Heat transfer to water droplets on a flat plate in the film boiling regime

UF00098225 ( .pdf )

HEAT TRANSFER TO WATER DROPLETS ON A

FLAT PLATE IN THE FILM BOILING REGIME


















By
KENNETH JOSEPH BAUMEISTER


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY











UNIVERSITY OF FLORIDA
December, 1964
























Dedicated


to my wife, Mary,


for the love and understanding
she has given to me throughout
my graduate education.










ACKNOWLEDGMENTS

The author wishes to express his sincere

appreciation to the members of his supervisory committee:

Dr. Robert E. Uhrig, chairman; Dr. F. L. Schwartz, co-

chairman; Prof. Glen J. Schoessow; Dr. G. Ronald Dalton,

and Dr. Robert G. Blake. Special thanks are due to Dr.

Schwartz for suggestion of the thesis topic and to Prof.

Schoessow for technical advice concerning the experimental

procedures.

The author wishes to thank the University of

Florida Computing Center for the aid given him. Gratitude

is also expressed to F. A. Primo, H. H. Moos, and Joseph

Mueller for their help in setting up the experimental

equipment and to Mrs. Gail Gyles for her helpful sugges-

tions while typing the thesis.

Thanks are also due to the staff of the Lewis

Research Center of the National Aeronautics and Space

Administration for the support given to the author while

carrying out this investigation. In particular, thanks

are due to Mr. Robert J. Usher, Chief, Training Branch,

and to Miss Gertrude Collins. Finally, very special

thanks are due to Mr. Harry Reilly, Chief, Reactor Analy-

sis Section, of the NASA Plum Brook Reactor Facility for

his guidance in the development of the author's technical

maturity and for his encouragement during the past two

years.
iii














TABLE OF CONTENTS


Page

ACKNOWLEDGEMENTS ..... . . .. .iii

LIST OF TABLES. . . . vi

LIST OF FIGURES . . . .. vii

LIST OF SYMBOLS. .. . . x

ABSTRACT . . . . . xii


Chapter

I. INTRODUCTION 1

II. METHOD OF ANALYSIS 8

General Approach. . .. 8
Momentum Equation . .. 13
Analog Solution of Momentum Equation 20
Steam Velocities . . .. 32
Mass Flow Rate . . .. 37
Energy Equation . .. 40
Macroscopic Energy Balance .. 48
Graphical Determination of Gap
Thickness and Evaporation Rate 52

III. FLOW DISTRIBUTION 55

IV. EXPERIMENTAL PROCEDURES 63

V. EVAPORATION RATES 79

Theoretical . . .. 79
Experimental . . ... 81
Comparison of Experiment to Theory 84

VI. OVERALL HEAT TRANSFER COEFFICIENTS 87

VII. GRAVITATIONAL EFFECTS 91

VIII. CONCLUSIONS 9q














APPENDIXES

A. REACTIVE FORCE . . .

B. SOLUTION OF THE MOMENTUM EQUATION
FOR P(r) . . .

C. ANALOG SYMBOLS . . .

D. PHYSICAL PROPERTIES . .

E. EXPERIMENTAL DATA . .

F. DROPLET SHAPE UNDER VARIABLE
GRAVITATION . . .


LIST OF REFERENCES

BIOGRAPHICAL SKETCH


Page


100


103

106

109

116


119


128

131















LIST OF TABLES





Table Page

1. ANALOG COMPUTER RESULTS . . 33


2. PARABOLIC FIT OF4) ANALOG RESULTS FOR
0(0) = 4.0 . . . 60


3. POLYNOMIAL COEFFICIENTS . 83


4. PHYSICAL PROPERTIES OF STEAM AT
ATMOSPHERIC PRESSURE . . 110


5. PHYSICAL PROPERTIES . . 111


6. DROPLET VAPORIZATION TIMES . 116












LIST OF FIGURES


Figure Page

1. Droplet States. . .. .. 5

2. Schematic Model of the Evaporation
of a Flat Spheroid. . . .. 9

3. Computer Diagram of Momentum Equations
for = 1, K= 1, and = 1. . ... .24
4. 0' as a Function of the Assumed 0 '(0). 25

5. Computer Diagram of Momentum Equations
for C= 0.1, K = 0.001, and /= 0.0015. 27
6. *as a Function of the Assumed (0). 28

7. (as a Function of the Assumed 4(0). 29

8. as a Function of the Assumed <0). 30

9. Graphical Simultaneous Solution of
Momentum and Energy Equations for V =
0.5 cc, Tp = 600 F, and (p = 0.5. ... 54
10. 6 as a Function of <(0) for V = 0.5 cc,
Tp = 600 F, and Cp = 0.5. . . 57

11. L( as a Function of 6 for V = 0.5 cc,

Tp = 600 F, and Cp = 0.5. . ... 58

12. Total Vaporization Time for Water Drop-
lets on a Flat Plate as a Function of
Their Initial Volume for Various Surface
Conditions at a Plate Temperature of
Approximately 600 F. .... .. 64


vii












13. Total Vaporization Time for Water Drop-
lets as a Function of Their Initial
Volume and Temperature of the Heating
Surface which Had a 10 Apex Angle. 65

14. Comparison of the Total Vaporization
Time for Water Droplets on a Flat Plate
and a 10 Conical Surface at Approxi-
mately 600 F. . . . 66

15. Schematic Cross Section of 304 as and
Graphite Test Plate. . . 68

16. Schematic Cross Section of Test Plate
with a 1 Conical Heating Surface. 69

17. Schematic Cross Section of Heating Area. 72

18. Schematic Diagram of Experimental
Apparatus . . . .. 73

19. Ejection Time of Water Droplet from
Pipette to the Hot Plate Surface as a
Function of the Volume of the Water
Droplet. . . .. . 76

20. Dynamics of Water Jet Ejected from
Pipette. . .. . 77

21. Theoretical Mass Evaporation Rate of a
Water Droplet as a Function of Volume
for a Plate Emissivity of 0.5 and Plate
Temperatures of 600 F and 1000 F. 80

22. Gap Thickness of the Water Droplet as a
Function of Volume for a Plate Emissiv-
ity of 0.5 and Plate Temperatures of
600 F and 1000 F. . . 82

23. Theoretical and Experimental Mass Evapo-
ration Rates of Water Droplets as a
Function of Droplet Volume, Plate Temper-
atures and Plate Emissivity. . 85


viii


Page


Figure









Figure


24. Theoretical Heat Transfer Coefficient
of a Water Droplet as a Function of
Volume for Plate Temperatures of 600 F
and 1000 F for a Plate Emissivity of
0.5. . . . . 88

25. Theoretical Evaporation Rates of a
Water Droplet in Both the Earth's and
Moon's Gravitational Fields for a Plate
Temperature of 600 F and an Emissivity
of 0.5. . . . 93

26. Specific Volume of Steam at Atmospheric
Pressure as a Function of Temperature. 112

27. Viscosity of Steam at Atmospheric
Pressure as a Function of Temperature. 113

28. Thermal Conductivity of Steam at Atmos-
pheric Pressure as a Function of
Temperature. . . .. 114

29. Schematic of Water Droplet. . ... 120

30. Path of Numerical Integration. ..... 123

31. Thickness of dater Spherid as a Func-
ton of its Volume for = 1.0 and
= 0.16. . . . .. 126

32. Thickness of Water Spheroid as a Func-
tion of its Volume for = 1... .127


Page













LIST OF SYMBOLS


Symbols

A

a

f



gc


hfg

k

I

M

N

P

q

ro

T

t

U


u

V

w


Area, ft2

Constant of proportionality, sec-1

Transformation variable, ft sec-1

Acceleration of gravity, ft sec-2

Dimensional conversion factor -
32.1739 ft Ib lb f- sec-2
m f
Latent heat of evaporation, BTU IbM -

Thermal conductivity, BTU hr-1 ft-1 Fl-

Average droplet thickness, cm

Mass, grams

Surface tension, dynes cm-1

Pressure, lbf ft-2

Rate of heat flow, BTU hr1-

Maximum radius of water spheroid, cm

Temperature, F or R

Time, sec

Overall heat transfer coefficient,
BTU hr~1 ft-2 F-1

Radial velocity, ft sec-1

Droplet volume, cc

Axial velocity, ft sec-l

x










Symbol

OC Thermal diffusivity, ft2 sec-1

r- Computer proportionality constant, volt-1

6 Steam gap thickness, in

C Emissivity for radiation

Dimensionless coordinate

S Computer proportionality constant, volt-1

Material constant, in sec-1 (lbf/lbr)- cm2 x
-3 -3
R cm

U Absolute viscosity, lb ft1 sec-1

Kinematic viscosity, ft2 sec-I

) Density, lbm ft-3

Computer proportionality constant, sec-1

S Computer time, sec

$ Dimensionless transformation variable

/ Dimensionless transformation variable

F Gravitational potential ~Earth=l), lbf lbm-

Computer transform variable, volts

S Material constant, grams sec-1 (Ib/lbf)1 x
cm R- cm9/4 volt-1

( Computer transform variable, volts
A/ Velocity correction factor

Subscripts
D Droplet Sat Evaluated at saturation
f film condition
f film
p plate 6 Evaluated at lower surface
of droplet
xi










Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy


HEAT TRANSFER TO WATER DROPLETS ON A FLAT PLATE
IN THE FILM BOILING REGIME

by

Kenneth Joseph Baumeister

December, 1964

Chairman: Dr. Robert E. Uhrig
Major Department: Nuclear Engineering


The mass evaporation rates and overall heat transfer

coefficients are determined both theoretically and experi-

mentally for water droplets which are supported by their

own superheated vapor over a flat hot plate.

The theoretical and experimental mass evaporation

rates are found to agree within 20 per cent over a drop-

let volume range of 0.05 cc to 1 cc and over a temperature

range of 600 F to 1000 F. In this parameter range, the

mass evaporation rate varies from 0.001 to 0.01 (g/sec),

and the steam gap thickness ranges between 0.003 and

0.008 in. Both the gap thickness and mass evaporation

rate increase for increased volume and temperature. The

overall heat transfer coefficient ranges between 70

(BTU/hr ft2 F) for 0.05 cc droplets and 40 for 1 cc drop-

lets in the temperature range considered. Also, the

xii











theoretical analysis yields the axial and radial velocity

distribution under the droplet and a velocity correction

factor which is applied to Fourier's equation for one-

dimensional steady state heat conduction across the

steam gap.

The water droplets are approximated by a flat

spheroidal geometry with a uniform steam gap beneath the

droplet and a saturated steam vapor cover on the top

surface of the droplet. The shape of the droplet and

the average droplet thickness are determined analyti-

cally. The analytical results compare favorably to

experimental measurements. The assumptions are made

that the bottom of the spheroid is at the saturation

temperature and that the evaporation takes place uni-

formly beneath the spheroid. The flow is shown to have

a Reynolds number of approximately 10; consequently, the

flow is treated as incompressible and laminar with

negligible energy dissipation. In addition, the con-

stant fluid property assumption is made, and because of

the large amount of time required for the evaporation

of the droplet, the droplet at any instant is assumed to

be in a pseudo steady state condition; that is, the flow

is approximated by a steady state solution at any

instant of time.


xiii











The analytical method of attack is to solve the

momentum, continuity, and energy equations simultane-

ously. The partial differential momentum and continuity

equations are reduced to ordinary non-linear differential

equations by the method of combination of variables.

Possible solutions to the non-linear equations are mapped

by means of an analog computer. Then, these physically

acceptable solutions are combined in a graphical manner

with the solutions of the macroscopic energy equation,

which is solved explicitly, to yield the mass evaporation

rate and steam gap thickness of the droplet as a function

of droplet size, plate temperature, and gravitational

potential.

The effect of the gravitational potential on the

mass evaporation rate is considered in detail in the

theoretical development. A reduction in the gravita-

tional potential from 1 (earth) to 0.16 (moon) is shown

to reduce the mass evaporation rate by approximately

half.


xiv













CHAPTER I


INTRODUCTION


The object of the study presented in this

dissertation is the determination of the overall heat

transfer coefficient from a heated flat plate to water

droplets which are supported by their own superheated

vapor. This is accomplished by a theoretical study of

the momentum, heat, and mass transport phenomena associ-

ated with this elemental two phase flow problem and by

an experimental verification of the theory. This

analysis differs from the semi-empirical and dimensional

approaches used in the past in that the analysis is

based solely on the solution of the relevant governing

equations involved. The analysis considers a wide range

of surface temperatures and volumes of water droplets.

Interest in this subject stems from the rapid

development of nuclear reactors as used in power and

propulsion systems. In particular, there has been a new

interest awakened in the general problem of heat trans-

fer with a change of state, such as occurs in the two

phase flow heat transfer in a boiling water reactor,

nuclear rocket, or in the more fundamental problem under









2

consideration in this dissertation. Also, an increased

understanding of the film boiling phenomenon, such as

occurs under the droplet, is of importance in nuclear

rocket technology, since film boiling heat transfer

occurs in both the rocket nozzle and core reflectors.

In addition, recent works by Adadevoh, Uyehara, and
*
Myers (1) end Borishansky, Zamyatnin, Kutateladze, and

Nemchinsky (2) indicate that there is still interest in

the subject of droplet vaporization in the fields of

internal combustion engines and metallurgy.

Investigations on this subject were begun as

far back as 1756, when Leidenfrost 3) first described

the phenomena of film boiling, and they have continued

up to the present time with much of the more recent

work found in the Russian literature. The most recent

and complete works are by Gottfried (4) and Borishansky

5) .

Gottfried presents both a dimensional and

semi-empirical correlation for the evaporation of small

water drops on a flat plate in the film boiling regime.


*The underlined numbers in parentheses in the text refer
to the number of the entry in the List of References. A
statement to the right of a comma within the parentheses
will give the location within the source to which the
reference is made. If only the underlined number is
given, the reference is to the source in general.








3

In his semi-empirical approach, the water droplet is

approximated by a frustum of a cone in which the upper

and lower areas are varied in such a manner so as to

best correlate the experimental data. From the upper

area, molecular diffusion is assumed to take place in

the absence of thermal convective effects or a

saturated vapor blanket. From the lower area, uniform

evaporation into a superheated vapor is assumed with

an outward flow through a uniform gap beneath the

droplet.

Borishansky has investigated the evaporation

of water droplets up to 4 cc in volume for plate tem-

peratures of 527 F and 662 F. He has used both

dimensional and semi-empirical techniques in correlat-

ing his data.

On the bases of Borishansky's experimental

results and the experiments performed in conjunction

with this dissertation, the general problem of water

droplet evaporation is broken down into the following

states, which are governed by the volume of the drop-

lets small spheroid, flat spheroid, and bubbly

spheroid. The small spheroid state, observed at 0.05

cc, is a perturbation of a purely spherical geometry

by the action of gravity working against the forces of

capillary tension in the surface of the droplet, as








4

shown in Figure 1. For larger size droplets, the

perturbation from the purely spherical state increases.

Finally, as the liquid volume approaches 1 cc, the

droplet teachers the flat spheroidal state in which the

thickness of the spheroid undergoes little change with

further increase in its volume. For volumes greater

than approximately 1.5 cc, the vapor formed beneath the

droplet tends to break intermittently through the sur-

face of the liquid which gives rise to the term bubbly

spheroid state, shown pictorially in Figure 1.

In this particular study, water droplets in the

volume range 0.05 to 1 cc are analyzed. In this volume

range, an analytical model based on a flat spheroidal

geometry reasonably satisfies the physical situation

and yet still has simple enough boundary conditions to

make the resulting boundary value problem tractable.

The shape of the droplet and the average droplet thick-

ness, i, are determined analytically. The analytical

values of t compare favorably to the measurements taken

by Borishansky.

The analytical method of attack is to solve the

momentum, continuity, and energy equations simultane-

ously. The partial differential momentum and continuity

equations are first reduced to ordinary non-linear dif-

ferential equations by the method of combination of

variables. A difficulty results from the fact that the























I


9,-
',
II01

\ ^
r
^ w


(a
rl
..
- 0



U2








6

non-linear equation cannot be solved in closed form.

The difficulty is further compounded because the

boundary conditions to the problem are unknowns. The

analytical approach to the problem is to map possible

solutions to the non-linear flow equations by means

of an analog computer and to use those particular

solutions which are physically acceptable. Then, the

solutions to the macroscopic energy equation, which

is solved explicitly, are combined in a graphical

manner with the results of the analog solutions to

yield the mass evaporation rate of the droplet as a

function of droplet size and plate temperature. The

overall heat transfer coefficient can be found

directly from the mass evaporation rate.

The theoretical mass evaporation rates are

found to agree within 20 per cent of the experimental

values over a temperature range of 600 F to 1000 F,

and a volume range of 0.05 to I cc. The analysis also

yields the axial and radial velocity distribution

under the droplet, the steam gap thickness, and a

velocity correction factor which is applied to Fourier's

equation for one-dimensional steady state heat conduc-

tion across the gap.

Also, with the possibility of the moon being

explored in the next decade and with the possibility









7

of operating with two phase flow in a low gravitational

field, the theoretical analysis takes into account the

effect of a variation of the gravitational potential on

the mass evaporation rate. In particular, the mass

evaporation rate is determined for the physical situa-

tion where the surface gravity is equivalent to that

of the moon's surface. The resulting values of the

mass evaporation rate are compared to the values

obtained on earth for a similar plate temperature and

droplet volume. The mass evaporation rate is approxi-

mately half that found on the earth for a given droplet

volume and plate temperature.













CHAPTER II


METHOD OF ANALYSIS


General Approach

Consider the flat water spheroid shown in

Figure 2. Heat transfer to this flat spheroid takes

place primarily by conduction and radiation through

the superheated film. Heat transfer and evaporation

from the upper surface are negligibly small in com-

parison to that beneath the droplet. Kutateladze

(k, p. 376) points out that the external surface of

the spheroid is covered by superheated vapor flowing

from beneath the spheroid. This vapor cover reduces

the energy transport from the upper surface to a near

zero value. However, even in the assumed absence of

a steam cover, both the free convective and radiative

heat transfer, and free convective evaporation are

negligible when compared to that occurring beneath the

droplet. The free convective evaporation was esti-

mated from a correlation presented by Wade (Z).

Thus the problem of heat transfer to a flat

water spheroid reduces to a problem commonly termed

mass transfer cooling (Q). Such a process is




















0

$4


Nk


(0 N


t
N
k






0
Io II
II N


a)
4J
Sra
S.4J
Q 'o
/Pe
/!
Sa)
/ ^
/- <


C-0
I
< -- ; -- > 0 ---








10

characterized by a mass flow through a porous surface

(transpiration cooling), by mass released from a

surface through evaporation or sublimation, or some

chemical reaction (film cooling, ablation cooling).

To determine the velocity, temperature, and mass

distribution for the above-mentioned problems, Hartnett

and Eckert (C) point out that in general it is neces-

sary to solve the continuity, momentum, energy, and

diffusion equations simultaneously. However, for the

superheated steam region under the water droplet, the

mass evaporating from the surface is the same as that

flowing beneath the droplet; thus, the diffusion equa-

tion is identically zero, as pointed out by Grober,

Erk, and Grigull (10, p. 416).

However, for the problem of evaporation of a

flat spheroid resting on its own superheated film, an

added condition of static equilibrium is required for

the solution to the problem. The pressure forces on

the bottom of the droplet must be sufficient to balance

the weight of the droplet minus the reactive force.

For this particular problem, the reactive forces are

negligible compared to the body force due to gravity,

as shown in Appendix A. Consequently, the reactive

force is neglected in this analysis.








11

Gottfried A4) and Kutateladze (6, p. 377) point

out that the flow under consideration is of very low

velocity and is well within the laminar range; thus,

the flow is treated as incompressible with negligible

energy dissipation due to friction. In addition,

because of the large amount of time required for the

evaporation of the spheroid, the droplet at any instant

is assumed to be in a pseudo steady-state condition;

that is, the flow is approximated by a steady-state

solution at any instant of time. Consequently, for

this case of axisymmetrical and incompressible laminar

flow with negligible dissipation and with constant

fluid properties, the momentum, continuity, energy, and

static equilibrium equations are as follows:


Momentum:

Bu bu gc iP 62u 1 bu u 62u
u-+ w- +V +--- (1)
br bz p 6r \,2 r br 2 bz2


bw bw g9 6P (2w 1 6w 62w
u-+ w + + + (2)
xr bz p z \r2 r br bz21


Continuity:

u + u + 0 (3)
br r 3z










Energy

bT ST
u + w = OCXV2T (4)
6r bz

Static Equilibriums

P(r,6) 27rrdr = VQ o (5)




The boundary conditions for the above equations areas

z = 0 u 0 w = 0 T = Tp (6)

z =6 u = 0 w= w(6) T = Tat (7)


The assumptions are made that the bottom of the

spheroid is at the saturation temperature, and that the

evaporation takes place uniformly beneath the spheroid.

The boundary condition on the axial velocity at the

upper surface is an unknown; in fact, at the present

time the gap thickness is also an unknown. The gap

thickness, 6, and w(6), are found by simultaneous solu-
tion of the above equations.

In this analysis, the determination of the
evaporation rate, heat transfer coefficient, and gap

thickness is in terms of the volume of the water droplet,

gravitational potential, and temperature of the heating








13

plate. Consequently, the transport properties of

viscosity and thermal conductivity, as well as the

specific volume of the steam, are expressed in terms

of the steam temperature. For the range of tempera-

ture under consideration in this dissertation, the

above properties are represented as linear functions

of temperature.


Momentum Equation

The logical beginning to this analysis is to

solve the momentum equation since, as a result of the

constant fluid properties assumption, the mutual

interaction between the equation of motion and the

energy equation ceases, and the velocity field no

longer depends on temperature. The usual approach (9)

is to evaluate the properties of the flow field at

the film temperature, as defined as:


T +T
p + Tsat (8)
Tf = 2
2


Therefore, the immediate problem is to solve the

continuity, Equation (3) and the momentum, Equations

(1) and (2), simultaneously. These equations form a

set of partial differential equations with two inde-

pendent and three dependent variables. Use of the








14
method of "combination of variables" at this time

reduces the above-mentioned equations into a set of

ordinary non-linear differential equations.

Consider for a moment the physical situation.
Defining u as a mean radial velocity,



u = u dz (9)



and the z component of velocity at the surface of the

water spheroid as w(6), then the conservation of flow

into and out of a cylindrical volume of radius, r,

under the flat spheroid results in


w(6)w r2 = i 2w r6 (10)

However, since w(5) is an assumed constant along the

bottom surface of the spheroid, it follows that


ua r (11)

Also, Gottfried (), working with the mean
radial velocity as defined above and with the Navier-

Stokes equation in the radial direction only, shows

that


P a (r2 r2) (12)








15

The above relationships indicate that a

combination of variables of the form used in the prob-

lem of three dimensional axisymmetry stagnation flow

(11, p. 83) will reduce the partial differential

equations (1), (2), and <3) into a set of ordinary

equations. The functional forms used in this conver-

sion are:

w = -2 f(z), (13)

u = r f'(z), (14)

P = 1 a2 ro2 r2 + F() (15)
2


Choosing these specific functional forms for w

and u satisfies identically the continuity equation,

as substituting Equations (13) and (14) directly into

the continuity equation (3) verifies.

The partial derivatives of the relationships

(13), (14), and (15) used in the combination of vari-

ables substitution are now listed for future reference.


= f' -r2 = 0, (16)

bu b2u
-- r f"- r f'' (17)


-= 0 ---2 0 (18)
br br2











-2 f' 2- 2 f (19)
bz 6z2

S= -a2 P- = -a2 F (20)
6r gc 3z 2 gc


Substituting the above relationships into
Equations (1) and (2) results in the following new

forms of the momentum equations:


f'2 2 ff" = a2 + f''' (21)


2 ff' a2 F' Lf'' (22)
4

Thus, the original partial differential
equations are by means of the above substitutions con-

verted into a set of non-linear ordinary differential

equations. Although these equations are not easy to

solve in the strict sense, they are much more easily

handled than the original partial differential equa-

tions.

Gottfried's pressure relationship (Equation
12) is generalized to include z variations in pressure,

since a trivial solution to the momentum equation re-

sults if the z variation in pressure is neglected, as

shown in detail in Appendix B. The boundary conditions,

Equations (6) and (7), now take on the following forms:










z = 0 f = 0 f"= 0 F = 0 T = Tp (23)

z = f = 6 f' = 0 T = Tsat (24)


The boundary condition on the function, F, is
chosen arbitrarily since the only interest is in the
relative variation of the function in the z direction.
The constant, a2, accounts for the absolute magnitude
of the pressure.
Equations (21) and (22) are now freed of the
constants a2 and V by making the following substitutions:

c= c1 2 f(z) = c20(0) P(z) = c3 (0 ). (25)

Substituting the above expressions into Equations (21)
and (22), the coefficients of these equations become
independent of a2 and I if


= z (26)


f(z) = Fa2L ( (27)

F(z) = () (28)










Therefore, it follows that


a


f = 7 a)U d a
dL dz


f I


f f (f) i


F' = V4a
_a 4d dz
a d dz


Substituting the above relationships into

Equations (21) and (22), the momentum equation takes

on the following forms:


2 -2 -" = 1 i + ,

2 I' = 'p, .


(34)

(35)


The boundary conditions on these equations


become


'=0


= =0


T=T
(36)

T= Tmat
(37)


(29)


(30)


(31)


(32)


(33)


=0


-w(5)


= a \I-
vU


_ 4U ,
a V~











Complications result because a closed solution

to the above non-linear ordinary differential equation

does not exist and because 6 and w(6) are unknowns.

The value of the parameter, 6 depends on the simultan-

eous solutions of all the governing equations. Conse-

quently, the method of solution is to assume many

reasonable values of 6 and to solve for the flow

distribution in each of these cases that satisfy the

static equilibrium condition, Equation (5). The results

are plotted and compared to the solutions of the energy

equation, Equation (4), for various values of 6. The

intersection of the results of the momentum and energy

equations represents the value of the gap thickness, 6..

The solution of Equations (34) and (35) in this

particular situation is most easily performed by means

of an analog computer. However, instead of assuming

values of 6, initial values of '' are assumed. The

values of 6 and w(6 J are determined from the output of

the computer.

The next section discusses in detail the analog

solution to Equations (34) and (35).










Analog Solution of Momentum Equation
The solution of ordinary differential equations
by use of a differential analog computer is discussed
widely in the literature (12), (13), and (14). Briefly,
the analog computer is capable of the basic mathematical
operations of addition, subtraction, multiplication, and
integration. The variable quantities of the differen-
tial equation are represented by voltages which may be
recorded by use of the proper recording equipment.

In setting up the momentum equations (34) and
(35) for solution on the analog, it is helpful to
rewrite these equations in the form:

"' = f'2 -200'' 1, (38)


S= 20' (39)

The first step in programing the above equations
for the analog computer is to change the variables in
the above equations to computer variables by making the
following transformations:


S= 7,. (40)

0(l = K (n). (41)

=(J =/?Ycn (42)








21

The derivatives of the above transformations

necessary for substitution into the analog equations

(38) and (39) are


d dr 1 de K do K .
= d= -.= (43)
dT dC a dr a dr



y" = (44)


K
,. .* (45)
0 3 3


= (46)



Substituting the above relationships into

Equations (38) and (39) results in the analog momentum

equations

"' = 2 Ko (47)
Ko- K


= -2 K 2 K (48)
8 m

The boundary conditions on the above equations

take on the form

T = 0 = 0 = 0 T = Tp (49)


S= = w() = 0 T = Tsat
a K 2/av (50)
(50)








22

The symbols a, and B are constant scale

factors. The symbol T represents the computer times

the time for the phenomena to occur on the computer.

The distance traversed from the plate is directly

related to the computer time, T. This relationship is

presented in the next section.

In attempting to find a solution to the boundary

value problem presented above, it is assumed that a

solution to the boundary value problem exists for every

assumed value of 6. Further, it is assumed that the

above-mentioned solutions are selectable from a set of

discrete solutions to the initial value problem of

Equations (47) and (48) having the initial conditions


T = 0 = 0 0 = 0 -- < i > T = 0. (51)


In order to check the above hypothesis, Equations

(47) and (48) are programmed for the analog computer for

values of a = 1, K = 1, and B = I, which imply for this

first program that


= T 4 = $ = Y (52)

The analog momentum equations become


S= ~ 22 1 -1 (53)


S= 2 (54)








23

Figure 3 shows the analog diagram for Equations (53)

and (54). The set of discrete initial conditions on

is selected over a sufficiently wide range to give a

reasonable topology of the total set of solutions to

this particular initial value problem.

The circuit shown in Figure 3 is programed on

the Applied Dynamics Analog Computer. In programing

this particular computer, the voltage to the quarter-

square multipliers must be greater than 10 volts in

order to keep the specified computer accuracy. For

this program, constant multipliers accomplish this.

The notation used in this diagram is standard analog

notation; however, the symbols are defined in Appendix

C.

Figure 4 represents the solutions for 4 '. The

requirement of satisfying the boundary conditions, Equa-

tions (36) and (37), limits the acceptable range of

S" (0) to


0 < "(0) < 1.31. (55)


It turns out that of the possible solutions the values

of physical interest are near


S" '(0) a 0.5. (56)

Figure 4 indicates that for the range of

interest, the analog equation requires rescaling so that






















0
I1
N

04



44

0

I r4
11






4J0


t;



14)





































































14 -4 0
yro y
r; a


0
O







a



0




0
43



0



0
0
CO








26
the analog output is at a greater voltage to reduce the

error involved. The precision of the analog computer

components used allows an accuracy of 0.1 per cent

based on a 100 volt output.

A trial and error procedure indicates that for
the range of interest scale factors of O0= 0.1, K =

0.001, and )= 0.0015 can be used. As a result, the

analog momentum equation takes on the following form:


S0. 01 0.02 1 (57)
100 100


=-6.6 0.02 (58)
100 .

The initial conditions on are found from
relationship (44). The four sets of initial conditions

used are


S= 0.5 = 5.0

= 0.4 = 4.0
(59)
= 0.3 = 3.0

= 0.2 2.0

The analog diagram for Equations (57) and (58)
is shown in Figure 5. The program results are shown in
Figures 6, 7, and 8, which represent and ,




























w
CO
0
V4

c4J
04 r


0








0-

0




14


0'
440
Sa


'I.













15.


12.5 --'.V




10.0


4o
r4i
0
7.5

(p(0)=4.0


5.0

-( 0)=3.0


2.5 -
(o)=2 .0



0 tIII I i I
0 2 4 6 8 10

7 (seconds)

Fig. 6. as a Function of the Assumed (0).

























60-

0


40 r


/(0)=4.0

20-

0)=3.0


10)=2 .0
0 2 4 6 8 10

7 (seconds)



Fig. 7. pas a Function of the Assumed (O).



















((0)= 5.0


2 4 6 8 1


T (seconds)




Fig. 8. Y as a Function of the Assumed (0).


I nn


(I)
4,
Ht
0


*A


r 1


- I


80

t-


60 -





40





20




or








31

respectively. The results are as expected. The

parameter, which is directly related to u, starts at

0, goes to a maximum value near the center of the steam

gap, and then returns to zero at the surface of the

droplet. The parameter, 4, which is directly related

to w, starts at zero at the plate and then reaches its

maximum value at the bottom of the droplet. The param-

eter, 4, which is directly related to the total pressure,

goes from a maximum value of zero at the plate to a

minimum value in the center of the channel. This is

because the total pressure head at the plate is par-

tially converted into a velocity head in the center of

the gap between the plate and the water droplet, thereby

giving rise to a lower pressure. The pressure then

returns nearly to its plate value at the surface of the

droplet. There is a negligible deviation due to the

small axial velocity head, which for this problem is

negligibly small compared to the radial velocity head.

Therefore, the pressure distribution at the surface of

the droplet is taken to be of the form


P(r,6) 1 a2 DP (r 2 r2) (60)
2 gc


The phenomena discussed above can be found in many







32
textbooks under the heading of Bernoulli's Equation

(5, p. 114).
Table 1 lists the important numerical values of
the end points of Figures 6 and 7. The next section is
concerned with the determination of the velocities and
evaporation rates from the data listed in Table 1.

Steam Velocities
The radial and axial velocities and the gap
thickness are determined from the analog parameters
listed in Table 1. These parameters are directly re-
lated to the axial velocity, w, by Equations (13), (27),
and (41) resulting in

w = 2 Vai K. (61)

The value of w at the surface of the droplet is given by

w(6) = 2 VTa K P,. (62)


where 4 is the value ofP at the surface of the drop-
let. The value of ~6 is tabulated in Table 1. The
parameter,C is directly related to the velocity, u,
by Equations (14), (30), and (43), resulting in


u = r a K (63)

















TABLE 1

ANALOG COMPUTER RESULTS


(N(o) -r,6 ,- 159

volts seconds volts volts


5.0 9.9 82.0 12.25


4.0 7.0 29.5 6.25


3.0 5.3 13.0 3.50


2.0 3.6 4.0 1.50



The output of the analog computer is read in
volts; however, the output, is considered to be
volts per unit time when used in the equations, in
order that the units will be consistent.







34
In a similar manner, the time required to satisfy
the boundary condition of 0, Equation (50), that is,
the time at which the curves in Figure 6 take to reach
their zero values, relates directly to the gap thickness,
6, by Equations (26) and (40), resulting in


6 = r -\- d 7. (64)


The parameter, a, still an unknown, is now determined
from the static equilibrium condition, Equation (5).
Substituting Equation (60) into Equation (5) and solving
for a2


1 PD 1
a2 c [ (65)
Sfo (ro2 r2)r dr

0

where

gc

Performing the required integration and solving for a,


a= gc V (66)
ITP o4







35
Therefore, Equations (61), (63), and (64) take
on the following forms:


u = r rgc j (67)



w 2 K 9.2 r (68)



S1(69)
4 rg V 4
T rPgc ro4


For a flat spheroid geometry, simplifying
Equation (66) by relating the radius to the volume
conveniently results in

V = A (70)

where
A = 7Tr2

Therefore
ro4 = (71)
2r 22







36
Substituting into Equation (66) results in


a = 472,g-c (72)
a ) v

or

ra = -V4 F f (73)


Therefore the equations (61), (63), and (64) take on
the following more convenient forms using the above
substitutions,


u= r47) f). (74)



w (647rgc) ,K (s)

T 7, v; py2)4
6= (76)
(4 7Tg p )0 rD

The next section deals with the determination
of the evaporation rate from the above velocity
relationships.








37
Mass Flow Rate
The mass loss required to satisfy the condition
of static equilibrium is calculated by the relationship


dM = lw(6) A (77)
dt I

Substituting Equations -(70) and (75) into the above
equation results in
3
= (64 7Tg)K ['( ( ) Y (78)


Or expressing

= (79)

S= / (80)


Equation (78) takes on the form


d = (64 rgc)K ( r (81)
dt 11b U


In this analysis, desiring to determine the
evaporation rate as a function of the plate temperature,
the temperature dependent parameters of density and
viscosity are expressed in terms of the film temperature







38
of the vapor. Figures 26 and 27 in Appendix D indicate
a linear relationship between the viscosity and the
specific volume with the absolute temperature of the
superheated steam in the range of interest.
Therefore

V = f (82)

= T (83)


Substituting Equations (82) and (83) into (81)
results in the following form of the evaporation equa-
tion:

dMi
S= (64 77g cb K r)e |T) .(84)
/ 3t

Defining the material quantity


= (647 ) (3.281x10-2) x453.6

x (3.531xl0-5) (85)

Substituting Equation (85) into (84) results in

dM irs ; 4 3
=eT V4 e (86)
dt f t

The last expression gives the evaporation as a function








39
of the material properties, Q, the gravitational constant,
F, thickness of the water spheroid, t, the absolute film
temperature, Tf, and the volume of the droplet under
consideration, V.

Equation (86) is to be evaluated for different
values of the gap thickness. Substituting Equations

(79), (80), (82), and (83) into (76), so as to relate
the gap thickness to temperature, results in the follow-
ing relationship for the gap thickness:


6= -- d r Tfr ., (87)

where

S(7 ) (3.281x10-2) x(.3531x10-4) x12 (88)

(47gc P/)

The parameter, A, is a function of material only as
is Q Equations (86) and (87) are now evaluated
separately to determine the required flow rate and gap
thickness which satisfy the momentum equation, contin-
uity equation, and the condition of static equilibrium.

What remains now is to determine the evaporation
rate from an energy consideration as a function of the
gap thickness. Therefore, the next two sections deal
with the solution of the energy equation and the
macroscopic energy balance.











Energy Equation

This section is concerned with the calculation

of the amount of heat transferred from the plate to

the water droplet by conduction. Previous work (4),

(5), and (6) used the relationship

kA
q (T T ) (89)


for the calculation of the amount of heat transferred

through the gap. However, the above relationship does

not consider the effects of the stream velocities on

the conduction heat transfer. Therefore, the above

equation is considered a first order approximation to

the energy equation, Equation (4). When the velocity

effects are neglected, the energy equation takes on the

form of the Laplace equation, of which Equation (89) is

the solution.

As a result of the work of the previous two

sections, the effect of the stream velocities on the

rate of heat transfer by conduction through the steam

gap can be determined. The linear relationship implied

by Equation (89) is perturbed by the ejection of the

saturated steam into the vapor stream.








41

For the problem under consideration, the

physical conditions indicate that


(T bT
<< (90)
br 0z


Therefore, Equation (4) can be simplified to

dT d2T
w -- = (91)
dz dz2


Substituting relationship (61) into Equation

(91) resullts in

dT d2T
2 a i2) PX -dT 0 d '(92)
dz dz2


However, combining Equations (26) and (40) results in


7 v 2 (93)



Differentiating the above with respect to z yields


dT (94)
dz


Therefore, the first and second derivatbis with respect

to temperature are written as follows







42

dT dT d 1 af dT
= --- -V --' (95)
dz dTdz (f d(

d2T 1 a d2T (9
.- =(96)
dz2 (f2 d,72

Substituting the above two relationships into Equation
(92) results in the following form of the energy equa-
tion:

d2T 2UKO 0 (97)
--- + = 0 (97)
d72 O daT

The parameter, (.? in the above equation is a
function of 7, and is conveniently approximated by the
following form (see Figure 7):


7. (98)


Therefore, the energy equation becomes

d2T 2 K O aT ?dT
d2T + 22K -c = 0. (99)
d72 O~( dT

Let

B = 2 VK 6 (100)
cX76








43

but the Prandtl number is equal to


Pr = (101)
0(


Therefore, the constant, B, takes on the form

2 PrK Kd 6
B = (102)
r6


and Equation (97) becomes

d2T dT
S+ B7dT 0 (103)
d-2 dT

The problem now is to integrate the above

differential equation and apply the thermal boundary

conditions shown in Equations (49) and (50). Equation

(103) is made readily integrable by substituting in

dT
y = (104)
dT

Substituting Equation (104) into (103) and integrating

yields

dT c= 4 exp ( -B72) (105)
d7y 2









Integrating the above results in


T = c4 expj- B72 dT + 5 .


(106)


The value for the above integral is given in reference
(16, p. 303) as


T = c4 ( i rff T)


But, reference (16, p. 297) shows that


+ C5


erf x =


(107)


(108)


n (-1)nn x2n+1
-- n (2n + 1)
VTn=0 n" (2n + 1)


Expanding Equation (108),

2 3 5
erf x = x (-- + --- ........... (109)
bstittin ( 3 10

Substituting Equation (109) into (107) results in


T = c4 (1 -2 + 4 ..........
6 40


+ C5 .

(110)


Defining


(111)


A= 2










results in


T = 1 + 7 ...... + c5.
1.1 2
(112

The value of the parameter, A, is approximately

0.05. Consequently, it follows that

2 < < A (113)


Since 7 is defined for the domain 0 = T 7
6

1. u. b. = 1 (114)



Therefore, from calculus (i7, p. 129), it is known that

an alternating series of the form (-i)n+l a where
1



remainder after n terms has a value between zero and

the first term not taken. Consequently, second order

terms and higher of A are neglected, since the maximum

error in the resulting series is less than 0.0025.

Therefore, the temperature is represented by


T = c4T 1 A ) + c (115)
7 5







46
Evaluating c4 and c5 from the boundary conditions,
Equations (49) and (50), yields

Tp Tsat)7 21
T = Tp -- i- (116)
6 -

However, in this particular problem, the
temperature distribution under the droplet is not of

great interest; rather, the heat flux at the droplet
interface is the important quantity. This is found
from the relationship


dT
q = AdT (117)
dz
'6

Using relationship (94) the above becomes


k A -a dT
Svy T = (118)


Differentiating relationship (115) results in a tempera-
ture gradient of the form

aT(T Tat 1r "
aT= ..- .- -1 3 (119)
ar 7-( -A) L '17 )


Therefore, the heat flux at the surface of the water
droplet is given by











q = k (Tp T ) -1 (120)
r 717 p sat-



but, substituting in Equation (64) results in

kA
q -A(T -T sat) 121)
6 P sat

where

-A = i 1 31 (122)



The parameter,A, represents a velocity
correction factor to the above equation. Bound up in

this correction factor is the consideration that some

of the heat leaving the plate goes into superheating the

vapor leaving the surface of the droplet. The value of

A is approximately 0.95; thus, the velocity correction
factor represents a 5 per cent correction on the energy

equation.

Next, the solution of the energy equation is
used in a macroscopic energy balance to determine the

allowable evaporation rate as a function of 6.










Macroscopic Energy Balance

The amount of mass transfer from the water

droplet is now calculated explicitly as a function of

6 by solution of the macroscopic energy equation. The

macroscopic energy balance for the water droplet model

as shown in Figure 2 takes the form


hfg = q + q (123)


Here qc is the conduction energy flux, q, is the net

radiative energy flux, hfg is the increase in enthalpy

during the vaporization of one pound mass of liquid,
QM
and 1 is the amount of liquid vaporized by the energy
dt
transferred by conduction and radiation through the

steam gap.

The conduction energy flux is represented by

Equation (121), while the radiative flux is given

approximately by the relationship (18, p. 64),


qr = o Fe Fa A (Tp4 sat4) (124k


From geometric considerations (12, p. 199, formula 6),


(125)








49

For the above geometry, where Fa = 1, the Fe factor

which considers the departure of the two surfaces from

complete blackness is represented as (18, p. 61):

1
F = (126)

Cp D


A water droplet at 212 F acts similar to a black body,

since the emissivity of water at 212 F is 0.963 (20,

p. 478). Therefore,


Fe p (127)


Thus, the radiative flux is written as


qr = ao Ep A (T4 Tsat4) (128)


The above equation is conveniently rewritten in the form


qr = tp A FT (Tp Tsat) (129)


where

0o Tp -Tsat4)
FT = (130)
(T Tsat


with the values of FT available in the literature (19,

p. 208) as a function of the body temperatures.







50
The absorption of some of the radiative energy
by the water vapor is neglected in this problem because
of the small path length between the droplet and the
plate (19, p. 214) and (21, p. 388).
Thus, the macroscopic energy balance takes on
the form


hfg kA (Tp satA+ pA FTTp Tat). (131)


Substituting in Equation (70) and solving for the
evaporation rate yields

dM V k c
....+ p FT (T T at) 132)
dt x hfg 6


with all the temperature dependent properties evaluated
at the film temperature, Equation (8).
The overall heat transfer coefficient, U, between
the plate and the water droplet is defined by

qc + r= U A (Tp Tat) (133)


Comparing the above relationship to the Equation (131)
results in the following form of the overall heat
transfer coefficient:

U = k + p (134)
6








51
Substituting the above into Equation (132) results in

dM 1
U A (Tp Tsa (135)
dt h p tfg
fg

Clearly, if the evaporation rate is calculated by

theory or experimentally measured, the overall heat

transfer coefficient is known directly from the evapor-

ation rates by the use of Equation (135).

Following the earlier procedure of expressing

the transport parameters in terms of temperature, the

thermal conductivity is expressed as a linear function

of temperature of the form


k = m + n Tf (136)


as shown in Figure 28 of Appendix D.

Therefore, the evaporation rate shown in

Equation (132) takes on the form

dM V 12(m + nTf) A ( X
= + FT x
dt t hfg 6
453.6
(Tp Tsat 3600 x (1.076x10-3) (137)











Graphical Determination of Gap Thickness
and Evaporation Rate

The evaporation of water vapor from a water

droplet has been determined in two ways. First, from

the standpoint of momentum required to produce static

equilibrium for a given gap thickness (repeated for

convenience),


dM = r T f Va (86)
dt f


and secondly, from a standpoint of energy transfer,

dM V 12(m + nTf)A
+-7 p FT x
dt th

453.6
(T Tsat x (1.076x10-3)
P sat 3600

(137)

Both equations shown above are solved explicitly for
different values of the gap thickness found from the

relationship:


6 = V (87)


When the evaporation rates calculated from Equations

(86) and (137) are equal for a given value of the gap








53

thickness, as calculated from Equation (87), the

governing equations and boundary conditions, Equations

(1) through (7), are satisfied concurrently.

For example, Figure 9 shows a graphical solution

of the momentum and energy equations for the case of a

0.5 cc water droplet resting on a plate at 600 F. The

point of intersection of the two equations on Figure 9

represents the conditions where all the governing equa-

tions are satisfied concurrently. The values of the

evaporation rate, which is directly related to the

overall heat transfer coefficient by Equation (135),

and the gap thickness are found directly from the

ordinates in Figure 9. Similar graphs were constructed

for different size droplets and for various plate

temperatures to determine the evaporation rates as well

as the overall heat transfer coefficient for a variety

of conditions.

It is shown in a later section that the

theoretical results are in excellent agreement with

theory.
















I. I I I I I I I I


Energy -
(137)



Solution
Point







Momentum -
(86)





I i_


0.002


I I I I I I


0.004


0.006


0.01


6 (inches)


Fig. 9. Graphical Simultaneous Solution of Momentum and
Energy Equations for V=0.5 cc, Tp-600 F, and C =0.5.


0.010

0.008


0.006 -

0.005-


0.004-


0.003-


0. 0021-


0.001'-
0.001













CHAPTER III


FLOW DISTRIBUTION

The velocity distribution and the Reynolds

number under the droplets are considered, since the

earlier assumption of laminar flow under the droplet

is now justified. In addition, it is important to

determine the magnitude of the radial velocity at the

edge of the droplet, since the droplet tends to move

slowly about when resting on a flat plate.

The velocities, u and w, are related to the

computer variables, and by Equations (61) and

(63). For a given set of physical conditions, such

as volume of the droplet and temperature of the heat-

ing plate, the velocity distributions vary directly

as and i. Consequently, the curves shown in Figures

6 and 7 represent the forms of the radial and axial

velocity distributions. Previously, the exact solu-

tion for the velocity distribution could not be

determined because the gap thickness, 6 was an unknown.

However, using the results of the previous section,

the velocity distribution can be evaluated since the

gap thickness is now a known quantity.

55







56
The curves shown in Figures 6 and 7 are
generated from known values of <(0) rather than specify-
ing 0. However, 0 is related to 9(0) by relationship
(87) and the curve shown in Figure 10 is constructed
from this relationship. Thus, for a known 0, ((0) can
be determined directly from Figure 10. However, a close
inspection of Figure 6 indicates that for all practical
purposes the curves representing are parabolas which
can be fitted by the equation


12 = 1 (138)

2

or using relationship (64)

2z 2
S- -i .(139)



The above equation can be rearranged to the form


1 | -( 1)2 (140)


The values of p -) in the above equation are
plotted in Figure 11 as a function of 6. This figure













I I I I I I


I I I I I I


0.002


0.004


0.006


(inches)


Fig. 10. 0 as a Function of 0() for V=0.5 cc,
T p600 F, and C =0.5.
P P


6.0





5.0





4.0


0


3.0


2.0





1.0





0












I I I I I


I I I I I I


0.004

6 (inches)


Fig. 11. a a a Function of for

V=0.5 cc, Tp=600 r. and -o=0.5.


15.0




12.5





10.0




7.5


5.0 -


2.5




0


0.002


0.006








59

is constructed from the analog results presented in

Figure 6 and the information presented in Figure 10.

The vertex of the parabola is taken as the anchor

point between the analog data and Equation (140).

The parabolic relationship very accurately

represents the curves in Figure 6 as seen in Table 2,

which presents a comparison of the ( calculated from

Equation (140) and from the actual data shown in

Figure 6, for the specific case of( (0) = 4.0. Thus,

the radial velocity distribution across the steam gap

can be considered to be parabolic in shape. Conse-

quently, for parabolic flow the average velocity is

two-thirds the maximum velocity (22, p. 624. There-

fore the average radial velocity, as defined in Equa-

tion (9), takes on the form


u =2/3 r a (141)


The maximum radial velocity occurs at the edge of the

droplet where r = ro; the maximum average radial

velocity is expressed as


Ur = 2/3 ro a T ) (142)
ro 0 2











TABLE 2

PARABOLIC FIT OF ANALOG RESULTS FOR (0) = 4.0




zr calc .Fig. 6


0


6/8

6/4

36/8

6/2

56/8

3 6/4

76/8

6


0.0

0.875

1.75

2.625

3.5

4.375

5.25

6.125

7.0


0.0

2.69

4.59

5.75

6.125

5.75

4.59

2.69

0.0


0.0

2.7

4.6

5.75

6.12

5.7

4.55

2.7

0.0








61

The maximum Reynolds number beneath the droplet is

calculated from the flow beneath the droplet by the

relationship

De U
Re (143)


where


S 4 x flow cross section44)
De = 4 x (144)
wetted perimeter

2 r 6
=4x
2 x 2TrO

= 26.

Thus



R a = 8/3 6 -- .(145)
emax 8/ 2



Consider, for example, the previous problem of

a 0.5 cc droplet on a 600 F flat plate. Figure 9

indicates that the gap thickness is 0.00475 inches.

From Figure 11, the value of( ) is equal to 13.25.

Using these values in Equations (142) and (145),results

in a Reynolds number of 10.6 and an average radial








62

velocity at the edge of the droplet of 5.25 ft/sec.

Thus, the flow is well within the laminar range and the

slight motion of the water droplet on the heating plate

is small compared to the average steam velocity leaving

the gap beneath the droplet.

An increase in the volume of the droplet

increases the exit radial steam velocity and the Rey-

nolds number, since both L and 6 increase with

increasing volume. Also, combining Equations (74),

(79), and (82) indicates that an increase in temperature

of the heating plate increases the exit radial steam

velocity and the Reynolds number in proportion to (Tf).

However, for the temperature range and volume range

investigated in this paper, the basic conclusion that

the flow is laminar is not affected by the volume and

temperature changes considered.













CHAPTER IV


EXPERIMENTAL PROCEDURES

In the theoretical analysis, the mass evaporation

rate for a droplet on a flat plate is shown to be a

function of the plate temperature and the volume of the

droplet. Hence, an experimental verification of the

theory requires that the evaporation rate be measured

for different plate temperatures and droplet volumes.

The evaporation rate is determined experimentally

from measurements taken on the total vaporization time.

The total vaporization time, that time required for the

entire volume of liquid which is placed on a heating

surface to vaporize completely, is measured as a func-

tion of droplet size for various plate temperatures and

surface conditions. The experimental data are listed

in Table 6 of Appendix E, while the plots of the data

are shown in Figures 12, 13, and 14. The slopes of

these curves, rate of change of volume with respect to

time, represent the evaporation rate of the droplet.

The determination of the slope of a curve V = f(t), when

a table of distinct sets of values (Vi,t ) are known, is

considered in detail in the next chapter.









64






5 I 1 1 1 1
O 304 ss, satin finish

Brass--highly polished
S 4 0 Graphite, satin finish
u 4 -










o o
4 2-









0 1
3-

41I

44.A



0 0







0 I I I I

0 200 400 600


Total Vaporization Time (sec)



Fig. 12. Total Vaporization Time for Water Droplets
on a Flat Plate as a Function of Their Initial
Volume for Various Surface Conditions at a
Plate Temperature of Approximately 600 F.
















1.0-




0.8




















0 I I I I
0.4




Pr 0.2






100 200 300 400 500

Total Vaporization Time (sec)



Fig. 13. Total Vaporization Time for Water Droplets as
a Function of Their Initial Volume and Temperature
of the Heating Surface which Bad a 10 Apex Angle.

















1.0-





S0.8-





0.6
0

4.4
0
S0.4-

0
I -
/ 7 10 Apex Angle

0.2 0 Flat Surface






0 100 200 300 400 500

Total Vaporization Time (sec)




Fig. 14. Comparison of the Total Vaporization Time
for Water Droplets on a Flat Plate and a 10
Conical Surface at Approximately 600 F.








67

The test sections used for vaporizing the water

droplets are shown in Figures 15 and 16. Both a 304

stainless steel and a graphite test plate, as shown in

Figure 15, were fabricated to allow a wide variation

in surface conditions. The stainless steel is a hard

metal impervious to the liquid, while the graphite

exhibits many small cracks across its surface. The

heating surfaces of the plates were machined to a

satin finish. A satin finish is equivalent to a

surface finish of approximately 125 microinches, rms.

A depth of cut of 0.001 inches with a cutting speed

of 0.0014 inches per revolution was used. The

machined surfaces were flat to 0.0005 inches, as veri-

fied by use of a dial indicator gauge mounted on the

lathe carriage which had performed the finishing cut.

In addition to the data taken with the above surfaces,

some data on a flat polished brass surface were avail-

able from reference (5).

The data shown in Figure 12 indicate quite

plainly that the surface condition has no noticeable

effect on the vaporization time. The volume range

below 1.5 cc in Figure 12 represents the small sphe-

roidal and flat spheroidal region, while for initial
























Is
2- %
- 1-3/4---- ---

I k-- 1


1-1/8


Heating Surface Machined
to a Satin Finish


rN 0.0005


II
3/16


T 7/16"


X Thermocouple locations


Fig. 15.


Schematic Cross Section of 304 ss
and Graphite Test Plate.








69











Material 304 Stainless Steel


Heating Surface
Machined to a
Satin Finish


X Thermocouple
Location


Fig. 16. Schematic Cross Section of Test Plate with
a 10 Conical Heating Surface.


I
^ --- 2


l-l/8'"








70

volumes greater than 1.5 cc the droplet is in the bubbly

spheroidal region, as depicted in Figure 1.

The surface condition does not noticeably affect

the vaporization time in either the flat spheroid or

the bubbly spheroidal region, because the droplet is

resting on its own vapor which prevents an interaction

of the surface with the droplet. This agrees with the

previous experimental observations 123, p. 191) that

the friction factor is independent of the surface

roughness under conditions of laminar flow.

A slight problem with the flat heating surface

results from the movement of the water droplet against

the barrier wall during the vaporization process. How-

ever, the interaction between the wall with the water

droplets in the volume range of interest, 1 cc or less,

is negligibly small, since only a very small fraction

of the droplet's periphery touches the barrier wall.

Nevertheless, to eliminate the effect of contact with

the barrier wall on the experimental evaporation rate,

the experimental data to be used in comparison with

the theoretical results were taken on a test section

with a 1 degree apex angle, as shown in Figure 16.

Figure 14 presents a comparison of the total vaporiza-

tion times as measured on a flat surface and a conical








71

surface with a 1 degree apex angle. As seen in this

figure, there is no noticeable difference in the

vaporization times, thereby confirming the earlier

observations that the side wall interaction is

negligibly small.

The test sections were mounted on the base

plate shown in Figure 17. Pyrex brand wool and glass

wool insulation were packed around the heating plate

to reduce the heat loss from the ends of the test

plate, thereby giving a more uniform temperature dis-

tribution across the test plate. A three inch aluminum

fence was built around the test section to reduce the

convection currents that would tend to flow over the

surface because of the free convective heat loss from

the test plate. In addition, the fence more nearly

produces the condition in which the water droplet is

surrounded by saturated vapor, thereby reducing any

mass transfer from the top of the droplet.

The test plate and base plate were mounted on

a 1200 watt, 220 volt electrical heating unit. The

amount of current to the heating unit was controlled by

a variac as shown in Figure 18.

Four 20 gauge Chromel-Alumel thermocouples were

embedded beneath the test section at positions indicated

























W)l \ ,o 0 *
1 o > oo e
0 o 0 oo 9*








> 0




4,4
SU
Il


4.)



0
4) 4 614
41I U



EE












94 0 PQ
1rl





a) ?D - tJ*















220 Volts


Four Chromel-Alumel Thermocouples


ference Junction
in Ice Bath


-I


Potentiometer Roneywell Rubicon
(Manually Balanced)


Fig. 18. Schematic Diagram of Experimental
Apparatus.








74

in Figures 15 and 16. The surface temperatures were

found by linearly extrapolating the upper and lower

thermocouple reading at the center point of the heat-

ing plate to the surface of the plate. The linear

correction applied to the center thermocouple was also

applied to thermocouples near the edge of the plate.

The thermocouples were fed through a selector switch

to a Honeywell manual potentiometer. A 32 F ice

reference junction was used.

Distilled water at its saturation temperature

was placed onto the surface by means of calibrated

pipettes. A 1 ml and a 10 ml pipette were used in the

experiment. The 1 ml and 10 ml pipette were read to

an accuracy of 0.005 ml, and 0.05 ml respectively.

Although the pipettes are calibrated for a liquid at

20 C, heating the pipettes to the saturation tempera-

ture of water (100 C) does not affect the accuracy of

the volume measurements due to the relatively small

amount of volumetric thermal expansion involved. The

change of volume due to an increase in temperature can

be estimated from the relationship VBf T. The volu-

metric expansion estimated by the previous expression

affects the results only if it were possible to measure

the volume to four significant figures.








75

The times in which the distilled water is ejected

from the pipette (shown in Figure 19) are short and do

not significantly affect the vaporization curves. How-

ever, because the ejection time is short, the water

leaves the pipette in a fine jet, When this jet was

allowed to impinge directly onto the surface of the

heating plate at 600 F or onto the top of a droplet

resting on a 600 F plate, the cooling effect of the jet

at the point of surface contact (see Figure 20, a and

b) initiates nucleate boiling which evaporates a con-

siderable amount of liquid in a very short time.

Consequently, the jet was prevented from impinging di-

rectly onto the heating surface by directing the jet

against a non-wettable surface in the manner shown in

Figure 20 c. When the heating plate temperature was

set at 1000 F, the cooling effect of the jet did not

initiate nucleate boiling; consequently, it was not

necessary to use the technique shown in Figure 20 c

at this higher temperature.

In collecting the data, the variac was first set

at a desired value and the equipment was allowed to warm

up slowly to a steady state value. Normally, this re-

quired 2 to 3 hours depending on the required surface

temperature. When a run was made, the thermocouple emfs



















6


5


S 4


E-4 3


4+ 2
o 0-1 ml
> Pipette
M 1 0-10 ml
Pipette

0 I I I I I
0 1 2 3 4 5 6

Ejected Volume (cc)




Fig. 19. Ejection Time of Water Droplet from
Pipette to the Hot Plate Surface as a Function
of the Volume of the Water Droplet.















4.1


41


&\
4)


40

I u

2


U


u

V4
41
4.


















9
I







oN
'0



4



U




u
r,



a














S.








78

were usually recorded from the Rubicon before the

droplet was placed on the plate, some time during the

vaporization process, and immediately following the

vaporization. Because of the relatively large amount

of time required for vaporization, there was ample

time to record all measurements by hand.

The results of the experimental measurements

are discussed next.












CHAPTER V


EVAPORATION RATES

Theoretical

The theoretical determination of the evaporation

rates for various plate temperatures and volumes

requires the solution of Equations (86), (87), and (137)

along with the construction of graphs similar to that

shown in Figure 9. Equations (86), (87), and (137) were

programed for solution on the IBM 709 digital computer

and the compiled results were used in the graphical

solutions. The material parameters used in the analysis

are tabulated in Appendixes D and F.

The theoretical mass evaporation rates are shown

in Figure 21 as a function of droplet volume and plate

temperature. This figure contains the locus of the

graphical solution points (see Figure 9) for various

droplet volumes and plate temperatures. As seen in

Figure 21, the evaporation rate increases with increas-

ing volume of the droplet and with increasing plate

temperature. The increase in the evaporation rate with

increasing droplet volume is due primarily to the

























0 .004- Tp 600 F




0.002 /
a 0.002
o /



> 0.001-



0.0006w r --
0 0.2 0.4 0.6 0.8 1.0

V (cc)




Fig. 21. Theoretical Mass Evaporation Rate of a Water
Droplet as a Function of Volume for a Plate Emmisivity
of 0.5 and Plate Temperatures of 600 F and 1000 F.








81

increase in heat transfer area associated with the

increase in the droplet volume, while the increase in

the evaporation rate due to an increase in temperature

is due primarily to the increase in the thermal con-

ductivity of the steam. The temperature dependence of

the viscosity and specific volume has a relatively

slight effect, as seen in Equation (34). Here it is

observed that the temperature effect is dampened by

the one-quarter power on the absolute film temperature.

The calculated gap thickness, as shown in

Figure 22, is relatively insensitive to volume changes,

but is affected by increased plate temperatures.


Experimental

In attempting to compare experiment to theory,

it is necessary to determine the slope of a curve

V = f(t) prescribed by a set of tabulated values (Vi,

ti). Pictorially, a tangent line is constructed to a

graphically fitted curve. However, Lipka (24, p. 234)

points out that exact or even approximate construction

of a tangent line to a curve is difficult and inaccu-

rate. Reilly (25) suggests finding the slope of

tabulated data by differentiating a polynomial fit of

the data.






























J 0.004-






0.003









0.002 I I 1 I I I
0 0.2 0.4 0.6 0.8 1.0

V (cc)



Fig. 22. Gap Thickness of the Water Droplet as a Function
of Volume for a Plate Emissivity of 0.5 and Plate Tempera-
tures of 600 F and 1000 F.







83

The total vaporization time, listed in Appendix

E, is used to determine a third order polynomial fit.

The fit is performed on the IBM 709 digital computer

using the UF-NILLS code which is currently in use by

the Nuclear Engineering Department of the University

of Florida. The polynomial is of the form


V = P(l)t + P(2)t2 + P(3)t3, (146)


where t is the time required to completely vaporize

a droplet of initial volume, V. The coefficients in

Equation (146) are listed in Table 3 as a function of

the plate temperature. They were determined by minimizing

the weighted squares of the residuals. Weights of V-1

were used.

TABLE 3

POLYNOMIAL COEFFICIENTS


Temperature
of Plate (F) P(l) P(2) P(3)


608 -7.2266-10-5 +4.8950-10-6 +4.40d0-10-9

1014 +4.0022-10-5 +1.0295-10-5 +5.4745-10-8



The curves shown in Figure 13 are drawn from

Equation (146) using the coefficients listed in Table

3. The mass evaporation rates are now determined

directly from the polynomial equation (146) by









84

differentiating it with respect to time and by

multiplying it by the density of the droplet. Thus,

dM
PD [P(1) + 2P(2)t + 3P(3)t2 1. (147)
dt


The evaporation rates as calculated from the

above equation are plotted in Figure 23 as a function

of initial droplet volume and plate temperature along

with the theoretical evaporation rates.


Comparison of Experiment to Theory

The theoretical and experimental results are

shown jointly in Figure 23. The emissivities chosen

in the theoretical calculations are based on data

tabulated in reference (2,0 p. 475). As seen in

Figure 23, excellent agreement exists throughout the

volume and temperature range considered. The devia-

tion of theory and experiment is less than 5 per cent

in the volume range of 0.5 to 1 cc, while approxi-

mately 20 per cent at a droplet volume of 0.05 cc.

The deviation seen at the lower droplet

volumes is probably a result of the increased devia-

tion of the flat spheroid model from the actual

physical situation. The droplet has a greater tendency

towards a spherical shape at these lower volumes.














1 I I I---- I I I


Tp=1000 F
(p -0.73
-p"


T =600 F
C =0.5
P


//
//
I/
'i
//


Ii

- Iii I, ,!


Theory

- Experiment


0.001 /

0.0008 /

_, iL


0.2


-- _I 1 I 1 I


0.4


0.6


0.8


1.0


V (cc)



Fig. 23. Theoretical and Experimental Mass Evaporation
Rates of Water Droplets as a Function of Droplet Volume,
Plate Temperatures and Plate Emissivity.


0.010

0.008

0.006


0.004





0.002


0.020

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PAGE 1

HEAT TRANSFER TO WATER DROPLETS ON A FLAT PLATE IN THE FILM BOILING REGIME By KENNETH JOSEPH BAUMEISTER A DISSERTATION PRE SE NT E D TO T H E G RADUATE COUNOL OF THE UNIV E R S ITY OF FLORIDA IN PARTIAL FULFILLM E N T OF TH E R E QUIR EME NTS FOR THE D EG REE OF DOCTOR O F PHILOSOPHY UNIVERSITY OF FLORIDA D ec emb e r 19 64

PAGE 2

UNIV ER S I TY OF FLORIDA 11 1 11 1111 11 111 111 1 11 11 11 111 111 11 11 1 11 111111 11 111111 111 1111 11 111 1 3 1262 08552 5656

PAGE 3

Dedicated to my wife, Mary, for the love and understanding she has given to me throughout my graduate education.

PAGE 4

ACKNOWLEDGMENTS The author wishes to express his sincere appreciation to the members of his supervisory committee: Dr. Robert E. Uhrig, chairman; Dr. F. L. Schwartz, co chairman; Prof. Glen J. Schoessow; Dr. G. Ronald Dalton, and Dr. Robert G. Blake. Special thanks are due to Dr. Schwartz for suggestion of the thesis topic and to Prof. Schoessow for technical advice concerning the experimental procedures. The author wishes to thank the University of Florida Computing Center for the aid given him. Gratitude is also expressed to F. A. Primo, H. H. Moos, and Joseph Mueller for their help in setting up the experimental equipment and to Mrs. Gail Gyles for her helpful sugges tions while typing the thesis. Thanks are also due to the staff of the Lewis Research Center of the National Aeronautics and Space Administration for the support given to the author while carrying out this investigation. In particular, thanks are due to Mr. Robert J. Usher, Chief, Training Branch, and to Miss Gertrude Collins. Finally, very special thanks are due to Mr. Harry Reilly, Chief, Reactor Analy sis Section, of the NASA Plum Brook Reactor Facility for his guidance in the development of the author's technical maturity and for his encouragement during the past two years. iii

PAGE 5

TABLE OF CONTENTS ACKNOWLEDGEMENTS LIST OF TABLES .. LIST OF FIGURES LIST OF SYMBOLS ABSTRACT . . Page iii vi vii X xii Chapter I. II. III. IV. v. VI. VII. VIII. INTRODUCTION METHOD OF ANALYSIS 1 8 General Approach. . . . . 8 Momentum Equation . . . . 13 Analog Solution of Momentum Equation 20 Steam Velocities . . 32 Mass Flow Rate . . . 37 Energy Equation . . . . . 40 Macroscopic Energy Balance 48 Graphical Determination of Gap Thickness and Evaporation Rate 52 FLOW DISTRIBUTION EXPERIMENTAL PROCEDURES EVAPORATION RATES 55 63 79 Theoretical . . . . . 79 Experimental . . . . . 81 Comparison of Experiment to Theory. 84 OVERALL HEAT TRANSFER COEFFICIENTS GRAVITATIONAL EFFECTS CONCLUSIONS iv 87 91 95

PAGE 6

APPENDIXES A. B. C. D. E. F. REACTIVE FORCE ..... SOLUTION OF THE MOMENTUM EQUATION FOR P (r ) . . ANALOG SYMBOLS . . PHYSICAL PROPERTIES EXPERIMENTAL DATA DROPLET SHAPE UNDER VARIABLE GRAVITATION ..... LIST OF REFERENCES BIOGRAPHICAL SKETCH V Page 100 103 106 109 116 119 128 131

PAGE 7

Table 1. 2 3. 4. 5. 6. LIST OF TABLES ANALOG COMPUTER RESULTS PARABOLIC FIT OF cp ANALOG RESULTS FOR q) ( 0) = 4. 0 . . . . . POLYNOMIAL COEFFICIENTS PHYSICAL PROPERTIES OF STEAM AT ATMOSPHERIC PRESSURE PHYSICAL PROPERTIES. DROPLET VAPORIZATION TIMES vi Page 33 60 83 110 111 116

PAGE 8

Figure 1. 2. 3. 4. 5. 6. 7. 8. LIST OF FIGURES Droplet States ........ Schematic Model of the Evaporation of a Flat Spheroid ........ Computer Diagram of Momentum Equations for (f = 1, K = 1, and /J = 1. . . . as a Function of the Assumed cjJ {O). Computer Diagram of Momentum Equations for cf= O .1, K= 0.001, and /J = 0.0015. <) as a Function of the Assumed
PAGE 9

Figure 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. Total Vaporization Time for Water Drop lets as a Function of Their Initial Volume and Temperature of the Heating Surface which Had a 1 Apex Angle .... Comparison of the Total Vaporization Time for Water Droplets on a Flat Plate and a 1 Conical Surface at Approximately 600 F. . . . .... Schematic Cross Section of 304 ss and Graphite Test Plate. . .... Schematic Cross Section of Test Plate with a 1 Conical Heating Surface ... Schematic Cross Section of Heating Area. Schematic Diagram of Experimental Apparatus .......... Ejection Time of Water Droplet from Pipette to the Hot Plate Surface as a Function of the Volume of the Water Droplet. . . . . . . . . Dynamics of Water Jet Ejected from Pipette. . . . . . . Theoretical Mass Evaporation Rate of a Water Droplet as a Function of Volume for a Plate Emissivity of 0.5 and Plate Temperatures of 600 F and 1000 F .... Gap Thickness of the Water Droplet as a Function of Volume for a Plate Emissiv ity of 0.5 and Plate Temperatures of 600 F and 1000 F .......... Theoretical and Experimental Mass Evapo ration Rates of Water Droplets as a Function of Droplet Volume, Plate Temperatures and Plate Emissivity ...... viii Page 65 66 68 69 72 73 76 77 80 82 85

PAGE 10

Figure 24. 25. 26. 27. 28. 29 30. 31. 32. Theoretical Heat Transfer Coefficient of a Water Droplet as a Function of Volume for Plate Temperatures of 600 F and 1000 F for a Plate Emissivity of 0 5 . . . . Theoretical Evaporation Rates of a Water Droplet in Both the Earth's and Moon's Gravitational Fields for a Plate Temperature of 600 F and an Emissivity of 0 5 . . . . . . . Specific Volume of Steam at Atmospheric Pressure as a Function of Temperature .. Viscosity of Steam at Atmospheric Pressure as a Function of Temperature .. Thermal Conductivity of Steam at Atmos pheric Pressure as a Function of Temperature. . . . . . . Schematic of Wat er Droplet. Path of Numerical Integration .. Thickness of Water Sphe~oid as a Function of its Volume for I = 1 ..... ix Page 88 93 112 113 114 120 123 126 127

PAGE 11

Symbols A a f g 1 M N p q T t u u LIST OF SYMBOLS Area, 2 Constant of proportionality, sec1 Transformation variable, ft sec-1 Acceleration of gravity, ft sec2 Dimensional conversion factor 32.1739 ft lbm lbf-l sec2 Latent heat of evaporation, BTU lbm-l Thermal conductivity, BTU hr-1 ft-1 F-1 Average droplet thickness, cm Mass, grams Surface tension, dynes cm1 Pressure, lbf ft2 Rate of heat flow, BTU hr1 Maximum radius of water spheroid, cm Temperature, For R Time, sec overall heat transfer coefficient, BTU hr-1 ft2 p-l Radial velocity, ft sec-1 V Droplet volume, cc w Axial velocity, ft sec-1 X

PAGE 12

Symbol ex_ K p cf r cp y; r cp _/\_ Thermal diffusivity, ft 2 sec1 Computer proportionality constant, volt1 Steam gap thickness, in Emissivity for radiation Dimensionless coordinate Computer proportionality constant, volt 1 Material constant, -3 3 1 -\ k in sec(lbf/lbm) cm 2 x R cm "iI Absolute viscosity, Kinematic viscosity, Density, lbrn ft3 lb ft-l sec 1 m ft 2 sec-1 Computer proportionality constan~ sec1 Computer time, sec Dimensionless transformation variable Dimensionless transformation variable Gravitational potential ~Earth=l), lbf lbm-l Computer transform variable, v o lts Material constant, grams sec-1 {lbn/lbf) x cm Rcm9/ 4 volt 1 Computer transform variable, v o lts Velocity correction factor Subscripts D f p Droplet film plate Sat Evaluated at saturation condition Evaluated at lower surface of droplet xi

PAGE 13

Abstract of Dissertation Presented to the Graduate Council in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy HEAT TRANSFER TO WATER DROPLETS ON A FLAT PLATE IN THE FILM BOILING REGI.MB by Kenneth Joseph Baumeister December, 1964 Chairman: Dr. Robert E. Uhrig Major Department: Nuclear Engineering The mass evaporation rates and overall heat tranfer coefficients are determined both theoretically and experi mentally for water droplets which are supported by their own superheated vapor over a flat hot plate. The theoretical and experimental mass evaporation rates are found to agree within 20 per cent over a drop let volume range of 0.05 cc to 1 cc and over a temperature range of 600 F to 1000 F. In this parameter range, the mass evaporation rate varies from 0.001 to 0.01 (g/sec), and the steam gap thickness ranges between 0.003 and 0.008 in. Both the gap thickness and mass evaporation rate increase for increased volume and temperature. The overall heat transfer coefficient ranges between 70 BTU/hr ft2 F} for 0.05 cc droplets and 40 for 1 cc drop le ts in the temperature range considered. Also, the xii

PAGE 14

theoretical analysis yields the axial and radial velocity distribution under the droplet and a velocity correction factor which is applied to Fourier's equation for one dimensional steady state heat conduction across the steam gap. The water droplets are approximated by a flat spheroidal geometry with a uniform steam gap beneath the droplet and a saturated steam vapor cover on the top surface of the droplet. The shape of the droplet and the average droplet thickness are determined analyti cally. The analytical results compare favorably to experimental measurements. The assumptions are made that the bottom of the spheroid is at the saturation temperature and that the evaporation takes place uni formly beneath the spheroid. The flow is shown to have a Reynolds number of approximately 10; consequently, the flow is treated as incompressible and laminar with negligible energy dissipation. In addition, the con stant fluid property assumption is made, and because of the large amount of time required for the evaporation of the droplet, the droplet at any instant is assumed to be in a pseudo steady state condition; that is, the flow is approximated by a steady state solution at any instant of time. xiii

PAGE 15

The analytical method of attack is to solve the momentum, continuity, and energy equations simultane ously. The partial differential momentum and continuity equations are reduced to ordinary non-linear differential equations by the method of combination of variables. Possible solutions to the non-linear equations are mapped by means of an analog computer. Then, these physically acceptable solutions are combined in a graphical manner with the solutions of the macroscopic energy equation, which is solved explicitly, to yield the mass evaporation rate and steam gap thickness of the droplet as a function of droplet size, plate temperature, and gravitational potential. The effect of the gravitational potential on the mass evaporation rate is considered in detail in the theoretical development. A reduction in the gravita tional potential from 1 (earth) to 0.16 (moon) is shown to reduce the mass evaporation rate by approximately half. xiv

PAGE 16

CHAPTER I INTRODUCTION The object of the st~dy presented in this dissertation is the determination of the overall heat transfer coefficient from a heated flat plate to water droplets which are supported by their own superheated vapor. This is accomplished by a theoretical study of the momentum, heat, and mass transport phenomena associ ated with this elemental two phase flow problem and by an experimental verification of the theory. This analysis differs from the semi-empirical and dimensional approaches used in the past in that the analysis is based solely on the solution of the relevant governing equations involved. The analysis considers a wide range of surface temperatures and volumes of water droplets. Interest in this subject stems from the rapid development of nuclear reactors as used in power and propulsion systems. In particular, there has been a new interest awakened in the general problem of heat trans fer with a change of state, such as occurs in the two phase flow heat transfer in a boiling water reactor, nuclear rocket, or in the more fundamental problem under l

PAGE 17

2 consideration in th is dissertation Also an increased und e r st andi ny of t.he fil m boilin g p he no me n o n, such as occurs under t he d r o plet, is of importance in nuclear rocket technolo g y since film boiling heat transfer occurs in both the rocket nozzle and core reflectors. In addi t ion, recent 1t.Orks by Adadevoh, Uyehara, and Myers OJ e n d Borishansky, Zamyatnin, Kutateladze, and Nemchinsky (1-) in d icate that there is still interest in th e subject of droplet vaporization in the fields of in t ernal combustion engines and metallurgy. Investigations on this subject were begun as far ba ck as 1756, when Leidenfrost (1) first described the phenomena of film boiling, and they have continued up to the present time with much of the more recent work found in the Russian literature. The most recent and complete works are by Gottfried (.1) and Borishansky (2_} Gottfried presents both a dimensional and semi-empirical correlation for the evaporation of small water drops on a flat plate in the film boiling regime. *The underlined numbers in parentheses in the text refer to the number of the entry in the List of References A statement to the ri g ht of a comma within the parentheses will g ive the loc at ion within the source to which the reference is ma e. If only the underlined number is given the reference is to the source in general.

PAGE 18

3 In his semi-empirical approach, the water droplet is appro x imated by a frustum of a cone in which the upper and lower areas are varied in such a manner so as to best correlate the experimental data. From the upper area, molecular diffusion is assumed to take place in the absence of thermal convective effects or a saturated vapor blanket. From the lower area, uniform evaporation into a superheated vapor is assumed with an outward flow through a uniform gap beneath the droplet. Borishansky has investigated the evaporation of water droplets up to 4 cc in volume for plate tem peratures of 527 F and 662 F. He has used both dimensional and semi-empirical techniques in correlat ing his data. On the bases of Borishansky's experimental results and the experiments performed in conjunction with this dissertation, the general problem of water droplet evaporation is broken down into the following states, which are governed by the volume of the drop let: small spheroid, flat spheroid, and bubbly spheroid. The small spheroid state, observed at 0.05 cc, is a perturbation of a purely spherical geometry by the action of gravity working against the forces of capillary tension in the surface of the droplet, as

PAGE 19

4 shown in Figure 1. For larger size droplets, the perturbation from the purely spherical state increases. Finally, as the liquid volume approaches 1 cc, the droplet reachers the flat spheroidal state in which the thickness of the spheroid undergoes little change with further increase in its volume. For volumes greater than approximately 1.5 cc, the vapor formed beneath the droplet tends to break intermittently through the sur face of the liquid which gives rise to the term bubbly spheroid state, shown pictorially in Figure 1. In this particular study, water droplets in the volume range 0.05 to 1 cc are analyzed. In this volume range, an analytical model based on a flat spheroidal geometry reasonably satisfies the physical situation and yet still has simple enough boundary conditions to make the resulting boundary value problem tractable. The shape of the droplet and the average droplet thick ness, i are determined analytically. The analytical values of 1 compare favorably to the measurements taken by Borishansky. The analytical method of attack is to solve the momentum, continuity, and energy equations simultane ously. The partial differential momentum and continuity equations are first reduced to ordinary non-linear dif ferential equations by the method of combination of variables. A difficulty results from the fact that the

PAGE 20

7 7777177717 7 77 7 7 7 7777 7777 7 7777777 Small Spheroids Flat Spheroids Bubbly Spheroids Fig. 1. Droplet States.

PAGE 21

6 non-linear equation cannot be solved in closed form. The difficulty is further compounded because the boundary conditions to the problem are unknowns. The analytical approach to the problem is to map possible solutions to the non-linear flow equations by means of an analog computer and to use those particular solutions which are physically acceptable. Then, the solutions to the macroscopic energy equation, which is solved explicitly, are combined in a graphical manner with the results of the analog solutions to yield the mass evaporation rate of the droplet as a function of droplet size and plate temperature. The overall heat transfer coefficient can be found directly from the mass evaporation rate. The theoretical mass evaporation rates are found to agree within 20 per cent of the experimental values over a temperature range of 600 F to 1000 F, and a volume range of 0.05 to 1 cc. The analysis also yields the axial and radial velocity distribution under the droplet, the steam gap thickness, and a velocity correction factor which is applied to Fourier's equation for one-dimensional steady state heat conduc tion across the gap. Also, with the possibility of the moon being explored in the next decade and with the possibility

PAGE 22

7 of operating with two phase flow in a low gravitational field, the theoretical analysis takes into account the effect of a variation of the gravitational potential on the mass evaporation rate. In particular, the mass evaporation rate is determined for the physical situa tion where the surface gravity is equivalent to that of the moon's surface. The resulting values of the mass evaporation rate are compared to the values obtained on earth for a similar plate temperature and droplet volume. The mass evaporation rate is approxi mately half that found on the earth for a given droplet volume and plate temperature.

PAGE 23

CHAPTER II METHOD OF ANALYSIS General Approach Consider the flat water spheroid shown in Figure 2. Heat transfer to this flat spheroid takes place primarily by conduction and radiation through the superheated film. Beat transfer and evaporation from the upper surface are negligibly small in com parison to that beneath the droplet. Kutateladze (._, p. 376) points out that the external surface of the spheroid is covered by superheated vapor flowing from beneath the spheroid. This vapor cover reduces the energy transport from the upper surface to a near zero value. However, even in the assumed absence of a steam cover, both the free convective and radiative heat transfer, and free convective evaporation are negligible when compared to that occurring beneath the droplet. The free convective evaporation was esti mated from a correlation presented by Wade (].). Thus the problem of heat transfer to a flat water spheroid reduces to a problem commonly termed mass transfer cooling CJ!). Such a process is 8

PAGE 24

+ cS z Heated Plate r 0 Liquid Volume V Steam Flow Fig. 2 Schematic Model of the Evaporation of a Flat Spheroid.

PAGE 25

10 characterized by a mass flow through a porous surface (transpiration cooling), by mass released from a surface through evaporation or sublimation, or some chemical reaction (film cooling, ablation cooling). To determine the velocity, temperature, and mass distribution for the above-mentioned problems, Hartnett and Eckert~) point out that in general it is neces sary to solve the continuity, momentum, energy, and diffusion equations simultaneously. However, for the superheated steam region under the water droplet, the mass evaporating from the surface is the same as that flowing beneath the droplet; thus, the diffusion equa" tion is identicallly zero, as pointed out by Grober, Erk, and Grigull (10, p. 416). However, for the problem of evaporation of a flat spheroid resting on its own superheated film, an added condition of static equilibrium is required for the solution to the problem. The pressure forces on the bottom of the droplet must be sufficient to balance the weight of the droplet minus the reactive force. For this particular problem, the reactive forces are negligible compared to the body force due to gravityr as shown in Appendix A. Consequently, the reactive force is neglected in this analysis.

PAGE 26

11 Gottfried ~) and Kutateladze (_, p 377} point out that the flow under consideration is of very low velocity and is well within the laminar range; thus, the flow is treated as incompressible with negligible energy dissipation due to friction. In addition, because of the large amount of time required for the evaporation of the spheroid, the droplet at any instant is assumed to be in a pseudo steady state condition; that is, the flow is approximated by a steady-state solution at any instant of time Consequently, for this case of axisyrnmetrical and incompressible laminar flow with negligible dissipation and with constant fluid properties, the momentum, continuity, energy, and static equilibrium equations are as follows: Momentum: ou O u u-+w= u r 2>z o w 6w u-+w-= r z _g_c _P + ( ciu + P dr 6 r2 1 au r Or Continuity: w ....:: 0 oz u + o 2 u l r 2 oz 2 + o 2 w ) oz 2 (1) (2) ( 3)

PAGE 27

12 Energy: (4) Static Equilibrium: ro 1 P ( r, c5 ) 2 7T rdr = V~ :L 9o (5) The boundary conditions for the above equations are: z = 0 u -= 0 w = 0 T = Tp (6) z = 6 u = 0 w = w(6) T = Tsat {7) The assumptions are made that the bottom of the spheroid is at the saturation temperature, and that the evaporation takes place uniformly beneath the spheroid. The boundary condition on the axial velocity at the upper surface is an unknown; in fact, at the present time the gap thickness is also an unknown. The gap thickness, 6 and w ) are found by simultaneous solution of the above equations. In this analysis, the determination of the evaporation rate, heat transfer coefficient, and gap thickness is in terms of the volume of the water droplet, gravitational potential, and temperature of the heating

PAGE 28

13 plate. Consequently, the transport properties of viscosity and thermal conductivity, as well as the specific volume of the steam, are expressed in terms of the steam temperature. For the range of tempera ture under consideration in this dissertation the above properties are represented as linear functions of temperature. Momentum Eguation The logical beginning to this analysis is to solve the momentum equation since, as a result of the constant fluid properties assumption, the mutual interaction between the equation of motion and the energy equation ceases, and the velocity field no longer depends on temperature. The usual approach (-2_) is to evaluate the properties of the flow field at the film temperature, as defined as: (8) Therefore, the immediate problem is to solve the continuity, Equation (3) and the momentum, Equations (1) and (2), simultaneously. These equations form a set of partial differential equations with two inde pendent and three dependent variables. Use of the

PAGE 29

14 method of "combination of variables" at this time reduces the above-mentioned equations into a set of ordinary non-linear differential equations. Consider for a moment the physical situation. Defining u as a mean radial velocity, u = l 0 u dz (9) and the z component of velocity at the surface of the w ater spheroid as w( i5} then the conservation of flow into and out of a cylindrical volume of radius, r, under the flat spheroid results in w (6) 1r r 2 = u 21r re (10) However, since w( o ) is an assumed constant along the bottom surface of the spheroid, it follows that u a r (11) Also, Gottfried (1), working with the mean radial velocity as defined above and with the Navier Stokes equation in the radial direction only, shows that (12)

PAGE 30

15 The above relationships indicate that a conbination of variables of the form used in the prob lem of three dimensional axisynunetry stagnation flow (11, p. 8 3) will reduce the partial differential equations (1), (2), and { 3} into a set of ordinary equations. The functional forms used in this conver sion are: w = -2 f(z), u = r f' (z) (13) (14) (15) Choosing these specific functional for~s for w and u satisfies identically the continuity equation, as substituting Equation~ (13) and (14) directly into the continuity equation (3 } verifies. The partial derivatives of the relationships (13), ( 14), and (15) used in the combination of vari ables substitution are now listed for future reference. Ou f' o2u 0 or = ::::: or 2 (16) ou o2u oz = r fl I o z 2 = r f' I I (17) ow 0 o2w 0 or ::::: -or 2 I (18)

PAGE 31

16 o w -2 f' () 2w -2 f I I = o z 2 = I z (19) OP -a2 p OP 1 p 2 or = r = ---a F' gc oz 2 gc (20) Substituting the above relationships into Equations (1) and (2 results in the following new forms of the momentum equations: f' 2 2 ff' = a2 + V t' ' 2 ff' = 1 a2 F' 4 f I I (21) (22) Thus, the original partial differential equations are by means of the above substitutions con verted into a set of non-linear ordinary differential equations. Although these equations are not easy to solve in the strict sense, they are much more easily handled than the original partial differential equa tions. Gottfried's pressure relationship (Equation 12 is generalized to include z variations in pressure, since a trivial solution to the momentum equation re sults if the z variation in pressure is neglected, as shown in detail in Appendix B. The boundary conditions, Equations (6) and (7), now take on the following forms:

PAGE 32

17 z = 0 f = 0 f' = 0 F = 0 T = Tp I {23) z = f = w(6) f' 2 = 0 T = Taat (24) The boundary condition on the function, F, is chosen arbitrarily since the only interest is in the relative variation of the function in the z direction. The constant, a2, accounts for the absolute magnitude of tn e pressure. Equations {21) and (22) are now freed of the constants a 2 and Vby making the following substitutions: F(z) = c 3 l/fC(). (25) Substituting the above expressions into Equations (21) and (22), the coefficients of these equations become independent of a2 and V if ( =~ z, f(z} = --{;;u<.() F(z) = :Vy;{() (26 (27) (28)

PAGE 33

18 Therefore, it follows that a( = -ffu (29) dz f' = a Substituting the above relationships into Equations (21) and (22), the momentum equation takes on the following forms: I = l + '', (34) 2 = y; I (35) The boundary conditions on these equations become ( = 0 = 0 = 0 1/1 = 0 T = Tp (36) ( =~6 = -w( 5 ) = 0 T = Teat 2 (37)

PAGE 34

19 Complications result because a closed solution to the above non-linear ordinary differential equation does not exist and because 6 and w(6) are unknowns. The value of the parameter, 6, depends on the simultan eous solutions of all the governing equations. Conse quently, the method of solution is to assume many reasonable values of o and to solve for the flow distribution in each of these cases that satisfy the static equilibrium condition, Equation ( 5). The results are plotted and compared to the solutions of the energy equation, Equation (4), for various values of o. The intersection of the results of the momentum and energy equations represents the value of the gap thickness, o The solution of Equations (34) and (35) in this particular situation is most easily performed by means of an analog computer. However, instead of assuming values of o, initial values of~'' are assumed. The values of 6 and w(o j are determined from the output of the computer. The next section discusses in detail the analog solution to Equations (34 ) and (35).

PAGE 35

20 Analog Solution of Momentum Equation The solution of ordinary differential equations by use of a differential analog computer is discussed widely in the literature (12) (13) and (14) Briefly, the analog computer is capable of the basic mathematical operations of addition, subtraction, multiplication, and integration. The variable quantities of the differen tial equation are represented by voltages which may be recorded by use of the proper recording equipment. In setting up the momentum equations (34) and (3 5) for solution on the analog, it is helpful to rewrite these equations in the form: = 2 -2" 1, = 2 It (38) (39) The first step in programing the above equations for the analog computer is to change the variables in the above equations to computer variables by making the following transformations: = 0 T, = K'j)cn, = /Jtj!
PAGE 36

21 The derivatives of the above transformations necessary for substitution into the analog equations (38} and (39} are 4> I = d4> dT l d4> = -K dci, K = --(43) C1 d T C1 d T C1 K 4> I I = ct> I (44) K (45) I I I = 4> a3 ct> B = (46) C1 Substituting the above relationships into Equations (38) and (39) results in the analog momentum equations 2 ri3 = KC1 ct, 2 1((14)~ (47) ct> K = -2 1(2 IC (48) -ci, -. ,, B a, The boundary conditions on the above equations take on the form T = 0 ct> = 0 4) = 0 'I' = 0 T = Tp (49) !~o l w{o) T = 4) = = 0 T = Tsat C1 \) K 2~ (50)

PAGE 37

22 The symbols a, K, and B are constant scale factors. The symbol 1 represents the computer time: the time for the phenomena to occur on the computer. The distance traversed from the plate is directly related to the computer time, 1. This relationship is presented in the next section. In attempting to find a solution to the boundary value problem presented above, it is assumed that a solution to the boundary value problem exists for every assumed value of o. Further, it is assumed that the above-mentioned solutions are selectable from a set of discrete solutions to the initial value problem of Equations (47) and (48) having the initial conditions 1 = 0 = 0 = 0 .. -co < > CD 'I'= o. (51) In order to check the above hypothesis, Equations (47) and (48) are programed for the analog computer for values of a= l, K = 1, and B = 1, which imply for this first program that l; = 1 The analog momentum equations become ... = 2 .. 2 .. 'I' = 2~ i l (52) (53) (54)

PAGE 38

23 Figure 3 shows the analog diagram for Equations l53) and {54). The set of discrete initial conditions on t is selected over a sufficiently wide range to give a reasonable topology of the total set of solutions to this particular initial value problem. The circuit shown in Figure 3 is programed on the Applied Dynamics Analog Computer. In programing this particular computer, the voltage to the quarter square multipliers must be greater than 10 volts in order to keep the specified computer accuracy. For this program, constant multipliers accomplish this. The notation used in this diagram is standard analog notation; however, the symbols are defined in Appendix c. Figure 4 represents the solutions for 4> The requirement of satisfying the boundary conditions, Equa tions ( 36) and (37), limits the acceptable range of { O} to 0 < ,II (0) < 1.31. (55} It turns out that of the possible solutions the values of physical interest are near ,''(0} = 0.5. (56) Figure 4 indicates that for the range of interest, the analog equation requires rescaling so that

PAGE 39

ll! -P +1op -10 P +1op 0 +10 0 2 -~ +2 ~p ~-.o+100 Fig. 3. Computer Diagram of Momentum Equations for (J =l, K=l, and jJ= 1. 10~ 4) ..., ,I:.

PAGE 40

2.0,--r-----r----,-----.----.----~ ''(0)=2.0 1.5 ''(0) ~ 1 ~ (0)=l.3 0.5 I (0) = l.0 -0.5.._ ___ _._ __ _ _.,__ ____ ___.__ __ ....._ _ __ ,.___ ___ __. 0 1.0 2.0 3.0 4.0 5.0 6.0 Fig. 4. as a Function of the Assumed ''(0).

PAGE 41

26 the analog output is at a greater voltage to reduce the error involved. The precision of the analo g computer components used allows an accuracy of 0.1 per cent based on a 100 volt output. A trial and error procedure indicates that for the range of interest scale factors of cf= 0.1, K = 0.001, and fJ= 0.0015 can be used. As a result, the analog momentum equation takes on the following form, 0.01 [ 4 2 ] 0. 02 1 'P 4 100 I 100 1 (57) '' = -6.6 f o.02 fi + i I 100 If '-58) .. The initial conditions on p are found from relationship ( 44). The four sets of initial conditions used are .. 1" = 0.5 C)= 5.0 = 0.4 = 4.0 (59) = 0.3 = 3.0 = 0.2 2.0 The analog diagram for Bquations (57) and (58) 1s shown in Figure 5. The program results are shown in Figures 6, 7, and 8, which represent p, f and

PAGE 42

.. -

$ 2
PAGE 43

U) -1,.l ,-f 0 > 2 8 1s.o .----.-----.---r---.---"T----.--~---,---r---t 12.5
PAGE 44

Ul +l 0 > 29 100....--------,.--...,......----T"-----------,.--------,--0 2 4 6 8 10 7 (seconds) .. Fig. 7. cp as a Function of the Assumed (_p(O).

PAGE 45

3 0 100 ,---.-----,---,--~------,------,--~~-~~-~ 80 cpo)=S.O < 0) =4. 0 I 40 I L 2 ol =3.0 0 ....._ _.....___ .._ 0 2 4 6 8 lt 7 (seconds ) Fig. 8. a s a Function of the Assumed ~O).

PAGE 46

31 respectively. The results are as expected. The parameter, cp, which is directly related to u, starts at 0, goes to a maximum value near the center of the steam gap, and then returns to zero at the surface of the droplet. The parameter, cp, which is directly related tow, starts at zero at the plate and then reaches its maximum value at the bottom of the droplet. The param eter,', which is directly related to the total pressure, goes from a maximum value of zero at the plate to a minimum value in the center of the channel. This is because the total pressure head at the plate is par tially converted into a velocity head in the center of the gap between the plate and the water droplet, thereby giving rise to a lower pressure. The pressure then returns nearly to its plate value at the surface of the droplet. There is a negligible deviation due to the small axial velocity head, which for this problem is negligibly small compared to the radial velocity head. Therefore, the pressure distribution at the surface of the droplet is taken to be of the form (60} The phenomena discussed above can be found in many

PAGE 47

32 textbooks under the heading of Bernoulli's Equation (15 p. 114). Table l lists the important numerical values of the end points of Figures 6 and 7. The next section is concerned with the determination of the velocities and evaporation rates from the data listed in Table l. Steam Velocities The radial and axial velocities and the gap thickness are determined from the analog parameters listed in Table 1. These parameters are directly re lated to the axial velocity, w, by Equations {13), ~7), and ( 41 ) resulting in w = 2 iJaV kc;f> (61) The value of w at the surface of the droplet is given by (62) where (p 0 is the value of cp at the surface of the drop let. The value of cp O is tabulated in Table l. The ,. parameter, cp is directly related to the velocity, u, by Equations (14), (30), and (43), resulting in (63)

PAGE 48

33 TABLE 1 ANALOG COMPUTER RESULTS .. cp co> volts seconds volts volts 5.0 9.9 82.0 12.25 4.0 7.0 29.5 3.0 5.3 13.0 2.0 3.6 4.0 The output of the~nalog computer is read in volts; however, the output,~' is considered to be volts per unit time when used in the equations, in order that the units will be consistent. 6.25 3.50 1.50

PAGE 49

34 In a similar manner, the time required to satisfy the boundary condition of= 0, Equation (50), that is, the time at which the curves in Figure 6 take to reach their zero values, relates directly to the gap thickness, 6 by Equations (2&) and ( 40), resulting in (64) The parameter, a, still an unknown, is now determined from the static equilibrium condition, Equation (5). Substituting Equation (60) into Equation (5) and solving for a 2 a2 = V PD lg T( p C 1 {65) 0 where Performing the required integration and solving for a, a = (66)

PAGE 50

35 Therefore, Equations (61), (63), and (64) take on the following forms: w = 2K(~ %1/lgc r: 4 r or For a flat spheroid geometry, simplifying Equation (66) by relating the radius to the volume conveniently results in where Therefore V = A i, A= 7Tr 2 0 rr2 1 2 (67) (68) ..(69) (70) (71)

PAGE 51

36 Substituting into Equation (66) results in (72) or a = (73) Therefore the equations (61), (63), and (64) take on the following more convenient forms using the above substitutions, w = Jr; ( 1 ) \ 1 V ,(75) (76) The next section deals with the determination of the evaporation rate from the above velocity relationships.

PAGE 52

37 Mass Flow Rate The mass loss required to satisfy the condition of static equilibrium is calculated by the relationship Substituting Equations ~70) and (75 r into the above equation results in Or expressing dM dt = p V l = u = p Equation ( 78 ) takes on the form : = ( 64 7f gc \K 9\ ri.(t1 (~ r 3 V4 I, (77) (79) (80 (81) In this analysis, desiring to determine the evaporation rate as a function of the plate temperature, the temperature dependent parameters of density and viscosity are expressed in terms of the film temperature

PAGE 53

3 8 of the vapor. Figures 26 and 27 in Appendix D indicate a linear relationship between the viscosity and the specific volume with the absolute temperature of the superheated steam in the range of interest. Therefore (82) (83) Substituting Equations (82) and 83) into 81) results in the following form of the evaporation equa tion: dM dt Defining the material quantity \l = { 647T gc A / K(t )" (3 .2s1x102 ) ~x453 .6 x {3.531xl0-S). (85) Substituting Equation 85) into 84) results in dM dt (86) The last expression gives the evaporation as a function

PAGE 54

39 of the material properties, 0, the gravitational constant, r, thickness of the water spheroid, t, the absolute film temperature, Tf' and the volume of the droplet under consideration, v. Equation (86) is to be evaluated for different values of the gap thickness. Substituting Equations (79 } (80 ) (82), and (83) into (76), so as to relate the gap thickness to temperature, resul t s in the follow ing relationship for the gap thickness: (87} where A = cf({T() (3. 28lxl02 )~x (. 353lxlo4 ) "' x l 2 (88) (47Tgc /Jo)\ The parameter, A, is a function of material only as is~ Equations ( 86) and (87} are now evaluated separately to determine the required flow rate and gap thickness which satisfy the momentum equation, contin uity equation, and the condition of static equilibrium. What remains now is to determine the evaporation rate from an energy consideration as a function of the gap thickness. Therefore, the next two sections deal with the solution of the energy equation and the macroscopic energy balance.

PAGE 55

40 Energy Equation This section is concerned with the calculation of the amount of heat transferred from the plate to the water droplet by conduction. Previous work{!), and (;) used the relationship k A q = 6 Tp-T ) sat (89..) for the calculation of the amount of heat transferred through the gap. However, the above relationship does not consider the effects of the stream velocities on the conduction heat transfer. Therefore, the abpve equation is considered a first order approximation to the energy equation, Equation (4). When the velocity effects are neglected, the energy equation takes on the form of the Laplace equation, of which Equation (89) is the solution. As a result of the work of the previous two sections, the effect of the stream velocities on the rate of heat transfer by conduction through the steam gap can be determined. The linear relationship implied by Equation (89 ) is perturbed by the ejection of the saturated steam into the vapor stream.

PAGE 56

41 For the problem under consideration, the ph y s i ca l conditions indicate that oT oT -< < Or Oz Therefore, Equation (4 can be simplified to dT d 2 T w = O'.:dz dz2 ( 90) (91) Substituting relationship (61) into Equation {91 res i tlts in dT d 2 T 2 K>=CXdz dz2 '1 92 ) However, combining Equations {26 f and (40 results in (93) Differentiating the above with respect to z yields (94) Therefore, the first and second derivattvas with respect to temperature are written as follows:

PAGE 57

42 dT dT dr l -vf dT == -= cf dz d T dz dT (95) d 2 T l a d 2 T = (f 2 dz 2 V d-f (96) Substituting the above two relationships into Equation (92) results in the following form of the energy equa tion: + M = 0 (97) dr The parameter, cp, in the above equation is a function of r, and is conveniently approximated by the following form (see Figure 7): Therefore, the energy equation becomes Let -+ d72 B == 2 V K (f (f\T dT C<. T dr 6 ( 98) = 0 (99) (100)

PAGE 58

43 but the Prandtl number is equal to Pr = V a Therefore, the constant, B, takes on the form B = and Equation (97 ) becomes + aT dT = di 0 {101} (102) (103) The problem now is to integrate the above differential equation and apply the thermal boundary conditions shown in Equations (49) and {50). Equation (103) is made readily integrable by substituting in y = dT ar (104) Substituting Equation (104) into (103} and integrating yields dT dt = c 4 exp (105)

PAGE 59

44 Integrating the above results in (106) The value for the above integral is given in reference (16, p. 303) as But, reference (16, p. 297) shows that erf x = _2 L -vffn=O Expanding Equation (108), n: 2n+l X (2n + 1) (108) erf x = .;,,, ( x x: + :: ) (109) Substituting Equation (109) into (107) results in T = Defining r( B 2 B 2 4 C4 1 -/ + -/ 6 40 !:). = B / 2 6 0 + C5 (110) (111)

PAGE 60

45 results in T = ( 1 6(.I_) 2 + Jj_ (.L) 4 ) + c 5 7c 1 1 7 0 (112 The value of the parameter, 6, is approximately 0 05. Consequently, it follows that Since T is defined for the domain 0 c:; T T 6 ( 113) 1. u b. ( : ) = 1 (114) o Therefore, from calculus (17, p. 129), it is known that CD an alternating series of the form \ ( l)n+l a where L... n 1 an~an+l>0 and where the limit an= 0 converges and the remainder after n terms has a value between zero and the first term not taken. Consequently, second order terms and higher of l are neglected, since the maximum error in the resulting series is less than 0.0025. Therefore, the temperature is represented by (115)

PAGE 61

46 Evaluating c 4 and c 5 from the boundary conditions, Equations (49) and (50), yields T = (Tp Tsat)T r 1 6,[ _L J 2] (1 6 T T 6 6 (116) However, in this particular problem, the temperature distribution under the droplet is not of great interest; rather, the heat flux at the droplet interface is the important quantity This is found from the relationship q = l: A dT dz I 0 Using relationship (94) the above becomes q = k A cf dT \ dT 6 (117) (118) Differentiating relationship {115) results in a tempera ture gradient of the form oT aT = (119) Therefore, the heat flux at the surface of the water droplet is given by

PAGE 62

q = 47 ( l 36 l (120) 1 6 but, substituting in Equation (64) results in q = ~ 121 ) where /\_ = ( l 36 l l 6 (122) The parameter A, represents a velocity correction factor to the above equation Bound up in this correction factor is the consideration that some of the heat leaving the plate goes into superheating the vapor leaving the surface of the droplet. The value of A is approximately 0 95: thus, the velocity correction factor represents a 5 per cent correction on the energy equation. Next, the solution of the energy equation is used in a macroscopic energy balance to detennine the allowable evaporation rate as a function of O.

PAGE 63

48 Macroscopic Energy Balance The amount of m ass transfer from the water droplet is now calculated explicitly as a function of 0 b y solution of the macroscopic energy equation. The macroscopic energy balance for the water droplet model as shown in Figure 2 takes the form = (123) Here~ is the conduction energy flux, qr is the net radiative energy flux, hfg is the increase in enthalpy during the vaporization of one pound mass of liquid, and Q.M is the amount of liquid vaporized by the energy dt transferred by conduction and radiation through the steam gap. The conduction energy flux is represented by Equation ( 121 ) while the radiative flux is given appro x imately by the relationship (18, p. 64), = ao Fe Fa A (T 4 T 4) P sat (124 } Fro m geometric considerations (19, p. 199, formula 6), l (125)

PAGE 64

49 For the above geometry, where Fa= 1, the Fe factor which considers the departure of the two surfaces from complete blackness is represented as (18, p. 61): 1 Fe = 1 + 1 1 (126) tp (D A water droplet at 212 Facts similar to a black body, since the emissivity of water at 212 Fis 0.963 (20, p. 4781. Therefore, (127) Thus, the radiative flux is written as qr = o o f p A ( T p 4 Ts at 4) (128) The above equation is conveniently rewritten in the form = (129) where = (130) with the values of FT available in the literature (19, p. 208) as a function of the body temperatures.

PAGE 65

50 The absorption of some of the radiative energy by the water vapor is neglected in this problem because of the small path length between the droplet and the p 1 ate ( 19 p 214) and ( 21 p 3 8 8 ) Thus, the macroscopic energy balance takes on the form Substituting in Equation (70) and solving for the evaporation rate yields dM dt = (132) with all the temperature dependent properties evaluated at the film temperature, Equation (8). The overall heat trasfer coefficient, U, between the plate and the water droplet is defined by = (133) Comparing the above relationship to the Equation (131) results in the following form of the overall heat transfer coefficient: u = (134)

PAGE 66

51 Substituting the above into Equation {132) results in dM dt = {135) Clearly, if the eva~oration rate is calculated by theory or experimentally measured, the overall heat transfer coefficient is known directly from the evapor ation rates by the use of Equation {135). Following the earlier procedure of expressing the transport parameters in terms of temperature, the thermal conductivity is expressed as a linear function of temperature of the form k = m + n Tf, as shown in Figure 28 of Appendix D. Therefore, the evaporation rate shown in Equation {132 ) takes on the form dM dt = X (136) (Tp Tsat) 453 6 x {l.076xl03 ) {137) 3600

PAGE 67

52 Graphical Determination of Gap Thickness and Evaporation Rate The evaporation of water vapor from a water droplet has been determined in two ways. First, from the standpoint of momentum required to produce static equilibrium for a given gap thickness (repeated for convenience), dM dt (86) and secondly, from a standpoint of energy transfer, dM dt = 1_2_(m_+_nT_f_) A_ ) + ( p FT t hfg 0 X {Tp Tsat) 3600 453.6 x {l 076xlo3 ) {137) Both equations shown above are solved explicitly for different values of the gap thickness found from the relationship: (87) When the evaporation rates calculated from Equations (86) and (13 7 ) are equal for a given value of the gap

PAGE 68

53 thickness, as calculated from Equation {87), the governing equations and boundary conditions, Equations (1) through (7), are satisfied concurrently For example, Figure 9 shows a graphical solution of the momentum and energy equations for the case of a 0 5 cc water droplet resting on a plate at 600 F The point of intersection of the two equations on Figure 9 represents the conditions where all the governing equa tions are satisfied concurrently The values of the evaporation rate, which is directly related to the overall heat transfer coefficient by Equation (135) and the gap thickness are found directly from the ordinates in Figure 9. Similar graphs were constructed for different size droplets and for various plate temperatures to determine the evaporation rates as well as the overall heat transfer coefficient for a variety of conditions It is shown in a later section that the theoretical results are in excellent agreement with theory.

PAGE 69

0 d) Cl) '-. t,"I d) +J "' C: 0 ..... +J "' 0 "' :> t/l Cll 54 0.01 0. 0080.006 Energy 0.005 (137) 0.004 Solution Point 0.003 0.002 Momentum {86) 0.001,__ ______ ____._ ____ .....__ __ _._ __ .___..______.__.....___.,____. 0.001 0.002 0.004 0 (.inches) 0.006 Fig. 9. Graphical Simultaneous Solution of Momentum and Energy Equations for V=0.5 cc, Tp=600 F, and tp=0.5. 0.01

PAGE 70

CHAPTER III FLOW DISTRIBUTION The velocity distribution and the Reynolds number under the droplets are considered, since the earlier assumption of laminar flow under the droplet 1s now justified. In addition, it is important to determine the magnitude of the radial velocity at the edge of the droplet, since the droplet tends to move slowly about when resting on a flat plate. The velocities, u and w, are related to the computer variables, and i by Equations (61) and (63). For a given set of physical conditions, such as volume of the droplet and temperature of the heat ing plate, the velocity distributions vary directly as~ and i. Consequently, the curves shown in Figures 6 and 7 represent the forms of the radial and axial velocity distributions. Previously, the exact solu tion for the velocity distribution could not be determined because the gap thickness, o, was an unknown. However, using the results of the previous section, the velocity distribution can be evaluated since the gap thickness is now a known quantity. 55

PAGE 71

56 The curves shown in Figures 6 and 7 are .. generated from known values of <)(0) rather than specify.. ing 0 However, 0 is related tocp
PAGE 72

0 6.0 s.o 4.0 3.0 2.0 1.0 0 57 0 0.002 0.004 0 (inches) .. 0.006 Fig. 10. () as a Function of <:i)(O) for V=0.5 cc, Tp=600 F, and tp=O.S.

PAGE 73

5 8 12.s 10.0 \-'O I N 7. 5 ~ 5.0 2.5 0.002 0.004 6 (inches) ( To ) Fi g 11. Cl) 2 as a Function of V = O.S cc, Tp=600 F, and ~=0.5. 0.006 6~or

PAGE 74

59 is constructed from the analog resul t s presented in Figure 6 and the information presented in Figure 10. The verte x of the parabola is taken as the anchor point between the analog data and Equation (140). The parabolic relationship very accurately represents the curves in Figure 6 as seen in Table 2, which presents a comparison of the

PAGE 75

60 TABLE 2 . PARABOLIC FIT OF cp ANALOG RESULTS FOR o) = 4. 0 z T cp calc
PAGE 76

61 The maximum Reynolds number beneath the droplet is calculated from the flow beneath the droplet by the relationship where Thus R e max = = Re = V 4 x flow cross section wetted perimeter = 4 X ----(143) (144) 8/3 ~r (i ) \ ( lJ (j' ,-, '/< c5
PAGE 77

62 velocity at the edge of the droplet of 5.25 ft/sec. Thus, the flow is well within the laminar range and the slight motion of the water droplet on the heating plate is small compared to the average steam velocity leaving the gap beneath the droplet. An increase in the volume of the droplet increases the exit radial steam velocity and the Rey nolds number, since both i and o increase with increasing volume. Also, combining Equations (74), (79), and (82) indicates that an increase in temperature of the heating plate increases the exit radial steam velocity and the Reynolds number in proportion to {Tf). However, for the temperature range and volume range investigated in this paper, the basic conclusion that the flow is laminar is not affected by the volume and temperature changes considered.

PAGE 78

CHAPTER IV EXPERIMENTAL PROCEDURES In the theoretical analysis, the mass evaporation rate for a droplet on a flat plate is shown to be a function of the plate temperature and the volume of the droplet. Hence, an experimental verification of the theory requires that the evaporation rate be measured for different plate temperatures and droplet volumes. The evaporation rate is determined experimentally from measurements taken on the total vaporization time. The total vaporization time, that time required for the entire volume of liquid which is placed on a heating surface to vaporize completely, is measured as a func tion of droplet size for various plate temperatures and surface conditions. The experimental data are listed in Table 6 of Appendix E, while the plots of the data are shown in Figures 12, 13, and 14. The slopes of these curves, rate of change of volume with respect to time, represent the evaporation rate of the droplet. The determination of the slope of a curve V = f(t), when a table of distinct sets of values (v 1 ,t 1 ) are known, is considered in detail in the next chapter. 63

PAGE 79

64 5 0 304 ss, satin finish L Brass--highly polished 0 Graphite, satin finish 0 4 0 (1) .... 1-1 A 3 1-1 Q) 4-4 0 2 ::, .... 0 > .... .... l .... C: H 0 0 2..00 400 600 Total Vaporization Time (sec) Fig. 12. Total Vaporization Time for Water Droplets on a Flat Plate as a Function of Their Initial Volume for Various Surface Conditions at a Plate Temperature of Approximately 600 F.

PAGE 80

65 1014 F 608 F 1 0 0 0 0.8 Q) .-I M Q M 0 6 Q) r'll 14,-4 0 Q) E 0 4 ::, .-I 0 > .-I rtS -.-4 0.2 .-I H 100 200 300 400 Total Vaporization Time {sec) Fi g 13. Total Vaporization Time for Water Droplets as a Function o Their Initial Volume and Temperature of the Heating Surface which Had a 1 Apex Angle. 500

PAGE 81

66 1.0 u 0.8 u (1) .-4 0 0 0.6 (1) rd 0 (1) 0.4 e ::, .-4 0 ::::,. 'v 1 Apex Angle rd "M 0.2 0 Flat Surface c:: H 0 .____~ __ _.__ __ ._ -.L __ ....__ ___,'--_---+__ ..,__ ___._ __ 0 100 200 300 400 Total Vaporization Time (sec) Fig. 14. Comparison of the Total Vaporization Time for Water Droplets on a Flat Plate and a 1 Conical Surface at Approximately 600 F. 500

PAGE 82

67 The test sections used for vaporizing the water droplets are shown in Figures 15 and 16. Both a 304 stainless steel and a graphite test plate, as shown in Figure 15, were fabricated to allow a wide variation in surface conditions. The stainless steel is a hard metal impervious to the liquid, while the graphite exhibits many small cracks across its surface. The heating surfaces of the plates were machined to a satin finish. A satin finish is equivalent to a surface finish of approximately 125 microinches, rrns. A depth of cut of 0.001 inches with a cutting speed of 0.0014 inches per revolution was used. The machined surfaces were flat to 0.0005 inches, as veri fied by use of a dial indicator gauge mounted on the lathe carriage which had performed the finishing cut. In addition to the data taken with the above surfaces, some data on a flat polished brass surface were avail able from reference {2_). The data shown in Figure 12 indicate quite plainly that the surface condition has no noticeable effect on the vaporization time. The volume range below 1.5 cc in Figure 12 represents the small sphe roidal and flat spheroidal region, while for initial

PAGE 83

I 1-1/8 6 8 II 2 Heating Surface Machined II 1-3/4 -lo,,. j / to a Satin Finish 1 / ~I _________ __ o _ o o o s / /, l y / 3/16 11 !./ X T 7/16 11 X X Thermocouple locations Fig. 15. Schematic Cross Section of 304 ss and Graphite Test Pla t e.

PAGE 84

1-1/ 8 Fi g 16. 6 9 Material 304 Stainless Steel 2 II 1/8 11 !-.(" r Heating Surface Machined to a Satin Finish X Thermocouple Location Schematic Cross Section of Test Plate with a 1 Conical Heating Surface.

PAGE 85

70 volumes greater than 1.5 cc the droplet is in the bubbly spheroidal region, as depicted in Figure l. The surface condition does not noticeably affect the vaporization time in either the flat spheroid or the bubbly spheroidal region, because the droplet is resting on its own vapor which prevents an interaction of the surface with the droplet. This agrees with the previous experimental o~servations { 23, p. 191) that the friction factor is independent of the surface roughness under conditions of laminar flow. A slight problem with the flat heating surface results from the movement of the water droplet against the barrier wall during the vaporization process. How ever, the interaction between the wall with the water droplets in the volume range of interest, l cc or less, is negligibly small, since only a very small fraction of the droplet's periphery touches the barrier wall. Nevertheless, to eliminate the effect of contact with the barrier wall on the experimental evaporation rate, the experimental data to be used in comparison with the theoretical results were taken on a test section withal degree apex angle, as shown in Figure 16. Figure 14 presents a comparison of the total vaporiza tion times as measured on a flat surface and a conical

PAGE 86

71 surface withal degree apex angle. As seen in this figure, there is no noticeable difference in the vaporization times, thereby confirming the earlier observations that the side wall interaction is negligibly small. The test sections were mounted on the base plate shown in Figure 17. Pyrex brand wool and glass wool insulation were packed around the heating plate to reduce the heat loss from the ends of the test plate, thereby giving a more uniform temperature dis tribution across the test plate. A three inch aluminum fence was built around the test section to reduce the convection currents that would tend to flow over the surface because of the free convective heat loss from the test plate. In addition, the fence more nearly produces the condition in which the water droplet is surrounded by saturated vapor, thereby reducing any mass transfer from the top of the droplet. The test plate and base plate were mounted on a 1200 watt, 220 volt electrical heating unit The amount of current to the heating unit was controlled by a variac as shown in Figure 18. Four 20 gauge Chromel-Alumel thermocouples were embedded beneath the test section at positions indicated

PAGE 87

Convection I Force ---Insula tion I o yo o 0 o r\ roo ~c, 0 0 D t1 0 0 0 o oO 'b 0 () 0 0 ,~ 06 oo o o I l)(IJ O ooa,, oO adoOi
PAGE 88

73 220 Volts Four Chromel-Alumel Thermocouples ,,. .,... Fi g 1 8 Test Section ..___ Reference Junction in Ice Bath Potentiometer Honeywell Rubicon (Manually Balanced) Schematic Diagram of Experimental Apparatus.

PAGE 89

74 in Figures 15 and 16. The surface temperatures were found by linearly extrapolating the upper and lower thermocouple reading at the center point of the heat ing plate to the surface of the plate. The linear correction applied to the center thermocouple was also applied to thermocouples near the edge of the plate. The thermocouples were fed through a selector switch to a Honeywell manual potentiometer. A 32 F ice reference junction was used. Distilled water at its saturation temperature was placed onto the surface by means of calibrated pipettes. A 1 ml and a 10 ml pipette were used in the experiment. The 1 ml and 10 ml pipette were read to an accuracy of 0.005 ml, and 0.05 ml respectively. Although the pipettes are calibrated for a liquid at 20 C, heating the pipettes to the saturation tempera ture of water (100 C) does not affect the accuracy of the volume measurements due to the relatively small amount of volumetric thermal expansion involved. The change of volume due to an increase in temperature can be estimated from the relationship VS u T. The volu metric expansion estimated by the previous expression affects the results only if it were possible to measure the volume to four significant figures.

PAGE 90

75 The times in which the distilled water is ejected from the pipette (shown in Figure 19) are short and do not significantly affect the vaporization curves. How ever, because the ejection time is short, the water leaves the pipette in a fine jet, When this jet was allowed to impinge directly onto the surface of the heating plate at 600 For onto the top of a droplet resting on a 600 F plate, the cooling effect of the jet at the point of surface contact (see Figure 20, a and b) initiates nucleate boiling which evaporates a con siderable amount of liquid in a very short time. Consequently, the jet was prevented from impinging di rectly onto the heating surface by directing the jet against a non-wettable surface in the manner shown in Figure 20 c. When the heating plate temperature was set at 1000 F, the cooling effect of the jet did not initiate nucleate boiling; consequently, it was not necessary to use the technique shown in Figure 20 c at this higher temperature. In collecting the data, the variac was first set at a desired value and the equipment was allowed to warm up slowly to a steady state value. Normally, this re quired 2 to 3 hours depending on the required surface temperature. When a run was made, the thermocouple emfs

PAGE 91

76 6 5 CJ Q) Ul 4 .... 8 3 .... 4-J 2 CJ 0-1 ml Q) Pip~tte ... 0-10 ml 1 Pipette 0 0 1 2 3 4 5 6 Ejected Volume {cc) Fig. 19. Ejection Time of Water Droplet from Pipette to the Hot Plate Surface as a Function of the Volume of the Water Droplet.

PAGE 92

Pipette Pipette (a) (b) Non-wettable Surface Pipette / l JJJ llll//lllllJ/7/llllJ/ (c) Fig. 20. Dynamics of Water Jet Ejected from Pipette.

PAGE 93

78 were usually recorded from the Rubicon before the droplet was placed on the plate, some time during the vaporization process, and immediately foll.owing the vaporization. Because of the relatively large amount of time required for vaporization, there was ample time to record all measurements b y hand. The results of the experimental measurements are discussed next

PAGE 94

CHA~ERV EVAPORATION RATES Theoretical The theoretical determination of the evaporation rates for various plate temperatures and volumes requires the solution of Equations (86), {87), and (137) along with the construction of graphs similar to that shown in Figure 9. Equations (86), (87), and (137) were pro g ramed for solution on the IBM 709 digital computer and the compiled results were used in the graphical solutions. The material parameters used in the analysis are tabulated in Appendixes D and F. The theoretical mass evaporation rates are shown in Figure 21 as a function of droplet volume and plate temperature. This figure contains the locus of the graphical solution points (see Figure 9) for various droplet volumes and plate temperatures. As seen in Figure 21, the evaporation rate increases with increas ing volume of the droplet and with increasing plate temperature. The increase in the evaporation rate with increasing droplet volume is due primarily to the 79

PAGE 95

0 (1) {/) '-,,. (1) a 0 -.-t Ill 1-1 0 a. rt1 > ral 8 0 0 021.--,.------,--r---r---r--T --.--~ ---0.010 0.008 0.006 0.00 0.002 0.00 0. 0006~ __.__ -~--~-......._-~-~ __.__ _____._ __..._______, 0 0.2 0.4 0.6 0.8 V (cc) Fig. 21 Theore t ical Mass Evaporation Rate of a Water Droplet as a Function of Volume for a Plate Bmmisivity of 0.5 and Plate Temperatures of 600 F and 1000 F. 1.0

PAGE 96

8l. increase in heat transfer area associated with the increase in the droplet volume, while the increase in the evaporation rate due to an increase in temperature is due primarily to the increase in the thermal con ductivity of the steam. The temperature dependence of the viscosity and specific volume has a relatively slight effect, as seen in Equation (34). Here it is observed that the temperature effect is dampened by the one-quarter power on the absolute film temperature. The calculated gap thickness, as shown in Figure 22, is relatively insensitive to volume changes, but is affected by increased plate temperatures. Experimental In attempting to compare experiment to theory, it is necessary to determine the slope of a curve V = f(t) prescribed by a set of tabulated values (V 1 ti). Pictorially, a tangent line is constructed to a g raphically fitted curve. However, Lipka (24, p. 234) points out that exact or even approximate construction of a tangent line to a curve is difficult and inaccu rate. Reilly (25) suggests finding the slope of tab ulated data by differentiating a polynomial fit of the data.

PAGE 97

82 0.008,------.---~--r--.------. --.----,-------.---T--0.007 0.006 0.005 0.004 0.003 0.002.__ ____._ __ ~----'---~---'--........_ __ _,_ ____._ __ _.__ ___ 0 0.2 0 4 0.6 0.8 1.0 V (cc) Fig. 22. Gap Thickness of the Water Droplet as a Function of Volume for a Plate Emissivity of 0.5 and Plate Tempera tures of 600 F and 1000 F.

PAGE 98

83 The total vaporization time, listed in Appendix E, is used to determine a third order polynomial fit. The fit is performed on the IBM 709 digital computer using the UF-NILLS code which is currently in use by the Nuclear Engineering Department of the University of Florida. The polynomial is of the form V = P(l)t + P(2)t 2 + P(3)t 3 (146} where tis the ti m e required to completely vaporize a d roplet of initial volume, V. The coefficients in Equation (146) are listed in Table 3 as a function of t h e plate tempera t ure. They were determined b minimizing d 1 w i h of v1 t he wei g ht e d squares of the resi ua s. e g ts were used. Temperature TABLE 3 POLYNOMIAL COEFFICIENTS of Plate (F) P(l) P(2) P(3) 6 08 -7.22665 +4. 8 9506 +4.4080 109 1 0 1 4 +4.0022lo5 +l.0295-5 +5.4745-s T h e curves shown in Figure l3 are drawn from Equation 146) usin g the coefficients listed in Table 3. The mass evapora t ion rates a re n ow determined dire ctl y from t he pol yn omial equation (146) by

PAGE 99

84 differentiating it with respect to time and by multipl y ing it by the density of the droplet. Thus, dM dt = PD [ P(l) + 2P 2}t + 3P(3)t2 ] (147} The evaporation rates as calculated from the above equation are plotted in Figure 23 as a function of initial droplet volume and plate temperature along with the theoretical evaporation rates. Comparison of Experiment to Theory The theoretical and experimental results are shown jointly in Figure 23. The emissivities chosen in the theoretical calculations are based on data tabulated in reference (20, p. 47S). As seen in Figure 23, excellent agreement exists throughout the volume and temperature range considered. The devia tion of theory and experiment is less than 5 per cent in the volume range of 0.5 to l cc, while approxi mately 20 per cent at a droplet volume of 0.05 cc. The deviation seen at the lower droplet volumes is probably a result of the increased devia tion of the flat spheroid model from the actual physical situation. The droplet has a greater tendency towards a spherical shape at these lower volumes.

PAGE 100

0.020 0.010 0 00 0 006 85 I ----0 .00 4 0.002 0.001 0.0008 I_ 0 I / > I II I I I I I I / / 0 2 0.4 0.6 V (cc) Tp=600 F c ;-:o.s Theor y E xperiment 0. 8 Fi g 23. Th eoretica l and E x perimental Mass Ev aporati on R ates of Wate r Droplets as a Function of Droplet Volume, Pl ate Temperatures and Plate Em issivity. 1.0

PAGE 101

86 During the vaporization process the droplet tends to vibrate, particularly at the higher plate temperature. The natural modes of vibration of the water droplets are similar to those exhibited by a ring in transverse vibration. However, in the case of a liquid water droplet vibrating, the droplet has a tendency to oscillate at one of its harmonics with the other harmonics being suppressed. The second, third, and fourth harmonic were observed in the exper iments. On the basis of the agreement of experiment and theory, it appears that the oscillations do not significantly affect the rate of evaporation.

PAGE 102

CHAPTER VI OVERALL HEAT TRANSFER COEFFICIENTS The overall heat transfer coefficient, as defined by Equation (134, is now determined from the theoretical evaporation rate. Solving Equation (135) for U u = dM h fg dt (148) where the area is expressed in terms of the droplet volume and thickness Equation ( 7 Q). The above equa tion is eval ua t e d for various droplet v o lumes and plate temperatures of 600 F and 1000 Fusing the theoreti cal mass evaporation rates shown in Figure 21. The results are shown in Figure 24. In the particular examples shown in Figure 24, it is interesting to note that there is a decrease in the overall heat transfer coefficient for an increase in temperature. This is a result of the manner in which U is defined. Although the mass evaporation rate is higher at 1000 F 87

PAGE 103

88 100 r--,-r----,--.----.--60 Tp=600 F N +J Ll,..j ..c: Tp = lOOO F 40 -8 CQ 20 o ~ -~ --~--~--~-~--~-~--~-~ -~ 0 0.2 0.4 0 6 0. 8 1 0 V (c c ) Fig 24 Theoreti ca l H ea t Transf er Coefficient of a Water Droplet as a Function o f Volume for Plate T em peratures of 600 F and 1000 F for a Plate Emissivity of 0.5.

PAGE 104

89 than at 600 F, the temperature difference in Equation { 14 8} is also higher resulting in a net decrease in the overall heat transfer coefficient. The overall heat transfer coefficient increases at lower volu m es as a direct result of the lower gap thickness associated with the smaller droplet volumes. However, there is a net decrease in the mass evapora t i on rate at these lower volumes, as shown in Figure 21, because of the smaller heat transfer areas associ ated with these low volumes. The amount of thermal radiation is calculated to be 2 (BTU/hr ft2 ) at 600 F and 4.4 at 1000 F. Consequently, radiation heat transfer at 600 F repre sents less than 5 per cent of the overall heat t ransfer coefficient; while at 1000 Fit represents less than 10 per cent. As the droplet volume decreases, the percentage of radiative heat transfer decreases s t i 1 further because of the relative increase of c onduction heat transfer which results from the small g a p thickness at the lower droplet volumes. Therefore, in the temperature range under consideration in this dissertation, the results are relatively inseneitive to the radiative parameters of the system.

PAGE 105

90 The magnitude of the heat transfer coefficient is one to two orders of magnitude below the heat trans fer coefficients associated with nucleate boiling which occurs when the temperature of the heated surface is slightly higher than the saturation temper ature of the liquid. The low magnitude of the coefficient accounts for the relatively long lifetime of the droplet on the hot plate.

PAGE 106

CHAPTER VII GRAVITATIONAL EFFECTS With man soon to be traveling to the moon in the next decade, the effects of reduced gravity on the mass evaporation rate were investigated for the particular case of evaporation in an enclosure located on the moon's surface. The environmental conditions inside the enclosure are assumed similar to atmospheric conditions on earth The effect of the gravitational force field on the mass evaporation rate is found by solution of Equations (86), (87), and (137) simultaneously as before. In the problem under consideration here, the value of rfor the moon's surface is 0.16 However, one complication is introduced, in that 1 is a func tion of the gravitational potential. Den Hartog {26, p. 86) points out that the shape of a droplet results as a compromise between the forces due to capillary tension in the surface of the drop and the forces of gravity on the liquid. The capi lary tension tends to make a drop purely spherical as in the case of a freely falli~g droplet, while the 91

PAGE 107

92 action of gravity tends to flatten the droplet as in the case of oil drops being placed on water. The problem now is to calculate the effect of a decrease in the gravitational field on t. The physical situation of a drop resting on a vapor film is analogous to the drop-shaped storage tank in which only membrane stresses are assumed to exist. In the case of the droplet, a thin surface film of uniform tension is formed which envelopes the liquid and prevents it from spreading over the sup porting surface. Timoshenko and Woinowsky-Krieger (27, p. 444 and 445) list the differential governing equations required for the solution of the droplet shape. The solution to this equation is not given explicitly; rather, a nu m erical integration is required. The differential equations, method of solution, and values oft on both the moon's and earth's surface are shown in Appendix F. Using the calculated values of t and r = 0 .16 the ma ss evaporation rate is calculated in the manner described previously. Figure 25 shows the mass evapor ation rate on the moon for a plate temperature of 600 F and an emissivity of 0.5. Also, the mass evaporation rate on the earth for an identical plate temperature and emissivity are shown for comparison purposes.

PAGE 108

0 co ......... O'I Q,) +l rd i::: 0 .... +l rd 1-1 0 rd tO Cl) 0.010 0.008 0.006 0.004 0.002 0.001 0.0008 0.0006 0.0004 0 0.2 93 0.4 0.6 0.8 1.0 V (cc) Fig. 25. Theoretical Evaporation Rates of a Water Droplet in Both the Earth's and Moon's Gravitational Fields for a Plate Temperature of 600 F and an Emissivity of 0.5.

PAGE 109

94 As seen in this figure, the mass evaporation rate is appro x i m ately half of that which exists on the earth for a given droplet volume. This effect is due to the decrease in the heat transfer area because of the smaller gravitational field. The droplet tends to remain spherical rather than pancake shaped as on the earth. It is hoped that experimental verification of the results shown in Figure 25 will be possible in the very near future.

PAGE 110

CHAPTER VIII CONCLUSIONS The mass evaporation rate for a water droplet supported by its own superheated vapor on a flat plate is determined by the simultaneous solution of the momentum, continuity, and energy equations, and by experimental measurements. The rate of vaporiza tion of a water droplet is shown to depend on the volume of the droplet, temperature of the plate, and the gravitational field. The theoretical and experimental mass evaporation rates are found to agree within 20 per cent over a volume range of 0.5 cc to 1.0 cc and over a temperature range of 600 F to 1000 F. In this parameter range, the mass evapora tion rate varies from 0.001 to 0.01 {g/sec The mass evaporation rate is found to increase for increases in either the volume of the water droplet or the plate temperature. Theoretically, the steam flow beneath the droplet is shown to be laminar with a radial velocity distribution which is parabolic in shape, and the 95

PAGE 111

96 steam gap beneath the droplet is shown to range between 0.003 and 0.008 inches. The gap thickness increases for increased volume and temperature, as did the evaporation rate. The shape of the droplet has been calculated by solution of the equilibrium equations based on membrane stress analysis. The theoretical results are in good agreement with experiments. The effect of the gravitational potential on the mass evaporation rate is considred in the theoreti cal development. A reduction in the gravitational potential from 1 (earth} to 0.16 (moon) reduces the mass evaporation rate by approximately half. A velocity correction factor to be used in conjunction with Fourier's steady state one-dimensional heat conduction equation is derived. This parameter takes into account the effect of the flow distribution on the energy transferred from the plate to the drop let. In the temperature and volume range under consideration here, this factor has a 5 per cent effect on the results. A relationship between the mass evaporation rate and the overall heat transfer coefficient is presented. The overall heat transfer coefficient

PAGE 112

97 r ang es between 70 (BTU / hr ft 2 ) for 0.05 cc droplets and 40 ( BTU / hr ft 2 ) for 1 cc droplets in the tempe r a t ure r ang e considered. Finally, it is c on c luded that the analytical m odel d e veloped in this dissertation can be used to predi c t the evaporation rate and overall heat t r ansfer c oeff ici ents for water droplets on a flat plate in t he f ilm b oili ng regime with a reasonable degree o f accu ra cy over the range of parameters investigated.

PAGE 113

APPENDIXES

PAGE 114

APPENDIX A

PAGE 115

REACTIVE FORCE The ejection of mass from the water droplet into the steam g ap below the droplet produces a force on the droplet itself much the same as that produced by a rocket engine. This force is not con sidered in the analysis, since it is negligibly small, as is now shown. Starting from Newton's law l d = ( momentum). (149) For a stationary droplet operating under pseudo-steady state conditions, the above equation takes on the form: w (6) dM = (150) The axial velocity, w{6), is related to the evaporation rate by the continuity equation, which is of the form: dM dt = A p w (6 ) 100 {151)

PAGE 116

101 Therefore substituting the above equation back into Equation ( 1 5 1) / results in 2 FR l (::) (152) = A P gc But, substituting in Equation (70) into the above rela ti onship resul t s in The va l ues of dM dt 2 = (:) (153) are found in Figures 21 and 23 while the other parameters are found in Appendixes D and F Substituting the numerical values for the 0.5 cc droplet example pro b lem of Chapter II into the above equation results in a calculated reactive force of 4 6 x 106 poun ds -force This force is clearly negligible when compared t o the weight of the water droplet.

PAGE 117

APPENDIX B

PAGE 118

SOLUTION OF THE MOMENTUM EQUATION FOR P(r) In attempting to solve a physical problem, it is customary to first use the simplest analytical model and then work up to more sophisticated models in successive steps However, if the pressure is assumed to be solely a function of the radial position, only the trivial solution for the velocity distribution is possible. Assuming P(r), Equation (22} takes on the form 2 f f' = V f 1 1 (154) Differentiating the above equation 2 f f I I + 2 f I 2 = i) t I I I (1.55) Plu g ging Equation (155) into Equation (21) results in 3 12 = a 2 (156) Or, rewriting, f' = constant. {157) The general solu ti on to the above ordinary differential equation is of course of the forms 103

PAGE 119

104 (158) The only solution to the above equation that can satisfy the boundary conditions, Equations (23) and (24), is the trivial solution f = 0 (159) It is really not surprising that the above aP has resulted, since the assumption that = 0 az prevents the operation of an axial pressure gradient which can act as a driving force for the z component of the velocity.

PAGE 120

APPENDIX C

PAGE 121

ANALOG SYMBOLS Summin g Amplifier and Constant Multiplier X ( t ) --~ X(t y ( t ---1 Scale factor Potentiometer Multiplying Device 106 X(t) Y(t ) kl

PAGE 122

X1 ( t )--------tc.: 1 I.C. (Initial Conditions) Integrating Amplifier .... 0 -..J

PAGE 123

APPENDIX D

PAGE 124

PHYSICAL PROPERTIES This appendix contains the numerical values of the physical properties necessary for the evalua tion of the equations presented in the body of this dissertation. Table 4 (19, p. 535) and Table 5 contain some of the properties necessary for the solution of the problem. From the listings in Table 4 Figures 26, 27, and 28 were constructed ehowing the relationship of specific volume, viscosity, and thermal conductivity of steam as a function of temperature. In addition, these figures display some of the numerical values of the constants necessary for substitution into Equations ( 8 2), {83), and (136). 109

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110 TABLE 4 PHYSICAL PROPERTIES OF STEAM AT ATMOSPHERIC PRESSURE T p k Pr (F ) (lbm/ft 3 } (lb /ft (BTU/hr m ft F } sec 212 0.0372 0.870 0.0145 0.96 300 0.032 8 1.000 0.0171 0.95 400 0.02 88 1.130 0.0200 0.94 500 0.0258 1.265 0.0228 0.94 600 0.0233 1.420 0.0257 0.94 700 0.0213 1.555 0.0288 0.93 800 0.0196 1. 700 0.0321 0.92 900 0.01 8 1 1. 8 10 0.0355 0.91 1000 0.0169 1.920 0.0388 0.91 1200 0.0149 2.140 0.0457 0.88 1400 0.0133 2.360 0.0530 0.87 1600 0.0120 2.580 0.0610 0.87 1800 0.0109 2.810 0.06 8 0 0.87 2000 0.0100 3.030 0.0760 0.86

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Quan ti ty hfg D T sat gc N B \I 111 TABLE 5 PHYSICAL PROPERTIES Value Units 970.3 BTU/lbm 59.83 lbr/cu ft 212. F 32.1739 (lbrn ft/lbf sec 2 ) 5 8 .9 (lOOC ). dynes/cm 1.2 x 106 1/C Refer ence (2 8 ) (19) { 28) (19) (29) (29)

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70 60 so E .a 40 ("') 30 20 10 0 0 200 400 600 800 1000 1200 T (R) Fig 26 Specific Volume of Steam at Atmospheric Pressure as a Function of Temperature. 1400 I-' I-' N

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CJ (I) c.o 4-1 e ..a ,-f ll'l 0 ,-f :::t )Q,------,-----r--.----,-----,----,-----r----r---,-----r---.----.---.---.---2 0 10-S /.1= T 10 760 o~----~-~~---~-~ --~ --....._~ __ _._ __ .....__~--~--~-~ 0 200 400 600 800 T (R) 1000 1200 1400 Fig. 27. Viscosity of Steam at Atmospheric Pressure as a Function of Temperature. ..... I-' w

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Ci. .., t 8 ..!.( 0.04~ -----r----.-----,--~--...---,.----,--...---,----,----r----.---,---.--0.03 0.02 k=0.0000305 T-0.006 0.01 o .__ __._ __ ......._ __. __ ___._ __ ....._ ___._ __ _._ __ ......._ __ ____._ __ _.__ __ ...__ ___._ __ ~----o 200 400 600 800 1000 1200 T (R) Fig. 28. Thermal Conductivity of Steam at Atmospheric Pressure as a Function of Temperature. 1400 1-J

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APPENDIX E

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Initial Volume (cc) 1.3 2~9 4.4 1.3 3.7 0.36 1.9 0.72 0.18 0.09 0.19 0.31 0.60 0.97 116 EXPERIMENTAL DATA TABLE 6 DROPLET VAPORIZATION TIMES Vaporization Time (sec) Initial Volume (cc) Vaporization Time (sec) Average Plate Temperature 602 F Plate Surface 304ss. flat, satin finish 417 0.52 294 547 1.0 374 636 2.6 535 430 3.3 586 615 4.2 602 264 0.045 101 480 1.6 479 331 2.25 522 181 2.0 493 140 2 9 567 193 3.75 578 236 0.055 115 321 0.04 89 400 0.92 390 Average Plate Temperature 605 F Plate Surface Graphite, flat, satin finish 0.44 250 1.5 403 0.95 370 3.5 577 0.51 280 2.4 499 0.80 342 1.75 453 1.1 375 4.5 602 2.8 550

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117 TABLE 6 (Cont'd) Initial Vaporization Initial Vaporization Volume Time Volume Time (cc) (sec) (cc) Average Plate Temperature 608 F Plate Surface 304ss, 1 deg. apex angle, satin finish 0.33 229 0.98 395 0.45 286 0.87 380 0.90 386 0.72 337 0.10 147 0 .61 303 0.60 314 0.11 140 0.20 205 0.045 100 0.05 100 0.05 102 0.83 345 0.42 282 Average Plate Temperature 1014 F Plate Surface 304ss, l deg. apex angle, satin finish 0.39 0.96 0.81 0.32 0 .60 0.14 0.08 0.06 0.05 0.04 143 213 198 139 186 96 75 61 57 55 0.59 0.80 0.97 0.33 0.26 0 .20 0.50 0.49 0.69 0.89 172 192 203 133 123 114 155 154 178 200

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APPENDIX F

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DROPLET SHAPE UNDER VARIABLE GRAVITATION The shape of a liquid droplet with a surface tension, N, in a gravitational field, r, can be found by solution of the equilibrium equation (27, p. 444) 1 1 6 P + = (160) N for the droplet shown in Figure 29. The notation used in reference (27) is altered slightly for conven ience in the problem at hand. By expressing r 1 and r 2 in terms of e and introducing the new variables j = sin e (161) t:iP {162) the equilibrium equation takes on the forms dj j p D r z + = (163) dx X N dz j = (164) dx j2/~ 1 119

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120 X 0 X--y -----.-t-----------t-----------~ d z tA Fig. 29. Schematic of Water Droplet.

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121 Equations (163) and (164) are integrated numerically starting from the upper point A shown in Figure 29. At this point, r 1 = r 2 from symmetry, and Equation {160} takes on the form 2N = (165) Po r z For a given value of the surface tension, N, the shape of the droplet is now solely determined by the pressure at point A which is directly related to the radius at point A by the above equation. The pres sure and radius at point A are set by arbitrarily letting z = d at point A. Therefore, 2N = (166) Po r a If dis chosen small, rA becomes large and the droplet takes on a pancake or flat spheroidal shape. If dis chosen large, rA becomes small and the droplet takes on a spherical shape By choosing numerical values of d and solving each problem, the shape of the droplet for various volumes is determined Further simplifications to the equations occurs if

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122 = (167) Starting with an assumed value of d, the integration is begun by making the first element of the meridian curve r 1 6 e = x. At the end of this arc, see Figure 30, = { (6x) 2 ) zl + 4E 2 (168) d 6 x jl = 2E 2 (169) After the values of z 1 and J 1 are found from the above equations, as shown in Figure 30, the com plete solution is marched out by means of Equations (163) and (164 as follows: j i+l = ji + = + dj dx i dz dx i dz dx i 6X {170) 1:t 6X 171)

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123 1 o i Fig. 30 Path of Numerical Integration

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124 where Equations (163) and (164) take on the forms dj I zi ji '{ 172) = dx i E2 xi dz I = dx i j. l. (173) dz I = dx i -j. l. (90c::: a -==:1ao }. (174) Note that in this numerical integration e =90 results in a calculational singularity; consequently, this point is excluded from the integration. The position xis defined by dz I xi+l = xi + dx i ~x (175) 1 : 1 l. The calculations are continued up to the point B, where the meridian curve has the horizontal tangent, BC.

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125 The volume of the droplet is readily computed during the integration b y the following summation: n V = r (176) i = l The average droplet thickness is calculated from the relationship V 2. = (177) ,,. (x } 2 i,max The preceding equations were programed on the IBM 709 The results are shown in Figures 31 and 32 In Figure 31, the calculated values of 2. are shown for values of r = l and r = O .16. Included in this figure are data points based on measured values of V and xL,max from reference (2.). As can be seen from this figure there is good agreement between the calculated and measured values of .e. for the case of r = 1. Con sequen t ly, it is expected that the calculated values of 2. for r = O .16 will be equally accurate. The r = l curve is replotted in Figure 32 for easier determina tion oft at lower volumes.

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0 o,l 1.0 0.8 0.6 0.4 0.2 0 0 1.0 126 r =O .16 r=l.O 0 o Measurements Reference UJ 2.0 3.0 V (cc) 4.0 Fig. 31. Thickness of Water Spheroid as a Function of its Volume for r =l. O and r =0 .16. 5.0

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= CJ ol 127 o.s 0.4 r; 1 0.3 0.2 0.1 0 0 0.04 0.08 0.12 0.16 V (cc) Fig. 32. Thickness of Water Spheroid as a Function of its Volume for r =l. 0.20

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LIST OF REFERENCES 1. Adadevoh, John K., Uyehara, O. A., and Myers, P. S. Droplet Vaporization Under Pressure on a Bot Surface A paper presented at the international summer meeting of the Society of Automotive Engineers June 10-14, 1963, 701B. 2. Borishansky, V. M., Zamyatnin, M. M., Kutateladze, S. s., and Nemchinsky, A. L., "Heat Exchange in the Quenching of Metal Parts in Liquid Media," Problems of Heat Transfer During a Change of State: A Collection of Articles, AEC-tr-3405, 1953. 3. Leidenfrost, J. G., De aguae communis nonnullis gualitatibus tractatus, Duisburg, 1756 (as cited by reference 10). 4. Gottfried, B. s., "The Evaporation of Small Drops on a Flat Plate in the Film Boiling Regime," Case Institute of Technology, Ph.D. Thesis, 1962. 5. Borishansky, V. M., "Heat Transfer to a Liquid Freely Flowing Over a Surface Heated to a Temperature Above the Boiling Point," Problems of Heat Transfer During a Change of State: A Collection of Articles, AEC-tr 3405, 1953. 6. Kutateladze, S. S., Fundamentals of Heat Transfer, Academic Press, Inc New York, 1963. 7 Wade, S. H "Evaporation of Liquids in Currents of Air," Transactions of Chemical Engineering Society, Vol. 20, 1942. 8. Eckert, E. R. G., "Research During the Last Decade on Forced Convective Heat Transfer," Lecture presented at the 1961 International Heat Trans fer Conference held August 28-September 1, 1961, University of Colorado, Boulder, Colorado, USA 128

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129 9. Hartnett, J. P. and Eckert, E. R. G., "Mass-Transfer Cooling in a Laminar Boundary Layer with Constant Fluid Properties," Transactions of the American Society of Mechanical Engineers, ASME Paper 55-A-108, 1955. 10. Grober, H., Erk, S. and Grigull, U., Fundamentals of Heat Transfer, McGraw-Hill Book Company, Inc., New York, 1961. 11. Schlichting, H., Boundary Layer Theory, McGraw-.Hill Book Company, Inc., New York, 1960. 12. Raven, F. H., Automatic Control Engineering, McGraw Hill Book Company, Inc., New York, 1961. 13. Korn, G. A. and Korn, T. M., Electric Analog Com puters, McGraw-Hill Book Company, Inc., New York, 1956. 14. Paynter, H. M., A Palimpsest on the Electronic Analog Art, George A. Philbrick Researches, Inc., Boston1 1955. 15. Prandtl, L. and Tietjens, o. G., Fundamentals of Hydroand Aeromechanics, Dover Publications, Inc., New York, 1957. 16. Abramowitz, M. and Stegun, I. A., Handbook of Mathe matical Functions, National Bureau of Standards, AMS-55, June 1964. 17. Soklonikoff, I. s. and Redheffer, R. M., Mathematics of Physics and Modern Engineering, McGraw-Hill Book Company, Inc., New York, 1958. 18. Brown, A. I. and Marco, S. M., Introduction to Heat Transfer, 2nd ed., McGraw-Hill Book Company, Inc., New York, 1951. 19. Kreith, F., Principles of Heat Transfer, InternationaJ Textbook Company, Scranton, 1963. 20. McAdams, William H., Heat Transmission, 3rd ed., McGraw-Hill Book Company, Inc., New York, 1954.

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130 21. Bennett, C. O. and Myers, J.E., Momentum, Heat, and Mass Transfer, McGraw-Hill Book Company, Inc., New York, 1962. 22. Bird, R. B., Stewart, W. E., and Lightfoot, E. N., Transport Phenomena, John Wiley & Sons, Inc., New York, 1963. 23 .. Vennard, John K., Elementary Fluid Mechanics, 3rd ed., John Wiley & Sons, Inc., New York, 1955. 24. Lipka, Joseph, Graphical and Mechanical Computation, 5th ed., John Wiley & Sons, Inc., London, 1918. 25. Reilly, H.J., Plum Brook Reactor Post-Neutron Tests: Part VII Temperature Coefficient, PBR-21, February 1962. 26. Den Hartog, J.P., Advanced Strength of Materials, McGraw-Hill Book Company, Inc., New York, 1952. 27. Timoshenko, Stephen P. and Woinowsky-Krieger, S., Theory of Plates and Shells, 2nd ed., McGraw Hill Book Company, Inc., 1959. 28. Keenan, Joseph H. and Keyes, Frederick G., Thermo dynamic Properties of Steam, John Wiley & Sons, Inc., New York, 1937. 29. Sears, Francis W. and Zemansky, Mark W., College Physics, Part 1, 2nd ed., Addison-Wesley, Inc., Cambridge, Mass., 1952.

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BIOGRAPHICAL SKETCH Kenneth Joseph Baumeister was born in Cleveland, Ohio, on July 7, 1935. In June, 1957, he received the degree of Bachelor of Engineering in Mechanical Engineer ing from Case Institute of Technology. After graduation from Case, Mr. Baumeister was employed at the NACA Lewis Research Center, Cleveland, Ohio, for two years. In 1959, he was transferred to the NASA Plum Brook Reactor Facility in Sandusky, Ohio. While employed from 1959 to 1962, he attended night classes at the University of Toledo to work towards the degree of Master of Science in Mechanical Engineering, which was granted him in June, 1962. From September, 1962, to January, 1963, he was employed as a part time instructor in the University of Toledo's Department of Physics. Mr. Baumeister enrolled in the Graduate School of the University of Florida in January, 1963, where he was awarded a Ford Foundation Fellowship. Until the present time, he has pursued his work towards the degree of Doctor of Philosophy, and has worked as a teaching assistant in the Department of Nuclear Engineering. Mr. Baumeister is a member of Phi Kappa Phi and is a registered Professional Engineer-in-Training in the State of Ohio. He is married to the former Mary Colette Reitz and is the father of three children. 131

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This dissertation was prepared under the direction of the chairman of the candidate's supervisory committee and has been approved by all the members of the committee. It was submitted to the Dean of the College of Engineer ing and to the Graduate Council and was approved as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December 19, 1964 c-/464: ~ .a: ~ Dean, College of Engineering Dean, Graduate School Supervisory Committee: