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, sec1
Transformation variable, ft sec1
Acceleration of gravity, ft sec2
Dimensional conversion factor 
32.1739 ft Ib lb f sec2
m f
Latent heat of evaporation, BTU IbM 
Thermal conductivity, BTU hr1 ft1 Fl
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, F or R
Time, sec
Overall heat transfer coefficient,
BTU hr~1 ft2 F1
Radial velocity, ft sec1
Droplet volume, cc
Axial velocity, ft secl
x
Symbol
OC Thermal diffusivity, ft2 sec1
r Computer proportionality constant, volt1
6 Steam gap thickness, in
C Emissivity for radiation
Dimensionless coordinate
S Computer proportionality constant, volt1
Material constant, in sec1 (lbf/lbr) cm2 x
3 3
R cm
U Absolute viscosity, lb ft1 sec1
Kinematic viscosity, ft2 secI
) Density, lbm ft3
Computer proportionality constant, sec1
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 sec1 (Ib/lbf)1 x
cm R cm9/4 volt1
( 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 nonlinear differential
equations by the method of combination of variables.
Possible solutions to the nonlinear 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 semiempirical 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
semiempirical 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 semiempirical 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 semiempirical 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 nonlinear 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
nonlinear 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 nonlinear 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 onedimensional 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)
/ ^
/ <
C0
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 abovementioned 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 steadystate condition;
that is, the flow is approximated by a steadystate
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 abovementioned equations into a set of
ordinary nonlinear 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 nonlinear 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 nonlinear 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
abovementioned 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,gc (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 ) (3.281x102) x453.6
x (3.531xl05) (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 ) (3.281x102) x(.3531x104) 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)
d2 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 (1.076x103) (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 (1.076x103)
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, Tp600 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
twothirds 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
Brasshighly 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 %
 13/4 
I k 1
11/8
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
ll/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 ChromelAlumel 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 ChromelAlumel 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 nonwettable 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
E4 3
4+ 2
o 01 ml
> Pipette
M 1 010 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 onequarter 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 UFNILLS 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 V1
were used.
TABLE 3
POLYNOMIAL COEFFICIENTS
Temperature
of Plate (F) P(l) P(2) P(3)
608 7.2266105 +4.8950106 +4.40d0109
1014 +4.0022105 +1.0295105 +5.4745108
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
