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[2] obtained from numerical solutions of integral equations.
Agreement between the present results and those of Lin
is seen to be good.
A presentation of E, results for a cone half angle of 5
is made in Fig 2. The structure of this figure is similar to
that of Fig. 1, and the findings are generally similar. As
before, the fluctuations exhibited by the petitioning results
are appreciably less than are those in the results from the
conventional solution method. Also, the partitioning results
converge more rapidly.
The findings presented above indicate that the use of the
energy partitioning approach may be highly advantageous
in problems where a portion of the energy content of a ray
bundle is governed by deterministic laws (e.g. geometrical
angle factors). Another interesting outcome of the present
work is the documentation of the fluctuations experienced
by the Monte Carlo results as the number of rays is increased.
This suggests that the output corresponding to a specific
pre-selected number of ray bundles may not always be a
proper representation of the results.
1. J. R. HOWELL, Application of Monte Carlo to heat
transfer problems, Advances in Heat Transfer, Vol. 5.
Academic Press, New York (1968).
2. S. H. LIN, Radiant interchange in cavities and passages
with specularly and diffusely reflecting surfaces. Ph.D.
Thesis. Department of Mechanical Engineering. t Ini
versity of Minnesota, Minneapolis, Minnesota (1964).
Inf. J. Heat Moss 7hmsfer. Vol. 16 pp. 694-696. Pergamon Press 1973. Prmted in Great Britain
HE T TR NSFER CROSS TURBULEINT F LLING FILMS
A F MILLS and D K CHUNG
School of Engineering and Applied Science, University of California, Los Angeles, California 90024, U.S.A.
Receiued 25 September 1972)
gravitational acceleration;
heat transfer coefficient ;
thermal conductivity;
Prandtl number
;
Reynolds number = 4F/p ;
Schmidt number ;
dimensionless velocity = u/&g&);
dimensionless coordinate measured from wall
Y (sS)
-.
v
flow
rate per unit width
;
film thickness ;
dimensionless film thickness ;
eddy diffusivity ;
= 1 + E/v;
dynamic viscosity ;
kinematic viscosity ;
density
;
surface tension.
=
Subscripts
D,
mass species diffusion ;
M,
momentum;
2
intersection of equations
(1)
and (4)
:
t,
turbulent.
RECENTLY
Chun and Seban [l] presented the results of an
experimental study into heat transfer across evaporating
turbulent falling films. They found h, cc Re04,whereas usual
analyses e.g [2,3] based on the conventional hypotheses
about turbulent transport that are consistent with pipe
flow, predict h, cc Re
in
the limit of high
Re see
Fig 1).
Our purpose here is to present an analysis which success-
fully predicts the Chun-Seban data, and which may also be
used for film condensation provided that vapor drag is
negligible.
Our starting point is the observation that conventional
hypotheses about turbulent transport do apply close to the
solid wall, as demonstrated by the electrolytk mass transfer
experiments of Iribarne et al. [4]. Any one of a number of
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695
eddy diffusivity profiles could therefore be used in the near
or for computational purposes,
wall region; we choose to use that of van Driest [S],
&vt &@s
~2 = 4 {I+ ../[I + W4y(1- exp(-y/26))*]).
(1)
&;=1+647xlo-* -----&6*-y+).
(4)
u
Next we suggest that the nature of turbulent trausport near
the interface should be reliably shown by gas absorption
experiments Lamourelle and Sandah [6j measured mass
transfer coefficients for gas absorption into a turbulent
falling fihn and deduced that a consistent eddy di~usivity
profile was
E,, = 7.33 x 10m6Reb8(6 - y) -m/s.
(2)
Accurate s~ci~~tio~ of the eddy di~sivi~ in the middle
region of the film is not required owing to the low thermal
resistance there. Thus the complete eddy diffusivi~ profile
can be obtained by simply running equations (1) and (4)
until they intersect, at yi+ say. The model of turbulent trans-
port is completed by specifying Pr, = 09 following recent
boundary layer practice, e.g. [S], and SC, = 1.0 in the
absence of reliable evidence to the contrary.
.
I , I
,
,I,
I
I
1, I,,,
I
- mm Dukl er
. _
em Chun- Seban corr elati on for ' %avy l m nar f l ow
Pr 410
-. 6
Present theory
e.
- -
L pn 0.4 -
. A*
_ - - - - -
1
I I
I I iI I
- - - - .
0.1
I I I I Ill 1
I
2
L
4 6
8
103
2
4 6 8
104
2
4
FIG. 1. Local heat transfer coefficient as a function of
Reynolds number; comparisons of theory with the experi-
ments of Chun and Seban.
Unfortunately equation (2) is not dimensionless, and was
The calculation procedure to yield the heat transfer
based only on results for water at 25C. Chun and Sebau
did obtain a set of evaporation data for water at ZfK, so
coefftcient invokes the usual Nusselt assumptions of
that equation (2) could be used directly in that case In order
negligible liquid acceleration and thermal convection, and
constant properties Since the shear distribution is linear
to generalize equation (2) to temperatures other than 25C,
and to liquids other than water, it must be rendered dimen-
there results for the velocity profile
sionless in a manner which recognixes the essential physics
PC
involved. Following Levich 17-Jwe view the damping of the
u+(y+) =
s
I-y/S +
---Y-- dy
,
(5)
eddy diiusivity represented by equation (2) as due to
%f
0
surface tension and thus suggest the dimensionhrss form
and the Reynolds number is defined by
E; = 1 + 647 x lo-BP &1*6s (6 _ y),
0
(3)
a+
Re
zs.
=4 u+dy+.
s
0
(6)
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Equations (1) and (4) are substituted in equation (5) and
to include the effects of vapor drag must await the av Pi
starting with nominal values of l&r and 6+, equations (5)
abifity of appropriate gas absorption experimental data.
and (6) are solved iteratively. The corresponding heat trans-
fer coefficient is obtained by integrating the energy conserva-
ACKNOWLKDGEMKNTS
tion equation
;
This work was sponsored by the Office of Saline Water
3 lif +
i)
Pr@*)*
on Grant No. 14-30-2678, and by the University of Cali-
fomia Water Resources Center on Project No. S142.
k 9
=-F(S+)
(7)
Computer time was supplied by the Campus Computing
Network of the University of California, Los Angeles.
where
1.
F@ +) = ---~ _
1
2.
Figure I compares our predictions with the Chun-Seban
3.
experiments. Data was obtained at four saturation tem-
peratures: 28, 38, 62 and 100C; the corresponding liquid
Prandti numbers were 5.7, 5.1, 2.91 and 1.77 respectivety.
4
The agreement is seen to be excetfent. At 28C we are in
effect demonstrating the consistency of the experimental
data on which our calculation procedure is based. The
good agreement at other temperatures confirms that the 5.
scaling with surface tension in equation (3) is appropriate.
We note further that (i) y: varied from 0.3 &+ to @85 6+ as 6.
Re increased through the range considered, and {ii) the gas
absorption experiments showed the mass transfer coe&ient
to be proportional to J@*;
thus the result 4 a Bee. is not
7
unexpected despite the fac0 that the near wah transport 8.
model suggests h, oc ReOa. Figure 1 of course appiies to
turbulent film condensation as well. Extension of this work
K
R. CWUN nd R. A. SEBAN,Heat transfer to evapomt-
ing liquid films, f. Neat Transfer 93, 391-396 (1971).
R. A. SEBAN, Remarks on tilm condensation with
turbulent Row. Trans. Am. Sac. Mech. Engrs 76,299-303
(1954):
A. E. DUKLER Fluid mechanics and beat transfer in
vertical falling film systems, Gem. Engng Prog. Synlp.
Ser. Nii6,
-10 (1960).
A. IRtBARNl?; . D. GOSMAN nd D. B. SPALDING, A
theoretical and experimental ~nves~~tion of diffusion-
controlled electrolytic mass transfer between a falling
liquid film and a wall, Znt. J.
Hear M ass Transfir $0,
1661-1676 (1967).
E. R. VANDRIEST,On turbulent flow near a wall, J. Aeru.
Sci. 23, ~~7--~~~ (1956).
A. P. LAMQURELLEnd 0. C. SANDALL,Gas absorption
into a turbulent liquid, Chem. En,snz Sci. 27, 1035-1043
(1972). -
_ _
V, C. twrcy Pk_~~~~cke~~~~~ydrQ~~}l~~~~~, . 691.
Prentice-Halt, Endewood Chffs, New Jersey (1952).
R. B. LANIX~ nd A. F. MILLS,The catcula&n of turbu-
lent boundary layers with foreign gas injection, Int. f.
Heat Mew Transfer X 5, 1905-1932 (1972).
K. C. CHKNG
and
JKNN-WUU OU
Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada
Br,
3,
t
NOMENCLATURE
Brinkman number, ~~~/~Te - T,) = &Ec;
specific heat at constant pressure
;
speciBc enthalpy and local heat transfer co-
efticient, respectiveIy
;
thermat conductivity
;
P>
local axial pressure :
49
heat flux vector ;
R, x,
radial and axial coordinates;
Ro,
pipe radius
;
I; T,, T, gas temperature, uniform gas temperature at
thermai entrance and constant wall tempera-
tare,
espectively;