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Transcript of Referencia 51
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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
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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
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C~l,-I
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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
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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)
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CV0
CV
toCVl
I.
S,
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oC) 43
0 v
o zCO 4-C) ::
oC1)
co
.4
e,
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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.
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Bureau of Standards A7, 288 1951.
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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
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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
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EXTEG
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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
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2 FORMAT (4 i;.). 4 EXTEU(J)r EXPF(O.7T'LOL0i- ('t'T(')3 ,
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APPINDI- V
LTGLE FUNCTIONS --
All numbers are to multiplied ,'by 10
Example: 0.12345678E - 01
:r b`.
= o0ol?-3.,,678
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