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'PARTIALW~TTJLLT,, E7NT 'I TE
FR
T1IE
TGc
,UF
DOCTOR F PHILUS,7IT/
at the
JIASSACHI-ST3T
I'TITUTE
OF TCTI-,TrOlIrGY
June 1960
Sign ~ataef
A4uthor ..........
De;
0C
'~r-i f-.
t'-%en
Mchsni cal
Meof
gn
p,
'
by .'-
14e
19T.I,60
.
T
AiX~~d., .....
:
T
,..::
m
V
SThesis
er7!sor
rt ...............
udents
Thy TE1
AF AY LTAlTeR
Oit2)RF
TIE LOVE OF ?.yENTIRE ±"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.
of Cincinnati in
In 1949 he entered the University
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 on
May14, 1960 in partial fulfillment of the requirements
for the degree of Doctor of Philosophy.
ABSTRACT
The fluid mechanics and heat transfer phenomenaoccurring during
condensationinside single-pass, horizontal and slightly inclined
condensertubeswere investigated
analytically
and experimentally.
Two independent analyses were developed for the condensate forming
on the wall. One was the development of the boundary layer equations for
a class of curved surfaces for which similarity solutions exist. These
equations could be solved
for
a circular
tube
only approximately. The
second methodwas the derivation of approximate solutions from the
momentum-energy
relations.
Both methodsyielded similar results indi-
cating that for ordinary refrigerants under normal operating conditions
the completesolution does not differ significantly fromthe simplest
approximation which is identical to the one developed by Nusselt.
or
liquid metals, however, both of these methods indicate a significant
reduction of heat transfer rates as compared to the simplified solution.
There is a discrepancy between the results of the two methods as to the
actual magnitudes; however, the boundary conditions for the second
method are more reasonable, and consequently, its results can be
considered more reliable.
The momentum equation was developed for the bottom condensate flow
and certain depth criteria were established. It was shown that a
relatively slight downward inclination will significantly increase the
heat transfer rates. Beyond a certain slope, however, heat transfer
willbe reduced
againdueto the changing flow pattern of the wall
condensate. A simple optimization procedure was developed for determining the slope for the highest heat transfer rate.
A fluid mechanics analogy setup was used with water
position, and a condensation apparatus was built for the
both fluid mechanics and heat transfer in horizontal and
using Refrigerant-ll3. These results check the analyses
experimental errors.
Thesis Supervisor:
Title:
Warren
. Rohsenow
Professor of Mechanical Engineering
for the horizontal
investigation of
inclined tubes,
well within the
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
.............
CHAPTERI INTRODUCTION
Definition of the Problem ........
Historical Background ..........
THE VAPOR PHASE.
CHAPTER II
. . 0
.
.
. .
.
.
.
.
.
.
The Momentum-Energy Eauation.
1
4
*
7
.........
10
...........
CONDENSATION ON THE TUBE WALLS
CHAPTER III
.
.
...............
10
Application of the Momentum-EnergyEquation to a
Round Horizontal Tube . . . .
.
16
First Approximation ...............
16
Second Aopproximation................
18
The Bomundary Layer Equations.
CILPTER IV
. . . . . . . . . . .
32
...............
THE AXIALLY FLOJ'INGCOIDEiSATE 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
Tubes...
.
Horizontal
44
47
54
66a
Condenser
. . . . . . . . . . . . . . . . . . .
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
44
. . .
. . ..
66b
67
Page
CHPTER
INTERACTION BETIJEN PHASES..
V
General Considerations.
..........
73
....
...
. . .
.
..
. . . . ..
73
.
74
The Effect of Shear on the Condensate Film in a
Horizontal Tube....... ..............
Numerical Solution
.. . .
. .
76
Estimate of Transition Flow Rate for Surface Tension
Effects .
CILAPTERVI
TEE
. . . .
. .. . . . . . .
E1Ei-INTAL
Fluid Mechanics
of the
Experim
80
...
.
82
TK. .
. . .........
Apparatus .
............
. . . . .
Procedure
.
OX
Discussion
82
.
.
82
83
84
. . . . . . . . . . .
84
. . . . . . . . . . . . . .
87
AND
DISCU'JSSION, CONCLUSIONS,
FOR FTU,
. .
. . .
Description of the Apparatus .
Exerimentl
.
. . . . . . . . . . .
ent Procedure
Condensation Experiments
VII
. . .
malogy Experiments . . . . . .
Description
CHPTER
. . .
_ECO2-TNDATIONS
. . . . . . . . . . . . . . . . .
89
. . . . . . . . . . . . . . . . .
89
NO0I'-C
. ......
Conclusions . . . . . . . . . . . .
Reconmendationsfor Future
ork .
.
.
. .. . . . . . . . . . . .
. . .
95
..
...
98
. . . . . . . . . .
.....
BIBLIOGAP
LIST FF
7?PENDIXI
. . . . . . .
............
107
.
.G'JES.
DETEFJ NATION OF T
P..P-IX..'.. II.,~ SAi2LE
CPLCUTA:IONS
11-~~~~~~
100
CA
L
,C ,,t.
.....
INTEPFACE BONDAPY
.q
.
.
.
.m
.m
.
.m
.
111
CONDITIONS
.m
.J
.,
.
.
.w
132
135
Page
APPENDiX IIl
EAE'.I
kL. DATA .IT
.1L
.................
Table A-1 Equipment Data
lWater Analogy Experiments.
.
Table A-2 Flow Depth Data
Water Analogy Experiments.
. . . . . . . . . .
Table A-3 Critical
.
. . . . .
. .
.
.
145
.
146
Flow Data
Water Analogy Experiments.
150
Table A-4 Eqauipment Data
Condensation Experiments .
Table A-5 Heat Transfer Data ....
. . . . . . . . . .
.
. .
.
.
. . .
.
. . .
151
152
Table A-6 Flow Depth Data
Definition of Symbols. .
Condensation Eeriments
APPENDIX IV CO:I°'UTER POGPRAMS .
APPE'.DIX V
ANTGLEFUNCTIONS ....
~PENDIX VI
E-Q T -I01T
D1:L~v',-N S
......
..
. .
...........
155
156
. . . . . . . . ...
.
158
. . . ..
.
165
..
........
. ..
^
. . . . . . . . . . . . . .
165
182
I
CHAPTER
INTURDUCTION
Definition of the Problem
The basic mechanism of condensation inside a horizontal tube is
the same as that occurring on the
of a tube;
but the geometrical
imposed upon the former make the problem considerably more
restrictions
complex.
outside
way to analyze the processis to follow the "life
The easiest
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
4-
The heat transfer through this condensate layer
5.
The axial
on the side walls of the tube
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,
it is obvious that the problem has to be solved ultimately by
but
inter-
relating all of them.
A qualitative
eaination
reveals that the most profnd
of the above phenomena imnmediately
difference betwen cdensation on the
cutside and on the inside of a tuberesults from the simple geomretr-ical
restriction forced upon the ltter
case.
This is that all the vapor
to be condensedhas to enter at the inlet, while all the liquid
condensedin the tube and any vapor to be condensed subsequently
to leave through
he outlet.
In other words, the total mass flow has
to pass through the relatively cnfined
itself.
has
volume created by the tube
T'his condition creates relatively strong interactions betwreen
the various
phenomena describedabove which usually do nt eist when
thevaporcondenses
on the outside of a toe.
Although in mst
the otside
cases it is more advantageaus to condense on
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 condensers
elcsed
prinmarilyin vehicles for outer space.
These tro
arnplic.ations, however. have a very basic difference between them.
namnelythe influence of gravity.
ifor an orbiting space vehicle the
terms horizontal and vertical have no real significance
-nless it is artificially
created:, plays no part at all.
andgravity,
In the case
of the "earth-bound" condensers, gravity exerts an al 1iT*tnt
influence on the henomenaoccurring.
In the foloing
chaptcrs this
latter problem wil be investigated almosti exclusively, first Oec se
of its
comercial
importance
endsecond because of is greater
complexcty,
The method of i-rvest- gation -1
gsien preusly;
tracd
"
foll essen tially
the
at is the "lih
e
h'istory" of the mediu -
cgnthe ,-ube.
or the :-juose
a-wna sis it
wil be
lin
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
condensate leaves the tube.
of
nly
As it will be seen later such a narrowing
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.
restricted
Consequently this investigation will
o the case of the pure saturated vapor.
be
/.
Histor'ical Backzgrmnd
The classical work of Nusselt (375 forms the basis of most
subsequent analyses of film condensation.
elaborated
on this
theory
results.An excellent
McAdams (34)
A number of investigators
and used it for comparisonto experimental
and exhaustive review of references is given by
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
on ans imrOvement of the originl
Nusselt
theory.
were based essentially
Peck and Reddie (38)
included an acceleration term, but their formulation of the equation
was incorrect.
ariation
Bromley,
round the tube,
t al (8) investigated the effect of temperature
then Bromley (7) and laterRohsenow(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,
relatively
high Prandtl Numbers there is no signific=1t
with
change from
the Nusselt solution; the heat ;,trnsfer rates ares-ignficantl
lered
forlicuid metals -,rithvery l
?randtl numbers.
Analysis of the condensation
uinsidehorizontal
Chalrl-> .," (10) who used the Nusselt analysis
togethr
uabes nasmade by
-ih
an emr -cal
relati-cn for the depth of the bottom condensate.
*
Iumb(ers in
aCrentheses refer
to numbers in the B1bliography.
R
5.
xperimental investigations of condensation inside horizontal, or
nearly horizontal, tubes were relatively few until recently.
et al (26) and Jakob (24, 25)
Jakob,
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)andDixon'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
velocities
existing in the tubes.
high vapor
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 15 o from horizontal found coefficients 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
thswork;
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 canlyfor 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.
trr.nsfer correlations
Until then, heat
ilthsuperheated 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
the
nclination.
ne-fourth power of the cosine of
7.
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.
information available on such flows.
I
There is not much
Analyses are practically non-
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 thnebottom
condensate.
T
the confined space available these waves have very
3
pronounced effect on the vapor flow.
In particular, they increase the
energy loss from the flow and, correspondingly
cause a greater pressure
The asymmetric configuration and the surface waves generated on
drop.
the bottom condensate indicate that the equivalent friction losses are
not mniformly distributed around the circumference of the flow.
Comparedto 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
lowerthanatthebottom
of the tube.
If the tube is inclined, most
of these surface waves disappear and the friction losses become more
uniform around the periphery.
4+
~
ccncept of friction factor
'or the purpose of calculations the
Ell be used with the numerical values taken
primarily from Gazley's (16) data.
As was mentioned earlier,thereis 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 maybe considered to be very nearly in thermodynamic
equilibrium with the vapor; that is, the temperature of this surface
maybe 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;
the pressure drop
and, i
t aken as
I
is small along the tube, temperatures may be
niform everywhere in the vapor phase.
the
n 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
discussed by several investigators
condition have been observed and
(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 tubewhichdependsnot only on the amount of superheat,
the magnitude of the subcooling of
the condensate.
but also
n
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 becamehigher 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.
;lat
~
knowledge the correlation of Balekjian and Katz (5),
!!;
~
Schrage's work (45),
To the author's
based on some of
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.
III
CHAPTER
CONDENSATION ON THE TUBE WALLS
The Momentum-Enery Equation
A solution to the problem of condensation on an nclined surface
I
A
may be obtained by writing the momentumand energy equations for the
condensate flow. By using the concept of similarity,
I
these equations
i
A
maybe reduced to a single ordinary differential
equation with only
.i
one dependent and one independent variable
.
4
5
I
i
S
I
..a
Consider the flow of condensate along the surface in the
positive x direction as shownin Figure 3.1.
condensate film thickness is ys
At some position x, the
The momentum equation in
the
x
i
-1
I
direction can be written
as follows.
d-y+#,T
f 1 4ax~dt+
0
5s
is
where
X
distance along plate, ft.
Y
distance perpendicularto plate into the fluid, ft.
liquid mass density, slugs/ft3
vapor mass density, slugs/ft3
a
velocity in x direction, ft/sec
liquid viscosity, lbf-sec/ft 2
/de gravitational acceleration, ft/sec2
surface inclination to horizontal
ml
*
an
L
~
I_
--
This approach is due to Mr. Michael
M. Chen.
(3.1)
71.
The enery
equation becomes,
AS
( Ys
~k,atl = ~-JA~zm ds -J:Pg (ts-t) p" a~y
.~~~~~~
0
k
where
(3.2)
o
thermal conductivity of liquid, Btu/sec-ft-°F
temperature,
OF
ts temperatureat the liquid-vapor interface,
°F
latent heat of vaporization, Btu/slug
Cpespecific 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
8
o
0s-t
'2.
I~~~~~~
c~fiM
, e+dyj / i+dy+
0
~~~~D
--a/Iw
,
Substituting these relations into the momentum equaation (3.1)
ignoring the contribution of the vapor phase,
since
pv
<
<
and
I for
ordinary applications,yields',
pl,/aions
/
.~~~~~~~~44
, R, ;(,?
(3.2) becomes,
Similarly the energy euation
'it
Defne, f
-
)~,
5
~~~~~~~~~~~~~~~~~~~~~c.~
t
dX 4tp
daxA~e4Y5
^y5sD
4
the volumetric liCuid
flw
a.s7
leng'h of tube, and
Thus equat-on (3.4)
C
(3.4)
rate per un
.
2
becomes,
(3.5)
?
It1-is
dx
(
in the e.n ondi:x""
shm-,m-.
* s Is.m
in t heacn...t.
~~~~~~~~,At
|
.1
.I
s
~~~~~~
+ C)
d
/e ai
this c
<<v
z
be done provided
i3;.
If there
s a sirmilaritysolution to these equations, then the
finctions of the velocity and temperature profiles, A
independent of x.
B, C, D
are
If, in addition, the temperature remains constant
along the wal, the equations maybe reduced to a single ordinary
differential
equation in the following manner.
From equation (3.5),
Ys '(kic ) S
Ys
ke^{tD
dx
(3.6)
dx
i,
dy 5s
dx
ket D
* .
_
Multiplying equation (3.3) by ys/
(3.6) and (3.7) together
I
s
dU
~-dx
.t
uaA and usin; euations
ith the definition o f
elds,
gr (pe - P,,)
FBP,#Y
df I C
RI-te
.I
e
(3.7)
t
peA +C ) d r
ys J. I
R
Y 2-,sim 6
lue ev,0
C(A Pg ) . Y sl"7
P./aU.
us
il
-I
BDket
d27
4 -
12;
4,-S6
5
CS)
,Al (I
z
' 1
pip- P
3
~
7
at
(3.8)
Rle f d
,i
w:
-1
Define a dim.ensionlessflow rate and distance as
,F
'.4
F f D3 (pt.-(o)k3431
F
.4
e is
(p ,)3
I _
dX
becomes,
= BDk4 -
1St
.
Ple A (C.)
sin
(3.9)
a characteristic length.
e*'
Thus equation (3.8)
-w
I
Rcf 1 3 Auo CS)3
where
ollows,
_
s=O= 3
+ FF'.
L
(3.10)
or
(1=8)
LL
+ FF"
(3.1oa)
(FJ
The -riht hnd side o- the equation represents the effect of
montu
. The results
:
I
I
I
t
i
i
_s
,=m.
'_
of .
.sse s aalysis
inc1Gicatetht
this efect
¥
Thus the rght
n
se
oi' the eq-atin
pert-ubat icn term,and t
Cr
appc~atins.
ce.te
may be cc sid-erad as a
ecaticn
may be soIved by successive
ns
r
The heat transfer coefficient maybe cmuted by considering the
heat tansferred at the wall.
<AA
where h
-k at} =~~~~kDkit
heat transfer coefficient
's'~~~~~
=.
.
Dkle
Pu~~~(+C:)
dS
Y7
en
_
,
-
_
_
I
-.
The local Nusselt nuber
I
5 . (F'
(3.11)
is
I
,A)A(I4 C-.
ke
-
-leke 4 LJ
i
I.
r
i
IIi
4
I
Ii
rpe
L(/se_ueke4 )A1
F
1
(3.12)
t
I
I
The mean 22usel
n=.Iber
beccmess
'}
,3
i
I
I4
f
I
~~~~~~~~~~~~I
I
It3
~NU =
' .F4pe- )I A F
L
i
!Lv
'
. ,,,
c(?,
4
.
-f,,)3,
-
I
,~17l)
,]
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
of the condensing surface, in the equations.
ry the radius
If the condensate film
thickness is very much smaller than r, the previously derived
equations
may be applied.
From equation
(3.1Ca)
i
,
~~~F(Ftt+_-s[
n
The boundary
condition
is,
( )
F=
0 at
8 F"
(3-14)
= 0 at the top of the tube.
Exp-ressF in polynorial form.
i
i
I
Ii
F=F + FE +F E .
i
(3-15)
I
lwhere6
it
was defined in equation (3.10a).
may be found by letting
"
O
II.=
E
approach zero.
F,-F-I-AInt6=Ot
-
Therefore,
V$~~
= |l
3
ction [finr'3' d#
Then
~
S
of
ubstitui
te
or angle
I-
g into
equa'ion
se .t1
l nuno er for a
from the
is tabulated i
(3.17)
AnooendixVI.
(3.13) yields the first ar-roximation
ound tube as a fLnctiono the -stance
opmost point.
7
.,
nuos k)~~~~4
*~
4jt
ke2
/f0
I
:p4
SrEC(-P
~N'Y
J
_
L
3
4
-
' ~~~~~~~~~~~~~~~~~~~
*
_-
iI<
..
(3.18)
X~~~~~~~~~~~~~~~~~~~~~I
.s.
If
the Nusselt number is to be based on the total area of the
=1+c3')
tube and not only on the p-ar where condensation acally
.
above eatation has to be multiplied by /~
N
4
3
0
L
(3
|~~~~~~~~~~~~r
(i
W
4r f1e(-a2
___
___
___
k4 A
R
r
(5'/3
i7
ccurs, the
0,14
ApL
-I41
(3.19)
As a first appromation,
Nusselts
parabolic velocity and
linear temperature distributions may be used.
i+
'That is,
3y+_y(3.20)
+
~*- y*
Then A
I
= 3 ,.B
=
6/5
C = 3/8, and D = 1.
(3.21)
The mean Nusselt nmber
1:
P gt/3i§40
Since
~~(3.22)
is usually very small, the equation may be eressed
in
a more convenient form as follow:s.
l
I
i
Ctin
.C
L
equation
are identical
The efect
V
J
,Leeket
iorsmallvaluesof 3
to
d
the numerical results
of the above
hose obtained by the iNusseltanalysis.
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).
3+ i ia1j(3.24)
(F.')
3 £iF,
(F.
+6J)L('+
)(ireF
Er')
If the terms containing
reduces to,
a3
and
are neglected, the equation
3 ation
(F25)
explicitly
olved
for F
can be s
as follows,
(3.25)
:Eqaion (3.25) can be solved expli citly f o F., as follcws~,
T 3
F,2)
E
F -3
F
LF.a-2
;E' -F
rq
= snr
I'
;_ .-
-
(i, f)
]_
2
_, .'
0
. - If
,14- 1
Fa
4'
)
-W
(I
,.k.
is
1)z
sin
0
40
,o,
Substitutinghe valuesof
and
fro
eation
(3.17) and simplifying yields,
4
I
14
-1,r
?~~~~~1LSJr
,V3~
dri~
0
Fo3i
Defining)
If
LI~mg
It
t ! g-ves,
andsb ti *`v
½~'
,
(3.26)
V'
-""C
S,
j---r
F-
_
Iz 0
I
a0
;nM? Cat
(3.27)
do
To
Define,
I
$Is
n3a 0 cos 0
-
di
(3.28)
0
Values for this acticn were obtained graphically.
Equation (3.27) becomes
I
F -lo£5
(3.29)
TWiththis solution the second approximation of the Nusselt
number is
4
x
(3.30)
12L+
x
o'+---I
or
I
3
X
,Uj A, A
3
.4
do
L
-C-
S+
(3.31)
F
?
-
To find improved velocity
21*~~~<_L
nd temperature profiles, the momentum
and energy equations may be solved for a fluid element between y and y.
The momentum equation is,
9
J
£
I r
i
,.
~fYsY
A
f
n,
c-dy + upj]- Udy 4. e
y
V
0
i
I
-Uoy
Ct
0g;0pg
Mfi
O's- Y) grOe -/7<) i' 5;(3.
2)
0
By an order of magnitude estimate of the vapor momentum equation,
which is developed in Appendix I, it can be concluded that
Hesyl dxJyeFer
0
Consequently, these tw.oterms effectively cancel each other,
although individually they are not negligible.
Using this relation
the momentum equation becomes in non dimensional form,
2
R Z +4yfe~
+
Pt C4 Y5
,
,u.lti-n
s~~e Ys F
y, ay,+
g by y/
d
~~~Yf
e5eg +2Ma'e
So dy
sYs/
y
Y+) t(Pe -p)Sl'7
pe ua,
nmdassuming again that the velocity
.nd te^-er-ture profiles are independent of x leads to the following
derivntion (si.il arly to equation 3.S),
i
F
I
Dkent
^;e
O ii+ c
+dXi
-"(+
I
0
I
d
3
.11
D kc At
e - /:W
(
'Idea
?e
c
I-
sin .
)
7)
or
dy+ = q -$son w
,.
dy+
A
r
{P4+
?.0A
( CI c
1
I
Y
-(F9*+?,]
= 90-
l)
PPuII
p r
- f
I
v
y
4
I
iC
I
,y
)-
0+
0~~
~-
-l(I-y)
+
Substituting the definition of B on the right hand side yields
t.
1'1
f
7t
.
dy
'~
---~('ty
Y) B~ +(,):
F~
+
I
y+
.
y*
+/c~
dY
Z
i
hs before, if E
4:
!V
+cd+tx4 y+(3.33)
+
CA
.
-04
.1
iz
+
o-
o
is small, the equation may be solved by succes-
sive evaluation of the terms in the series expansion for u* and A.
+
R7-Be,
+
+R6i.
Let
l
.Ao(/-x"')
*
I
i
[Ir
!
with the boundary condition
I
+*gy+tI
IV
0
therefore,
therefore,
t
U,@y2i
with
A0 = 3,
0
(3.6)
(d
dy
u
3y+-..
= 6/5,
I
2)
D = 1
++
(3 34)
y1#4+ 9 y
,f
Y~
0
0
-
Y-
4s Y i~+
4 Y S
(3-35)
T:
Substiut~itg into (3-33) gives the second approximation of the
velocity profile.
-.
-
y+) +,q+C°SY+z
{
1
os
4+
:
i ..
3yf4
co-hy
i
Integrating,
1 . R,
l
I Cos
'
(y . /4
_a.
S .
I, Cos 4
l
+
4
]y
[
3 +
2
Ico$
q 4.' v 30
1
y
+Z'
4
(3.36)
V
25.
Aply the bcunda conditions.
I
f
U
dy o
=
)dy
44
- -
U
I
I
+duy
Ju
5
Since the first t erm is equal to unity,
I
0
I, cos 6
a
'3l
'5
2
L
-
II
C
+ £It
P_00
I
+
IZ, coss
1J
.
I
+
7+.
+- 105
C.
'0
.s it'77
If
'9
t- 71
Cas
04f
¢
7S'
k4
(3.37)
Y
+
+3II CO
C9
14+
_
y+"
y+-
a
y 2
cs si+
Ii
-7
,
5
v tY
.,+r
(3 38)
-p
The enerj
eat-icn
ecomes,
A1~
V.
12I,
.4
. /L
ly
I-
+
0o
-
d
C0
uy
(3 ,)
0
y
or iunnon-dimensional form
I
k4
iYs
Z-
d
oh
= d Aplu,,,y
I'd ~i
la
U.
Mr
0
fw
dt/.
0
y*
d cAle(Ate&K t/7
Osl*d
c_ _
0
!
Jlo
1- 4Yi
y,
I
-j'e()
+vct)fL
eudy
y1
0
0 7
y~p(.
IL -
t). t
il
/-ke _~
'
ke
ti/,d~y _
0
t
ke
J(
Y
't (A dy~3
(3.40)
0
T,
27.
Substituting he value of ys from euation (3.6),
Ay+
Id d~ay
D /+.
_
DCYr
-/'-C '
)Y
F,
!+
*+
O
/
, a+y
-{
I
___
/+csL s
4
(3.41)
+1I
+4
4y
or,
Y+
fD
S
MI+
lCS
D lCT
I
d/(
+
t
++
(3-42)
+r /
Let
c+ Is +
C
Co+ SC, +...
D
Do+
~o
tY
with boundary condition
1
*1Y
. I
t.
at,++
'o
0
y
+
= 0 and
/+
Substituting into (3.42)
+
D, +...
= 0 at
,DoaI
tou y+
dt.
st,+
3
6
C, -
results,
LI
y2) y +_
>
_ Y
+SD,
-^43
0
'I3(y
/
YF
3I+ 41.
= 1 at
Y+>3)
d+
/
I Y
_OL
After integrating and collecting terms
3 .1~
+
4
1
Y+51)
II
To satisfy the boundary condition at the interface,
=
1+
. I
I
+1: -!; I _1
,
I+ !t
p
Ia
+
*
(3.43)
a
I
0O
L
I-
Y
S
/o1Im+]?
3'
)
04w
_
(I
4
1l-+.a
[
T'nis result is identical
(34/4)
+ _
B
,
I
_
s)
(3.45)
to the one obtained by Rohsenow (40).
Next find
I
C - LLt+dy+
0
/
Substituting from equations (3.6),
higher order terms containing
(3.38),
gives,
(3.45),
andneglecting
T
I 0fI(
'y-
)
Y+
+
0
-
(r
+5y'-yL] ;
29.
Y
-
a Y1
I
10(+.,7
- ,kY, It+
]Fy
l( a v
___I
J+Z3
a
I
y
40
Y+5
) dy+
or
3
C= I.
I-
-+6
-r
lo +I )
'i--4
Ir.
1 'r4
9
L
- A 4 I*-
I q.O
c -7 -,
a
+
Y6
(346)
sr40
30.
____
____
C = I-]
___3
I
+
l_!~'
104)
t , BM~~~~~~~
lO~~z+2!)_10(1+I
[,.
.~~~~~~~~~
$ S~at27
c050)
0 ao^46I,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
DD=/4
I+ /O
=
i_
10 +
nt
ke dr
(3.48)
jos4)
_(3-49)
-
ctt
a e
t
(3a 50)t
In the derivation of the basic equation it was assumed that
A, B, C, and D are independent of $.
obvious
then,
that
terms containing
the
second
I, cos
From the above equations it is
approximation is valid only where the
are
small compared to the others.
The results indicate that a small Prandtl number decreases the
Nusselt number; while a large Prandtl number tends to increase it
slightly.
The effect of the Prandtl number as given by ts
is sho-wnin
igure 3.3.
analysis
Here the ratio of the Nusselt number obtained
from these equations to the number calculated from the original
Nusselt analysis (which is essentially the first approximation
31
described above) is plotted as a function of the temperature difference
and the
Prandtl
number.
nese caJcuLaxlons
The angle term containing
snce
its
2 was ignored in
va&lueis negigible up to an angle or
about 1500. .or ordinary fluids
approximations are unnecessary.
is usually very small and further
The last approximation already shows
that similarity does not exist for a circular tube except at
4
= 0.
The change of the velocity profile is very small up to an angle of
about
.
~~~~.
.
.
X
.
1500.
'"
-arrrr
o-=U~e:Uly
Based on these observations, Chen (11) concluded that
..
1....
,uuo
!J
wyi~rwu~rae~zlu
aLy Deu VUL"1nu
Lj.yasunVgsn
velocity profile remains similar to the one occurring at
means that
FF"/('t)
=
ne
= 0. This
0 and equation 3.10a becomes,
(I + ) F(F')3
S;n
(3.51)
with k'(0)= 0, the solution becomes
.
(3.52)
The velocity profiles
now are similar and can be solved by either
the perturbation method or by successive substitutions in equation 3.33.
The results of this solution are also shownin Figure 3.3 together with
the results of Sparrow and Gregg (47) which will be discussed in the
next section.
!1
.Y, e
I
.,
The Boundary Layer .Equations
_
~~~~~~~~~~~~~~~~~~~~~_
The problem of film condensation on a surface curved in two
dimensions may be approached from the exact Navier-$tokes equations.
If it is assumedthat the thickness of the condensatelayer 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 sameangle to the horizontal. Thus
the Navier-Stokes equations maybe written in the following form:
<,a4
| ils
C-)
a
SC
th.
aC) av
+U,.--
g ax
+
g'pt-,2)sin
t x
Pe
g(p. PO,)Cos/ p
a2t
~ay
axe
__
(3.51)
d)
y
e +(e
de aJ
(3.52)
.(3.53)
r
~L(4~
.ax
Oy@X
(dh
~y
direction;
The x direction is along the plate in the downward
the y direction is perpendicularly into the fluid; and
is the down-
ward inclination of the plate to the horizontal as shomnin igure 3.1.
This notation is identical with the ne used in the momentum-energy
method.
The boundary conditionshave to be established. -.t the
wmll no
slipping occurs, consequently:
at
y =
u
=
=
O
(3-54)
.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.
-u---------------- --
I-r-----in
--
The order of mamnitude considerations anlied
the derivation
of the ordinary boundary layer equations can also be used in this
au
Du
@q+
case.
As a result, the Navier-Stokes euations reduce to the
following form,
56)
_______(3
-~
aX
-
a
(3.53)
ay
These equations describe the flow without any heat transfer
effects.
To include these, the energy equation is needed in addition.
The derivation of allthese equations maybe found in the references
(43).
or steady state conditions, the complete set of equations
aescriol-ngtne i±ow ancd one neat transler
=. ......+...
-+
oaX
.
-
Oy
Decome,
"'
0
"--
(3 57)
(3%.s)
34.
.3
I'
11P
;
p-t §t(- F a kedC
'}~
-
The boundary conditionsare:
u = v
at y =
at
(3.5)
=Y
=
and t = t
(3.59)
V
sy
andt = t
y
(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
Analternative approachis to
abovepartial differential equations.
reduce the partial differential equations to ordinary differential
equations by the use of properly selected stream functions.
latter
text.
method is described in the following
This
Denoting differen-
tiation by a letter subscript, the stream function,
, is defined in
the usual mannerthat will automatically satisfy the continuity
equation.
u = ~'
(3.61)
y
= -x
The remaining two equations
-y-P
-
y-"
yt -
.
(3.62)
become:
/e
ty =ty
-+
yyY
(3.63)
(3.64)
(3.64)
ke
where
conditions are:
The corresponding bamundary
at y =
=y 0,
, t
g
~~x
tV
(3.65)
=0
. t=t
aty= 'ys
(3.66)
yys
gQ,}
9(PY
-Pe)S'(3.67
G(
)
Let
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
,y)
where
(X)-
which will indeed yield this result.
I+
j _tf
I) a(
(J)
J ) ' L4r
+
4
LidJS
-
- '
+v +~+j
aby
2:
07
17Pry- -
9/r:a]
j gy
g ~~, +3j3j ~~~~
+J(3~
Z 3j~~~~~~~'¶~~~
=G+1,czF~~~~~~ci~~~(l(÷jbI
=
L.
qF~
-
%- -
7
%f
| +3J
J'a
(giJ
L
Substituting relations (3.67)
T
q 7-J + J
Ix
7
|
(3.68)
(3.63) gives,
and (3.68) into
atJ a
O- Y(xy)
-
,,r
i(jI)j
- _ I
(3.69)
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
(3.71)
= N(x)
Substituting into (3.69) produces 3
G va J ( E'q )
(3.72)
.Divide both sides by a2 (jj'E2 N 2 ).
NjJ'JEN
Nil
The coefficients of J
2
a
IE)
(3.73)
and J"', in parentheses, are constant.
Therefore:
NNi=
jSubstituting
from (3.75)
Bet, a constant
= $.
into (3.74)
jj" = o(')*
a constant
(3.74)
(3.75)
gives,
(3.76).
37,
all functions of
satisfying relation (3.76) will, therefore,
differential equation, provided
reduce the equation to an odinary
that the term containinrgG is also dimensionless.
Dividig
(3.76)
by jj' results in an exact differential -*iththe following solutions:
iO+
X
!
when B,
F3X
J
1
(3.77)
w
when B= 1
with the relation between Boand n:
1
1 -Bo
n
(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~(3.79a) 4~
.;
T -1-..
NlI~4ExIt
N "I I E. · f
I
'
(3.79b)
1
17,
ItE* 4Egl t I
(3.79c)
.1y
n3
(3.79d)
ElY- I 4
For the second class,
G G-xEs
-- Es.e
(3.80)
E'I
(3.80a)
N,',e
(3.80b)
E_
(3.80c)
, e- I ..y
(3 .8d)
Equation
(3.73),
defining J, now can be written in the 'folloing
form:
I
q,
(iiI
)11
)
G
YQ
.ED' -i(J
(3-81)
J3
This is a variation of the well-knownFalkner-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,
- ' "'",
I
39.
I7
such that
- B,Ej'y
7
(3.82)
= 0 when y = 0
(3.83)
Substituting into equation (3.64) yields:
et8
:- .
t,-(
8 C'ii
f
E J1 y= ft7
(3.84)
In order to render this equation dimensionless, define,
ts-
tst
t S- tw where
t may be a function
of x only.
(3.85)
At
Thein, if t
remains constant,
equation (3.84) becomes
( 8Ea;,..j '(- ,Ej'yT'A- TAt,)-(, , Eaj j $y+
4 aEJj')(-,Ej'Tt)=-),E'
(3.86)
ATat
Performing the operations
and makingthe equation dimensionless
yields,
ci
(L
j tJ
,
Accordingly,
if l
A t'
t
(3.87)
J
is a constant,
theenergy
equation
will also be reduced to an ordinary differential equation by the
previous transformations.
Thus equtions
(3.63) and (3.64) may be treansformed into the
following ordinary differential euations,
4
.i
+J
J
(
v
L,40.
(3.88)
i°
+
E7) (-'l
(3.89)
TIt a [J - EJ T] ( :.~i"
whereE 7 3-7
j,)22 E -
G
E
3(.
1and
~
-9
j
At'
t
At
(3.90a , b, c)
The boundary conditions are:
at
7= 0
at 7 =
, T =1
=
J,
J
(3.91)
,T = 0
(3.92)
These equations can be solved for any known value of
so In
order to find ts along the surface, an energy balance equation
may
to crosssections of the condensate layer,
between
be written
x =x 0 and x = x.
x
Ys
kf
tyyo
(3s
e/7t
uJy+fc uC(t t) c'
(3.93)
·
This euation is the sameas (3.2) the energy equation of the
method. Substituting the previous dimensionless
momentam-energy
relations leads to the following derivation,
d¢(4+)]d
- kj'(o) [4,i . At(4=)
pe
Ot I( + T) 2
.
I(oJBtJ j cx-keT
j (*)] W
() p j-,di ckxo)
BfkeT
Y
'
4E
*A>J
pi
L,,
')V,
J(Q5
- ITd0
XS
(3.94)
41-
The integral can be eliminated by using equation (3.89) together
with its boundary conditions (3.91)
I
-
(3.92).
and
=
[ 44i f 7( j()
k'() AP0
++El
J
If
--APea
.T(?,) +
] + ,4b6,J[(J~~~T)°+]J
'U.
0
8: ke'(o)
A .t(o) Cj(xo)
1+ Eq
(s I +~~~
R VkeA
A/l 92i
~
This equation then determines
7
s
(3.95)
If A t(xo) or f(xo) is
equal to zero, however, further simplification is possible which
results in the following expression:
(7s)
8, ke t
(3.96)
I
"-
I
It is clear now that
if At
is ccstant,
the above ecuatin
a may o
rat tne consan-
n equ$ation 3.88 is unity.
arbitrary.
nhdeendentof x.
is
set eualected
be
n
such
B
is essenti:ally
The equations then become,
J
1J+4 ='
+
I
+
~
-
+ I 30
t.
I
' I[ ¶'l]
(3.8Sa)
(3.s9a)
0
sy
w|-
f~~~
EJ)
O
(1
-79]
_
r'c$~~
1+
(3.9ea)
Examiningthe 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
at
p
5
=
1/2.
Since no similarity exist at this point, while
= 0 it does, this approximation is certainly no better than
Chen's described before.
Comparing the energy-momentum method to Sparrowls results, as
shownmin Figure 3.3, indicates that the previous one yields consistantly lower values for the heat transfer rate.
The discrepancy is
due obviously to the difference in the interface boundary conditions.
k~~~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.
fluids the corrections are insignificant,
Sinceforordinary
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.
in
the fact
that
Experimental proof may be found
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. Nowgravity is the controlling factor at least in the downstream portion of the tube.
This is the problem to be investigated
in the following section.
Free-Surfaceklow . . . 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
sides.
n the
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.
established
If the death could be
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
They
the civil engineers.
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
The similarity to the condensate flow
is discharged through one end_
problem is obvious.
Upon closer examination, however, some rather
differences occur.
imnortant
I-,
t'irstof these is the size.
Comoared to
even the largest condenser tube used in practice, the smallest lateral
As a result, the
spillway or any similar structure is enormous.
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
46.
1
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 seconaaecreases 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
What are exactly the characteristics
st.
of the phenomenaoccurring
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 crosssection changes along the length.
latter variation:
Of primary interest here is the
the establishment of the surface profile.
are actually two parts to this problem:
There
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 amountof workdone, in particular,
needed methods for predicting
by the civil engineers who
open channel flows (23)(32).
_ I__~~~~~~~_
____
_·~~~~~~
.... __I···
:········_··_···___I
_I _·?I_~~~~~~~
_~~~~~
_ _ C__I~~~~~~~~_I_
47.
The Crit ical Deoth of
re-eSu4rface "'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
ninimum
stream will go through a
energy of the
The specific energy of
he stream
may be defined as follos:
V.~
|(
9
~g
5
(
n~~~~~~~~~~~~~~~~I
I
specific head, ft
where
p
density, slugs/ftt3
V
velocity
A
cross-sectionel area of flow, perpendicular to the stream
ft/sec
lines, ft2
pressure, lbffft2
gravitational acceleration, ft/sec 2
$
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
the depth of the liquid by h, the
Defining
straight.
essentially
above equation becomes,
-V
(.2)
.. elevation of the channel bottom above a datum, ift
where
depth of flow, ft
Q
For a given flow rate
3
and flow rate per unit width
t /sec,
of the channel, q = b ft3 /sec-ft, the velocity may be expressed as
bD..
V Substituting
into
(4-3)
h
bh
(4.2)yields:
h+ s-4)
h
H
For the given cross-section taking s
0
M-II"J
.-
I
parameter.
.. Ln
n
t.A
(LA.
..A.I
L L J.r V
^Ir
.J
A.. ItLP
1J
A typical graph is shonvmin
4JItPIL
heat versus
= 0, secific
I l-:.J
I
IA (:
<.aJ
C-
1"
I
I
U
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.
4
~(4.5)
h;~~~~
Some physical significance may be found in this expression if
the corresponding critical velocity is calculated.
'13 h,
i'
_. ...
.... _-
-. - --_-.-
-- -
-
--
t(4.6)
I
49'
This expression is identical with the velocity of wave propagation
Thus the critical depth and the
deep.
on the surface of a liquid h
C
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
~
MnM--Tn" i
Tc (s
flhT
J
34IN
~
-TAr
a
,-T- 7
ad-
n
|
%3r_-
Id_*~
T
T
T
nn
T
T
n
|
s
1ni n; Il Ia+
n
iIM11
....
-0
Il
IAJ i
.
_1
M
show
....
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
f
4I-
i
50.
not separate from the tube but will adhere to the lip and flow don-i-
~i
~
By increasing the pressures in the flow at the outlet, this
ward.
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
+
(4.7)
3h
where
/3XQH
(4
3dA
·e ----Ai
(4.8)
JA
I
+
V
Thus OXis a function of the velocity distribution;
a function of the velocity distribution,
while
3 is
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 hydrostatic,
3 = 1.
curve upward
If-the streamlines curve docwtrard 3 < 1, if they
3 > 1. For fully developed, laminar flow in a pipe
= 2.
Now the minimum energy principle may be stated as:
3hZq
L -
ah
ah
1
51.
If it is assumedthat the variatian of o. and ( are small near
the critical depth, the aboveequation simplifies to:
iIi
i
II
I
~ ah
dz
QL + . O
(4.22.)
ZA&I
i
iI
Again investigate the two-dimensional flow.
i
II
T,5
-rh.
T P. i9 o
9(4+
=°
i
I
_A5%; +1=0
i
h4L a
3
a
..E
(4.22)
A 9
Thus non-uniform velocity distributions
tend to increase the
depth.
From the foregoing discussion the following general conclusions
may be drawn:
a.
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
V~~~~~~~~~~~~~~~~~~~~~~~~~5.
Now consider the circular cross-section of Zigre
A =~r 0
.;f-sshncos&)
r. (l-
h
3.2
(4.13)
cose)
(4.24)
(4-15)
__
Substituting into equation (4.11), and simplifying yields,
~i~~~
!~~~~~~~~,_ .z(~e ~~a9~'~ ,,,J's
i
3
gr. 5
a
-
S
a
Ae'
'
I Vr
v
c
,.
(4.16)
Sn
i
j
This function is plotted together with some of the experimental
//3 is assumedto be equal to unity, the
results in Figure 7.1. If
correlation with experimentaldata is rather goodat high flow rates,
above
/Vgro
=
0.15,andbecomesprogressively
worse at lower flow
rates. If, for laminarflows, Re < 3,000, it is assumedthat
!i,
= 2, the correlation
deviation still
becomes more satisfactory;
althoughthe
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.!?)
53.
where
Z
actor, ft2 ' 5
section
A cross-sectional area of flow,
btop
width
_it 2
of flow, ft
.'orthe circular cross-section at the critical point
j
.ICa
1.
i
ze_
IC
i
i
kI
(Of.
-sin).
c A
If
If Z
r
n
°
(4 8)
)
5
2/r.
is expressed in terms of h/d,the
can be approximated for the range 0.02 C
<
hdo
resulting relation
0.9 by the following
relation:
(4.19)
This expression is practically exact for the range 0.04
0.85, and it is 12% too high at 1do=
0.02, 7
too low at
0
i d
o
<
'd=
0.9.
~~~~~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^
= 74
g
or
A6
S
-
2 6 ( d<)
(4.20~~~~~~
a(l42
(4.2oa)
54.
I1
These equations then relate theoretically the flow rate
Ior the range investigated in the
critical depth for a circular tube.
experiments,
v/ii
he
ad
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 adhered 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 maybe neglected, and the flow rate
varies linearly along the length.
effect
His estimates of the friction
indicated that the calculated upstream depth would have increased by
not more than 10%in the most extreme cases.
55.
Let us examinethe problem of condensate flow inside a circular
tubeby developing the generalized one-dimensional continuus flow
equations for a two-phase mixture with condensation.
Equations of State:
For the liquid density
=
(4.21a)
constant
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 gAet
(4.22a)
b
where C Velocity of surface wave, ft/sec
gGravitational acceleration, ft/sec 2
AtLiquid cross-sectional area, ft 2
b Widthof the liquid free surface,
ft
In differential form
to
r
cZ
rt~~~
~Definition
dAe
A)
At
bi
of the Froude Number:
?r~-Qe -
-2
[F
25
FeI1
I
i
I
.A-
(4.22b)
It
Qe~b
c
Atc/M
3
Z +Ge
(4.23a)
6
At3
(4.23b)
56.
Froude Number
E
where
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:
c~-(4.24a)
%X
r',r',,,~
i
rt
I
+rw
* ~~
I-,,zF
r,,
."
de r
elo,.
where
1 , and
(4-24b)
.
dr',.
(4.25b)
(4.26)
- P at2t
PftVapor and liquid mass flow rates,
Q.Vanor volumetric flow rate,
slugs/sec
ft 3 /sec
Energy Equation for Overall Heat Transfer:
Ax4t = , (
ht)
Heat transfer coefficient, Btu/sec ft2
where
A,
(4.27)
°
F
Heat transfer area., ft2
ean temperature differential between vapor and surface,
Mt,
to be considered as a constant,
Total mass flow rate
,
and h.0
°F
at any cross section,slugs/sec
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
doe tm
+
/~ i~quati
:
S. +
+
A i
(4.2&a)
- A.
Q
P
29.4t!,
At
(4.2Et)
MomentumEuations:
For the vapor phase,
-A, dp -4 P,dz
(.4,29)
Av Vapor cross-sectional area, ft2
where
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
r P ,2
-w-
,(pegAe5Js+
I
rc-A- '
zo
S
44H 4J.
wall conjensat
.}
-
-
'I F--"'
,'.
t
where S
centrroid
f
of
Depth
ft
Af below free surface,
Licuid shear stress on walls, lbf/ft
2
P., 1'etted perimeter of liquid on walls, ft
I
w
it
Velocity components of condensate layer on walls wher
reaches the bottom condensate
I
The shear stresses may be expressed as follows
!
.i
1%
C.
.-I...
LS I
!I
r
Ircrl'-6-
Q-1--
(4.31a)
2-Av
a
Lw
-'c'ds
where 1, snd ~e are frictlin
2Ag
(4-37b)
p2
factors for the vapor and the licuid
m. cn be apro.-imated by the usuaI expressions in terms of the
.- R
h. -±er.
(L32a)
sCt(8c
4t,
iet
p/
Ba-sedon Gazleyts data (16), the corc-an
1¢
follr5
~ ~ ~~~o.
for Re <
for
o
>
Gecmet.;
ows
ma
be
L
*
ve
-,3
(l.33a)
= 0.085, b = , C= 10
(433b)
= 18
3000
..
3000
b. = -, C,=
'"v -
cf F'low:
'.~-.....'esnd½o
Ced
I
may b
-ren-tr cal fLazcticnsm'. be o.resced in terms of
¢,.,-
*
by the
ot.co-densate
e
layer.
h--'
59.
(4
2 .
u0
A~~s~~u
,, ^,3r4
62N(I-5ino.Zc
et() sn c](43
i%='a(W-_
ArO(
49c
s$
dC
(4.34c)
4 )(4-34g)
(4.34e)
2in
^w
e4$4)rO
~~(4-3Lf)'
(l-
P- t-7rO c -&j'ne)
-(4-34g)
pW~ro 9C
A1e5s 0
(4-34h)
CoC5s<e
'
~11
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
0
of the first approximation (see page 23) and Rohsenow's (40) value of
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
naA3 4
r
3trtC3
J3 L
S
0
1RJ
V
hI
J
aJ
-
d3(1J0.683
(4.36)
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).
U cry
P
12
0
WQ/co-iensae
dz
t
A,
.
60.
i
i
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
the magnitude of the term.
reduce
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)
has to be solved.
r
3
p gro3
j~~~~~~~~~~~~~
Dividing this expression by
and defining
ds /dz results:
9
(/tt)
w
[AR
Aeo
w
Ts
( I~ ~(I~
,7egr.3
2<tizc
-l {X)
dZ(ig) j
Aec
(4.38)
The numerical solution of this equation may be simlified
folloing
assumptions.
First the vapor friction factor, fv' may be
Since the
considered a constant, 0.0085, even in the laminar region.
vapor shear is very small for R.e
v <
not introduce a serious error.
3000, this simplification ,rill
Second, the liquid
feed rate
=Ld, /d z may be taken as constant along the tube.
evidence, as shown in the data in Appendix III,
that for the range of flo; rates examined
uniform
L ..
by the
or most prrt of the tube.
Table
Experimental
A-6,
indicates
the liquid level ws
ves'
For the inclined tube the liquid
61.
depth was observed to be essentially constant after the initial licuid
ramp reported in Appendix III, Table A-5.
Examining the angle function
tabulated in Appendix VI, reveals that ta is not too
in equation (.36),
°
°
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,
where the
I
(4c3.
4
C was taken as unity.
factor
Qeu XI=
Su. Subtittn
=u - yield)
- i.
(L.40)
tt =) int
into (k.35)y~_elds:
Substitutingff
,
F.. =
=~~~~~~~~~ -=
fi~~~~~~~~~~
I
n
AepefI
Ar
g~rt'
_ -b
0004__~
Aer
3
a
.,g_
__
1l A 4 -Ra- .'
3
L
The licuid
actua:lly the cse
04oo2sR(L-z)
Ab
I1
d
3
j
c0.
At
}
3
flow was assumed to be laminar ever,,here.
for all condensazatonexoeriments.
(4.42)
This was
.
-•~~~~~~~~
~~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,
tranfer
atevaris
heat transfer
rate
varies ascos/4
as cosl/(sinO
infinite circular tube.
As long as 2 r
1 ) during condensation on
tan(sin
X
)
<
n
L, this
relation is the best available estimate of the influence of
on the heat transfer through the wall1 condensate.
he
nclination
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
but tilting
= 0.010,
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.
such conditions the flow
depth starts at
One may argue that
zero at the inlet.
ncler
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 exoperimentssuggest another type of boundary ccndition, which not
5~~~~~~~~~~~~~~~
011-L
",
es
VJ.LVVIZI
t:
~
9VUU
extremely simple.
T-t :;:1UU-LU--9
i31C--/
.UtAU
Ullt,
A*e*
:iiLlr75V"
Lin
es
-me
IUU-lr
hbe
..........
BrloQa1A
n1$o db
UII
63.
m~Again
the experimental observations revealed that the flow deDths
i
I
were very uniform along most of the tube.
i
I
ccurred near the entrance
uniform depth
tube, very close to the outlet end.
The only departure from near
region and, in the horizontals
This observation suggests
or a
boundary condition that the depth should remain very nearly constant in
the downstream portion of the tube.
I.
Such an assumption will lead to the
elimination of the terms containing area variations after equation (4.42)
is integrated, and the resulting relation becomes:
L
Ae
/
LrL)L
~(u
-z*-
... gro.
J
00417IleSY
7
L
h~~~~~
where
i
L..~
4. !.
i
Ia~~~~~~~~~~~~~~~~~
~
Sim]
6,
~~~~niae
itro.
i-iso ntarto.
CR
(4.43)
(p3
limits o interatiC.
I ] indicats
[
InIctsimtofitgaon
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
t
,,roc- ellureP+o fi ndl+1
wrt
+kheI
e-Y gmon
hihs a
t
he.1attr"ns-fer rt.
has been shovm that the heat transfer coefficient varies as
1/30 d
d
inl/3
13
plotted against 0 (or
scs/4
1n
-l
and as cosl/(sin
1
Therefore, when c
o).
is
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
]41
I/n
0
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
values of 0 for test runs with various flow rates.
These values were
commuted by integrating step-by-step starting from a knom
outlet end.
The agreement between
measured and calculated
mean values for
only fair. but the obto-ained
[vzc~ i'-n
hf.n p
due to the insensitiveness of relation (4.44)
L
of the
0 over the
depth At the
ngles is
ength of te
,l..;fenrou"
to variations of ~ in the
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
.1 L'..-
I , m,
-
\C 0
*
r-~
ro
'tC~
U)
o .o
sc0
* -4 -.
O
00
I
o
H
r-4
o
Ho
Ho
HH
H
i
oH
O',
xO
r-
'£c
I'
~D
-~+ c
0
to
0
cv
C~
rI
H
00
(N
cv
II*tr*
H
H 0H
C~
00O
14o
afi
* HH
a
NH
0
Oa"
0
00
OH
H
rH .O
r-
E..
cv
0
Ox cr
0o
4
0
0"%
OHH
o o
1H
00
C0
0** 0g-
r~
-Ct
0
Hx
m
zc9
0
H
o
o
H
Uc'c
*
OI H
H
CO
i
cy
C~
cc
oH o
C"
cv
o C'
o
UL
H
o o
3
c~
o
0 r-
-4c\c
s
r-
c,
o
0
-I
0
*
CO
U)
E
I
0
cv
cv
cN
OH H
co
r4
o
CVL
H
C HOHo~
Cco0
I
H oH
0
c\
H
0r-4
C'H
0CO
O OH
I
00
O
o
o
r-
H
H 0
n - -.
0
cv
cv
C)
C)
bi)o 0~ 0 ~
~-
_ cD o
I
C)
C)
+ 40
4-*4
C;
w
I
-.
C+
-P
Ch
o:
O
,)
~
q
i
0h Q*l
4-)
^
S !--.
2>
-
rnC' ¢,
C,
ca
o
U
U
O
4CQ2
&
U
O
t
~
HC)
C
.
5s
C)
C)
to
r-3
+~
O
>C
>2
U
C
4-4
-)
E4-'
C)
10
C.)
4-)
+~
C) o cqJ
oo
ca
o
4
C) P U
C) C) -4-H 4- t* H 4C)
C.)
Cd
r-.
G) :
O
0
c.c
u
-p
m
:
t1
C) :
C
Ci 0J
C)
4
m
66a.
r
66b.
o
0
0
tOs
H
r
n
to
H
ot-*
O
<M
,.-I
C)
to
-4
~-
Os
0
Em-~
Cr/
CVl
o0o
.
O tO
CV H
CV
H
H
0
O
0".
0
0
C--'V
o
r-I
00
H H
.
*
CN
H
r-I
0
tO
CV
* C)
cm
*
to
C-
r-iH
o0
0
CV
C'¢ H
r--I
CY
S
CV
0C,
0
0
0
CV
CV
0
rXn 0H
c~
C\
HoCY
r--I
H
r--I
,--I
r-t
H
E-
0
0o
0o
r-"'l
C1
r4Hr1.
I
0 r-0
PI,
L)
nq
r2
¢
\ D0
O
0
Os
00
C+
0
I
C'H
r-i
I.
CY'\
H
0
0H
0O
0
0
E-)o3
:I.,
*N
r)
EA
pqH
0
'H
I-H
I
,--I
r-4
r- H
co
-'
*
0
OH
0
bjD
0
0
C~l
,-I
H-H
I
t£
0 H0
O
o+
0
tC\ t(l\
(NCV
C)
-4-')
oC1)
H
d
.4-
r
co
4
Iz
.4
e,
Cl
0
ul cd ,0
^,
Q4
o C)0
o-<
o H H
-~
cJ
-H43
aJ
0
o
4-> --
0
Co
'
S,
-'
-4-3 0
J.,,S
4-'>
CD
-H
C.)
0
44
c-)
co
H
C
0
U)
0
ao'
C)
H
(n
O)
r.-4
o
43
0
v
o
CO
4-
C)
0
4) ,QU
0.3
CI-4
C)
u
n
O
0c~
a)
C)
C)
z
::
67.
Heat Conduction Through the pLxi y l
oyn
Condensate on the Bottom
of the Horizontel Tube
2aronly heat transfer through the thin condensate layer on
o
I
the walls of the tube
has been considered.
To complete the
nvestiga-
heat transfer through the axially flowing condensate on the bottom
^tion,
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
ferred across this condensate by conduction only.
cross
section
is shown in Figure
corner A shown.
3.2,
rans-
The geometry of the
with an enlarged view of the
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,
ture
anduniform wall tempera-
t w . In order o further simplify the solution, it maybe assumed
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
=.
The bund-ry
at- r
conditions
at r cs
= r
dta
°(4.45)
re:
cos
(44Z6)
tto
=
r
art
c
t = t
t to he edge of
the film
(4 47)
68.
In this form the problem becomes one of
ure conduction.
The
particular configuration involved suggests a solution by conformal
mapping.
sho*
The equation of the tube circle in the x-y coordinate system
is
X _+aY+?.c'
0 -a
x+iny2+ycos
+y2.yr cose
I|Divngby2
ividingby
Diviing
2
byyil
sinc
2
r2as $,
rOsne
and defining X
,
c
(4.4a)
- /rosin fc and
C
Y y/rsin
&~c
yields:
y/r.
i
r
X'+ Y+
Y cot e-
The moon-sha.ped cross-section
I
ABCDEF
of the bottom condensate may
be mappedinto the infinite slab A'BtC'D'EtF shomwn
in Figure 3.2b by
using the complex transformation equation:
.
In~- I
~~w;
where
= u + iv and
=
+ iY.
Consequently,
W-nx-,
Y
Multiplying both the numerator and denominator by
+
simplifying yields:
In~jTX+
W
Since w
'
2Y
a
4
u + iv,
e ConsV
= equ(cosVentlyv)
<_j +Yy -
Consecquently,
e COSY
-
Xa+yaI
U
X )Z2?
and
e
(A
Sr
_2y
tyA
- iY and
69.
tcan V
YZY I
X
The temperature distribution in
between t
w
along A'Bt Ct and t
s
ttw
en. #
the complex u-v plane is linear
along B'EtFt.
Mathematically:
(450)
t =
or
Suibstitutingthe value of v from equation (4.49) gives the
temperature distribution in the X-Y plane:
2Y
,
+
X
(4.51)
+Y
The heat transfer through the condensate may be computed from:
- s(t, j
-
ay5o
Ys
)
-Q
where q' - heat transferred through bottom condensate, Btu/sec.
r
I
Substituting the dimensionless relations and using equation (4.51)
yields:
(
dA
YS
~'-(1
- rsi ' tt ?' '5 d
L-
t-xj
ro6i'
f -
/-+ I
±-S
Y so
rX0
V5,
=Zkg$
a Jrn I -
I
s
£~~~~ ia
''-_ain.Z
i-
_Zke~t
L
Yo
2
3ke4t
'/r= 7Cr
_
,)
[Lso
- I
(4.52)
71.
of Heat Fluxes through
Comarison
he Bottom Condensate and through
the Film on the Tube Walls
Theheatflux per unit length through the condensing film may be
computed from equation (3.18) with the constants taken as
B = 6/5,
R
3,
C = 0.68, and D = 1.
L
i
ki
v
3
Dividing equation (4.52) by (4.53)
(4-53)
yields:
I
(4.54)
Fromequations (3.6), (3.9), and (3.17), with the sameconstants
as given before equation (4.53-),
do
a.;
Yso
/#
i
i.
I
F
i
i
L.
setting
df
X
A(I+.68s)
kAeLt
1n'
r
and
dx
9ee( e-A,)(1 +O68Y)r.31
,Uaek
4 t
(4.55)
'
72.
2
The ratio of heat fluxes, then, depends on the fluid properties,
the quantity
t/r3( + 0.68 )
t]/4
,
and the depth of the bottom
?~]1'
0
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
l h-
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
-I,,h
4i-i
4mr
4not
o: nol+o
hnv+.n9+
~ ~ ~~~~~~of~~~~ ~ ~ilio~~
,,
--
-t-
-
- - -…-i
show that the bottom condensate effectively insulates part of the
I
condenser tube.
The ratio expressed by equation (4.54) goes through a
I
I
meximum near
infinity as
0
= 170° because the theoretical film thickness goes to
approaches
80,
and the assumiedgeometry 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.
situation is impossible.
Physically this
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.
73.
I
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 momentumof 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
heat transfer since the liquid acts as an effective
vapor pressure drop is negligible.
effect on the
insulator
and the
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.
interface were not considered at all.
Instabilities of the
The liquid film was treated as
laminar everywihere,and the vapor was assumed to flow
Another effect arising from the presence of to
tension.
]:
~
- a, _
. ~
phases is surface
It influenced the flow pattern of the liquid primarily in
w
n t1 A. _ _ Ah
4*-zoV
4 | OU
__ | _ 49; Ad
peso 1i';
el=No | Aft t~~nW1asroe
abutY-L
J.±:QLafg
can-
densate from breaking away at the outlet into a so-called free nappe;
and,
'~.
axially.
consequently, raised the surface above that predicted by the
.1
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
approximation momentum changes may also be ignored.
Then the force
balance for a small control volume, (y
-
)r0 d Odz, can
first
be written
as
follows.
In the ~ direction:
whichcn be integrated to find the velocity
which can-be integrated to fnd
I -.e
.
.. L
~~
svelocity,ua
~
~
9
and the(5)distribution
mean
the velocity distribution and the mean
75.
--9(_
,06Pe
si
i
(yt iAt
_)
(5.2)
-
Uy
()
In the axial, z direction:
..J -d
Jw
(54)
which can be integrated to obtain
(5.5)
I
i
i
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:
5s
A~n
uss
dl . ~gAto
where
=
MJ
(5.7)
mass flow rate of condensate.
The mass flow rate may be expressed in terms of velocities.
drf--
.d4,'Z
58
76.
Substituting
equations
(5.3),
(5.6), and (5.8) into (5.7)yields:
/ -Jz
Ysr
aZ
iA*c a-- (59)
_ '
.
.t
A·
.
l*
cA
2
v
.a
-
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 conformwith these last two requirements.
The vapor shear stress, tv,
consequently,
depends
on the mass flow rate and,
on z; but it may be assumed to be independent of
angular position.
The temperature difference,
, the
t, maybe taken as
constant.
Numerical Solution
Expanding equation (5.9) results:
go,* tea-/ ;
Y
w
ySzi
C) 4
/
ampe
3
+
7Z_ ket
dz.
Xi
(5.l0)
77.
Substituting
i CW
I
ap
V
= s3 and rearranging yields,
7_
_~
Z]
)2
,A
(52.1)
The shear stress maybe expressed as a function of the Reynolds
numberor the mass flow rate in the following form (Equations 4.31,
4.32, 4.33)
hi'
W-'-4C4
-
37;
,r. 4
(5.12)
d.
J
Is
.;z
6,i-A
Z-+b-i
%0%
%66'rn
I P
a1z
(5.-3)
where the subscript ( ) refers to the vapor phase of the flow
dh
hydraulic diameter, ft
ai b constants depending on the range of the Reynolds
number
Since at any cross section
ri 4
r
tinlet
a constant
ed(.
dro
(5.-14)
From equations (5.4) and (5.7)
; pi ,,At 10d
V ~~~~~~~~~~-
U-
(5.1 5)
78.
For a numerical solution equations (5.11), (5.12), (5.13), and
(5.15) can be written in finite difference form.
known, P- at a small distance Az
Since
0o
away can be calculated.
can be repeated until the end of.the tube is reached or
zero.
o at z = 0 is
Care has to be taken in the selection of
The procedure
V becomes
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 J~
tA
}
e
s;n
Cd<°sS;3
-
P.3 'I(* +
(5.1 a)
.2
"3iR-i:>^.+§zz] .tat
|
2kS
~~~~~~~~dr.
(5.14 a &b)
(5.15a)
A4
Equations (5.12) and (5.13) remain the same.
Since the computer could not start with Po
50
=
thicknesses were assumed for the first value of p.
O, several small
The magnitude of
these were determined as arbitrary fractions of the thickness at the
top of the round tube without shear; that is,
wherei~
isa)btr
KK
where E is an arbitrary
I
II
constant.
(5.16)
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 flowratewas
given
f~
and the corresponding tube length could be determined from the
calculations.
~Flow
rates
werevaried 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
and the increments were taken as r10
and 3
r/20
F;
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.
Figure 5.1.
i
'i4
i
.i
6L: _.
A typical set of surface profiles are shown in
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~
~(5-16)
et At
The horizontal component of the surface tension at the lip of the
tube outlet is:
P,,
Pw wetted
where
C
'"'
SX'
-~O~Jat
~(5.17)
tube perimeter, ft
surface tension of liquid film, lbf/ft
contact angle
Equating the two expressions yields:
, 9,P Aa,,sint
Qa.
s_
or in dimensionless form for a round tube:
Qt~~
"'2c<r
i~cS)tm
__ __ _
~_.__
_ _
_ _ _
. , s
_
_
is
(5.1.8)
'
For the water-plastic combination a contact angle of approximately
45° was measured.
The resultant flow rate is, assuming
eC
= 450° ,and
81.
taking
Or
= 493
x 10
3
lbf/ft corresponding to the approximate water
temperature used of 80 OF,
Qt
g.
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.
eld.LJ~~1.L
i~~~~~~~~~~~~~~~~~~~~~~T
11 G,~tH:
L5.Lb
V.~82.,
T
THE EXT.. NTALVOI
Fluid MechanicsAnalor_ Exoeriments
It was recognizedat the beginningof this workthat 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
1
inches long with an internal diameter of 1.10 inches.
At twelve points
along the tube, equally spaced at 4.5 inches, meanswere provided for
admitting liquid on both sides and for letting vapor or air out at the
top. This arrangement is shownin 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 somallstainless steel
tubes through which the fluid had to flow to reach the test section.
I
I+.
I
-i-
These
nressure
differences were read an manometers.
~be adjusted by small needle valves.
The flow rates could
The-distributor chamberon top of the
833.
block hs
distributor
a
and
a me0
filter,
Caloh
and it h'.ad a valve
This valve
and air bleed.
on top which served both as a low regulator
was cornected to a line whnichret'urned the excess liquid to the col ecting
FYrom
this tk_ a small Monel
tanle at the outlet of the test section.
back
the fluid through a standard commercial filter
Metal pumpcirculated
Although means were provided for supDlying air at
into the distributor.
the entr=ancef the test section, nd for bleeding it off at twelve
points along the tube, this pat of the setun was not u
it was discovered hat the riples
the first trials
caused by he fid
the suroly points were xge enough to be lcu:crt?
strears entering
b- very low flows
The ensu g wave attern
f air.
could nt
in the second phase cf te eerirents
be ecccted
process.
Since
to the one occurring during, a condensation
to be siilar
Duing
ized
he actual condensation flow
patterns were to be observed. the analogy studi es ,ith a high vanor shear
stress were abandned
T'husthe
irst hase f he errerimental
.rk
endeavoredprimarily to establsh fIoT deoths near te end f the
hori zontal tube amdto fd
fro
transition
the criticni
laminmarto turbulent
oerc-.med using distilled
used effective
for i
low occurs.
water as the fluid.
d
tube -e
flows the lcuid s-reass entering at te
stron;
I
oblique stand
aes
,'~'- ': as sarte
5'r,
'- -d
-~- b
-Lrcl
rC_
Reynlds n'bes
that mae
.
the fo,
at which
These exre- -ents were
This se+tu could not be
mnens because in sercritical
-el-ve suply
oints
ener-ed
y depth measurementvmenin!
rates were adjusted at the
'¢-. tc th
c' d ar-r
e
-tr
7- "ei
n I
fateoC Te
byreJatJ-t-e-'-'--~ t~ nu),o. ... cr
cr- v__ c "
'rt
-
?s.
8A.
the
rn
steadiest at
orked
ump
required flow rates the
with the
After
were regulated by the small1 needle valves mentioned before.
..,e
m
,14,
t~~~~VLiJ
, T-ri.mp
,+nl
+.. rP-...,
_...
w
'JL>,
s
.1.
rate
' f-i,
.
-- > --
**
s
unp
supply points
Individual
The flow rates to the
for the
a relatively low speed.
T.ns, most of the controlling was achieved by the valves
running at the optimum.
that
as found
It
or at the distributor bleed.
atlet
the
a.t
mneasured
a
- _-
-lV
uvu-
-
The depth of the flow at
tube outlet wirtha graduate and a stop watch.
any point was established by wrapping a strip of paper around the tube,
of
and marking both the circumference and the points where the srface
the licuid met the walls, viewed diagonally across the tube.
The
temperature of the liquid was measured by a mercury-in-glass thermometer
by
Turbulence was determined
of 0.2 OF.
graduated in increments
injecting ink axially into the stream through a small stainless steel
syringe.
CondensationExeriments
Since very little experimental data was available in the literature
that could be used directly to check the validity of the an-alysis,
an e:perLmental setuapws
became clear that
transfer data.
needed to furnish heat
In addition, it was deemed important that the apttratus
in
conditions existing
should also furnish some information on the flow
the tube.
it
Refrigerant 113 (C2 C13 F 3 ) was chosen as the fluid bec.use of
relationships.
its favorable presmure-esmDerature
Descri-tion of the P!-Dratus
The equiprmentis sho.n in
..
in:zgl.re
-
1
-O-
l
1!
q3
..-
O
-;
_..--
~~U1n'...,-...
e%
C
9
-
-i-F-]
IVJU1
i-- V-
of 5 1/2" standad
heaer.
-
.andschematically
'1gres 6-5, 6-6, 6-7,
-
.1
Xri
--
-
Ln
e
L1
L
(2
£I?
el
.
f
'-,-
r
…ill
brass pipe, set on top of a 2000 w.att flat plate
Inside the top of the boi' er
.s a circular slash
pla te.
A 7/8
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
The entrance tube contained a stainless
heat-resistant flexible hose.
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
and for illumination of the interior of the equipment.
observation
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
which also
the mercury manometerconnection,
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.
seal at the top,to
m iid
The overflow line had a liquid
prevent the escape of any vapor, and a shut-off valve.
r
3/8" o.d. copper line.
to a ne liter
caai-tv
Plass renpiPVertrouna
The receiver supplied the boiler through a short
was connected to the charging valve too.
copper tubing w._hich
I
''1
.,j
a
All
arts,
86.
except the test section, the observation windows, and manometers,were
insulated with glass wool insulation covered with aluminumfoil.
The
burette insulation was divided into several sections which could be
movedup and downto 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 wattmeterconnected to a switch box, which
with the
made it possible to measure the powergoinginto both heaters
same wattmeter.
Temperatures were measuredat 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
right behind the lower edge of the test section.
unction was located
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 poiLntson the bottom.
......
I|
~
were one-sixth of the tube length from each end and two in the miccue.
The thermnocoule wires were connected to a Leeds and Northrup semi-
potentiometer through a ten-point selector switch.
precision
-ki
tour
,.....q,
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
The
was evacuated, then it was charged with distilled Refrigerant 113.
test section was leveled by adjusting the four bolts on which the framework 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.
I.
.
~.!~
'~ .
-.~
was used for the test runs.
The following procedure
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.
velocities around the condenser were checked next.
L
-
1-
-
-..
..-
I
.-
I-
.-
I.
.--
II-
.-.-
-
were turned on and set at the required power level.
The air
The purge line vacuum
1-
-
-1
-.
.
,-
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 minimizedthe amountof condensation
occurring outside the test section.
purge line was opened very slightly.
After an initial warm-up period,
The bleed flow rate was kept at
minimum by adjusting both a needle valve and a pincheck
line.
After equilibrium was reached, data were taken in the following
1.
Barometric pressure
2.
Mercury and refrigerant levels in the manometers
3.
Thermocouple millivolt readings
4.
#aw
5.
Visual observations and measurements of the flow pattern
6.
Photographs of the interior flow pattern at various
rates in the burette
points along the test section
7.
iI
a
on the vacuum
order, unless some special circumstances arose:
I
I
II
the
Heater wattages and variac settings.
89.
CHAPTER VII
DISCUSSION, CONCLUSIONS, AND RECOUENDATIONS
FOR FUTURE VORK
Discussion
The fluid mechanics analogy setup was built for the purpose
of
It could
providing direct access for measurements on the liquid used.
not fulfill all the requirements of a really close analogy to the condensation process, but it provided very useful quantitative and
qualitative results that could not be obtained from a condensation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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
Re
>
= 1 may be used for the turbulent range,
3,000; the dashed line, with
flows, Re <
3,000.
with the curves, don
/
is applicable to laminar
= 1414
The experimental points for both fluids agree well
to a flow rate of about Qt
=
0.0.
Below
this point, the measuredflow depth remained essentially constant at
cc
=
90°0 deviating from the theoretical
curve.
One, minor reason for
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
I
90.
Another, much more important reason is the
yield greater depths.
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 backward 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 determination 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
F91.
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
WWhenthe tube
patterns to the ones actually found in a condenser 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 0 cm = 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
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 upstream 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
the mean.
at the highest.
Room temperature fluctuated within 1.5
The accuracy of volumetric flow rates was about 2.
of
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
The biggest problem in performing the experiments
be discussed here.
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
The bleed was regulated at such a rate that the
of the test section.
visual observations showed no reduction in the wall condensate film
This change in thickness could be detected as a hardly
thickness.
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.
lower flow rates
At the
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
In the horizontal position waves caused by the vapor shear
to observe.
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
qI
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
disanearance
the
-------,-,--------
I
shear
waves.
of
---------------
At the very highest flow rates very strong disturbances were
observed at the entrance.
In the horizontal tube no distinguishable
bottom condensate existed here.
on the wall.
In the inclined tube ripples appeared
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 Remynoldsnumber of about 35,000,
which was approximately the highest value encountered in these
The surface profiles,
experiments.
when not considering the waves, werefairly
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
entrance then continued rather uniformly to the end.
-I.'
I i~j._.
11-;.
at the
2
Conclusions
Heat transfer rates for laminar condensation of a pure vapor in
horizontal and inclined tubes can be predicted by the use of the
Pren+.P(I_
-
For
vE fluid of
n
P
[Veds
vra
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
For liquid metals, a correction factor
(40) value of 0.68 used for C.
from Figure 3.3 is needed, although there is no good experimental data
oni nhlh
t-
nhr'vn_
k +.h+.n
…
Th ri-n-M of' +.hi li.+.om
be established for horizontal or essentially subcritical flows by a stepby-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
flow rates a constant
Figure 7.1.
outlet
angle
of
/
0c
fgr>
0.08.
For lower
= 90 ° can be used as shown in
As a matter of fact, for fluids similar to Refrigerant-113,
0cme
a constant mean depth of
120° may be assumed over the entire
J
range up to 1
5
/gr
0
= 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
ecm
+
= 0.5 to z = 1) for calculating
=ci2
cm/2Chaddock (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.,>^
~
tueXa sl<,l
..
:q
~
~
ao.i5wl-±ra
___
_nl
4;xvorlrvlcln
ohnnhsel+-o1o
¶JoJe
wal
enr±1±ice ne
>
urui eru.
rTar nE l
-. 7Leouu cra
i|
96.
r-IVYUI
U--o
·I·VLI-
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
-- I-- vanor
-
w rtes
."
abvr
- n en+eying
vni'
Paimnold nmbr
-
1--
-UY
of about 35,000, these relations are no longer valid because gravity
effects are negligible and the orientation of the tube becomes unimportant.
terms
In such a case, heat transfer data can be correlated in
of a Reynolds
number, irrespective of slope (2).
Consequently,
for condensation inside tubes, the horizontal and inclined positions
nya
wrar1M+.hn-n+o
tn+w
rnms
better.
-r~+.
-
~NA;n
en
_
-
,
,
a:+. rno
. -
, ....
---
+c
-
.rG,
-
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.
specified:
In each case the
ollowing quantities are
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.
more aproximate
cm = 120° .
The quicker and
one is to assume a constant mean condensate angle of
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 assurmea cm
and a corresponding O, calculate
cm
as before, then from
flow rate there.
the flow rate
Figure 7.1 find an outlet depth corresponding to the
Starting from this end calculate the surface
rofile
I
by integrating equation (4.42) sten-wisealong the tube.
value of
I:: t
.I--,
I
.
.SI
II
A.
; ...a
over
the length of the tube
nd conare
Find the
it to the assumed
ean
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
be estimated from equation (3.23)
For the inclined
$cm
and its
flow
rate,
the iTusselt
number can
as above.
Atube first
corresponding
,
assume a constant mean condensate angle
. UsingO, calculate
from equation (4.36).
the constsant change of
Substitute this value into
from equation
(4-36).~~~~~~~~~~~4
equation (4.43) and set the limits between z
the
0.5 and z
= 1.
Using
angle functions (based on equations (4.34)) tabulated in Appendix VI,
find several values of
Plot the
corresponding to different assumed angles.
C-- 0 curve, as shown in Figure
corresponding to the given slope.
.3, and find the angle
The agreement between this value of
end the originally assumed one should be within about 3
If it is not, assume a new value for
the firnalvplue of
, and A
,
in this case.
, and repeat the procedure.
With
the flow rate and the Nusselt number can
be evaluated as before from equations (4.36) and (3.23), wi-ththe latter
erresslon zultipliedby cosl/4(sin
and the angle integral fnction
or ). Actually both this function,
appearing in (3.23) coanbe read off
directly from the chart in Figure 4.3.
for the given fl, find the point on the
product
of te
To estimate the optium
o- 0 curve for which the
other two coordinates, relation (A.4:), is rnamtum.
For both horizontal and inclined tubes, the entering vor
nuber
.- ,A
II f
.t
slope
should be ascertained to be less than 35,000.
Reynolds
ecomm.enda:ionsfr FaLtare
The many comple
deiork
processes occt'ring during condensation iside
tube suggest a number of investigations, some of
practical
while
thers
might
a
hrich would be highly
Here a few
have only academic interest.
of what seem to be the most important leads will be mentioned.
1.
Exoerimental data is needed
checking the eisting
.~~
~ ~ ~ ~ -
n liquid metals for
analyses.
_.
I
1 1
1-_
. T£neeect c superheat snouLcaae 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-diometerratios.
4. The vapor flow inside a ube ^ith a liuid
occupyring
the bottc. section could be investigated theoretically.
la.minar
fl6o,
and neglecting
of the vapor may be clculated
cross-secion.al
at
the interface cn
di stri butic'.
I
I
I
A
I
I
are rtios.
licuid
veloc 'ies,
Assu=.ing
the flow pattern
fordifferent
vasor-licuid
Then the bmdarvy ccndicins found
be used to solve for
the liquid
velocity
5.
Te
in a tube or channel
robem of condensation
gravity actLng should be exomined.
Here the problem o
move the condensate becomes imortant.
th n
how 'to
ossibility to be
One
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.
visual observations
In all of the experimentailinvesti..gat-.ions
should be cnsidered e:xremelyimDortnnt
conditionsccurring.
to estaoblish
the
exact flow
I ("
NOMNICLATJRE
a
constant in boundary layer equations
a.
constant in friction fctor
1.
(4
enations
~a
u+I
w
RO
perturbation terms of A
A
cross sectional area of flow, ft2
A
heat transfer area, ft2
h
liquid cross sectional area, ft2
A
vapor cross sectional area, ft2
b
width of liquid surface, ft
b.
exponent in friction factor equations
B
o +2dy
V
.
B
0
1
constants in boundary layer euations
0,Y1,
c
velocity of surface waves, ft/sec
cP
licuid
secific heat, Btuft4/lbf-sec2-OF or Btu/sug-°F
C
1 -
u t dy
0
.
C0,
perturbation terms of C
C
correction factor for velocity distribution
d
licuid
dhv
vapor hydraulic diameter, ft
d
diameter of tube, ft
v
ht
0
hydraulic diameter, ft
D+
0
..
-
DI 1
,%. . .
tV
perturbationterTisof D
10i.
base of natural logarithms
e
quantities defined in boundary layer equations
UaYs, volumetric wall condensate flow rate per unit length,
f
ft3 /sec-ft
fe
liquid friction factor
fv
vapor friction factor
_-_
F
U
-1/4
)-14 dimensionless flow rate
ay'1s C)
~
b__.
I
gAj=
Ae
, Froude number
Z-
g
gravitational acceleration, ft/sec2
g(Y)
function in boundary layer equations
g(&,, -p,)sin
0
-wint^i -%'
_+_y-v- +tri-n r i _-ra~jnr__.7- +..qnn
.
G(x)
depth of flow, ft
h
c
h
critical depth, ft
h
heat transfer coefficient, Btu/sec-ft2-OF
h.an
enthalpy of mass entering tube, Btu/slug
in
out
H
enthalpy of mass leaving tube, Btu/slug
specific energy of liquid flow, ft
i
s 3~~~~
-4/3 0 |Snf/3
I1
4
I2
JT2
sin 1 30
cos
0 do
do
o
J (x)
function
in boundary layer equations
J(73
function
in boLndary layer equations
102.
thermal conductivity of liquid, BtW/sec-ft-°F
kt g
!
K
t
|
constant
~~~~~kea
D
1,
)% o0e~kAD
PC
I~~~~ e
{I
~
1caratersti lengh, f
characteristic length, ft
L
length of tube, ft
m
exponent
n
exponent
N(x)
function in boundary layer equations
h l
XNu
h
,
k
,mNusselt number
k
NIu
mean Nusselt number based on arc
Nu
mean Nusselt number based on total circumference,
p
pressure, lbf/ft2
P
Ys
PV
r ( + sin 0), vapor wetted perimeter, ft
P
w
rOOc, liquid wetted perimeter, ft
0
-3
ta
Pr
k
k
Prandtlnumber
' 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
svolumetricflow rate, ft3 /sec
C
licuid volumetric flow rate, ft3/sec
Qedet
;
V
::
.!.
tvv; A -t
iv
l
Iv
V
-
-h miUl
vanor volumetric flow rate, ft3 /sec
V
%
nU
i
/qtiArk
103.
r
radial distance, ft
rO~
radius of tube, ft
Ae
TF hydraulic
rhe
radius of liquid, ft
rpx
1
v
hydraulic radius of vapor, ft
rhv
p
Re
4Q
Re&
4 eP g-9"liquid Reynolds number
/-IPw '
V
Reynolds nmmber
~~~~~~~I
.1
'
V
n/hvpv
s
elevation, ft
s
elevation of channel bottom, ft
s
depth of centroid of A
t
temperature,
ts
saturated vapor and condensate surface temperature,
t
wall temperature,
O
c
i~~~~~
vapor Reynolds number
W
t
t
s
below surface, ft
°F
OF
- t
w
t - twr
+
tt
~t
At
t
-t
s At
at
T(5)
u
velocity in x direction, ft/sec; except in complex trnems-
'I
formation of bottom condensate cross section where
I
.~~~~
.
u
I
.',
.1
I
.:
.I
k
.,
"
ua
'
~
_
Ys
V
fudy,
S
,-
.. ,-
m
coordinate in complex u-v plane
'~~~~~~~~~~
:'
uU/
F
mean velocity, f/sec
I.
ut
velocity in y direction, ft/sec
v
plane
coordinate in corn.olex
V
velocity, ft/sec
.
...
V
C
-
ft/e
t...
v oc
critical velocity, ft/sec
mean velocity, ft/sec
V
m
velocity in z direction, ft/sec
w
+ iv, complex variable
x
distance, ft
X
dimensionless distance in
y
distance, ft
Ys
wall condensate film thickness, ft
direction
i
i
Ys~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Z
Yso
wall condensate film thickness at surface of bottom flow, ft
Yso
y atc*
y4.
+
i
I
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
i
.I
. i'4
-i
-I
-iil,6
'.4
,.-I
At
=
r 2 5($c
sin
lin
2,
section fctor,
ft2 ' 5
105.
Greek Letters
--;;
Cf.
V3dA, velocity correction
AV3
mA
2/sec
ft
diffusivity,
'thermal
. ,icp/9,
ke
1
3
+ s) VdA, pressurecorrection
-
contact angle
mass flow; rate, slugs/sec
|F rtotalmass
In
Ira?e--
'I
m,
-iE~
;9
'Iw~
~
flow rate in tube, slugs/sec
-"
----
llZ<
A
vapor mass flow rate, slugs/sec
BDke A t
C )n
dAtio(1
-
I
I
. dimensionlesstemperaturedifference
transform variable for boundary layer equations
(xy)
1
at Ys
angle in condensate
Q
c/2, condensate subtended h-lf
angle
2
4
latent heat of vaporization, Btu-ft /lbf-sec or Btu/slug
;X (1 + C),
'
viscosity,
ue
generally C = 0.68
bf-sec/ft2
liquid viscosity,
}''c r
.....
bf-sec/f
4
+.~_n
--yn. bf-se/-L2
2
kinematic viscosity, ft /sec
I
I
te, sec
106.
liquid mass density, lbf-sec2/ft4 or slugs/ft3
vapor mass density, lbf-sec2/ft4 pr sigs/ft
r
3
ds
('I
0
-o,
dz'
slope
surface tension, lbf/ft
TS
vapor shear stress at interface, lbf/ft2
Pw
liquid shear stress on wlls,
0
surface inclination to horizontal.
lbf/ft
2
For round tube it is the
angle from top of tube to a point on wall
2e
Ol
, 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
Y'
T,
)i
(
)xy,.
(
)
s
( )
()w
(
I
4,
I
Ir
-md:
.4ks,,,,
differentiation with respect to variable in subscript
at the liquid surface
at the wall
(
(
(
)"
derivatives
derivatives
r
107.
2
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Tables of FLuctions," J.
Abra"oitz,
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1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
rv
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r
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f
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a
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i
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48.
deutsch In.;
i..
of
81, 13, 1959.
Transfer on a Horizontal Cylinder," Trans AS,
I
reatment
Condensation of Alcohol Vapors,'"Zeitschrift de. Ver.
, 959, 1941.
Young, . L. and W. J. Woblenberg, "Condensation of Saturated
Frecn-12
Vapor on a Bank of Horizontal Tubes, "Tran m s. PS.
6LA,
787, 1942o.
r
LIST OF FIGUPES
Page
FIG. 3.1
Condensate Flow on an Inclined Surface .
. . . .
. 113
.
3.2a
Geometry of Condensate Flow Inside the Tube . . .
3.2b
Geometry of Condensate Flow on Complex u-v Plane.
3.3
Theoretical Heat Transfer Results . . . . . . . .
.
.
14
. 114
· 115
.
Specific Head vs. Depth for Free Surface Flows
.. . .
4.2
Geometry of Bottom Flow Along the Tube. . .
. . . 116
4.3
Slope Optimizing Chart for Condenser Tubes. . .
4.4
Ratio of Heat Quantities Transferred Through the
FIG. 4.1
. .
Bottom Condensate and the Wall . . .
..
117
*
118
·
. .
$
119
FIG. 5.1
Calculated Surface Profiles at Tube Entrsnce
FIG. 6.1
~luid Mechanics Analogy Apparatus . . . . . . . . . .
6.2
....
. . . . . . . . .
Schematic Diagram of Setup for
luid
.
122
Detail of Test Section for Fluid Mechanics
mAnalogyExperiments .........
. . . . . .
123
6.5
Condenser Setu
6.6
Inlet Side View - Condenser Setup . . .
6.7
Discharge Side View - Condenser
Setup. .
6.8
Schematic
diagr-n
. . . .
. .
. . . . .
124
. . . . . . .
125
...
126
. . .
127
of Setup for Condensation
Exeri ments
6.9
121.
Mechanics
AnamlogyExperiments ...............
6.4
120
Liquid Distributor - Fluid Mechanics Analogy
ApDaratus
6.3
116
.
..
. . . . . . .
.
. .
Typical Photographs ShowingFlow Patterns in the
Condenser
Tube - Upstrenm and Dowmstream ends.
128
~~~~~~~~~112.
,
Page
FIG. 7.1
7.2a
Discharge Depth vs Flow Rate in Horizontal Tubes...
Condensation of Refrigersnt-113 - Selected
Heat Transfer Results . . . . . . . . . . . . ..
7.2b
-A
130
Condensation of Refrigerant-113
Heat Transfer Results .
I
129
. . . . . . . . . . . .
131
r
CONDENSATE
FLOW ON AN
FIG. 3.1
I
'if
INCLINED
SURFAC"
IN4
I
w
H~~~
nn~~~~~~~-
m
<
C-
~~~X
7
wd
-
-_
0~~~~~~07:
7
-
--
f It
C
Li2
I-%~~~0
%
0
i
0
0
3~ m
_,
0
LL) L
N
U
*0
4-
C:
0
WO
.
u
LLi
-Li
I
0
>
e:_
0
0
C.",
I
IL
--
>H
0
cO
r)
II
.
(1
C::-L
H
-
V
~~~
~
<~
~
~
6
~
~
0Lo
C)
I
~~~~~~~~~~~
0
W
0
i
-
·
115
r
0
I
(I)
-J
Ct
L:
W
C:
Li
9
:-7
1---;
I
H-
a
Un
Q
k.
0
00
L_
0
LU
FI_
-
0
00
C)1
C)
'
C,
_
0
Ca,
I4-
0
4-
1
z;
I
I
'i.-
p .
. to
L
r
H
h
SPECIFIC HEAD VS. DEPTH
FOR FREE SURFACE FLOWS
FIG. 4.1
I
appe For High
-
Rates
I
OUr1CL
rr
LOW rilOW
Rates in Horizontal
Tubes
I
I
GEOMETRY OF BOTTOM FLOW ALONG THE T-
FiG. 4.2
-
...
l ....;-:----CO
I:~....
117
.......
-:-
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~/
~~ ~
0,95;-,''t
0.98'::- .. 0.97":'! ,6:
..
":'-..0995'"-~::.::~~~~~~~~~~---4-_::,0":::!..
:t:"-
!:::Z314
L
..............
~
....~.............
:......
i;.......
...................
~.................
-........
t ...... ~~~~~~~~~~~~~~~......
: .............
~~~~~.
~.... .......
... .....
t...'"...:
..
..... 1~.......
.... i ... ~
. .7
I . 1--'I .-I-.L l....l~l
I t..i~ ....·.......
1'l~l '- -'" I.. 1.......-'
. .1
1' 6 :7t
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
....
*I,-
----
~
-_I;--~.L.
:
-i.:- ":---
:
.
I-
-I~-j.
---
_ --.
L
.-
.
----
.... i.....
~~~~~~~~~~~~~~~~~~~~~~~~~......
t....r....
:. ...
:-_· .... L
-.-.
...
...
t~;~......
Ie
8
-
..- ....
.; ...... . -II.-...-i----.
'~~. ..... : .....
T-?....,---m-:.....:
.....
:- ....... ....' ................
:
: - ._=
;~'
'~
-
;. :'~.....-
- -_ ,-
,.--
---........
.........3_-':
...-....
-- ....
..
:........
=== == ======== ............
':---:~----:-:====
.
. ------ :---,:
..... :--+4
--.------ ...:
:.:::.:±:.i.:: ..:
....... --...
1'e93. 4
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~...
---------_---
:
1-944::
.
. . .
. .' I:''
.i_'"-+ rlirii
. .
....
'r-'
- - - - -.
I, -
J
..--.-
_
.....-
.
.$I
I
.--
--
r....
_-_li
~..
....
'.......~--..
..-
.II--;
-----
~4
-....... -·....i
:
.:.97.:--
__~ _ -.-:':
.....
.:
- .: ., -:---:-L _--:3
is-
1
- :1'~'
.7::.::.:
____,
.-. :---I---.
:
::::::::::::::::::::::::
_.-:
7..:;...-_
_
:.-
ii--
-.
i ..
-i
--
.. :-_.-
'..,- ...
==:t-=
....
p:--::
: .-.:
------- .
-
-----·------
,-i;__
---
i:-----.
j
l
-
;,...
'~..
L_
f
__-.-:---
?....!--
.
i__
-
. .. ..
--T...
-;i-..i-.
-
......--
"-'-
i
:
..........
..... ...........: -
f-- ... ;4--
.
i-'_:j
--------·
-
-·
_i._._.__
i-
' 7"
TT...'
- _...
t:::
.....t0...
'-:_-
T---_,-.
L
--_
!--~
__-
--
T-----.
..
.....
'----'-------
...
S C' ET_<JP_
1it tN:::HAR
0
........
;......
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
- - -. ,-:.
'~
- ;.
i
::O NDENS ER-TU B ES:: -- -:::::::
.-
I
F~~~~~~~~~~~~
C
Z
1-4
I
I
1-
0
0o
0
I
CD
LIJ
~ ~ ~
0 0
L
n
C.D_
0
~~~~~~~~~~~-W
~O
0
IO
0
pIC
(/)Q
s
o
LI
0:
H
C7
0
o
_p
d~
e-pW,
r"
0
9
o
o
-'
!b
~3J-SNV8JJ I-V3Ft
6L
o0
-O0
l.LV
~
0
11-11,
<
NW,
1.8
0
x
-
I
I .
1.6
_p
1.4
_
0=
1.2
X
I
111
Xx
_
II
i-.
X
Cl
1.0
z
0.8 _
I
0.6 _
0
0.4
k
~,
I
A-A>
, //~,'
_
0
T--'~
+0=30
Refrigerant
113
ro 0.02384 ft
At= 12°F
Inlet Mass Flow Rate =2.4 X 10
_
..I
.'3
1
t
7.r
"i
A
1
slugs /sec.
-
_
I
.1
*.2
.3
I
f
.4
.5
I
.6
AXIAL DISTANCE,
I
Nusselt
-J
0.2
I'
x
NusseltI
#X
'4-
en
en
I
CALCULATED
.7
.8
I
I
.9
;.0
1.1
S/o
SURFACE PROFILES AT TUBE ENTRANCE
FIG. 5.1
.. ...,
"'.
I
i"
·- .~ni
i'?
j')
?'
LICUJID DSTM
? .'YD
. ECI{ANICS
i
,
L-
IBUT2Rap
A2.Mi
t
APPAAiTUS
0
a
0
.-0
c
0
0,-
:o
I-
1- o
o
0
f
o-i 0 .
> =
4
-
~-
0
:
:Rt
>
e
C'
i.--
C,
Ld
0
P
z
2
0
c
0).
q,-j
co
ME a>
V)
0
Y l
c
o .0
0 0
au
0o
o
>a
0Ci
.2:
0
w
tJ
CD
0Ct
-J
cn
Z
Z
C)
00
UiJ
w
rr)
(0
-J
L.
-
C a
: 0
0
LL
C-
o o
_f--
O-
C
L
C
0c
0z
<:
.
CS
II
{,
I
0
I.
I
*I
-1
,!
w1
0C')
r
2
Throttle
Venturi
Connections
_
Liquid
Supply
Liquid
Supply
II
I
i
DETAiL OF TEST SECTION
FOR FLUID MECHANICS ANALOGY Ev-EREMNNTS
-
'I
i
FiG.
.-
1
i
- I:-1
,
.
.,
.,
.
.
.
..
l
:
0 .
.
.'
.
:,
.
. .
:
:
i -.
.
.,
.
.
: -'.
....
:
,-
-
.-,
.
. .
.,
.
.
.
-
':
\, \,..
.,.
s
": -
. W-.'-
:['.
Ei,
1'-<
.l
D
1'.
.
I': F
'
.
4
,..
:c
' IT'S
: Of.:
.-.
i
.
-.:
:
-:
' '''' '
'
-
'.
'
.!
.,
') i '
,
'..
_
.
.:,
.
.
- r--i
, '
_E . '' ',
'..'
.:
ZA...
.'
.
'.
.,'
.
...
j
I.
I
.
.
~~'.,:
''~~
CONDENSER
SETUP
FtI.
i
6
5
123.
INLET SIDE VIEWF
CONDENSER SETU'
FIG. 6.6
..- :
@
s.
-i .
~~~~~~~~~~~~~~~~~~~~
6
..~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.
I
.
DICRIGE
. VT-...A
rDE
DT±SCHE 1G
i Z
I-~~~
I
~.:. ;~,~
7''- I
i ,
i
.1
I ,"
i7l-L,
CCONDEISE
R SETP
FIG.
127
I
U)
z
10
0
0
i0:
z
x2QX
LLJ
0
z0
U)
2
ff
0Cz:
00
LL
0-
C:)
.0
LI
0
0
Cd
U)
-u0
_
i
o0>
i
a
,2
128.
I
:
:
:
:'
:
:
TYPICAL
PHOTOGRAPHS SHOWING FLOW PATTERNS
IN THE CONDENSER TUBE
UPSTREAM END ON TOP
-..
~~~~
~DOWNSTREAM
END ON BOTTOM
FIG. 6.9
I'
_
i
I
I
I
~
i0
0
129
I
co
I
Q
I
0
I
i
i
II
i
cn
oQ
-
Oo
0
i
-J
<
z0Z
0N 0
°lCt
W
Nt
-)0)-
I
-
N
6
\1
Z
I
o
z
._
W
c
!- o >
>.,
~w~
>z t
I
0
C
i
O
J
(0
9o
m
C
0 ..L
H,
L
0
,
0
>
c
4,
<
0
C
(./)
a
N
0
II
0.
a-
0
O
0
o
0
I
I
A
0---
0
0I
0
-_
I
F .
au
I
S "V 93'1NV
,
N3 c
S338931-0 + 3-9NV 3lV8-N30
0
I
0
'i
0
A
0
Ir
130
Slope of Tube (sine)
Sym bo I
0o
0
A
0.010
I
-
I
Exp. Variation of 0cm
.120°-1300
80 ° - 950
0.1736
600° Theoretical Curves for Vapor at 130 °F
70'
400
300
10 ?
I
-200
w
()
(n
z
100
0.02
3 0 .0
DIMENSIONLESS
4
0.05
0.06
0.08
TEMPERATURE-DIFFERENCE-
CONDENSATION OF REFRIGERANT
SELECTED
HEAT TRANSFER
FIG. 7.2a
Cp lAt
x
113
RESULTS
0.10
T
131
Symbol
0
A
Slope of Tube(sin e)
Exp. Variation of Ocm
120°-130°
0
0.002
0.005
110°
90°-100o
0.010
800- 95 °
0.040
0.1736
70° - 80°
70 ° - 80°
0.020
0
lo
*
+
60 ° - 70°
Theoretical Curves for Vapor at 130°F
40
30
20
0
1.
z
I-
w
(n
z
I0C
0.02
0.03
DIMENSIONLESS
0.04
0.05
. .~~~~~~~~~~~~~~~~~~~~~~~~~~~
TEMPERATURE
0.06
DIFFERENCE-
CONDENSATION OF REFRIGERANT
HEAT TRAN SFER RESULTS
F IG. 7.2b
0.08
CpI At
113
.10
lr
LX
-.Lj2,.
I~~~~~~~~~~~~~~~~~~~~~~~~~~~s
APPENDIXI
DETEI.NATION
OF TE
ITE
FACE BOUADRYs CONDITIONS
133.
I
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
y s and y-~o
equation for 'anelement of the vapor phase between y
Ys
is,
y
~~~~+ dblZf~e
aayV
yV
|
0
Y
(A.l)
pA ed> J Uddx~pv~tdY - O
Ys
Y/
In dimensionless form, using the samedevelopment that led to
equation(3.-8),
P, )4yt y+]
- +,^,:"dy
f -j
J
P'[
n0
I L
fdx
- X
(,
(A.2)
dy+- 0
4Yf
di
If
p<
C 1, terms containing this ratio maybe neglected.
Using equation (3.6) yields:
.,,'yt; ?1
At, ~J'
ktD
T
-.
)
UV.,
~~~~(A.3)
(A_+3)
.(I
134.
The solution of this equation is:
(A-4)
Uv X4 LI5 e- II
equation for an element starting in the
Nowwrite the momentum
liquid surface
,
ay i
ny*
,
+
+i 4
Y;
+Pd
ciW
ae ly
(A.5)
As
tawf
(
aJy o
/
J
4A
In K z AS
s
fd>fdXj
0d~ q
s'SfYdx
?U', /t
(A.6)
F-
-I
135.
APPENDIXII
SALVE
CALCULATIONS
36.
HORIZONTAL TUBE
RUN NO.
82
Given Data:
Comparative Experimental Data:
r = 0.02384 ft.
t
L
Qo
= 2.354 ft.
cr =0
F
= 0.750 x 10 -4 ft3 /sec
Q o
t.
1 = 141.8
= 0.1507
OF
At = 23.2 OF
Fluid:
= 129.1
_
0cm
115
°
Refrigerant-113
Properties
p
g
=
k = 1.293 x 10 5 Btu/sec ft °F
92.17 lb/ft3
?g = 0.6904 lb/ft3
A/g
$
PC
= 0.9150
x 10- 5 lbf sec/ft2
= 61.15 Btu/lb
hin/g = 99.62 Btu/lb
= 0.0890
hou/g
= 35. 5L Btu/lb
(,cg)
Assume cm = 1200, e
= 60 ° ,
= 93.-30 lbf/ft3
= 1200
From euation (4.36):
n=: 2
[,1.48
2
=
3
L
3
x (1.293)
3
x
0.10x10
0.3375 x 10 -4
ft
~ 0 15
x (0.02384)
3
x (92.17 x 61.15 x
2
3A
x (23-)
3
/sec
ft
00)-
1
x 1.240 x 1.564
,.
_,-
13 7.
.3375x 10- 4 x 2.35L = 0.1596
L
>/32.17 x (0.02384) 5
_7o
5=
-tI
0.02545
=
0.0946
gr 0O
Integrating
0=
0c = 106 °, ec = 530,
From Figure 7.1:
equation
(4.L2) from z=
127°
1 to z = 0.8 yields
on the
right hand side the terms of equation (4.43):
I
or =
0
0.7964
II -0.001417x 133.4 x 0.02545 x 98.7 x (2.6972)2x (0.008)
1
= -0.000417
(~~~0.208
III -1.125 x 0.02545 x 0.0946 x ( 240)2 x (0.64 - 1) = 0.01690
IV
133.4 x 0.02545 x (2.6972)2 [0.04- 0.001417
=
V
x 98.7 x
:00
0.00779
1
-2.67 x
0.02545(0.64 - 1)
Jx
= 0.05505
The left hand side becomes:
Ae Sc]2
=
3J
- f
-
Afs c
rp 3-
0.07214
o
1
2
o
Equating the two sides
Aes c
r
3
=
0.07214 - 0.000417 + 0.01690 + 0.00779 + 0.05505
=
0.1514
Correspondingly
0c
125°
= 62.5°
,
= 117.5°
Ic
Following the same procedure yields:
109°
from z
= 0.8 to z
= 0.5
0
=142 °, e
from z
= 0.5
= 0.2
0
= 158 °, ec = 79 ° , 0 = 10
0m c
o z
=710=
°
M assumed = 1200
113
Assumed value of 0m is satisfactory.
The heat transfer coefficient can be calculated from equation (3.23)
or by the following
n,,e
h
equivalent
(
+ 0.68
relation:
)
0.3375 x 10-4 x 92.17 x 61.15 x 1.060
21C x 0.02384 x 23.2
2 7Cr At
0
m
=
0.0580 Btu/sec ft2 OF
= 208.8 Btu/sec ft2 OF
=
hm(experimental)
et
(hin - hout)
2r 0 L 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
ke!
0.0560 x
0.04768
1.293 x 10
= 206.4
r
1 7-)Ois .
INCLINED TUBE
RUD NO. 119
iI
Given Data:
I
I
Comparative Experimental Data
ro
=
L
= 2.354 ft.
0.02384
ft.
'
=
0.1736
ti
=
127.8 OF
At
=
94
Fluid:
=
t
124.7
=
Qe0
F
0.4955
x 10- 4 ft3 /sec
Qeo
= 0.0995
OF
0cm
cm
T 580
Refrigerant 113
Properties:
94.41 lb/ft 3
k
e
=
5
1.332 x 10--5
Btu/sec-ft-OF
v0g
=
0.5496 lb/ft3
Pe
A/g
=
62.32
hin/g = 97.56 Btu/lb
=
0.0347
Btu/lb
=
hout/g
0.9941 x
10- 5 lbf/sec/ft2
= 34.-54 Bu/lb
(peg~u)
O = 93.68lb/ft3
I
I
I
1Lo.
7I
As sunme
580,
cm cm
2
.a
=29 °
= 1510, ec
~~~c
[92.86x (1.332) 3
3 x 0.9941
x 10o- 1 5 x (0.02384)
x 10-5 x (93.1
i
i
0.2057 x 10 4 ft 3 /sec-ft
=
I
.2057 x 10- 4 x 2.354
gL
~,,5
Jgr
/32.17 x (0.02384)5
= 0.0970
I
f 2 L2
=-
gro 5
i
II
I
= 0.00943
= 0.1664
One
3
x
x 62.32 x 1.024)3
4 x 1.240 x 1.859
--"9r'1
I
Substitute the above quantities together with the angle functions
i
corresponding to the ass-umed depth into equation (4.43), and set the
I
i
i
i
I
II
98.7 x 0.08212 x o x (0.5)
-0.001417 x 170.0 x 0.00943
=
III
0.1664
4.050
x (-0.125)
(3.
(.484)81 x
(1 - 0.25)
-0.2013
(-0.25
170 x 0.00943 x 0(3.08212
(3.060)2
=
V
=
0.00145
-1.125 x 0.00943 x (0011)2
=
IV
nd z 2 = 1.
= 0.5
limits at
+
0.001417 x
98.7 x 0.125)
0.490
-0.00301
-2.67 xo0.00943 (1 - 0.25)
0.08212
=
4.0500'
Or
=
-0.2300
+ 0.00145 - 0.2013 - 0.00301 - 0.2300
0.1069
=
0
142.
i
f
i
i
II
f
Following the same procedure yields:
Ii
=
300,
=
1500
a
=
0.0824
eC
=
270, 0
=
1530
O
=
0.1620
ecc
=
250,
=
155°
o
=
0.2658
=
23,
=
157 °
or =
C
C
Plot on chart of Figure (4.3) and read 0
O'
=
0454
=
153.30 corresponding to
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
Croptt = 0.2
143.
I-
The heat transfer coefficient can be calculated from equation (3.23)
or by the following,
fl
h
equivalent
, X (1 + 0.68 )
21Xr
m
relation:
-
0.2057 x 93.12 x 62.32 x 1.024
At
271Cx 0.02384 x 9.4
=
0.0870 Btu/sec-ft2 -OF
=
313.2 Btu/hr-ft2 -OF
Qe..r.. t (h n - hout)
21rc0 L At
hm(experimental)
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-ft 2 -OF
=
317.9 Btu/hr-ft2 -°F
Difference between heat transfer coefficients is 1.5%
Nusselt Number based on diameter is:
hd
m o
k
-
0.0883 x 0.04768
1.332 x 10-5
= 316.0
4
I.4
----, -
- ,
144.
YInpINDTX iTT
-
EPI-1,17PUT! DP,L
-
-;;:m
145.
TABLE A-1
EQUIP,NT DATA
ANALOGY
WATER
Test Section
EXPERIMENTS
.
.
M-lateriel: iethacrylate plastic
Length:
54 in.
~'ieaninternal
diameter:
1.101 in.
12, equally spaced
Numberof liquid feed locations:
12, ecually spaced
Numberof air bleed locations:
~~~~~~~~...
5,.,'''11U
Pump
S~~~
..uIJ
Eastern
Industries,
Model E-7, Monel Metal Pump with Obmite
Control Rheostat
-?.
~Liquid used wasdistilled water
Note on all experimental
data
There were a number of eloratory
between actual test runs.
Also, in
tests made both before ald
some of the emxperiments
errors or other difficulties were discovered.
rums had an assigned number.
Each of these
In the following tables only those
data are included )Jnichhave direct and meaningful relation to
the topics covered in the text.
146.
TABLE A-2
FLOW DEPTH DATA
WATER ANALOGY EXVERI'TNTS
Flow Bate
No.
_
d
__
Agle
ml/sec g
_
11.70
0.1616
21.28
0.2938
25
_
c
Degrees
CIII
4 39.26
0.5419
5
30.18
6
7
25.45
37.45
0.4166
0.3514
36.72
0 . 5069
9
34.54
0.4768
10
31.25
0.314
1
12
13
6.093
8.508
12.48
0.084
88.0
14
17.66
15
16
17
18
19.45
10.90
15.04
19.13
0.1175
0.1722
0.2437
0.2685
0.1505
0.2076
98.1
103.9
120.0
123.9
104.7
108.7
115.0
137.0
19
20
8.6o
33.04
21 36.57
29 4.17
23
24
38.58
31.19
0.5170
0.264
0.3948
0.4561
0.5048
0.5684
0.5367
o0.4305
137.1
143.2
144.0
142.5
134.5
25
23. 09
0.3188
123.7
26
19.11
09.2638
118.7
27
1Z..33
28
10.90
0.1978
0.1505
112.5
0 .0454
89.0
120.2
_I_
1.5
1.5
1.5
1.5
1.5
1.5
I·_
___I___
83.6 p
Water tem.:
85.8 OF
1.5
1.5
1.5
Water temp.:
80.8
1.5
1*5
1.5
1.5
1.5
1.5
Water temp.:
82.5 OF
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1-5
30
31
4.015
1.591
0.0554
92.5
0 .0220
32 1.709
0.o236
87.8
89.6
1.5
33
2.810
0.0388
4.600
0.0635
0.3324
1.5
34
89.3
93.0
123.0
____·__·_il·
Water temr.:
3.296
24.10
Comments
1.5
1.5
29
35
,
d
from Otlet
_11_
109.3
127.1
139-6
133.5
131.0
138.0
142.2
136.8
134.7
2
L -z
Distance
byr Liquid
Qt.
___
SubtendedLocatio
OFl
Water temp.: 83.0 OF
1.5
1.5
1.5
1.5
Water temr.i:
80.
OF
ist
147.
TABLE A-2
( catinued)
Angle Subtended
l1ow Rate
Rxun
No.
__%b_
Qet
2ec,
ml~~sec
s/r
r/
by Liquid
I_ _
111~~~_
36
2.380
n
n
t
0.0328
It
Unn
0.0875
it
it
it
6.340
TI
if
it9
it
38
9.200
it
0.1270
It
if
n
n
1T
it
39
13.96
It
it
It
II
it
if
n
0.1927
if
I!
itg
n
n
40
20.167
0.2855
It
if1!
t
II
It
11
if
it
24.53
if
97.4
105.6
98.5
102.2
104.3
102.9
111.0
122.2
131.0
104.1
114.5
115.1
111.7
124.0
135.0
140.0
115.8
127.0
131.0
14o.
151.8S
158.0
120.3
136.3
143.4
144.3
157.0
171.-5
0.3382
In
it
97.4
166.0
n
41
95.0
129.8
if
II
Y?
d
Distance £
124.2
144.9
152.1
154.3
165.9
176.7
179.0
Comments
from Outlet
_
I
119.1
121.0
1!
37
Degrees
LocationL-2
-
·---_ _
_
_I
_
__
1.5
8.17
16.33
32.67
40 . 3
48.5
1.5
8.17
16.33
Flowis ltxinar
although streamlines
are wavy.
24.5
32.67
40.83
48.5
1.5
8.17
16.33
24.5
"low is laminar
although streamlines
are wavy
32.67
40.83
48.5
1.5
8.17
16,33
Xlow is redominantly
turbulent
24.5
32.67
40.83
48.5
1.5
8.17
16.33
24.5
32.67
40.83
48.5
1.5
8.17
16.33
24.5
32.67
40.83
48.5
Flow is turbulent
148.
TABLE A-2
(continued)
flow Rate
Angle Subtended
by Liquid
Run
No.
2
mi/see
___
42
1_1·--11
30*80
1---
-·l-··ICii
04250
11
11
it
it
1!
it
it
if
n
43
42.45
It
It
ll
ifn
145
42.60
'I
t
179,O
it
187.5
ifn
U
190.*5
it
I!
211.7
216.4
if
1t
tI
if
1I
if
it
46
0,588
T!
1I
38.87
if
n
0.5365
Ut
lt
n
19
It
M
it
t11
47
35.5e3
It
IT
0.4905
I!
it
19
tt
it
i1
48 26.94
I
it
if
tf
19
It
it
n1
it
if
I
201.9
138,0
174.8
187.8
197.2
209.3
2124.
fl
if
I!
129.5
157.0
165.0
169.6
179.0
188.3
189.4
216.7
138.7
173.0
187.0
191.7
202.1
208.0
211.7
139.9
171.4
I--I-
---
---
8.17
16.33
24.5
32.67
40.83
48.5
1,5
8.17
16.33
24.5
32.67
40o83
48.5
1.5
8.17
16.33
24.5
32.67
40.3
48.5
1.5
206.2
40.83
48.5
1.5
195.2
··_··
48.5
1.5
206.3
g190.7
I
8.17
16.33
24.5
32.67
40.83
186.8
200,0
139.6
162.0
170.2
174.0
180.9
_-
1.5
8.17
16.33
24.5
32.67
156.0
Comments
from Outlet
-_--·.1I·1-·^··-
0.5855
It
n
, Degrees
Location d z
Distance 0
8.17
16.33
24.5
32.67
40.83
48.5
In the following
experiments the water
was obserred to cling
to the tube rather
strongly, Consequently
allobserved angles are
somewhathgh.
149.
TABLE
A-2
(cotinued)
Angle Subtended
F;low Rate
Run
No.
by Liquid
2 , Degrees
m!/sec
Q&
L
11_
49
111_
18.10
la
I
_
0.2500
It
___l_·___a
165.4
ti
I
172.8
174.0
119.1
126.0
132.7
136.0
142.5
0.1231
f!f
if
it
11
· _II
32.67
40.83
48.5
1.5
8.17
16.33
24.55
32.67
40.83
48.5
tf
U
· __··_I___
132.0
145.3
153.2
159.1
11
11
from Outlet
1.5
8.17
!6.33
tI
If
j
_I
I!
80.92
Comments
Qf0
_
1I
50
Location L
d
Distance'
11
11
I1
I!
150.'5
II
ll'
155.6
245
__ W____
__·yl
150.
TABLE A-3
CRITICAL FLOW DATA
WATER
Fiow Pate
lI¾ALT-JOGY
XPTERIKENTS
Description
of
ml/sc
·
24.10
23.76
23 * 30
O*332,4
rfl
0.3101
Ft2
+64'
21.96
0.328o
0.3286
0.3032
0.3114
2 SQ
.22.56
n
I
23.738
24.50
23.62
24 00
0.3230
0.3283
0.3382
0.3260
n SS3 '
Commenrts
at Outl.et
gro
Q0
____1_1
low
1_11·
1_1__
_I___
__I
·
-LIYIUYULIIIIYU··-
*bun No. p2
TurbL ent
Laminar
Laminar
Laminar
Laminar
Laminar
Luid suprv neares to
outlet was closed in order
to reduce turbulence.
Water Temperature was 80.4 F
Laminar
Laminar
Transition point is at
LLmnar
Turbulent
=
Mostly T'rbulent
Tarbulernt
.33
Equivalent Critical Reynolds
Number
Re
= 3580
Bc-3
0
Run No. 44
15.44
0.2131
17.35
19.26
18.50
0.2395
0.2739
0.2659
0.2554
1i.838
0.2602
17.23
17-94
0.237
19.84
0.24'77
Laminar but Disturbed
Partly Laminar
Turbulent
Turbulent
Turbulent
Liquid was suplied uniforuly
through all inout locatians,
including the one nearest to
outlet
OF
Water temper3ature
was 0.2
Transition. point
s at
Turbulent
Partly Laminar
Turbulent
Equivalent Critical Rey-olds
Numb
- er
rte
2,330
A
I
-
. 5L
:..-.
TABLE A-4
EQUIPMENT DATA
CONDENSAIGN E
RIIv
ITS
Test Section
;~.;o-
Material:
-<.
Length:
-...
Std. 5/8" o.d. Aerofin conper tubing
28.25 in.
Internal diameter: 0.71
+ 0.003.
0.000
D
.>-7 ·
0.3527 ft 2
heat Ctransfer
vInternal
area:
Number of fins
Fin
er inch:
utside diameter:
'in thickess:
8
1.375 in.
0.008 in.
-Thermoceuples
..
Materials:
No. 30, Leeds &
orthup
copper-constantan wires
(Emf)
CaOlbration:
0.
(Ef
-0.0048
bs.
Locations: One in the vapor stream entering
test
section;
three
pairs soldered nto slots at the top and bottom of the tbe
at 1/6 L, 1/2 L, and 5/6 L distance a.longthe tube; one
directly behind bottom point at outlet end.in the condensate
stream discharging from test section; to movable junctions.
,
-.
i.
-..
. .
Potentiometer:
Leeds& Northup Semiprecision Potentiometer
??Variacs:General Radio Company, Type
-
pairs
Wattmeter:
Weston
Leveling
nstrument:
,,Licuid
/,---!'due
.,
... -.
used
00-R
s~General
Radio Cmpany, TlypeW54T
310,
Model
No. 3760
Wild (Heerbrugg) Switzerl.nd N2-59138
was distilled Refrigerant-13
~~islldRfiem-1
152.
TABLE A-5
NOM.MCLATURE YOR HEAT TMNSFER
DATA
0r
slope of tube
tr
room temperature,
t.
vapor entering temperature, °F
J1
t
condensatedischarge
tt
average temperature at top of tube,
tb
average temerature
At
mean temperature difference between vapor and wall,
0
F
temperature,
OF'
F
at bottom of tube, OF
dimensionless
discharge flow rate
mean central
0cm
Shd
no
k1
I
angle subtended by condensate,
degrees
c eAt
dimensionless temperature difference
Nusselt number
F
r
t 0 N O O
O
C N CE- N O 0\o r(
.0
ot -o
H to0N H\O 00 N LN
N o GI U\ t
n r- O
> 1 H N -- N
0 0i 0N
0O
C N N , -4 N N CZ O'>rc-o
t
-'"H '
n N --.
0O N - 0c
N
C N Ot
N
N
Ot N N N N Nt H N N N N N N N N N
N N N N N N \ NiN Oi
N N N N
N N N
m
01 rl-A ct0 N N n OO\
Hf1
WN-(0
644
0toC
to NC\4 -44
4
4
m
r-
00
C(
00x t:-(,.
4
ONO
4
L(1 t0w O0 -4 ' c CmrH
LroUJ
4
r
'
E0
wr
0Oo
u0 O-NcN4& NQ
¢N
-NcN\ C
H4 00
*
&
*4 4 4
4 &
4
X
- \ t - 0(t
3 C"
m L- 000 b3 ->U4 - rR(0
O , 4 W -4
'A,N-H
4
*
N H H H'-o 0 N N- 0
0
o
Oi0
b
5 - otC' LNtON
t
0
Dn- )
v."
'0
00
D
M
M
H
m
a)H
1 =i" = 1:= ::
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N. k(
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= = =rH 101111i10
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TABLE A-6
DEFINITIONSOF SYMOLS
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dimensionless flow rate
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156,
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TABLE A-6
LOW DEPTH DATTA
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A
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