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LI2T'AP -,PNATL -_, 72TSTDJ TiP-TZC~

I1,1D TCLTTT TT'-,TT3

by

jokul 0 C ,t

T/r,?.. U-niversity f ~'', ' 'I

L L.7 ...,~.

Io,~ Un9 i versi y of l!!oi is

LTUBMITTED IN 'PARTIAL W~TTJLLT,, E7NT 'I TE

PE( ,TJiRENTS FR T1IE TGc ,UF

DOCTOR F PHILUS,7IT/

at theJIASSACHI-ST3T I'TITUTE OF TCTI-,TrOlIrGY

June 1960

Sign ~ataef A4uthor ..........

De; t'-%en Meof Mchsni cal gn p, T 14e 19T.I,60

0C '~r-i f-. by .'- ' .T Thesis S er7!sor

AiX~~d., ............... ,. .:: .....: m rt udents

V

-.. . I I

1, I

. -, .. .. - I . .

Thy TE1 Oit2)RF AF AY LTAlTeR

TIE LOVE OF ?.y ENTIRE ±"i,4LY

BIOGRAPHICAL SKETCH

The author was born in Budapest, Hungary on December 28, 1929.

After finishing high school, he attended the Hungarian Institute of

Technology for a short while before emigrating to the United States in

1948. He went to his parents and brother in Dayton, Ohio, where he

attended the University of Dayton. In 1949 he entered the University

of Cincinnati in hio, where he graduated in 1954 as the outstanding

engineering seniors receiving the degree of Mechanical Engineer. In

August of 1954 he married Elizabeth Owens, also a graduate of the

University of Cincinnati and a Canadian by birth. They then moved to

the University of Illinois where the author held a Visking Corporation

Fellowship. In 1955 he received the Master of Science degree there.

The same year they came to M.I.T. to accept a Whitney Fellowship.

The author started his teaching career as an assistant in 1956. He

was promoted to Instructor in 1957 and to Assistant Professor in 1958.

During the summers he worked in industry as engineer and consultant.

At present their family consists of a daughter, Christine, and a son,

David.

LAI:INAR CNDENSATION INSIDE HORIZN AL AND INCLINED TUBES

by

John C. Chato

Submitted to the Department of Mechanical Engineering onMay 14, 1960 in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy.

ABSTRACT

The fluid mechanics and heat transfer phenomena occurring duringcondensation inside single-pass, horizontal and slightly inclinedcondenser tubes were investigated analytically and experimentally.

Two independent analyses were developed for the condensate formingon the wall. One was the development of the boundary layer equations fora class of curved surfaces for which similarity solutions exist. Theseequations could be solved for a circular tube only approximately. Thesecond method was the derivation of approximate solutions from themomentum-energy relations. Both methods yielded similar results indi-cating that for ordinary refrigerants under normal operating conditionsthe complete solution does not differ significantly from the simplestapproximation which is identical to the one developed by Nusselt. orliquid metals, however, both of these methods indicate a significantreduction of heat transfer rates as compared to the simplified solution.There is a discrepancy between the results of the two methods as to theactual magnitudes; however, the boundary conditions for the secondmethod are more reasonable, and consequently, its results can beconsidered more reliable.

The momentum equation was developed for the bottom condensate flowand certain depth criteria were established. It was shown that arelatively slight downward inclination will significantly increase theheat transfer rates. Beyond a certain slope, however, heat transferwill be reduced again due to the changing flow pattern of the wallcondensate. A simple optimization procedure was developed for deter-mining the slope for the highest heat transfer rate.

A fluid mechanics analogy setup was used with water for the horizontalposition, and a condensation apparatus was built for the investigation ofboth fluid mechanics and heat transfer in horizontal and inclined tubes,using Refrigerant-ll3. These results check the analyses well within theexperimental errors.

Thesis Supervisor: Warren . Rohsenow

Title: Professor of Mechanical Engineering

ACKNOWLEDGEIENTS

The author wishes to thank the following people for their

help, advice, discussion, or encouragement during this work:

The members of the thesis committee, Professors

A. L. Hesselschwerdt, Jr., W. M. PRohsenow, and A. H. Shapiro.

Professor P. Griffith, Messrs. M. M. Chen, and

E. M. Dimond.

Miss Alice Seelinger for an excellent typing job.

Last, but not least, my entire family and especially my

wife, Beth, whose moral support was indispensable.

The project as sponsored in art by the A.S.R.E., now

called A.S.H.R.A.E., and by the Whirlpool Corporation.

The experimental work was done at the MIT Refrigeration

and Air Conditioning Laboratories, which are under the direction

of Professor A. L. Hesselschwerdt, Jr.

Part of the work was performed at the MIT Computation

Center.

TABLE OF CONTENTS

Page

CHAPTER I INTRODUCTION .............

Definition of the Problem ........

Historical Background ..........

CHAPTER II THE VAPOR PHASE. .........

CHAPTER III CONDENSATION ON THE TUBE WALLS ...........

The Momentum-Energy Eauation. ...............

Application of the Momentum-Energy Equation to a

Round Horizontal Tube . . . . . . . . . . . . . . . .

First Approximation ...............

Second Aopproximation ................

The Bomundary Layer Equations. ...............

CILPTER IV THE AXIALLY FLOJ'ING COIDEiSATE ON THE BOTTOM

OF THE TUBE . . . . . . . . . . . . . . . . . . . .

Free Surface. . . An Old Problem with a New Twist . . . . .

Critical Depth of Free-Surface Flows . . . . . . . . . . .

The Profile of the Free Surface . . . . . . . . . . . . . .

Table 4.1 Comparative Results in Inclined Condenser Tubes.

Table 4.2 Comparative Results in Horizontal Condenser

Tubes . . . . . . . . . . . . . . . . . . . ....

Heat Conduction through the Axially Flowing Condensate

on the Bottom of the Horizontal Tube .........

Comparison of Heat Fluxes through the Bottom Condensate

and through the Film on the Tube 'alls . . . . . ..

1

4

7

10

10

16

16

18

32

44

44

47

54

66a

66b

67

. . . . . . . .

. . 0 . . . . *

Page

CHPTER V INTERACTION BETIJEN PHASES.. .......... 73

General Considerations. . . . . .. . . . . .. 73

The Effect of Shear on the Condensate Film in a

Horizontal Tube....... .............. . 74

Numerical Solution .. . . .... . . 76

Estimate of Transition Flow Rate for Surface Tension

Effects . . . . . . .. . . . . . . . . . . . . 80

CILAPTER VI TEE E1Ei-INTAL TK. .. . ......... ... . 82

Fluid Mechanics malogy Experiments . . . . . . . . . 82

Description of the Apparatus . . . . . . . . . . . . 82

Experim ent Procedure ............ . . 83

Condensation Experiments . . . . . . . . ...... 84

Description of the Apparatus . . . . . . . . . . . . 84

Exerimentl Procedure . . . . . . . . . . . . . . . 87

CHPTER VII DISCU'JSSION, CONCLUSIONS, AND _ECO2-TNDATIONS

FOR FTU, OX . . . . . . . . . . . . . . . . . 89

Discussion . . . . . . . . . . . . . . . . . 89

Conclusions . . . . . . . . . . . . . . . . . . . . . . 95

Reconmendations for Future ork . . . .. ... 98NO0I'-CLATU .. .. . . . . . . . . . . . . . . . . . . . . . 100

BIBLIOGAP .. ... ....................... 107

LIST FF .G'JES. . ..... 111

7?PENDIX I DETEFJ NATION OF T INTEPFACE BONDAPY CONDITIONS 132

P..P-IX..'.. II.,~ SAi2LE CPLCUTA:IONS 13511-~~~~~~ CA L ,C ,,t. .q . . . .m .m . .m . .m .J ., . . .w

Page

APPENDiX IIl EAE'.I .IT kL. DATA ................. .1L

Table A-1 Equipment Data

lWater Analogy Experiments.

Table A-2 Flow Depth Data

Water Analogy Experiments.

Table A-3 Critical Flow DataWater Analogy Experiments.

Table A-4 Eqauipment Data

Condensation Experiments .

Table A-5 Heat Transfer Data ....

Table A-6 Flow Depth Data

Definition of Symbols. .Condensation Eeriments

APPENDIX IV CO:I°'UTER POGPRAMS . ......

APPE'.DIX V ANTGLE FUNCTIONS .... ..

145

146

150

151

152

........... 155156

. . . . . . . . ... . 158

. . . .. .. . .. . 165

.... .... ^ 165

. . . . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

182~PENDIX VI E- Q T -I 01T D 1:L~v',- N S

CHAPTER I

INTURDUCTION

Definition of the Problem

The basic mechanism of condensation inside a horizontal tube is

the same as that occurring on the outside of a tube; but the geometrical

restrictions imposed upon the former make the problem considerably more

complex. The easiest way to analyze the process is to follow the "life

history" of the medium as it passes through the tube. Thus, the follow-

ing phenomena can be found as distinct but interrelated phases of the

overall picture:

1. The flow of vapor in the tube

2. The mass and heat transfer at the vapor-condensate

interface

3. The flow of condensate on the side walls of the tube

4- The heat transfer through this condensate layer

5. The axial flow of the liquid on the bottom of the tube

discharging at the outlet

6. The heat transfer through this layer

7. The interactions between these various phenomena.

Each of the above items can be studied separately at first, but

it is obvious that the problem has to be solved ultimately by inter-

relating all of them.

A qualitative eaination of the above phenomena imnmediately

reveals that the most profnd difference betwen cdensation on the

cutside and on the inside of a tube results from the simple geomretr-ical

restriction forced upon the ltter case. This is that all the vapor

to be condensed has to enter at the inlet, while all the liquid

condensed in the tube and any vapor to be condensed subsequently has

to leave through he outlet. In other words, the total mass flow has

to pass through the relatively cnfined volume created by the tube

itself. T'his condition creates relatively strong interactions betwreen

the various phenomena described above which usually do nt eist when

the vapor condenses on the outside of a toe.

Although in mst cases it is more advantageaus to condense on

the otside ocf the tube, there are certain applications where conden-

sation on the inside is inherently required. Two outstanding examples

of this latter condition are the air cooled condensers, sed primarily

in air conditioning and refrigeration, and the radiation cooled con-

densers elcsed prinmarily in vehicles for outer space. These tro

arnplic.ations, however. have a very basic difference between them.

namnely the influence of gravity. ifor an orbiting space vehicle the

terms horizontal and vertical have no real significance and gravity,

-nless it is artificially created:, plays no part at all. In the case

of the "earth-bound" condensers, gravity exerts an al 1iT*tnt

influence on the henomena occurring. In the foloing chaptcrs this

latter problem wil be investigated almosti exclusively, first Oec se

of its comercial importance end second because of is greater

complexcty,

The method of i-rvest- gation -1 foll essen tially the lin

gsien preusly; at is the "lih e h'istory" of the mediu - be

tracd " cgn the ,-ube. or the :-juose a-wna sis it wil be

3.

assumed that film-ise condensation occurs, which is the usual case..

In order to define the problem more precisely, the flow will be

investigated primarily for the case of a single pass condenser tube.

Thus the end conditions will be that only vapor enters and nly

condensate leaves the tube. As it will be seen later such a narrowing

of the scope is necessary in order to eliminate some variables from a

very complex process.

Suerheated vapor introduces some problems of physical chemistry

not yet fully understood. Consequently this investigation will be

restricted o the case of the pure saturated vapor.

/.

Histor'ical Backzgrmnd

The classical work of Nusselt (375 forms the basis of most

subsequent analyses of film condensation. A number of investigators

elaborated on this theory and used it for comparison to experimental

results. An excellent and exhaustive review of references is given by

McAdams (34) who also summarizes the experimental results. The measured

values of the heat transfer coefficients run from about 36% below to 70%

above the predicted values for condensation of pure saturated vapors n

the outside of single horizontal tubes.

Until recently the analytical investigations were based essentially

on ans imrOvement of the originl Nusselt theory. Peck and Reddie (38)

included an acceleration term, but their formulation of the equation

was incorrect. Bromley, t al (8) investigated the effect of temperature

ariation round the tube, then Bromley (7) and later Rohsenow (40)

exanuned the problem of subocooling of the condensate and suggested

correction factors. These results indicated that for the usual

operating conditions all these effects are negligible for pure saturated

vapors. Recently Sparrow and Gregg (46,47) presented solutions of

bond:ary-layer type equations developed for the condensats layer.

Their results indicate that although for the usual flids, with

relatively high Prandtl Numbers there is no signific=1t change from

the Nusselt solution; the heat ;,trnsfer rates are s-ignficantl lered

for licuid metals -,rith very l ?randtl numbers.

Analysis of the condensation uinside horizontal uabes nas made by

Chalrl-> . ", (10) who used the Nusselt analysis togethr -ih an emr -cal

relati-cn for the depth of the bottom condensate.

* Iumb(ers in aCrentheses refer to numbers in the B1 bli ography.

R

5.

xperimental investigations of condensation inside horizontal, or

nearly horizontal, tubes were relatively few until recently. Jakob,

et al (26) and Jakob (24, 25) condensed steam and found good agreement

with the Nusselt theory even though the effect of the axially flowing

bottom condensate was ignored. Trapp (50, 51) found heat transfer

coefficients for ammonia and alcohol vapors which were generally lower

than the theory predicted. Chaddock's (10) and Dixon's (15) experi-

ments indicated that the variation of the depth of the bottom condensate

is small in tubes up to a length-diameter rtio of about 30.

A nmber of experLments were done with relatively high vapor

velocities existing in the tubes. Under such circumstances the vapor

shear has greater importance and the results sho., marked deviation from

the Nusselt theory. Tepe and iueller (49) condensing benzene and

methanol inside a tube inclined at 15o from horizontal found coeffi-

cients 15 times higher than the Nusselt results. Akers, et al (2, 3)

correlated a great number of local heat transfer data as a function of

a density-corrected local Reynolds Number, and his points fall for the

most part above the Nusselt values, particulaxly at high flow rates.

ksimkLmits (18) results with Refrigerant 12 agreed well with kers'

correl.tion and were about 30% above theoretical values. None of these

results, hwever. cover the situation that was to be investigated in

ths work; ncly where the condensate layer on the bottom is deep

enoa -to seriously influence the heat transfer rates, and the vapor,-

A. I-shear is relatively low for most part of a relatively long condenser

tube.

i',sra and Bonilla (36) did condensation experiments with liquid

metals. Teir results -,ere markedly below the theoretical 'values

:.

predicted by any of the analyses and the scatter was very large. This

was due primarily to the lack of control of the vapor shear conditions

at the interface. Consequently this data is good canly for qualitative

comparison.

Recent investigations have throm some light n the phenomenon

observed quite a long time ago, among others by Jakob (24), that when

superheated vapor is condensed the liquid interface temperature is

lower than that corresponding to thermodynamic equilibrium. Schrage (45)

did basic analyses on this problem, and Balekjian and Katz (5) used

some of his results to correlate their data. This field is still quite

unexplored and more investigations will be needed. Until then, heat

trr.nsfer correlations ilth superheated vapor have to be treated very

carefully.

The roblem of condensation on an inclined tube was investigated

by Hassan and Jakob (19), ho found analytically that for a long tube

the heat transfer rate varies as the ne-fourth power of the cosine of

the nclination.

CW~TER II

THE VAPOR PHASE

In the usual case where the specific volume of the vapor is

considerably greater than that of the liquaid, the medium enters at a

relatively high velocity. As more and more is condensed along the

tube, the vapor slows down. In the case of single-pass condenser

tubes the vapor velocity becomes essentially zero at the outlet end.

It is clear that both entrance and exit conditions have to be considered

for the complete definition of the problem. The most profound aspects

of such a process as condensation inside a tube are that there is no

fully developed flo, it is constantly changing spatially; and that the

flow cannot get far enough from end effects to escape their influence.

As a matter of fact end effects exert all important control on the

entire process.

Since in a horizontal tube condensate collects at the bottom, the

cross-section of the vapor flow is asymmetric. There is not much

information available on such flows. Analyses are practically non-

I existent and experimental data are rather scarce. The most complete

and detailed work is that of Gazley (16) whose experiments were done

~y ~ on two-phase mixtures without change of phase. for lack of better

information it may be assumed that the friction factors found in this

reference can be applied to the constantly varying vapor floa in a

condenser tube. In the horizontal tube the flow pattern is further

complicated by the fact that, with the exception of the very lowest of

flow rates, there are always surface waves generated on thne bottom

condensate. T the confined space available these waves have very

7.

3

pronounced effect on the vapor flow. In particular, they increase the

energy loss from the flow and, correspondingly cause a greater pressure

drop. The asymmetric configuration and the surface waves generated on

the bottom condensate indicate that the equivalent friction losses are

not mniformly distributed around the circumference of the flow.

Compared to the wavy surface of the bottom condensate, the liquid on

the walls is smooth if the condensation is film-wise, as is the usual

case. Consequently, the friction losses at the wall are probably

lower than at the bottom of the tube. If the tube is inclined, most

of these surface waves disappear and the friction losses become more

uniform around the periphery. 'or the purpose of calculations the

4+ ~ ccncept of friction factor Ell be used with the numerical values taken

primarily from Gazley's (16) data.

As was mentioned earlier, there is a rather important difference

between the condensation of a saturated vapor and that of a superheated

vapor. In the former case the surface of the condensate in contact

with the vapor may be considered to be very nearly in thermodynamic

equilibrium with the vapor; that is, the temperature of this surface

may be taken as the saturation temperature, the same as that of the

vapor. Thus essentially no heat transfer occurs in the vapor.

However, there is a slow outward movement towards the condensing

surfaces. Under such circumstances it is reasonable to consider the

vapor as uniform in temperature and in pressure at any cross-section;

and, i the pressure drop is small along the tube, temperatures may be

t aken as niform everywhere in the vapor phase.

I n the vapor entering the tube is superheated, then the entire

condensation process becomes much more complicated. Some of the most

9.

important effects arising from this condition have been observed and

discussed by several investigators (5, 24, 45).: The first and most

obvious difference is that the vapor has to be cooled before it can

condense. Consequently, a temperature gradient exists in the core of

the tube which depends not only on the amount of superheat, but also n

the magnitude of the subcooling of the condensate. The peculiarities

of this dependence have been observed and discussed by akob (5), who

found that, as the subcooling increased, the temperature of the core of

the vapor became higher at the outlet end instead of lower. He

explained this observation by pointing out that at higher subcooling,?

the vapor-liquid interface is at a lower temperature; and, consequently

any vapor molecule hitting the surface will have a much greater tendency

to condense instead of returning to the core. At lower subcooling

greater number of molecules can approach the interface, then return to

the vapor core at a lower temperature level and cause an overall re-

~4 ~ duction of the vapor temperature. Another important effect of superheat!

occurs at and near the interface. Thermodynamic equilibrium does not

exist anymore, and the condensate surface temperature is different from.,;

that of saturation. As it was mentioned above, the condensate is sub-

cooled, and the surface temperature is also lowered. To the author's

;lat ~ knowledge the correlation of Balekjian and Katz (5), based on some of

!!; ~ Schrage's work (45), is the best available on this phenomenon.

Data in this investigation were all taken with saturated vapor,

therefore, the effect of superheat did not appear.

i

10.

CHAPTER III

CONDENSATION ON THE TUBE WALLS

The Momentum-Enery Equation

A solution to the problem of condensation on an nclined surface

may be obtained by writing the momentum and energy equations for the

condensate flow. By using the concept of similarity, these equations

may be reduced to a single ordinary differential equation with only

one dependent and one independent variable .

Consider the flow of condensate along the surface in the

positive x direction as shown in Figure 3.1. At some position x, the

condensate film thickness is ys The momentum equation in the x

direction can be written as follows.

f1 4ax~dt + d-y +# ,T 5s an (3.1)0 is

where X

Y

a

/de

distance along plate, ft.

distance perpendicular to plate into the fluid, ft.

liquid mass density, slugs/ft3

vapor mass density, slugs/ft3

velocity in x direction, ft/sec

liquid viscosity, lbf-sec/ft2

gravitational acceleration, ft/sec2

surface inclination to horizontal

* This approach is due to Mr. Michael M. Chen.

ml L ~ I_ --

I

A

I

i

A

.i

4

5I

i

S

I..a

i-1

I

71.

The enery equation becomes,

A SYs (

~k, a tl = ~-JA~zm ds -J:Pg (ts-t) p" a~y (3.2)0 o.~~~~~~

where k thermal conductivity of liquid, Btu/sec-ft-°F

temperature, OF

ts temperature at the liquid-vapor interface, °F

latent heat of vaporization, Btu/slug

Cpe specific heat of liquid, Btu/slug-OF

To non-dimensionalize the above equations, define the following

quantities as special properties of the velocity and temperature

profiles.YS

where t is wall temperature

Y. _,U

+ (it.,~~~~~~~~~~~~.

I

o80 s-t

'2.

I~~~~~~

0c~fiM , e+ dy j / i+ dy+, ~~~~D --a/Iw

Substituting these relations into the momentum equaation (3.1) and

ignoring the contribution of the vapor phase, since pv < < I for

ordinary applications, yields',

~~~~~~~~~~~~~~~~~~~~~c.~pl,/aions / R, , ;(,? )~, .~~~~~~~~44 5

Similarly the energy euation (3.2) becomes,

'it ^y5sD daxA~e4Y5 dX t 4tp 4 C (3.4)

Defne, f - a.s7 the volumetric liCuid flw rate per un

leng'h of tube, and 2 .

Thus equat-on (3.4) becomes,

(3.5)

( + C)? dx d

It1- is shm-,m-. in the e.n ondi:x""

* I s s.m in t he acn...t. this c be done provided

~~~~~~~~,At <<v z

| ~~~~~~ /e ai

.1

.I

s

i3;.

If there s a sirmilarity solution to these equations, then the

finctions of the velocity and temperature profiles, A B, C, D are

independent of x. If, in addition, the temperature remains constant

along the wal, the equations may be reduced to a single ordinary

differential equation in the following manner.

From equation (3.5),

Ys '(kic ) SYs ke^{tD dx

dx

dy 5s

dx

i,

_ t

d rket D

peA +C )

(3.6)

Multiplying equation (3.3) by ys/ e uaA and usin; euations(3.6) and (3.7) together ith the definition o f elds,

s dU ys J. I

FBP,#Y df I CRI-te ~-dx

C(A Pg ) . sl"7 YP./aU. us

(3.7)

.I.t

* .

gr (pe - P,,) Y 2-,s im 6

R lue ev, 0I

il

-I

BDket,Al (I CS)

d 4 27 -1 2; 4,-S6 5

z pip- P ' 1

3

Rle f -d

Define a dim.ensionless flow rate and distance as ollows,

F f D3 (pt. -(o) k343 1 -w

F IRcf 1 3 Auo CS)3

where e is a characteristic length.

e*' dX

(3.8)

(3.9)

Thus equation (3.8) becomes,

= BDk4 - +Ple A (C.) L

F F'.(1=8)

or

L + FF"L (FJ

(3.10)

(3.1oa)

The -riht hnd side o- the equation represents the effect of

montu . The results of . .sse s aalysis inc1Gicate tht this efect

_s ,=m. '_

7~

at

,i

w:-1

,F

'.4

.4

1 -St

(p ,)3

sin I _ . _

s= O= 3

:

I

I

I

tii

¥

Thus the rght n se oi' the eq-atin may be cc sid-erad as a

pert-ubat icn term, and t ce.te ecaticn may be soIved by successive

appc~atins.r C r ns The heat transfer coefficient may be cmuted by considering the

heat tansferred at the wall.

<AA -k at} = ~~~~kDkit

where h heat transfer coefficient

's'~~~~~ =. Dkle

. Pu~~~(+C:) dSY7 -en _ , _

_ I

-. . ( 5 F' (3.11)

The local Nusselt nuber isI

ke

, A)A (I 4 C -. --le ke 4 LJ

I .r i

4 rpe (/se )A1 1L _ue ke4 F (3.12)

The mean 22usel n=.Iber beccmess

I ~~~~~~~~~~~~IIt3

'} ~NU =,3

' .F4pe - ) I A FL ,~ I 17l)

i ,,, - 4 ' . . -

!Lv c (?, -f,,)3, ,]

�I

i

IIi

I

Ii

tII

iII-

4

fI

T

A .1ication of the Momentu-En-er y Eauation to a Round Horizontal Tube

First Approximation.~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Fror condensation on a horizontal tube define 1 ry the radius

of the condensing surface, in the equations. If the condensate film

thickness is very much smaller than r, the previously derived

equations may be applied. From equation (3.1Ca)

, ~~~F(F n tt+_-s[ ( ) 8 F" (3-14)

The boundary condition is, F= 0 at = 0 at the top of the tube.

Exp-ress F in polynorial form.

F=- F + FE +F E . (3-15)

it lwhere 6 was defined in equation (3.10a).

" may be found by letting E approach zero.

OI.=I F,-F-I-AInt6=Ot -

Therefore,

V$~~ = | l (3.17)

Then ction [finr'3' d# is tabulated i Anooendix VI.3 ~ S ubstitui g into equa'ion (3.13) yields the first ar-roximation

o f e t se l .t1 nuno er for a ound tube as a fLnction o the -stance

or angle from the opmost point.

I-

i

i

i

I

Ii

i

I

7

* ' ~~~~~~~~~~~~~~~~~~~~~~~~-7 -.,

3

4 nuos k)~~~~4*~ ~N 'Y

_-

iI<

..

.s. X~~~~~~~~~~~~~~~~~~~~~I

_ I

L SrEC(-P :p4 J /f0 ke2 4jt

(3.18)

If the Nusselt number is to be based on the total area of the

tube and not only on the p-ar where condensation acally ccurs, the

above eatation has to be multiplied by /~ .

N 4 W f1e(-a2 4r 0, 14= 1 0 3 (i +c3') ___ ___ ___

(3 L R ApL k4 A

r -I 41|~~~~~~~~~~~~r (5'/3 i7

(3.19)

As a first appromation, Nusselts parabolic velocity and

linear temperature distributions may be used. 'That is,

i+ +3y+_ y(3.20)~* - y* (3.21)

Then A = 3 ,.B = 6/5 C = 3/8, and D = 1. The mean Nusselt nmber

I

1:

P gt/3i§40 ~~(3.22)

Since is usually very small, the equation may be eressed in

a more convenient form as follow:s.

l .C Ctin d

L ,Leeket V J

ior small values of 3 the numerical results of the above

equation are identical to hose obtained by the iNusselt analysis.

The efect of ; and the Prandtl number may be found by successively

evaluating the higher order terms in equation (3.15) and by refining

the velocity and temperature profiles.

Second Approximation

To find F:, substitute the first t terms of equation (3.15)

into the left hand side of equation (3.14).

(F. + 6J)L('+ Er')3 £iF, ) ( ireF (F.')3+ i ia1j(3.24)

a3If the terms containing and are neglected, the equation

reduces to,

I

i

3 ation (F25) can be s olved explicitly for F as follows,

:Eqaion (3.25) can be solved expli citly f o F., as follcws~,

F,2) T 3 E F -3 F LF.a-2 ;E' _, -F .'

r= snr q -

2 ]_ (i, f0 )

I' .;_ - Fa sin

Substituting he values of , o, and fro eation

(3.17) and simplifying yields,

4

I Fo3i

Defining)

0 14 -1,r?~~~~~1LSJr ,V3~ dri~

If

LI~mg ½~' ,

It

and sb ti t ! *`v g-ves,

(3.25)

(3.26)

. - If) is,.- 1) zk.

0

,14 - 14'

-W (I

40

I _ a 0 ;nM? Cat doTo

Define,

I - $Is a n3 0 cos 0 di0

Values for this acticn were obtained graphically.

Equation (3.27) becomesI

£5F -lo

TWith this solution the second approximation of the Nusselt

number is

x

or

3

3.4

do

V' -""C

(3.27)

(3.28)

(3.29)

12L+o'+---I

(3.30)

I

X

L

(3.31)

F- S,j ---rIz 0

4

x

,Uj A, A

-C- S+

F?

- 21*~~~<_L

To find improved velocity nd temperature profiles, the momentum

and energy equations may be solved for a fluid element between y and y.

The momentum equation is,

9 J £ ,. ~fYs YI r f A n,c-dy + upj]- Udy 4. e Vy 0

-U 0g;0pg Ct oy Mfi O's - Y) gr Oe -/7<) i' 5;(3. 2)

0

By an order of magnitude estimate of the vapor momentum equation,

which is developed in Appendix I, it can be concluded that

Hesyl dxJyeFer0

Consequently, these tw.o terms effectively cancel each other,

although individually they are not negligible. Using this relation

the momentum equation becomes in non dimensional form,

2 ~~~YfR Z +4yfe~ + +2Ma'e e5eg

Pt C4 Y5 d So dy

, s~~e Ys F sYs/ Y+) t(Pe -p)Sl'7 y, ay,+ y

,u.lti-n g by y/ pe ua, nmd assuming again that the velocity

.nd te^-er-ture profiles are independent of x leads to the following

derivntion (si.il arly to equation 3.S),

i

i

i

I

F

Dkent^;e O + c ii

I

-"(+ I +dXi

0

I

son w r

* -(F9-+?,]Y 4

iC

p rIIPPu

I

,y 0+ y 0~~

dI3-)

c 7)sin . I-

or

dy+dy+ ?.0A ( CI )- c

{P4+

I

= 90- l) -l(I-y) +

I

.11

= q -$1

,. A

- f

v ~-

�e - /:W D kc At?e 'Idea (

Substituting the definition of B on the right hand side yields

'~ --- y ~('t Y ) B~ + (,):~F +dy +

y+ y*

+/c~ dY + 4 + cd+tx y+(3.33)-04 o- o

hs before, if E is small, the equation may be solved by succes-

sive evaluation of the terms in the series expansion for u* and A.CA .+ +

R7 -Be, +R 6i.Let

* .Ao(/-x"')

with the boundary conditionI

+*gy+ tI0

therefore,therefore,

U ,@ y2i 2) (3.6)

with A0 = 3, = 6/5, D = 1

I (d dy u 3y+-.- .0 y1#4 + 9 y ,f

4s Y i~+ 4 Y S

(3 34)

(3-35)

Y~

0 + + - Y-

0

t.

1'1 f

7t .

I

.

Zi

.1

iz

4:!V

Ii[Ir!

t

IV

l

T:

Substiut~itg into (3-33) gives the second approximation of the

velocity profile.

4- os 1

: i ..

-. - y+) - { ++,q+C°SY+z

co--hy i

3yf4

Integrating,

1 . R, (y

la. _

'. /4

I, Cos 4l ]y [

+ 4 vq 4.' 30

I Cos S .

1

32

+ Ico$

(3.36)

+Z'y 4

V

25.

Aply the bcunda conditions.f

oI

Ju

I

U

I

duy+ -5

Since the first t erm is equal to unity,I

0

2 L I I

IZ, cossI, cos 6'5

I + £It C J P_0 1 0

.s it C .'0 +- 105 7 + .'77

Y +¢7S'

14+

_

+3II CO a

c si+ s-7 Ii

y+" y+- , .,+r 5 v tY

'3 l a -

+

I+ If '9

(3.37)

2y -

(3 38)

.

t- 7104f Cas k4C9

dy - U - - 44 )dy =

-p

The enerj eat-icn ecomes,

V.

o0

C00

uy

A1~

.4 /L . lI- y +

(3 ,)

y

or iun non-dimensional form

0

doh

cAle(At e &K t/7

I

I'd ~i la U. Mr fw dt/.

0y*

Osl*d0

!

1- Jlo 4Yi

y,

0 y1

I

eudy-j'e() +vc t)fL

ke

0 7

ti /,d~y _0

ke JY

(

t't (A dy~3 0

d-

k 4iYs

dc_ _

y~p( t) 'IL - . _~ .il t ke /-

12I,

= d Aplu,,,yZ-

(3.40)

27.

Substituting he value of ys from euation (3.6),

Ay+

)Y _ D CYr *+ -/'-C ' O

Id d~ay !+

___ F,/+csL s

D lCTMI+ lCS

+1I

/

, - { a+y 4

Y+ I

fD +4 4 ++S d/( + t y +r /

Let

c+ Is +

C Co+ SC, +...

D Do+ D, +...

~o - 'otY 1 0*1Y. I

with boundary condition t. = 0 at

tou y+Substituting into (3.42) results,

LI _ Y -^43 > y2) y +_0

/ + SD, + 'I3(y /

3I+ 41.YF

Y +>3) d+

/I Y

or,

(3.41)

(3-42)

+y = 0 and

, DoaI

= 1 at

+dt.

D /+.

st, +

C, - 36

at,++

T,

I

After integrating and collecting terms

+

a

To satisfy the boundary condition at the interface,

= 1+ ,I+ !t

Ia p

I

0O 04w )

I- S/ o 3'1Im+]? [

4Y_ _ (I + _

l- 1 +.a B

T'nis result is identical to the one obtained by Rohsenow (40).

Next find

C C -0

/

I

LLt+dy+

Substituting from equations (3.6), (3.38), (3.45), and neglecting

higher order terms containing gives,

(3.43)+ * . II

(34/4)

L I, _

s)(3.45)

_O L

3 ~ .1- 1 4

II

Y+51)

+ 1: -!; I _1

T

29.

'y- Y+ ) + -

I

10(+.,7l( a v

___I I J+Z3 ya 40

a +

Y6

(346)

I 0fI(0

]Fy

or

C= I. 3I -

lo + I ) L

9 I q.O

(r Y - a Y 1 - ,kY, It+

Y+5 ) dy+

-r

- A 4 I* -c -7 -,

+5y'- yL] ;

Ir. 'r-+ 6 1 4'i --4

30.

____ ____ ___3 sr40TC = I I-] [,.

BM~~~~~~~ I l_!~' +t , lO~~z+2!)_10(1+ 104).~~~~~~~~~

$ S~at27 0 ao^46I, c050) J (3-47)

Using the series expansion form of the constants and substituting

equations (3.37) and (3.44) gives the additional quantities to be used

in equations (3.30) and (3.31).

Rz_3 +E(4.4S#A51 jos4) (3.48)

D = I+ /O i_ _(3-49)D=/410 + -

n t dr ke ctt a (3a e t 50)t

In the derivation of the basic equation it was assumed that

A, B, C, and D are independent of $. From the above equations it is

obvious then, that the second approximation is valid only where the

terms containing I, cos are small compared to the others.

The results indicate that a small Prandtl number decreases the

Nusselt number; while a large Prandtl number tends to increase it

slightly. The effect of the Prandtl number as given by ts analysis

is sho-wn in igure 3.3. Here the ratio of the Nusselt number obtained

from these equations to the number calculated from the original

Nusselt analysis (which is essentially the first approximation

31

described above) is plotted as a function of the temperature difference

and the Prandtl number. The angle term containing 2 was ignored in

nese caJcuLaxlons snce its va&lue is negigible up to an angle or

about 1500. .or ordinary fluids is usually very small and further

approximations are unnecessary. The last approximation already shows

that similarity does not exist for a circular tube except at 4 = 0.

The change of the velocity profile is very small up to an angle of

about 1500. Based on these observations, Chen (11) concluded that

~~~~. . . . . '" .. 1.... !J

X o-=U~e:Uly ,uuo wyi~rwu~rae~zlu aLy Deu VUL"1nu Lj.y asunVgsn ne

velocity profile remains similar to the one occurring at = 0. This

means that FF"/('t) = 0 and equation 3.10a becomes,

(I + ) F(F')3 S;n (3.51)

with k'(0) = 0, the solution becomes

(3.52)

The velocity profiles now are similar and can be solved by either

the perturbation method or by successive substitutions in equation 3.33.

The results of this solution are also shown in Figure 3.3 together with

the results of Sparrow and Gregg (47) which will be discussed in the

next section.

-��a�rr�rr

.

.Y, e

The Boundary Layer Equations., ~~~~~~~~~~~~~~~~~~~~~_ . _

The problem of film condensation on a surface curved in two

dimensions may be approached from the exact Navier-$tokes equations.

If it is assumed that the thickness of the condensate layer is much

smaller than the radius of curvature at any point, then the equations

describing the condensate flow at that point .will be the same as that

for a flat plate inclined at the same angle to the horizontal. Thus

the Navier-Stokes equations may be written in the following form:

| ils <,a4 g'pt-,2)sin + (dh d) (3.51)SC C-) a Pe t x axe y

th. aC) av g(p. PO,)Cos / p e de+U,.-- __ +(e aJ (3.52)

g ax ~ay a2t O y@X

~L(4~ r .(3.53).ax ~y

The x direction is along the plate in the downward direction;

the y direction is perpendicularly into the fluid; and is the down-

ward inclination of the plate to the horizontal as shomn in igure 3.1.

This notation is identical with the ne used in the momentum-energy

method.

The boundary conditions have to be established. -.t the wmll no

slipping occurs, consequently:

at y = u = = O (3-54)

!1

I

.V- 7 ~~~~~~~~~~~~~~~~~~~~I

At the vapor interface the conditions are not simple as it was pointed

out before. In order to obtain a similarity transformation, it may be

assumed as an approximation that

aty=ys · = (3.5C)

This assumption leads to higher heat transfer rates than those

predicted by the energy-momentum method.

The order of mamnitude considerations anlied in the derivation

of the ordinary boundary layer equations can also be used in this

case. As a result, the Navier-Stokes euations reduce to the

following form,au Du _______(3 56)

-~ @q+ - a (3.53)

aX ay

These equations describe the flow without any heat transfer

effects. To include these, the energy equation is needed in addition.

The derivation of all these equations may be found in the references

(43). or steady state conditions, the complete set of equations

aescriol-ng tne i±ow ancd one neat transler Decome,

= ...... .. ... + "' "-- (3 57)

-- + - 0 (3%.s)oaX Oy

-u-------- -------- -- I-r----- --

34..3

I'

11P

;

p- §t( t - F a ke dC (3.5)

'}~ - The boundary conditions are:

at y = u = v = and t = t (3.59)V

at y =Y sy and t = t (360)

These are actually the standard boundary layer equations with

the additional assumptions that the buoyancy forces within the liquid

and the viscous dissipation energy are negligible.

The problem now arises, how to solve these equations in particular

cases. One obvious way is the numerical method applied directly to the

above partial differential equations. An alternative approach is to

reduce the partial differential equations to ordinary differential

equations by the use of properly selected stream functions. This

latter method is described in the following text. Denoting differen-

tiation by a letter subscript, the stream function, , is defined in

the usual manner that will automatically satisfy the continuity

equation.

u = ~' (3.61)y

= -x . (3.62)

The remaining two equations become:

-y -P - -y /e + -" yyY (3.63)

y t - ty =ty (3.64)(3.64)

where ke

The corresponding bamundary conditions are:

at y = = 0, g , t t (3.65)y ~~x V

aty= 'ys .=0 t=t (3.66)yys

Let gQ,} G( ) 9(PY -Pe)S'(3.67)Then the question is, for what types of functions of G(x) will the

above relations reduce to ordinary differential equations. If this

reduction can be accomplished, there is a transformation of the form:

.,-.t= T J(l)'j (X)- ,y) where O- Y(xy) (3.68)

which will indeed yield this result. Substituting relations (3.67)

and (3.68) into (3.63) gives,

atJ a a( I) j _ tf I + + T 4

Ix 7 q 7-J + J -LidJS - '

| (J) ' L4r J ) aby +j 2: 17Pry - - 07 9/r :a]

= + v Z qF~ +J(3~ g + 3j ~ j gy=G+1,czF~~~~~~ci~~~(l(÷jbI 3j~~~~~~~'¶~~~7 ~~, +3j ~~~~

L. - % - - %f - -,, r _ - I

| +3J (g J iJ 'a i(jI)j (3.69)

L

r -

36.

Obviously all coefficients of the function J and its derivatives

must be of the form: G (constant), in order to render the expression

reducible to an ordinary differential equation. I these coefficients

are examined and it is noted that G must be a function of x only, then

the following relations are found to exist,

g(y) = E, a constant (3.70)

|y 7Y = N(x) (3.71)

Substituting into (3.69) produces 3

G va J E'q ( ) (3.72)

.Divide both sides by a2 (jj'E2N2 ).

NjJ'JEN a IE) (3.73)Nil

The coefficients of J 2 and J"', in parentheses, are constant.

Therefore:

NNi= Bet, a constant (3.74)

j- = $. a constant (3.75)

Substituting from (3.75) into (3.74) gives,

jj" = o(')* (3.76).

37,

all functions of satisfying relation (3.76) will, therefore,

reduce the equation to an odinary differential equation, provided

that the term containinrg G is also dimensionless. Dividig (3.76)

by jj' results in an exact differential -*ith the following solutions:

iO + X ! when B , 1

J w F3X (3.77)when B= 1

with the relation between Boand n:

1n 1 -Bo (3-78)

Substituting these results into (3.73) shows the types of

functions for G that will allow a transformation of this kind to be

performed.

When B 1:

G + (3.79)

When B = 1

G; g E3 (3-80)

It is convenient to express all functions in terms of the

variation of gravity along the plate, G.

For the first class of allowable functions,

C 4 El1. xI (3.79)

~j | ~+El~XI 4~ ~(3.79a)

.;

N "I I E. · f I 'NlI~4ExIt1

17, I tE* 4Egl t I .1y

n3ElY- I 4

For the second class,

G -xEsG --Es.e

E'I

N,',e

E_ , e- I .. y

Equation (3.73), defining J, now can be written in the 'folloing

form:

I , q (ii- I ) 11 ) (3-81)G

YQ .E D' -i(J J3

This is a variation of the well-known Falkner-Skan equation

appearing in bundary layer theory.

The energy equation c now be investigated separately. From

(3.68), (3.71),. and (3.75), the following relation cn be found,

(3.79b)

(3.79c)

(3.79d)

(3.80)

(3.80a)

(3.80b)

(3.80c)

(3 .8d)

T -1- ..

39.

- B,Ej'y such that 7 = 0 when y = 0 (3.82)

(3.83)

Substituting into equation (3.64) yields:

(3.84)

In order to render this equation dimensionless, define,

ts- tstt S- tw - At (3.85)

where t may be a function of x only. Thein, if t remains constant,

equation (3.84) becomes

( 8 Ea ;,.. j '(- , Ej 'y T'A -TAt,) -(, , Ea j j $y +

(3.86)4 a EJj ')(-,Ej'Tt)= -),E' ATat

Performing the operations and making the equation dimensionless

yields,

(3.87)ci , (L j tJ J

A t'Accordingly, if l t is a constant, theenergy equation

will also be reduced to an ordinary differential equation by the

previous transformations.

Thus equtions (3.63) and (3.64) may be treansformed into the

following ordinary differential euations,

4

.i

- ' "'",I

I7

et8 :- . t,-( 8 C'ii f E J1 y= ft7

J + v J ( E7) (-'l + i°

T It a [J - EJ T] -

3- :.~i" G - ~ At'whereE E7 ( 2 E - and E t (3.90a7 j,)2 3(. 1 -9 j At

The boundary conditions are:

at 7 = 0 J, = , T = 1

at 7 = J , T = 0

These equations can be solved for any known value of so

order to find ts along the surface, an energy balance equation

be written between to cross sections of the condensate layer,

x = x and x = x.0

(3.88)

(3.89)

, b, c)

(3.91)

(3.92)

In

may

x Ys (3s

kf e/7t uJy +fc uC(t t) c' (3tyyo ·

This euation is the same as (3.2) the energy equation of the

momentam-energy method. Substituting the previous dimensionless

relations leads to the following derivation,

- kj'(o) [4, .i At(4=) d¢(4+)]dY '-keT I(oJ BtJ j .cx- pe Ot I ( + T) 2BfkeT () p j-,di ckxo) j (*)] W

4 E (3.94)

* A>J pi J(Q5 ') V, XS - IT d0

L,40.

.93)

L,,

41-

The integral can be eliminated by using equation (3.89) together

with its boundary conditions (3.91) and (3.92).

- k'() [ 4 j () =I J AP0 4i f 7( ++El

If--A Pea .T(?,) +

] + ,4b6,J[(J~~~T) °+ ]J 'U.

0

8: ke'(o) A .t(o) Cj(xo)1+ Eq

A/l 92i R (s VkeA ~ (3.95)I +~~~

This equation then determines 7 s If A t(xo) or f(xo) is

equal to zero, however, further simplification is possible which

results in the following expression:

(7s) 8, ke t (3.96)

It is clear now that if At is ccstant, i s nhdeendent of x. n

the above ecuatin a may o be set eualected such

rat tne consan- n equ$ation 3.88 is unity. B is essenti:ally

arbitrary. The equations then become,

J +4 =' 1J (1 + I 30 (3.8Sa)+ EJ) -79] t. OI ~ I[ ' ¶'l] I (3.s9a)

+ - sy 0f~~~ w |- _

r'c$~~ 1+ (3.9ea)

Examining the set of permissible functions reveals that the flat

plate problem w.ith constant G can indeed by solved by this method, but

the circular tube with G varying as sin x cannot be solved directly.

Sparrow presented the flat plate solution, and later used Hermcnn's (21)

approximate function to obtain a solution to the circular tube problem

(46, 47).

In finding an approximate solution, Hermann matched the const.nts

of a set of four equations such that these equations were satisfied

exactly at 5 = 1 /2. Since no similarity exist at this point, while

at p = 0 it does, this approximation is certainly no better than

Chen's described before.

Comparing the energy-momentum method to Sparrowls results, as

shownm in Figure 3.3, indicates that the previous one yields consis-

tantly lower values for the heat transfer rate. The discrepancy is

due obviously to the difference in the interface boundary conditions.

k~~~-

I -"

I

111

I

The ones assumed in the momentum-energy method conform closer to the

usual physical situation existing, such as an essentially stagnant

vapor condensing on a surface.

Since for ordinary fluids the corrections are insignificant, the

results of all these methods will have to be compared to those obtained

from data on liquid metals. It was mentioned before that isra and

Bonilla's results qualitatively show the marked reduction in heat transfer

rates for liquid metals.

/1

CHAPTER IV

THE ATILLY FLOVING CONDENSATE ON T BTTOii OF THE TUBE

It is evident that all the condensate formed inside the tube has

to leave through the outlet end. If there is a considerable amount

of vapor passing through the tube, such as in the first pass of a

multi-pass condenser, then the shear exerted on the surface of the

liquid by the relatively fast-moving vapor controls the flow pattern

and the effect of gravity becomes negligible. Experimental proof may be found

in the fact that the data obtained in horizontal and in vertical tubes

can be correlated by the same curve above a certain Reynolds number (2).

If, however, the tube is part of a single-pass condenser, then the

vapor shear is zero at the outlet and negligible for a certain distance

upstream. Now gravity is the controlling factor at least in the down-

stream portion of the tube. This is the problem to be investigated

in the following section.

Free-Surface klow . . . An Old Problem with a New Tist

If vapor shear is relatively low, the condensate forming on the

sides of the tube collects on the bottom; then it has to flow out

axially and is discharged at the outlet. The depth of this axial

flow has a marked influence on the heat transfer. This depth is

considerably larger than the thickness of the condensate film n the

sides. Therefore, the axial condensate flow reduces the effective

heat transfer area in the tube. It will be shown later, quantitatively,

that the amount of heat passing through the bottom condensate layer is

negligible compared to the heat transferred on the side walls.

·"I,. ·

f 45.

Consequently, as far as the overall problem is concarned, the most

important factor is the depth of the flow. If the death could be

established the effective heat transfer area could also be fnd.

The problem of liquid discharge from a partially filled tube, or

the more general problem of liquid discharge from any open channel

has been an old one, in particular, with the civil engineers. They

have been confronted with such problems ever since they undertook

the task of controlling rivers, harbors, and building hydraulic

structures such as channels or dams. n examining their work in this

field, one can find a problem which is very closely related to the one

at hand. This is the discharge from a so-called lateral spillway.

Here the liquid is fed into the open channel all along its length and

is discharged through one end_ The similarity to the condensate flow

problem is obvious. Upon closer examination, however, some rather

imnortant differences occur. t'irst of these is the size. Comoared to

even the largest condenser tube used in practice, the smallest lateral

spillway or any similar structure is enormous. As a result, the

lateral spillway contains usually turbulent flow, while the condensate

flow can be laminar, transient, or turbulent, with the laminar type

usually dominating a great part of the tube. In the case of turbulent

flow the velocity distribution is much more uniform than in the

laminar or even transient case. Thus, the lateral spillway problem

is much more amenable to a one-dimensional analysis, although the

methods used can be applied in either case. nother important

difference arises from the axial vapor shear acting on the surface of

the condensate layer in the upstream portion of the tube. This affects

I-,

46.

the flow pattern in several ways, to of which are the tendency to

reduce the depth and to induce rather strong traveling waves on the

Ia. lne zlrs01 to nese .ncreases, one secona aecreases One

effective heat transfer area. A third difference between the two

types of phenomena is that the amount of condensate feeding into the

bottom layer at any point along the tube depends on the depth, while

in a lateral spillway such an interdependence usually does not st.

What are exactly the characteristics of the phenomena occurring

on the bottom of the tube? In order to answer this question as

thoroughly as possible it is not enough to investigate the axial flow

all by itself. The interactions at the boundaries, such as vapor

shear at the free surface, and their possible effects on the flow

should also be considered, at least qualitatively.

The axial flow of condensate is non-uniform in two respects.

First the quantity flowing increases downstream; second the cross-

section changes along the length. Of primary interest here is the

latter variation: the establishment of the surface profile. There

are actually two parts to this problem: one is to find a control

point or cross-section at which the depth may be predicted; the other

is to determine the change of depth, starting from the control

section, all along the tube. On both of these problems there was a

great amount of work done, in particular, by the civil engineers who

needed methods for predicting open channel flows (23)(32).

1

____ I_ _~~~~~~~_ _ _ _ C__I~~~~~~~~_I_ _~~~~~ _ I _·?I_~~~~~~~ :········_··_···___I __I··· _·~~~~~~....... ....

47.

The Crit ical Deoth of re-eSu4rf ace "'lows

The problem of establishing a control oint hs been approached

through the use of the so-called critical depth theory° This states

that in a free surface flow; at some oint te secf:c energy of the

stream will go through a ninimum The specific energy of he stream

may be defined as follos:

| ( 9 5 (

V. ~ ~g I n~~~~~~~~~~~~~~~~I

where specific head, ftt3p density, slugs/ft

V velocity ft/sec

A cross-sectionel area of flow, perpendicular to the stream

lines, ft2

pressure, lbffft2

gravitational acceleration, ft/sec2

$ elevation above a given datum level, ft

subscript m - mean value

To illustrate this theory and its implications, first the flow

in a rectangular channel will be investigated with the assumptions

that the velocity and density distributions are uniform and that the

pressure distribution is hydrostatic, that is, the streamlines are

essentially straight. Defining the depth of the liquid by h, the

above equation becomes,

- V (.2)

where .. elevation of the channel bottom above a datum, ift

depth of flow, ft

3For a given flow rate Q t /sec, and flow rate per unit width

of the channel, q = b ft3/sec-ft, the velocity may be expressed as

bD..

V - (4-3)bh h

Substituting into (4.2) yields:

H h h+ s-4)

For the given cross-section taking s = 0, secific heat versus0

M .- -II"J .. Ln I .A (LA. n t ..A.I .J ^Ir L L J.r V 1J A.. ItLP 4JItPIL I l-:.J I IA (: <.aJ C- I 1" U I

parameter. A typical graph is shonvm in igure 4.1.

The minimums of the specific head curves may be found by

differentiating eqution (4.4) with respect to h and setting the

result equal to zero.

h;~~~~~~~ 4 ~(4.5)Some physical significance may be found in this expression if

the corresponding critical velocity is calculated.

'13 h, t(4.6)

i'

_. ... .... -. -_- -- - - --_-.- - --

49'

This expression is identical with the velocity of wave propagation

on the surface of a liquid h deep. Thus the critical depth and theC

corresponding critical flow is analogous to the sonic flow occurring at

the throat of a nozzle or at the outlet of a duct.

I{-~ ~ Equation (4.4) and the corresponding curves in Figure 41 show

i flhT Tc (s J ~ a , 7 ad | Id_*~ T T T T T | 1ni n; Il Ia + .... Il IAJ i _1 ....

depths may exist, one greater and one smaller than the critical depth.

Correspondingly the velocities are either sub- or supercritical. W.hich

one of these flows exists in a given channel depends on how the fluid

enters at the upstream end, the losses occurring in the channel, and

the slope of the channel. In a long horizontal tube, the flow will

always become subcritical provided no shear stress exists at the free

surface to impart energy to the fluid. If at the outlet end there is

a change to a steeper slope or a so-called free discharge exists, the

control point in the form of critical depth must occur near this end.

The exact location depends on how the transition from subcritical to

supercritical flow occurs. This transition in turn is influenced

primarily by the geometry of the channel near the outlet. It has been

established experimentally in hydraulics that the critical depth actually

occurs a very short distance (about 3 or 4 times the depth) upstream of

a so-called free overfall. If the transition geometry is gradual, such

as in the case of an elbow at the outlet, the critical depth would

occur somewhere in the elbow. This is one of the possible reasons

for lower surface profiles found (15) in horizontal pipes with elbows

at the outlet. At very low flow rates surface tension effects become

more important and tend to raise the top of the liquid layer.

In the case of the free overfall, at low flow rates the liquid will

4I-f

i

I

MnM--Tn" 34IN -TAr -T- - n %3r_- nn n s n iIM11 -0 . M 1

50.

not separate from the tube but will adhere to the lip and flow don--i-

~i ~ ward. By increasing the pressures in the flow at the outlet, this

effect also tends to raise the surface level.

The effect of non-uniform velocity distribution on the critical

depth should be investigated nextb If the bottom of the channel is

taken as datum, equation (4.1) may be written (23.) as:

iwheres~c + 3h (4.7)

where

·e ----- Ai 3dA (4.8)

JA

I /3XQH(/P + V (4-9)Thus OX is a function of the velocity distribution; while 3 is

a function of the velocity distribution, and it also depends on the

curvature of the streamlines, that is, the deviation of the pressure

from hydrostatic. In particular, if the pressure is exactly hydro-

static, 3 = 1. If-the streamlines curve docwtrard 3 < 1, if they

curve upward 3 > 1. For fully developed, laminar flow in a pipe

= 2.

Now the minimum energy principle may be stated as:

3h Zq ah ah

� L -

51.

If it is assumed that the variatian of o. and ( are small near

the critical depth, the above equation simplifies to:

dz + QL . O~ ah ZA&I (4.22.)

Again investigate the two-dimensional flow.

T -rh. T i9 oP.,5 9(4+ = °_A5%; +1=0

(4.22)

Thus non-uniform velocity distributions tend to increase the

depth.

From the

may be drawn:

a.

foregoing discussion the following general conclusions

In order to be able to use the critical depth theory

successfully, a horizontal channel or tube with a sharp

cut-off should be used. Then the critical depth .,ill

always occur in the horizontal part near the outlet end.

If the length-diameter ratio of the tube is large, the

exact location of this control point becomes unimportant.

b. The critical deth theory is expected to predict depths

th:.t are too low when the flows are small, particularly

if the velocity distribution is neglected in the

calculations.

L

1

iIi

iIII

iiI

i

II

iI

i

ah4L a 3 ..E

A 9

V~~~~~~~~~~~~~~~~~~~~~~~~~5.

Now consider the circular cross-section of Zigre 3.2

A r0=~ .;f-sshn cos&) (4.13)

h r. (l- cos e) (4.24)

__ (4-15)

~i~~~ Substituting into equation (4.11), and simplifying yields,

!~~~~~~~~,_ .z(~e ~~a9~ '~ ,,,J's ,.a - a ' I Vr c v (4.16)

3 gr.5 S Ae' Sn

This function is plotted together with some of the experimental

results in Figure 7.1. If //3 is assumed to be equal to unity, the

correlation with experimental data is rather good at high flow rates,

above /Vgro = 0.15, and becomes progressively worse at lower flow

rates. If, for laminar flows, Re < 3,000, it is assumed that

! i, = 2, the correlation becomes more satisfactory; although the

deviation still remains considerable at the lowest flow rates.

Equation (4.16) can be expressed in another form which allows a

very accurate approximation in terms of the depth-diameter ratio, h/D.

Define the section factor as:

ZMA (47)(4.!?)

i

ij

53.

where Z section actor, ft2 '5

A cross-sectional area of flow, _it 2

b top width of flow, ft

.'or the circular cross-section at the critical point

c A (Of. -sin). (4 8)If n r ° )

If Z 2/r.5 is expressed in terms of h/d,the resulting relation

can be approximated for the range 0.02 C hdo < 0.9 by the following

relation:

(4.19)

This expression is practically exact for the range 0.04 i d o <

0.85, and it is 12% too high at 1do= 0.02, 7 too low at 'd= 0.9.0 ~~~~~0

Since Z goes to infinity as bidoapproaches one, the approximate

relation should not be used beyond the limits specified.

Substituting (4.19) into (4.18) yields:

ff^ g = 74 2 6 ( d<) (4.20~~~~~~0a)

or a(l42A6 (4.2oa)

S -

j

1.i ze_ IC

iikI

.IC a

54.

These equations then relate theoretically the flow rate ad he

critical depth for a circular tube. Ior the range investigated in the

experiments, v/ii may e assumea o De L.±:. or e v ~uu ase an

the hydraulic diameter of the flow) and 1.1 for Re > 3000.

The experimental work showed excellent agreement with these relations

for high flow rates. For low flow rates of Q1grr_ 5 4 0.08, however,

the correlation became progressively worse, primarily because surface

tension prevented the formation of a free nappe, and the liquid ad-

hered to the lip of the tube. The experimental results indicate that

for these lower flow rates the central angle subtended by the condensate

near the outlet remains constant at 1 = 90° (see Figure 7.1).

The Profile of the Free Surface

In hydraulics the problem of predicting free-surf4ce profiles, the

so-called backwater curves, has been studied for a long time. As a

result, there is an extensive literature available as reference; and

all text books in hydraulics treat this problem more or less in detail

(23)(43). Their analyses, however, do not consider any shear acting on

the surface of the flow; and they are usually one-dimensional. Also,

most of the investigators are dealing with uniform flow rates along the

channels, with relatively few examining the effect of such a variation.

Li (33) derived equations for surface profiles in rectangular and tri-

angular horizontal channels, with the assumptions that the velocity

distribution is uniform, friction may be neglected, and the flow rate

varies linearly along the length. His estimates of the friction effect

indicated that the calculated upstream depth would have increased by

not more than 10% in the most extreme cases.

I1

55.

Let us examine the problem of condensate flow inside a circular

tube by developing the generalized one-dimensional continuus flow

equations for a two-phase mixture with condensation.

Equations of State:

For the liquid density

= constant (4.21a)

For the vanor of an ordinary refrigerant under the usual operating

conditions the pressure drop along a condenser tube is small compared

to the absolute pressure. Consequently, the density,

p= constant (4.21b)

Velocity of Surface Waves on the Bottom Condensate Layer:

C2 g Aet (4.22a)b

where C Velocity of surface wave, ft/sec

g Gravitational acceleration, ft/sec 2

At Liquid cross-sectional area, ft 2

b Width of the liquid free surface, ft

In differential form

to r dAe A) (4.22b)c Z At bi

rt~~~ ~Definition of the Froude Number:

?r ~ --2 -Qe Qe~b[F 25 Atc/M c (4.23a)

- It Z + - 3 (4.23b)Fe- Ge 6 At3

I1

I

i

I

.A-

56.

where E Froude Number

Liquid volumetric flow rate, ft3/sec

With the above definition a Froude Number of unity corresponds

to the critical depth in any cross-section.

Continuity:

r', %X r',,,~ c~-(4.24a)

i r,, I-,,zF (4-24b)

I rt + rw * ." . ~~~~~~~~~~~~~(4-25a)de r dr',. (4.25b)

elo,. - P at2t (4.26)

where 1 , and Pft Vapor and liquid mass flow rates, slugs/sec

Q. Vanor volumetric flow rate, ft3 /sec

Energy Equation for Overall Heat Transfer:

Ax 4t = , ( ht) (4.27)

where Heat transfer coefficient, Btu/sec ft2 °F

A, Heat transfer area., ft2Mt, ean temperature differential between vapor and surface,

to be considered as a constant, °F

Total mass flow rate at any cross section, slugs/sec

, and h.0 Enthalpies at inlet and exit, Btu/slug

57.

Energy Equation for Liuid Flow:

Equation 4.2 has to be modified to include the effect of a variable

pressure in the vapor phase. Thus,

~:---P +h S. + (4.2&a)

do e tm + A + i P Q - A. (4.2Et)/~ 29.4t!, i~quati : At

Momentum Euations:

For the vapor phase,

-A, dp -4 P, dz (.4,29)

where Av Vapor cross-sectional area, ft2

P Wetted perimeter of vapor, ft

Z Axial direction, ft

7$ Vapor wall shear stress, lbf/ft 2

Correction factor for velocity distribution

For the liquid phase

-A dp d ,(pegAe5Js + S -

r P ,2 4 A 44H J.-w- rc- I - ' .} -

z owall conjensat

where S Depth f centrroid of Af below free surface, ft

Licuid shear stress on walls, lbf/ft 2

P., 1'etted perimeter of liquid on walls, ft

w Velocity components of condensate layer on walls wher it

reaches the bottom condensate

The shear stresses may be expressed as follows

(4.31a)

Lw a p2 2Ag (4-37b)

where 1, snd ~e are frictlin factors for the vapor and the licuid

m. cn be apro.-imated by the usuaI expressions in terms of the

.- R -'c'ds h. -±er.

iet Ct(8c 4 s t,

Ba-sed on Gazleyts data (16), the corc-an ma be vefollr5 ~ ~ ~~~o. 1¢ may b ows

for Re < 3000

for o > 3000

= 18 b. = -, C,= -,3

.. = 0.085, b = , C= 10'"v -Gecmet.; cf F'low:

-ren-tr cal fLazcticns m'. be o.resced in terms of e h--'

¢ ,.,- '.~-.- ....'e snd½ o Ced by the ot. co-densate layer.

'I F-- "'

t

I

I

!.i

!I

( L32a)

p / * * I

(l.33a)

(433b)

,'.

1%C.

.-I... r Q-1--LS I Ircrl'-6- 2- Av

�L

59.

u0 2 . (4.34e)i%= 'a(W-_ 49c s$ 4)(4-34g)

A~~s~~u ,, r4 (I-36 o. et() sn c](43^, 2N 5in Zc (4.34c)

A- rO ( dC 2in (4.34e)

^ w rO (l- e4$4) ~~(4-3Lf)'

P- 7rO t- c -& j'n e) -(4-34g)

pW~ro 9C (4-34h)

A1e5s ' 0 ~11 CoC5s<e 4;2 ) (4.34i)

Some of these functions are tabulated in Appendix VI.

Additional Relations:

Based on equations (3.9) and (3.17), using the constants Ao, Bo, and

D of the first approximation (see page 23) and Rohsenow's (40) value of0

0.68 for C, and noting that the change of volumetric flow rate dQe/dz = 2f,

the following relation may be established.

d rs(12ep)ke na A3 4 r S hI V 1RJ aJ3trtC3 d3(1J0.683 J3 L J - (4.36)

0

The last term of equation (4.30) may be estimated by integrating the

velocity profiles of equations (5.2) and (5.5) together with relations

(3.6), (3.9), and (3.17).

12 U P cry dz t

0WQ/ co-iensae

.A,

60.

The last expression, the momentum influx of the wall condensate,

is based on the assumption that the condensate on the bottom and on the

walls meet at an angle. Physically this situation is impossible;

surface tension will create a smooth transition which will considerably

increase the thickness of the wall condensate and, consequently reduce

the magnitude of the term. Since this effect is small to begin with,

it is reasonable to omit it entirely.

For the determination of the liquid surface profile equation (4.30)

3has to be solved. Dividing this expression by p gro3 and definingj~~~~~~~~~~~~~r ds /dz results:

9 (/tt) w [AR w Aeo Ts

( I~ ~ (I~ ,7egr.3 Aec dZ (ig) j 2<tizc {X) -l

(4.38)

The numerical solution of this equation may be simlified by the

folloing assumptions. First the vapor friction factor, fv' may be

considered a constant, 0.0085, even in the laminar region. Since the

vapor shear is very small for R.ev < 3000, this simplification ,rill

not introduce a serious error. Second, the liquid feed rate

=L d, /dz may be taken as constant along the tube. Experimental

evidence, as shown in the data in Appendix III, Table A-6, indicates

that for the range of flo; rates examined the liquid level ws ves'

uniform or most prrt of the tube. For the inclined tube the liquid

L ..

ii

61.

depth was observed to be essentially constant after the initial licuid

ramp reported in Appendix III, Table A-5. Examining the angle function

in equation (.36), tabulated in Appendix VI, reveals that ta is not too

sensitive to variations of 0 in the range of 120° to 160°. Consequently,

the error caused by this assumption can be easily minimized by selecting

a good mean value for the condensate depth when determining n1. The

resulting relations become3

_dp o. O4zp'Q ,n , 4 (4c3.

where the factor C was taken as unity.

Qeu XI= (L.40)

Subtittn i. Su- -=u tt int . - =) yield)

Substitutingff into (k.35) y~_elds:

, .F. ==~~~~~~~~~ -=Aer _ 0004__~ -b .,g_ __

fi~~~~~~~~~~ 3 a 3 1l A 4 -Ra - .'

AepefI n 04oo2sR(L-z) 1 d I 3 } (4.42)

I g~rt' L Ab j c0. At Ar 3

The licuid flow was assumed to be laminar ever,,here. This was

actua:lly the cse for all condensazaton exoeriments.

I

.

-•~~~~~~~~ ~~62.

This equation may be integrated numerically, provided that the

geometry is known at one point. For a horizontal tube or for a very

slightly downward tilted tube, where the downstresm liquid flow is

definitely subcritical, a critical depth wrll exist at the outlet. As

it was pointed out previously, at low flow rates the outlet end still

controls the flow, but the depths are considerably increased due

primarily to surface tension effects. The depth of flow becomes less

as the inclination is increased; the exposed wall area, and consequently

the effective heat transfer increases. This improvement, however, is

eventually counteracted by the effect the inclination produces on the

wall condensate. Hassan and Jakob (19) showed that, theoretically, he

tranfer atevaris ascos/4 1heat transfer rate varies as cosl/(sin O ) during condensation on n

infinite circular tube. As long as 2 r tan(sin X ) < L, this

relation is the best available estimate of the influence of nclination

on the heat transfer through the wall1 condensate. The experimental data

Appendix III, Table A-5, show a considerable decrease in condensate

depth as the tube is tilted a small amount of the order of = 0.010,

but tilting he tube further did not affect the depth very much.

For inclined tubes the establishment of a control section becomes

very difficult, if not well nigh impossible. One may argue t hat ncler

such conditions the flow depth starts at zero at the inlet. This

assurmtion leads to a numerical step-by-step solution of equation (4.42).

On the other hand, careful consideration of the flow patterns observed

in the exoperiments suggest another type of boundary ccndition, which not5~~~~~~~~~~~~~~~ .- A*e* Lin es .......... n1$o db -me hbe Brlo Qa1A

extremely simple.

011-L ", es J.LVVIZI V t: ~ i31C--/ 9VUU T-t :;:1UU-LU--9 UtAU Ullt, :iiLlr75V" IUU-lr UII

63.

i m~Again the experimental observations revealed that the flow deDths

were very uniform along most of the tube. The only departure from near

uniform depth ccurred near the entrance region and, in the horizontals

tube, very close to the outlet end. This observation suggests or a

boundary condition that the depth should remain very nearly constant in

the downstream portion of the tube. Such an assumption will lead to the

I. elimination of the terms containing area variations after equation (4.42)

is integrated, and the resulting relation becomes:

L Ae

/

L rL)L

... gro. ~(u -z* -

7 00417IleSY L Sim] 6, CR (4.43)

where 4. !.

J

i itro.h~~~~~ (p3

I ] indicats limits o interatiC.L..~ ~ ~~~~niae i-iso ntarto.Ia~~~~~~~~~~~~~~~~~

[ InIctsimtofitgaon

Ii

I

i

F 6

The results were actually not very sensitive to the limits. As

Table 4.1 shows, integrating between three sets of limits did not change

the value of the calculated condensate angle, e , by more than 2; nd

all results were reasonably close to the observed one, with the limits

+ +z = 0.5 to z = giving the best value.

The method developed here also lends itself to a simple optimizing

roc- ellureP +o fi ndl+1 t e-Y gmon wrt +kheI hihs a he.1at tr"ns-fer rt. t

has been shovm that the heat transfer coefficient varies as

1/30 d 3 1 1n scs/4 -linl/3 d and as cosl/(sin 1 o). Therefore, when c is

plotted against 0 (or c), the scale for 0 can be also marked for the

values of the first function, and the scale for C' for the values of the

second. The optimizing procedure becomes smply a matter of plotting

of, then finding the point along the curve for which the product

Co5 I/n ]410

is maximum. Such a chart is shown in Figure 4.2 with the C curve

plotted for Run No.,1l9.

For the horizontal position Table 4.2 shows a comarison of the

values of 0 for test runs with various flow rates. These values were

commuted by integrating step-by-step starting from a knom depth At the

outlet end. The agreement between measured and calculated ngles is

only fair. but the obto-ained mean values for 0 over the ength of te

[vzc~ i'- p ,l..;fenrou" n hf.n

due to the insensitiveness of relation (4.44) to variations of ~ in the

L

,,

65.

range considered. The step-by-step integration of equation (4.42)

involves small differences of the angle function on the left hand side.

Consequently, errors in estimating the friction factors influence these

calculations rather strongly, and the discrepancies found are not

I| surprising.

I

.1L'..-

\C 0

* O

00

r-~ r r-4o o

-4c\cHo HHH oH

xO O',

r- o o'£c C"

~D Hx

* -0 0

to-~+ c

* go o

Ox cr C~

r- r- rH .O o

a N a"H O00H OH

* 0

HOHC o~H oH

O OC'H H

0 0

oH U)

.oo sc0* -4 -.

o oH ' tC~r- -0 cc

o oH UL C'

0 0

oH Hoo oH Uc'cI * 0O H H

00 0

C0 0"%OHH

0I 0*tr*

0 0

H H HI 00O14o afi* HH

s

cyC~

c~

cv

-Ct

0cv

rI

(N

0I 0

O H Ho

H CVLI 0 0Cco0r-4H

o oI H 0n - -.

O r- H

r-30 ~ +~bi) o ~- ~ 0

_ D 40 cc- + C) oo a U

4 *4 U OC) - 4 CQ2 &C; -. U C) w C+ O t

-P O

C ^ h 0h Q >4-) *l ~ Cr ¢, C' C) Uo O ! 2> --. H >2: ,) i S n

q ~ - C , C . C

cvC~

H00

c,

cv

cvcvcN

c\

cv

cv

C)

C)to

E4-'

C)

10C.)4-)Cd

r-.

c.cu

0

5sC)

4-4

-)

o oo +~ ca o 4C) cqJ U C) P C) C) -4--H 4- t* H 4-C) C.)

G) : : C ) :O -p t1 C 40 m Ci 0J mC)

I , , m -

I

i

3

I'

66a.

CO

m

zc9

0or~

1H

4 -I

COE .. U)E

cor4

0CO

i

I

I

0

tOsHr

too- H

cm 0

00

t-* C)O S

rXn Hc~ 0 ~-* C)O s

* .

0 r-0

O

CY'\0 0

0OH

r2\ D0 ¢C+Os

-'* 0

bjD t£

0 0O H

o+ H

ul cd ,C)0 0^, Q4o o-<o H H

-~ cJ C

U) aJ '

0

n<M,.-I0CV

CV0

to-4Hr-I0tOCV

H0

,--Ir-tH0

r-"'l

r-i

0H0

,--Ir-4

0

C~l,-I

.4-

Cl

0-H-43

C.)

04 4

o

H CYHor--I r--I

0 0C'¢ r--ICY Hr-iH

o o0 0o* .

O tOCV CVH H0 0

C--'V

O o0". r--

H H

o o0 0

C1 r1.r4H0 0

C'H

H 0

0 O

0

r- H

0 0tC\ t(l\

(N CVH-H

C)-4-')

d r

Iz

0 0 4->o Co

o' a (n 4) ,QC) U O)H r n.-4 O0.3

r

66b.

Em-~Cr/

E-

PI,

L)nq

E-)o3:

'HI-H

0I.,*NpqHr)EA

co

C\0C,

C-CN

4

c-)co0

-'

a)

C)

CV0

CV

toCVl

I.

S,

-4-3 0

oC) 43

0 v

o zCO 4-C) ::

oC1)

co

.4

e,

--H

0

c~0

J.,,S4-'>CD

-HCI-4C)0u

I

I

I

I

I

67.

Heat Conduction Through the pLxi y l oyn Condensate on the Bottom

of the Horizontel Tube

o 2ar only heat transfer through the thin condensate layer on

the walls of the tube has been considered. To complete the nvestiga-

I ̂ tion, heat transfer through the axially flowing condensate on the bottom

must aso be evaluated. It was shown previously that at least the

upstream portion of this condensate flow must be lminar since the

liquid velocities increase from zero at the entrance end. If the flow

is laminar and secondary flows are neglected, then heat will be rans-

ferred across this condensate by conduction only. The geometry of the

cross section is shown in Figure 3.2, with an enlarged view of the

corner A shown. or the sake of obtaining a well-defined boundary, it

is assumed that the bottom condensate and the film meet at an angle as

shown. The thermal boundary conditions may be assumed to be uniform

saturation temperature, ts, at the interface, and uniform wall tempera-

ture t . In order o further simplify the solution, it may be assumedw

that the variations of condensate depth along the tube are gradual

enough to be neglected at any particular cross-section. Thus, the

governing equation can be simplified into Laplace' s equation for heat

conduction in t-o dimensions.

7 2 t = 0 or in cylindrical coordinates

=. art dta °(4.45)

The bund-ry conditions re:

at r- r t = to (44Z6)at r cs = r cos c t = t to t he edge of (4 47)

the film

68.

In this form the problem becomes one of ure conduction. The

particular configuration involved suggests a solution by conformal

mapping. The equation of the tube circle in the x-y coordinate system

sho* is

X _+aY+?.c' 0 -a

x+ y + 2. yr cos e 2 r2as $,I|Divngby2 iny2+ycos rOsne (4.4a)

ividing by 2 sinc , and defining X - /rosin fc andDiviing by yil c CY y/rsin & yields:i y/r. ~c

X'+ Y+ Y cot e- I

The moon-sha.ped cross-section ABCDEF of the bottom condensate may

be mapped into the infinite slab A'BtC'D'EtF shomwn in Figure 3.2b by

using the complex transformation equation:

. ~~w; In~- Iwhere = u + iv and = + iY. Consequently,

W-nx-, Y

Multiplying both the numerator and denominator by + - iY and

simplifying yields:

X+ aW In~jT ' 4 2YSince w u + iv,

e ConsV = equ(cosVentlyv) <_ j + Y -y tyA

Consecquently,

U Xa+ yaI (A _2y

e COSY - X )Z2? and e Sr

r

69.

tcan V X YZY I

The temperature distribution in the complex u-v plane is linear

between t along A'BtCt and t along B'EtFt. Mathematically:w s

t -tw en. or t = # (450)

Suibstituting the value of v from equation (4.49) gives the

temperature distribution in the X-Y plane:

, 2Y

+ X + Y (4.51)

The heat transfer through the condensate may be computed from:

ay5 o- (t, s j -Ys )-Q

where q' - heat transferred through bottom condensate, Btu/sec.

Substituting the dimensionless relations and using equation (4.51)

yields:

dA( YS ~'-(1 - rsi tt ?' '5 ' dL- t-xj

ro 6i'

f - Y s o

/-+ I rX0 V5,

a n Jr I - I si- ±-S''-_ ain.Z£~~~~ ia

_Zke~t

3ke4t'/r- = - _

7Cr ,)

L Yo

[L so

2

- I(4.52)

rI

= Zkg$

71.

Comarison of Heat Fluxes through he Bottom Condensate and through

the Film on the Tube Walls

The heat flux per unit length through the condensing film may be

computed from equation (3.18) with the constants taken as R 3,

B = 6/5, C = 0.68, and D = 1.

i ki L

3 v (4-53)

Dividing equation (4.52) by (4.53) yields:

I

(4. 54)

From equations (3.6), (3.9), and (3.17), with the same constants

as given before equation (4.53-), and setting X

do a.; A(I+.68s) dfYso kAeLt dx

1n' /# 9ee( e-A,) (1 +O68Y)r.31 ,Uaek4 t (4.55)

r

i

i.I

F

i

i

L.

'

72.

The ratio of heat fluxes, then, depends on the fluid properties,

t]/4the quantity t/r3( + 0.68 ) , and the depth of the bottom

0 ?~]1'

condensate expressed in terms of the central half-angle 0. Figure 4.4

shows that this ratio is always very small for such vapor velocities

where the bottom condensate still exists as a distinctly separate flow

regime. For the highest flow rates encountered in the experiments with

Refrigerant-113, the heat loss through the bottom condensate in a

horizontal tube was about 2.5% of the heat transmitted through the

condensate film on the wall according to these estimates. Turbulence

of ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~- 4mr 4not -I ,,h 4i-i ilio~~ hnv+.n9+ o: nol+ol h- -- -t- - - - -…-i

show that the bottom condensate effectively insulates part of the

condenser tube. The ratio expressed by equation (4.54) goes through a

meximum near 0 = 170° because the theoretical film thickness goes to

infinity as approaches 80, and the assumied geometry canmot be

satisfied near this point.

One more comment is in order here. For the above calculations a

sharp angle was assumed to exist where the condensate film on the wall

meets the liquid flowing on the bottom of the tube. Physically this

situation is impossible. A smooth surface profile connecting the two

flow regimes increases the liquid thickness in this vicinity where

most of the heat transfer through the bottom condensate occurs.

Consequently, the actual heat transfer becomes even less than predicted

by the above relations.

2

,,

I

I

I

73.

CHAPTER V

INTERACTIONS BErkWEN PHASES

General Considerations

Due to shear at the interfaces and the geometrical restrictions

imposed by te tube, the flow patterns of the two phases are strongly

interrelated. The liquid collecting in the tube restricts the vapor

flow area. 'aves generated at the surface increase the effective shear

stress and cause an additional pressure drop in the vapor phase. On

the other hand, the high velocity vapor flow draws the liquid along and

causes surface instabilities. If the vapor enters the tube at a down-

ward angle instead of axially, the bottom condensate layer will be

lowered by the momentum of the incoming stream. Vapor shear was taken

into account in the previous Chapter to determine the depth of the

bottom condensate. Surface waves have insignificant effect on the

heat transfer since the liquid acts as an effective insulator and the

vapor pressure drop is negligible. However, the thin layer o wall

condensate might be seriously affected by the vapor drag, especially

near the entrance region. To investigate the magnitude of this

phenomenon, a simple analysis was developed. Instabilities of the

interface were not considered at all. The liquid film was treated as

laminar everywihere, and the vapor was assumed to flow axially.

Another effect arising from the presence of to phases is surface

tension. It influenced the flow pattern of the liquid primarily in

- a, _ w peso 1i'; No | Aft t~~nW1 4* 4 | __ | _ 49; Ad n t1 A. _ _ Ahel= -zoV OU asroe abutY-L J.±:QLafg can-

]: ~ densate from breaking away at the outlet into a so-called free nappe;

and, consequently, raised the surface above that predicted by the

'~.

. ~

I

74.

critical depth theory. A rough order of magnitude estimate was developed

for the maximum flow rate above which this effect becomes negligible.

A second, rather insignificant, effect of surface tension was the

smooth transition of the free surfaces where the wall condensate met

the liquid flowing axially on the bottom. As it was found before, most

of the heat transfer through the bottom condensate occurred near this

region; and, consequently, this effect reduced this heat transfer rate

considerably, while the wall condensate heat transfer was not influenced

very mach.

The Effect of Shear on the Condensate Film in a Horizontal Tube

Consider the flow of condensate on the walls of the horizontal

tube. Due to gravity, the condensate will tend to flow downward and

collect on the bottom of the tube. The axially flowing vapor, however,

will tend to drag the liquid along due to the shear stress at the

interface. Since the liquid layer is very thin compared to the tube

radius, the curvature of the wall may be neglected. As a good first

approximation momentum changes may also be ignored. Then the force

balance for a small control volume, (y - )r0d Odz, can be written as

follows.

In the ~ direction:

which cn be integrated to find the velocity (5)distribution and the mean

which can-be integrated to fnd the velocity distribution and the mean

I -- svelocity ,ua

.e ~ ~ ~ ~ 9

.

.. L

.1

-- ,06 si9(_ Pe i (yt iAt _) (5.2)

- ()Uy

In the axial, z direction:

.. J d - J w (5 4)

which can be integrated to obtain

(5.5)

Y >As (5.6)

where w is the velocity in z direction.

For laminar flow of the condensate, heat transfer is by conduction

only and the temperature distribution may be assumed to be linear.

Then the energy balance for an element ysro $dz becomes:

5sA~n uss dl . ~gAto MJ (5.7)

where = mass flow rate of condensate.

The mass flow rate may be expressed in terms of velocities.

drf -- .d4, 'Z 58

75.

I

ii

76.

Substituting equations (5.3), (5.6), and (5.8) into (5.7) yields:

_ l* ' . ·A 2 v .a -/ -Jz Ysr aZ * iA- -- c a (59).t . cA t

where A = (1 + 0. 68 -- )

The boundary conditions may be selected by careful examination of

the flow. At the entrance of the tube an initial condensate layer

thickness may be prescribed. For example, if the vapor flow can be

assumed to be axial at the entrance, then a zero initial thickness may

be considered. The flow must be symmetrical with respect to the

vertical through the center of the tube. From the physical standpoint

the thickness is finite at the top of the tube and the liquid surface

continuous. It must be realized, however, that certain mathematical

solutions might not be able to conform with these last two requirements.

The vapor shear stress, tv, depends on the mass flow rate and,

consequently, on z; but it may be assumed to be independent of , the

angular position. The temperature difference, t, may be taken as

constant.

Numerical Solution

Expanding equation (5.9) results:

go,* tea-/ ; w C) ySzi 4 +

Y / 3 7Z_ ket (5.l0)am pe dz. Xi

Substituting = s3 and rearranging yields,

ap I_~ 7 _

,A )2 Z] (52.1)

The shear stress may be expressed as a function of the Reynolds

number or the mass flow rate in the following form (Equations 4.31,

4.32, 4.33)

37;

hi'

W -'-4C4 ,r. 4

where the subscript ( ) refers to the vapor phase of the flow

dh hydraulic diameter, ft

ai b constants depending on the range

number

Since at any cross section

ri 4 r tinlet a constant

dro e d (.From equations (5.4) and (5.7)

of the Reynolds

(5.-14)

pi ; ,, A t 1 0dV ~~~~~~~~~~- U-

(5.1 5)

V

(5.12)

(5.-3)d. J

77.

- Is %0% %66'r n I P6, a1z.;z i-A Z-+b-i

i CW

78.

For a numerical solution equations (5.11), (5.12), (5.13), and

(5.15) can be written in finite difference form. Since o at z = 0 is0o

known, P- at a small distance Az away can be calculated. The procedure

can be repeated until the end of.the tube is reached or V becomes

zero. Care has to be taken in the selection of oints. since the

thickness tends to infinity at the top of the tube at the outlet end.

In finite difference form the governing equations can be written

as follows:rat. . , - f ..-

41 Z L A J~ t e s;n P.3 'I(* +

} Cd<°sS; -3 (5.1 a)

"3iR-i:>^.+§zz] .tat .2 (5.14 a & b)

| ~~~~~~~~dr. 2kS A4 (5.15a)

Equations (5.12) and (5.13) remain the same.

Since the computer could not start with Po = O, several small50

thicknesses were assumed for the first value of p. The magnitude of

these were determined as arbitrary fractions of the thickness at the

top of the round tube without shear; that is,

wherei ~ KK isa)btr (5.16)

where E is an arbitrary constant.

I

II

79.

The bottom condensate depth was assumed to be constant at

!i~fi = 120°. The fluid properties used were those of Refrigerant-13,

and the tube radius taken was that of the experimental condenser. In

order to avoid time consuming trial and error procedures, the tube

length was not specified. Instead the entering vapor flow rate was

given and the corresponding tube length could be determined from the

calculations.

f~ ~Flow rates were varied from 0.2 x 10 4 to 4.8 x 10 4

lbf sec/ft or slugs/sec; temperature differences were from 2 to 25 F;

and the increments were taken as r10 and 3 r/20 and 20.

The results showed that even for flow rates and temperature

differences considerably higher than the ones encountered in the

experiments, the film thickness reached essentially the same value as

predicted by the no-shear theory in less than three radius lengths

along the tube. Consequently shear has negligible effects as far as

the thinning out of the condensate film is concerned. The experimental

observations, however, indicated that the film had a wavy surface at

the entrance of the tube at the highest flow rates. Such an instability

problem was, of course, not included in the above analysis.

The numerical solutions became unstable after the thickness

reached the no-shear values; and, consequently, the calculations could

not be continued to the end of the tube. The purpose of this analysis,

namely the estimate of the effect of shear near the entrance, was,

however, fulfilled. A typical set of surface profiles are shown in

Figure 5.1.

i

'i4i

.i

6L: _.

80.

Estimate of Transition low Rate for Surface Tension Effects

An order of magnitude estimate may be mde of the minim flow

rate below which the nappe cannot break away by considering the minimum

momentum required to balance the surface tension of the solid liquid

boundary formed as the nappe springs free. If uniform velocity distri-

bution is assumed, the momentum of the flow in the horizontal direction

is,

PC1 M^A~ et At ~(5-16)

The horizontal component of the surface tension at the lip of the

tube outlet is:

P,, '"' SX' -~O~Jat ~(5.17)

where Pw wetted tube perimeter, ft

C surface tension of liquid film, lbf/ft

contact angle

Equating the two expressions yields:

, P A a,,sintQa. 9, s_

or in dimensionless form for a round tube:

Qt~~ " '2c<r i~cS)tm is_ _ _ _ _ ~_.__ _ _ _ _ _ . , s _ _ ' (5.1.8)

For the water-plastic combination a contact angle of approximately

45° was measured. The resultant flow rate is, assuming e = 450° ,andC

taking Or = 493 x 10 3 lbf/ft corresponding to the approximate water

temperature used of 80 OF,

Qtg. f = 0.1113

For the Refrigerant-113 and copper combination a contact angle of

50°, taking ec = 450 and e = 1.302 x 10- 3 lbf/ft, yields,

= 0.0914

These flow rates are both in the regime where the observed outlet

depths begin to deviate from the critical depths predicted by the

theory, and where the above effect became noticeable in the experiments

with water.

* Personal co.munication from Professor P. Griffith of CIT.

... !,

81.

:-.

eld.LJ~~1.L L5.Lb V.~82.,

i~~~~~~~~~~~~~~~~~~~~~~T 11 G,~tH: T

THE EXT.. NTAL VOI

Fluid Mechanics Analor_ Exoeriments

It was recognized at the beginning of this work that one of the weak

points of previous analyses was the determination of the d eoth of the

axially flowing condensate. The first part of the experiments was set

up in order to gain more insight into the problem of a liquid flowing

I along a horizontal tube with the flow rate varing along the length of

the tube. The importance of the outlet end conditions has been pointed

out before. In these experiments special attention was given to the flow

conditions existing in this region.

Description of the Aparatus

The apparatus is shown in Figures 6-1 and 6-2, and schematically in

Figure 6-3. The test section consisted of a transparent plastic tube 54

inches long with an internal diameter of 1.10 inches. At twelve points

along the tube, equally spaced at 4.5 inches, means were provided for

admitting liquid on both sides and for letting vapor or air out at the

top. This arrangement is shown in Figure 6-4. Liquid was supplied to

these points from a distributor which also served as a flow-rate measuring

device for each supply point. The flow rates were determined on the

Venturi principle by measuring the liquid static pressure differential

between the top of the distributor block and in the somall stainless steel

tubes through which the fluid had to flow to reach the test section.

These nressure differences were read an manometers. The flow rates could

I ~be adjusted by small needle valves. The-distributor chamber on top of the

I+.

I

-i-

1

833.

distributor block hs a me0 and a Ca a loh filter, and it h'.ad a valve

on top which served both as a low regulator and air bleed. This valve

was cornected to a line whnich ret'urned the excess liquid to the col ecting

tanle at the outlet of the test section. FYrom this tk_ a small Monel

Metal pump circulated the fluid through a standard commercial filter back

into the distributor. Although means were provided for supDlying air at

the entr=ance f the test section, nd for bleeding it off at twelve

points along the tube, this pat of the setun was not u ized Duing

the first trials it was discovered hat the riples caused by he fid

strears entering the suroly points were xge enough to be lcu:crt?

b- very low flows f air. The ensu g wave attern could nt be ecccted

to be siilar to the one occurring during, a condensation process. Since

in the second phase cf te eerirents he actual condensation flow

patterns were to be observed. the analogy studi es ,ith a high vanor shear

stress were abandned T'hus the irst hase f he errerimental .rk

endeavored primarily to establsh fIoT deoths near te end f the

hori zontal tube amd to fd the criticni Reynlds n'bes at which

transition fro laminmar to turbulent low occurs. These exre- -ents were

oerc-.med using distilled water as the fluid. This se+tu could not be

used effective for i d tube -e mnen s because in sercritical

flows the lcuid s-reass entering at te -el-ve suply oints ener-edstron; oblique stand aes that mae y depth measurementv menin! ?s .

,'~'- ': as sarte . the fo, rates were adjusted at the

I 5'r, tc th d r c' '¢-. oC Te fate ar--Lrcl'- -d --~ b "ei -tr 7- '--~ t~n I ... . cr by reJatJ- t-e -'-- 'rt

rC_ - nu),o e cr- v__ c " -

8A.

the rn atlet or at the distributor bleed. It as found that for the

required flow rates the ump orked steadiest at a relatively low speed.

T.ns, most of the controlling was achieved by the valves with the unp

running at the optimum. The flow rates to the Individual supply points

were regulated by the small1 needle valves mentioned before. After

,14, m ..,e _ rP-..., ... , +.. ,+nl T-ri.mp f-i, ' rate a mneasured a.t thet~~~~VLiJ .1. s 'JL>, w . ** -- > -- s uvu- -lV - _- -

tube outlet wirth a graduate and a stop watch. The depth of the flow at

any point was established by wrapping a strip of paper around the tube,

and marking both the circumference and the points where the srface of

the licuid met the walls, viewed diagonally across the tube. The

temperature of the liquid was measured by a mercury-in-glass thermometer

graduated in increments of 0.2 OF. Turbulence was determined by

injecting ink axially into the stream through a small stainless steel

syringe.

Condensation Exeriments

Since very little experimental data was available in the literature

that could be used directly to check the validity of the an-alysis, it

became clear that an e:perLmental setuap ws needed to furnish heat

transfer data. In addition, it was deemed important that the apttratus

should also furnish some information on the flow conditions existing in

the tube. Refrigerant 113 (C2C13 F3) was chosen as the fluid bec.use of

its favorable presmure-esmDerature relationships.

Descri-tion of the P!-Dratus

The equiprment is sho.n in '1gres 6-5, 6-6, 6-7, .and schematically

- .. ; _ e% 1n'...,-... O ..- ~~U 1 Xri Ln L1 (2 £I? el . '-,- r f

in:zgl.re -O- IVJU1 i-- V- …ill

of 5 1/2" standad brass pipe, set on top of a 2000 w.att flat plate

heaer. Inside the top of the boi' er .s a circular slash pla te. A 7/8

1!

.. - 1 l - - - C 9 - -i-F-] - . -- - e L

q3

85.

o.d. copper tube served as riser from the boiler. A heater wire ,rapDped

around an insulated portion of the tube served as superheater. The

vapor passed from the riser into the 1/2n o.d. entrance tube through a

heat-resistant flexible hose. The entrance tube contained a stainless

steel thermocouple well and the connection to the first mercury manometer.

The entrance tube joined the top of the test section at an angle of

approximately 45° to allow the installation of a glass reticle at the

front end of the test section. This window served both for observation

and for illumination of the interior of the equipment. The condenser

section consisted of a 28.25 inch long 5/8? o.d. standard finned copper

tube.

The condensed liquid was discharged into a 100 cc burette. This

burette was sealed into the bottom leg of the copper tee into which the

outlet end of the test section was soldered. The other two legs of the

tee, one directly opposite to the outlet of the test section and one on

top contained observation windows. The burette was sealed with two

O-rings and several layers of Glyptal compound. The observation windows

were glued to a brass bushing which was sealed in the tee by two -rdings.

The tee also contained the mercury manometer connection, which also

served as the purge line, the condensate outlet thermocouple, and a

liquid overflow connection. At the bottom of the measuring burette

a needle valve was installed to serve both as a flow regulator and a

shut-off device for the measurements. The overflow line had a liquid

seal at the top,to prevent the escape of any vapor, and a shut-off valve.

m iid r to a ne liter caai-tv Plass renpiPVer trouna a

3/8" o.d. copper line. The receiver supplied the boiler through a short

copper tubing w._hich was connected to the charging valve too. All arts,

I

''1

., j

86.

except the test section, the observation windows, and manometers, were

insulated with glass wool insulation covered with aluminum foil. The

burette insulation was divided into several sections which could be

moved up and down to allow for readings to be taken at several intervals

along the tube.

The electrical input to the heaters was regulated by to Variacs.

The power was measured by a wattmeter connected to a switch box, which

made it possible to measure the power going into both heaters with the

same wattmeter.

Temperatures were measured at the entrance of the test section, at"

six points along the tube, at the outlet, and at any two other points as

required. All thermocouples were made of No. 30 gauge copper-constantan

thermocouple wires with the reference junctions kept in an iced thermos

bottle. The inlet unction was housed in a stainless steel well located

along the axis of the short inlet tube. The outlet unction was located

right behind the lower edge of the test section. This unction was silver

soldered protruding outside the stainless steel tubing through which the

connections led to the outside. The location of this thermocouple forced

at least part of the condensate to flow along the steel tube until it

reached a splash guard which directed it into the measuring burette.

The six thermocouples along the test section were soldered into grooves

cut into the tube parallel to the fins. Three of these were located on

the top, three at diametrically opposite poiLnts on the bottom. tour...... ,.....q,

were one-sixth of the tube length from each end and two in the miccue.

I| ~ The thermnocoule wires were connected to a Leeds and Northrup semi-

precision potentiometer through a ten-point selector switch.

-ki

87.

The air velocities around the test section were controlled by two

fans which were adjusted to provide a reasonably uniform flow distribution

along the tube. The air velocities were measured with an Anemotherm Air

Meter.

Photographs of the flow conditions inside the test tube were taken

through the outlet observation window, while illumination was provided

with a photographic flood light through the reticle at the entrance.

Because of condensation on the reticle, pictures could not be taken through.

it, although visual observations were possible. A single-lens reflex

camera was used with special lens attachments that provided closeup

pictures at relatively restricted depth of focus. Thus the approximate

location of the flow pattern picture could be determined. Visual measure-

ments of the central angle subtended by the bottom condensate were also

made with the aid of a glass reticle marked with diagonals 30° apart. To

reduce the blow-out hazard, a power cut-off switch was installed at the

too of the outlet manometer on the atmospheric side.

Experimental Procedure

The apparatus was placed in a constant temperature room. The system

was evacuated, then it was charged with distilled Refrigerant 113. The

test section was leveled by adjusting the four bolts on which the frame-

work rested. An optical level was used to establish the horizontal

position of the tube. To obtain any desired slope, up to about 10 °,

accurately machined steel blocks were placed under the two supporting

bolts near the entrance of the test section. The following procedure

was used for the test runs.I.

.

~.!~

'~ . -.~

88.

First the room temperature was established. In order to reduce the

possibility of air leaking into the system, this temperature was set at

such level that during the runs the system pressure always stayed above

atmospheric pressure. This meant that for low flow rates of vapor higher

room temperatures had to be used than for high flow rates. The air

velocities around the condenser were checked next. The purge line vacuum

were turned on and set at the required power level. For most runs the

liquid overflow valve was kept closed, and the burette outlet needle

valve was adjusted such that during equilibrium the burette was nearly

full of liquid. This arrangement minimized the amount of condensation

occurring outside the test section. After an initial warm-up period, the

purge line was opened very slightly. The bleed flow rate was kept at a

minimum by adjusting both a needle valve and a pincheck on the vacuum

line. After equilibrium was reached, data were taken in the following

order, unless some special circumstances arose:

1. Barometric pressure

2. Mercury and refrigerant levels in the manometers

3. Thermocouple millivolt readings

4. #aw rates in the burette

5. Visual observations and measurements of the flow pattern

6. Photographs of the interior flow pattern at various

points along the test section

7. Heater wattages and variac settings.

I

L - - -..1 - I ..- .- I- .- I. .-- II- .-.- - - 1- -1 -. ,- .

IIII

iI

89.

CHAPTER VII

DISCUSSION, CONCLUSIONS, AND RECOUENDATIONS

FOR FUTURE VORK

Discussion

The fluid mechanics analogy setup was built for the purpose of

providing direct access for measurements on the liquid used. It could

not fulfill all the requirements of a really close analogy to the con-

densation process, but it provided very useful quantitative and

qualitative results that could not be obtained from a condensation~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~m.

experiment.

The data for these water analogy experiments are tabulated in

Appendix III, Tables A-2 and A-3, for a range of flow rates of

0.02 Q / / 7 0.6. The results of flow depth measurements at the

outlet of a horizontal tube are plotted in igure 7.1 for both the fluid

mechanics analogy and condensation experiments. The curves in the figure

indicate the variation of critical depth as calculated from equation (4.16).

The solid line, .i.th = 1 may be used for the turbulent range,

Re > 3,000; the dashed line, with / = 1414 is applicable to laminar

flows, Re < 3,000. The experimental points for both fluids agree well

with the curves, don to a flow rate of about Qt = 0.0. Below

this point, the measured flow depth remained essentially constant at

c = 90°0 deviating from the theoretical curve. One, minor reason forc

this discrepancy is the fact that at lower flow rates the actual position

of the critical depth moves closer to the outlet end of the tube where

the surface profile varies rather rapidly. Thus the measurements, taken

at an essentially fixed distance upstream from the end, would tend to

90.

yield greater depths. Another, much more important reason is the

increasing importance of the surface tension at the lower flow rates.

It was observed during the experiments with water that while at high

flow rates the liquid broke away horizontally from the end of the tube

to form a free nappe, at low flow rates the liquid clung to the lower

lip of the outlet and flowed downward and sometimes even slightly back-

ward from the end of the tube. This flow pattern resulted in a sharp

downward curvature of the surface profile. The forces due to surface

tension became large enough to effectively dam up the flow and, together

with the change in pressure distribution at the overflow, caused an

increase in depth. Under such circumstances the relatively simple

critical depth theory was unsatisfactory since it accounted only for

gravity. An order of magnitude estimate of surface tension effects was

discussed in Chapter V.

i

The estimated accuracy of the measurements was 3 for the flow rates

and 3° for the subtended angles at the point of measurement approximately

1 1/2 diameter distance from the outlet end.

The fluid mechanics analogy experiments also allowed the determina-

tion of the critical Reynolds number where transition from laminar to

turbulent flow occurs. These data are shown in Table A-3 of Appendix III.

The critical Reynolds number was found to be 3,580 with a minimum of

disturbances and 2,830 with strong disturbances near the measuring point

at the outlet.

What the fluid mechanics analogy setup could not simulate was the

effect of vapor shear and the proper flow pattern in the inclined position.

Both of these shortcomings arose from the local disturbances caused by the

I

91.

relatively concentrated liquid feeds. Even the smallest of air flows in

the tube caught these local ripples and made the entire surface wavy.

Consequently, it was considered meaningless to try to compare these flow

patterns to the ones actually found in a condenser tube. WWhen the tube

was inclined and the flow became rather fast, these feed points generated

comparatively large, standing oblique waves which made any depth

measurements meaningless.

The condensation experiments provided a great deal of quantitative

and qualitative information. The most important numerical results were

the ones for condensate flow depths and heat transfer rates.

All these data are also tabulated in Appendix III, Tables A-5 and

A-6. Flow rates varied in the range 0.007 C Q, /5 . 0.2; the

slopes ranged from O' = 0 to Or = 0.1736.

The outlet depth data for the horizontal position was discussed above,

together with the water analogy data. The measured surface profiles in

the horizontal position were compared to the calculated profiles in

Chapter IV (p. 65), and representative values were shown in Table 4.2.

The agreement was fairly good, provided that the actual outlet depth was

used as the starting point for the step-by-step integration procedure.

The data also indicate that the mean depth along the horizontal tube

remainued essentially constant for the entire range of flows at a value

of 0cm = 120°.

For the inclined positions, a marked decrease of depth was observed

up to a slope of about 0.01; a further increase in slope decreased the

depth rather slowly. The ranges of mean condensate angles observed for

each inclination are summarized, together with the heat transfer data,

in Figures 7.2a and 7.2b. These mean angles were compared to the results

IF-

92.

calculated from equation (4.43) in Chapter IV, and comparative values

were tabulated in Table 4.1. The agreement was considered good.

The measurements of the subtended angles are estimated to be within

5° of true values in the downstream portion and progressively worse up-

stream becoming probably about twice as much near the inlet. The flow

rates are accurate to 3%.

The heat transfer data are presented in Figures 7.2a and 7.2b.

The theoretical lines and experimental data correlate rather well.

The increased heat transfer coefficients for the inclined tubes are

clearly distinguishable. A few of the low points are mainly due to

small quantities of air in the apparatus particularly at the flow rates

where the inside pressure was nearly the same as the atmospheric. The

greatest source of error in these measurements lies in the determination

of the mean wall temperature and, consequently, the magnitude of the

temperature difference. There was a strong variation of temperatures

from the top to the bottom of the tube, although axial variations were

small. The temperature deviated from 9 to 21% from the mean with the

inclined tubes, particularly the one with the greatest slope, yielding

the smaller values. The temperatures for the inlet vapor and the

discharged condensate could be taken within 0.2 F, but the wall

temperature fluctuations varied from practically zero at the low flow

rates to 1 at the highest. Room temperature fluctuated within 1.5 of

the mean. The accuracy of volumetric flow rates was about 2. The

heat transfer results are estimated to be within about 15% of true values.

The manometer readings indicated that the maximum pressure drop in

the condenser tube was of the order of 5 mm Mercury. The two,

independently mounted manometers did not allow accurate readings of

93.

such small magnitudes.

Some qualitative observations do not show up in the results nd will

be discussed here. The biggest problem in performing the experiments

with the condensation setup was to eliminate the effect of air in the

system. In spite of the precautions taken small amounts of air always

remained in the system, in part because the refrigerant contained some

of it itself. Finally the purge line had to be installed at the outlet

of the test section. The bleed was regulated at such a rate that the

visual observations showed no reduction in the wall condensate film

thickness. This change in thickness could be detected as a hardly

noticeable ring at some cross section of the tube. By increasing the

bleed rate such a ring could be made to travel to the downstream end of

the tube and disappear. In order to eliminate the air from the liquid,

it was allowed to boil for at least two hours before the first test of

a day was started. In addition, the room temperature was kept at such-

a level that the system pressure was always above atmospheric. At the

lower flow rates nd pressure levels it was sometimes quite difficult

to keep the aforementioned condensate ring at the outlet end of the

tube, and a few of the data points are low quite probably because of

this phenomenon.

The variation of wave patterns and ripples was very interesting

to observe. In the horizontal position waves caused by the vapor shear

on the bottom condensate appeared at relatively low flow rates. First

they were only near the entrance region; then, with increased flow rates,

they spread over the entire surface. These large shear waves, however,

disappezred for the most part along the surface profiles as the tube was

inclined. Instead, very tiny, oblique standing ripples occurred near

r"' I

94.

the walls; and at higher flow rates occasional large traveling gravity

waves were generated near the point where the liquid ramp existing in

the entrance region joined the rather uniform surface along the tube.

While the shear waves were rather uniform and periodic with a size up

to about one-quarter of the depth; the gravity waves appeared rather

randomly, traveling all by themselves, and their size was comparable

to the depth. These observations indicate, among other things, that

the vapor pressure drop should be less in an inclined tube not only

because the vapor cross-sectional area increases, but also because of

the disanearance of shear waves.

At the very highest flow rates very strong disturbances were

observed at the entrance. In the horizontal tube no distinguishable

bottom condensate existed here. In the inclined tube ripples appeared

on the wall. Once vapor shear becomes dominant, the analytical results

based on the existence of a primarily gravity controlled free-surface

flow cannot be used. It is recommended, therefore, that these results

should not be applied above a vapor Remynolds number of about 35,000,

which was approximately the highest value encountered in these experiments.

The surface profiles, when not considering the waves, were fairly

uniform along the tube for most runs. In the horizontal tubes the

greatest variations occurred near the outlet, and also at the entrance

due to he momentum of the vapor stream. In the inclined position,

except for the smallest inclinations, the flow formed a rp at the

entrance then continued rather uniformly to the end.

-I.'I i~j._.

11-;.

qI

I ---- ----- ,-,-------- -- ------- --------

Conclusions

Heat transfer rates for laminar condensation of a pure vapor in

horizontal and inclined tubes can be predicted by the use of the

- vE [Veds Pren+.P(I_ For fluid of P v ra n r ma one.

the first approximation of the momentum energy solution, quation (3.23),

which is equivalent to Nusselt's results, can be employed for estimating

film coefficients as a function of the condensate angle, with Rohsenow,'s

(40) value of 0.68 used for C. For liquid metals, a correction factor

from Figure 3.3 is needed, although there is no good experimental data

oni nhlh t- k +.h+.n nhr'vn_ Th ri-n-M of' +.hi li.+.om …

be established for horizontal or essentially subcritical flows by a step-

by-step integration of equation (4.42) starting at the outlet. For

horizontal tubes with a straight-cut discharge, the critical depth can

be assumed to exist at the outlet if Q / fgr > 0.08. For lower

flow rates a constant outlet angle of 0 = 90 ° can be used as shown inc

Figure 7.1. As a matter of fact, for fluids similar to Refrigerant-113,

a constant mean depth of 0cm e 120° may be assumed over the entireJ

range up to 1 /gr 05 = 0.2. These methods will yield conservative

figures if applied to a condenser tube with an elbow at the outlet.

For inclined tubes equation (4.43) together with equations (4.34)

4- +can be used (integrating from z = 0.5 to z = 1) for calculating

= cm/2-ecm ci2

Chaddock (10) showed that a horizontal tube is better for heat

transfer than a vertical one. The results of these investigations

indicate that when condensation occurs inside a single-pass condenser

3n.,>^ ~ ~ ~ ___ _ 4;xvorlrvlcln nl ohnnhsel+-o1o rTar nE l

tueXa sl<,l ao.i5wl-±ra ¶JoJe wal enr±1±ice ne > urui eru. -. 7Le ouu i| cra

. .:q

2

96.

slope for a given flow rate can be estimated from equation (4.43)

together with Figure 4.3, as described in Chapter IV. This relation is

expected to yield good results even if there is an elbow at the outlet.

At hi-gh vanor w rtes abvr n en+eying vni' Paimnold nmbr

of about 35,000, these relations are no longer valid because gravity

effects are negligible and the orientation of the tube becomes unim-

portant. In such a case, heat transfer data can be correlated in

terms of a Reynolds number, irrespective of slope (2). Consequently,

for condensation inside tubes, the horizontal and inclined positions

nya t n+w rnms wrar1M +.hn-n +o -r~+. ~NA ;n _ en a:+. rno +c .rG-

better. The only exception is the case of extremely short tube with

a length-diaemeter ratio of less than about 6.

The following are brief descriptions of the calculating procedures

to be used for predicting the performance of horizontal and inclined

single pass condenser tubes. In each case the ollowing quantities are

specified: the entering vapor condition, the temperature differential

between vapor end wll, the dimensions and slope of the tube..

For the horizontal tube two procedures may be used. The quicker and

more aproximate one is to assume a constant mean condensate angle of

cm = 120°. Consequently, with 0 = l80 ° - ($c2) = 1200°, the flow rate

can be calculated by integrating equation (4.36); and the Nusselt number

can be evaluated from equation (3.23) with C 0.68. The more accurate

method is to assurme a cm and a corresponding O, calculate the flow ratecm

as before, then from Figure 7.1 find an outlet depth corresponding to the

flow rate there. Starting from this end calculate the surface rofile

by integrating equation (4.42) sten-wise along the tube. Find the ean

value of over the length of the tube nd conare it to the assumed

--- -- I-- - r- ." IV- -YUI U--o -·IV ·1-- -UY LI-·U-- - , , , .... . - --- - , -

I

I�:: t

.I--, I.SIII

.�A.

�; ...a

o7.

value. If the two are within lO0 the agreement may be onsidered

satisfactory; otherwise a new mean angle should be assumed and the

procedure repeated. n:ith the final value of , the iTusselt number can

be estimated from equation (3.23) as above.

For the inclined Atube first assume a constant mean condensate angle

$cm and its corresponding . Using O, calculate the constsant change of

flow rate, from equation (4.36). Substitute this value intofrom equation (4-36).~~~~~~~~~~~4

equation (4.43) and set the limits between z 0.5 and z = 1. Using

the angle functions (based on equations (4.34)) tabulated in Appendix VI,

find several values of corresponding to different assumed angles.

Plot the C-- 0 curve, as shown in Figure .3, and find the angle

corresponding to the given slope. The agreement between this value of

e nd the originally assumed one should be within about 3 in this case.

If it is not, assume a new value for , and repeat the procedure. With

the firnal vplue of , and A , the flow rate and the Nusselt number can

be evaluated as before from equations (4.36) and (3.23), wi-th the latter

erresslon zultiplied by cosl/4(sin or ). Actually both this function,

and the angle integral fnction appearing in (3.23) coan be read off

directly from the chart in Figure 4.3. To estimate the optium slope

for the given fl, find the point on the o- 0 curve for which the

product of te other two coordinates, relation (A.4:), is rnamtum.

For both horizontal and inclined tubes, the entering vor Reynolds

nuber should be ascertained to be less than 35,000.

.- ,AI I f

.t

ecomm.enda:ions fr FaLtare deiork

The many comple processes occt'ring during condensation iside a

tube suggest a number of investigations, some of hrich would be highly

practical while thers might have only academic interest. Here a few

of what seem to be the most important leads will be mentioned.

1. Exoerimental data is needed n liquid metals for

checking the eisting analyses.

.~ ~ ~ ~ ~ ~ - - _. - I 1 1 1-_

. T£ne eect c superheat snouLcaa e nvesiga1ea te n view

of the newer theories predicting a lowering of heat tr-ansfer

rates wi,th hgh superheats.

3. Condensation experiments could be done with different

outlet end conditions, particularly with elbows of different

diameter to radius of curvature ratios, ad ,ith tubes of various

length-diometer ratios.

4. The vapor flow inside a ube ^ith a liuid occupyring

the bottc. section could be investigated theoretically. Assu=.ing

la.minar fl6o, and neglecting licuid veloc 'ies, the flow pattern

of the vapor may be clculated for different vasor-licuid

cross-secion.al are rtios. Then the bmdarvy ccndicins found

at the interface cn be used to solve for the liquid velocity

di stri buti c'.

III

A

I

I

5. Te robem of condensation in a tube or channel th n

gravity actLng should be exomined. Here the problem o how 'to

move the condensate becomes imortant. One ossibility to be

explored is condensation on a porous wall through which the

condensate is sucked at varfous rates.

6. _rther eeriments will be necessry ith high va0or

vapor velocities to study the transition recime where gravity

becomes unitmoortsnt.

In all of the experimentail investi..gat-.ions visual observations

should be cnsidered e:xremely imDortnnt to estaoblish the exact flow

conditions ccurring.

I ("

NOMNICLATJRE

a constant in boundary layer equations

a. constant in friction fctor enations1- .

(4 ~a u+Iw

R O perturbation terms of A

A cross sectional area of flow, ft2

A heat transfer area, ft2h

liquid cross sectional area, ft2

A vapor cross sectional area, ft2V

b width of liquid surface, ft

b. exponent in friction factor equations

B +2dy. o0

B 1 constants in boundary layer euations0,Y1,

c velocity of surface waves, ft/sec

cP licuid secific heat, Btuft4/lbf-sec2-OF or Btu/sug-°F

C 1 - u t dy0

C0, . perturbation terms of C

C correction factor for velocity distributionv

d licuid hydraulic diameter, ftht

dhv vapor hydraulic diameter, ft

d diameter of tube, ft0

D+ 0 tV

DI 1 perturbation terTis of D.. - ,%. . .

10i.

base of natural logarithms

quantities defined in boundary layer equations

UaYs, volumetric wall condensate flow rate per unit length,

ft3 /sec-ft

liquid friction factor

vapor friction factor_-_ -1/4

U ~ ay'1s -14 C) ) dimensionless flow rate

b__.Ae gAj= Z- , Froude number

gravitational acceleration, ft/sec2

function in boundary layer equations

depth of flow, ft

critical depth, ft

heat transfer coefficient, Btu/sec-ft2-OF

enthalpy of mass entering tube, Btu/slug

enthalpy of mass leaving tube, Btu/slug

specific energy of liquid flow, ft

3~~~~4 s -4/3 0 |Snf/3 0 do

JT2 sin 1 30 cos do

ofunction in boundary layer equations

function in boLndary layer equations

I

e

f

fe

fv

F

g

g(Y)

G(x)

h

chh

in

h.an

out

H

i

I1

I2

J (x)

J(73

g(&,, -p,)sin 0-wint- _+_ ^i -%' y-v- +tri n r i ra~jnr- +..qnn- _-- __.7- .

102.

ktg thermal conductivity of liquid, BtW/sec-ft-°F

K constant

t ~~~~~kea D1, )% o0e~k ADPC 1caratersti lengh, f

I~~~~| e characteristic length, ft

L length of tube, ft

{I ~ m exponent

n exponent

N(x) function in boundary layer equations

h l h

XNu , ,m Nusselt numberk k

NIu mean Nusselt number based on arc

Nu mean Nusselt number based on total circumference, 0 ta

p pressure, lbf/ft2

- 3P Ys

PV r ( + sin 0), vapor wetted perimeter, ft

P rOOc, liquid wetted perimeter, ftw

Pr k Prandtl numberk ' 1

q heat flow through wall condensate, Btu/sec

qt heat flow through bottom condensate, Btu/sec

volumetric licuid flow rate per unit width of channel,

ft3/sec-ft

C svolumetric flow rate, ft3/sec

licuid volumetric flow rate, ft3/sec

Qedet tvv; A -t l Iv -V -h miUl V % U n i /qtiArk

; iv vanor volumetric flow rate, ft3 /secV

::.! .

!

103.

r radial distance, ft

rO~ radius of tube, ft

Aerhe TF hydraulic radius of liquid, ft

rpx

v hydraulic radius of vapor, ftrhv p

V

Re 4Q Reynolds nmmber

1 Re& 4 eP g-9" liquid Reynolds number/-I Pw '

.1 ~~~~~~~I '

V n/hvpv vapor Reynolds number

s elevation, ft

s elevation of channel bottom, ftO

s depth of centroid of A below surface, ftc

t temperature, °F

ts saturated vapor and condensate surface temperature, Fi~~~~~t wall temperature, OFW

t t - ts w

+ t - twrtt~t At

t -t

AtT(5 ) s at

u velocity in x direction, ft/sec; except in complex trnems-

formation of bottom condensate cross section where

u coordinate in complex u-v plane

.', ' YsV'~~~~~~~~~~

:' " ua _ fudy, mean velocity, f/sec

.: ,- S.~~~~ ... ,- ~ uU/

'I

I

I

.1I

.Im

�k �.,

ut velocity in y direction, ft/sec

v coordinate in corn.olex plane

V velocity, ft/sec

. ... t... v o c f t/eV critical velocity, ft/sec

C

V mean velocity, ft/secm

velocity in z direction, ft/sec

w + iv, complex variable

x distance, ft

X dimensionless distance in direction

y distance, ft

Ys wall condensate film thickness, ftYs~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Z

Yso wall condensate film thickness at surface of bottom flow, ftYso

y4. y atc*

+y Y/Ys

Y dimensionless distance in y direction

Y. ~ Quantities defined in equation (3.38)

z distance in axial direction, ft

z+ z/L, dimensionless distance in z direction

Z At = r 5($c 2 sin lin 2, section fctor, ft2 ' 5

I. -

i

i

i

I

,

i

. I

. i�'4

- i

- I

-iil,6'.4

,.-I

105.

Greek Letters

Cf. --;; V3dA, velocity correctionAV3m A

ke. ,icp/9, 'thermal diffusivity, ft2/sec1

3 - + s) VdA, pressure correction

contact angle

mass flow; rate, slugs/sec

|F rtotal mass flow rate in tube, slugs/secIn

ra ? 'I ;9 m, ~ 'Iw~ -iE~ llZ< AI e-- -" ----vapor mass flow rate, slugs/sec

BDke A t

dAtio(1 C )n

I - . dimensionless temperature difference

1(xy) transform variable for boundary layer equations

at Ys

angle in condensate

Q c/2, condensate subtended h-lf angle

latent heat of vaporization, Btu-ft4/lbf-sec2 or Btu/slug

X ' ; (1 + C), generally C = 0.68

viscosity, bf-sec/ft2

ue liquid viscosity, bf-sec/f

.....}''c 4--yn. +.~_n r bf-se/-L2

kinematic viscosity, ft2/sec

I I te, sec

I

106.

liquid mass density, lbf-sec2/ft4 or slugs/ft3

vapor mass density, lbf-sec2/ft4 pr sigs/ft 3

ds0-o, slopedz'

surface tension, lbf/ft

vapor shear stress at interface, lbf/ft2

liquid shear stress on wlls, lbf/ft 2

surface inclination to horizontal. For round tube it is the

angle from top of tube to a point on wall

2 e , central angle subtended by bottom condensate

stream function

dQe /dz, axial rate of change of condensate flow, ft3/sec-ft

Subscripts and Superscripts not Defined in Nomenclature

( )xy,. differentiation with respect to variable in subscript

( ) at the liquid surfaces

( ) at the wall

( )w ( ( )" derivatives( Y' T, ( )i derivatives

r('I

TSPw0

Ol

I

4,

I

--Irmd:

.4ks,,,,

r107.

BIBLIOGRAPHY

1. Abra"oitz, M., Tables of FLuctions," J. cf Research, National

Bureau of Standards A7, 288 1951.

2. Akers, W. W., H. A. Deans, and 0. K. Grosser,"Condensing Heat

Transrer w1tlitn iorlzontal 1lubes," econL at'.L neat ransierConference, AIChE-ASIE Preprint 1, 1958.

3|. kers, W. W. and H. R. Rosson, "Condensation Inside a HrizontsiTube," Third Nat'l Heat Transfer Conference, ASIIE-AIChE Preprint

114, 1959.

4. Altman, M., F. W. Staub, and R. H. Norris, "Local Heat Transfer andPressure Drops for Refrigerant-22 Condensing in Horizontal Tubes,"Third Nattl Heat Transfer Conference, ASMAE-AIChE Preprint ll5, 1959.

5. Balekjian, G. and D. L. Katz, "Heat Treansfer from Superheated Vaporsto a Horizont.-l Tube," Paper No. 57-Ht-27, ASIE-AIChE Joint HeatTransfer Conference, August 11, 1957.

6. Benning, A. F. and R. C. McHarness, "The Thermodynamic Propertiesof 'Freon-113' (CC12 Y-CC1F2 )," E. E. du Pont de Nemours & Co., Inc.,Bulletin No. T-113A.

7. Bromley, L. A., "Effect of Heat Capacity on Condensation,"Ind. EnR. Chem., L No. 12, 2966, 1952.

8. Bromley, L A., R. S. Brodkey, and N. Fishman, "Effect of TemperatureVariation around a Horizontal Tube, nd. Enu. Chem, 4, No. 12,2962, 1952.

9. Carpenter, F. G. and A. P. Colburn, "Effect of Vapor Velocity onCondensation Inside Tubes," General Discussion on Heat Transfer,London, England, 1951, U.S. Section I.

10. Chaddock, J. B., "Film Condensation of Vapor in Horizontal Tubes,"

Sc.D. Thesis, I',T, 1955; and Refri;.:ersating Engineerin-, 6, No. 4,37, April, 1957.

11. Chen, M. N.., " nmalytical Study of Lazinar ilm Condensation;Part I, Flat Plates, and Part II, Single and iulti ple HorizontalTubes," To be nresented at the semi-annual meeting of the AS'E.,19.60, at Buffalo, N. Y., AS Paper Nos. 60-l-.A44 and B.

12. Chow, V. T., "Integrating he Equnation of Gradually Vari ed Flow,"Poc.c ASC,, 1, No. 8o, Nov. 1955. Discussion by A. S. H-arrison,

-roc. i.SCE. o32, No. 1010, June, 1956.

1:,.

.,,

2

e

.1

.1

R

� P, �

AL-1. -_

1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ rv

13 12.*T ~i f atpl Cc ffc -nt of C nrde-s nVapors," I ,f A : Ch... 7 .o. _ 195

oIban_ A .tP-oblems L Design -nct Research cn Condensers fVao-or s ' c Va r 7 i>ures rr' n. f , I4AS 1951.

15. Dixon: J. R.,"Condensate F- in Torizontal uoes, S Theis,PET, 1953.

164 Gazley, C., Jr., et , 'Co-Carrent Gas-Lia -d f-low," Parts ITa-d III Heat Transfer and luid o ics !nstitute, 19Lg,

17. Crivai, U., "Heat Tr ser in t'iTim Condensaion:" csc. n.ies. S No s 1, 10, 1952.

1. Ha1LnS N.P "Ro-fger' t -_ Coondensatiln si-e a Hrz-C2 Tube,"S.M- Tnesis, : .T 1959,

19. Hassan, . nd .. Jakob, "Laminar 1m Condensation of PureSatuarated Vaors on Inclined Crcular Cylinders,'" k84Ie ?er .T57I35 19 57.

20. Hasssm K.' :"L Amnar-Fi-m Cndensation of Pure ~ .. -Var-r- rsat Rest cn Ncn-Tscthermsl Surfaces." ASZ Paor _lo. 5-A-232, 955.

21. ITHrn-n , P., '"Hea Tra^nnsfer by FTree Convect in from H.orizon.l-Cylinders in Diatoric -Gases," 4NACA T 1366! 1954

2 2 Hinds, 4, "id Channml 8 Sn .i .G rn . T C . ns V.- 831, i926.

, a.e.gr,.' C., .1nieern ~Tid e'chanics I. .... , ,S LC, .Londcn, 1956.

2L. Jakob, N, at T r, nr i'.ley &sons, Inc., ew YC"r, 19L9.

I ,u T7.-,>o~~~~7 . .,~ _ < O 1O9~~~~~~X, tt-- 'X-Xi _ _ _ _ _ _____,,___________')4

2 6. Jaob, N., S. arn, and .Eck, Zs-itchr- d. Var. d;utsc. In,

27. -at- T . * a "Condensal-ticn of Frcn Cn-12 th innd Te 2'7:' 51 19297.--- . -- : -........3151 ,7

0- .

: -'_. '.7 ,~ . . . .. S -C,.

, Q C ,.t,_ _ _ _ _ _

i

109.

30, Tnith, E. H. "The Mechanics cf Iim Cooling - Jt i Jet r- ion.24, 359, 1954; Part II, _? No. 1, 16, 1955.

31. Lnmb, Sir- H., ?vd d cs Dover Publicatirons To- , 195.

32. Lehtinen, J. A., "Film Condensation in a Vertical Tube -ec toVarying Vapor Velocity," Sc.D. Thesis, iT June 1o7

33. Li, W., "Open Channels -with Non-Uniform Discharge," Proc. ACE, 80,No. 381, January 1954.

34. - McAdams, W. H. Heat Trsmission, cGraw-Fll Book Co., Inc.New York 3rd. Editian 154

35 F-ar waod, . . nd A. . Bennng, "Thermal Conductances and HeatT-ranssissi.c Coefficients of Freont Refriger.t" E. I. du Pontde Nemours & Co. 'Uetin No. B-9, 1942.

36. Misra, B. nd C. F. ' oniia, "Heat Transfer in the Condensation of..etcl Vc.-Ors, '.C.h-,. C en, Progress Sposium Series ,. 18,

37. Nusselt. I., "The Surface Condensation of Steam," Zeitsch d.Ver. eutoc. Ing.. AO, 541 and 569, 116.

38. Peck, R. . and N. A. 'eddise "eat Transfer Coefficie nts .orVaor Condcns .g on Horizontal &Tubes, Ind. n. -c-., 43No. 12 2926 151.

9 Potter, . C. and S. P. Patel, "Condensation of :Freon-12" InsideHorizont al Tube" Refr. ng.., 6L, No. 5: 45 1956.

40. Rohb'.cn . M.. "Heat Trnsfer nd Temerature Distribution inLarnar :i C tcnd ion.," ASY P aper o-. 54-A-144, 1954.

4.. Rohsencw, ' . : .. . Webber, and A. T. Ling, "ffect of VapzorVelocity on. LaIrin.r and Turbulent F'iLnm Condensation," ASIPaper No. 5/.4 -k;, 14954

12. Rouse, H., Eeermn Hfdrauli cs, Proc. of the .ourth HydraulicsConference, Iwa Inst. of ydraulic Research, 1949, John Wiley &Sons, Inc.

43- Scblichting, H., Boanmdar Layer TheorZ, 1VcGraw-Hill Book Co., Inc.,New York, 1955.

44. Schmidt, T. E., "Heat Transfer during Condensation in Containersand Tubes," Kaltetechlmik, 282, Nov. 1951.

45. Schrage, R. W., A Theoreticnl Stuy of Interrhase Mass Transfer,Columbia University Press, New York, 1953.

I

i

i

iiJ

iiiIIi

. 1-I

!10.

46. Sparrm, E. M. and J. L. Gregg, "A Boundary-Layer reatment of

Laminar-Film Codensation," Trans. ASE, Series C, J. of HeatTransfer, 81, 13, 1959.

47. Sparrow, E. M. and . L. Gregg, "'Lainar Condensation HeatTransfer on a Horizontal Cylinder," Trans AS, Series C.J. of Heat Transfer, 81, 291, 1959.

48. Staub, L. B., et al, "Open-Channel Flow at Small Reynolds Nmbers,Trans. ASCE, 123, No. 2935, 1958.

49. Tepe, J. B. and A. C. Mueller, "Condensation and Subcooling Insidean Inclined Tube," Chem. E_. Prog., 43, 267, 1947.

50. Trapp, A., "Heat Transfer for the Condensation of Ammonia"Warmle und Kltetecbmhnik, 42, 161, 1940.

51. Trapp, A., Condensation of Alcohol Vapors,'" Zeitschrift de. Ver.deutsch In. ; , 959, 1941.

52. Young, . L. and W. J. Woblenberg, "Condensation of SaturatedFrecn-12 Vapor on a Bank of Horizontal Tubes, "Tran m s. PS. 6LA,787, 1942o.

I

J

i..

m

i -6w.-.

LIST OF FIGUPES

FIG. 3.1

3.2a

3.2b

3.3

Condensate Flow on an Inclined Surface . . . . .

Geometry of Condensate Flow Inside the Tube . . .

Geometry of Condensate Flow on Complex u-v Plane.

Theoretical Heat Transfer Results . . . . . . . .

Specific Head vs. Depth for Free Surface Flows .

Geometry of Bottom Flow Along the Tube. . . . . .

Slope Optimizing Chart for Condenser Tubes. . .

Ratio of Heat Quantities Transferred Through the

Bottom Condensate and the Wall . . . ..

. . . 116

. . . 116

. . 117

118

FIG. 5.1 Calculated Surface Profiles at Tube Entrsnce ....

FIG. 6.1 ~luid Mechanics Analogy Apparatus . . . . . . . . . .

6.2 Liquid Distributor - Fluid Mechanics Analogy

ApDaratus . . . . . . . . . .

6.3 Schematic Diagram of Setup for luid Mechanics

Anamlogy Experiments ...............

6.4 Detail of Test Section for Fluid Mechanics

mAnalogy Experiments .........

6.5 Condenser Setu . . . . . . . . . . . . . . . . .

6.6 Inlet Side View - Condenser Setup . . . . . . . . . .

6.7 Discharge Side View - Condenser Setup . . ...

6.8 Schematic diagr-n of Setup for Condensation

Exeri ments . .. . . . . . . . . . . . . .

6.9 Typical Photographs Showing Flow Patterns in the

Condenser Tube - Upstrenm and Dowmstream ends.

Page

. 113

. 14

. 114

· 115

FIG. 4.1

4.2

4.3

4.4

119

120

121.

122

123

124

125

126

127

128

.

.

.

· * $

r

, ~~~~~~~~~112.

Page

FIG. 7.1 Discharge Depth vs Flow Rate in Horizontal Tubes... 129

7.2a Condensation of Refrigersnt-113 - Selected

Heat Transfer Results . . . . . . . . . . . . .. 130

7.2b Condensation of Refrigerant-113

Heat Transfer Results . . . . . . . . . . . . . 131

I

-A

CONDENSATE FLOW ON AN INCLINED SURFAC"FIG. 3.1

I

'if

r

wm <H~~~nn~~~~~~~-

C-

~~~X

7 -_ wd Li20~~~~~~07:7 - -- I-%~~~0

_,0

U*04-

C:0 .

u

-Li

0

03~ m

WO(1

C::-L> HcO

>- -- ILH 6

0 0Lo< C)- ~ ~ ~ ~ ~ ~ ~ ~ ~ I

V ~~~~~~~~~~~~~~~~~~~~~~~~~I|

0W0

0

LL) LN

LLi

0II0

IN4

I

-

f It C %

i

I

.

I

r)e:_0

C.",

i ·

-

- 0

0I4-

r 115

0

(I)

-JCt

L:

W

C:Li

H-Un

00

k.0

L_

0LUF-

I_

00C)1

' C) p . . to L

C,

Ca,_

04-

1 z;

I

I

I

:-71---;I

9a

Q

'i.-

r

H

SPECIFIC

h

HEAD VS. DEPTH FOR FREE SURFACE FLOWSFIG. 4.1

appe For HighRates

I OUr1CL rr LOW rilOWRates in Horizontal

Tubes

GEOMETRY OF BOTTOM FLOWFiG. 4.2

ALONG THE T- -I

I

-

I

117I:~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~/ ....;-:-------- -:- ....... ..CO .......l .- ......~::.::~~~~~~~~~~---4-_::,0":::!.. :t:"- ":'-.. 0995 '"-- .. 0.98 ': ..:- 0.97":'! ,6: 0,95;-,''t

L . .7 ..... 1 .......~ .... i ... ~ I . 1--'I .-I-.L l....l~l I t..i~ 1' l~l - -'- I . .1 ..' ....·....... 1.......-' ".............. : ...... ; .......~ .................~ ....-~ .............i ................... -........ t ...... : .............~~~~~.........~ ~~~~~~~~~~~~~~...... .......~.... ... .....t...'"...: ...... ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ *I,- ----

~ ~~~~~~~~~~~~~~~~~~~~~~~~~........:...... ...t....r.... ... t~;~......

-_I;--~.L. : . I - -I~-j. --- _ --. L .- ---- . i..... .... :. :-_ ·- L -.-. ... .... ....; ..- ....... . -II .-...- -.i----

!:::Z314

1' 6 :7t

Ie 8 -

is- 7..:;...--

-i.:- ":--- : '~ .~ ..... : .....T-?....,---m-:.....: .....:- ....... .... ' ................ :

: - ._= ;~ ' - - ,- ,. -- '~ ;. :'~- .....- - --- ........ ........_ .3_-': ...-.... -- .... . .:.:::.:±:.i.:: --.------ : ..... :--+4 .: ------ :---,: . ... . . ':---:~----:-:==== === == ======== ............:........

--... -.......~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.........:.......- ~ ...---------- _- -- :

. . . . . .' I:''

.i_'"-+�r�l�iri�i�. . r-' -- - - -. '

.... , I -..--.- _ .....-

J .$ I I .-- -- r....

__- -.~ _ -:': .. ... .:: ., -:---:-- - .-

-----_-_li .II--; ....'.......~--.. ..- ~4 ~.. ...... i -.- ..- .. -· :

_ _ :.- -.

: - _...

==:t-=

'..,- ...

t: ::

.

: .-. -

ii- -- -

f

__-.-:---

.... ..i

p:--::.. :-_.-

-i --

?....!--L_

L _ - ' --:3 :1'~

i:-----. j

. i__

------- .

---,-i;__ l

'~..

i-

i-'_:j

1 ____,

----- ·-- �- ---

;,..._i._. _.__

�--------·

---;i-�..i-. -·

i

.-. :---I�---�.: :::::::::::::::::::::::: _.- :

T... . . . . .

.

: T---_,-.

TT...'' "- 7-

-- ......-- "-'- ..........

f ... -- ;4--

'-:_-L --_ __- --

.... . ..... ......: -

!--~ T----�-. .. .....

'----'-------

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~- - - -. ,-:. ... '~ - ;. i ........ ;......

1'e93 . 4

1-944::

.:.97.:--

.7::.::.:

.....t0...

S C' ET_<JP_ 1it t N:::HAR 0 ::O N DE NS ER-TU B E S:: -- -:::::::

F~~~~~~~~~~~~

C.-Z

1-0

o00

CD

LIJ ~ ~ ~ L0 0 n

~~~~~~~~~~~--C.D_W ~O0 0

IO

pI-s Co (/)Q

0 : LI0 H

C7

0 _p

o 0

d~ ~

0 9 6L o o o0 <-' ~3J-SNV8JJ I-V3Ft -O0 l.LVb!

I

1-4II

I

W,r" e-p 11-11,

I . I I I I IX i-. #X

0= 111 Xx

//~,' ~, T--'~ A-A> , +0 I =30

Refrigerant 113ro 0.02384 ftAt= 12°FInlet Mass Flow Rate =2.4 X 10

xX NusseltI

Nusselt

slugs /sec. -

I I f I I I

.1 *.2 .3 .4 .5 .6 .7 .8 .9 ;.0 1.1

AXIAL DISTANCE, S/o

CALCULATED SURFACE PROFILES AT TUBE ENTRANCEFIG. 5.1

NW,

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0x-'4-

Clenen

z0I

-J

0.2

0

I..I

.'3

1t7. �r

"iA

_p

_

_

_

k

_

_

_

_

I'

1

. , ... . I"'. i"

.~ni ·-i'?

j')

?'

LICUJID DSTM IBUT2R ap

? .'YD . i ECI{ANICS A2.Mi APPAAiTUS

L-

, t

a0

- >

0

0,-:o1- of 0> =4 ~-:Rt

e P

c0I-

0

0:

0).-j 0q, 0

co > ME a >V) a Y l

C a: 0

.-0

C,o o .0-i

0c c

o .0a- uo 00 Cio .2:

C'i.--z2Ld

0

wtJ

CD0Ct-J

cnZ

ZC)

0UiJ0w

0LL

C-

_f--LC0z<:.

{,

II

I.

I

*I-1

,!

rr)

(0

L.

0

w1

0C')

0

-J

o oO-

C0c

-

CS

I

Connections _

LiquidSupply

Throttle

Venturi

LiquidSupply

DETAiL OF TEST SECT IONFLUID MECHANICS ANAL OGY Ev-EREMNN TS

- FiG. . -

r

FOR

'I

2

II

I

i

i

1 i- I:-1

~~'.,: ''~~ CONDENSER SETUPFtI. 6 5

,

..,

., .

.

. ..l:

0 .:,.

. . .

.': .

:

i -...,

. .

: -'.

.... :

,- -

.-, .

. ..,

.

. .

- ':

\-\, ,...,.s

": -. W-.'-

: ['.Ei,

1'-<. l D

1'. .

I': -F '.4

,..:c

' IT'S

: Of.:.-.

i .

-.:: -:

.! '' '''' '' - '.

') i '.,, '..

_

. .:,

. .- r--i

, '

_E . '' ',

'..'.:

Z A...

.'. '.

.,' .

....

.

i

j

I.

I

123.

INLET SIDE VIEWF

CONDENSER SETU'

FIG. 6.6

..- : @ -i . ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1 6 s...~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.

DICRIGE . ...DT±SCHE 1G i Z rDE VT- A

I-~~~ C CONDEISE R S ETP

~.:. ;~,~ FIG. .

I

I

7'' - I

�i ,

i .1

I ,"

i7l-L,

127

10

0

0

z

-u0_ a

o 0>

U)

z

i0:x2LLJQX0U)z02

0C:z00

LL

0-

C:)

Cd.0LI

00U)

,2

i

f f

I

i

128.

TYPICAL PHOTOGRAPHS SHOWING FLOW PATTERNSIN THE CONDENSER TUBE

UPSTREAM END ON TOP

-.. ~~~~ ~DOWNSTREAM END ON BOTTOM

FIG. 6.9

I

::

:

:':

:

_ I I I I

I'

i0 ~ 129

0

oQ

O o0

0 0NCt°l W

-) 0) - -

\1

IozW

c>.,

C

mC4,c

C

a

0.a-0 aO u0

o I I 0--- I -_ 0 I I I0 0 0 0 0 0

S 93- '1NV , "V c 'i N3S338931-0 + 3-9NV 3lV8-N30

Q

0

cn

-J<Zz0

Nt N6

Z

._!- o >

~w~> tz0 0 ..L

(0 09o >O J ,H,L0

<0(./)

0N

0

F .

i

I co

I

I

i

iII -i

i

I

I

i

II

A A

r130

Sym bo I

0o

A

-

Slope of Tube (sine)00.0100.1736

- Theoretical

400

300

10 ?I

-200w()(n

z

100

Curves for Vapor at

Exp. Variation of 0cm.120°-130080°-600° -

130 °F

95070'

3 .0 0 4 0.05 0.06 0.08 0.10

Cp lAtDIMENSIONLESS TEMPERATURE-DIFFERENCE- x

CONDENSATION OF REFRIGERANT 113SELECTED HEAT TRANSFER RESULTS

FIG. 7.2a

I

0.02

I

I

T131

Slope of Tube(sin e)00.0020.0050.0100.0200.0400.1736

Exp. Variation of Ocm120°-130°

110°

90°-100o800- 95°

70° - 80°

70° - 80°

60°- 70°

Theoretical Curves for Vapor at 130°F

0.03 0.04 0.05. .~~~~~~~~~~~~~~~~~~~~~~~~~~~

DIMENSIONLESS TEMPERATURE

CONDENSATIONHEAT TRAN

F

0.06 0.08

DIFFERENCE- CpI At

OF REFRIGERANT 113SFER RESULTSIG. 7.2b

Symbol0

A

0lo*+

40

30

2 001.

zI-

w

(n

z

I0C

0.02 .10

l r

-.Lj 2,.LX I~~~~~~~~~~~~~~~~~~~~~~~~~~~s

APPENDIX I

DETEI.NATION OF TE ITE FACE BOUADRYs CONDITIONS

133.

DETEPtINIATION OF THE I2TERFACE BJ'DARY CONDITIONS

To find the interface boundary conditions the vapor velocity

profile has to .be established at least approximately. The momentum

equation for 'an element of the vapor phase between y y s and y-~o

is, Ys

| yV aayV ~~~~+ dblZf~e y Y 0

pA ed> J U ddx~pv~tdY - O (A.l)

Ys Y/

In dimensionless form, using the same development that led to

equation (3.-8),

P, )4yt y+] P'[

- +,^, :"dy n0 f -j J I L

fdx- X

4Yf

di

( , (A.2)dy+- 0

If p< C 1, terms containing this ratio may be neglected.

Using equation (3.6) yields:

At, ~J' ktD (A_+3)UV., ~~~~(A.3).,, 'yt ; ?1 .(I .-T)

I

134.

The solution of this equation is:

Uv X4 LI5 e- II (A-4)

Now write the momentum equation for an element starting in the

liquid surface

Y;ciW e a ly

fd>fdXj0d~ q

s dx'SfY-

+ Pd

?U', /t

/aJy o

As

+ i 4

+ (

(A.5), ay i

ny* ,

s

tawf J 4 A

In K z A S

(A.6)

135.

APPENDIX II

SALVE CALCULATIONS

-IF -

36.

HORIZONTAL TUBE

RUN NO. 82

Given Data: Comparative Experimental Data:

r = 0.02384 ft.

L = 2.354 ft.

cr =0

t. = 141.8 OF1

At = 23.2 OF

Fluid: Refrigerant-113Properties

p g = 92.17 lb/ft3

?g = 0.6904 lb/ft3

A/g = 61.15 Btu/lb

$ = 0.0890

t = 129.1 F

Qo = 0.750 x 10-4 ft3/sec

Q o= 0.1507

0cm _ 115 °

k = 1.293 x 10 5 Btu/sec ft °F

PC = 0.9150 x 10-5 lbf sec/ft2

hin/g = 99.62 Btu/lb

hou/g = 35. 5L Btu/lb

(,cg) = 93.-30 lbf/ft3

Assume cm = 1200, e = 60 ° , = 1200

From euation (4.36):

3 ~ 15 3 2 3An=: 2 [,1.48 x (1.293) x 0 x (0.02384) x (23-) 12 3 x 1.240 x 1.564L 3 0.10x10 x (92.17 x 61.15 x 00)-

= 0.3375 x 10-4 ft 3 /sec ft

13 7.

L .3375 x 10-4 x 2.35L = 0.1596

_7o >/32.17 x (0.02384) 5

5 = 0.02545gr 0O

-tI = 0.0946

From Figure 7.1: 0c = 106°, ec = 530, 0 = 127°

Integrating equation (4.L2) from z= 1 to z = 0.8 yields on the

right hand side the terms of equation (4.43):

I or = 0

0.7964II -0.001417 x 133.4 x 0.02545 x 98.7 x (2.6972)2 x (0.008) = -0.000417

1III -1.125 x 0.02545 x 0.0946 x ( 240)2 x (0.64 - 1) = 0.01690(~~~0.208

IV 133.4 x 0.02545 x (2.6972)2 [0.04 - 0.001417 x 98.7 x :00

= 0.00779

1V -2.67 x Jx 0.02545(0.64 - 1)

The left hand side becomes:

Ae Sc]2 Afs c3J = rp 3- 0.07214

- f - o1 2

Equating the two sides

Aesc

r 3o

= 0.05505

= 0.07214 - 0.000417 + 0.01690 + 0.00779 + 0.05505

= 0.1514

,. _,-

Correspondingly 0c 125° Ic = 62.5° , = 117.5°

Following the same procedure yields:

from z = 0.8 to z = 0.5

from z = 0.5 o z = 0.2

0m c 113

0 =142 °, e = 710= 109°

0 = 158 °, ec = 79 ° , 0 = 10 °

M assumed = 1200

Assumed value of 0m is satisfactory.

The heat transfer coefficient can be calculated from equation (3.23)

or by the following equivalent relation:

n,,e ( + 0.68 )

2 7Cr At0

0.3375 x 10-4 x 92.17 x 61.15 x 1.06021C x 0.02384 x 23.2

= 0.0580 Btu/sec ft2 OF

= 208.8 Btu/sec ft2 OF

et (hin - hout)hm(experimental) = 2r 0L At

0.750 x 10- 4 x 93.30 x (99.62 - 35.54)21T x 0.02384 x 2.354 x 23.2

= 0.0560 Btu/sec ft2 °F

201.6 Btu/hr ft2 °F

Difference between heat transfer coefficients is 3.6%

Nusselt Number based on diameter is:

hd_n.o = 0.0560 x 0.04768

ke! 1.293 x 10

= 206.4

hm

r1 Ois

7-) .

INCLINED TUBE

RUD NO. 119

Given Data: Comparative Experimental Data

= 0.02384 ft.

L = 2.354 ft.

' = 0.1736

t = 124.7 F

Qe0 = 0.4955 x 10- 4 ft3/sec

Q eo= 0.0995

ti

A t

= 127.8 OF

= 94 OF

Fluid: Refrigerant 113

Properties:

94.41 lb/ft 3

v0g = 0.5496 lb/ft3

A/g = 62.32 Btu/lb

= 0.0347

-5k = 1.332 x 10- 5 Btu/sec-ft-OFe

Pe = 0.9941 x 10-5 lbf/sec/ft2

hin/g = 97.56 Btu/lb

hout/g = 34.-54 Bu/lb

(pe g~u) O = 93.68 lb/ft3

0cmcmT 580

iI

ro

II

I

I

I

1Lo.

As sunme

cm 580, = 1510, ec =29 °cm ~~~c

.a 2 [92.86 x (1.332) 3 x 10o-1 5 x (0.02384) 3 x 4 x 1.240 x 1.859

3 x 0.9941 x 10-5 x (93.1 x 62.32 x 1.024)3

= 0.2057 x 10 4 ft 3 /sec-ft

gL~,,5Jgr

.2057 x 10- 4 x 2.354

/32.17 x (0.02384)5

= 0.0970

f 2L2=-

gro 5

One

= 0.00943

= 0.1664

7I

ii

I

I

i

II

I

I '1

Substitute the above quantities together with the angle functions

corresponding to the ass-umed depth into equation (4.43), and set the

limits at = 0.5 nd z2 = 1.

I 98.7 x 0.08212 x o x (0.5)

II -0.001417 x 170.0

= 4.050

x 0.00943 (3. x (.484)81 x (-0.125)

= 0.00145

0.1664III -1.125 x 0.00943 x (0011)2 (1 - 0.25)

= -0.2013

IV 170 x 0.00943 x 0(3.08212 (-0.25 + 0.001417 x 98.7 x 0.125)(3.060)2 0.490

= -0.00301

V -2.67 xo 0.00943 (1 - 0.25)0.08212

= -0.2300

4.0500' + 0.00145 - 0.2013 - 0.00301 - 0.2300 = 0

Or = 0.1069

--"9 r-

i

Iiii

if

iiIIfIi

Following the same procedure yields:

= 300, = 1500

= 270, 0 = 1530

= 250, = 155°

= 23, = 157°

Plot on chart of Figure (4.3) and read 0

a = 0.0824

O = 0.1620

o = 0.2658

or = 0454

= 153.30 corresponding to

O' = 0.1736

Assumed value of 0 satisfactory

A maximum product of the other two coordinates (relation 4.44) is

1.874, and it occurs at about

Cr t = 0.2opt

142.

C

eC

ecc

C

143.

The heat transfer coefficient can be calculated from equation (3.23)

or by the following, equivalent relation:

fl , X (1 + 0.68 )

21Xr At- 0.2057 x 93.12 x 62.32 x 1.024

271C x 0.02384 x 9.4

= 0.0870 Btu/sec-ft2-OF

= 313.2 Btu/hr-ft2-OF

hm(experimental)

Qe..r .. t (h n - hout)21rc L At0

0.4955 x 10 - 4 x 93.68 x (97.56 - 34.54)2 x 0.02384 x 2.354 x 9.4

= 0.0883 Btu/sec-ft2-OF

= 317.9 Btu/hr-ft2 -°F

Difference between heat transfer coefficients is 1.5%

Nusselt Number based on diameter is:

hdm ok

- 0.0883 x 0.047681.332 x 10-5

= 316.0

hm

4

I.4

I-

144.

YInpINDTX iTT

- EPI-1,17PUT! DP�,L

----, - - ,

145.

TABLE A-1

EQUIP,NT DATA

WATER ANALOGY EXPERIMENTS

Test Section . .

M-lateriel: iethacrylate plastic

Length: 54 in.

~'iean internal diameter: 1.101 in.

Number of liquid feed locations: 12, equally spaced

Number of air bleed locations: 12, ecually spaced~~~~~~~~...

Pump5,.,'''11U

Eastern Industries, Model E-7, Monel Metal Pump with Obmite

Control Rheostat

-?. ~Liquid used was distilled waterS~~~ ..u IJNote on all experimental data

There were a number of eloratory tests made both before ald

between actual test runs. Also, in some of the emxperiments

errors or other difficulties were discovered. Each of these

rums had an assigned number. In the following tables only those

data are included )Jnich have direct and meaningful relation to

the topics covered in the text.

- -;;:m

146.

TABLE A-2

FLOW DEPTH DATA

WATER ANALOGY EXVERI'TNTS

Flow Bate

Q t.

ml/sec g

Agle Subtended

byr Liquid

25 Degreesc

L -zLocatio ,d

Distance

from Otlet

2 11.7021.28

4 39.265 30.18

6 25.457 37.45

36.729 34.5410 31.251 6.09312 8.50813 12.4814 17.66

15 19.4516 10.9017 15.0418 19.1319 8.6o20 33.0421 36.5729 4.1723 38.5824 31.1925 23. 0926 19.1127 1Z..3328 10.9029 3.29630 4.01531 1.59132 1.70933 2.81034 4.600

35 24.10

0.16160.29380.54190.41660.35140.51700 . 50690.47680.3140.0840.11750.17220.24370.26850.15050.2 0760.2640.39480.45610.50480.56840.5367o0.4305

0.318809.26380.19780.15050 .04540.05540 .02200.o2360.03880.06350.3324

109.3127.1139-6133.5131.0138.0142.2136.8134.788.0

98.1103.9120.0123.9104.7108.7115.0137.0137.1143.2144.0142.5134.5123.7118.7120.2112.589.092.587.889.689.393.0123.0

1.51.51.51.51.51.51.51.51.51.51.51.51.51.51.51.51.51.51.51.51.5

1*51.51.51.51.51.51.51.51.51.51.5

1.51.51.51.51-51.51.51.51.51.51.5

Water temr.:

Water tem.:

Water temp.:

Water temp.:

83.6 p

85.8 OF

80.8 OFl

82.5 OF

Water temp.: 83.0 OF

Water temr.i: 80. OF

No. Comments

_ __ ___ _� _ CIII� _1�1_ _I_ �____·�__·_il· I·_ ___I___

d

147.

TABLE A-2

( catinued)

l1ow Rate

__%b_Qet /ml~~sec r s/r

Angle Subtended

by Liquid

2ec, Degrees

L-2Location -d

Distance £

from Outlet

36 2.380

n

n

t

1!

it37 6.340TIif

38 9.200it

1T

it

39 13.96ItitItII

it

ifnn

40 20.167It

It11

n

41 24.53

it

if

0.0328It

nUn

0.0875it

it

it9

it

0.1270It

ifnn

0.1927if

if

I!gitn

0.2855

1!if

it

0.3382In

IIY?

tII

if

95.097.497.4

105.6119.1121.098.5102.2104.3102.9111.0122.2131.0104.1114.5115.1111.7124.0135.0140.0115.8127.0129.8131.014o.151.8S158.0120.3136.3143 .4144.3157.0166.0171.-5

124.2144.9152.1154.3165.9176.7179.0

1.58.17

16.3332.6740 . 348.51.58.17

16.3324.532.6740.8348.51.58.17

16.3324.532.6740.8348.51.58.17

16,3324.532.6740.8348.51.58.17

16.3324.532.6740.8348.51.58.17

16.3324.532.6740.8348.5

Flow is ltxinaralthough streamlinesare wavy.

"low is laminaralthough streamlinesare wavy

Xlow is redominantlyturbulent

Flow is turbulent

Rxun

No.Comments

111~~~_ _ I_ I _ · _ _ _ _I _ ---- _ _

ist

148.

TABLE A-2

(continued)

Angle Subtended

by Liquid

2 , Degrees

Location zd

Distance 0

from Outlet

42 30*8011

it

1!

it

43 42.45It

Itll

ifn

It

n

145 42.60'IT!

tI

1I

1I

it

46 38.87if

ifltI!n19

t11

47 35.5e3

It

IT

19

it

48 26.94I

if

tf

itn1

0425011

it

it

if

n

0.5855t

it

nifU

itI!

0,588

1tif

if

ifn

fl

0.5365Ut

ItM

it

0.4905

I!

it

tt

i1

it

19

It

it

if

129.5157.0165.0169.6179.0188.3189.4

179,O187.5190. *5

201.9211.7216.4138,0174.8187.8197.2209.32124.

216.7138.7173.0187.0191.7202.1208.0211.7139.9171.4156.0186.8200,0206.2

206.3

139.6162.0170.2174.0180.9g190.7

195.2

1.58.17

16.3324.532.6740.8348.51.58.17

16.3324.532.6740.8348.51,58.17

16.3324.532.6740o8348.51.58.17

16.3324.532.6740.348.51.58.17

16.3324.532.6740.83

48.51.58.17

16.3324.532.6740.8348.5

In the followingexperiments the waterwas obserred to clingto the tube ratherstrongly, Consequentlyall observed angles aresomewhat hgh.

flow Rate

RunNo.

mi/see

Comments

I

___ 1_1·--11 1--- -·l�-··ICii -_--·.1I·1-�·^··- _- I� ··�_·· I--�I�- --- ---

149.

TABLE A-2

(cotinued)

F;low Rate

m!/secQ& Q f0

Angle Subtended

by Liquid

2 , Degrees

Location Ld

Distance'

from Outlet

49 18.10 0.2500la It1I

I!

tI

ti

If

11

tf

IU

50 80.92 0.1231f!fif

11

11

11

II

it

11I1I!ll'

RunNo.

Comments

132.0145.3153.2159.1165 .4172.8174.0119.1126.0132.7136.0142.5150.'5

155.6

1.58.17

!6.33

24532.6740.8348.51.58.17

16.3324.5532.67

40.8348.5

L 11_ 111_ _ I _ _�I �___l_·___a · __··�_I_�__ · _II __ W____� __·yl�

j

150.

TABLE A-3

CRITICAL FLOW DATA

WATER lI¾ALT-JOGY XPTERIKENTS

Fiow Pate

ml/sc groQ 0

Description

of lowat Outl.et

TurbL entLaminarLaminarLaminarLaminarLaminarLLmnarLaminarLaminar

TurbulentMostly T'rbulent

Tarbulernt

Laminar but DisturbedPartly Laminar

TurbulentTurbulentTurbulentTurbulent

Partly LaminarTurbulent

*bun No. p2

Luid suprv neares tooutlet was closed in orderto reduce turbulence.Water Temperature was 80.4 F

Transition point is at

= .33

Equivalent Critical ReynoldsNumber Re = 3580

Bc-3 0

Run No. 44

Liquid was suplied uniforulythrough all inout locatians,including the one nearest tooutletWater temper3ature was 0.2 OFTransition. point s at

Equivalent Critical Rey- oldsNumb - er

rte 2,330

Commenrts

24.1023.7623 * 30rfl 4'Ft2 +621.96

2 SQ.22.56n I23.738

24.5023.6224 00

15.4417.3519.8419.2618.501i.838

17.2317-94

O*332,40.328o

0.32860.31010.30320.31140.32300.32830.33820.3260n SS3 '

0.21310.23950.27390.26590.25540.26020.2370.24'77

____1_1 · 1_11· 1_1__ _I___ __I · ·-�-LIYI�UYULIIII�YU�·�-�

IA

. 5L- :..-.

TABLE A-4

EQUIPMENT DATACONDENSAIGN E RIIv IT S

Test Section; ~. ;o-

Material: Std. 5/8" o.d. Aerofin conper tubing-<.

Length: 28.25 in.-. ..

Internal diameter: 0.71 + 0.003.- 0.000 D

.>-7· C vInternal heat transfer area: 0.3527 ft2

Number of fins er inch: 8

Fin utside diameter: 1.375 in.

'in thickess: 0.008 in.

.. -Thermoceuples

Materials: No. 30, Leeds & orthup copper-constantan wires

(Emf)CaOlbration: (Ef 0. -0.0048

bs.

Locations: One in the vapor stream entering test section; threepairs soldered nto slots at the top and bottom of the tbeat 1/6 L, 1/2 L, and 5/6 L distance a.long the tube; one-.directly behind bottom point at outlet end. in the condensatei.stream discharging from test section; to movable junctions.-... . ,

Potentiometer: Leeds & Northup Semiprecision Potentiometer

??Variacs: General Radio Company, Type 00-R- pairs s~General Radio Cmpany, Tlype W54T

Wattmeter: Weston Model 310, No. 3760

Leveling nstrument: Wild (Heerbrugg) Switzerl.nd N2-59138

,,Licuid used was distilled Refrigerant-13

..-,/,---!'due ~~ islldRfiem-1.. -.

152.

TABLE A-5

NOM.MCLATURE YOR HEAT TMNSFER DATA

slope of tube

room temperature, F

vapor entering temperature, °F

condensate discharge temperature, OF'

average temperature at top of tube, F

average temerature at bottom of tube, OF

mean temperature difference between vapor and wall, F

dimensionless discharge flow rate

mean central angle subtended by condensate, degrees

dimensionless temperature difference

Nusselt number

0r

tr

t.

J1t0

tttb

At

0cm

c eAtS-

hdnokI1

ct0 N N n O\ O m r t 0 N O O O N CE- N O C r( 0\ o t -o - o .0 H to0 N H\O 00 N LN N o GI U\ tn O r- > 1 H N -- N 0 i 0 0N 0 C N N -4 N N > t O rc-o -'- , " H ' CZ N --. O' n 0 N -O 0cN C N N N Ot N N N N Nt Ot H N N N N N N N N N N N N N N N \ Ni N N N N N Oi N N N

WN-(0 0toC to N- C\ -4 ONO C( 00 00x t:- (, O ' A, N-H W , - Oo wr O- 0 u0 N cN NQ --¢N N cN\ C H 0044 6 4 4 4 4 4 4 . 4 * 4 4 * & & * 4 4 4 & 4 4 4 4 X

L( 1 t0 w O0 -4 ' c Cm r H N H H H'-o 0 N N- 0 - \ t - 0( t C" 3 m L- 00 0 b3 -> U4 - r R(0m LroUJ r- Oi 4 r ' E0 o 5 0 b 0 - otC' LN tON t 0 v." Dn - ) '0 00 D M M

k( 1 i" 1 101111i10 i 00000 Q0 U 2 0 0aN. " Oj = = := :: = = = rH 9 C4 tY-b -11 N~ C 94 g=

0000 0 00000

C=ID

H4 4c~ 4~4 0 c cNc" m 0c'-N-rl z t: C . -->' CC I)-t, 0

C r rHI r- r- rI NH rH .C H rN l HH rt rH f II I I I I I I i / I I VC-000000000000000000 O C O O O 4O o o ooHC Z D H D D C0CD H O N O N O O 0 0

C~

ra0

-1 ' ,'i H H aH 'i HH H HHIIH'<"-H N2.-.-.. ' <- '.. -..- --. -. .H ,, H Hq Hq Hq H H H H H- j] H < r~~~~l r-4;}B r-{| c,,N oc r-[t r-I '- ri' -ll ,24 r-- _' '- o--4.. ,- o ', ' ,

=I =I t \- = = 2- -= r = I I I I I I - I = I 1 I Htf r- D N rC I C ' I t I i r- -I l rl r-I rt r- H H O 111111 OI O IC -D F00 H N- 0 00 Hd H- H- N N N Hd Hi Ni H3 Hi H H- H H H H- H 00000 00 H H 00 O 0 -~

0 0 Nt N 0 0 O to -tO 0 t O 0 C 0 N O'C O tO tO O Ot 0D O O0 O 0 O" 0 0D tr- 0 0t tO -0 L- 0 OH ON Cf ' - N N0 H H N -- r C, 0 C - O 0 0 0 t t to boO w to 0s tO ! t 1-O D t > t \Q `0

r- r1 c rd r i r-l rt rrl r'"! rl r- i r- ' H r- r- l I rd

(00 '\ 00L r H N N - cC In * tO c1 0 O O tO D C m tf O tO 0 O -U On -t 00 tO00 -tC 0 C,o Ol L- p.0 j O E t L 1 Cr ;t r -l O to Ch .- t O". r O o L-0i' 0o N0 O

xi 4' L-0 NtO o: clE L 1E P 0 ©D r O O - O' - 4-c , -IT Xo',, oh- , \ D P',@ t\)- tA o, o C Oi r ' ' O N C to t O'~ - OO H H H 0 0 0 Hi Hn H H H H H H I H Hi Hi i H -H H H 0 0 O H HI H 0 0 H H H H H H H 0 H

4 4 4 4 4 4C44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 .4.4 0 4 4 8 . 4 4 00000000000000000000000000000000000000000ooooooo

0 -4o O\H 0 -' 'D N O C C.- 0 C o 0 No N r- \o (co \ -0 r' -) C'- OH U O N, t- t to - H -- H r i ) st

rH - 0(H c2- O 4 0V 000 ( -c. "C i c,.OO N NQ O O N 4; r- OO C -,oo N '- OrHq r r r'l -H HH . C l ,1 C C N r- r ', r N N H H H r H ri r r H rH-l N r-i r

( C) O ON I' -+ r -- N - C- r - N 0' '- 4 44) U4 C, 444040o t4 4 4 , r

o 0 ON 0> pC N 0; H H H - 0H H N 0 rH N C O r- C 0N N N ' rH rH O rHr- H H H H H H H H Hr r-Hl r r r r r- r H Hd H

O O C O " O, C C \N N O E- 0 0(-0 N 4 4 4 4 4 4 74 40i 4 r- I ' 4W 4 - 4y t o 4 4

-- 4 ON H . 0 " O N " H N 0 (0 0 -0 0 '-.H CN, o c H N H N H C rN H N r

N (0 N o q C0 N N H Nd - ( 0 C00 N qNO' N-O -40 o4 N H H -0 -- N C O' 0 CO H [ -V Hrd Ct7 e < .- OX rv c< C^ < rd (< r-q C:'\ o ¢',/ ('~ '<-rr' r-i rd r'- rdt rd r- ri r-- r'- r-' rd r r-m r- rd rd r-- r-i r--- rd

r- C) - On 0 C- 0 I0 -0 H

rH r rd N rH r I r- -

H 00 -r·', ", qO.O

r- r- H1 H Her rdrS '- I 0r-I r-- rl r1 r'-

00 C) O", 0)0 O C r-D (I0 O 'n ONO 0\-) t O, ri ,C 0 Hd N rd H -rH ' Hr4 4 4 4 * 4 4 4 4 4 4 4 4 4 4 4 4

to c) O-r * &4\ o0 to r 1 InH N- N N OH N O0 - N N H 0 H NrH r r- r ri r r- H H H H H H H H Hr

CI t 0l f C0 0 Io 0 00- cN r- N N.N N C- 00C-) O C\, 0 C) 0(D O 0* 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

01 (I cr m t-rQr 10 o7 E-1 0 r-i N 1*0r. OO Hd( dN' O&% O H N Y CO-4'd -OCr-t r rN rd r r rd r l r r-I (\ N r- r r IH HI H H H H H H H H H H H H1 H- H H

rY %~ O C .e Xd rd' '-. '~ ° C- tC ~ oL)C00 cOl OH C -Ho - o r0( -I 0Or- 1 - rN- r q -t -- r-, - t N (0 NO r

rH H H H H4 H H H H H H H H H H

r- N -, r0 - I H C -i ( -4 0. c % -.t r- N -1- rk ) ) - C CO ¢ -04 H ri H t N ( 0(0( Ce C -;4 4 4 4 4 4 4 4 4 4 & 444 I4 4 4 44 4 444444 Jr oCD C C-o r- o O rQ -t t C2e t H \rd - N O U-\ mA o r- CY ON r- ol CJN rI Cb t C," a, rl rQ 0

su e 4> r~~~~~~~~~~~~~t C- C"\ -4- Ut- W> * ei o tl (" C,- H} N N¢ o-- rdT N,( -- M. c\l < ): a '1 CH H H H H H H H H H H Hd H Hq H H H H H H H H H H H H H H H H H H H Hi Hi H H H H H H H H

q- ~

otA cq o'~NtN 0 NC:V 2 - - _ tO r 2 = = C = - _ = = D- N = - = r ,. C = = C = = -.+=¢'4 u' O OO

C O S

O O 0C; O-

0 0 C0 0 0 0

r N C - C O r- N H u ) (0 C0 rH N ' "-t r- to O o rN N (t 0 r-Ho C O N 0 H N 0-r- r- r- N- N - - -N N to to tO tO tO tO ( o 0 C)0 0 C 0 CO O0 0 0 O 0 0 0 0 0 0 0 0 0 0 H r - HrH H - H - r- r r r- r-- i r- H H r-I H

m Ha) H

-t d

a)IC')

R

C>

-1

H

V

CEfE

,-o

3p

,04,-

4,

04,'

4,r

4,

b

01 rl-AHf1

O a to M 0C N (' C)

0 ) CM 0 0 '-0 C0 0 C CMC-- 0 H - H or- C- 0 M -40

a' 0 tX) 0' -< a' 0 CM H 0 0wrv~~~r Lro w qomt

C Cl C- C er - Ol o , -tO 04 o-

CMMC I I CMC 1 00000 :: 0000000-4r-l , - 4 s

H CIN (I-) LfN I

0 0 0 0 0 c ' C HO H OH t -p0 ) t -=PCM r-t r-t tCC cd trCf 1r>b£ gO~' O OOO1$tf -i

'C' r- CM 0 to 0 CM r- 0C" 0 CMC- C- t- - ) - '. 0 .O '.0 ',0 -- ,

N U- cr CM(\ 0 0C H 0 0 0 0HO tO --- 1 O H O-, in.) Oa

C u O '- r-! -cr- -t C- 0rHH H 0 0 H r-I r- l (H I rC00000000000CM C) - C -i-C(M CM to '0 0 t- 0n rto 0 tO C , .0') '. H 0-'HH H H H H H C C H

C 0 oh 0 ' 0 0 - - 0 - 0'-'t 0 )r H r-c r-t Cr - 0

~-~ ,--,, o ro · oo;.q 990% 0CM 0 C- '0 ' 0 a' I C-'0 0r"CM 0 C 0 H Cl Cr- o-0o H C rHr r r- r r r r r r- rd rH H - H H Hr 0 cr- C) C- 0 C- LI- 1 - cr-\ CC O CM ' 00 0 H- r 0 O' C C .

c l'q t - t r r0- 0 o H H- CM 0 , ! C - 4 0 Hr- CM H -

H H H H H]

r-l r -r -r -r- r N r r-t r--I rHcH) H H H H r- H , - -H -H -H

toHC--C4toC)-4to H --0 - --t o r- 0o o t o 0 0 H

L,,o r- Cw , r ( rl e% --, m U'N,H -r-i r- N tO O ,. C i -0. rH r" l r-H r r r -H H H H H H rqC-t C-

0 0O - C C-- o C H C 'H) HD H C CMr CM CM CM CM C')

H- H- Hi H H H- H- H Hd H H- rH

*g"_- - i- C-i ;..- -r; ;;;I;~~;_,

0,

o

-iP,

OV

0

.,o

*rtt4

H

-P-q

.n

4-

0-p

-p4

.

0 of4 t;

155.

TABLE A-6

DEFINITIONS OF SYMOLS

condensate discharge rate, ml/sec

dimensionless flow rate

anigle subtended by condensate, degrees

approximate dimensionless distance from outlet end,

fraction of tube length

Q

2e = 0C C

L-&L

3tr�

es�1.

156,-I !

TABLE A-6

LOW DEPTH DAT TA

Ca'NPDIS iTN ¶,'?? I R Si

OPiZCTT POSITICT

0e___ 2e L-zJl

Com.merl.t

8 0,O36079 0 477

10 , '{

!6 0.o301 .2 'A 'A Q-,.

X 1 0 65

1 5 73, S

16 i.7987 0 .8o5

18 1,01919 1,0o55Si,{) 1 b17 21 1,263'2 1

-~ 56831l 1 c"C"

t{3A6 1.72%32 0,913'21

1 PlO 4

It

0 N0'7O .3 l'Lte7lw0.02174

0.02894

0,03290,0 278'6

0 Oq 3 (0 ?.

0 .0" 7]a0 o,]3

C~0 /,. .) 2, 0 ,* S ...,

{,i,,{ ,dP0.

0 tSe.

015200.07'3

0.0S6950,!0207 12C

0 , 1 6"t

85

9591

90

36

0192.9

93

'2)!z .- 'P

97

103

100

1021, f

913

111, L.- 4

114'!I_011a

129'-4fA

N-ar Quiwst a Iu t!

Y! titt It

11 {,i

it f fII It

1J9MI tI t

tltlIt it

11"79

0.91Near ,~tle

0,.3

0.7It 9 ttt II

It tt

0,95

o950,a3

0 ,

RlunNo. C;e

DubiousIII

____qLEOjri�·________LYi�l�ll�l�·I ii�i-j�-CI=-Di�i���-WIY�-�-E-�I·�-_-LiC�

i

157.

TABLE A-6

(continued)

Quo

-~L-z

2&aComments

35 2.020t,

It

1t36 2.307IT

37 0.470

T!

38 0.526tI

I1

39 0.733II

it

40 1.224

It

t

41 2.4141t

42 2.816It

M.

0.1432

:tI!

0.1637n

0.0333TI

It

t

0.0373R, .

TI

tIT0.0519TI

0.0868

ft

O._11

0.1996Itti

107118123120110.51211221309497

118115

9599

118120

9310310610998

102102101113122120113117112

Near Outlet0.30.7

o.95Near Outlet

0.30.70.95

Near Outlet0,30.70.95

Near Outlet0.30.70.95

Near Outlet0.30.70.95

Near Outlet

0.7

Nea~r 13utlet0.95Near Outlet

0.30.7Near Outlet0 .30.7

No. Q&

Dubious

s w--

_; r

- i5St

.fe.0'

.' l'GF v _y: ,'a

9. . .

. .':-'

. ',_

¢w'F

a;

Fe!o"

kW

V$S,bzj

V

@,,_,_

@1;

ft

_S, 5;,

gr,=,,,r,!5,,

¢$ �s

. r.'

tt'

ot¢,4X.s'.

ki,_ .

ja: M

....,.,:'

:..,

,J''

'm'\ ::

D _ . '

. .t3

d

tt.

A i

_ ___�C�__ I____·I· _I·__I·__ _· _

iPPFhDIX V

"Y071UTFR PROGDR-ISS

I 5, 8.

fNO[IENCLATURE OR COIIFUT'R PROBGRP1 No M

(In order of appearance)

- 3p _yP :-- Ys

A p in z direction

A p in 0 (or x) direction

0

g(p~e - fzv,)

r0

[i 1/3s in 0 P (1j

931-3

4/3 + cos0

n~~~~~~ n

1.5 Pn ) z

r0

rAz

constants to determine total arc and numbers to

be printed

g( eC - /v)

A v

kt

dhvro

AV

159.

P

DPZ

DPX

x

Y

GFP.V

R

DTENiP

DZ

DPHI

M,N

G

VISLI

VISVA

COND

EVA

HYD

r 40

K

code for indicating last set of input data

START, CORUNT

REV

A

B

GPMEXP

HYDEXP

VISEXP

TAU

DRIVE

codes for determining printing sequence

ReV

a.1

b.12+b.r

dni

v

b.M 1

-C

3V 4 kt At

initial ys of condensate layerCONST

SUM

DEGAM

HFATL

DNUSSL

M

I-

1

Ys

dr'V

dz

rin A '/2 rroA t hLm

hLm

Q

JORhE

160.

GAM2.IN

161.

? 931-3

.,JOHN CHATO 763CONDENSATI TN A HORTIZONTAL TUBEDMF NS ON P(70),O (PZ (0 TO),O PX(70) ,X( 9 7 ,GRA V( ,S A -R(7 )READ .12EFL.'UtD,RDTE Mf PGAMVREAD 14DZDPHT lN ,GREAD 1.9 V I SL ,VI SVA ;,COND VAP

READ 8 ,HYD '-O l, 40R .FORMAT(7H FLtJID=IT43H R=Eia8'H. DTEP Ei0,b~6 GAMVE1,iD3)FOXRMAT- H DZ:E15, ,6H DPH =ES , 3 H MI ,3- :k=H T 37,'H GE Ii,,)FOPMAT(7H VI SL =EL 1 ,7H iSA=E S ,86 . CO.DI = - ,6H V A O= 5,OFRM AT {: 5 H 'Y - 1 8 3- =5 i ,3, Ff~et>4t n (DE!b. 1- .... _ . H. E~', ,

WIT. OTU TP ?4 DZ t TAP , I R i GAVF'.R E, , T T TA P - Z4 P H 1 ) N

WiRE OUTPUT. TAPE 2 ,12,VILI t DTISv4CrE ,CvA~,z!V.,kj r : ,. T D It3'3!' T AP',l ' '2 F 7 9 o H' E V f' v.r !)

GAM I NGAMV

ST:, 0A '- T *DO 68 I1 ,-- : S tX(T )=S*fDHI-OPHI/2'"'1/>, ~i~*HO f:, ... I- D 2

TF (EV-3000.) 21,23,23AIS:

A TO 25/A=0.072R-0,243

CA.MFXP=E.XPF( ( 2 ,+B .*LOGF1 G A A V) HYDFXP=EXPE(B*LOGF(HV/R)+49 r 5 X P p F P ( 3 -* t 0, l--'; r V i?\TVSEXPEXPE(B*LOGF(ISVA )T AUJ=A*HYDE MAP *GAM'EXP/ ( FP I SEXP*C* ( p -- ) )IF (S'TAPT) 27,27 7 rFn IV\. e T I <D*Dr TMP/- EVAPCONST=--=*XP.P f LOc-F (D V 'F*R /G) )/)

vf t nN z -tl-TlACC2 1',1 ·@4R ~- ,,D~ .... -£l . / y T>§' "; ;i5l

.S T T T P . T1 AY ,.,.')=CONCTI~ fl C 0 ADI''1, 2E T-START=,

DO 3,2 I=i2DPX )=-3 * - t )+,*P( I+)-P( T4-,)I. . · ,-m '2, , , L ; t,.

DC 4 / !=3 9 MfPX( I P)3,*P ) ,-,'P( .- )+P -- ,?)U .,M = 0 00 4n J=x

.UM.SUM"+i */Y (.3f F G t-4 2* C ON?.D D T fiEM' O p H -SU/',AP

nO 38 1 ,iRPk. (I)=G/ R' ( S I X ( I ) *y fT) *DPX(i T.' ./?,PHI)

+Pi (I Y T .T )*. r'5Fr(X(, )T

SLFAR ( I..) ]. 5* Pt I 3 At * ( 2, +' ) A.eFC;,P2,+A f4 ,/il VDPZ( I = iVE-rFAV\ )+SHFAP( I ) *D Z/TAUDO 41 I IP( I I !P 7 P 7 (t)

P T T ., P

I

4

t

I�1

4II

I

4

I,

"P

* *3

.4

0

2.4$

.9

rBr�o

'1

[5

i0

!2

.

4

-.6

A

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162.

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163.

NOMENCLATURE FOR COMPUTER PROGPRA No. M 931-5

(In order of appearance)

ANG, THE central half angle, &c for liquid, for vapor

Ae AA

r2 or r2ro ro- 0 ~ 0

rhe /ro

rh/ro

Ae sir 03

3

f sinl/3 do

0 3 / 4

(TEG)34

All numbers are to be multiplied by lOE

Example: 0.12345678E - 01 = 0.012345678

A

AREA

RPM

PRE

FRO

TEG

EXTEG

: .. ..

C .:: . JOHIN. CHMT, :C A c-. ' i-UC. ' ..QN-' Ci-i. : ''IEN L ON ANG i440- TH:t- I 44o ), AiLA '4.U , nn±+4U l+uriv i ±44u,

1PRE i440 ... FRO 1440: j, 'F-G 0 3- L'A iA- Jo R t ki j) 7

r 10. READ 1i, A CR i '2 FOR ;AT .( H a AC=.-..,3H Mvt i4)

g~~~~D i ;14 i -= i11i

:'.. AN' i S.'=$'*168.0./,R -T Ht I -5*J, 14j 59b, / RCARitA ( i ) = FH (i - i ( .' id i i 3

.iA.. !1HL iiARI ( / (i 4 * in I 'RH V ± A LA 11/ (2 * - ( [ C1.

-- ' ( i PR=1 ':t...oi 6: ( ( P,-1* i tU3 i ) ) Isr LR14 FRO (I) S6w TF d -'' o,_ ( 1 i'rA i i- i;

,-D "O 24 J=l 90READ 22 'T'EGi(J)

2 FORMAT (4 i;.). 4 EXTEU(J)r EXPF(O.7T'LOL0i- ('t'T(')3 ,

DO 26 g=9i 1/9a $1s'> C =K 8 O J

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-: i. 1 (ANG i 1=1,i) , A({} (AP( ii 11* ri )I=i Jt ,ji(ANG{I;PR'i),1=,t),(A-AGU )'FRO(i) ivi)

.' 28 -QRi'T 13( o. r A /);AliVE uUtPUF TAPE 7, 10 (LAiLu{(),o±;±cw

A FORAT {(4(i4,Ei4. j

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ST '77 7 7. ` STO j-f' itw it.: E ND (UsJhusUU)

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165.

1�

APPINDI- V

LTGLE FUNCTIONS --

All numbers are to multiplied ,'by 10

Example: 0.12345678E - 01

:r b`.

= o0ol?-3.,,678

II F --1, 1

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