mills heat transfer

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SHORTER COMMUNICATIONS

[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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696


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;

mills heat transfer

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