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Perro .en 'I _p UNNERSITY OF MINNESOTA St. Anthony falls (h~. G.lie Laboral
ST. ANTHONY FALLS HYDRAULIC LABORATORY
LORENZ G. STRAUB. Direclor
Technical Paper No. 26. Series B
Two-Phase Flow Studies in Horizontal Pipes with Special Reference to
Bubbly Mixtures
by
WALTER JAMES and EDWARD SILBERMAN
Seplember 1958
Minneapolis. Minnesota
UNIVERSITY OF MINNESOTA
ST. ANTHONY FALLS HYDRAULIC LABORATORY
LORENZ G. STRAUB. Director
Technical Paper No. 26, Series B
Two-Phase Flow Studies in Horizontal Pipes with Special Reference to
Bubbly Mixtures
by
WALTER JAMES and EDWARD SILBERMAN
September 1958
Minneapolis, Minnesota
Reproduction in whole or in part is permitted
for any purpose of the United States Government
PREFA CE
The studies described in this paper were sponsored by the David
Taylor Model Basin under Contract Nonr 710(21). The contract was effective
January 1 , 1957, and was scheduled to terminate September 30, 1958. The con
tract has actually been extended, but this report covers the work through
the original completion date. . The contract called for "a literature search
and experimental and analytic studies on the mechanics and limitations of con
veying air-water mixtures through closed conduits," and can be looked upon
as a resumption of an earlier contract, Nonr 710(07), which terminated in
December 1954.
The entire program was under the general direction of Dr. Lorenz
O. Straub, Director of the St. Anthony Falls Hydraulic Laboratory. The ex
perimental work and data reduction were performed at various times by Tuncay
Aydinalp, Athanasios Patitsas, Darrell E. Anderson, Robert L. Steele, and
Hsing Chuang. Special thanks is given to Mr. Chuang who was associated with
the project longer than any of the others.
The manuscript was prepared for printing by Delores Grupp and Mary
Anne Peterson under the general supervision of Loyal Johnson. Thanks is giv
en to these people as well as to the shop men who built the apparatus and to
the draftsmen who worked on the report.
iii
ABSTRACT ----- - --
An investigation has been made of the flow of bubbly mixtures in
horizontal pipes, together with some related work on other flow patterns.
Pressure drop is calculated through the use of a conventional friction factor.
In the bubble-flow regime, it is found that the friction factor is approxi
mately equal to or Slightly greater than the friction factor for liquid flow
ing alone in the pipe. The bubbles move at nearly the mean velocity of the
liquid while their size is inversely proportional to the liquid velocity and
directly proportional to the square root of the pipe diameter.
The use of a friction factor for calculating pressure drop for other
than bubble flow may also be useful, but more study is necessary to deter
mine friction factors.
iv
Preface .... co ...... ..
Abstract .......... .. List of Illustrations List of Tables List of Symbols
CONTENTS --------
• • • • •
• •
I. INTRODUCTION ..... "000 .................. .
II. SOI1E PREUMINARY CONSIDERATIONS
III.
IV.
PRESSURE DROP AND FRICTION FACTOR · . . · . . A; The Basic Equation for Pressur~ Drdp B. Experimental Data co .. .. .. .. co ..
1" First Experilnental Apparatus 2. Second Experilnental Apparatus 3. Chisholm and Laird E.'CjJt3r:i1nents
C. Friction Factor for Air-I-/ater Mixtures
ANALYSIS AT A CROSS SECTION .. .. .. .. .. .. .. .. .. .. A. Phase Fraction and Phase Velocity. • • •
1. Homogeneous and Nonhomogeneous Flow
•
2. Experilnental Evaluation of Ro/RL alid B. Experilnents on Bubble-Size Distributiori
V.
1. Apparatus.. .. .. .. .. .. .. .. .. .. .. 2. Data and Discussion of Results
C. V~locity Traverses in Bubble Flow • • 1 . Apparatus and Data .......... .. 2. Analysis and Discussion of Results
S~~Y AND CONCLUSIONS .- • • . List of References . • • . • • • • Figures 1 through 15 . . . . • Appendix A - Tables I through IV • • • • • • Appendix B - Lilniting Cases of Equation (17)
v
•
•
•
•
• • • VcJVL
•
• • •
•
• • • • •
• • • •
Page
iii iv vi
vii viii
1
3
6 6
12 12 13 15 16
18 18 18 19 20 20 21 22 22 24
28
31 35 53 61
Figure
1
2
3
4
S
6
7
8
9
10
11
12
13
14
IS
LIST OF IL LUSTRATIOdS
Flow-Pattern Regions ~ 0 • " .. • .. • • • • • • • •
Schematic Drawing of First Experimem;al Apparatus
Friction Factor at Small Air-Hater Flow Ratios • •
Schematic Dral;ing of Second Exper imental Apparatus
Chisholm and Laird Friction-Factor Data
Fri ction-Factor "Rati o as a Funct ion "of Compressibility Parameter . " .. " "
Gas-Liquid Fraction
Velocity Ratio from Chisholm and Laird Data
Sampling Apparatus for Bubble Si ze
Bubb2e- Slze Distribution 0
Characteristi~ Bubble Size
Velocity Profnes in B'.lbbly l'!:ixttn'e
Semilcgarithmic Plotting of Vel ocity Profiles in Pure-Water Flows 000 ..... "0 .. 000.0 0 "<> ... .. 0 •••••
Semi.logarithmic Plotting of Velocity Profiles in Air-,Iater Mixture flows "........ .. ..... .. .. .. " .. ~ . .. . .. " . .
Velocity Profiles Compared Hith Universal Lal"' of the ;Jall for Smooth Pi pe ~ " .. " " .. .. .. .. .. " .. .. " " .. .. . . . .
vi
Page
3S
36
37
38
39
40
41
42
43
44
4S
46
48
49
So
. able lio .
LIS T OF TAB L E S
I Pressure-Drop Data and Computations £or First Installation
II Pressure-Drop Data and Computations for Second Installation . • • • • • . . . • • ... .. • • • • .. . . . . · . .
ill Chisholm and Laird Pressure-Drop and Liquid-Fraction Data and Computations •• • • • • • • • • •• •
Bubble-Size Data • • • • • • • • • • • • • •
i Flow Properties for Velocity Traverses • • •
Diameter of Four-Inch Pipe • •
TI: Results o£ Velocity Traverses • • • •
vii
Page
53
54
55
57
24
25
26
LIST OF SYMBOL S ---- - ------
A - Ar ea of flm·,.
a,b - Constants .
D - Pipe diameter.
d - Bubbl e diameter, measured at atmospheric conditions.
d' - Dimensionless bubble diameter, d' = Ved,jp/u D.
f - Fricti on factor.
G - Weight per unit time.
g _ Gravitational constant.
P - Pressure.
Q - Volume per unit time.
R - Gas constant.
~ - Liquid fraction, ALIA; RG - gas fraction, Aa/A.
Re - Reynolds number, Re = VD pi 1-1.
r - Pipe radius.
T - Temperature.
V - Mean velocity.
Ve = Q/A.
V* = J Tip,
w - Specific weight.
x - Variable distance along pipe.
y - Distance from wall.
a = GG/GL' mass flow ratio.
f::J. - Increment.
e - Compressibility parameter, [al(l + all
viii
2 (gFIT/Ve ).
). = jO~:75
11 - Viscosity.
x ---, 62.3
Baker's parameter [2] (w in pounds per cubic foot) •
n(x ) Pressure parameter, QL/QG(x) = P(x)laPL'
p = wig, Density.
rr - Surface tension.
TO - Local mean wall shear stress around perimeter.
73.3 ILL
~ubscript L - Liquid.
ubscript G - Gas.
(
62.3
"L )J
l/3,
Baker'sparameter [2] (IT in dynes per centimeter, ILL in centi-
poises, "L in pounds
per cubic foot ) .
} No subscript means mixture.
ubscript m - Mean or average value.
TWO-PHASE FLO w, S '''!' U DI E S IN HORIZONTAL --- ----- --- - -------
PIPES WITH SPECIAL REFERENCE T 0 - - - - - - - - - - "' """" - - - ,.- -BUBBLY MIXTURES ------ --------
1. INTRODUCTION
Gas- liquid ,mixture flows are subject to a fairly complex series of
flow patterns. For the purposes of this paper, these flow patterns may be
described as follows,
(1) Bubble~, in.which separate bubbles of gas move along
the pipe with approximately the same velocity as the liq
uid. These bubbles can be uniformly distributed in the
pipe or they can move alo.ng in the upper region of a hori
zontal pi;>e with pure liquid flowing in the lower region.
(2) Plug flow, in which bubbles in the upper part of a hori
zontal pipe aglomerate to form large bubbles or plugs.
Plug flow occurs at low ratios of gas-to-liquid flow.
(3) Slug~, in >Thich a more or less well- defined inter
face separates liquisi and gas. The level of the inter
face rises and fall~, and frothy slugs pass regularly a
long the pipe at a much greater velocity than the average
liquid velocity.
(4) Annular Q£!!" in >Thich the liquid flows in a film around
the pipe wall and the gas flows at high velicity through
the central core. The film may contain gas bubbles.
(5) }list flow, in which liquid droplets are entrained more or -- . less uniformly throughout a gas flow. Spray ~ and ~~
perse .!12!! have also been applied to the regime after
annular flow breaks down.
(6) Separated flow, in which liq~d flows along the bottom of
the pipe and gas flows above. This type of flow occurs
in a horizontal pipe at small liquid flow rates. If low
gas-to-liquid flow ratios exist, the flow occurs with a
2
relatively smooth interface (stratified flow) and has
characteristics approacl)ing those of open-channel flow.
If the ratio is .higher, the interface is disturbed by
waves (~flow).
This study is primarily concerned with bubble flow in horizontal
pipes, but in order to delineate the limits of the bubble-flow regime, some
consideration has also been given to the other patterns.
}Iost of the scientific investigation of two-phase flow has been
conducted in the last twenty years. An extensive bibliography is contained
in a summary by Santalo [1]*. The flow-pattern descriptions given above are
in essential agreement with Santalo, who took his classification from Baker
[2]. Baker prepared a chart, drawing on the experiments of many investi
gators, showing the relation of f low pattern to rate of flow. This chart is
reproduced herein as Fig. lea) (using the nomenclature of the present paper).
Baker observed that the lines on his chart actually represent re'gions of over
lap of two flow patterns.
The bulk of the studies in horizontal pipes occurred at the Univer
sity of California and resulted in the well-kno>m Lockhart-Hartinelli corre
lations [3, 4, ;;, and 6] for pressure drop. Apparently these correlations
have dominated much of subsequent two- phase, flow stv:di'8'S. The Lockhart-·
Martinelli correlations are highly empirical, however, and are not uni ver
sally applicablE' as has been indicated, for example, by Baker [2] and H. S.
Ibsen in his discussion of a recent paper by Chisholm and Laird [6]. In the
present study, improved correlating devices have been sought, especially those
involving less empiricism.
There are two· major parts in this report. The first, contained in
Section III, following, considers gas-liquid mixture flows from the view
point of mean through-flow properties. In this part pressure drop and fric
tion factor in two-phase flows are discussed.
The second part of the report, Section IV, considers details of the
gas-liquid flow at a given cross section of a pipe. These details are impor
tant in applications which involve heat and mass transfer, for example. In
~~ Numbers in brackets refer to the List of References on p. 31.
3
the first part of Section IV, some consideration is given to the problem of
determining the relation between the pattern of flow and division. of the cross
section between liquid and gas. In the remainder of Section IV, some exper
imental details associated with the bubble-flow pattern are presented. These
involve bubble-size distribution and velocity-profile measurements.
II. SOBE PRELIMINARY CONSIDERATIONS
Tbe present investigation is concerned with steady mass flow of
gas and liquid in a horizontal pipe of uniform cross section. The following
assumptions are made throughout:
(1) No absorption or evolution of gas occurs.
(2) The gas obeys the perfect gas law.
0) Isothermal conditions exist, the temperature of the liquid
being imposed upon the gas.
(4) Vapor pressure and surface-tension effects may be ignored.
Since the weight rate of flow along the pipe
are the liquid and gas components of weight flow~ GL
and
where
is a constant for a given flow o
G is constant, as
G . 1 ,~ G3 respectlve y,
(1)
The total volume rate of flow Q at any position along the pipe
x is
Q(x) • ~ + QG (x) ~ ~ [1 + l/n(x)]
(2) where
but
,~
Symbols are summarized in the List of Symbols in addition to being de-fined where they first occur.
4
where w is the specific weight of tile fittid, and "'..nee, for an ideal gas,
where P is the absolute pressure, T toe absol1!te tE!t:perature of the liq
rod, and R the universal gas constant,
(3) where
It is seen that n is a dimensionless pressure parameter (n corresponds to
lip of Reference [7]).
The specific weight of the mixture is
w{x) = G/Q{x) = wL
(l + u)/[l + l/n{x)] (4)
and the mixture velocity is
V{x) = Q{x)/A ~ V~[l + l/n{x))
(5) where
and A is the cross-sectional area of the pipe; V~ is a fictitious veloc
ity of the liquid that would exist if there were no gas Dow.
If an instantaneous cross-sectional view of the two-phase Dow is
taken, a part of the Dow area (~) will be occupied by liquid and the re
mainder (AG) by gas. The liquid fraction Il:r, and gas fraction RG are de
fined as:
(6)
The mean velocities of the liquid and gas portions of the Dow are
Using Eqs. (6), (7), and (3),
5
1 (8) • ----
11 (X)
In the preceding expressions (x) has been used to make clear the
dependence on position along the pipe of certain of the variables. This sym
bolism will be dropped in most of the remainder of the paper, but the depend
ence on x of t hese variables should not be for gotten.
Two additional parameters (which are empirical) used by Baker [2]
and appearing in Fig. 1 also require definition. These are:
62.3
73 .3 aoo 'li ---
:'he w's ;Ire in pounds per cubic foot, the surface tension (j in dynes per
centimeter, and the viscosity ~ in centipoises in these parameters. It
should be noted than when water nea~ roam temperature is used as the liquid,
'" • 1 and X· 28.8 .,.r;;;;. Hence, with water near room temperature, the
abscissa in Fig. lea) can be written in the alternative forms
a a 1 .F; .. X ._--X '"
X 28.8 2 11(28.8)
(lOa)
while the ordinate can be writ ten
In Fig.
~ ~ 1.
GG a GL a
GL
.---= (lOb)
AX X A X '/I A
l(b), Baker's diagram has been replotted with GLiA as ordinate using
It can be seen from the figure that a large part of the overlap region
6
between bubble flow on the one hand and slug or plug flow on the other occurs
for ~/A ~ ,00 Ib/seC-ft2
• In fact, all lines of constant ~/A for 1/1 ~ 1
in Fig. l(a) make 4,-degree angles with the abscissa.
III. PRESSURE DROP AND FRICTION FACTOR
In a project report dealing with an earlier portion of the two
phase flpw studies [7] it was found that for very small ratios of gas-to-liq
uid flow, the pressure drop in a bubbly mixture flowing in a pipe could be
calculated approximately by treating the mixture as incompressible and using
the corresponding incompressible friction factor. There is some mixture ratio,
however, beyond which the mixture can no longer be treated as incompressible.
Then the pressure drop >Iill vary with the pressure in addition to its varia
tion with pipe wall friction.
A. The Basic Equation for Pressure Drop
With application to bubble flow in mind, it is ass\lllled that the
gas-liquid mixture is homogeneous. A free-body analysis is carried out on
a section of mixture of length ~x flowing in a circular horizontal pipe of
constant cross-sectional area A and diameter D. Applying Newton I s first
law to this free body in the direction of flow produces the equation
AP(x) -~~XT (x+6x/2)-AP(x +Clx). D 0
w(x + [}'x/2) Al>x-----
g
V(x + /:).x) - V(x)
6t
Here T (x) is the average unH shear stress around the perimeter Of the o
cross section at any x-position.
arranging the terms, !'!lplacing
and letting {l,x - 0, this
After
[', xl (l,t canceling out the. common factors, re
by V(x + t;x/2) , dividing by {I, x,
dP(x)
dx
equation reduces to
4 T (x) o .. (x) v(x)
+----+----D g
dV(x) ---=0
dx
This differential equation describes the local conditions at x.
(11)
7
If there were only liquid flowing in the pipe, the third term in
Eq. (11) would .be zero and the equation would reduce to the condition for
steady, uniform, incompressible flow. For incompressible flow, the pressure
drip is conveniently expressed in terms of a friction factor f such that
dP f wLV,,2
-+- • 0 (12) dx D 2g
with 8g T
0 f • (13a)
wLV" 2
and where g is the acceleration of gr avity. In compressible flow of a sin
ble fluid, the same friction factor defined as
8g T (x) o
f(x) - -------
is applicable for finding the frictional wall shear stress [8, p . 414). Now
let it be assumed that for homogeneous mixture flow, the friction fact or can
a gain be used for finding the wall shear stress. In this case
8g T (x) o
f(x) = ----2 w(x)V (x)
(13c)
In all of Eqs. (13), f depends on the local Reynolds number and surface
!'Oughness of the pipe only.
Upon substituting relations (5) , (4), 0), and (13c) into Eq. (11) ,
the latter can be put in the form
r, . a - 1
La - -n-(x-)-+-l- 2D
f(x)dx dn(x) + --- = 0 (14)
.. here
8
• gRT
V 2 $
(15)
This is the basic differential equation that must be integrated to determine
the pressure drop along a pipe carrying a homogeneous gas-liquid mixture flow.
Before integration can be performed, the variation of the friction factor f(x)
with x must be established.
It has already been observed that f varies only with Reynolds num:
ber and surface roughness. Reynolds number may vary along a pipe, however,
being given by the following equation:
DVp DG Re = -- • -- • - (1 + a) (16)
gIlA gIlA
Here p is the density and ~ the coefficient of viscosity of the mixture.
Apparently the mixture viscosity is larger than the liquid viscosity and de
pends on the viscosity of the liquid and the geometric proportions of the gas
liquid mixture [9]. The geometry will change as the pressure along a pi pe
changes, but in this analysis it ,!ill be assumed that the change in geometry
relative to the initial geometry is negligible. Since the temperature remains
essentially constant at the value determined by the liquid temperature, it
follows that the mixture viscosity remains essentially constant. Hence, th~
Reynolds number remains approximately constant UllJier isothermal conditions,
and since pipe roughness also remains constant for a given pipe, it may be
assumed thet the friction factor does not vary with x.
Under the above conditions Eq. (14) is readily integrated. The
boundary conditions are:
At x· 0, p(o) m P, nCo) = n o 0
At x m '1.'
The solution is
9
fx:t (17)
2D
Since this equation consists of three dimensionless groups, any consistent
set of units can be used. The friction factor group ~/D demands no expla
nation. The n group is the pressure group defined by Eq. (3); in fact, n o
- ~ measures the pressure drop over the length x:t. The dimensionless num-
ber 6 has not been explained before. Since the values of 6 range from zero
for pure liquid flo>1 to ir4'inity for pure gas flo>1, it can be regarded as
a degree of compressibility parameter, i.e., a measure of how far the homoge
neous mixture deviates from incompressibility (pure liquid flo,,). It should
be noted from Eq. (15) that 6 remains constant along a pipe and does not de
pend on local pressure o
In the developmerrt of Eq. (17), bias in favor of pure liquid flO>1
was maintained by factoring. out the properties of the liquid. If gas bias
had been maintained instead, the same equation as Eq. (17) would have been
obtained except for substituting 9 - 1/9'. 'Here 9' is zero for pure gas
flo1-1 and infinite for pure liquid flow. It is shown in Appendix B that EqD
(17) reduces to the incompressible result when GG - 0 and to the isothermal,
compressible result 1-Then Dr,- O.
To obtain, pressure drop from Eq. (17), it is necessary to estimate
the :friction factor, which is a f1ll1ction of Reynolds number and surface rough
ness. Reynolds number is given by Eq. (16); from Fig. 1, it is seen that in
bubble flo;l, Cl < 10-2 >.0/1 so that if the liquid is water and the pressure
is only of the order of a f"" atmospheres, Eq. (9) shows that Cl can be neg
lected in Eq. (16). Hence, if it could be assumed that ~ is given by its
value for pure liquid, the appropriate Reynolds number would be that for the
liquid component flo1-Ting alone in the pipe.
Define ( Dct roughness\
g~A ' iJ
7hen (18)
10
if fJ. == i'I. in a given pipe. This c <~ on~..1l. be assumed to hold. Since
it is known that fJ. > i'I. 19], tnerev'..J.l be a tetxlency to overestilnate Rey
nolds number and, therefore, to unde~st-aate the friction factor except for
completely rough flow. Obviously , the above method for estilnating f will
not be correct for large u. (In fact, if" is sufficiently large, Reynolds
number can be defined as
DGG Re = -- (1 + lin)
gIlA
so that the Reynolds number of the gas flowing alone might be more appropriate
for finding f than the Reynolds number of the liquid flowing alone.)
Equation_ (17) is trapscendental. For a given friction factor, the
calculation of PI' given Po' can be obtained by a trial-and-error method
from Eq. (17), or computer techniques can be applied. A more readily avail
able and silnpler method is deSirable. Approxilnation formulas are useful for
this purpose. Two of these are developed below by carrying out the logarith
mic expansions of Eq. (17).
Since P, and therefore n, must be known in advance, Eq. (17) o 0
can be written
--- • ell n + $n $n (19) 2D
"here ~,,~ "0 - "l' In using the logarithmic series expansion
x2 x3
$n (1 - x) • -x - - - -- - .00 , -1 < x < 1 2 3
it is necessary that 6n/n + 1) < 1 and lln/n < 1. These inequalitie~ o 0
ilnply nl
+ 1 > 0 and "l > 0 respectively, >1hich are true since n ~ 0,
always. The rapidity of convergence depends on the smallness of these ratios.
In the data analyzed later in this project,
n + 1 o
and lin
n o
II
<: 0.5
and most ratios were substantially less than 0.5. The expansion is carried
out to t>!o places only, resulting in a quadratic equation in the unkno>ln lin.
2(en 2 _ 1) ~ (lIn)2 ___ 0 ___ li n + --.0
n (n + 1) o 0
D (20)
This equation is solved by the quadratic formula. In order to obtain the
correct incompressible formula as GG- 0, the minus sign must be taken.
The result is
where
f~ (no + 1)
p - p • lip = p r 100
1 - 1-------
(en 2 _ 1) (n + 1) o 0
D f( en 2 - 1) o
r a -------
en 2 + 2n + 1 o 0
(21)
If the second term under the radical is considerably less than unity, a fur
ther simplification can be made by expanding the binominal term under the
radical sign. The result is:
P (n +l)f'c o 0 --J. II p = ------ (22)
2D( en 2 _ 1) o
For very small gas-to-liquid flo>! ratios, Eq. (22) becomes the in
compressible equation
lip =--D 2g
V 2 e
12
B. Experimental Data
Experiments were conducted for two purposes; namely, to determine
the liJniting conditions for bubble flow and to verify the assumptions lead
ing to Eqs. (17) and (18) for calculating pressure drop in bubble flow. The
latter experiJnents were conducted first and will be described first. Some
additional experiJnental data obtained recently by Chisholm and Laird [6J and
covering higher ga s-liquid ratios have been used to supplement the present
experiJnents. All of the experiments used air for the gas and water for the
liquid component.
1. First ExperiJnental Apparatus
Figure 2 is a diagrammatic sketch of the apparatus used in the in
itial experiJnental work. Actually, two pipe sizes were used in this study;
2-1/2-in. and 4-in. galvanized pipes. The 2-1/2- and 4-in. aspirators (air
water mixers) were designed and constructed for a previous project and their
construction details and operating characteristics are given in the report on
that project [7 J. The discussion that follows will be for the 2-1/2-in. sys
tem only. The description and operation of the 4-in. system was siJnilar ex
cept that it had no Lucite viewing sections before and after the test section.
The system was nonrecirculating. The water orifice was calibrated
in place by running tru, discharge into a rectangular enclosure. The air noz
zle was not calibrated, and the flow rate through it was computed from a the
oretical formula. Some runs were made by forcing in air from the compressed
air system. In these cases the air flow .las measured by a calibrated rota
meter. Air was always introduced into the system through the aspirator. The
test section consisted of approxiJnately the last 17 ft of a 23-ft continuous
length. of pipe. Pressure-measuring connections were made at the beginning,
middle, and end of this test section. Each pressure connection consisted of
four symmetrically placed holes around the circumference of the pipe, all
connected to a copper-tubing F.LIlg which was connected to the manometer sys
tem. Initial pressures in the test section were measured by an open manometer
and a calibrated Bourdon gage. The manometer readings were used in the compu
tations unless the initial pressure exceeded the manometer range. The pressure
drops along the first half and along the whole test section were measured by
a mercury U-tube manometer. The discharge valve was partially closed for some
runs so that data were obtained at various levels of average system pressure,
All data-taking points were preceded by sufficient lengths of pipe se that
13
any upstream disturbance would not affect readings taken at tnese points and
so that turbulent flow would be fulls" developed.
All data necessary to compute the friction fact·or f as defined by
Eq. (17) were taken. The data, as well as the more important computed num
bers, are given in Table I (in Appendix A). When GG equals zero, as in runs
27 and 34 of Table I, for example, f == fL. The range of variables covered
by these data is indicated by the shading in Fig. 1. Pattern observations of
the flow confirmed that all of these data were in tne bubble flow regime.
in Fig.
0./"'"
The bulk of the friction-factor data from Table I has been nlotted
3, along with other data, in the form f/fL versus Baker ' s abscissa,
for several values of ~/A.
2. Second Experimental Apparatus
The purpose of these additional expe riments was twofold. It was de
sired to determine the <offects of a flow disturbance occurring between two
straight lengths of pipe, and it was also desired to investigate the limits
of the bubble-flow regime. The only disturbance employed was a lBo-degree
return bend. Since the average test-section pressures before and after the
bend would be different because of pressure drop in the bend, the effect of
system pressure on pressure drop and flo" pattern >muld also be under study.
By forcing in air from an air compressor, the upper limit of the compressi
bility parameter e was increased from 2 for the previous experiments to a
bout 200 for these experiments.
A sketch of the second system is shown in Fig. 4. It consisted
basically of two identical 2-1/2-in.-pipe test sections placed before and
after the return bend. Each test section consisted of the last 12 ft of a
continuous 22-ft length of pipe. Air flow was measured using a calibrated 01'
if;i.ce. The manometry system was altered halfway through the tests. Original~
ls", each test section was provided with a mercury manometer on:q. Many of
the readings were small enough So that it was felt that some error in the
pressure-drop readings resulted. For the second half of the data, the manom
etry system "'IS redesigned (details not shown) by connecting needle valves
as double-pole, double-throw switches. This valving permitted the pressure
drop in either test. section to be measured by a Mariam No.3 or mercury manom
eter. Sight glasses were installed before and after the return bend to ob
tain visual data on the pattern of flow.
Data were taken for a series of air flOl-lS at a constant water flow
and constant initial pressure. The average system pressure of the initial
test section was naturally higher than that in the final test section. As
the air or water flow ,""s changed, the pressures in both sections changed,
but not in a readily comparable manner. Since it was disquieting to have the
test-section pressures change in unknown manners, it was decided to hold the
initial pressure of each test section .at a nearly constant value for each
constant water-flow series. This was done in the following manner: The first
test of a constant water-flow series wa s conducted with maximum air flow, the
back pressure valve being completely open. This test produced the highest
system pressure and established the value of P for each test section for o
this series. Data from both sections were recorded. Since the air flow for
the next run was less, pressures throughout the system dropped. The back
pressure valve was adjusted until the value of P for the initial test sec-o
tion was restored to the same value as for the first run. Data from the ini-
tial test section were recorded. This process was repeated for the second
test section. The total operation was repeat<ld for the complete range of air
settings and constant Hat er settings. In order to obtain data from each test
section under "before" and !lafter" conditions" and in order to insure that
any unusual phenomena arising from these studies were due to a natural func
tioning of the system, rather than due to any errors in the system, the test
ing program was repeated with the test sections interchanged.
The data and computations for the second set of experimental studies
are tabulated in Table II (in Appendix A). Only data taken after interchanging
the test sections and improving the manometer system are shown in the table.
As before, f was computed from the measured pressure drop using Eq. (17);
some of these data have been plotted in Fig. 3 along with the earlier data
from the first apparatus.
Flow-pattern observations were made when taking data for the initial
test section (minor adjustments in the back pressure valve in regulating the
initial pressure of the final test section did not seem to affect the flow
pattern). The pattern observations are shown in detail in Table II and are
plotted in Fig. 1. The viewing sections immediately before and after the re
turn bend represent extreme nonsymruetry in obsero-ing the flow pattern since the
pattern in the initial yiewing section is reached under maximum equilibrium
establishing conditions, while the flow pattern after the bend is in a highly
disturbed condition. Even so, the pattern rare1;y changed significant1;y be
fore and after the disturbance. The most serious effect of the bend occurred
in a series (162-16B) where several of the runs suffered a change from bubble
flow to slight slugging flow. However, for this particular series the flow
pattern changed from bubble flow for low air content to slug flow for higher
air content in the upstream section. Hence, this series passed through a crit
ical pattern change and it is not surprising that several intermediate runs,
under a sudden reduction in pressure, cha nged from bubble flow in the initial
section to slight slugging flow in the second section. It is believed that
t his shift in patte rn, then, was not a result of the disturbance but rather
of the pressure drop. The pattern observations tend to confirm, in general,
Baker's region of transition between bubble and slug flow.
It may be noticed in Table II (in Appendix A) that in the bubble
flow regime -, f/f1
was approximately the same both upstream and dOlffistream
of t he bend. This was not true, however, outside the bubble regime. For some
flOl' S in this category f/f1 was substantial1;y smaller in the upstream test
section. The exact cause of this phenomenon is unknown. Several poss ib1li ties
exist. One is that the pressure being materially different upstream and down
st ream of the bend, a dilferent patte-rn of flow occurred (perhaps approaching
separated flow) in the upstream section even though this was not apparent in
the viewing section which was located at the end of this test section. Another
possibi).ity is that the flow pattern actually changed in the upstream test
section, perhaps not having settled down to an established pattern at the be
ginning of the test section. Because of the uncertainty, no attempt has been
"",de to ana1;yse these data at this tilne.
Using the second apparatus, Some data were taken of the pressure
drop around the lBO-degree bend. The data showed considerable scatter, but
tbe bend coefficient defined as the pressure drop in the bend divided by the
dynamic head of the flowing mixture was rough1;y constant as it is for pure
water. Since only one bend was used, and that under limited conditions, more
investigation of loss due to bends is required.
3. Chisholln and 1aird Experiments [6 J
A recent paper by Chisholln and Laird made available data covering
a IlUch wider range of variables than that cOVllred in this project. According
Ul the authors, the studies i.twolved bubble, slug, and annular flows under
16
essentially atmospheric conditions. However, after studying the data fully,
it was found that apparently very few were in the bubble-flow regime and a
few more were in the transition from bubble to slug flow. Although Chisholm
and Laird used nominall-in.-diameter pipes of several roughnesses, only their
data for smooth and glavanized pipes were considered.
The Chisholm and Laird data, together with some computed numbers,
are reproduced in Table III (in Appendix A). The friction factor f was com
puted from the measured pressure drop using Eq. (17), even though the mixtures
in only a very few of the runs could be considered as homogeneous. Flow pat
tern observations were not recorded by Chisholm and Laird.
The ratios f/~ were computed using tabulated values of fL in
the Chisholm and Laird paper. Figure 5 portrays the f/~ data as contours
of constant f/fL
on Baker's flow-pattern diagram. The contour lines in the
bubble-flow regime of Fig. 5(b) have been drawn with the aid of data from
Tables I and II. Some of the Chisholm and Laird data in the bubble-to-slug
flow transition region are also plotted in Fig. 3 for comparison with the pres
ent data,
C, Friction Factor for Air-Water ~lixtures
The friction-factor data from Tables I, II, and III (in Appendix A)
have already been plotted in Figs. 3 and 5 in the fom f/f1
as a function of
Baker ' s parameters. The galvanized-pipe data from all three tables seem to
be in good agreement with each other "here they overlap. For the upper dia
grams of Fig. 3, where the flow is almost entirely bubble flow,
0.98 < f/fL
< 1.25,
the higher values of the ratio occurring at th .. higher gas-liquid flow ratios.
A variation of :2 per cent in f/fL
can be explained by experimental error;
but the remainder of the rise above f/fL
• 1.0 is systematic and possibly
results from the fact that Il. > Il.r. and becomes increasingly so as the air
water ratio increases. (It may be possible to calculate Il. for the mixture
from these data.)
Figure 5 gives a picture of the general trend of f/fL
outside the
bubble-flow regime. Even though the homogeneous assumption and the assumption
of constant Reynolds number used in developing Eq, (17) no longer hold, the
17
concept of friction factor appears to be a useful one for calculating pressure
drop. Both the smooth- and gal vanized-pipe data of ChiS):lOlm and Laird, when
plotted as the ratio f/f1' exhibit quite similar trends, f/fL being rela
tively constant in bubble flow and decreasing steadily and with increasing
rapidity as t he stratified portion of the diagram is approached. Outside the
bubble-flow regime, f/fL
appears to be slightly larger in the galvanized
pipe than in the smooth pipe, other conditions being the same. Perhaps a
better referen~e friction factor than fL could be devised for the Chisholm
and Laird data, but 'no attempt has been made to do so in preparing this paper, •
.uso, Bakel" s diagram may not be t he best base on which to plot these data,
especially in view of the presence of the empirical parameters ~ and '" •
Same data ,should be obtained with other pipe sizes to check the method of pre
sentation. Further work toward improving Baker ' s diagram and correlating fric
tion factor with it may be useful.
In any event, it appears that t here is a smooth change in friction
factor with -the flow parameters even. across the transition regions bet"een
the various flow patterns. Therefore, a knowledge of the flow pattern does
not appear to be essential in predicting f/fL
or pressure drop (within a
certain margin of error). The same assumpt i on regarding pressure drop was
~de in reaching t he Lockhart-Martinelli correlations [4].
In view of the approximate independence of f/fL
from flow pattern,
it may be possible to find a single parameter which will correlate f/fL
for
use in predicting pressure drop. Several parameters have been tested with
this object in mind, and the best one appears to be the compressibility param
eter e. defined by Eq. (15) and already appearing in the pressure-drop e
quation, Eq. (17). In Fig. 6 all the data from Table s I, II, and III (in
Appendix A) have been plotted in the form of f/fL
versus e. Figure 6(a)
shows these data for e ::; 3 using a natural scale for e while Fig. 6(b)
shows the data for e ~ 1 using a logarithmic scale for e in order to
accommodate the wide range of the data.
For Fig. 6(a) , where all the data lie within the bubble-flaw re-
(23)
!'its all the data within !4 per cent. This is as close as fL itself can
18
be predicted. When extended to the higher values of e shown on Fig. 6(b) ,
Eq. (23) still fits all the data within about ~10 per cent to e· 20. Equa
tion (23) has been plotted on Fig. 6.
At still greater values of e, in Fig. 6(b) to a minimum near e • 10$
f/fL
values decrease again as shown
and then appear to rise again. Both
the galvanized- and smooth-pipe data of Chisholm and Laird for several values
of air-water flow ratio appear to be correlated by the plotting shown in Fig.
6(b). It may be noted that the spread . in f/fL
at any e is not any worse
than the spread in pressure drop given by the Lockhart-Martinelli ' correlations
for one pipe surface only, using the same data (6). The form of plotting used
in Fig. 6 needs to be checked using data for other pipe sizes at large e;
s)lCh checking is not within the scope of this pap(lr. At the lower values of
e, however, especially in the bubble-flow regime, the plotting used in Fig.
6 has correlated data for several pipe sizes obtained by different investi
gators and the correlation may be considered quite satisfactory.
It may be observed in Fig. 6(b) that there is a transition between
nSlllg f/fL
for e < 35 and falling f/fL
beyond; and that the transition
is near the transition from bubble flow to slug flow. In the present experi-
ments bubble flaw always existed for. e < 6.
e > 45. In the intermediate region, the flow
Slug flow always
could be either
existed for
essentially
bubble or slug flow, depending on other parameters. There is a possibility
that ·.the transition in slope of f/fL is related to the change in flow pat
terno
IV. ANALYSIS AT A CROSS SECTION
A. Phase Fraction and Phase Velocity
1. Homogeneous and Nonhomogeneous Flow
Liquid and gas fraction have been defined by Eq. (6) and liquid
and gas velocities by Eq. (7) of Section II. The relation between these e
quations was given by Eq. (8).
For steady bubble flow (and perhaps for steady mist flow) in hor
izontal pipes where the mixture may be considered pseudo-homogeneous, V G = VL
19
since the gas bubbles are trapped in the surrounding liquid flow (or the mist
droplets are trapped in the surrounding gas flow). Then Eq. (8) becomes
(24)
This is the basic equation for phase fraction when the flow can be assumed
homogeneous 0
In nonhomogeneous flow patterns, it is not so simple to predict the
relation between VG and VL
• In annular flow, where the gas component is
completely isolated from the pipe walls by the liquid component, V G > VL
•
In separated and slug fiow_ it is not possible to make a definite statement
without further information, but because of direct wall friction of the gas
in these cases it is unlikely that the ratio VG/VL will ever exceed that
for annul"r flow. Hence, as tha gas flow rate increases at constant water
flow rate, it can be surmised that VG/VL will rise from near unity for bub
ble fiow, reach a crest for annular flow, and recede toward unity for mist
flow. Corresponding changes in RG/R:r, in accordance with Eq. (8) can be ex
pected.
2. Experimental Evaluation of RGII\. and ValvL
Phase fraction and velocity measurements were not obtained in the
.,resent experiments, but were obtained by Chisholm and Laird [6] in connection
with the friction-factor experiments ' discussed earlier. They used the method
of trapping the flow between two simultaneously closing valves and measuring
the liquid fraction. The Chisholm and Laird data for liquid fraction were
already tabulated in Table III (in Appendix A), as were the computed results
for ValvL.
The two quick-closing valves in the Chisholm and Laird experiments
were located 8 ft apart in a l-in. pipe, so that RL has been averaged over
a reach whose average pressure is not precisely known--only the pressure at
the midpoint of the reach is given and this is recorded in Table III in terms
of the parameter n as lin (where the subscript denotes measurement at the m
midpoint of the test length).
The Chisholm and Laird data, plotted as log (RalR:r,) versus log
(l/nm) are shown in Fig. 7. Also shown in Fig. 7 is the line of Eq. (24),
20
the homogeneous flow theory result. The data for both the smooth and galva
nized pipes appear to correlate along a fairly well-defined cm:ve that is
bounded for small air-water flow ratios by Eq. (24). Furthermore, for large
air-water flow ratios (large l/nm) the data . apP!!ar to be limited by another
straight line parallel to that given by Eq. (24), indicating that a maximum
value of V r/VL has been reached.
The velocity-ratio data ax:e plotted on Baker's diagram in Fig. 8.
As was the case with friction factor, there appears to be a smooth change in
V r/VL independently of flow pattern. Again, the smooth- and galvanized-pipe
results are qualitatively similar with the velocity ratio somewhat larger for
the galvanized pipe than for the smooth pipe, other conditions being the same,
except near the bubble-flow regime.
Comparison of Figs. 5 and 8 shows that there is no correlation be
tween f/fL and VC/VL.
The assumption that VC/VL - 1 in the bubble-flow regime appears
to be substantially supported by the Chisholm and Laird data.
B. Experiments on Bubble-Size Distribution
1. Apparatus
The apparatus used to measure bubble size in the 2-1/2-in. pipe is
shown in Fig. 9. The mixture of air and water is generated as described in
Section III. After an appropriate length of run, the mixture passes through
the sampler where it is split into two parts. The smaller part passes through
a narrow, transparent test section that permits photographic determination
of bubble size. The remainder of the flO'i' is turned through a bypass line and
rej oins the sampled material immediately behind the test section. The pressure
in the transparent section is measured by a manometer in comparison Idth the
pressure just upstream from the sampler.
The pipe-test length consisted of four 5-ft sections of 2-1/2-in.
pipe. These lengths were aligned by using pre drilled aligning holes in the
flanges and snug-fitting aligning pins. The sampler could be installed at
either end of the 20-ft test run or at the quarter points. The details <4 the sampler construction and the techniques used in photographing, sampling,
measuring, and counting the bubbles are given in Reference (7).
2l
The same sampler, with ilppropriate 4-in. fittings, was used for the
4-in. pipe. In this pipe, hOl'19Ver, the vertical height of the sampled section
consisted only of a half 9iameter of pipe instead of a "hole diameter as in
the 2-1/2-in. pipe. Hence, a photograph of a sample would show the bubbles
in the upper half or 'the lower half of the pipe, depending upon which way
the sampler was installed. This ~eant that a nonuniform bubble distribution
would cause errors in the size distribution. For this reason the sampler was
installed on the 4-in. pipe at the entrance to the first 5-ft section only
'''here the separating process would have the least time to occur. For this
report, the samples were taken in the upper half of the 4-in. pipe only.
2. Data and Discussion of Results
In the 2-1/2-in. pipe, sampler
ends and at the middle of the test pipe.
data were actually obtained at both
As fo r the pressure-drop data dis-
cussed in Section III, only small air-water ratios were used (e::; 1.5) so
that the flow ,/as in the lower part of the bubble regime.
A number of preliminary experiments, not detailed herein, showed
that within the limits of experimental error, the bubble-size distribution
at the various positions of the sampler were reducible to each other using
the perfect gas laws. Hence, all bubble-size data were eventually reduced
to atmospheric conditions using the perfect gas laws. Most careful attention
was given to counting the bubbles and measuring sizes for the runs with the'
sampler at the middle of the test pipe. The data for these runs are given
in Table IV (in Appendix A) with the designation M following the run num
ber. All of the 4-in.-pipe data are also included in Table I V, the letter
A denoting that the runs were taken with the sampler at the beginning of the
test length as already noted. Typicaldata from Table IV are plotted on log
arithmic probability paper in Fig. 10(a).
Dimensional analysis indicated that the bubble-size data might be
correlated by introduction of a dimensionless bubble diameter d' given by
the modified Weber number,
(25)
Here d is the bubble diameter measured at atmospheric conditions and (]'
is the surface tension of the water. Computed values of d' are given in
Table IV and plotted in Fig. lO eb) on semilogarithmic paper for both the 2-
1/2-in. and 4-in. pipe. Although the data scatter somewhat, a mean line can
be drawn through the data; the equation of this line is:
22
d' = 7.50 - 1.h9 tn (per cent larger) (26)
For the 50 per cent size, whi ch may be called the geometric mean
size and labeled d50 ' the straight line of Fig. lOeb) gives d' = 1.67.
Using this value (together with the numerical values p = 1.9h and cr ~ 0.005
in the pound-foot-second system) in Eq. (2S) yields
Direct measurements of d50
for each run from graphs like those in Fig.
lOra) have been plotted in the form of Eq. (27) in Fig. 11 and compared with
the equation. The scaoter of data in Fig. lOeb) shows upas a deviation from
Eq. (27) in Fig. 11, but there is a very definite trend for mean bubble size
to increase with the square rootof the diameter of the pipe and inversely as
the liquid flow rate per unit area. This means that as the slug-flow r egime
in Fig. 1 is approached, the bubbles increase in size, and vice versa.
The data in Table IV (in Appendix A) as typified by Fig. 10 (a),
indicate that there is little variation in size distribution. The bubble
diameter exceeded by only 5 per cent of the bubbles, dS
' has also been
plotted on Fig. 11 as a function of .j D/V ?' It is seen that these large , bubbles have approximately three times the d~ameter of the mean bubbles. The
range of variablesin the present investigation has been quite small, however.
A wider range might bring ne';- factors to light.
C. Velocity Traverses in Bubble Flow'
1. Apparatus and Data
Both the 2-1/2-in. and h-in. test pipes used in the pressure-drop
experiments of Section III were ~ovided with velocity-probe stations at or
near the positions of the downstream pressure taps shown in Fig. 2. A Pitot
cylinder was used as a measuring probe for the total head. Each station con
sisted of two pairs of holes drilled in the pipe wall, the holes of each pair
beingat opposite extremities of a vertical or horizontal diameter. The Pitot
cylinder passed through both holes of a pair; in the h-in. pipe the cylinder
could be traversed across the pipe from wall to wall, but in the 2-1/2-in.
pipe, it was possible to traverse only from the wall to the center. In the
2-1/2-in. pipe a 0.075-in. OD cylinder was used, while in the 4-in. pipe.
the cylinder was 0.125-in. 00. Traversing was by screw thread, the least
reading in the smaller pipe being 1/80 in. and in the larger pipe, 1/96 inch.
The stagnation hole in each cylinder was a No. 76 drill hole (0.020 in.).
Static head was measured at the pipe wall opposite the Pitot cylin
der by manifolding the two probe holes not in use for the cylinder. Static
head and total head were compared in a mercury U-tube manometer to read ve
locity head. Before and after each traverse, the static-head readings were
compared with the static head at an upstream ring of taps, with the Pi tot
cylinder both in and out of the flow. The information so obtained was used
to correct the measured velocity head to the value it would have had without
the blockage caused by the presence of the Pitot cylinder.
A few velocity-head measurements were made for small air-water flow
ratios using this apparatus. Velocity head was determined in exactly the same
manner for both pure water flows and air- water mixture flows. In the mixture
flows, no bubbles entered the stagnation hole of the Pitot cylinder because
of the curved streamline pattern around the cylinder. Occasionally, bubbles
entered one of the manifolded static lines (transparent tubing was used for
these lines so that bubbles could be observed); when this occurred, the lines
were flushed before proceeding with the experimental work. Actually, the ve
loci ty head of the water phase of the misture was measured by the process de
scribed, but it was already known from the analysis of Part A preceding, that
the entrained bubbles moved at approximately the same mean speed as the water.
Hence, it was assumed that the measured velocity headand the velocity derived
therefromwas applicable to the mixture asa whole. Integrationof the velocity
profiles and comparison with the measured input of air and water, as shown
subsequently in Table VII, indicated that there was some error in this assump
tion; however, the error is probably unimportant insofar as the purposes to
which the profiles will be put are concerned.
Four different sets of velocity data were obtained, two sets in :the
4-in. pipe and two in the 2-1/2-in. pipe. Table V lists the general flow
properties associated with each of the runs . In every case, each diametral
or radial traverse in the air-water mixture was preceded by a similar trav
erse in pure water. The Pitot cylinderwas not withdrawn between these pairs
of traverses. This procedure permitted the zero of the tube in the mixture
24
flowto be well established, as will be demonstrated below. Velocity profilE!!
for the air-water mixture flows are plotted in Fig. 12. In the figure, y i.
the distance from the pipe wall, and I' is the radius of the pipe.
TABLE V
FLOW PROPERTIES FOR VELOCITY TRAVERSES
Air-Water ~jJ[ture
Pipe Dia. Flow Rate Measured Flow Rate Gr/A
Run in inches Water Only Water Air* Total Ib/sec C1
(nominal) cfs cfs cfs cfs ft2 x 103 lin }.
I 4.0 1.861 1.740 0.198 1.938 0.28 0.229 0.114 1.29
II 4.0 1.850 1. 730 0 .256 1.986 0.39 0.320 0.148 1.3" III 2.5 0.805 0.679 Q.128 0.807 0.52 0.407 0.189 1.3l
IV 2.5 0.860 0.806 0.335 1.141 1.09 0.720 0.415 1.20
*Converted to probe- station conditions using the perfect gas law.
2. Analysis and Discussion of Results
It is immediately apparent from Fig. 12 that the velocity distri
bution is materially affected by the presence of the air bubbles, particularly
near the top of the pipe. (Similar profiles for the pure-water runs show a
bout the same scatter as is shown between the horizontal traverse points of
Fig. 12.) In particular, at the higher air concentrations, the velocity dis
tribution in the vertical is asymmetrical, while that in the horizontal is as
symmetrical as in the case of pure water. The velocity profiles in the upper
vertical traverse" are generally not so flat as those in the other traverses;
-this would indicate that the pipe with air-water mixture flow is effectively
rougher near the top than el sewhere. It is to be expected that there will
also be lack of symmetry of the wall-shear str~ss distribution and. per haps,
secondary currents in the mixture.
In order to investigate the wall region in air-water mixture flows
more completely, the data leading to Fig. 12 were replotted using a logarith
mic scale for the wall distance. The replotted data are shown in Fig. 14.
while corresponding data for pure-water flO>IS are shown in Fig. 13.
The zero of the wall- distance scale in these figuresis not a meas
ured value, but lies somewhere within the rough-pipe surface. The origin
25
was determined from the pure->later traverses for each run by assuming that
there exists a universal law of the wall of the form [10].
where V is the local velocity,
V* = Jr:ft is the shear velocity,
TO is the wall-shear stress~
p is the fluid density,
y is the distance from the wall,
a is a universal constant given as 5.6 by Clauser [10] , and
b is a lumped constant including the roughness effect of the
walls and depending on the units used for other terms.
(28)
The law, Eq. (28), was assumed to hold from very near the wall toward the
center as far as a value of y of 15 or 20 per cent of the pipe radius. By
uniformly increasing or decreasing the values of y readon the Pitot-cylin
der scale until the points approached as closely as possible to satisfying
Eq. (28), the pure-water traverses shown in Fig. 13were obtained. The effec
ti ve diameter of the 4-in. pipe between virtual zeros turned out to be con
siderably greater than the measured diameter between roughness tips, as shown
in Table VI.
Vertical, in.
Horizontal, in. ,
TABLE VI
DIAMETER OF 4-INCH PIPE
Measured by Distance Between Calipers Run I
4;036 4.077
4.000 4.097
Virtual Origins
Run II
4.066
4.076
Also obtainable from Fig. 13 are values of the shear velocity, V *'
For pure- water flows , the shear velocity is related to the friction factor,
f, by Eq. (13a) which may be written
26
(29)
The values of fL obtained from V * in Fig. 13 using the value $.6 for a
in Eq. (28) are shawn in Table VII and compared with the corresponding values
from pressure-drop measurements obtained further upstream. The friction fac
tors by the velocity traverses appear to be generally gre ater than those by
pressure drop. The difference may be real because of the different measuring
locations .
Having determined the virtual origin for each traverse from the
pure-water data, the same origin was used for the corresponiing air-water
mixture traverse, (which, as already noted, was taken immediately after the
pure-water traverse without withdrawing the tube). The air-water mixture
graphs of Fig. 14 were obtained from Fig. 12 by this means. It may be ob
served that there is a strong tendency for the air-water mixture profiles to
follow Eq. (28) near the wall, also, albeit with different values of the co
efficients aV* and b than for pure water.
TABLE VII
RESULTS OF VELOCITY TRAVERSES
Integrated Flow Rates Friction Factor f/fL
Air-Water Mixture 1-later % Diff.
Water Only, fL Air-Water Mixture
Only from Total %Dift.
Run cfs Measured cfs from Pressure Vel. Pressure Vel.
Measured Drop* Trav~ Drop** Trav ..
I 1.866 +0.3 1.910 -1.4 0.0144 0.0178 1.00 1.046
II 1.792 -3.1 1.896 -4.$ 0.0144 0.01$1 1.01 1.026
III 0.790 - 1.9 0.772 -4.3 0.0186 0.0178 1.02 1.013
IV 0.840 -2 .3 1.0$4 -7.6 0.0178 0.0198 1.06 1.009
*Computed from measured pressure drop between the two upstream stations shown in Fig. 2.
**Average values from Fig. 3.
That the air-water mixture traverses follo>1 the law of Eq. (28) is
not surprising if it can be assumed that a bubbly mixture at small air-water
ratios is locally homogeneous in the wall region of each traverse. Assuming
that Eq. (28) applies, a value for V* can be determined from the slopes of
27
the straight lines for each traverse if a is assumed constant (at, say,
5.6). It is readily seen from Fig. 14 that aV*, and hence the wall-shear
stress, is generally greatest near the top of the pipe and least near the
bottom. This result appears somewhat anomalous; it might, at first, be ex
pected that the presence of air would tend to reduce the shear stress as in
the central parts of the pipe. To explain the wall-shear-stress distribution
around the periphery of the pipe, it may be conjectured that there is a sec
ondary current flowing from the center toward the top, down around the sides
to the bot tom, and thence up to the center. Such a current would tend to
thin the viscous region near the top and thicken it near the bottom of the
pipe, thus increasing the effective roughness near the top and decreasing it
near the bottom. Increased effe cti ve roughness .near the top of the pipe is
in accord with the velocity profiles of Fig. 12.
The ratio f/fL may be determined from Figs. 13 and 14. If in pure
liquid, the measured slopes of the straight lines representing the velocity
traverses in Fig. 13 are designated as aV * = SL' while in the mixture, the
average of the measured slopes is designated as aV * = SM' the following
relations apply:
2 - 2
T = pV = peS/a) 0 *
(0)
a 2 T a (SJa) o,L
fL = = 2 2
(31)
PLVe Ve
a T o,M 8(~/a)2
f = = (32) 2
V 2 2
pV (1 + lin) L
f/fL = [(SMfSL)(ve/VL)(l/(l + 1/n))]2 (33)
Friction-factor ratios have been computed from the data in Figs. 13 and 14
using Eq. (3), and the results are shown in Table VII; comparison is made with
the pres sure-drop results from Fig • 3 • Numerically, the comparison is probably
28
as close as can be expected; the trends of the data appear to be in opposite
directions, however.
In Fig. 15 the velocity traverses for Rilll IV of Fig. 14 have bee::
replotted in the more conventional form
V/V = a log yV Iv + b ' * *' (34)
and compared with an accepted formula for smooth bOillldary-layer flow. Also
shminis one of the pure-water traverses for Rilll IV from Fig. 13. It appears
that the bottom of this galvanized pipe behaves like a smooth surface while
the top acts like a pipe of increased roughness in the presence of air bubbles.
The conjecture on secondary currents offered earlier would seem to explain the
relative locations of these curves.
The large discrepancy in integrated air-water flow rates for Rilll IV
as shown in Table VII could be reduced by taking V rJVL
> 1 in the integra
tion. For Vr/VL = 1.22, for example (which appears reasonable from Fig.
8(b», the percentage error would be the same as for the pure-water flow.
V. SUMMARY AND CONCLUSIONS
In bubbly mixtures flowing in horizontal pipes, it has been shown
that pressure drop may be calculated by using the conventional friction fac
tor for flow of a homogeneous fluid in the pressure-drop equation, Eq. (17).
2D
no e(no - ~) - tn -
~
(17)
In order to obtain the . friction factor f, the Reynolds number and pipe
roughness must be known. As a rough approximation,
f = f L
where fL is the friction factor for the liquid component of the mixture
flowing alone in the given pipe. The present experiments show that a some
what better estimate for f can be obtained using the empirical relation given
by Eq. (23)
29
f/fL = 1 + 0.035 J9 (23)
Equation (23) indicates that the friction factor is actually increased by the
presence of bubbles . The increase is possibly attributable to an increase in
viscosity.
Whether or not bubble flow occurs depends on the gas-liquid flow
~atio and on ·the liquid or gas flow rate per unit area. Baker ' s diagram re
;>roduced in Fig. 1 might be used for estimating the limits of the bubble-flow
,..,gime. Fromthe diagram it appears that bubbleflow will occur only for water
!'low rates greater than about 500 Ib/ft2 - sec. This limit has been verified
roughlyin the present experiments.
Ole flow always existed for e < 6
!or e as great as 45.
The present experiments
and essentially bubble
showed that bub
flow might exist
Outside the bubble regime, Eq . (17) is still useful for calculating
~ssure drop. However f < fV generally. Figures 5 and 6 give some in
formation regarding the ratio f/fL for these flows, but more study is re
quired before a gerieral rule for finding f can be given. It appears that
flow pattern, in itself, is of little importance in predicting f.
In the bubble regime, the bubbles move at approximately the same
speed, or very slightly faster than the mean liquid velocity. In other re
gimes, the gas component moves faster than the liquid, the mean velocity ratios ·
reacing a maximum in annular flow. Again, flow patterr: outside the bubble
regime has little direct influence on relative liquid and gas velocity.
The velocity profiles in bubble flowin a horizontal pipe are quite
similar to those in flow of a single fluid. However, the profiles are not
symmetrical about the pipe axis; the profiles seem to indicate a secondary
current upward in the center of the pipe and dowm,ard around the walis. The
upper partof the pipe, where the bubbles are more concentrated, is effiOctive
ly rougher than the bottom.
Bubble size in bubble flow is proportional to the square root of the
pipe size and inversely to the liquid flow rate per unit area. The bubbles
become larger. as the flow goes from the bubble- flow pattern toward the slug
flow pattern.
30
-. As a tentative conclusion, it appears that disturbances of the flo>',
such as produced by bends, do not alter the conclusions already given above;
the pressuredrop is given by Eq. (17) on both .sidesof the bend, proper values
of the parameter n being used on each side. The effect of disturbances of
other types is still under study.
31
LIS T 0 F RE F E RE N C E S
[1] Santalo, M. A. "Two-Phase FlO1<." Applied l'rechanics Reviews, Vol. 10, pp. 523-526. 1958.
[2] Baker, O. "Simultaneous Flm< of Oil and Gas." Oil and Gas Journal, Vol. 53, No. 12, pp. 185-195. July 26, 1954-. - --
[3] Boelter, L. M. K., Martinelli, R. C., Morris, E. H., Taylor, T. H. M., and Thomsen, E. G. "Isothermal Pressure Drop for Two-Phase, Two-Component Flowin a Horizontal Pipe." Transactions of the AmericanSociety of Mechanical Engineers, Vol. 66, pp. 139-151." 1944. --
[4] Lockhart, R. W., Martinelli , R. C., and Putman, J. A. "Two-Phase, TwoComponent Flow in the Viscous Region." Transactions of the American Institute of Chemical Engineers, Vol. 4, pp. 681- 705:' 1946. -
[5] Lockhart, R. W., and Martinelli, R. C. "Proposed Correlation of Data for Isothermal Two- Phase , Two-Component Flow." Chemical Engineering Progress , Vol. 45, pp. 39-48. 1949.
[6] Chisholm, D., and Laird, A. D. K. "Two-Phase Flow in Rough Tubes." Transactions of the American Society of Mechanical Engineers, Vol. 80, No. 2; pp. 276-286. FebruarY1958.
[7] Silberman, Edward, and Ross, James A. Generation of Air- Water Mixtures in Closed Conduits.5 Aspiration. University of 11innesota, St. Anthony Falls Hydraulic Laboratorv Pro:iect Report No. 43, December 1954. (Available only on lliiversity inter-library loan. )
[8] Schlicting, H. Boundary Layer Theory. New York : McGraw-Hill Book Com-pany, 1955. .
[9] Weining, F. S. "Some Propertiesof Foam and the Possible Use of Foam for Model TestsEspecially in HypersonicRange." Proceedings, Third Mid-WesternConference on Fluid Mechanics, lliiversity of MiriIiesota, pp. 515-527. 1953.-
[10] Clauser, F. H. "The Turbulent Boundary Layer." Advances in A~lied Mechanics, IV, New York : Academic Press, pp. 1-51-. 19.
FIGURES -------(1 through IS)
10
"'--, 0 ., on ..... 10 ,Q
.s
.<
~ " '" ,.. -·u 0
~ on 0
'" .,
10 "0 ., -= '6 0 ::;
., 10 "' 10
., 10
Modified
Disperse
----Annular
Slug
o. As Dependent on Gc;/A~
· 2 -I 10 10
Gas-Liquid Flow Ratio by
-----/
/
Stratified
Weight
.-' .-'
10
Nos:: b 104 ,---------~----------,_---------,-----------,----------,
Q) .Area Covered by Data of Table I ~ ~ ,Q
.E
0 ., -<t
~ ::J -., a.
~
~ ;0 .2 lJ.. -.c: .S?' ., ~ "0 ·S 0-:J
, 10
10
I 10"
Bubble Flow Pattern Observations, Tobie II
Bubble Flow Bubbles on Top of Pipe only Bubble.~ with some slugs Slug Flow
~~~ " Disperse "
Stratified N~ '-', ..............
't-_
Gas - Liquid Flow Ratio by Weight oj ~
Fig . 1 - Flow-Pottern Reg ions
10
35
(Orifice
____________ -,1 4 11 I .t'~ .... -<
(" I I I V'l"'----.., Water Control Valve
Inlet Water
Pump Water Flow Manometer
4" Pipe
6"
2112" Aspirator j-2112" Galvanized Pipe Test SectionJi VieWingSectio'r-B~5 7/8" I B~5 5/B"~ I 2 1/2" Pipe
Calibrated Gage ~Air Control Valves
Air Flow '=Air Inlet Inclined Manometer
Open Manometer
Pressure Drop Manometer
Fig. 2 - Schemat ic Drawing of First Experimental Apparatus
Bock Pressure Valve
w
'"
37
:~ I II~t~fTItftttI1Irl 111111111 0.9
1.1
1.0
.9 _ 1.2
Ii. 5 t.I -o ~ c o
10
~ 1.3 ~
lJ..
1.2
1.1
1.0
<D GL/A" 1028 <D <D o GL/A" 1152 <D fD t:, GL/A-123?
!Of'> ",0 ~ )"
" IT IL.>~
<D ~
<D GL/A-684 <D 0 GL/A" 700-825
0 IJ O 0 ~
<D GL/A"59I 0 GL/A"594
Present Data Table I, Table II <D[ 2 112 inch Galvanized Pipe 0 <D 0 nP,JJ 4 inch Galvanized Pipe t:,
Chisholm and Laird Data Table m [ 0 I inch Galvanized Pipe 0
-, 10 -,
10 -, 10
Modified Gas - Liquid Flow Ratio a/>'>f
Fig. 3 - Fricti on Factor at Small Air- Water Flow Ratios
, --= ~
Venturi Meter 4" p. - Ipe Water Pump
Air FlOW-""'" .,...... ----Open
h Manometer Manometer
'-./ Water Flow Manometer
-'--' .1.. ';I<' *" I ,---remDerature
orifice~~ )Jressure
Initial presOre Fi'O Pressure Initial -rest Section
2112"Pipe 2 1/2"Pipe 12"'011
Air-Wate~ Mixer
Back Pressure Valve
........-.,...... Pressure Drop
Manometer
( ) Flnol
Pressure
12 1-0" Second -rest Section
I-- Pressure Drop Manometer
'-....- -.-1
COpen '--' Manometer
Initial Pressure
Fig. 4 - Schematic Drawing of Second Experim ental Apparatus
Viewing Section
~~
eetlon
Wate Inle1
Air Inlet
r
\.0.> (»
100 r------,----------"T-----,-----,
a. Smooth Pipe Disperse
c:
..< 10 f------+------+ 00 0.. ............ 0 __ .0-
~ '" (!)
Bubble
Stratified
0 .1 L..mzID-DiliU?L._-.L ____ L ___ ~ .0001 .001 .01 .1
Modified Gas-Liquid Flow Ratia a/X';
100 r------,------,-------,r------,
N ~ -I., ~ .£l
c:
>-u o ~ on o
(!)
"0 .!!! -"0 o
::;;
b. Galvanized Pipe
Disperse
10 '------'-----+-If--
0.1 L~2:2;~~!....f.....-:.H~--,J.,__-----L----.0001 .001 .01 .1
Modified Gas-Liquid Flow Ratio a/X';
Fig. 5 - Chisholm and Laird Friction-Foctor Doto
39
uO
..., "--0" -0 cr: ~
0 -0 0 u.. c: 0 -0 ~ u..
:~Ftmt¥t;~o rO t\+·, ~om~; 1 .09
1.3 1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.5
Oil
0.3
o 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.2 2.0 2.2 2.4 2.6 2.8 3.0
I 10
Compressibility Parameter, a
Present Data Table I Table II 2 112 inch Galvanized Pipe 0 CD 4 inch Galvanized Pipe 6.
Chisholm and Laird Data Table m. I inch Smooth Pipe I inch Galvanized Pipe
lSI o
--
-,y ~~ip"-h~ <J DDr u[jJ
IN ~~ l'!: ~1SI1Sk, b.a>1
102 10J 10"
Compressibility Parameter, e 10' 10'
Fig. 6 - Friction-Foctor Ratio as a Function of Compressibility Parameter
41
100r---r---r-'-I"---'---'--""--"--"-'-r.~--'---'--'" I I I I ,
Chisholm and Laird Data [6]
o l>
Smooth Tube Galvanize Tube
1/ v v O~
/ t Ji'.: .c:
~tQt 1.0 1----t---t-+-+-v7f7\-, ux.M :=L--+--t-t-+---+---+-+-++---+---+--+--t--1
/,~ &'<J->
Ol~~ 0.1 1.0
o
Fig. 7 - Gas-liquid Fraction
100 1000
42
N
f o CI>
.!t!
.0
c:
VI o (!)
"0 CI> -"0 o
:::!:
N
~ /, CI>
.!t! a c:
.<
~ " (!)
'" :!:: 0 0
~ VI 0 (!)
"0 CI> -= "0 0
:::!:
100~---------.-----------.-----------,-----------,
a. Smooth Pipe Disperse
10~----------~--------~~--
joool
100
10
.1.0001
Bubble
.001 .01
Modified Gas-Liquid Flow Ratio
b. Galvanized Pipe
Disperse
Bubble
o
Stratified
.1
a/)..y
Stratified
.001 .01 .I
Modified Gas -Liquid Flow Ratio a/A",
Fig. 8 - Velocity Ratio from Chisholm and laird Dato
Air F tow Inclined Manometer=::
( C
2 112" 'Pipe
Water Flow Manometer
Air
~ /"_--!!-:.
2.28"
Section A-A
Pressure Drop Manometer
Fig. 9 - Sampling Apparatus for Bubble Size
Transparent Test Section A
:;l
Pressure Control Valve
Discharge
e;
44
"0 II
1.0 ~
!! 9 '" E 8 o o 7
'" ::a 6 .0
0.01
01
%~J V
5
1.0 ~
'" 2.0 '" ~ .3 30 4.0 - 5.0 <:
'" 60 u ~
7.0
'" 8.0 a. 9.0
95
99
l6 ~ ~ ,.{ til fJl
f 11) A
IW fI --r+-
/- ... 01
2 112" Galvanized Pipe >-lJ"J d'J 7 o Test 2M
11/ K o Test 9M 4 II Galvanized Pipe
1/ Jf r; Test 3A
Jo/' 999
99.990.0 I 002 0.04 0.06 0.1 0.2 .0.4 0.6 Oll 1.0
Bubble Diameter in inches (Reduced to Atmospheric Pressure)
a. Typicol Bubble Size Distribution
I O~
I I 7.5 -d' ~per Cent Larger = exp' lA9
o 0 n "'" d' ~ ~8Q h ~ I?>
~l; rl
rfJ;;:. ~ 0 ~'O ~
::J 5 m '" 4 '" '" 3 <: o 2 '" <:
'"
f-- 0
f-- r;
2r Pipe
~ 9!
4" Pipe nO
~ E o .0
Col 0.5 5
Per Cent Lorger 10
b. Dimensionless Size Distribution
Fig. 10 - Bubbl e-Size Distribution
5.0 100
.1
.0
.0
.0
.0
~ .0 .c .., c::
0
9
8
7
6
5
.04 c:: .. Q) -Q)
E .0 o
3
o .!! ..0 ..0
" <D .02
.0 1
1//
.01
//
//
/ /
/
45
o
0 ltt V ~/ (l V
p5~ V ~/ V '"' /'- d5 '" 3./0
9) VL
1/ /
[7 dso ~./5/VL ~
~ 0
~ ~-
0
<6
0 I " 2 ~2 Galvanized Pipe
" 4" Galvanized Pipe
.02 .03 .04 .05 .06 .07 .oS .09.l0
Fig. 11 - Characteristic Bubble Size
46
1.0.
0..9
0.8
0.7
0.6 ~
~ 0.5
Q) <.> c: c - 0..4 '" i:5
r--
~ 0..3
'" '" 0.2 Q)
c: 0 ·in c: Q) 0..1 E
0
0. h-rc
6 6 6 6
10. 12 14
I-
o Top ~ 0- Horizontal -0 Horizontal Q Boitom -e
\;f
-0.
~ ~ :I
< ~
~ 6-0 ~
;J ~
A .n ~
Q;J Run I W ~ i"" 6 ~ r.y
6 -0 fit:
RunlI 6 -<: 'V
6 !:fQ 0 , 4 Inch Pipe
6 Qf9'Q ~-uJj -n;
16 18 20. 22 24 26 28 Velocity V in fps
Fig. 12 - Veloc ity Profil e s in Bubbly Mixtu res
10
-- 0.9
0.8
Q.7
0.6
0.5
0..4
0.3
0..2
0..1
0. 30.
~
"->-
.0
0 .9
0 .8 6 Top 0- Horizontal
-0 Horizontal 9 Bottom
--+---4---4-~rMh~--{f-!
0 .7
.6 ~ 0 ~
Q) u c: 0 -'" is
~
'" '"
0 .5
0 .4
0 .3
-0 v (5)-
I
~ 0.2 f---+--+--I----+--+-----jr'O-~c_+_-_d_.._-+-_I .~ -{~-,v-
E Run m 6 0 is 0.1 1----1----1-----j---,4+---.o----ol--{<2>-~-'4---+---+---+~-I
d~ f90-~9 0- (f;)-00 !:-----,1"'2-----'~14,:u'-..:'16~--=----,1-!:-8 ----,2:00:--::'2::-2 -~,-n1---+----j---
0B-
b. 2 112 Inch Pipe
1.0
09
0.8
0.7
0.6
0.5
0.4
0.3
0.2
-'- ~~~~6i CAl--0--' -q+c;8---+~~--~---:-Ir::? ~~~9'i">+=-+---+-+--+---+---I 0.1
~2~O~~2~2~~2L4--2L6--2L8----~L----3L2----3L4--~~L----3L8--~400
Velocity V in fps
Fig. 12 (conc luded) - Velocity Profiles in Bubbly Mixtures
47
~
" >,
V> a. - I .£0
>1
>-'w o Q)
>
, l
r
3
j
~
3 -D.
, /
~ ,flY
h> ~
~ ( kP I'-AV. ° 5.63 a. Run!
L '" .£') 1-
0..0.1 0..0.2 0..0.4 o..o.S 0..1 0..2 0..4 0..6 0.8 La
2 :6
2 V> .e-2 c:::
> 2
>-u o Q)
>
4
;>
a
3
- ./
'~ ~ -' v ,
Wall Distance y in inches
/. 91
~ W
I9i L ~ V
~ I-AV.o6.40.
c. Runm
1-0..0.1 0..0.2 0..0.4 0..06 0..1 0..2 0..4 o..S o.B 1.0
Wall Distance y in inche.s
19
lv.lo509~br" 18
V> a. - I
l ~~ r ~ .£0
>1 3 l>;l
- -'~ >-u .Q
~
) .c~ b. RunlI
t Iff 3 -o.cY - AV.o5.11
'// 1 1 1-0..0.1 0..0.2 0..04 0..06 0..1 0..2 0..4 0..6 0..8 1.0.
Wall Distance y in inches
~S
~ ~4
£1f1t P" ~2
./, 1<1' i:9P ~o.
d-!J P 18
~ d RunN .J:.
16~ f" rAV.o7.22
~rcr 1 - -- - 0..4 0..6 0..8 1.0. 0..0.1 0..02 0..0.40..06 0..1 0..2
Wall Distance y in inches
Fig. 13 - Semi logorithm;c Pl otting of Velocity Profiles in Pure-Water Flow,
ceo
22 ::Jo-r<:
"
~ 1f 6 /
)
~ ¢--/ ~ l
f5' ¢>-/
2.0
UI .e- 18
~ ...d 5I--AV.=5.42
f&' ~jy > 16
~ 14 '0 o ~ 12
.0
,
6
;g:2 [6'-A'4=56 /
a, Run I /
j. ~AV. =7.25
" / 111
S.ol .0 . .02 .0.04 0.06 .oj .0.2 .0.4 .0.6 DB 1..0
26
24
UI .e 22 .5 > 2.0
~ 18 u o 0; 16 > ""
I~ ~ I u ",,/
14
Wall Distance y in inches
~ AV.= 5.57~ / W'
AV.=6.2~ ~ ~ ~:6~
~ 1.,/" j,J
~AV.=8 . .o5 c. Runm
" <5
I~~ .0 . .02 .0.04 .0.06 .0.1 .0.2 .0.4 .o.6.oB 10
Wall Distance y in inches
22
2.0 UI a. ..... 18 .5 > 16
.~ 14 g ~ 12
1.0
801
32
3D UI .e- 28 .5 > 26
~ 24 u o
- 2 ~
9~ / I
I~ . .ol
.~ />"
~,;r 6 /
/
AV.=4.62\.- ~~ \?' / 6 /
,cr~ /
6" 1-/
.,... AV.= 5.1.0 ~9/ b. Runll
J.
...c ~V.=7.37 />
.-I I
u .0 . .02 .0 . .04 .0.015.0.1 .0.2 .0.4 .o.6.oB L.o
Wall Distance y in inches
d. Runlll
.0 . .02 .0 . .04 .0.015.0.1 .0.2 .0.4 .o.6.oB 1..0
Wall Distance y in inches
Fig . 14 - Scmilogarithmic Pl o tt ing of Veloci ty Profil es in Air- Water Mixture Fl ows ~
30
28
24
v V* 20
16 Q
12
8 10 20
~Q y -Q Q I .......... .-
..... --b ~ Q--~.> QQ~~"'9
\0 ~ -f.OI
Q -~ :7 ~.<; ,6~?~ ~
-@ 0-
o .....5l.: ~1'\ge 'I.
.--i.- S«,oo\'(I lc@~
... -U'
~ 'SJ'
~ f
W -0
~ ::;;-,
-r:J'6'-
. ..,.-v "V
l ... rl&fO" ru -::: o Mixture
o Water only ...c:
~ .--l 1 I I 40 60 80 100 200 400 600 1000
f::!;.
2000 4000
v
Fig. 15 - Velocity Profiles Compared Vlith Universal law of the Wall for Smooth Pipe
• k
10000
'8
APPENDIX A --------(Tables I through IV)
z 0 t= « ...J ...J « .... <J> ~
.... <J> a: u.
a: f(
<J> Z
~ 0 t=
'" ~ ...J ., ::>
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0
1i
~ « 0
a. 0 a: 0 , '" a: ::> <J> <J>
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;,'
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~ 000000000 000000
'1"i9.9.9."!."'l "''''''"-''>0-" '<1 ........ 40.., ~":l"';"i ~~~ \1\0""'''''-''
!~BE~S~ ;i"~~~' ';~~~~ii <."§ · i~;::i~i~~~ i~~~~~ ~~ ..,..,~~ ............. 33~ "~" or' .... ~ ......... ~~p.~ .... "'0 ........ "' ..... ..,00000'" NO~ ...... '" "I"' ...... ..< ":":;i'i""l'" &~,,~;;~ ~"~~Ii" <~'~ ~~" ,; i~M'~~lR~ :8:8 ~ 5:tlHX ~~M" ~ ~~~ 3 ~3~ ... ~ a",::t ~~ 0 ............ ~ .... ::l\::l\ 1:;:::l .... ~:;ISI.
~'~ ~~~g~~ !~S"'~~ .~~ ~ , ~'P.~oo",~ U~!'i ~m~m~ ~E;~s§ ~"I'1~~q • . . . . . . . .. ~ .. ~~ . 0000000 000000000000 0000 000 .- 000000000 000000
,; 'i""c~" ~~~~~q ~~~~~o a~H d < .... *~IX~ ..... ~ 3!;~~80 . ,;,,:!i~~o ""0 ~ ~o~d~~~~o 000000 00000000000 DODo 00 ~ 00000
mm~ ~::t::t~~~ ::m5::t::t~ °ao~ m ;. ~~~~S~::Sll;'o ~S!~~::I0 ........ ~S~ ............... !L9 • ~ ~~:::I:::I~:::IS :::I::1::l::l~~
1 !
~I'!.~I'!.~~ ~~~~~~~ .... .., ........ .., ..........................
~~~~ ~~~ :l ~~~~~~~~~ S;7kS;~~~ ~S;~~~~ ~~~~~~ ........................ ........................................ "' .... "' ............ """ ,; ~
1XG:G(~:5I,t:\!R ~!i:a$$;3 :S:8!O:!8.t1i!. l=!~;::!~ ~j!:!.1:: .; l!?~SiCI:t,:all;';S t":8:;~~S:
~~I::§~i '.eS J::;S$'5: ~q~§~§ ~"'I<;~ ~""~ ~~~~m ~~~~~ ~~'lU • :=! .qq ~":":":"!": 000000 .~~":": "!~"'l • 000000 000000 00000 0000 0000000 0000000000
~:SG\~::i::: lG\j!:!.i~s ~~§~;;:$ ~~m ~&~§ ~~~~~~~ ~~ ... ~G: g:<,e~t§ .":-,!'; . ": . ~q .~. ":'1~"':": ~-'!';": • 00000000 .... 00'" "<0"<000 ........ .-1 ... 0 0000 ....... 00000 ........... 0000000
j:~~3~:S ",,,,,,,,'h ~~~~~ ~~~I ~ ~
q~~~~~ 33~§~~: ~~~~~ ~~~~ I ~ ... ..< ...... ,.; ,.;,.;,.;,.;,.;,.; ............... .-1 .... .-I .............. """ .................... ................
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~ .~§" .• "~.:!~~ ~o:!~5£a ~~t"~;8 .... ~~~ 5;:;-~$.-;l:.l=! ~;;;O:::~';!1 S§2~~ ~00:S 00
~ ~~c:El~El .................... asaoa~l ass 9 sa .~~~ij~~ ... ~"'oooo ......
c:qc:qqq ~~~~~ o~~~~ 000000 000000 000000 00000 0000 0000000
~"'... ~'" ~~~~~~ ~~~~~~ ~~~~~~ .s 000000000000 000000
~ ~ ~~~~q~ ~~~~~~ ~~~~~~ Q" 000000 00 ... 000 000000
• "
~~~~~ ~~~~o 000000000
53
ii~ ~~~~lm go 00000000
~§ ~~~ ".~,,§ 00 0';0000' ';';
~~~ ~~I:i~ I ~I .... .................... g;fi §.§~un ...;..; ..;..;..; o,,; ....... ci
~~ mm~~ co 000 00000
"';"';
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< m~~~~o ~o 0 000 0000
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g.
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~~~ ... ~~~ q,,:~*! .... 000 000 ~
~~~~ 535 ............... t;~GtGt qc:qq
~~ . ~.~ ............. """
~~ .. ~ m aas 0000 000
! ~~~~~§ ~~m~ .~~~~~ m§§ §m ~m:m ~~~~~ g~~~ m~ ~~~ ~ .s ~!!~~~~ ~~~~~~ !~.~~~~ ~~~~~ ~~~~ ~~~;1~~~ a~~i3,g ,g,g,gi3~ 1 ~~~~~~ ~~~!R~:IS:IS ~~~~ :::I~:::l~~ ::::~:: ~"''''~''''''''' ~:::l~~?i ~?i?i?i:::l
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mfnmi§!~~ . ...... . ...... 00000000000000
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........................ "' ..........
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oo~~~q~~~~~s:~~ oo ....... ...................... -=>
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:!$~a~-a !~HHi~ 0000';"; 00000
;~m~a ~~~~§§ ... ........ gO 0';0000 ~~~~~~ ~~~~~
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~~~~~ ~~1'1~~ 0 ........... "'NNNN
g;$~~/j ~~;.~~ 0";0000';";0';
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gg~~::l ........ ..l ........
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q~~~~ ~~i!~~ Oooco ... N .... "''''
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~~~~ ~~~~ 00 ............... ...
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............. N.<J "' ... ..., ...........
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~~~I::!.I::!. j::!1::!.j::!1::!.1::!.j:!. j:!.j:!.j:!.j:!.j:!.j:!. ~j:!.j:!.;:!~ j:!j:!\:!,j:!j:!. j:!.j:!.j:!~ ¢¢¢J;! j::!j::!j::!j::!
~~~-';-'; "'. "!.~~"':~ .., ...... =-""'" ""''''''''''''' ~~..:;..:;'!. "!.":;"l~ ..:;"!.-';"!. -';'!.-';"!. 222" 8:~'li~' §~~:§~~ t ~~"~ 3&O!Oi~ .~ 0 ~ ~~$R l1g:~~ .., ......... .., ... ....
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~ • ~~~~~ ~~~~~~ ~~~q~~ ~o;~~~
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< -" j::!I!P;:! ~j!!.I:'je.
0 0 ,..; ......... ............
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s~ri5~~ ;"':3§g~~ . . . . .. • '1 •• .•. 0000000000000
~~o:o:~~ ~~q~~~q ...................................... *~~B~ E:~~iil@n 0000000000000
~~i~i:i ~a~;i~:§ 00000'; 0000000
mm~ 0000000
·
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'!.'!."t'!.'!.'!. ..:; ~'!."t"!."':"!.
S!€~!5 ~~~~!~~ ~~~!:;~~ ":":"':;;"l'1"!. N"'..::I"' ''' ~ ~~:::S~j!!.!li 2
~ 0
3 ~ .;
§ .;
• ~
~
· ~ • ~ 0
• ~ ~ ~
a~g~&:s «"'0.., ~"~~'l~ ." ." ... ..::1 -" .. ...
... ..; ~...;.s:j ~:Jeirria~!L,,-.. "' i
.. ... .... "" ... ::1 ::i::\~:RliS~ Nt • • • ! ~~~~~~ ~~~'4~rt": ! ~~5~~j g~IRl!:I!..::I"" 00000 ... "'..,"'''' ... ~~ ":"''''~~~2 ~ "
.; N. . , ,
~ " " 0 • " "-0' :;
...... fit .. Cen~ D1.oo ., in . "''''' .-
'" 00" " • O. 6l.CI, .. • O.\,CQ II lO' }
~ • L SI' , ','24.1
' .<nJ .... '-'S ,.~ " .6 l.?} 0.0:1:7 55.' .. " 0 . 0)) »., 2. 11 1 ,."" >5.7 ' .4' , .... ••• 4.0> 0 . 05) 4.' 4.'" ,."'" ' .1 S~1 ' .066 >.6 5. 14 G.on >~ 6. )) ' .m , .. 6." ,."" ,., 7.1,,7 ,.'" , .. '"
Ilso ~ 0.0255 1 • - 0 .470
Doll. • O .2~ ... ~ 0 .20) " 10, 3
~ • 1.1) . VI " 11 .2
,.m ~9.s ,." D,ma 1) .5 >~. ,."" 6J . 6 >.56 0 .0)0 45 .' >.9S ... " n .' 7.", ,.W "., 2 .7) , .... "., ,-" ,."'" U .' ,.~
,."'" .... ,.'" ,.066 .... 4.'" ,.om 6. '" 4.70 0. 078 S." S." 0. 0Il~ ,.'" S.47
'-"'" .. '" S." ,"" >.'" 624 ' .UR > ... 6." ,.'" 0. 61 .... ,.W 0.)) .. "" ' .U7 , .. '"
dSO
• 0.0268 r .. 0.620
V I. ' M~ • • ~o.ni" 10' ) 1 • 1.)75, f, . 18. 5
MU 8,. ) ' .m 0.011 ".S ,.'" 0.022 "J >." 0.028 .... 1 .45 0.0)4 "'., 1.14 0 .0}9 n .o .. '" '."" 29.S .. " ' .<>SO 2) . ) .. '" ,."" 17,0 2 .89 ,."" U. ' ) .16 0 .061 u., ) .117 O. 07J 10. 1 ) . 16 0 .0'1'8 .. " 4.oS ,."'" I." 4.'" ,."'" 6. '" 4. 6) O.09S S." 4." 0. 101 S.'" 5. 21 0.106 ) .15 S.SO '.ill '.S> S.'" ' .ill 1.26 6." ,-", 0. 6) 6.J1 ' .W , 6.65
~ . 0.0289 ( • • O.72S
...,. I'Wr c..nt ,-Da., 111 . "' .... '" OJl . Mlb. .. - 0 .218 .. 10· )
~ - lSS, ", - 25 . 0
'.m2 78 . B >.U ,.'" '"., >." 0 . 02$ 118 .8 •• n 0 , 0)1 29 . S ' .n 0 .0)7 " .. ) . ) 1 ,."-' "., ) . 81 a,Gsa .. " 4.4) ,."" 6.", 4." ,.'" S.'" ,." , ... MS 6.w ' .0!4 2. 57 6." ,."" >.n .. " 0.0117 '.M .. " 0.093 ,." .. '" ,.'" , .~ e. 81 ,~ 0 .11 .. ~ , .W , 9.95
~ • 0.0224 (. - 0.59)
TABLE III
BUBBLE - SIZED DATA
-. ""- ,- ..... DU., in . .....u 01 • . , 111. ..
"'r c..n~ "'~.
"' VI.· 0 . )05, .. - 0.l5'2 z UI·J Gr:/,- • OSl O, .. • G.l ab
~ - 1.5) , " - 25.11 ), - 1.Ia9 , ', - 21 .) ' .m2 ".' 1.G) ' .<llJ ~ .. 0 . 019 6U 1. 55 ,.W 61 .1.. 0.02$ s> .6 ' .06 0 . 021. " .4 0 , 0)1 29. 0 2SS 0 . 0)1 1oO . ~ 0, 011 m~ '"'. 0 .037 211 . ~ ,."-' " .. ,.'" ,."-' " .1 0 , 01,9 w •• 4.>4 0 ,01.9 U.6 0.055 I.' 4.65 0.055 .... ,.'" S.6 S~S ' .06> '"" , ... 4~ S." 0,067 6.'" ,."'" ... 6.>0 0.07} 4.4) ,.""' ... 6.n ' .m ,.'" ,."" >~ 7.22 0.0115 .... ,."" ,.'" !." ,."" loLB ,."" ' .n ." ,.'" >.W ,.WS , ... B.71 G. lOb MS ,.W ,." '.)0 D.UD ,." '.ll, 0.18 "'., D,u6 , 0.129 , "' ..
'\n • O.(2)b
i" . D.5r.l
dSO
• 0 .0212
'" • • 0.576
,-
0 . 9) > ... >." 2 . )) .. " ) .26 '.n 4.>0 h.65 S-" 5. 59 6.oS 6." 6." 1.4) .. '" .. " U
.1 " Gd" ' 01. .. Or/' . o .~ ... ,'"0.n8 " ro') Or/' • o.l81 , .. • 0. 156" 10') .. • 0.1)2 " 10' )
~ • 1.)2 , v, - ~.6 ~ - 1. J:2. ." • 1.9 .0 ), - 1 ,16. V, ' U.s
' .m2 M.7 '."" a.au "'., '.~ 0 .011 97.a O.h~ 0 ,018 15. 6 >.'" 0 ,018 75 .4 1. 22 0 . 017 93 . 7 , .... 0 .024 .... 1 . 57 0 . 02" ," .6 1.6) ,."" 89.0 0 . 9OS 0.0)0 SO., L 99 0 .0)0 48 .0 .... 0 , 028 " .. >.U 0 .0)6 ~ ., .. " 0.0)6 ,.., 2.45 0 .0)) 56.' >.)6 ,.W "'., ' .n ,."" 27.5 2 .85 0 . 0)9 46 •• > ~ ,."" 26. 8 ,." 0.0108 n.5 ) .28 ,."'" 40.5 >.n ,."'" "., , . SO ,."'" U.' ,." ' .<>SO ,,~ 2 .0] ,."'" " .4 ,.W ,."'" •• m 4." 0.055 )) . J 2.26 , .066 >l.' 4.)5 ' .066 .. '" 4.40 ' .06> U . 4 2.'" 0 . 072 .. '" 4." 0 ,072 4.23 4. DB , .066 27 .8 ' .n ,.'" ..,.5 S.16 O.07a , .~ ,." 0.072 2} . 0 ,." ,.""' 5.37 S .~ ,."" ' .~ S." o.on " .> ,~.
,."'" l . U S." ,."'" • • U 6.U 0.(11) "'., '.40 ' .096 J.21 6 . n ' .096 1 . 76 6 .~ ,."" U .S ,.67 ,."" 7.'" 6. 77 0 .102 >.06 6." '-".4 u .> ) . 85
,.'" >.O! .. " ,~ ' .n .. " ,.'" ••• 4m ,~ ' .n . . SO ' .m 0.)$ US ' .WS ••• ~.29 ,.W , .... '""'"
, , ,= •• > 4. S> 0.116 4.' 4." ,.W 4.' 4." ,-". 7.4 ,.'" ' .U> ,." ' .4) '.>Ll , , ...
150 • O.O)ll1 dSO
• 0 .0279
• • 0.6SIi ( • • o . na d
SO • 0 ,0402
I' • . 0.610
V'" o.l, " I " Ga''' . o.J), .. • ~.1.61 " 10' ) .. • 0 .D1 " 10' 0a" • 0. )01.," • 2U " 10' )
l ' 1 . )6, f , • 1.8.' l ' 1 . bTS.', · 21.4 l ' 1.LJ., v, • n . b
0,011 04.6 ~: i!i ' .m2 8,S ' .m ,.OU as. s 0 . 67 0 ,017 "'., ,. 0.018 ,,~ 1 .05 0.017 "' .. >." 0. 022 ... , >." ,."" 6>.' >."" 0.022 ... , >.'" O.02a SO., 1 . 1.8 0.029 45.' 1 . 75 0 .028 49 .1 1.67 0 .0)) U.' 1 .17 0 ,0)5 J) . 3 , .U 0.0).b " .4 7." 0.0) 9 )1 . ) • • 06 ,."" 25 .1 '"'" 0.0)9 29 . 5 .. '" '-"'" 26. ) 2.}S 0 .01&7 n •• '.M ,."" "' .. .... '-"SO " .. ' .65 0,05) ".6 '~S ' .<>SO " .. , .0> ' .0S6 11.1 7 .... 0.OS9 "'., )." ,."" 16.4 ) . )5 ,.W U.' ,." ,."" W •• ),as ,.W U.' ).66 0.061 U .S ).$) ' .on • • SO 4.22 0 .061 w.> 1...0) 0 .0f) .. " , .~ 0.016 4,57 4." O. C1Jl .. '" 4. )S
'.'" 6.'" 4.U ,.'" 4.24 4." 0.07ft 6. '" 4." ,."'" 4.95 4.U ,.'" '-'4 S. 28 0 ,0lIl, 5.67 ,.m ,.'" 4.95 4." 0. 094 2 . 28 ,.'" ' .m 4." S.'" 0,095 ).8S S." 0.100 1 .6) S." ' .W6 ' .>S 6.)6 '.700 7." S." 0.ll2 >.'" 6.65 O. ll~ 7. S> 6." ,.'" 1 .6S 5 . ~' 0.118 '.65 .. " '.ll, 1.a, 7 .)5 0 .112 >.W S.89 0 .129 0.)) •. n ,~'" 1.26 .. '" ' .U4 O. SS •• 06 0.1)5 , .. '" ,.'-"> , .. '" 0.17) , • .w
4so • 0.02«1 (' . '."''' <Iso • 0.0280
i. · 0.70) • • 0 .610 t, · 0.1OJ
.. Coo ..... cted t.o .~pbo.rlc oooncUU ........ ina per!."" , .. 1t. ... d
SO • Oo_tr1c .... dl"II,fLr, lnoll .. .
• • LorviUoo.1a ,t.andard oSt...uu ... .
57
....b hr Cent ,-Du. , in . .....,. .. IY/' • 0.178, .. • 0.128 " 10- )
), · 1.Ll , " - n .}
' .<nJ 8) .2 > ... 0 .019 n.' >.SO 0 . 025 ,.., '." 0 . 0)1 "J • • SO 0 , 0)8 26 .7 ,. .. ' .... 16.8 '.SO '.<>SO "' .. M' ,."" "' .S 4."-0 .06) .. " S." ,."" 6." ' . SO 0 .015 h.'" 6." ' .OM ,.'" 6." ,.'" .... .... ,."'" ... , •• SO 0 . 100 1. 28 .... ' .W6 0 ,26 '.SO ,.W , '.00
jso . 0.0251 • • 0.615
0r/" ' o.Js. "~0.J25" 10')
l ' 1.16." - 12.2
0 .01.: 94 .0 0 .482 0 ,017 n .' 0 .72) 0 ,022 66.' 0 . 51'6] 0 .028 "'., '.n o.on 46.' >.44 0.0'9 ... , >.69 , .... n~ 1. 93 0 . 050 26. 0 2 .17 0.055 "., 7.4> ' .06> "., US ' .066 >'-' 2. 89 ' .on 16.0 ) .17 0 . 0Il} 10.0 ,.67 ,.'" 6.' ,." 0.105 4.' 4. SO ' .m '-' S.'" 0.1)4 , 6.'"
~ • 0.0)20
t.· 0 . 718
V" . 0.1, ,,~ oms" to')
1 . 1.)45. v, • n .4
' .W "-.. o.6S 0 ,016 76 .2 '." ,."" "., >.'" 0.027 ~ .. 1.6) 0 .0)) 45.4 >." 0.0,)8 "., 2. U ' .044 "., .. '" '-"'" "' .. ..... 0.as5 "' .. ) . 25 ,."" .. '" ,.'" ' .on .~, 4.'" 0.016 6.'" h.:;a ,.""' 4-" 4.'" '."" l.'" , ... , .... >.6S 6.m ,."" ' .~ 6." O,12S , •• SO
~ • 0 .0261.
I' . 0. 6SS
A P PEN D I X B -------- -LIMITING CASES OF EQUATION (17)
61
APPE N DI X B
LIMITING CASES OF EQUATION (17)
A. Pure Liquid Flow (GG- 0)
For the gas content to approach zero implies a __ e __ O. Upon
establishing the following limits
lim G --0
G
e n n
n + 1 o ------~ + 1
"--2
it is readily shawn that the limiting case of Eq. (17) as GL __ 0 is
-------g 2D
This is the established result obtained by integrating Eq. (12).
B. Pure Gas Flow (GL -- 0)
(A-I)
For the liquid content to approach zero implies a -- e - (l) • The
following limits are required in this development
n = 0, lim GL-O
en --- ~ 0,
no Po I n -- = In--
nl PI
e n 2 n
RTG 2 G
(A-2)
62
a ( nn ) m ~ 0, m ~ ), 4, ••• 1" ~
Equation (17l can be rewritten as:
no -- = a(no - ~l - ~n - - aen
2D ~ ~ n -- n.j o ].
1+-1'_ + 1
.L
n + 1 o
.. In----
~ + 1 (A-)
let GL be so small that (no - ~l/(~ + 1) < 1. Apply-lng the logarithmic
expansion series
x 2 ~ tn (1 + xl = x - - + - - • • ., -1 < x < 1 (A-4)
2 3
to -aen [1 + (no - ~l/(nl + 1l) and combining the first two terms of this
expansion with a(no - ~), Eg. (A-3) can be put in the form
where
------2D
no n + 1 o
tn -- + en ---- +
~ + 1
(fa ( n )] ., n = 3" 4, ~ .. .. L 1 + n1
(A-S~
as 0L--O. Letting ~ -- 0 t~'s "--'ting case of Eq. (17) becomes
Now
and
Po -- - ------ - $n --
2D
2RTG 2 2V 2p w Goo 0 ___ c __ _
gA2 g
2ID'G 2 G
(The subscript 0 has been dropped on the right)
--=-
Hence, Eq. (A-6) can be put in the form
P 2 2 o - PI • ----
g
63
(A-6)
(A-7)
(A-B)
This is the established result for isothermal compressible flow in pipes obtained by integrating Eq. (11) using Eq. (13b).
6$
ru ~ PArLR NO. 26,SERIES B
or tbe t."~ Falls ~aulic Laboratory
Copies Organization
100 Commandin- Jr!'icer and Director, David Taylor Model Basin, ,/ashington 7, D. G., Att : Gode 513 .
9 Ghief, 3ureauof Ships, Departmentof the Navy, 'washington 2$, D. c. 5 - Technical Library (Code 312) 1 - Technical Assistant te Chief of ths Bureau (Code 106) 1 - Preliminary Design (Code 420) 1 - Hull Design (Code 440) 1 - Research and Development Program Planning (Code 320)
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3 Chief of Naval Research, Department of the Navy, Washington 2$, D. C., Att : Mechanics Branch (Code 438).
1 Director, U. S. Naval Research Laboratory, Washington 2$, D. C., Att : Code 2021.
1 Commanding Officer, Office of Naval Research, Branch Office, The John Crerar Library Building, 10th Floor, 86 East Randolph Street, Chicago 1, Illinois.
1 Commander, U. S. NavalOrdnance Laboratory, White Oak, Silver Spring, Maryland.
1 Commander, U. S. Naval Ordnance Test Station, 3202 East Foothill Boulevard, Pasadena, California.
1 Commanding Officer and Director, U. S. Navy Underwater Sound Laboratory, Fort Trumbull, New Lopdon, Connecticut.
1 Superintendent, U. S. Naval Postgraduate School, Monterey, California, Att: Librarian.
1 Chief of Research and Development, Department of the Army, Washingten 2$, D. c.
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Director, U. S. Waterways Experiment Station, p. O. Box 631, Vicksburg, Mississippi.
Office of the Chief of Engineers, Department of the Army, Gravelly Point, Washington 25, D. c.
Director, National Aeronautics and Space Administration, 1512 H Street, N. W., Washington 25. D. C.
Director, Langley Research Center, NASA, Langley Field, Virginia.
Director, Levis ResearWl Center, NASA, 21000 Brookpark Road, Cleveland 35, Ohio.
Director, lI;ydraulic Laboratory. Bureau of Reclamation, Denver Federal. Center, Denver, Colorado.
Commander, Armed Services Technical Ini'ormation Agency, Arlington Hall Station, Arlington 12, Virginia, Att : TIPDR.
Director, National. Bureau of Standards, National. lI;ydraulic Laboratory, Washington 25, D. c.
2 Newport News Shipbuilding and Dry Dock Company, Newport News, Virginia. For distribution as follows :
1 - Assistant Naval. Architect 1 - Director, lI;ydraulics Laboratory
1 Chief, Engineel-ing Re!!earch Division, Colorado State University, Fort Collins. Colorado.
1 California Institute of Technology, Division of Engineering, Pasadena 4, California, Att . Dr. M. S. Plesset, Professor of Applied Mechanics.
2 Dr. Jobn Breslin. Davidson Laboratory. Stevena Institute of Technology, 7ll Hudson Street. Hoboken, New Jersey.
1 Professor L. J. !looper, Director, Alden lI;ydraulic: Laboratory, Wor-cester Po~clm!.c Inlltitute. Worcester 2, Massachusetts.
1 Dr. A. T. Ippen. Direcl=. ~CB Lahoratory, Massachusetts Institute of Technology, Gambridge 39. Massacb.usetts.
1 Professor Laurens Troost, 1l9.9li. Dspartment of Naval. Arclrl.tecture and MarineEngineering. Masssehu3e tB Institute of Te::lmology, Cambridge 39, Massachusetts. .
1 Director, Woods Hole Oceanographic Institute. Woods Hole, Massachusetts, Att : Dr. C. O. Iselin. Senior Oceanographer.
1 Dean M. p. O' Brien,Department of Engineering, University of California, Berkeley 4, California.
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D:-. I!c::;+.e:r 2c<:se, Director. Iowa Institute of lIydraulic Research, S!.a-:.e tb.:!.Tes_tj' o~ Iowa, Iowa City, Iowa"
S. Loga:: (err and Company, 1520 Bethlehem Pike, nourton, Pennsylvania, Att.: ~. S. Logan Kerr.
AllisChal.JIersColtpaDy, M1lwaukee,Wisconsin, Att, Mr. Wm. J. Rheingans.
1 Director, Engineering Societies Library, 29 West 39th Street, New York 18, New York.
2 Library, California Institute of Technology, Pasadena, California.
1 Librarian, Massachusetts Institute of Technology, Cambridge 39, Massachusetts.
2 Librarian, Library of Congress , Washington 25, D. c.
1 Librarian, School of Engineering, University of Texas, Austin, Texas .
2 Director, Ordnance Research Laboratory, Pennsylvania State University, University Park, Pennsylvania.
3 Serials Division, University of Minnesota Library, Hinneapolis, Hinnesota.
1 Dr. W. D, Baines, Head, lIydraulic Laboratory, National Research Council, ottawa, Ontario, Canada.
1 Director, Netherlands Ship Model Basin, Haagsteeg 2, Wageningen, The Netherlands,
1 Director of Research, British Shipbuilding Research Association, 5 Chesterfield Gardens, Curzon Street, London, WI, England,
1 Dr. H. W. Lerbs, Director, Hamburg Model Basin, Branfelder Str. 164, Hamburg 33, Germany.
1 Dr. Hans Edstrand, Director, Statens Skeppsprovningsanstalt, 14 01-braltargatan, Df>teborg C, Sweden,
1 Professor J, K, Lunde, Director, Skipsmodelltanken, Skipsbygging, Trondheim, Norway.
1 Mr, M, G. Hiranandani, Director, Central Water and Power Research Station, 20 Bombay Poona Road, Poona 3, India.
1 Directeur, Bassin d ' Essais des C~nes, 6, Boulevard Victor, Paris Dei France"
1 Director, Canal de Esperiencias Hidrodinamicas, Carretera de la sierra, El Pardo, Madrid, Spain.
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Presidenza, Italian Moael Basin, Via della Vasca Navale 89, RomaSeda, Italy.
Chief of Cavitation Tunnel, WerkBtad, Aktiebolaget Karlstads Mechaniska, Kristinehamn, Sweden.
Mr. Ovid Baker, Magnolia Petrolewn Company, Natural Gas Department, P.O. Box 900, Dallas 21, Texas .
Mr. Richard Anderson, Reactor EngineeringDivision, Argonne National Laboratory, Box 299, Lemont, Illinois.
Professor A. D. K. Laird, University of California, Berkeley, California ..
1 Dr. D. Chisholm, 4 Mount Road, Cosby, Leicester, England.
