Momentum Heat Mass Transfer MHMT1 Introduction and rules (form of lectures, presentations of papers by students, requirements for exam). Preacquaintance

Momentum Heat Mass TransferMHMT1

Introduction and rules (form of lectures, presentations of papers by students, requirements for exam). Preacquaintance (photography). Working with databases and primary sources. Tensors and tensorial calculusRudolf Žitný, Ústav procesní a

zpracovatelské techniky ČVUT FS 2010

Rules, database, Tensorial calculus

sourceDt

D

3P/2C, 4 credits, exam, LS2015, room C-434• Lectures Prof.Ing.Rudolf Žitný, CSc. (Wednesday 9:00-11:30)• Tutorials Ing.Karel Petera, PhD. First tutorial at 25.2.2015

• Rating 30(quise in test)+30(example in test)+40(oral exam) points

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3P/2C, 4 credits, exam, LS2015, room C-434


MHMT1 Momentum Heat Mass Transfer

• Textbook: Šesták J., Rieger F.: Přenos hybnosti tepla a hmoty, ČVUT Praha, 2004

• Monography: available at NTL Bird, Stewart, Lightfoot: Transport Phenomena. Wiley, 2nd edition, 2006Welty J.R.: Fundamental of momentum, heat and mass transfer,Wiley,2008

• Databases of primary sources Direct access to databases (WoS, Elsevier, Springer,…)knihovny.cvut.cz

Jméno DUPS

Literature - sourcesMHMT1

Information about peoples (publications)

database of papers used in presentations.

EES Electronic sourcesMHMT1

Key words

Full text available in pdf

Science DirectMHMT1

Prerequisities: Tensors

Bailey

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Transfer phenomena operate with the following properties of solids and fluids (determining state at a point in space x,y,z):

Scalars T (temperature), p (pressure), (density), h (enthalpy), cA (concentration), k (kinetic energy)

Vectors (velocity), (forces), (gradient of scalar), and others like vorticity, displacement…

Tensors (stress), (rate of deformation), (gradient of vector) , deformation tensor…

Prerequisities: Tensors

u

f

T

u

Scalars are determined by 1 number.

Vectors are determined by 3 numbers

Tensors are determined by 9 numbers

333231

232221

131211

321 ),,(),,(

zzzyzx

yzyyyx

xzxyxx

zyx uuuuuuu

Scalars, vectors and tensors are independent of coordinate systems (they are objective properties). However, components of vectors and tensors depend upon the coordinate system. Rotation of axis has no effect upon a vector (its magnitude and the arrow direction), but coordinates of the vector are changed (coordinates ui are projections to coordinate axis). See next slide…

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Rotation of cartesian coordinate systemMHMT1

Three components of a vector represent complete description (length of an arrow and its directions), but these components depend upon the choice of coordinate system. Rotation of axis of a cartesian coordinate system is represented by transformation of the vector coordinates by the matrix product

'1 1 2 3

'2 1 2 3

'3 1 2 3

cos(1',1) cos(1', 2) cos(1',3)

cos(2 ',1) cos(2 ', 2) cos(2 ',3)

cos(3',1) cos(3', 2) cos(3',3)

a a a a

a a a a

a a a a

a1

a2

a’1

a’2

1

2

1’

2’

2 cos(1 ', 2 )a

1 cos(1 ',1)a

'1 1'2 2'3 3

cos(1',1) cos(1', 2) cos(1',3)

cos(2 ',1) cos(2 ', 2) cos(2 ',3)

cos(3',1) cos(3', 2) cos(3',3)

a a

a a

a a

[ '] [[R]][ ]a a

Rotation matrix (Rij is cosine of angle between axis i’ and j’)

a

Rotation of cartesian coordinate systemMHMT1

Example: Rotation only along the axis 3 by the angle (positive for counter-clockwise direction)

1

'1 1'2 2

cos(1',1) cos cos(1', 2) sin

cos(2 ',1) sin cos(2 ', 2) cos

a a

a a

1[[R]] [[R]]=[[I]] [[R]] [[ ]]T TR

2

1’

2’

2,'12

1 ',1

( ) 2 ',12

2 ', 2

Properties of goniometric functions sin))2

(cos( sin)2

cos( cos)cos(

2 2

2 2

cos sin cos sin

sin cos sin cos

cos sin cos sin sin cos

sin cos cos sin sin cos

1 0

0 1

therefore the rotation matrix is orthogonal and can be inverted just only by simple transposition (overturning along the main diagonal). Proof:

a

Stresses describe complete stress state at a point x,y,z

ij

x

y

z

x

y

z

x

y

z

zz

zy

zxyz

yy

yxxz

xy

xx

Index of plane index of force component (cross section) (force acting upon the cross section i)

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Stress tensor is a typical example of the second order tensor with a pair of indices having the following meaning

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Later on we shall use another tensors of the second order describing kinematics of deformation (deformation tensors, rate of deformation,…)

Nine components of a tensor represent complete description of state (e.g. distribution of stresses at a point), but these components depend upon the choice of coordinate system, the same situation like with vectors. The transformation of components corresponding to the rotation of the cartesian coordinate system is given by the matrix product

Tensor rotation of cartesian coordinate system

[[ ']] [[R]][[ ]][[R]]T

cos(1',1) cos(1', 2) cos(1',3)

[[ ]] cos(2 ',1) cos(2 ', 2) cos(2 ',3)

cos(3',1) cos(3', 2) cos(3',3)

R

where the rotation matrix [[R]] is the same as previously

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Orthogonal matrix of the rotation of coordinate system [[R]] is fully determined by 3 parameters, by subsequent rotations around the x,y,z axis (therefore by 3 angles of rotations). The rotations can be selected in such a way that 3 components of the stress tensor in the new coordinate system disappear (are zero). Because the stress tensor is symmetric (usually) it is possible to anihilate all off-diagonal components

Tensor rotation of cartesian coordinate system

'1

'2

'3

0 0

0 0 [[R]][[ ]][[R]]

0 0

T

Diagonal terms are normal (principal) stresses and the axis of the rotated coordinate systems are principal directions (there are no shear stresses in the cross-sections oriented in the principal directions).

ji

ji

ij

ij

for 1

for 0

100

010

001

Special tensorsKronecker delta (unit tensor, components independent of rotation)

Levi Civita tensor is antisymmetric unit tensor of the third order (with 3 indices)

In terms of the epsilon tensor the vector product will be defined

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Scalar productMHMT1

a

b

Scalar product (operator ) of two vectors is a scalar

3' '

1 1 2 2 3 31

i i i i i ii

a b a b a b a b a b a b a b

aibi is abbreviated Einstein notation. Repeated indices are summing (dummy) indices.

' '

' ' ' ' ' ' ' '1 1 2 2 3 3

i im m j jk k

Ti i im ik m k mi ik m k mk m k m m

a R a b R b

a b a b a b a b R R a b R R a b a b a b

Proof that

cos|||| baba

Remark: there were used dummy indices m and k in these relations. Letters of the dummy indices can be selected arbitrary, but in this case they must be different, so that to avoid appearance of four equal indices in a tensorial term in the following product a.b (there can be always max. two indices with the same name indicating a summation)

' 'i i i ia b a b

Scalar productMHMT1

Example: Scalar product of velocity [m/s] and the normal vector of an oriented surface [m2] is the volumetric flowrate Q [m3/s] through the surface

u

sd

usdQ

sd

u

Example: scalar product of velocity [m/s] and force [N] acting at a point is power P [W] (scalar)

u

F

i i x x y y z zP u F u F u F u F u F

jijiiji fnnfn

3

1i

Scalar product

x

y

z

nf

Scalar product can be applied also between tensors or between vector and tensor

i-is summation (dummy) index, while j-is free index

This case explains how it is possible to calculate internal stresses acting at an arbitrary cross section (determined by outer normal vector n) knowing the stress tensor.

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Scalar product - examples

Dot product of delta tensors

Scalar product of tensors is a tensor

Double dot product of tensors is a scalar

Trace of a tensor (tensor contraction)

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ijmjim 1 0 0 1 0 0 1 0 0

0 1 0 0 1 0 0 1 0

0 0 1 0 0 1 0 0 1

im mj ij

: km mk

( ) example ( ) 3mmtr tr

233223132321231 babababac

kjijkj k

kjijki babac

abbac

3

1

3

1

)(

Vector product

Vector product (operator x) of two vectors is a vector

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Important relationship between the Levi Civita and the Kronecker delta special tensors

for example

b

a

c

sin|||||| bac

jminjnimkmnijk

check for i=1,j=2,(k=3),m=1,n=2 123 312 = 11 22 - 12 21 = 1

check for i=1,j=2,(k=3),m=2,n=1 123 321 = 12 21 - 11 22 = -1

Vector productMHMT1

F

r

Moment of force (torque) FrM

umF 2

Coriolis force

uF

application: Coriolis flowmeter

Examples of applications

i j ijab a b

Diadic product

Diadic product (no operator) of two vectors is a second order tensor

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Differential operator (Nabla)MHMT1

Bailey

ixxyx

i ),,(

Differential operator (Nabla)

Symbolic operator represents a vector of first derivatives with respect x,y,z.

applied to scalar is a vector (gradient of scalar)

ii x

TT

z

T

y

T

x

TT

),,(

applied to vector is a tensor (for example gradient of velocity is a tensor)

i

jj

zyx

zyx

zyx

x

uu

z

u

z

u

z

uy

u

y

u

y

ux

u

x

u

x

u

u

i

GRADIENT – measure of spatial changes

MHMT1

),( yxu

( , )u x dx y

dxx

uudu xxx

xu xu

dx

source 0 u

3

1 i

i

i i

izyx

x

u

x

u

z

u

y

u

x

uu

sink 0 u

Differential operator

The vector fi represents resulting force of stresses acting at the surface of an infinitely small volume (the force is related to this volume therefore unit is N/m3)

Scalar product represents intensity of source/sink of a vector quantity at a point

i-dummy index, result is a scalar

jizzyzxzzyyyxyzxyxxx

zyxzyxzyxf jif ,,

DIVERGENCY – magnitude of sources/sinks

Scalar product can be applied also to a tensor giving a vector (e.g. source/sink of momentum in the direction x,y,z)

onconservati 0 u

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ii xx

T

z

T

y

T

x

TTT

2

2

2

2

2

2

22

Laplace operator 2

Scalar product =2 is the operator of second derivatives (when applied to scalar it gives a scalar, applied to a vector gives a vector,…). Laplace operator is divergence of a gradient (gradient of temperature, gradient of velocity…)

),,(23

12

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

ii

j

i i

jzzzyyyxxx

xx

u

x

u

z

u

y

u

x

u

z

u

y

u

x

u

z

u

y

u

x

uu

Physical interpretation: 2 describes diffusion processes (random molecular motion). The term 2 appears in the transport equations for temperatures, momentum, concentrations and its role is to smooth out all spatial nonuniformities of the transported properties.

i-dummy index

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Divergency of gradient – measure of nonuniformity

Laplace operator 2MHMT1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

)exp()( 2xxT

T2

Negative value of 2T tries to suppress the peak of the

temperature profile

Positive value of 2T tries to enhance the

decreasing part

Integrals – Gauss theorem

V S

Pdv n Pds

Variable P can beScalar (typically pressure p)

Vector (vector of velocity, momentum, heat flux). Surface integral represents flux of vector in the direction of outer normal.

Tensor (tensor of stresses). In this case the Gauss theorem represents the balance between inner stresses and outer forces acting upon the surface,

Divergence of P projection of P to outer normal

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V S

pdv npds

V S

udv n uds

V S

dv n ds

Volume integrals with nabla operator can be converted to surface integrals (just only by replacing nabla with unit normal vector )

V ds

n

n

Physical interpretation: accumulation in volume V is overall flux through boundary

TranscriptionsMHMT1

Bailey

Symbolic notation, for example , is compact, unique and suitable for definition of problems in terms of tensorial equations. However, if you need to solve these equation (for example ) you have to rewrite symbolic form into index notation, giving equations (usually differential equations) for components of vectors or tensors, which are expressed by numbers (may be complex numbers).

T2

2T f

Symbolic indicial notation

General procedure how to rewrite symbolic formula to index notation

Replace each arrow by an empty place for index

Replace each vector operator by (-1)

Replace each dot by a pair of dummy indices in the first free position left and right

Write free indices into remaining positions

Practice examples!!

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Coordinate systemsPrevious conversion procedure can be applied only in a cartesian coordinate systems

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Formulation in the x,y,z cartesian system is not always convenient, especially if the geometry of region is cylindrical or spherical. For example the boundary condition of a constant temperature is difficult to prescribe on the curved surface of sphere in the rectangular cartesian system. In the case that the problem formulation suppose a rotational or spherical symmetry, the number of spatial coordinates can be reduced and the problem is simplified to 1D or 2D problem, but in a new coordinate system.

In the following we shall demonstrate how to convert tensor terms from symbolic notation (which is independent to a specific coordinate system) into the cylindrical coordinate system.

1 1 1 1

T T r T T z T T sc

x r x x z x r r

2 2 2 2

T T r T T z T T cs

x r x x z x r r

Coordinate systems (cylindrical)

Cylindrical (and spherical) systems are defined by transformations

rx1

x2

(x1 x2 x3)

(r,,z)

zx

rsx

rcx

3

2

1 1

2

3

0

0

0 0 1

dx c rs dr

dx s rc d

dx dz

where sin coss c

1

2

3

0

0

0 0 1

c sdr dx

s cd dx

r rdz dx

Using this it is possible to express partial derivatives with respect x1,x2,x3 in terms of derivatives with respect the coordinates of cylindrical system

3 3 3 3

T T r T T z T

x r x x z x z

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dzz

xd

xdr

r

xdx iii

i

r

c

xr

s

xs

x

rc

x

r

2121

, , ,

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In the same way also the second derivatives can be expressed

2 2 2 2 22

2 2 2 21

2( )

T T cs T T T s T sc

x r r r r r r r

2 2

2 23

T T

x z

2 2 2 2 22

2 2 2 22

2( )

T T cs T T T c T cs

x r r r r r r r

giving expression for the Laplace operator in the cylindrical coordinate system

2 2 2 2 2 2

2 2 2 2 2 2 21 2 3

1 1T T T T T T T

x x x r r r r z


(use goniometric identity s2+c2=1)

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Previous example demonstrated how to solve the problem of transformations to cylindrical coordinate system with scalars. However, how to calculate gradients or divergence of a vector and a tensor field? Vectors and tensors are described by 3 or 3x3 values of components now expressed in terms of new unit vectors

( ) ( )( ) zrm m r zu u i u e u e u e

Einstein summation applied in the cartesian coordinate system

unit vectors in the cylindrical coordinate system

( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ...z z

mn m n

r r r r zrr r rz zz

i i

e e e e e e e e

Unit vectors in orthogonal coordinate system are orthogonal, which means that( ) ( )( ) ( ) ( ) ( )1 0 ... 1zr r r ze e e e e e

and therefore the components in the new coordinate system are for example( ) ( ) ( )( ) ( ) ( ) r z r z

m m rz m mn nu u e u e e e e e

( ) ( )( ) zri r i i z iu u e u e u e

izz

i iee

)()( this is i-th cartesian component of the unit vector


( )1

( )2

( )3

0

0

0 0 1

r

z

i c s e

i s c e

i e

Transformation of unit vectors

( )1

( )2

( )3

0

0

0 0 1

r

z

e c s i

e s c i

e i

Example: gradient of temperature can be written in the following way (alternatively in the cartesian and the cylindrical coordinate system)

( ) ( ) ( )1 2 3

1 2 3 1 2 1 2 3

( ) ( ) ( )

( ) ( )

1

r z

r z

T T T T T T T TT i i i c s e s c e e

x x x x x x x x

T T Te e er r z

rx1

x2

(x1 x2 x3)

(r,,z)( )e

1i

2i

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( )re


Nabla operator in the cylindrical coordinate system

( ) ( ) ( )1r ze e er r z

1

T T T sc

x r r

substitute by using previously derived

MHMT1 Coordinate systems (cylindrical)

Example: Divergence of a vector ( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

( )

r zr m m z mm

m m

r zr zm m mr zm r m m z

m m m m m m

u e u e u euu

x x

ue e eu ue u e u e u

x x x x x x


components of unit vectors follow from the previously derived ( )

1( )

2( )

3

0

0

0 0 1

r

z

e c s i

e s c i

e i

1 0 0

0

0

)(3

)(3

)(3

)(2

)(2

)(2

)(1

)(1

)(1

zr

zr

zr

eee

ecese

esece

Note the fact, that the partial derivatives of these components with respect to xm are not zero and can be calculated using the previously derived relationships

( )1

1 1 1

re c c ss

x x x r

1 2

, s c

x r x r

giving

( )1

1 1 1

e s s sc

x x x r

…and so on


Substituting these expression we obtain

r

csu

r

sucs

r

u

r

su

r

csuc

r

u

x

ur

rr

22

2

1

1

2 222

2

r rr

u uu u u cs c c css u cs u

x r r r r r r

3

3

zu u

x z

Summing together, the final form of divergence in the cylindrical coordinate system is obtained

z

uu

rr

u

r

uu zrr

1


Example: Gradient of a vector

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )

...r z r z

r n n z n r n n z nr r rrr m n r m n

m m

u

u e u e u e u e u e u ee e e e

x x


Substituting previous expressions for unit vector and their derivatives results to final expression for the velocity gradient tensor in a cylindrical coordinate system

m and n are dummy indices (summing is required)

z

u

z

u

z

u

u

ru

u

ru

u

r

r

u

r

u

r

u

u

zr

zr

r

zr

1

)(1

)(1

MHMT1 Coordinate systems (general)

Procedure how to derive tensorial equations in a general coordinate system.

2. Define transformation x1(r1,r2,r3),… r1(x1,x2,x3),… and therefore also the first and the second derivatives of scalar values, for example

3. Define unit vectors of coordinate systems r i and express their cartesian coordinates Calculate their derivatives with respect to the cartesian coordinates

4. In case that the result is a vector, for example the gradient of scalar, calculate its components from

1. Rewrite equation from the symbolic notation to the index notation for cartesian coordinate system, for example j

iji

uu

x

),,( 321 rrrfx

rij

j

i

11 21 311 1 2 3

,...j j j ju u u uf f f

x r r r

( ) ( )

1 2 3( , , )ri rim me g r r r

( )31 2

1 1 2 3

r r r r rm m m m m

i i i i

e g g g g rr r

x x r x r x r x

In the case that result is a tensor calculate its components

( ) ( )m mu u e u e

( ) ( ) ( ) ( )r z r zrz i ij je e e e

Remark: This suggested procedure (transform everything, including new unit vectors, to the cartesian coordinate system) is straightforward and seemingly easy. This is not so, it is „crude“, lenghty (the derivation of velocity gradient is on several lists of paper) and without finesses. Better and more sophisticated procedures are described in standard books, e.g. Aris R: Vectors, tensors…N.J.1962, or Bird,Stewart,Lightfoot:Transport phenomena.

EXAMMHMT1

Tensors

What is important (at least for exam)MHMT1

You should know what is it scalar, vector, tensor and transformations at rotation of coordinate system

[ '] [[R]][ ]a a [[ ']] [[R]][[ ]][[R]]T

3

1i i i i

i

a b a b a b

Scalar and vector products

kjijkj k

kjijki babac

abbac

3

1

3

1

)(

(and what is it Kronecker delta and Levi Civita tensors?)

(and how is defined the rotation matrix R?)


Nabla operator. Gradient

ixxyx

i ),,(

Divergence

3

1 i

i

i i

izyx

x

u

x

u

z

u

y

u

x

uu

Laplace operator

ii xx

T

z

T

y

T

x

TTT

2

2

2

2

2

2

22


V S

Pdv n Pds

Gauss integral theorem

(demonstrate for the case that P is scalar, vector, tensor)

Documents

Momentum Heat Mass Transfer MHMT1 Introduction and rules (form of lectures, presentations of papers by students, requirements for exam). Preacquaintance