. 7

Bulletin 47 CONVECTIVE HEAT AND MASS T RANSFER FROM WATER SURFACES J. Taylor Beard Charl es S. Chen Chandri ka Prasad

CONVECTIVE HEAT AND MASS TRANSFER FROM WATER SURFACES

] . Taylor Beard Department of Mechanical Engineering

University of Virginia

Charles S. Chen Department of Mechanical Engineering

University of Virginia

Chandrika Prasad Department of Mechanical Engineering

University of Virginia

The work upon which this report is based was supported by funds provided by the United States Department of the Interior, Office of Water Resources Research, as authorized under the Water Resources Act of 1964.

OWRR Project B-021-VA

Water Resources Research Center Virginia Polytechnic Institute

and State University Blacksburg, Virginia

November 1971

VPI-WRRC-BULL 4 7

-,D 2 01 v~-7 'vl01 47 c~ rz

TABLE OF CONTENTS

INTRODUCTION

BACKGROUND

THEORETICAL ANALYSIS

ANALYTICAL RESULTS

EXPERIMENTAL INVESTIGATION

DISCUSSION OF RESULTS

CONCLUSIONS

RECOMMENDATIONS

NOMENCLATURE

REFERENCES

APPENDIX

ACKNOWLEDGEMENTS

ill

.. 3

..... 5

.. 7

.13

.... 21

.... .•. 27

.31

.33

.35

.37

.41

.43

I.

II.

LIST OF TABLES

Various Cases Considered for Analytical Solutions . . . . . . . . . . . .

Summary of Analytical Results

LIST OF FIGURES

.14

.28

1. Mathematical Model of Air-Water System . . . . . . 7

2. Stream Function, Velocity, and Velocity Gradient in Air for A.= 0.05 . . . . . . . . . . . .15

3. Stream Function, Velocity, and Velocity Gradient in Water for A.= 0.05 . . . . . . . . .16

4. Concentration and Concentration Gradient of Water Vapor in Air for A.= 0.05 . . . . . . . . . .1 7

5. Temperature and Temperature Gradient in Air for A.= 0.05 . . . . . . . . . . . . . . . .18

6. Temperature and Temperature Gradient in Water for A. = 0.05 . . . . . . . . . .19

7. Schematic of the Experimental Apparatus .22

8. Cross Section of Double Channel System .23

9. Effect of Interfacial Motion on Heat, Mass, and Momemtum Transfer . . . . . . . .29

lV

ABSTRACT

A study of convective heat and mass transfer between horizontal flows of air and water was undertaken. Analog computer solutions of the laminar boundary layer equations for both the gaseous and the liquid phases were developed. Profiles of velocity, temper­ature, and concentration and their gradients were obtained from these solutions.

The experimental verification of these results was carried out in a special air-water facility. Concentration of water vapor in air was obtained through the use of a Mach-Zehnder interferometer and the thermocouple-measured temperatures. A Helium-Neon gas laser was used as a light source for the interferometer.

Analytical and experimental results are in good agreement and indicate a marked influence of the interfacial motion on heat and mass transfer rates. In co-current flow the increase in the inter­facial velocity increased the heat and mass transfer rates. This is due to increased convective heat and mass removal resulting from the finite velocity of the interface itself and the increased velocities in the boundary layer close to the interface.

1

INTRODUCTION

Analysis of heat, mass, and momentum transfer from a gas-liquid interface is of major importance in many problems of engineering interest. Such an analysis from an air-water interface is of particu­lar interest for water thermal pollution control where an air stream flowing parallel to the surface of a water reservoir, creek, or river is used to dissipate heat. This investigation was undertaken to study forced convection heat and mass transfer between air and water set in laminar parallel motion and to examine the influence of the interfacial motion on these transfer rates. Fluid mechanics of the contacting phases, which plays an important role in convec­tion heat and mass transfer, was also studied.

The results of these studies may be extended to other industrial gas-liquid contacting operations such as film coolers, falling film absorption towers, condensers, wet cooling towers, and distillating towers.

3

BACKGROUND

Conservation and optimum utilization of water resources have been of utmost concern to society. A rapid increase in the popula­tion, the industrial activity, and the standard of living has placed a great demand on the available water supply. Waste heat discharge to a water body is a pollutant because it causes an increase in the water temperature which may be hazardous to many important aquatic species.

The familiar methods of reducing waste heat rejection to water bodies involve the use of cooling ponds and cooling towers. These work on the principle of air-water contact with heat energy re­moved by radiation, evaporation, and convection.

The mathematical formulation of the physical heat transfer process which occurs is not a simple matter. It is necessary to consider both heat transfer in water and heat transfer between water and the atmosphere in order to describe temperature distri­bution mathematically. For effective management control tech­niques, specific predictions of the temperature throughout the water system are desirable. It is necessary to predict how the waste heat will be dispersed in the water body and how it will be trans­ferred into the atmosphere.

The effects of wind on water standing in a laboratory channel were studied by Keulegan. 1 He found a curve which related the ratio of the water surface velocity divided by the average velocity of air in the channel to a Reynolds number based on the water surface velocity and the depth of water. The curve was confirmed by Masch2 and Van Dorn.3 Recent studies of Hidy and Plate4

indicate values of this ratio to be 203 lower than those of Keulegan and Masch. The difference has not been explained ex­cept that Hidy and Plate used channels of quite different sizes, thus the average velocity in the entire channel had no clear mean­mg.

5

The results of Hidy and Plate indicate that up to a wind speed of 3 m/sec, no waves appear on the water surface. However, small oscil­lations of the entire surface in this range of air flow could be detected by watching variations in the light reflected from the water surface. At velocities higher than 3 m/sec, waves appear on the surface and grow with increasing air speeds.

Plate and Goodwin5 carried out experiments to study the influ­ence of wind on moving water. The experimental setup was the same as that used by Hidy and Plate. For moving water the surface velocity will also depend on the flow of water in the channel. Their results also indicate a significant deviation from the general trend indicated by Keulegan.

The velocity distribution for the laminar parallel flow of a viscous incompressible fluid over another fluid of different density and viscosity was obtained by Lock6 and Potter. 7 Most of their analy­sis was limited to the case where the lower fluid was stationary.

Beard and Hollen8 used Reynolds analogy and empirical relations for wind-induced stresses to estimate convection heat transfer for use in the energy budget method to calculate evaporative losses from a water body. Such an analogy is well proven for solid-gas and solid-liquid interfaces, but such a justification is not available for a gas-liquid interface.

Byers and King9 studied mass transfer from a gas-liquid interface under isothermal conditions. Their studies indicate that co-current flow increases the mass transfer rate while counter-current flow decreases this rate. Since heat transfer and mass transfer are gov­erned by similar mechanisms, it is believed that heat transfer will be likewise influenced by the motion of the interface.

In most of the engineering applications, heat and mass transfer occur simultaneously and often these are strongly coupled. A com­prehensive analysis of simultaneous heat and mass transfer from a gas-liquid interface has not been found in the available literature.

6

THEORETICAL ANALYSIS

A schematic of the physical system under study is shown in Figure 1, which also shows the coordinate system and dimensional no­menclature. The origin of the coordinate system is taken at the point at which the two fluids come in contact. The fluids are considered in parallel and co-current motion and the flow is as­sumed to be laminar, two-dimensional, and incompressible. Before contact, the velocity and temperature of both fluids are uniform and correspond to the free stream conditions. It is assumed that the fluid properties are constant for the range of temperatures considered and the flow is such that the usual boundary layer approximations are valid.

FIGURE 1

MATHEMATICAL MODEL OF AIR-WATER SYSTEM.

y

L. Air

Water

7

The upper fluid . is a mixture of water vapor and air, and the concentration of water vapor in the free stream air is expressed by its mass fraction, mvoo· Due to viscous, heat-conduction, and mass diffusion processes, velocity, temperature, and vapor concentra­tion profiles are formed in the boundary layer of the upper fluid. The diffusion of air into water is neglected; hence, only velocity and temperature profiles are considered in the boundary layer of the lower fluid. Under these conditions, the governing con-servation equations are:

au· 1

av· 1

Continuity: + 0 [1] ax ay

au. au· a2u· 1 1 1

Momentum: u· + v· = vi [2] 1 1 37 ax ay

at · 1 at·

1 a2t·

1 Energy: u· + V· = Q'.· [3] 1 1

1Ty2 ax ay

amv amv a2m v Diffusion: u1--- + v1--- =Dv-- [ 4]

ax ay ay2

where i = 1 for air, i = 2 for water, and the subscript v denotes water vapor. The continuity, momentum, and energy equations occur in pairs, one each for the gaseous and the liquid phase. The di ffusion of water vapor into the air stream is given by equation [ 4] .

The appropriate boundary conditions are:

At y = o [5]

[6]

[7]

8

au 1 au 2 µ1 = µ2 [8]

ay ay

at· at2 amv 1

K1-=K2- + P 1 Dvhfg ay ay ay

[9]

At y = + 00 : [ 10)

[11]

[ 12]

At y = - 00 : [ 13]

[ 14]

Equations [ 5, 6, 7] denote the interfacial velocity, temperature, and vapor concentration. Equation [8] is based on the continuity of shear stress and equation [9] is obtained from an energy balance at the interface taking mass transfer into consideration. Equations [ 10] through [ 14] denote the free stream conditions of air and water.

The above set of partial differential equations are reduced to ordinary differential equations by defining a set of dimensionless variables and stream functions as follows:

Upper fluid-air, ( o<y<oo):

u u 771 = ( _l_ ) i'2 y

VlX

'1'1 =(v1U1x)Y2F1(771)

Lower fluid-water, (-oo<y<o):

772 = -( ~) Y2 y VlX

9

It can be shown that these stream functions satisfy the continuity equations.

Momentum, energy, and diffusion equations (2, 3, 4] may be transformed to the following dimensionless forms.

I I

F I fl + Y2 F1 F1 = 0 1

[ 15]

Vl + v2 ( -) v2 F2F2 , = o [ 16]

V2 V1 I

+ V2 - F1e1 =O [ 17]

O'.l

V1 I

+V2--F1C =O D v v

(19]

m -m and C = vo v

v

and the superscript prime ( ' ) denotes differentiation with respect to fl.

Boundary conditions, equations [ 5-14], are also reduced to the following dimensionless forms:

10

[20]

(21]

[22]

I

F1(oo)=l [25]

81(00)=1 [26]

Cv(oo) = 1 [27]

I VI Y2 F2 (oo)=A (-) [28] v2

e 1 (oo) = 0 [29]

where subscript o denotes the interfacial conditions and 00 the free

stream.

Since the flow is steady and the interface between air and water is taken as the streamline 'Ir = o, an additional boundary condition is imposed as:

[30]

A detailed explanation for the use of this boundary condition is given in the appendix.

11

ANALYTICAL RESULTS

The differential equations, [ 15] through [ 19], with the boundary conditions, [20] through [ 30], were solved on an analog com­puter. An examination of the boundary conditions reveals that many of the interfacial conditions are not explicitly known. The objective here is to select the unknown interfacial conditions in such a way that the free stream conditions are satisfied. This is achieved by adjusting the initial conditions on the analog com­puter until the end conditions are satisfied. The initial conditions selected on the analog computer theref9re correspond to the inter­facial conditions of a physical problem. This correspondence was made possible through the proper selection of dimensionless vari­ables used in converting the partial differential equations to ordi­nary differential equations.

A number of different cases were considered for the analytical solutions. These are listed in Table I. The analytical solutions for Case II are plotted in Figures 2 through 6. The analytical solutions for the other cases are given elsewhere. l 0 .

Cases I through XI correspond to various values of the velocity ratio (A. = 0 to 0.9) and for an air-water system with the following conditions: free stream air temperature = 7 5° F; free stream water temperature = 125°F; and relative humidity in free stream air = 803.

Case XII was considered to study velocity profiles for flow of air over stationary water under isothermal conditions. In Case XIII the value of A was zero but air and water temperatures were differ­ent than those of the other cases. In Case XIV the air was set hotter than water and the lower fluid was considered stationary, i.e. A.= 0.

13

TABLE I

VARIOUS CASES CONSIDERED FOR ANALYTICAL SOLUTIONS

Velocity Air Water Relative humidity ratio temperature temperature in free stream dir.

Gase A. Tl, OF T2, OF 3

I 0 75 125 80 II 0.05 75 125 80 III 0.1 75 125 80 IV 0.2 75 125 80 v 0.3 75 125 80

VI 0.4 75 125 80 VII 0.5 75 125 80 VIII 0.6 75 125 80 IX 0.7 75 125 80 x 0.8 75 125 80

XI 0.9 75 125 80 XII 0 75 75 80 XIII 0 30 50 80 XIV 0 90 60 80

14

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n2

EXPERIMENTAL INVESTIGATION

The experimental setup consists mainly of a Mach-Zehnder interferometer, a test-section located in the optical path of the interferometer, temperature measuring thermocouples, and velocity anemometers. A schematic diagram of the air-water facility used for the experimental investigation is shown in Figure 7. The test-section consists of a rectangular channel 5 3/4 in. high and 8 in. wide with vertical sides made of optically flat glass. During preliminary runs it was found that the meniscus formed by water on the glass walls tended to "black-out" interferometric fringes close to the interface. To eliminate this meniscus effect, a center channel was built as illustrated in Figure 8. Water flow was confined to this center channel ( 2 5 / 8 in. high and 6 in. wide). Flows were controlled by the inlet and outlet valves such that the center channel was always slightly overflowed. Overflowed water was removed through a bottom drain.

The entire test section was mounted on rigid supports and on wheels riding on two tracks. This arrangement was necessary so that the interferometric measurements could be made at different locations of the entire channel length.

Water was circulated through the channel in a closed loop by means of a pump and reservoir. Air was drawn through the channel above the water surface by a fan mounted on the exit side of the test section. Electric heaters and temperature controllers were provided to regulate air and water temperatures. To minimize vibration, the water pump was mounted on a separate platform and flexible connectors were used wherever possible. Smooth water surface was obtained by proper adjustment of guide vanes at the inlet.

Measurement Technique

The facility was instrumented for taking profiles of velocity, temperature, and concentration. The velocity in air was measured by means of a hot-wire anemometer. Average velocity of water

21

~ ~

~

FIG

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7

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MA

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OF

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TU

S.

Fan

Tan

k

23

was measured by mass flow rate. The temperature in air and water was measured by thermocouples. A He-Ne gas laser provided the monochromatic light source for the Mach-Zehnder interferometer. Photographed interferograms were used in conjunction with the temperature measurements to obtain water vapor concentration in the air. A micrometer head was used for vertical translation of probes in the boundary layer.

The interferometer was used to measure the refractive index, which is related to density. Density is in turn related to temperature and concentration. It has been used by many for temperature measurement in airll ,l 2 and liquid. l3,l 4 It has also been used for vapor concentration measurement in an isothermal flow by Lin, Noulton, and Putman.15 But the use of the interferometer in simultaneous heat and mass transfer processes presents some difficulties. A two wave-length technique was used by Ross and El-Wakil16 and by Sadovnikov, Smol'skiy, and Schitnikov1 7 for simultaneous heat and mass transfer. Difficulties arose in this technique due to insufficient independency of the equations.

El-Wakil, Myers, and Schilling18 used a one wave-length technique for simultaneous heat and mass transfer assuming that normalized temperature and concentration profiles were similar, and thus were somewhat restrictive. One way to avoid these difficulties is to measure the temperature by a non-interferometric technique and calculate concentration from the interferogram. Such a technique was used by Adams and McFadden 19 to study heat and mass transfer during sublimation of p-dichlorobenzene from a heated vertical plate undergoing free convection. Detailed description on the use of one wave-length technique to study heat and mass transfer from an air-water interface was discussed by Prasad, Chen, and Beard. 20

For simultaneous heat and mass transfer in a multicomponent system, the fringe shift is related to the temperature and concentration by the following equation:

24

L S=±

w T [L· Y. N. l (noo - 1) ~ I AJ. I - 1

T N00

[ 31]

In the present analysis both temperature and concentration measurements are required. In a two-component system consisting of air and water vapor, only the concentration of one species needs to be measured. Equation [ 31] can then be simplified to

Na N00 T [ w S l ~= -(--) - 1 ± - [32]

Na - Nv Na - NV T00 L (n00 - 1)

where Na = molar refraction of dry air, Nv = molar refraction of water vapor, and N00 = molar refraction of free stream air which contains moisture. The plus sign in . equations [31 and 32] corresponds to n>n00 and minus sign to n<n00•

25

DISCUSSION OF RESULTS

Three sets of experiments were performed during this investi­gation. The first set involved the flow of air over stationary water and the other two sets were concerned with the flow of air over moving water. The air velocity was varied from 0.52 fps to 2. 7 fps and water velocity was varied from 0 to 0.156 fps to obtain vari­ous values of the water/air velocity ratio. Three values of the velocity ratio used were 0, 0.05, and 0.30.

For velocity ratio A. = 0.05, the air velocity measured in the boundary layer is plotted in Figure 2. Due to lack of instruments the water velocity profile was not measured. Experimental data for diffusion of water vapor in air are plotted in Figure 4 in terms of vapor concentration. The experimental temperatures in air are plotted in Figure 5 and those in water in Figure 6. The agreement between the theory and the experiment in general was quite good. The deviation appearing in Figure 6 was amplified due to non­dimensionalizing of the temperature. In terms of Fahrenheit tem­perature, the maximum difference between the theory and the experiment was less than 2 .53.

Theoretical velocity profiles for an unheated air and water system were in good agreement with the results of Lock6 and Potter. 7

Theoretically calculated interfacial velocity was 2.143 of the free stream air velocity for the case of stationary water (A.= 0), which is within 1.53 of the values obtained by Lock and Potter. The interfacial velocity will vary with the movement of water and the viscosities of the fluids under consideration.

Though the experiments were run for only three values of the velocity ratio, A. = 0, 0.05, and 0.30, the theoretical analysis was extended up to A.= 0.9 and the results are summarized in Table II. As the water velocity is increased with a constant air velocity (up to A. = 0.9) the shear stres$ or friction factor decreases but heat and mass transfer rates increase. The effect of velocity ratio or the interfacial motion on shear stress as well as heat and mass transfer is reflected by the changes in velocity, temperature, and concen-

27

tration gradients at the interface and are plotted in classical dimen­sionless forms of friction factor, Nusselt number, and Sherwood number in Figure 9.

TABLE II

SUMMARY OF ANALYTICAL RESULTS

1 2 3 4 5 6 Velocity Surface Velocity Concen- Surface Tempera-

ratio, velocity, gradient, tration temp., ture gradient, gradient,

u uo '' < (o) Case A.=_2 ·u1 F1 (o) e 1 (o) e{ (o)

U1

0 0.0274 0.3247 0.2894 0.098 0.2884 II 0.05 0.0646 0.3246 0.3028 0.058 0.1960 III 0.10 0.1106 0.3178 0.3122 0.0567 0.3242 IV 0.2 0.2072 0.3126 0.3300 0.0517 0.3572 v 0.3 0.3064 0.2953 0.3450 0.0465 0.3766

VI 0.4 0.400 0.2607 0.3600 O* 0.3974 VII 0.5 0.500 0.2302 0.3760 0 0.4154 VIII 0.6 0.600 0.1933 0.3910 0 0.4306 IX 0.7 0.700 0.1503 0.4050 0 0.4440 x 0.8 0.800 0.1006 0.4172 0 0.4576

XI 0.9 0.900 0.0535 0.4370 0 0.4694 XII 0 0.0214 0.3248 0.2730 0.984 0.0015 XIII 0 0.0192 0.3245 0.2948 0.096 0.2720 XIV 0 0.0223 0.3228 0.2780 0.0733 0.2960

*For values of A greater than 0.30, the lower fluid velocity was found to be effectively equal to the interface velocity. Under this condition heat transfer in water will be by conduction only and convection effects may be neglected. To obtain solutions for cases of A.= 0.4 to 0.9, the dimensionless air temperature was defined as

e1 = t1 - to

T1 - to

where t0

is the interface temperature.

28

0.4

0.3

0.2

0.1

FIGURE 9

EFFECT OF INTERF ACIAL MOTION ON HEAT, MASS, AND MOMENTUM TRANSFER.

0.2 0.4 0.8

f F ~' (o) = 2

(Rexl)l/2

' ' ' 1.0

29

CONCLUSIONS

1. In the co-current gas-liquid flow experiment, the interfacial motion enhances the heat and mass transfer rate which is also predicted by the theoretical calculations. This is due to increased convective heat and mass removal resulting from the finite velocity of the interface itself and the increased velocities in the boundary layer close to the interface.

2. For higher velocity ratios, the velocity in the lower fluid may be assumed uniform, and the energy convection in the lower fluid may be neglected. This will result in a linear temperature profile in the liquid.

3. Analog computers may be used to solve certain classes of boundary layer equations, especially where similarity trans­formation can be used to reduce the partial differential equations to ordinary differential equations.

31

RECOMMENDATIONS

This study was limited to laminar gas-liquid flow. The maximum velocity of air used was 2. 7 fps. In practice the velocity of air may be many times greater than this value, and the flow of air may be turbulent. Study of turbulent flows will be a logical extension of this study.

Only co-current flow of air and water was investigated. However, counter-current flow occurs more frequently in industrial appli­cation. It is a general notion to assume that heat and mass transfer rates in counterflow are greater than those in co-current flow. The effect of interfacial motion as studied in this investigation casts some doubts on this procedure. Hence, extension of this study to counterflow and other gas-liquid systems is recommended.

33

NOMENCLATURE

English letter symbols

c dimensionless concentration D mass diffusivity F dimensionless stream function

hfg latent heat of vaporization of water K thermal conductivity

L optical path length m mass concentration M molecular weight n index of refraction N molar refraction

s fringe shift t temperature in the boundary layer T free stream temperature u velocity in the x direction u free stream velocity

v velocity in the y direction w mass transfer rate per unit area x horizontal coordinate x mole fraction y vertical coordinate

Greek letter symbols

w

thermal diffusivity U dimensionless coordinate, (-) 112 y

vx stream function ratio of free stream water /air velocity wave length of light

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µ v p ().

1

absolute viscosity kinematic viscosity density dimensionless temperature,

Subscripts

1 alf

2 water v water vapor o interface ex:> free stream

Superscripts

36

Prime (') denotes differentiation with respect to r/ and (

0) denotes differentiation with respect to time.

REFERENCES

1. G. H. Keulegan, "Laminar Flow at the Interface of Two Fluids," J. Res. Nat. Bur. Std. (U.S.A.), Vol. 32, 1944, p. 303.

2. F. D. Masch, "Mixing and Dispersive Action of Wind Waves," Int. ]. of Water and Air Pollution, Vol. 7, 1963, p. 697.

3. W. G. Van Dorn, "Wind Stresses on an Artificial Pond," ]our. of Marine Research, Vol. 12, No. 3, 1953, pp. 249-276.

4. G. M. Hidy and E. J. Plate, "Wind Action on Water Standing in a Laboratory Channel," J. of Fluid Mechanics, Vol. 26, pt. 4, 1966, pp. 651-687.

5. E. ]. Plate and C. R. Goodwin, "The Influence of Wind on Open Channel Flow," Coastal Engg., Santa Barbara Specialty Conference (ASCE), Chapter 17, October 1965.

6. R. S. Lock, "The Velocity Distribution in the Laminar Boundary Layer between Parallel Streams," Quarterly J. of Mech. and App. Maths., Vol. 4, 1957, p. 81.

7. C. E. Potter, "Laminar Boundary Layer at the Interface of Co-current Parallel Streams," Quarterly J. of Mechs. and App. Maths., Vol. 10, 1963, p. 302.

8. J. T. Beard and D. K. Hollen, "Influence of Solar Radiation Reflectance on Water Evaporation," Bulletin No. 30, Vir­ginia Water Resources Research Center, Virginia Polytechnic Institute and State University, Blacksburg, Va., August 1969.

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9. G. H. Byers and C. ]. King, "Gas-Liquid Mass Transfer with a Tangentially Moving Interface," Part 1, AICHE J our., Vol. 13, No. 4, 196 7, pp. 628-636.

10. C. Prasad, "Convective Heat and Mass Transfer from an Air­Water Interface," Ph.D. Thesis, University of Virginia, Char­lottesville, Va., August 1971.

11. R. J. Goldstein and E. R. G. Eckert, "The Steady and Tran­sient Free Convection Boundary Layer on a Uniformly Heated Vertical Plate," Int. J. of Heat and Mass Transfer, Vol. 1, 1960, pp. 208-218.

12. ]. Winckler, "The Mach-Zehnder Interferometer Applied to Studying an Axially Symmetric Supersonic Air Jet," Review of Sc. Inst., Vol. 19, No. 5, 1948, p. 30.

13. S. A. Fisher and D. Wilkie, "Preliminary Report on an Ex­perimental Investigation of Heat Transfer in a Fluid Con­taining a Distributed Heat Source," U.S. Atomic Energy Authority, DEG Report No. 214(W), 1960.

14. D. Wilkie and S. A. Fisher, "Natural Convection in a Liquid Containing a Distributed Heat Source," Int. Heat Transfer Conf., Paper 119, 1961, University of Colorado, Boulder, Colorado.

15. C. S. Lin, R. W. Noulton, and G. E. Putman, "Inter­ferometric Measurement of Concentration Profile in Turbu­lent and Streamline Flow," Industrial and Engg. Chemistry, Vol. 45, 1953, pp. 640-646.

16. P. A. Ross and M. M. El-Wakil, "A Two Wave-Length Inter­ferometric Technique for the Study of Vaporization and Combustion of Fuels," Progress of Astronautics and Rocket­ry, Vol. 2, 1960, pp. 265-298.

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17. G. V. Sadovnikov, B. M. Smol'skiy, and V. K. Shchitnikov, "Investigation of Simultaneous Heat and Mass Transfer Using an Interferometer," Heat Transfer, Soviet Research, Vol. 1, No. 1, 1969, pp. 32-38.

18. M. M. El-Wakil, G. E. Myers, and R. J. Schilling, "An Inter­ferometric Technique for Mass Transfer from a Vertical Plate at Low Reynolds Number,'' ]our. of Heat Transfer, Trans. ASME, Series C., Vol. 89, No. 3, Nov. 1966, pp. 399-406.

19. J. A. Adams and P. W. McFadden, "Simultaneous Heat and Mass Transfer in Free Convection with Opposing Body Forces,'' AICHE ]our. Vol. 12, No. 4, July 1966, pp. 642-647.

20. C. Prasad, C. S. Chen, and J. T. Beard, "An Interferometric Technique for Temperature and Concentration Measurement for an Air-Water Interface," Transactions of the ASME, Journal of Basic Engineering, Vol. 93, Series D, No. 2, June 1971, pp. 185-191.

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APPENDIX

DISCUSSION OF THE BOUNDARY CONDITIONS, EQUATION [30]

The boundary condition as given by equation [30] implies that V 1 = V 2 = 0 at the interface. The vertical velocity component is

a'l' related to the stream function by the relation V=- - .For steady

ax flows and no mass transfer from the interface, the interface be­tween the water and air may be considered as the streamline '11 = 0 passing through the origin. But this is not true where condensation and evaporation take place. The vertical velocity at the interface is connected to the mass transfer by the following equation:

[33]

or,

[34]

The equations [ 33] and [ 34] indicate that the normal velocity can be obtained if (i) the concentration gradient at the interface is obtained from the solution of the diffusion equation and (ii) the interface vapor concentration is obtained from the solution of the energy equations, as the interface vapor concentration is related to the interface temperature. But solutions of the diffusion and energy equations require a prior knowledge of the velocity profiles. At the same time the energy equation is coupled to the diffusion equation through the boundary condition as given by equation [24]. In view of the highly coupled nature of these equations, a simplified boundary condition as given by equation [30] was used in the present analysis.

Another reason for neglecting the normal velocity at the interface is that its magnitude is quite small compared to the horizontal component of the velocity. An order of magnitude analysis based on the solution for A. = 0.3 indicates that the normal velocity at

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the interface is less than 1 % of the horizontal component of the interface velocity, while that in water is much less than 1 %.

42

ACKNOWLEDGEMENTS

This work was done at the University of Virginia under the spon­sorship of the Virginia Water Resources Research Center through Grant No. B-021-VA. The use of the analog computer facility of the Department of Mechanical Engineering at the University of Virginia is acknowledged.

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