kinetic speed of direct reaction; Hmax (mol m22) �
molar surface concentration freely connecting the position of specific places on the
probe; h (m) � height of liquid layer; Dmol (m2 s21) � molecular diffusion
coefficient.
It expresses the ratio of the maximum direct reaction rate to the maximum nor-
mal diffusion rate. It characterizes biochemical processes, for example, those in
DNA microorganisms related to the hybridizing process and diffusion and convec-
tive motions. In the case of convective hybridization ðDabiol{1Þ; the hybridizing
time dependencies (in the probe covering place) are almost identical to the average
value of the relative molar superficial concentration of created hybrid pairs on the
probe surface. For Dabiol$ 1, pure diffusive hybridization occurs. Microbiology.
Microorganisms.
Info: [B86].
Gerhard Friedrich Damkohler (1908�1944), German physical chemist.
414 Dimensionless Physical Quantities in Science and Engineering
9.2.9 Deep Parameter Pd
Pd 5
ffiffiffih
L
r
h (m) � depth of surface water; L (m) � characteristic length of surface gradient.
It expresses the relative depth of the surface water on an observed surface sec-
tion. Geophysics and geomorphology. Ecology.
Info: [B65].
9.2.10 Depletion Number Dp
Dp 5εdεl
5DpðΨ ;Φ;ΩÞ
εd (kW, MJ year21) � exergy depletion rate; εl (kW, MJ year21) � exergy loss rate,
exergy destruction; Ψ (2) � exergy cycling fraction number (p. 415); Φ (2) � exergy
efficiency number (p. 416); Ω (2) � renewed exergy fraction number (p. 422).
It expresses a dimensionless indicator of exergy consumption for the loss exergy
unit. The depletion number Dp is the function of three dimensionless indicators:
the exergy cycling fraction number Ψ (p. 415), the exergy efficiency number Φ(p. 416) and the renewed exergy fraction number Ω (p. 422).
Info: [B25].
Catherine Preston Koshland (p. 416)
9.2.11 Exergy Cycling Fraction Number Ψ
Ψ 5 rm rq 5mRΔε1mCΔε2
; where rm 5mR
mC
; rq 5Δε1Δε2
rm (2) � mass recovery factor; rq (2) � quality recovery factor; mR (kg s21) � mass
flux of resources recovered from output of consumption process; mC (kg s21) �mass flux of consumption process; Δε1 (kJ kg21) � specific exergy difference of
recovered resources; Δε2 (kJ kg21) � specific exergy difference of consumption
process.
It expresses the ratio of the velocity of the exergy return to sources to that trans-
ferred from the sources. Ecology.
Info: [B25].
Catherine Preston Koshland (see page 416).
415Geophysics and Ecology
9.2.12 Exergy Efficiency Number Φ
Φ5ε1 2 ε2
ε1
ε1 (kW, MJ year21) � exergy removal rate; ε2 (kW, MJ year21) � exergy loss rate.
It expresses the ratio of the difference between transferred and lost exergies and
converted exergy or, alternatively, the ratio of the exergy, distributed into other
sources, to the transferred one. Ecology.
Info: [B25].
Catherine Preston Koshland (born 11.5.1950), American
engineer and ecologist.
She concentrates mainly on the problems of combustion
and arising emissions. Among other things, she is engaged
in research related to reducing the influence of temperature,
oxygen and other substances on the environment, especially
in relation to older industrial facilities, such as incinerators
and incineration furnaces, among others. She is engaged in
work related to increasing plant efficiency and to reducing
atmospheric contamination. She investigates the areas that
are the origins of toxic industrial by-products. Above all,
she tries to optimize industrial systems to achieve signifi-
cant reductions of emissions.
9.2.13 Foppl�von Karman Number NFK, γ
NFK 5EL2
k
E (Pa) � modulus of elasticity; L (m) � characteristic length (radius of roundel,
virus etc.); k (Pa m2) � flexural rigidity.
In the case of Newtonian fluids, it characterizes, for example, the behaviour of
the cytoplasm of erythrocytes flowing through a circular cross section of a capillary
tube. It also characterizes changes in spherical swirl walls, among other things. Its
value is NFKAh103; 108i for elastic walls. Medicine. Biology.
Info: [B82],[B68].
Theodore von Karman (p. 67).
9.2.14 Froude Dam-Breach Number Frdb
Frdb 5qdffiffiffiffiffiffigV
p
416 Dimensionless Physical Quantities in Science and Engineering
qd (m2 s21) � maximum outflow rate for unit width of breach; g (m s22) � gravita-
tional acceleration; V (m3) � volume outflow from hydrograph.
It characterizes the outflow wave propagation from a breached dam.
Geophysics. Ecology. Floods.
Info: [B89].
William Froude (p. 63).
9.2.15 Froude River Inflow Number Frin
Frin 5wiffiffiffiffiffiffiffig0R
p 5Qiffiffiffiffiffiffiffig0R
pAi
; where g0 5ΔRiR0
g
wi (m s21) � inflow velocity; g0 (m s22) � corrected gravitational acceleration;
R (m) � hydraulic radius of the underflow; Qi (m3 s21) � inflow volume rate;
Ai (m2) � inflow area; ΔRi (kg m23) � density difference between inflowing river
water and surface lake water; R0 (kg m23) � average density of water in the lake;
g (m s22) � gravitational acceleration.
It characterizes the river water inflow into a lake or similar water system. It is
an important hydrodynamic criterion in describing solid particle sedimentation. The
critical value is Frin5 1. For Frin, 1, inflow strokes of discharge occur.
Geophysics. Hydrodynamics. Ecology.
Info: [B52].
William Froude (see above).
9.2.16 Froude River Outflow Number Frout
Frout 5QoutffiffiffiffiffiffiffiffiffigH5
pQout (m
3 s21) � outflow volume rate; g (m s22) � gravitational acceleration; H (m) �total depth of lake.
It characterizes the lake water outflow into a river or other water system. It is
often applicable as an important hydrodynamic criterion in describing solid particle
sedimentation and dispersion. The critical value is Frout5 1. Geophysics.
Hydrodynamics. Ecology.
Info: [B52].
William Froude (see above).
9.2.17 Heating Agrobiological Temperature Qheat
Qheat 5cpTτa
417Geophysics and Ecology
cp (J kg21 K21) � specific heat capacity; T (K) � temperature; τ (s) � time;
a (m2 s21) � thermal diffusivity of ground.
It characterizes temperature penetration into soil which depends on specific ther-
mal capacity and diffusivity, and on action time. It is used in the cultivation of
agricultural soil using electrical field technology.
Info: [B113].
9.2.18 Human Inhalation Number Ninh
Ninh 5DQ
m5
D
Dexp
D (kg m23) � personal dose or exposition to concentration; Q (m3 s21) � volume
air flux in input; m (kg s21) � mass flux intensity of polluted gas; Dexp (kg m23) �
fully mixed gas concentration.
It expresses the ratio of the inhaled concentration dose to the fully mixed gas
concentration. Pulmonology. Ecology.
Info: [B51].
9.2.19 Imberger Number Im
Im5h
L012
1
Rk
ð12 e2Rk Þ�
1h
L1; where Rk 5 kh
h (m) � reference depth (e.g. daily thermocline); L0, L1 (2) � Monin-Obukhov
length for solar wave radiation and for remaining heat fluxes at the lake
surface (p. 21); Rk (2) � function of surface water mixing; k (s21) � damping
coefficient.
On a water surface, it expresses the surface water tendency to mix. The critical
value is Im5 0.2. With Im5 0.3, the tendency appears to cause stratification and
mixing at the surface. Geophysics. Ecology.
Info: [B52].
Jorg Imberger (born 10.9.1942), Australian engineer and
limnologist.
He is engaged in research on fluid dynamics, especially
in mixing and streaming in lakes, at river mouths and at the
sea coast. His work consists of designing computer models
to investigate the movement of water with biochemical
activity. It concerns interactions between physical, biologi-
cal and chemical processes. With the range of his work and
the standard of its results, he has contributed markedly
to the improvement of the environment in many countries
of the world.
418 Dimensionless Physical Quantities in Science and Engineering
9.2.20 Internal Mixing in the Metalimnion RS
RS 5τsτi
5h2
h2 2 h1Lak; where h1 , h2
h1 (m) � depth of the upper layer; h2 (m) � depth of the lower layer; τs (s) � time
of internal wave period for reserved water system (e.g. lake or pond); τi (s) � time
of free wave period; Ts (K) � temperature of surface water layer; Ti (K) � temper-
ature of lower water layer; Lak (2) � lake number (p. 420).
It expresses the time of one strong wave period. With h2. (h2�h1) and
Ts/Ti, 1, shear instability may be expected with increased mixing in the top part
of the fluid surface. With h2/(h1�h2). 1 and Ts/Ti, 1,26 vehement oscillation
occurs (Lak, 1), which is connected to bottom water rising to the surface. The
critical value is RS5 1. Geophysics. Ecology.
Info: [B52].
9.2.21 Joule Agrobiological Number Joargo
Joagro 5L2E2γλT
L (m) � characteristic length; E (V m21) � electric field intensity; γ (S m21) �specific electric conductance; λ (W m21 K21) � thermal conductivity; T (K) �temperature; ω, ωcr (s
21) � supply and critical frequency of electric field.
It expresses the ratio of the total energy of a low frequency ðω{ωcrÞ electricfield to the generated heat. It describes the amount of energy dispersed during the
process of soil cultivation using the electric field method, which deactivates various
detrimental microorganisms in the soil. Agrobiology.
Info: [B113].
James Prescott Joule (p. 298).
9.2.22 Knudsen Dispersion Number Kn
Kn52λd
54Ma
Re
λ (m) � mean free path; d (m) � particle diameter; Ma (2) � Mach dispersion
number (p. 421); Re (2) � Reynolds dispersion number (p. 423).
This number expresses the ratio of the mean free path of molecules to the char-
acteristic length (i.e. a particle diameter). It is a modification of the Knudsen num-
ber (1.) Kn (p. 69) for colloidal systems with dispersed solid or liquid particles.
Ecology. Meteorology.
Info: [A0].
419Geophysics and Ecology
Martin Hans Christian Knudsen (15.2.1871�27.5.1949),
Danish physicist and oceanographer.
He was engaged especially in research related to statisti-
cal mechanics, continuum mechanics and fluid mechanics.
Above all, he is known because of his studies in molecular
gas flow and for the development of the Knudsen cham-
bers, which is the basic component of molecular epitaxial
systems. The problems connected with the Knudsen number
are related, for example, to dust particle movement in the
atmosphere, to satellite movement and to the aerodynamics
of aircraft wings.
9.2.23 Lake Number Lak
Lak5SSchðH2 htÞ
w2A32LðH2 hV Þ
SSch (m5 s22) � Schmidt stability; H (m) � total depth of the lake; ht (m) � depth
of diurnal thermocline; w (m s21) � water shear velocity due to wind; AL (m2) sur-
face area of the lake; hV (m) � height of water centre.
It expresses the ratio of the acting forces prohibiting mixing and the forces
delivering the energy for system mixing. In the expression, the first number (SSch)
expresses the stability, which depends on the water system size. With Lak. 1,
there is no flow of a lower water layer to the surface. With Lak, 1, the cold and
the depth cause a flow towards the surface. The critical value is reached with
Lak5 1. Geophysics. Ecology.
Info: [B52]
9.2.24 Littoral Water Exchange Nlit, CV
Nlit 5τfτc
5Vh
13
w tan ϕL73τc
;where w5
ffiffiffiffiffiffiffiffiffiffiffiffiffiαqAghcR
3
s
τf (s) � flushing time; τc (s) � time over which night cooling acts; V (m3) � vol-
ume of the lake; h (m) � total depth of the lake; w (m s21) � convective flow
velocity; ϕ (2) � bottom slope angle in littoral area; L (m) � characteristic
length (equivalent radius of the lake bottom); α (K21) � thermal expansion coeffi-
cient of water; qA (W m22) � surface heat flux; g (m s22) � gravitational accelera-
tion; c (J kg21 K21) � specific heat capacity of water; R (kg m23) � water density.
It characterizes the influence of the night cooling of shallow littoral waters on the
exchange of this colder water with the warmer water from superficial layers of the
water system in its deeper parts by free convection. The critical value is Nlit5 1.
420 Dimensionless Physical Quantities in Science and Engineering
When Nlit, 1, in the shallow littoral zone the cooled water will flow into deeper
waters of the lake because of natural (gravitational) flow, and the warmer water takes
its place from the superficial layer at the deeper areas of the lake. When Nlit. 1 in
the littoral zone, the night cooling destroys the thermal equilibrium of the water, but
this water exchange is not influenced significantly. Geophysics. Ecology.
Info: [B52]
9.2.25 Mach Dispersion Number Ma
Ma5wp 2wf
�� ��cf
wp, wf (m s21) � velocity of particles (p) and fluid (f); cf (m s21) � sound speed.
It expresses the ratio of the difference between particle and fluid velocities to
the sound velocity under equal conditions of temperature and pressure. Ecology.
Meteorology.
Info: [A0].
Ernst Mach (p. 73).
9.2.26 Manning Roughness Coefficient Nn, n
Nn 5A53S
12wini
QO23
A (m2) � flow cross section; S (m m21) � inclination of water level; wini (m s21) �inlet velocity; Q (m3 m21 s21) � water volume flux on 1 m ground width; O (m) �wetted perimeter.
It characterizes the irrigation of furrowed clayey soil at diverse inflow velocities
and surface inclinations with various kinds of grain crops. Geophysics. Biosystems.
Info: [B104]
9.2.27 Peclet Solar Number Pesol
Pesol 5H2
aτ
H (m) � water height in solar reservoir; a (m2 s21) � thermal diffusivity; τ (s) �daytime duration of solar radiation.
It expresses the convective-to-conductive heat transfer ratio as water flows into
a solar reservoir. Ecology.
Info: [B48].
Jean Claude Eugene Peclet (p. 180).
421Geophysics and Ecology
9.2.28 Radiation Weighting Factor wrad
wrad 5H
D
H (Sv5 keV μm21) � equivalent dose in sievert (Sv); D (Sv) � absorbed dose by
the biological tissue.
The radiation weighting factor is used because some types of radiation are more
harmful biologically than others, provided the absorbed dose is equal. It charac-
terizes the influence of various kinds of radiation on biological damage to body tis-
sue. By means of the wrad, the dose absorbed by biological tissue can be
recalculated to an equivalent radiation dose by distinguishing the harmful effects of
diverse kinds of absorbed radiation. wrad5 1 holds for photons and electrons,
wradAh5; 20i for neutrons, wrad5 5 for protons, and wrad5 20 for alpha particles.
Info: [C50].
9.2.29 Ratio of Spot Radius R
R5r
h
r (m) � spot radius; h (m) � height of liquid layer.
It expresses the ratio of the spot radius to the fluid layer height. It represents a
geometric parameter, of which a large value corresponds to a large target, and vice
versa. Microbiology. Microorganisms.28
Info: [B39],[B86].
9.2.30 Renewed Exergy Fraction Number Ω
Ω5εren
εren 1 εnon
εren (kW, MJ year21) � exergy from renewed resources; εnon (kW, MJ year21) �exergy from nonrenewable resources.
This number expresses the exergy from renewable sources which is multiplied
by the inverse value of the sum of the exergies from both renewable and nonrenew-
able sources. It influences the depletion number Dp (p. 415) by expressing supplies
to the system which have been derived from renewable sources. The renewable
resources in question are, for example, solar radiation, wind or tide energy, and the
supply sources include biomass, among other sources.
Info: [B25].
Catherine Preston Koshland (p. 416).
422 Dimensionless Physical Quantities in Science and Engineering
9.2.31 Resuspension Number Nres, NRS
Nres 5ws
wb
5d2g0pLH
18νffiffiffiffiffiffiCD
phlwi
where ws 5d2g0p18ν
; wb 5
ffiffiffiffifficD
pwihl
LH; g0p 5
ΔRpR
g
ws (m s21) � sedimentation rate at the bottom; wb (m s21) � shear velocity at the
bottom; d (m) � effective sedimentation size; g0p (m s22) � corrected gravitational
acceleration; L (m) � characteristic length; H (m) � total depth of the lake;
ν (m2 s21) � kinematic viscosity; hl (m) � depth of upper water layer; wi (m s21) �phase speed of the internal wave; ΔRp (kg m23) � density difference between the
particles and water; R (kg m23) � water density; g (m s22) � gravitational accelera-
tion; CD (2) � drag coefficient (p. 60).
This number is an important ecological criterion determining whether a sedimen-
tation will settle on the bottom or be dispersed during the activity of waves in a lake
or other similar water systems. In compliance with the data in [C36], the value
Nres, 1 holds for the particle diameter d5 0.1 mm, as does the value Nres, 1024
for the diameter d5 1 μm. The critical value is Nres5 1. Geophysics. Ecology.
Info: [A30],[B25].
9.2.32 Reynolds Dispersion Number Re
Re5wp 2wf
�� ��dν
54Ma
Kn
wp, wf (m s21) � velocity of particles (p) and fluid (f); d (m) � particle diameter;
ν (m2 s21) � kinematic viscosity; Ma (2) � Mach dispersion number (p. 421);
Kn (2) � Knudsen number (1.) (p. 69).
It is a modification of the Reynolds number Re (p. 81) for colloidal systems with
dispersed solid or liquid particles. Ecology. Meteorology.
Info: [A0].
Osborne Reynolds (p. 82).
9.2.33 Schmidt Dispersion Number Sc
Sc5νD
5nfλd4
4
ν (m2 s21) � kinematic viscosity; D (m2 s21) � diffusivity; nf (m23) � numerical
gas density; λ (m) � mean free path; d (m) � particle diameter.
It is a modification of the Schmidt number Sc (p. 263) for colloidal systems with
dispersed solid or liquid microparticles. Ecology. Meteorology.
Info: [A0].
Ernst Schmidt (p. 264).
423Geophysics and Ecology
9.2.34 Sediment Number Eis
Eib 5qbffiffiffiffiffiffiffiffiffiRgD
pD
ð1Þ;
Eit 5qtffiffiffiffiffiffiffiffiffiRgD
pD
ð2Þ;
where qt 5 qb 1 qs; R5RsR21
qb (m2 s21) � volume bedload transport rate per unit width of cross-section diame-
ter; qt (m2 s21) � total volume bed material transport rate per unit width;
qs (m2 s21) � volume suspended load transport rate per unit width; R (2) � sedi-
ment submerged specific density; g (m s22) � gravitational acceleration; D (m) �characteristic sediment size; Rs (kg m23) � sediment density; R (kg m23) � water
density.
It expresses the sedimentation process with water flow in rivers, channels and
other water courses. It is also called the Einstein sedimentation number.
Geophysics. Ecology. Morphodynamics.
9.2.35 Smith Number Smi
Smi5εwτL
ε (2) � scale factor, its value ε{1; w (m s21) � effective rain velocity; τ (s) �time connected with measurable surface changes; L (m) � characteristic length.
It characterizes the influence of rain in changing the character of a land surface.
Geophysics and geomorphology. Ecology.
Info: [B65].
9.2.36 Sperm Number Sp
Sp5 Lηωk
� �14
L (m) � characteristic fibre length; η (Pa s) � dynamic viscosity; ω (s21) � angular
frequency of motion; k (N m21) � linear stiffness.
This number expresses the viscous-to-elastic tensions ratio on microbiological
fibres, such as bacteria and many eukaryotic cells moving by means of a hair
structure (flagellum). For the so-called simple swimmers, internal elasticity domi-
nance holds with a low value of Sp. With high values of Sp, the influence of vis-
cous friction predominates. Microbiology. Microphysics.
Info: [B32].
424 Dimensionless Physical Quantities in Science and Engineering
9.2.37 Tank Solar Number Ntank
Ntank 5RcpVSατ
R (kg m23) � water density; cp (J kg21 K21) � specific heat capacity of water;
V (m23) � water volume in reservoir; S (m2) � surface area of reservoir;
α (W m22 K21) � heat transfer coefficient; τ (s) � daytime duration of solar
radiation.
It expresses the ratio of the amount of water in an exchanger to the absorbing
area. It is an important quality in designing of solar collector equipment utilizing
natural water convection. Ecology.
Info: [B48].
9.2.38 Timescale Parameter Pτ
Pτ 5L
wΔτ
L (m) � declivity length of surface; w (m s21) � mean declivity velocity of surface
water; Δτ (s) � time interval of surface change.
It characterizes the development changes caused by hydrodynamic action of sur-
face water on a landscape. Geophysics and geomorphology. Ecology.
Info: [B65].
9.2.39 Volumetric Concentration of Target DNA C 00
C00 5
h
Hmax
C0
h (m) � height of liquid layer; Hmax (mol m22) � molar additional value of volume
concentration of free bonded parts in spot place on the probe; C0 (mol m23) �initial volume concentration on target DNA in the solution.
It characterizes the ratio of the number of DNA targets, represented over the
spots, to the number of tied up sides in a spot in the instant τ5 0. Explicitly, it
expresses the parameter determining the need to demarcate a target from outside
of the spot surface. Its value acts on the relative molar surface concentration of
formed hybrid pairs. Microbiology. Microorganisms.
Info: [B86].
425Geophysics and Ecology
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