Steadystate temperature of an evaporating water droplet with a monolayer coatingGlenn O. Rubel Citation: Journal of Applied Physics 67, 6085 (1990); doi: 10.1063/1.345168 View online: http://dx.doi.org/10.1063/1.345168 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/67/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in SteadyState Transducer Calibration in a Reverberant Water Tank J. Acoust. Soc. Am. 54, 303 (1973); 10.1121/1.1978187 Theory of SteadyState Evaporation of Alloys J. Appl. Phys. 42, 3697 (1971); 10.1063/1.1659672 Theory and Practice of Steady-State Evaporation of Alloys J. Vac. Sci. Technol. 6, 568 (1969); 10.1116/1.1315685 SteadyState Burning of a Liquid Droplet. II. Bipropellant Flame J. Chem. Phys. 25, 325 (1956); 10.1063/1.1742880 SteadyState Burning of a Liquid Droplet. I. Monopropellant Flame J. Chem. Phys. 23, 1928 (1955); 10.1063/1.1740607

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Steady ... state temperature of an evaporating water droplet with a monolayer coating

Glenn O. Rube! u.s. Army Armament, Munitions, and Chemical Command, Chemical Research, Development, and Engineering Center, Aberdeen Proving Ground, Maryland 21010-5423

(Received 4 December 1989; accepted for publication 6 February 1990)

A mathematical model is derived for the steady-state temperature of an evaporating water droplet with a monolayer coating. The model uses the flux-matching arguments of Fuchs to describe the concentration and temperature fields outside the evaporating droplet. Assuming the mean molecular speed, the gas-phase diffusion coefficient, and the gas thermal conductivity to be temperature independent, a closed-form expression is derived for the steady-state temperature of the droplet as a function of the monolayer accommodation coefficient. Comparison between model predictions and recent fluorescence thermometry experiments shows good agreement.

l. INTRODUCTION

As a droplet evaporates, it releases latent heat of evapo­ration, reSUlting in a droplet temperature below that of the ambient temperature. This process is commonly referred to as self-cooling. Fuchs 1 derived an expression for the steady­state temperature of a droplet that is evaporating according to Maxwell's law of diffusive evaporation. The effect of con­vection and radiation on the evaporation process was ne­glected. Furthermore, the dependence of the gas thermal conductivity and the diffusion coefficient on the tempera­ture and vapor concentration was ignored. Fuchs showed that when the temperature-dependent diffusion coefficient was considered, the droplet evaporation rate was corrected by a factor that equalled the ratio of the droplet to ambient temperature. Thus at an ambient temperature of 25°C and for a 5 °C temperature difference between droplet and am­bient; the droplet evaporation rate with a temperature-inde­pendent diffusion coefficient differed from the rate with a temperature-dependent diffusion coefficient by 2%. It was also shown that for a droplet the size of which is much greater than the gas mean free path, the steady-state tem­perature is independent of droplet size.

Fukuta and WaIter2 derived growth equations for pure and solution droplets based on the flux-matching arguments similar to those of Fuchs. Condensation and thermal accom­modation coefficients were introduced in order to account for the effect of surface coatings on the transport of vapors to the droplet. Under the assumption that the droplet tempera­ture differed from the ambient temperature by a small amount, the growth rate of droplets the size of which exceed­ed 1 f.lm was derived. In their derivation, the diffusion coeffi­cient, thermal conductivity, and the thermal velocity were considered temperature independent.

It is the objective of this paper to develop an expression for the steady-state temperature of an evaporating pure droplet that is coated with a monolayer. The model uses the flux-matching arguments of Fuchs to describe the concen­tration and temperature fields outside the evaporating drop­let. Consistent with the arguments of Fuchs, the temperature dependence of the gas thermal conductivity, the gas-phase

di.ffusion coefficient, and the mean molecular speed is neg­lected.

II. MODEL

Fuchs argued that vapor transport to a droplet is de~ scribed by kinetic theory within one mean free path of the droplet surface and by continuum theory for distances ex­ceeding one mean free path. To mode! the vapor transport rate, the kinetic and continuum rates are matched one mean free path from the droplet surface, and this boundary condi­tion is used to solve for the concentration field outside the droplet. Using this argument, the water transport rate to a droplet is given by

dnw 2 del --=41TD(a+A) - , dt dr a-j-A

(1)

where nw is the number of water molecules, D is the gas­phase diffusion coefficient of water, a is the droplet radius, It is the mean free path of the gas surrounding the droplet, c is the water-molecule concentration, and r is the radial coordi­nate. Under steady-state conditions, the molecular concen­tration satisfies Laplace's equation !J,.c = 0 subject to the boundary conditions that c = c(a + A )at r = a + A and c = c( (0) as r-+ 00. The solution for c(r) for r> a + A is then given by

c(r) = c( (0) + (a + /l)[c(a + A) - c( (0) ]lr. (2)

At r = a + ),' the kinetic water transport rate is matched to the continuum diffusive transport rate [Eq. (I)] to give

41Tam a2[vl (a}c(a)/4 - VI (a + A)c(a + ,1)/4]

(3)

where am is the mass accommodation coefficient of water describing the fraction of water-droplet collisions that result in collection, and v! is the mean molecular speed ofthe water molecule. Under isothermal conditions, Eqs. (2) and (3)

are sufficient to define the concentration field c( r) complete­ly.

Similar arguments can be applied to determine the tern-

6085 J. Appl. Phys. 67 (10), 15 May 1990 0021-8979/90/106085-03$03.00 @) 1990 American Institute of PhYSics 6085

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perature field outside the droplet. Under steady-state condi­tions, the temperature field T(r) satisfies Laplace's equation t:..T = 0, and the solution for T(r) takes the same functional form as that ofEq. (2). Matching the kinetic and continuum heat transport rates one mean free path from the droplet surface gives

41Ta T a2 [5KT(a)v2 (a)nz/8 - 5KT(a + A)v2

X(a+A)nz/8] = _41rK(a+ lly dT/ ' dr a+A

(4)

where the specific heat of an air molecule is set to S/2K, V2 is the mean molecular speed of the air molecule, nz is the num­ber concentration of air molecules, K is the gas thermal con­ductivity, and aT is the thermal accommodation coefficient describing the fraction of air-molecule-droplet surface colli~ sions that result in complete energy transfer. Here it is as~ sumed that the transport of heat is controlled by the air mol­ecules because at one atmosphere the air-molecule concentration is over a magnitude greater than the water~ molecule concentration. It is also assumed that the air-mole~ cule number concentration is independent of temperature.

In general, the mean molecular speed v, the gas-phase diffusion coefficient D, and the thermal conductivity K are temperature dependent. However, the temperature depen­dence of these parameters is mild, varying as the square root of the temperature. Thus to a fair approximation, v, D, and K

can be considered temperature independent for small differ­ences between the droplet and ambient temperature. For ex­ample, for as °C temperature difference between the droplet and ambient, the thermal conductivity and thermal velocity change by 1 % when the temperature dependence is consid­ered. This approximation permits the decoupling of Eqs. (3) and (4). Simplification ofEqs. (3) and (4) leads to an evalu­ation of the unknowns c (a + A) and T( a + A.) in the form

(5)

and

4K(a + A) T( 00) + 5a yv2 Kn 2 a2 T(a)/2 T(a+A)= 2

4K(a + A.) + 5aT v2 Kn2 o /2

At steady state, the rate of heat release by the droplet through evaporation is equal to the rate of heat gain by con~ duction, Le.,

dnw j 2 dT I L--=41T(a+/.) K- , dt dr a+A

(6)

where L is the latent heat of evaporation of water per mole­cule. Combining Eqs. (1 ), (2), (5), and ( 6) the temperature depression of an evaporating water droplet with a monolayer is given by

T( 00) - T(a)

2DLv i a m [4K(0 +..1) + 5a T v2Ka2n2/2]

5v2 KKn 2 a T [4D(a + A) + a m a2vt ]

X [c(a) -c(oo)]. (7)

Using the ideal-gas law and the auxilliary expression for the droplet surface pressure PC a) as

6086 J. Appl. Phys., Vol. 67, No. 10, 15 May 1990

Pea) = [PC 00) ]Oexp{ - (L /K)

x[T(oo) - T(a)]lT(a)T(oo)}, (8)

the steady-state temperature of a monolayer-coated droplet is given by

T(a) - T( 00) + A([P( 00 )]0

xexp{ - (L IK)[T( 00) - TCa)]

X [ T( a)] T( 00 ) I - I} I KT( a) - B} = 0, (9)

where

and

A = 2DLvj am [ 4K(a + A) + 5a"j"v2Ka2n2 12 ]

5v2 KKn2 a T [4D(a+A) +am a2vj ]

B = PC 00 )/KTC 00).

III. DISCUSSION

Vis-a-vis the classical expression for the temperature depression of an evaporating droplet, Eq. (9) exhibits a dis­tinct droplet radius dependence. However, in the limit of very large droplet radius, the droplet radius dependence dis­appears and Eq. (9) takes the form of the classical self-cool­ing expression. Figure 1 shows the dependence of the tem­perature depression /j on the mass accommodation coefficient as predicted from Eq. (9) assuming complete thermal accommodation. The calculations are conducted for a series of droplet radii: 1, 10, 100, and 1000 /-lm. The following data set was used to generate the curves in Fig. 1: D = 0.239 cm2 Is, L = 6.74 X 10 - 13 erg/molecule, VI = 5.92 X 104 em/s, v2 = 4.75 X 104 cm/s, K = 2.38 X 103

erg/em s K, n2 = 2.46 X 1019 molecules/cm3, A

=6.5xlO- 6 cm, [P(00)]o=1.82X104 dyn/cm2, PC 00 ) = 1.07 X 104 dyn/cm2, and T( 00 ) = 289 K.

All curves show the same trend that as the mass accom­modation coefficient decreases, the temperature depression decreases. This is expected because as the mass accommoda-

6.0

sz .... <0 5.0

Z Q 1 -1.0 p.m til til 4.0 2 - 10.0 p.m !OJ a: :; - 100.0 p.m I\.

~ 4 - 1000 /Lm !OJ 3.0 a: ::I

~ a: ... 2.0 "" :t w ~

1.0

1.0 2.0 3.0 4.0 5.0 6.0

-Log am

Fl G. I. Dependence of the steady-state temperature depression of an evapo­rating water dropiet on the water mass accommodation coefficient a for droplet radii of I, 10, 100, and 1000 Itm.

Glenn O. Rubel 6086

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tion coefficient decreases, the amount oflatent heat released due to evaporation decreases too. The temperature depres­sion takes the largest value of 5 ·C for a mass accommoda­tion coefficient of unity, and asymptotically approaches zero as the mass accommodation coefficient decreases. The larger the droplet the greater the reduction in the mass accommo­dation coefficient for a given reduction in the temperature depression. For example, while a l-,um droplet requires a two order of magnitude reduction in the accommodation coefficient to decrease the temperature depression to 0.75 ·C centigrade, a lOOO-flm droplet requires a five order of magni­tude reduction in the accommodation coefficient to decrease the temperature depression to the same value.

Recently Seaver and Peele3 demonstrated a noncontact thermometry technique to measure in situ and in real time the temperature depression of an evaporating water droplet. In the experiment, a single 2-mm diameter droplet doped with Eu + 3 ethylenediaminetetraacetic acid (RDTA) was suspended in an acoustic field, and the fluorescence spec­trum thereof measured. The dopent shows a temperature­dependent fluorescence spectrum that can be accurately cal­ibrated and used to determine the temperature of the droplet from the measured fluorescence spectrum. In addition to the fluorescent dopent, a fatty alcohol surfactant was added to the water droplet to study the effect of the surfactant on the temperature of the evaporating droplet. They observed that the droplet temperature discontinuously increased at a criti­cal point in the evaporation process, coincident with a dra­matic decrease in the droplet evaporation rate. For a I-mm radius water droplet, the temperature of the droplet in­creased from an average temperature of 12 ·C to an average temperature of 16°C at the critical change in the evaporation rate. For an ambient temperature of 16 ·C, this temperature change translates to a reduction in the temperature depres­sion from 4 ·C to less than 1 dc.

Rubel and Gentry4 studied the effect of surfadants on the evaporation and growth of solution droplets using single­particle electrodynamic trapping. It was shown that both the evaporation and growth rate of the droplet discontinuously changed at a critical point in the dynamics of the droplet associated with a phase transition in the monolayer. In their studies, the effect of hexadecanol on the transport rate for a 50-pm droplet was investigated, and it was determined that the mass accommodation coefficient for water decreased by more than an order of magnitude at the monolayer phase transition. For hexadecanol on an acidic subphase, the water mass accommodation coefficient decreased from 7.0 X 10 - 4

to 2.0 X 10 5 as the hexadecanoI monolayer experienced a phase transition from the liquid to solid condensed mono­layer. Using these values for the water mass accommodation coefficient, the change in the temperature depression can be

6087 J. Appl. Phys., Vol. 67, No. 10, i5 May 1990

calculated at the onset of octadecanol monolayer phase tran­sition. As seen from Fig. 1, the temperature depression would decrease from 4.45 ·C to 0.75 °C as the accommoda­tion coefficient decreased at the hexadecanol monolayer phase transition. This calculated change in the temperature depression is in good agreement with the experimental re­sults obtained through :fluorescence thermometry, providing further evidence that the discontinuous decrease in the tem­perature depression is associated with a phase transition in the octadecanol monolayer and its concomitant reducti.on in the droplet evaporation rate.

Furthermore, it is noted that better agreement between experiment and theory can be obtained if the difference in the hexadecanol and octadecanol monolayer evaporation re­sistance is taken into account. LaMer, Healy, and Aylmore5

showed that as the alkyl chain length increased, the evapora­tion resistance (accommodation coefficient) increased (de­creased). Thus octadecanol with two additional carbon atoms as compared to hexadecanol would exhibit a smaller accommodation coefficient than hexadecanol. This would slightly reduce the temperature depression calculations and result in better agreement between theory and experiment.

IV. CONCLUSIONS

A mathematical model is derived for the steady~state temperature of an evaporating droplet with a monolayer coating. The model uses the flux-matching arguments of Fuchs to describe the concentration and temperature fields outside the evaporating droplet. Assuming the mean molec­ular speed, the gas-phase diffusion coefficient, and the gas thermal conductivity to be temperature independent, a closed-form expression is derived for the steady-state tem­perature of the droplet as a function of the monolayer ac­commodation coefficient. As opposed to the classical expres­sion for the temperature depression of an evaporating droplet, the temperature depression of a monolayer-coated droplet is droplet-radius dependent. Comparison between model predictions and recent fluorescence thermometry ex­periments shows good agreement.

IN. A. Fuchs, Evaporation and Droplet Growth in Gaseous Media (Perga-mon, New York, 1959), p. 8.

'N. Fukuta and L. A. Walter, J. Atmos. Sci. 27,1160 (197D). J M. Seaver and J. R. Peele (unpublished). 4G. O. Rubel and J. W. Gentry, J. Phys. Chern. 88, 3142 (1984). 'V. K. LaMer, T. W. Healy, and L. A. G. Aylmore, J. Colloid Int. Sci. 19, 676 (1964).

Glenn O. Rubel 60S7

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