Title: Heat transfer to water droplets on a flat plate in the film boiling regime
CITATION THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00098225/00001
 Material Information
Title: Heat transfer to water droplets on a flat plate in the film boiling regime
Physical Description: xiv, 131 leaves : illus. ; 28 cm.
Language: English
Creator: Baumeister, Kenneth Joseph, 1935-
Publication Date: 1964
Copyright Date: 1964
 Subjects
Subjects / Keywords: Water -- Thermal properties   ( lcsh )
Heat -- Transmission   ( lcsh )
Nuclear Engineering Sciences thesis Ph. D   ( lcsh )
Dissertations, Academic -- Nuclear Engineering Sciences -- UF   ( lcsh )
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis - University of Florida.
Bibliography: Bibliography: leaves 128-130.
General Note: Manuscript copy.
General Note: Vita.
 Record Information
Bibliographic ID: UF00098225
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000559275
oclc - 13453514
notis - ACY4724

Full Text












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




University of Florida Home Page
© 2004 - 2010 University of Florida George A. Smathers Libraries.
All rights reserved.

Acceptable Use, Copyright, and Disclaimer Statement
Last updated October 10, 2010 - Version 2.9.9 - mvs