A11102 5 1 1 1 7 5

NA

j'llillJInillPr STANDARDS & TECH R.l.c

Evaporation of A Water Droplet

Deposited on A Hot High Thermal

Conductivity Solid Surface

Marino di Marzo

Mechanical Engineering Department

University of Maryland

College Park, MD 20742

David D. Evans

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Gaithersburg, MD 20899

August 1 986

U S. DEPARTMENT OF COMMERCE

national bureau of standards

-QC

inn

.1156

86-3384

1986

C.2

NBSIR 86-3384

EVAPORATION OF A WATER DROPLET

DEPOSITED ON A HOT HIGH THERMAL

CONDUCTIVITY SOLID SURFACE

NBS

RESEARCH

INFORMATION CENTER

&UOO

f,

IH%

L.l

Marino di Marzo

Mechanical Engineering Department

University of Maryland

College Park, MD 20742

David D. Evans

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Gaithersburg, MD 20899

August 1986

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler. Director
-

'

.
TABLE OF CONTENTS

Page

LIST OF FIGURES iv

Abstract 1

1. INTRODUCTION 2

2. PHENOMENOLOGY 3

2.1 Radius of the Wetted Region * 4

2.2 Droplet Shape 6

3. THEORETICAL EVAPORATION TIME 8

3.1 Solid Surface Temperature 9

3.2 Liquid-Vapor Interfacial Molar Fraction 9

3.3 Model Formulation 11

4. MODEL VALIDATION 12

4.1 Evaporation Time 12

4.2 Detailed Features 13

5. CONCLUSIONS 15

6. ACKNOWLEDGMENTS 16

7. REFERENCES 16

8. NOMENCLATURE 17

-iii-
LIST OF FIGURES

Page

Figure 1. Typical droplet evaporation behavior 19

Figure 2. Characterization of the parameter 20

Figure 3. Solid-liquid angle of contact versus the wetting

parameter 0^ 21

Figure 4. Characterization of the convective heat transfer

coefficient 22

Figure 5. Evaporation time as a function of the initial solid

surface temperature and droplet volume 23

Figure 6. Evaporation time: theory versus experiments 24

Figure 7. Evaporative heat flux comparison 23

Figure 8. Liquid water inventory versus time 26

Figure 9. Temporal heat flux behavior

Figure 10. Spacial heat flux behavior 28

-iv-
EVAPORATION OF A WATER DROPLET DEPOSITED ON A HOT HIGH

THERMAL CONDUCTIVITY SOLID SURFACE

David D. Evans and Marino di Marzo

Abstract

A model is presented that predicts major features of the evaporation of

water droplets deposited on a hot non-porous solid surface. In the temperature range of interest, nucleate boiling heat transfer is fully suppressed,

hence the model is only concerned with the evaporative process. In the model,

the solid material is assumed to have high thermal conductivity and

diffusivity, so that the surface temperature under the water droplet can be

considered uniform. The temperature of this portion of a larger solid surface

covered by the liquid is calculated from the classic solution for contact

temperature between two semi-infinite bodies. The liquid-vapor interfacial

temperature and the water-vapor molar fraction in the air at the exposed

surface of the water droplet are deduced from the coupled heat and mass

transfer energy balance at the interface. Spatial and temporal integration of

the overall droplet energy equation is used to predict the droplet evaporation

time and the instantaneous evaporation rate. Model predictions for the total

evaporation time and temporal variation of the droplet volume agree well with

experiments performed using a heated aluminum block. The model is used to

quantify spacial and temporal heat fluxes distribution at the exposed surface

of the water droplet.

Key words: cooling, droplets, evaporation cooling, extinguishing, spray

quenching, sprinkler system, water sprays.

- 1 -
1 . INTRODUCTION

Many studies have been performed to quantify the vaporization process for

both single droplets and multiple droplet arrays impacting on hot surfaces.

For the studies found in the published literature, the full span of the

droplet vaporization processes including boiling phenomena are usually

reported. These would include evaporation, nucleate boiling, film boiling,

and Leidenfrost transition. This present investigation is more limited in the

span of vaporization processes studied, being only concerned with the evaporation of a single droplet on a hot surface. The study does report very

detailed results for spatial and temporal variation of the heat flux at the

exposed surface of the droplet, and for temporal variation of the droplet

volume. Exclusive study of the evaporative process implies that conditions

are maintained under which nucleate boiling is fully suppressed.

This detailed study of water droplet evaporation is intended to form the

basis for a simple thermal model of solid fuel fire extinguishment. Depending

on the type of solid fuel that is burning, the surface may take many different

configurations having greatly different responses to water droplets impact.

Common natural solid fuels produce low thermal conductivity chars. Some

manufactured materials such as thermoplastics may form a shallow liquid film

on the solid surface, others produce a non-porous surface. This work Is a

first phase of an extensive research program concerned with a series of conditions that must be evaluated in order to construct accurate droplet cooling

models for the many different surfaces that may be generated by burning of

solid fuels.

-2 -
The thermal conductivity/diffusivity of common solid fuels is very low

compared to those of metals. This implies that the local cooling induced by

the droplet evaporation is more severe. The heat fluxes from the solid under

the droplet are not large enough to sustain nucleate boiling, hence evaporation is the mode of heat transfer to be considered. In this first investigation, aluminum was selected as the solid material for the purpose of comparing

the experimental results with those of other investigators. At present,

abundant information is available only for droplet deposited on metal bodies.

Results of several previous investigations of droplet evaporation have

been drawn upon in the process of formulating analytical portions of the model

for droplet evaporation presented in this study. Seki [1] has demonstrated

good agreement between the theoretically based computations of the solidliquid contact temperature and experimental measurements. The works of

Michiyoshi and Makino [2,3,4] establish the thermal behavior of the droplets

during the evaporative process. Zhang [5] describes the droplet shape as

spherical segments, which proved to be a rather accurate description for

droplets observed in this study. The model for the droplet evaporative

process presented in this paper represents improvements to the previous models

formulated by Bonacina [6] and Rizza [7].

2PHENOMENOLOGY

A spherical droplet upon impacting a solid surface spreads on it. The

final configuration of the liquid varies a great deal and depends on a multitude of parameters. For the case of liquid water impinging a surface at near

water saturation temperature, the shape can be regarded as a spherical segment

-3 -
[5J. Note that, in this study, the droplet is considered to be homogeneous in

the liquid phase at all times, and evaporation at the liquid-vapor interface

is the only means of generating the vapor phase. Experimentally, small

stationary bubbles are observed at times on the solid surface under the

droplet spherical segment [2]. However, these small bubbles are minimal in

volume and don't seem to affect the evaporation process significantly.

2.1 Radius of the Wetted Region

The radius of the wetted area is a very important parameter, as

recognized by Bonicina [6] and other investigators. In the literature, the

radius of the wetted area is non-dimensionalized with the radius of the

equivalent initial liquid volume spherical shape forming the parameter $ . o For droplets "softly" deposited on the surface (droplet release from height

less than 1 centimeter), a value of 3 q

in the range 1.2 to 1.5 is observed.

However, for sprayed water, several investigators have reported values of 6 o up to 4.5 [6,7]. In this study the interest is on "softly" deposited droplet.

It is crucial to note that these values of 6 are referred to the initial

o condition at deposition before evaporation begins (t = 0). Values for the

more general parameter 3 based on the liquid volume at any time during the

evaporation process can be calculated at any stage of the evaporative process

using as a reference the radius of the spherical droplet of equal volume.

Figure 1 depicts the various phases of the evaporation of a "softly" deposited

droplet. Note that the radius of the wetted area is constant throughout most

(90 to 95 percent) of the evaporative process .

-4 -
The importance of the parameter 3 is that it accounts for the effects of

a number of variables that characterize the surface conditions (wettability,

roughness, etc.), the fluid (surface tension, wetting ability, etc.) and the

droplet deposition process. All the experimental data were obtained for pure,

degassed and de-ionized water droplets deposited on an aluminum block with the

exposed source coated with a very thin (- 0.06 pm) chromium layer. The

chromium was vapor deposited on the aluminum to obtain a polished (roughness

of 0.1 - 0.15 pm), scratch resistance surface. Prior to each droplet deposition the surface was cleaned with ethyl alcohol in order to remove particles

and grease. In spite of this careful procedure some scatter in the experimental data was observed. This scatter is due mainly to the deposition technique, in which there was some uncertainty in positioning the hypodermic

needle used to dispense the water above the surface. Due to the

advancing/ receding angles hysteresis, when the needle is retracted, in some

occasions the angle of contact between the liquid and the solid was less than

the advancing angle (e.g., the largest angle possible). In order to obtain a

meaningful information for the value of 3 q

, a large data base was compiled.

For each temperature considered (7 values between 75 and 100°C) and for each

—9 3 volume considered (5 between 10 and 50 x 10 m ) ten values of 3 were

o collected for a total of 350 data points. No systematic variation with volume

was found because, in the limited range of droplet volume of interest, the

values of 3q

obtained for the various volumes were not clearly distinguishable .

The temperature dependence of 3 q was more evident and, as illustrated in

Fig. 2, the mean values and the standard deviation of the wetting parameter

are plotted versus the initial solid surface temperature. Most of the data

- 5 -
especially in the low temperature range lies in the five percent band about

the linear regression of the average values which is given by

B os= 0.009 T + 0.6 (1) v ' o

This analytical representation will be used in the following to provide values

for the wetting parameter 3 q

. The range of droplet sizes of interest to this

analysis is dictated by the actual size of the droplets delivered by the fire

suppression equipment.

2.2 Droplet Shape

An analytical formulation defining the surface of a spherical segment in

cylindrical coordinates is given by

y = (B

2 - r2

) 1/2 - A (2)

where A and B are constants. By equating the volume of the droplet in

spherical configuration to the volume of the droplet in spherical segment

configuration (and by imposing that at the droplet outer edge its thickness is

zero), the thickness of the droplet at its apex, a, can be written as

a = R [4/B

3 + (l + 16/86

) 1/2

] 1/3 + [4/g

3 - (l + 1 6 / 0 6

)

1 /2

]

1 /3 (3)

With these two conditions obtain

(A)

- 6 -
and

B = (|) + 1r ) (5)

where Y equals a/R. Therefore, by introducing non-dimensional coordinates,

the equation for the droplet shape can be written as

(1/Y + Y ) 2_2

s -

5

z

where s is y/R and z is r/R.

(1/Y - Y)

( 6 )

In order to evaluate how well the droplet shape agrees with the assumed

spherical segment shape, the solid-liquid angle of contact a is calculated and

compared with the measured values from photographs of water droplets "softly"

deposited on a polished aluminum surface. By differentiating Eq. (6) with

respect to z, at z = 1, and substituting the value for Y obtained from Eq.

(3), the angle a is evaluated as

0 = arctg

1 (1/Y + Y ) 2 4

(7)

Figure 3 shows that good agreement is found between measured and computed

values for droplets just deposited on the surface and during the evaporation

process. This occurrence was also reported by Zhang [5] in his optical

investigation of the droplet behavior.

- 7 -
3. THEORETICAL EVAPORATION TIME

The high thermal conductivity and diffusivity of the metal solid suggest

that the temperature of the solid surface under the droplet can be assumed

constant during the evaporative process. This assumption implicitly uncouples

the solid substrate properties from the evaporation process. For the case of

a low conductivity/dif fusivity material it is anticipated that such uncoupling

will not be possible.

The assumption that B q

is a function of the temperature only is less

justifiable, however, two points need to be mentioned in this respect:

(a) the measured dependence of 3 q on the volume is rather erratic.

Variations are introduced by different local surface conditions, and

by different droplet deposition procedures. In spite of numerous

experiments, a clearly defined dependence of 3 q on any parameter but

temperature.

(b) the evaporation time is not an extremely strong function of B q

. A

rationale for this can be explained by noting that, as the liquid

evaporates, 3 increases and its value quickly becomes independent of

the initial conditions. The evaporation rate in the initial portion

of the process constitutes a minor contribution to the overall

phenomenon (as will be shown); it will become clear that any initial

variations of the parameter 3 are rapidly eliminated and do not

influence the overall process significantly.

- 8 -
3.1 Solid Surface Temperature

Seki [1] gives an interesting method to evaluate the solid surface

temperature in the wetted region. By referring to the semi-infinite body

solution for sudden surface temperature change, he evaluates the surface

temperature as

T /C p k + T /Cpk so s S S WO WWW

/C p k + /C p k

s s s WWW

( 8 )

The temperature for water near saturation (on metallic surfaces) behaves in

very good agreement with this model as documented in Seki's work [1]. Note

that Michiyoshi confirms that the liquid-solid interfacial temperature can be

considered constant in the evaporation process [2,4].

3.2 Liquid-Vapor Interfacial Molar Fraction

In order to evaluate the liquid-vapor interfacial molar fraction, mass

transfer as well as heat transfer must be considered. The heat balance at the

interface is

W A M = h (T - T ) w

v i a'

(9)

where W is the molar flux of the vapor at the water-air interface, the molar

flux is due to the contribution of the diffusive and overall fluxes; that is

W = hm

(Pa/M

a Hx

i x ) + x,Wa

J i

( 10 )

- 9-
By rearranging and by noting that the heat is transferred through the liquid

only by conduction, one obtains

Rs

X, - X

i a

.

1 " X

i_

p n ii a m M

L a J

(ID

The Chilton-Colburn analogy [8] relates the mass transfer coefficient hm

to

the convective heat transfer coefficient h by the following expression

in pc

a a

By combining Eqs. (11) and (12) and noting that the water-air molecular weight

ratio is 0.624, yields

2/3

( 12 )

k^ ( t t ^ — n di/i D

2/3

"A x. - X

i a Rhs l s V °* 6 " 4

a c 1 - x

a L aj L ij

Note that the only unknown is x^, since the interfacial temperature is the

saturation temperature of steam at the pressure corresponding to its molar

fraction. For a given liquid layer thickness (R • s), the molar fraction can

be determined via an iteration procedure. Furthermore, the left hand side of

this equation is finite at r = R if the temperature at the droplet exposed

surface coincides with the solid surface temperature. This condition seems

reasonable and it is used to determine the vapor molar fraction in the air at

the outer edge of the droplet.

The convective term accounts for less than 2 percent of the evaporative

heat transfer, therefore, it could be neglected; however, the convective heat

transfer coefficient is paramount because it determines the mass transfer

- 10-
coefficient. The heat flux through the block is determined measuring the

temperature gradient across a layer of glass of known thermal conductivity

inserted under the aluminum block. The lateral surface of the glass layer and

of the aluminum block are insulated to insure one dimensional heat flux in the

vertical direction. A silicon heat sink compound paste is used to guarantee

effective contact between the glass and the aluminum block. With this

procedure the heat flux is measured within 5-6 percent accuracy. The block

surface temperature and the ambient temperature are measured to determine the

convective heat transfer coefficient once the heat flux is known. The experimental measurements for the convective heat transfer coefficient are

illustrated in Fig. 4. The linear regression of these data is given by

Note that this is the solid surface to air heat transfer coefficient and it is

assumed that the droplet does not affect greatly the convection pattern

associated with the much larger heated solid surface. The radiation contribution is in the order of 4 percent and it is accounted for in the experimental

determination of h.

An overall heat balance on the droplet can be written for the small time

interval dt as

h = 0.098 T + 2.8 (14)

s o

3.3 Model Formulation

l2nR.2 j q z dz} dt = - A p dV

A e

J w

(15)

- 11 -
where q e is the evaporative heat flux, that is

q = 0.624

e

x

a

The instantaneous rate of evaporation is then

dV 2 rrR2 (0.624) h D 2/3 1

f x, - X

i a dt P c a

J w a a 0 L iJ

is function both of z and t.

( 16 )

(17)

The initial volume being known, a numerical technique allows one to

determine the total evaporation time from Eq . (17). Note that at each time

step the evaporative mass flux is obtained by computing the integral in the

right hand side of the equation.

4. MODEL VALIDATION

The details of the experimental set up and of the data acquisition are

described in detail in reference [9].

4.1 Evaporation Time

Figure 5 illustrates the comparison of results between the computations

of the model developed in this study and experiments. Good agreement Is

observed. The discrepancies at the high temperatures and Large droplets

diameters are due to a number of factors. The influence of the droplet on the

convective heat transfer coefficient and slight variation in the solid-liquid

- 12-
interfacial temperature coupled with uncertainties in the various experimental

data acquisition are the most probable factors.

Figure 6 shows a comparison of the total evaporation time measured and

calculated for a large number of experiments conducted in this study as well

as in others. In all these experiments, the droplet volume range between 10

—9

and 50 x 10 cubic meters, and the initial surface temperature is in the

range 75 to 100°C.

In Fig. 7 experimental data by Michiyoshi and Makino [2,3] are compared

with experimental data of Trehan [9] and with the theoretical computations.

—9

The data by Michiyoshi and Makino are for droplet volumes between 9 x 10 and

9 3 —9 3 19 x 10 m , the other data are for droplets in the range 20 x 10 m to

—9 3 50 x 10 m in volume. All this information is concerned with the average

heat flux for various initial surface temperatures. The most important

finding is that the solid material (e.g., aluminum, copper, steel) do not have

a strong influence on the phenomena. The theory provides an extremely good

representation of the data by Michiyoshi and Makino. The exponential behavior

of the heat flux (hence of the time) with respect to the initial surface

temperature must be noted. The volume dependence is more complex but less

strong.

4.2 Detailed Features

To further validate the model, the normalized liquid volume is plotted

versus the time, non-dimensionalized with respect to the total evaporation

time, in Fig. 8. Very good agreement is found for the data which were

obtained from measurements of the photographic sequence shown in Fig. 1.

- 13 -
It is interesting to deduce information concerning the temporal and

spacial distribution of the heat flux on the droplet-to-air interface from the

model calculations. Figure 9 shows model predictions of the heat flux variations during the evaporation process. The corresponding spatial heat flux

distribution is presented in Fig. 10. The high heat flux at the droplet outer

edge is evident. Note that for three quarters of the total evaporation time

the central portion of the droplet liquid-vapor interface does not contribute

much to the process; in the last quarter, the dramatic flux increase is

produced by the contribution of most of the droplet surface because the

reduced thickness of the droplet enhances the heat transfer by conduction in

the liquid layer. The predominant contribution of the outer edge of the

droplet to the evaporative process is well documented in the literature and it

is often associated with the Maragoni effect [10].

Michiyoshi [4] observed that the temperatures at the solid surface and at

a fixed point above it (in the liquid) are constant during the evaporation

process. This face implies that the heat flux is also constant. The results

just shown confirm that during most of the evaporation process, in the region

close to the droplet symmetry axis, the heat flux does not vary significantly.

These results also show that the heat flux at the droplet edge can be as

high as three times the spatial average heat flux at the beginning of the

evaporative process. As the phenomenon progresses, the spatial average heat

flux increases but the flux distribution in the radial direction becomes less

important. Note that the droplet under consideration has a radius of about

3 x 10“3 m and that a gradient of 1 °C across that distance produces a heat

flux of more than 40,000 W/m-2 in aluminum. It follows that the hypothesis of

- 14 -
uniform temperature distribution under the droplet seems reasonable since the

temporal and spacial average heat flux for this case is of about 50,000 W/m .

The application of this simple boundary condition to the case of low thermal

conductivity/dif fusivity solid materials is not possible and a significant

temperature distribution in the radial direction under the droplet is

expected. To solve this problem the droplet energy equation must be coupled

with the solid energy equation at the liquid-solid interface in order to

obtain a solution that accounts for the temperature distribution under the

droplet

5. CONCLUSIONS

The evaporation of droplets in contact with hot, high thermal

conductivity/diffusivity solid surfaces is studied to develop a model of the

evaporative process. A model based on the assumption of uniform solid surface

temperature is formulated for applications to droplets deposited on solids

with large thermal conductivity and diffusivity. The droplet shape is

successfully characterized using a spherical segment. The energy equation

written for the droplet and at the liquid-vapor interface allows the determination of the evaporation time in reasonable agreement with experimental

data. Further validation of the model is presented to describe the spatial

and temporal behavior of the evaporative heat flux.

- 15-
6. ACKNOWLEDGMENTS

This research is supported by a grant of the Center for Fire Research of

the National Bureau of Standards. The authors are deeply indebted to

Professor Michiyoshi for his prompt and courteous comments on the experimental

data reported in his papers. The Computer Science Center of the University of

Maryland provided funding for the numerical computation.

7. REFERENCES

[1] Seki, M., Kawamura, H., and Sanokawa, K., Transient Temperature Profile

of a Hot Wall Due to an Impinging Liquid Droplet, J. Heat Transfer, 100,

176 (1978).

[2] Michiyoshi, I. and Makino, K. , Heat Transfer Characteristics of

Evaporation of a Liquid Droplet on Heated Surfaces, Int. J. Heat Mass

Transfer, 21 , 605 (1978).

[3] Makino, K. and Michiyoshi, I., Effects of the Initial Size of Water

Droplet on Its Evaporation on Heated Surfaces, Int. J. Heat Mass

Transfer, 22 , 979 (1979).

[4] Michiyoshi, I., private communication (1985).

[5] Zhang, N. and Yang, W.J., Natural Convection in Evaporating Minute

Drops, J. Heat Transfer, 104 , 656 (1982).

[6] Bonacina, C., Del Giudice, S., and Comini, G., Dropwise Evaporation,

J. Heat Transfer, 101 , 441 (1979).

[7] Rizza, J.J., A Numerical Solution to Dropwise Evaporation, J. Heat

Transfer, 103

501 (1981).

[8] Chilton, T.H. and Colburn, A.P., Mass Transfer (Absorption)

Coefficients - Prediction Data on Heat Transfer Fluid Motion, Ind. Eng.

Chem., 26_, 1183 (1934).

[9] Trehan, A.K., diMarzo, M., and Evans, D.D., Transient Cooling of a Hot

Surface by Droplet Evaporation, Proc. Eastern Session Combustion

Institute, Fall Technical Meeting, 61-1 (1985).

[10]

Scriven, L.E. and Sternling, C.V., The Marangoni Effect, Nature, 18/ , 186 (1960).

- 16-
8 . NOMENCLATURE

a

A,B

C

D

h

hm

k

M

q

r

R

s

t

T

V

W

x

y

z

a

0

Y

0

droplet thickness at its apex

shape function parameters

specific heat

mass diffusivity

solid-to-air heat transfer coefficient

water-to-air mass transfer coefficinet

thermal conductivity

molecular weight

heat flux

radial coordinate tangent to the solid surface originating

at the center of the wetted area

radius of the wetted area (constant during the evaporation

time: R = R

Q

)

non-dimensional thickness of the droplet (y/R)

time

temperature

droplet volume

vapor molar flux

vapor molar fraction

coordinate normal to the solid surface originating at the

center of the wetted area

non-dimensional radius (r/R)

thermal diffusivity

wetting parameter (the ratio of the wetted area diameter to

the diameter of a spherical droplet of equivalent volume)

shape parameter

solid-liquid contact angle

- 17 -
A latent heat of vaporization

p density

Subscripts

a

c

e

i

o

s

w

air

cumulative or total

evaporative

interface

initial value (at t = 0)

solid or solid surface

water

-18
figure one

Initial Droplet Vol. = 30x 1 0" 9 m

3 Initial Surface Temp.= 98°C

Total Evap. Time (meas.)=57 s Total Evap. Time (calc.) = 51 s

(The spacing between two grid marks on the scale is 5x10“ 4 m)

19
SHAPE

PARAMETER

p

INITIAL SOLID SURFACE TEMPERATURE (°C)

FIGURE TWO

20
SOLID

LIQUID

ANGLE

OF

CONTACT

(degrees)

WETTING PARAMETER p

FIGURE THREE

21
o

o

o

05

O

00

O

N-

(Oo S w/M) ±N3l0ldd300

d3dSNVd± 1V3H 3AI103ANO0

oc3

2

U3

Ci' pO

22

INITIAL

SOLID

SURFACE

TEMPERATURE

(°C)
EVAPORATION

TIME

(s)

FIGURE FIVE

23
MEASURED

EVAPORATION

TIME

(s)

FIGURE SIX

24
AVERAGE

EVAPORATIVE

HEAT

FLUX

(kW/m

2)

INITIAL SOLID SURFACE TEMPERATURE (°C)

FIGURE SEVEN

25
NON

DIMENSIONAL

VOLUME

(V/V

0

)

FIGURE EIGHT

26
NON

DIMENSIONAL

HEAT

FLUX

(q/q

c

)

FIGURE NINE

27
NON

DIMENSIONAL

HEAT

FLUX

(q/q

a

t

z

= l>

FIGURE TEN

28
NBS-1 14A (REV. 2-80

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See / nstruction s)

1. PUBLICATION OR

REPORT NO.

NBSIR-86/3384

2. Performing Organ. Report No. 3. Publ ication Date

August 1986

4. TITLE AND SUBTITLE

Evaporation of a Water Droplet Deposited on a Hot High Thermal Conductivity

Solid Surface

5. AUTHOR(S)

Marino di Marzo and David D. Evans

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions) 7. Contract/Grant No. NATIONAL BUREAU OF STANDARDS

DEPARTMFNT OF COMMERCE

Gaithersburg, MD 20899

Mechanical Engr. Dept.

University of Maryland

College Park, MD 20742

8 . Type of Report & Period Covered

9.

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP

10.

SUPPLEMENTARY NOTES

2] Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here) A model is presented that predicts major features of the evaporation of water droplets

deposited on a hot non-porous solid surface. In the temperature range of interest,

nucleate boiling heat transfer is fully suppressed, hence the model is only concerned

with the evaporative process. In the model, the solid material is assumed to have high

thermal conductivity and diffusivity, so that the surface temperature under the water

droplet can be considered uniform. The temperature of this portion of a larger solid

surface covered by the liquid is calculated from the classic solution for contact

temperature between two semi- inf inite bodies. The liquid-vapor interfacial temperature

and the water-vapor molar fraction in the air at the exposed surface of the water

droplet are deduced from the coupled heat and mass transfer energy balance at the

interface. Spatial and temporal integration of the overall droplet energy equation

is used to predict the droplet evaporation time and the instantaneous evaporation rate.

Model predictions for the total evaporation time and temporal variation of the droplet

volume agree well with experiments performed using a heated aluminum block. The model

is used to quantify spacial and temporal heat fluxes distribution at the exposed

surface of the water droplet.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

cooling; droplets; evaporation cooling; extinguishing; spray quenching; sprinkler

system; water sprays

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