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The Cooling Effect of A Single

Evaporating Droplet on A Hot

Semi-infinite Metal Body

Marino di Marzo

Anil K. Trehan

University of Maryland

College Park, MD

David D. Evans

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Gaithersburg, MD 2C3S9

January 1987

Issued April 1987

Sponsored in part by:

The Computer Center

iversity otf Maryland

QC liege Park, MD

100

IJ56

NO. 87-351

1987

NBSIR 87-3517

THE COOLING EFFECT OF A SINGLE

EVAPORATING DROPLET ON A HOT

SEMMNIFINITE METAL BODY

Marino di Marzo

Anil K. Trehan

University of Maryland

College Park, MD

David D. Evans

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Gaithersburg, MD 20899

January 1987

Issued April 1987

Sponsored in part by:

The Computer Center

University of Maryland

College Park, MD

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

TABLE OF CONTENTS

Page

LIST OF FIGURES iv

NOMENCLATURE v

Abstract 1

1 . INTRODUCTION 2

2. MODEL FORMULATION 3

3. EVAPORATION TIME PREDICTION 7

4. SOLID THERMAL BEHAVIOR 10

5. CONCLUSIONS 12

6. ACKNOWLEDGEMENTS 12

7. REFERENCES 13
FIGURE CAPTIONS

Page

Figure 1. Typical droplet evaporation phenomenology 14

Figure 2. Model and coordinate system 15

Figure 3. Predicted and measured evaporation time versus initial

solid surface temperature for various droplet volumes 16

Figure 4. Average evaporative heat flux versus initial solid surface

temperature for various solid materials and droplet volumes... 17

Figure 5. Spatial temperature distribution in the semi-infinite

solid at time t = 1.0 t c 18

Figure 6. Volume of influence at various times (progression) 19

Figure 7. Volume of influence at various times (regression) 20

Figure 8. Maximum non-dimensional volume of influence versus initial

surface temperature for various droplet volumes 21

Figure 9. Non-dimensional volume of influence versus predicted

evaporation time 22

IV
NOMENCLATURE

A, B , C

,

A', B', C'

c

D

f

h

k

L

q

r

constants

specific heat

mass diffusivity (steam-air)

generic function

convective heat transfer coefficient

thermal conductivity

latent heat of vaporization (water)

heat flux

radial coordinate tangent to the solid surface originating at the

center of the wetted region

R

t

x

y

z

a

6

Y

wetted region radius

time

temperature

initial solid surface temperature prior to the droplet deposition

droplet volume

volume of influence

molar fraction of steam in air

coordinate normal to the solid surface originating at the center of

the wetted region upward oriented (in the droplet region)

coordinate normal to the solid surface originating at the center of

the wetted region downward oriented (in the solid region)

thermal diffusivity

ratio of the wetted region radius to the radius of the spherical

droplet of equivalent liquid volume

square root of the product of specific heat, density and thermal

conductivity

V
6 shape parameter

p density

Subscripts

a air property

c cumulative or total

i at the droplet exposed surface

o initial value

s solid property or at the exposed solid surface

u at the solid surface under the droplet

w liquid water property

VI
THE COOLING EFFECT OF A SINGLE EVAPORATING

DROPLET ON A HOT SEMI-INFINITE METAL BODY

Marino di Marzo, David D. Evans and Anil K. Trehan

ABSTRACT

Transient cooling of solid surfaces by water droplet evaporation has been

investigated through controlled experiments using a large heated aluminum

cylinder. Quantitative prediction of droplet evaporation time and in-depth

transient temperature distribution in the solid have been made. In the case

studied, a single droplet is deposited on a horizontal non-porous surface with

initial temperatures in the range of 75°C to 100°C. 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 energy balance at

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

equation is used to predict the instantaneous evaporation rate and the droplet

evaporation time. The boundary conditions for the wetted region proposed by

Seki are used to obtain the transient temperature distribution for a semiinfinite solid. The region of the body affected by the droplet cooling is

identified and its volume (normalized with the volume of the droplet) is

plotted against the evaporation time. All data, regardless of the droplet

volume or of the initial body surface temperature, lie within a narrow band

about a straight line. This finding is the first important step to obtain a

simple model for spray cooling based on local accurate description of the

droplet-solid interactions. Modeling of spray cooling phenomena is the

foundation for the construction of a thermal model for solid fuel fire

extinguishment

.

- 1 -
1 . INTRODUCTION

The thermal behavior of a semi-infinite solid subjected to local surface

cooling induced by a single evaporating droplet has been described in detail

by Seki [lj. Numerous theoretical and experimental studies are found in the

literature concerning the mechanism of evaporation of droplets impinging on a

hot surface. A simple model is derived by Bonacina [2] which identifies an

important geometrical parameter for the droplet-surface interaction. Grissom

[3] investigated the cooling effect of a spray on a heated surface. The

evaporation of single droplets is the objective of the works of Michiyoshi and

Makino [4,5] which give a detailed insight into the thermal features of the

evaporation process. Sadhal [6] provides an interesting analysis of the

droplet-solid interactions for evaporation and condensation. The optical

studies of the droplet configuration on a solid surface reported by Zhang [7]

allow one to conclude that a spherical segment is a good geometrical representation of the actual droplet shape. In a previous paper di Marzo and Evans

[8] derived a rigorous analytical model to predict the evaporation time, the

evaporation rate and the local heat flux distribution at the droplet exposed

surface during its evaporation.

In many of these studies, the full range of vaporization processes

(including evaporation, nucleate boiling, film boiling, and transition

boiling) are reported. This detailed study of the water droplet vaporization

is intended to form the basis for a thermal model of solid fuel fire extinguishment, so this study is limited to investigations of the evaporation

phenomena only.

-2-
The objective of this paper is to show that a simple relationship exists

between the evaporation time and the volume of the portion of the solid

affected by the cooling process induced by the droplet evaporation. In order

to illustrate this finding, the basic heat transfer governing equation for the

semi-infinite solid will be derived and merged with the previously developed

model for the prediction of the evaporation time [8] to determine the cooling

effect in the solid.

The high thermal conductivity of the heated aluminum block used in this

study allows one to assume that the temperature, T

u

, at the solid surface,

under the droplet is constant in both space and time. This hypothesis was

successfully tested by Seki [1] against experimental data. The experimental

findings by Michiyoshi [4] infer also that the temperature of the solid

surface under the droplet is constant during the evaporation process. This

information coupled with the finding reported by Seki [1] provides a simple

boundary condition for the wetted region of the solid surface.

2. MODEL FORMULATION

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

[7,8].
In Fig. 1, a typical sequence of photographs of an evaporating droplet is

presented. It is important to note that the wetted region of the semiinfinite solid surface remains constant in size throughout most of the

evaporative process.

The ratio of the wetted area radius to the radius of an equivalent liquid

volume spherical droplet is identified by the parameter 3 [2], This parameter

is the only experimental input used in the analysis. The complexity of the

parameters influencing the initial droplet shape (surface tension, surface

wettability, etc.) is such that this semi -empirical approach was selected to

formulate a reasonably simple analysis. The effect of the droplet kinetic

energy (before impact) on the parameter 3 is negligible for droplet release

height of less than 1.7 centimeters. This study is concerned only with

droplets released at less than 1.7 centimeters from the solid surface. These

droplets are referred to as "softly" deposited droplets [2,3].

The cooling effect induced by the droplet evaporation is complex to model

analytically. The boundary conditions of the energy equation written for the

droplet should match the boundary conditions of the energy equation written

for the solid. Namely, the heat flux and the temperature should be set equal

at each location and at each time at the liquid-solid boundary.

The model proposed by Seki [1] is based on a simplifying assumption. He

considers the solid-liquid boundary to be at uniform constant temperature.

This assumption finds its justification in the large values of the thermal

conductivity of the metals. The heat flux required for the droplet evaporation is induced by small temperature gradients in the solid. Since Michiyoshi

-4 -
[4,5] confirmed experimentally that the solid-liquid interfacial temperature

does not change appreciably in time, one can conclude that a uniform, constant

temperature, T

u , under the droplet is a reasonable boundary condition. The

advantages of such hypothesis is that now the energy equation for the solid

can be integrated independently from the droplet behavior and only the overall

evaporation time is required to obtain information on the solid cooling.

The solid thermal behavior is described by the energy equation and its

boundary conditions, as previously derived by Seki [1]. In summary, for the

system configuration depicted in Fig. 2, one can write

3T _ 3 2T . 1 3T . 3 2

T

3t

' “s ~2 +

r ir + ~2

3z 3r

with the following boundary conditions:

( 1 )

r = 0 3T/3r = 0 ( 2 )

r + °° 3T/3r = 0 (3)

Z + oo q ha (T - Ta) « - k 3T/3z

o s (4)

z = 0 and r > R hfT - T ] = - k 3T/3z

a

K s aJ s (5)

z = 0 and r < R for t < 0 and t > t

h (T - T ) = - k 3T/3z

a

K s a

J s

( 6 )

z = 0 and r < R for 0 < t < t - c T = T (7)

-5 -
The evaporation time, t c

, is obtained from the model predictions as will be

shown in the following section. Equation (1) is numerically integrated and

the temperature distribution in the solid is obtained at each instant of time

during the evaporation process and thereafter in the thermal recovery

t ransient

.

The spatial and temporal temperature distribution information generated

by this process is overwhelming. In order to compact this information and to

deduce useful correlations between the magnitude of the cooling effect and the

droplet evaporation phenomena, a new quantity is introduced, namely the volume

of influence. The volume of influence is the volume of the region where the

temperature variation with respect to the steady state conduction temperature

distribution prior to the droplet deposition exceeds a given value. The

maximum possible temperature variation occurring in the solid is the difference between the solid surface temperature prior to the droplet deposition and

the contact temperature (between the water and the solid) given by Seki [1] as

T

u

T Y w

1 w

+

+

T yo ' s

( 8 )

The volume of influence will be defined as the volume where the temperature

variation exceeds ten percent of the maximum possible temperature difference.

The results of the computations will be shown in terms of volume of influence

and in particular in terms of the maximum volume of influence which is

observed at the end of the evaporation process.

-6 -
3. EVAPORATION TIME PREDICTION

In an earlier paper, di Marzo and Evans [8] showed that the droplet

geometrical configuration is given by

y

7 1/2

( 1/6 + 6 ) 2

4

r (9)

where 6 is

6 = r { [4/b

3 + (i + i6/e

6

)

1 / 2

]

1 / 3 + [4/e

3 - (i + i6/e6

) 1/2] 1/3

} do)

and e is defined as the ratio of the radius of the wetted area to the radius

of an equivalent liquid volume spherical droplet. The droplet configuration

can be approximated to a spherical segment throughout all the evaporative

process. It must be noted, however, that in the last portion of the process

(t > 0.95 t ) the liquid film becomes very thin and the wetted surface suddenly receeds. The temperature at the exposed surface of the liquid droplet can

be evaluated by a heat balance at the liquid vapor Interface. By considering

that the heat conducted through the liquid layer to the interface is equal to

the energy required to evaporate the liquid droplet plus the small amount of

energy convected away, one can write

w

0.624 h

a L_

ca

( 11 )

where the mass transfer coefficient is evaluated by making use of the ChiltonColburn analogy [9]. Note that the only unknown is x^ since the interfacial

temperature, T

i> is the saturation temperature of steam at the pressure

corresponding to its molar fraction.

- 7 -
The overall heat balance on the droplet allows one to calculate the

instantaneous evaporation rate as

dV _ 2 tt (0.624)

h

a D 2/3 R x . - x

r 1 a

/ rdr ( 12 ) dt pc a w a a o

x.

l

where x^ is function both of y and t.

By integrating this equation over the droplet exposed surface at each

instant of time and by reassessing the liquid inventory (and subsequently the

shape of the droplet) at each time step, the overall evaporation time can be

determined. Figure 3 illustrates the results obtained with the model in

comparison with experimental data for water droplets evaporating on an

aluminum cylindrical block. For a more complete description of the experimental apparatus and procedures, refer to Trehan [10]. A complete derivation

of the evaporation time model is given by di Marzo in the aforementioned

reference [8]

Comparison of the model results with experimental data

obtained by Michiyohsi and Makino [4,5] show that the solid material does not

influence the results provided that its thermal conductivity is such that

uniform temperature at the solid surface under the droplet can be regarded as

a reasonable assumption. In Fig. 4, various experimental data for different

materials are compared with the theoretical calculations. Note that the

evaporative heat flux is directly related to the evaporation time.

This droplet evaporation model provides detailed information on the local

heat fluxes at the solid liquid-boundary during the evaporation. Therefore, a

coupled solution of Eq. (11) and Eq. (1) could yield the desired results

without imposing the constant temperature condition at the liquid-solid inter-

-8 -
face. The drawback of this approach is that an iterative computation scheme

needs to be set up to solve both the solid and droplet energy equations

simultaneously.

By making use of a closed form analytical solution described by Carslaw

and Jaeger [11], it was determined that the temperature nonuniformities in the

liquid solid interfacial region amount to about 1°C. This result was obtained

imposing, as a boundary condition of the solid energy equation, heat fluxes of

the order of those required for the droplet evaporation.

In conclusion, one can argue that, with respect to the heat and mass

transfer phenomena, three resistances in series can be envisioned where the

first resistance in the solid is negligible with respect to the other two.

The heat transfer resistance in the droplet is comparable to the mass transfer

resistance in the air-vapor region. For the case of low conductivity solids,

these three resistances are all of the same order of magnitude; therefore, the

full coupled problem must be solved.

To elaborate further on these results, it is interesting to note that the

evaporation time has an exponential behavior with respect to the initial solid

surface temperature. This means that

In t = - A' T + f(V ) (13)

c o o

Furthermore, in order to find the intercept, f(V

Q

), one can observe that also

the volume show an exponential behavior hence:

-9-
( 14 ) f (V ) = B

1 ln(V ) + C' o o

By combining these two equations, the following result is obtained

t = A V

B cooe - C T (15)

The numerical values for the constant A, B, C were found to be respectively

880, 0.7 and 0.05 for water droplets on heated aluminum surfaces. The scatter

of the experimental data about the model predictions is in the order of 15

percent

.

4. SOLID THERMAL BEHAVIOR

The results of the numerical integration of Eq. (1) are illustrated in

Fig. 5 through 7. The effectiveness of the volume of influence in compacting

the cooling effect information is evident if one compares Fig. 5 and Figs. 6

and 7

.

The volume of influence as a function of the initial solid surface

temperature for various droplet volumes is depicted in Fig. 8. Again, an

exponential pattern is observed both for the volume of influence curves as

well as for the various droplet volumes. Consider, therefore, the plot of the

volume of influence versus the evaporation time for the same droplet volume

and initial solid surface temperature, as shown in Fig. 9. The uncertainties

of the various parameters determining the evaporation time and the volume of

influence are such that the points can be considered to lie in close proximity

to the linear regression of the point values (for more details on the order of

- 10-
approximation, see di Marzo [8]).

The expression for the linear fit is given by

V . l = V r 0.021 t + 3)

o'- c

J ( 16 )

By combining Eq. (15) and (16), one obtains

V. = 18 V

1,7

e - 0.05 T + 3 V (17)

l o o o

This closed form, simple, analytical result illustrates the power of the

analysis. Note that the specific numerical constants are for aluminum. The

same governing equations and boundary conditions are common for high thermal

conductivity materials. Therefore, similar trends in the volume of influence

are expected, thus leading to a linear dependence of the non-dimensional

volume of influence with the evaporation time. The temperature spatial and

temporal uniformity under the droplet is the condition that allows the

uncoupling of the droplet evaporation process from the solid thermal

behavior. In order to obtain this result the thermal conductivity of the

solid must be larger than the thermal conductivity of the liquid water. The

volume of influence is the basic information that allows one to predict the

cooling effect in a semi-infinite body subjected to droplet evaporation.

Further research based on these results will provide information on the

cooling effect of an array of droplets impinging a surface, thus leading to a

comprehensive model to predict spray cooling.

- 11 -
5. CONCLUSIONS

The energy equation, written in cylindrical coordinates for the semiinfinite metal body, provides information on the spatial and temporal temperature distribution in the body. More compact information is given in terms of

the volume of influence. The boundary conditions for the energy equation are

formulated by using the liquid-solid contact temperature suggested by Seki [1]

coupled with the evaporation time calculated by di Marzo [8]

.

This analytical

model to predict the evaporation time requires that the wetting parameter 8

must be determined by experiments

.

The volume of influence is found to correlate linearly with the

evaporation time and to be independent of the droplet volume and the initial

solid surface temperature. A consequence of this finding is that a simple,

closed form expression for the volume of influence can be obtained. This

information is the first step towards the analysis of multi-droplet cooling

effect thus leading to the modelling of spray cooling processes.

6. ACKNOWLEDGEMENTS

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

the National Bureau of Standards. The Computer Center of the University of

Maryland provided partial funding for the numerical computations.

- 12-
7. REFERENCES

[1] M. Seki, H. Kawamura, K. Sanokawa, "Transient Temperature Profile of a Hot Wall Due to an Impinging Liquid Droplet", Transaction of the ASME

,

Journal of Heat Transfer, Vol. 100, pp. 176, 1978.

[2] C. Bonacina, G. Comini, "Dropwise Evaporation", Transaction of ASME,

Journal of Heat Transfer, Vol. 101, pp. 441-446, 1979.

[3] W.M. Grissom, F.A. Wierum, "Liquidspray Cooling of a Heated Surface",

International Journal of Heat and Mass Transfer, Vol. 24, pp. 261-271,

1981.

[4] I. Michiyoshi, K. Makino, "Heat Transfer Characteristics of Evaporation

of a Liquid Droplet on Heated Surfaces", International Journal of Heat

and Mass Transfer , Vol. 21, pp. 605, 1978.

[5] K. Makino, I. Mikiyoshi, "Effect of Initial Size of Water Droplet on Its

Evaporation on Heated Surfaces", International Journal of Heat and Mass

Transfer, Vol. 22, pp. 979, 1979.

[6] S.S. Sadhal, M.S . Plesset, "Effect of Solid Properties and Contact Angle

in Dropwise Condensation and Evaporation", Transaction of ASME, Journal

of Heat Transfer , Vol. 101, pp. 48-54, 1979.

[7] N. Zhang, W.J. Yang, "Natural Convection in Evaporating Minute Drops",

Transaction of the ASME, Journal of Heat Transfer , Vol. 104, pp. 656,

1982.

[8] M. di Marzo, D.D. Evans, "Evaporation of a Water Droplet on a Hot High

Thermal Conductivity Solid Surface", NBSIR 86-3384, 1986.

[9] T.H. Chilton, A.P. Colburn, "Mass Transfer (Absorption) Coefficients

Prediction Data on Heat Transfer Fluid Motion", Industrial Engineering

Chemistry , Vol. 26, pp. 1183, 1934.

[10] A.K. Trehan, M. di Marzo, D.D. Evans, "Transient Cooling of a Hot

Surface by Droplet Evaporation", Proceedings of the Eastern Session of

the Combustion Institute, Fall Technical Meeting, pp. 61-1, 1985.

[11] H.S. Carslaw, J.C. Jaeger, Conduction in Solids, Oxford University

Press, Oxford, England, 1946.

- 13-
14

FIGURE

ONE
FIGURE TWO

15
EVAPORATION

TIME

(s)

75 80 85 90 95 100

INITIAL SOLID SURFACE TEMPERATURE (°C)

FIGURE THREE

16
AVERAGE

EVAPORATIVE

HEAT

FLUX

(kW/m

FIGURE FOUR

17
18

RADIAL

DISTANCE

(mm)
t/tc

19

RADIAL

DISTANCE

(mm)
t/tc

20

RADIAL

DISTANCE

(mm)
NON-DIMENSIONAL

MAXIMUM

VOLUME

OF

INFLUENCE

(Vj/V)

FIGURE EIGHT

21
FIGURE NINE
FORM NBS-1 14A (REV 11-84)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

1. PUBLICATION OR

REPORT NO.

2. Performing Organ. Report No. 3. Publication Date April 1987

SHEET (See instructions) NBSIR 87-3517

4. TITLE AND SUBTITLE

The Cooling Effect of a Single Evaporating Droplet on a Hot Semi-Infinite Metal Body

5. AUTHOR(S)

Marino di Marzo, David D. Evans, Anil K. Trehan

6. PERFORMING ORGANIZATION (If joint or other than NBS. see in struction s) 7. Contract/Grant No.

NATIONAL BUREAU OF STANDARDS

U.S. DEPARTMENT OF COMMERCE *• Type of Report & Period Covered

GAITHERSBURG, MD 20899

9.

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

The Computer Center

University of Maryland

College Park, MD

10.

SUPPLEMENTARY NOTES

J 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) Transient cooling of solid surfaces by water droplet evaporation has been investigated

through controlled experiments using a large heated aluminum cylinder. Quantitative

prediction of droplet evaporation time and in-depth transient temperature distribution

in the solid have been made. In the case studied, a single droplet is deposited on a horizontal non-porous surface with initial temperatures in the range of 75°C to 100°C.

The liquid-vapor inter-facial 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. Spacial and temporal integration of the

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

and the droplet evaporation time. The boundary conditions for the wetted region

proposed by Seki are used to obtain the transient temperature distribution for a semiinfinite solid. The region of the body affected by the droplet cooling is identified

and its volume (normalized with the volume of the droplet) is plotted against the

evaporation time. All data, regardless of the droplet volume or of the initial body

surface temperature, lie within a narrow band about a straight line. This finding is

the first important step to obtain a simple model for spray cooling based on local

accurate description of the droplet-solid interactions. Modelling of spray cooling

phenomena is the foundation for the construction of a thermal model for solid fuel

fire extinguishment.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) conduction; droplets; evaporation; extinguishment; fire models; spray quenching

13.

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