Evaporation and Condensation of Large Droplets in the Presence of

Evaporation and Condensation of Large Droplets in the Presence of Inert AdmixturesContaining Soluble Gas

T. ELPERIN, A. FOMINYKH, AND B. KRASOVITOV

Department of Mechanical Engineering, The Pearlstone Center for Aeronautical Engineering Studies, Ben-Gurion University of theNegev, Beer-Sheva, Israel

(Manuscript received 28 October 2005, in final form 11 July 2006)

ABSTRACT

In this study the mutual influence of heat and mass transfer during gas absorption and evaporation orcondensation on the surface of a stagnant droplet in the presence of inert admixtures containing noncon-densable soluble gas is investigated numerically. The performed analysis is pertinent to slow droplet evapo-ration or condensation. The system of transient conjugate nonlinear energy and mass conservation equa-tions was solved using anelastic approximation. Using the material balance at the droplet surface theauthors obtained equations for Stefan velocity and the rate of change of the droplet radius taking intoaccount the effect of soluble gas absorption at the gas–liquid interface. The authors also derived boundaryconditions at gas–liquid interface taking into account the effect of nonisothermal gas absorption. It isdemonstrated that the average concentration of the dissolved species in a droplet strongly depends on therelative humidity (RH) for highly soluble and for slightly soluble gaseous atmospheric pollutants. Therewiththe difference between the average concentration of the dissolved species in water droplets attains tens ofpercent for different values of RH.

1. Introduction

Wet removal of atmospheric polluted gases by clouddroplets is involved in many atmospheric processessuch as polluted gases scavenging, cloud microphysics,etc. Heat and mass transfer during gas absorption byliquid droplets and during droplets evaporation and va-por condensation on the surface of liquid droplets isimportant in various fields of modern environmentalengineering and atmospheric science. Clouds representan important element in self-cleansing process of theatmosphere (Flossmann 1998). The consequence forthe aerosol climate forcing is that the cooling can beintensified with increasing atmospheric amount of wa-ter-soluble trace gases such as NH3 and SO2, counter-acting the warming effect of the greenhouse gases(Krämer et al. 2000). All these phenomena involveevaporation of droplets suspended in a multicompo-nent gaseous mixture. Scavenging of atmospheric pol-

luted gases by cloud droplets is a result of gas absorp-tion mechanism (Pruppacher and Klett 1997; Floss-mann 1998). Sources of soluble gases presented in theatmosphere are briefly reviewed by Macdonald et al.(2004), Sutton et al. (1995), Fraser and Cass (1998),Van der Hoek (1998), and Elperin and Fominykh(2005).

Gas scavenging by atmospheric water droplets in-cludes absorption of SO2, CO2, and NH3, and someother gases. Presence of soluble gas in the atmospherecan affect the dynamics of evaporation and condensa-tion of water droplets in an atmospheric cloud. Averageconcentrations of CO2, NH3, and SO2 in the atmo-sphere can be found, for example, in Seinfeld (1986)and Liu and Lipták (1999). Comprehensive study ofcoupled heat and mass transfer during gas absorptionby liquid droplets and droplets evaporation and growthis a necessary step in an adequate predicting of atmo-spheric changes under the influence of hazardous gases.

Different facets of the problem of evaporation ofdroplets in the flowing or stagnant gases were discussedin numerous theoretical and experimental studies, andseveral comprehensive reviews are available (see Sirig-nano 1993; Chiu 2000).

Soluble gas absorption by noncirculating droplets

Corresponding author address: T. Elperin, Dept. of MechanicalEngineering, The Pearlstone Center for Aeronautical Engineer-ing Studies, Ben-Gurion University of the Negev, P.O. Box 653,Beer-Sheva 84105, Israel.E-mail: [email protected]

MARCH 2007 E L P E R I N E T A L . 983

DOI: 10.1175.JAS3878.1

© 2007 American Meteorological Society

JAS3878

was investigated experimentally by Hixson and Scott(1935), Bosworth (1946), and Taniguchi and Asano(1992). Conditions of noncirculation for falling liquiddroplets were achieved by using high-viscosity liquids inexperiments of Hixson and Scott (1935) and Bosworth(1946) or by using small water droplets in the experi-ments of Taniguchi and Asano (1992). In these experi-mental studies droplet diameter was larger than 1 mm.Taniguchi and Asano (1992) used water droplets withSauter mean diameter equal to 0.185, 0.148, and 0.137mm in experiments with CO2 absorption. Bosworth(1946) compared experimental results with error func-tion solution of nonstationary equation of diffusionwritten in spherical coordinates for a liquid droplet.Taniguchi and Asano (1992) correlated experimentalresults with Newman’s solution (see Newman 1931).

Dispersed-phase controlled isothermal absorption ofa pure gas by stagnant nonevaporating liquid dropletwas investigated analytically by Newman (1931), andgas absorption in the presence of inert admixtures whenboth phases affect mass transfer was analyzed byPlocker and Schmidt-Traub (1972). Liquid and gaseousphase controlled mass transfer during soluble gas ab-sorption in the presence of inert admixtures by a stag-nant nonevaporating liquid droplet was studied also byChen (2002) by solving the coupled time-dependent dif-fusion equations for gas and liquid phases. Vesala et al.(2001) solved the problem of trace gas uptake by drop-lets under nonequilibrium conditions numerically andanalytically and derived simple formulas for the gasuptake coefficient. State of the art in gas absorption byspheroidal water droplets is presented by Amokraneand Caussade (1999).

Effect of vapor condensation at the surface of stag-nant droplets on the rate of mass transfer during gasabsorption by growing droplets was investigated theo-retically by Karamchandani et al. (1984), Ray et al.(1987), and Huckaby and Ray (1989). In particular theeffect of vapor saturation ratio on sulfur S(IV) accu-mulation during sulfur dioxide absorption by evaporat-ing (growing) droplets was specified by Huckaby andRay (1989). Liquid-phase controlled mass transfer dur-ing absorption was investigated by Karamchandani etal. (1984) and Ray et al. (1987) in the case when thesystem consists of liquid droplet, its vapor, and solublegas. Droplet growth was assumed to be solely con-trolled by vapor condensation, and influence of gas ab-sorption on the rate of vapor condensation was ne-glected. The assumption of uniform temperature distri-bution in both phases was used by Karamchandani etal. (1984). The results of Karamchandani et al. (1984)indicate that for fast droplet growth rates, induced byrelatively high degrees of supersaturation, the absorp-

tion rate is significantly enhanced, leading to highersolute concentration near the surface than that ob-served for nongrowing droplets. Ray et al. (1987) ana-lyzed gas absorption by a stationary growing dropletin a stagnant supersaturated medium by solving thecoupled nonstationary mass and energy balance equa-tions for the gaseous and liquid phases. The obtainedresults showed that in the case when heats of conden-sation are low, the droplet growth can increase signifi-cantly the rate of absorption. For high heats of conden-sation, the rapid increase of surface temperature in-duced by high growth rates can significantly reduceabsorption rates for those gases whose solubility de-creases with temperature increase. Huckaby and Ray(1989) employed the approach suggested by Ray et al.(1987) to investigate a system consisting of a liquiddroplet, its vapor, soluble and inert gases. Diffusionresistances in both phases during gas absorption weretaken into account. Numerical calculations were per-formed for water droplet evaporating (growing) in gas-eous mixture containing sulfur dioxide (SO2). It shouldbe emphasized that in the above mentioned studies (seeKaramchandani et al. 1984; Ray et al. 1987; Huckabyand Ray 1989), only the influence of condensation onthe rate of gas absorption by stagnant liquid dropletswas investigated.

In the present study we investigate the interrelatedeffects of gas absorption and evaporation (condensa-tion) at the surface of a droplet on transient heat andmass transfer in the presence of inert gases. We alsotook into account the following effects that were ne-glected in all previous studies: (i) effect of gas absorp-tion on Stefan velocity, and (ii) thermal effect of ab-sorption on droplet evaporation or condensation. Incontrast to previous studies (see, e.g., Karamchandaniet al. 1984; Ray et al. 1987; Huckaby and Ray 1989) inour calculations we considered the molecular transportcoefficients of the gaseous phase as functions of tem-perature and concentrations of gaseous species. Conse-quently the suggested model can be used over a widerange of parameters such as temperature, relative hu-midity, etc.

2. Mathematical model

a. Governing equations

Consider a spherical droplet with the initial radius R0

immersed in a stagnant gaseous mixture with tempera-ture Te,�. The gaseous mixture containing K compo-nents includes noncondensable soluble species that isabsorbed into the liquid. The first component of thegaseous mixture is formed by the molecules of the vola-tile species of a droplet. The soluble species is absorbed

984 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 64

into the liquid, and other K � 2 components do notundergo phase transition at the droplet surface. Thedroplet is heated by a conduction heat flux from thehigh temperature surroundings and begins to evapo-rate. In the further analysis we assume spherical sym-metry and neglect effects of buoyancy and thermal dif-fusion. Under these assumptions, the spherically sym-metric system of mass and energy conservationequations for the liquid phase 0 � r � R(t) reads

r2�T �L�

�t� �L

�

�r �r2�T �L�

�r �, �1�

r2�YA

�L�

�t� DL

�

�r �r2�YA

�L�

�r �. �2�

In the surrounding gaseous medium, r � R(t), the mass,species, and energy conservation equations read

r2��

�t�

�

�r�r2��r� � 0, �3�

r2�

�t��Yj� �

�

�r��r r2Yj� �

�

�r ��Djr2

�Yj

�r �, �4�

r2��cpTe�

�t�

�

�r��r r2cpTe� �

�

�r �ker2�Te

�r �. �5�

In Eqs. (1)–(5) is the gas density, j is the number ofgaseous phase species (subscript 1 denotes the volatilespecies, j � 1, . . . , K � 1, j A1), Yj and Mj are themass fraction and the molar mass of the jth gaseousspecies, Y (L)

A is the mass fraction of the absorbate in theliquid, Dj is the diffusion coefficient, Dj � (1 � Xj)/�kj(Xk/Djk), Xj is the mole fraction of the jth species,and Djk is the binary diffusion coefficient for species jand k.

Estimation of the characteristic values of the terms�/�t and �(v) in the continuity Eq. (3) shows that inthe considered problem 2/c2 K 1, where is the gasvelocity and c is the speed of sound. Therefore for thesolution of the system of energy and mass conservationequations instead of Eq. (3) we can use anelastic ap-proximation:

�

�r�r2��r� � 0. �6�

The radial flow velocity r can be obtained by integrat-ing Eq. (6)

��r r2 � const. �7�

The system of energy and mass conservation Eqs. (3)–(5) must be supplemented with the momentum conser-vation equation. However, in the case of small flowvelocities the pressure gradient is negligibly small(�p � 2), and the pressure can be assumed constant.The gaseous phase properties can be related throughthe ideal gas equation of state:

p � p� � �RgTe �j�1

K �Yj

Mj�, �8�

where Rg is the universal gas constant, p� is the gaseousmixture pressure far from the droplet.

b. Stefan velocity and droplet vaporization rate

Equations (1)–(5) must be supplemented with equa-tion for determining gas flow velocity. Consider thecase when effect of droplet volumetric expansion isnegligible. The continuity condition for the radial fluxof the absorbate at the droplet surface reads

jA | r�R � �YA�s � DA��YA

�r�

r�R�

� �DL�L

�YA�L�

�r�

r�R�

, �9�

where the signs “�” and “�” denote values at the ex-ternal and internal surfaces, respectively. Since otherK � 2 nonsoluble components of the inert admixturesare not absorbed in the liquid the integral fluxes vanish;that is, Jj � 4�R2jj � 0, ( j 1, j A), and

jj � �Yj�s � Dj��Yj

�r �r�R�

� 0, � j � 1, j � A�.

�10�

Taking into account this condition and using Eq. (7) wecan obtain the expression for Stefan velocity:

�s � �DL�L

��1 � Y1�

�YA�L�

�r�

r�R�

�D1

�1 � Y1�

�Y1

�r�

r�R�

.

�11�

The material balance at the gas–liquid interface reads

dmL

dt� �4�R2�s ��R, t� � R�. �12�

Equation (12) yields

�s �dR

dt �1 ��L

�s�. �13�

1 In general, Eqs. (4) are written for all j � 1, . . . , K. However,only K � 1 of the values Dj can be specified independently. There-fore the Kth species can be treated differently from the others andmay be found consistently using the identity �K

j�1 Yj � 1.


Substituting Eq. (13) into Eq. (11) and assuming thatL k we obtain the following expression for the rateof change of droplet’s radius:

R �DL

�1 � Y1�

�YA�L�

�r�

r�R�

��D1

�L�1 � Y1�

�Y1

�r�

r�R�

.

�14�

As can be seen from Eqs. (11) and (14) in the case whenall inert admixtures are not absorbed in the liquid, weobtain the following expressions for Stefan velocity andrate of change of droplet’s radius:

�s � �D1

�1 � Y1�

�Y1

�r �r�R�

, �15�

R ��D1

�L�1 � Y1�

�Y1

�r �r�R�

. �16�

Expressions (11), (14) imply that the absorption ofsoluble admixture decreases Stefan’s velocity andevaporation rate. However, since absorption is accom-panied by thermal effect and solubility of differentgases in a liquid strongly varies with temperature, theinfluence of gas absorption on the rate of evaporation isquite involved.

c. Initial and boundary conditions

The system of conservation Eqs. (1)–(5) must besupplemented by initial conditions and the boundaryconditions at the droplet surface. The initial conditionsfor the system of Eqs. (1)–(5) read

At t � 0, 0 � r � R0, T �L� � T0�L�, YA

�L� � YA,0�L� ,

�17�At t � 0, r R0, Te � Te,0�r�, Yj � Yj,0�r�.

At the droplet surface the continuity conditions for theradial flux of nonsoluble gaseous species yield

Dj��Yj

�r �r�R�

� �Yj� | r�R�. �18�

For the absorbate this condition assumes the form ofEq. (9). The vapor concentration at the droplet surfaceY1,s(R, t) is the function of temperature Ts(t) and can bedetermined as follows:

Y1, s�R, t� � Y1, s�Ts � ��1, s

��

p1, s�Ts �M1

p�M, �19�

where M1 and M are the molar mass of volatile speciesand gaseous mixture, respectively. Partial pressure at

the droplet surface p1,s is determined by the followingequation (see Reid et al. 1987):

ln�p1, s pc� � �1 � ��1�a1� � a2�1.5 � a3�3 � a4�6�,

�20�

where � � 1 � Ts /Tc is the critical temperature, and pc

is the critical pressure, and the values of the coefficientsai are presented in Table 1.

The droplet temperature can be found from the fol-lowing equation:

ke

�Te

�r�

r�R�

� �LL�

dR

dt� kL

�T �L�

�r�

r�R�

� La�LDL

�YA�L�

�r�

r�R�

.

�21�

The last term in the right-hand side of Eq. (21) arisesdue to a heat released at the gas–liquid interface dur-ing gas absorption in liquid (see, e.g., Elperin andFominykh 2003). The equilibrium between soluble gas-eous and dissolved in liquid species can be expressedusing the Henry’s law. Consequently the boundary con-dition for the Eq. (2) reads

CA � HApA, �22�

where CA is the molar concentration of the species dis-solved in liquid, HA is the Henry’s law coefficient, andpA is the partial pressure of species A in the gas. Thefunctional dependence of the Henry’s law constant ver-sus temperature reads

lnHA�T0�

HA�T��

�H

RG� 1

T�

1T0�, �23�

where �H is the enthalpy change due to transfer ofsoluble species A from the gaseous phase to liquid, RG

is the gas constant. The dependencies of Henry’s lawconstant on temperature for aqueous solutions of dif-ferent soluble gases (ammonia, carbon dioxide, and sul-fur dioxide) are shown in Fig. 1. Inspection of Fig. 1shows that Henry’s law constant strongly varies withtemperature and hence the dependence of Henry’s con-stant on temperature must be taken into account.

At the gas–liquid interface

Te � T �L�. �24�

TABLE 1. The coefficients ai used in Eq. (20).

Coefficient a1 a2 a3 a4

Value �7.764 51 1.458 38 �2.775 80 �1.233 03


In the center of the droplet the symmetry conditionsyields

�YA�L�

�r�

r�0� 0,

�T �L�

�r�

r�0� 0. �25�

At r → � and t � 0 the soft boundary conditions atinfinity are imposed:

�Te

�r �r→ �

� 0,�Yj

�r �r→ �

� 0. �26�

3. Method of numerical solution

The presence of two computational domains [0 � r �

R(t) for the liquid phase, and r R(t) for the gaseousphase] for the system of Eqs. (1)–(5), which are sepa-rated by a moving boundary r � R(t) complicates thenumerical solution. Moreover the characteristic timesof the heat and mass transfer in the gaseous phase andheat transfer in the liquid phase are much less than thecharacteristic time of the diffusion of soluble compo-nent in liquid. To overcome these problems we intro-duced the following dimensionless time variable:

�DLt

R02 . �27�

Then the governing equations can be rewritten for timevariable � using the following coordinate transforma-tions (see, e.g., Ray et al. 1987):

x � 1 �r

R�t�, �28�

for the domain 0 � r � R(t), and

w �1� � r

R�t�� 1� �29�

for the domain r R(t). The parameter � was chosensuch that the concentrations of the gaseous species andthe gas phase temperature achieved their initial valuesat the distance �R(t).

In the transformed computational domains the coor-dinates x ∈ [0, 1], w ∈ [0, 1], and can be treated iden-tically in numerical calculations. The gas–liquid inter-face is located at x � w � 0. The system of nonlinearparabolic partial differential Eqs. (1)–(5) was solvedusing the method of lines developed by Sincovec andMadsen (1975). The spatial discretization on a three-point stencil was used in order to reduce the system ofthe time-dependent partial differential equations to asemidiscrete approximating system of coupled ordinarydifferential equations. Thus the system of partial para-bolic differential equations is approximated by a systemof ordinary differential equations in time for functionsTe, T (L), Y (L)

A , and Yj at the mesh points. The meshpoints were spaced adaptively using the following for-mula:

xi � �i � 1N �n

, i � 1, . . . , N � 1, �30�

so that they cluster near the left boundary where thegradients are steep. In Eq. (30) N is the chosen numberof mesh points, n is an integer coefficient (in our cal-culations n is chosen equal to 3). The resulting systemof ordinary differential equations was solved using abackward differentiation method. Generally, in the nu-merical solution, 151 mesh points and an error toler-ance �10�5 in time integration were employed. Vari-able time steps were used to improve the computingaccuracy and efficiency.

During numerical solution of the system of Eqs. (9)–(10) the properties , cp, Dj, and ke were evaluatedsimultaneously at each grid point at each time step. Thecompilation of the formulas for calculating these prop-erties is presented by Reid et al. (1987), Ben-Dor et al.(2003a,b). Properties of the liquid and the gaseousphases were adopted from Reid et al. (1987). Calcula-tions were terminated when the condition R/R0 � 0.1was fulfilled.

4. Results and discussion

The above model of droplet evaporation in the pres-ence of inert admixtures containing noncondensable

FIG. 1. Henry’s law constant for aqueous solutions of differentsoluble gases vs temperature.


soluble gas was applied to study evaporation of waterdroplet immersed in a stagnant gaseous mixture com-posed of the ternary combinations of nitrogen (N2),soluble ammonia (NH3), soluble carbon dioxide (CO2),sulfur dioxide (SO2), and vapor of the water droplet.

For the purpose of validation of a computer code thepresent theoretical model was compared with the ex-perimental results obtained by Ranz and Marshall(1952), and with our previous model developed formoderately large droplets which assumed uniform tem-perature distribution inside the droplet (Ben-Dor et al.2003a). The dependence of droplet-squared diameterversus time for stagnant water droplet evaporating in astill dry air is shown in Fig. 2a. In these calculations the

effect of gas absorption was excluded. In this plot thesolid line indicates theoretical results obtained usingthe present model, the dots indicate experimental re-sults and the dashed line present the results obtained inBen-Dor et al. (2003a). The model suggested by Ben-Dor et al. (2003a) was developed for the case of smalldroplet Biot number when the temperature distributioninside the liquid droplet can be assumed to be uniform.As can be seen from this plot the present model dem-onstrates better agreement with experimental results ofRanz and Marshall (1952).

Using the present model and taking into account gasabsorption we calculated the dependence of dimension-less droplet radius R/R0 versus time for evaporating aswell as growing water droplets with the initial radii 25�m immersed into a gaseous mixture containing solublegas (sulfur dioxide; see Fig. 2b) and different concen-trations of water vapor. The curves were plotted forwater droplet with the initial temperature 278 K sus-pended in N2/SO2/H2O gaseous mixture with ambienttemperature equal to 298 K and concentration of sulfurdioxide of 0.1 ppm. The value of relative humidity var-ied in the range from 50% to 115%. The concentrationof sulfur dioxide equal to 0.1 ppm is pertinent to highlypolluted atmosphere.

In the following calculations we accounted for theeffect of gas absorption during droplet evaporation.The results obtained using the above described theo-retical model were compared with the experimental re-sults by Taniguchi and Asano (1992). The dependenceof the relative dissolved carbon dioxide concentrationin a droplet versus Fourier number is shown in Fig. 3.

FIG. 2. (a) Temporal evolution of the radius of the evaporatingwater droplet in dry still air. Solid line is the present model,dashed line is the nonconjugate model (Ben-Dor et al. 2003a),circles are experimental data (Ranz and Marshall 1952). (b) Tem-poral evolution of droplet radius for a droplet suspended inN2/SO2/H2O gaseous mixture with T0 � 278 K, Te,� � 298 K,R0 � 25 �m, [SO2]� � 0.1 ppm.

FIG. 3. Comparison of the calculated dimensionless averageaqueous concentration with the experimental data (Taniguchi andAsano 1992) and analytical solution for stagnant nonevaporatingdroplet.


The relative absorbate concentration is determined asfollows:

� �YA

�L�� YA,0

�L�

YA,s�L� � YA,0

�L�, �31�

where Y(L)A is the average concentration of the ab-

sorbed CO2 in the droplet:

YA�L�

�1

Vd� YA

�L��r�r2 sin� dr d� d�. �32�

In Eq. (32) r, �, and � are spherical coordinates. Tan-iguchi and Asano (1992) performed measurements forthe CO2–water spray system with the local mass flowrates of liquid in the range from 4 � 10�3 kg s�1 to8 � 10�3 kg s�1. The dashed line in the figure repre-

sents the analytical solution in the case of aqueous-phase controlled diffusion of carbon dioxide in a stag-nant nonevaporating droplet (see, e.g., Seinfeld 1986):

� � 1 �6

�2 �n�1

� 1

n2 exp��4�2n2Fo�. �33�

As can be seen from Fig. 3 for small Fourier numbersthe obtained results show good agreement with the ex-perimental data obtained by Taniguchi and Asano(1992) and analytical solution (33).

The typical concentration of carbon dioxide in atmo-sphere varies in the range from 300 to 360 ppm (Liu andLipták 1997) while concentration of sulfur dioxide var-ies from 0.01 ppb at the ground level in clean air to 0.2ppm and higher in a polluted air (Liu and Lipták 1997;Seinfeld 1986). The dependence of average aqueousSO2 and CO2 molar concentration versus time is shownin Figs. 4a,b and 5. The curves were plotted for differentvalues of the relative humidity (RH). The calculationspresented in Fig. 4a were conducted for a large dropletimmersed into a highly polluted atmosphere with theconcentration of SO2 equal to 0.1 ppm (Seinfeld 1986).Numerical results shown in Fig. 4b are typical forcleaner atmosphere (10 ppb of SO2). As can be seenfrom these plots the average concentration of the dis-solved species increases with decrease of RH. Since thesolubility and Henry’s constant of the dissolved admix-ture increase with temperature decrease, the evaporat-ing droplet with lower surface temperature can absorba higher amount of the dissolved species.

The dependence of the relative concentrations of thedissolved carbon dioxide and sulfur dioxide, �, in adroplet versus time is shown in Figs. 6–7. The curves areplotted for droplets with different radii evaporating

FIG. 4. (a) Temporal evolution of the average aqueous SO2

molar concentration for various values of relative humidity([SO2]� � 0.1 ppm). (b) Temporal evolution of the average aque-ous SO2 molar concentration for various values of relative humid-ity ([SO2]� � 0.01 ppm).

FIG. 5. Temporal evolution of the average aqueous CO2 molarconcentration for various values of relative humidity.


into the ambient N2/SO2 and N2/CO2 gaseous mixtureswith RH � 0. In calculations the concentrations of sul-fur dioxide and carbon dioxide were set equal to 0.1 and300 ppm, respectively (see, e.g., Pruppacher and Klett1997; Gravenhorst et al. 1978). These values of SO2 andCO2 concentrations are typical for atmospheric condi-tions. Inspection of these plots shows that for a dropletwith the radius 10 �m the saturation is reached after�0.01–0.02 s while several seconds are required for thedroplet with the radius 100 �m. We performed calcu-lations for evaporating and growing droplets. The de-pendencies of the normalized concentration of the dis-solved CO2 versus time for the case of droplet evapo-ration into the dry gaseous mixture and for the case of

droplet condensation in gaseous mixture with RH �105% are shown in Fig. 8. The calculations were per-formed for a water droplet with the radius of 25 �mevaporating in N2/CO2 gaseous mixture with a givencarbon dioxide concentration (300 ppm) at infinity. Theobtained results show that saturation is reached morerapidly in the case of droplet condensation. The latterphenomenon is associated with the magnitude of drop-let surface temperature. During condensation droplettemperature increases while during evaporation dropletsurface temperature decreases. Since Henry’s law con-stant depends exponentially on temperature, the in-crease of the surface temperature strongly affects equi-librium aqueous phase concentration. Therefore in thecase of droplet condensation the aqueous phase satu-ration is reached faster although the droplet volumeincreases.

It was emphasized above that Henry’s law constantand, consequently, concentration of the dissolved spe-cies at the droplet surface strongly depend on tempera-ture. Therefore droplet surface temperature is an im-portant parameter for investigating gas absorption byevaporating (growing) droplets. Figure 9 shows thedroplet surface temperature as a function of time. Thecalculations were performed for water droplet with theinitial radius 100 �m evaporating in N2/CO2 gaseousmixture with the different initial values of carbon diox-ide mass fraction (from 0.1 to 1.0) far from the droplet.Water vapor appears in the gaseous phase due to drop-let evaporation, and it constitutes the third species ofthe ambient gaseous mixture. It is assumed that initiallythe mass fraction of water vapor outside the droplet isequal to zero. Inspection of Fig. 9 shows that in the caseof large concentrations of carbon dioxide during the

FIG. 7. Temporal evolution of the dimensionless average aque-ous CO2 concentration (dry ambient gaseous mixture) for variousinitial sizes of evaporating droplet R0.

FIG. 6. Temporal evolution of dimensionless average aqueousSO2 concentration (dry ambient gaseous mixture) for various ini-tial sizes of evaporating droplet R0.

FIG. 8. Temporal evolution of the dimensionless averageaqueous CO2 concentration (R0 � 25 �m).


transient period of droplet evaporation the droplet sur-face temperature as a function of time passes throughthe maximum. This well-pronounced nonlinear behav-ior of the droplet surface temperature stems from theinteraction of several different phenomena. In particu-lar, thermal effect of dissolution and Stefan flow are thecause of the maximum of droplet surface temperatureduring the transient period of droplet evaporation (seeFig. 10). As can be seen from Fig. 10 neglecting theseeffects we obtain the result that is typical for dropletevaporation in gaseous mixture containing nonsolublegaseous species. Calculations were also performed forwater droplets evaporating in N2/SO2 and N2/NH3 gas-eous mixture (Figs. 11 and 12). As can be seen fromFigs. 10–12 in the case of CO2 dissolution thermal effectof dissolution is essential for large concentrations ofcarbon dioxide in the ambient gaseous mixture while inthe case of SO2 and NH3 dissolution this effect is es-sential for significantly smaller concentrations of SO2 inthe gaseous phase. Inspection of Fig. 11 shows that inthe case of NH3 dissolution the thermal effect is detect-able when ammonia concentration is of the order of 103

ppm. The essential differences in surface temperatureat steady-state stage in the case of water droplet evapo-ration into N2/CO2 gaseous mixture (see Fig. 9) areassociated with large differences between heat andmass transport coefficients for carbon dioxide and ni-trogen. Figure 13 shows the droplet surface tempera-ture as a function of time for droplets with the radius100 �m growing in N2/CO2/H2O gaseous mixture(YH2O,� � 0.011). The curves were plotted for waterdroplets growing in N2/CO2/H2O gaseous mixture with

different initial concentrations of CO2 at infinity. Ascan be seen from Fig. 13 the droplet surface tempera-ture rises above the ambient temperature. Similar tothe case of droplet evaporation the essential differencesin surface temperature at the steady-state stage duringdroplet condensation are associated with large differ-ences between heat and mass transport coefficients forcarbon dioxide and nitrogen.

Summarizing the above findings it can be concluded

FIG. 11. Temporal evolution of the surface temperature for awater droplet evaporating in N2/NH3/H2O gaseous mixture, R0 �100 �m, T0 � 274 K, Te,� � 288 K.

FIG. 9. Temporal evolution of the surface temperature for awater droplet evaporating in N2/CO2/H2O gaseous mixture, R0 �10 �m, T0 � 274 K, Te,� � 288 K for various ambient concentra-tions of CO2.

FIG. 10. Effect of Stefan flow and thermal effect of absorptionon droplet surface temperature for a water droplet evaporating inN2/CO2/H2O gaseous mixture, YCO2

� 0.9, R0 � 100 �m, T0 � 274K, Te,� � 288 K. Line 1 includes effects of Stefan flow and thermaleffect of absorption, for line 2 the thermal effect of absorption isneglected, and for line 3 the effects of Stefan flow and the thermaleffect of absorption are neglected.


that droplet temperature, interfacial absorbate concen-tration, and the rate of droplet evaporation (condensa-tion) during gas absorption are highly interdependent.Gas absorption by evaporating (growing) droplets isaffected by simultaneous heat and mass transfer pro-cesses: diffusion of soluble gas into the droplet and heateffect of absorption. A number of physical parameters,such as, Stefan velocity, Henry’s law constant and drop-let surface temperature, have a strong effect on increas-ing (decreasing) evaporation (condensation) rate, inter-facial absorbate concentration, etc. The scheme of theinterrelation between heat and mass transport duringtransient period of droplet evaporation with gas ab-

sorption is shown in Fig. 14. The weak effects associ-ated with the dependence of heat of absorption anddiffusion coefficient of soluble gas in a liquid phase ontemperature were neglected in our model. In contrastto droplet evaporation during droplet condensation thevapor flux and the hydrodynamic flow (Stefan flow) aredirected toward the droplet surface. Vanishing of thesurface temperature maximum during transient periodof condensation (see Fig. 13) is a result of these phe-nomena. Nevertheless the above analysis demonstratesthat in the case of absorption by evaporating dropletsthe heat and mass transfer processes are also highlyinterdependent.

The results of numerical calculations (see Figs. 9–13)show that small amounts of soluble gases in a gaseousmixture, corresponding to their typical concentrations

FIG. 12. Temporal evolution of the surface temperature for awater droplet evaporating in N2/SO2/H2O gaseous mixture, R0 �100 �m, T0 � 274 K, Te,� � 288 K.

FIG. 13. Temporal evolution of the surface temperature for awater droplet condensation in N2/CO2/H2O gaseous mixture withYH2O � 0.011, R0 � 100 �m, T0 � 274 K, Te,� � 288 K.

FIG. 14. Schematic description of the interrelation between heatand mass transport during the transient period of droplet evapo-ration with gas absorption.


in a clean atmosphere, do not have essential influenceon evaporation or condensation of liquid droplets. Atthe same time soluble gases with higher concentrationsaffect droplet evaporation or condensation. Conse-quently, in order to analyze this effect we performednumerical calculations of droplets evaporation or con-densation for the concentrations of soluble gases,higher than those for typical atmospheric conditions. Itshould be emphasized that even for small concentra-tions of soluble species taking into account the mutualinfluence of absorption and condensation (evapora-tion) allows us to analyze correctly the influence ofcondensation (evaporation) on gas absorption. The lat-ter stems from the nonlinear character of heat and masstransfer equations describing gas absorption and drop-let evaporation (growth).

5. Conclusions

In this study we developed a model that takes intoaccount the mutual effects of gas absorption and evapo-ration (condensation) of liquid droplet in the ambientatmosphere composed of liquid droplet vapor and inertnoncondensable and nonabsorbable gas or noncon-densable and soluble gas. The results obtained in thisstudy can be summarized as follows.

1) The suggested model of droplet evaporation (con-densation) in the presence of soluble trace gasestakes into account a number of effects that wereneglected in all previous studies, such as effect of gasabsorption on Stefan velocity and thermal effect ofabsorption on droplet evaporation (condensation).It is demonstrated that droplet evaporation or con-densation rate, droplet temperature, interfacial ab-sorbate concentration, and the rate of mass transferduring gas absorption are strongly interdependent.We performed the detailed analysis of the interre-lation between heat and mass transport during tran-sient period of droplet evaporation (condensation)with gas absorption. The scheme of interrelation be-tween droplet evaporation (condensation) and gasabsorption is presented in Fig. 14.

2) During droplet evaporation the interfacial tempera-ture as a function of time shows a maximum thatincreases with the increase of the ambient concen-tration of absorbate. For CO2 dissolution the inter-facial temperature as a function of time shows amaximum for large concentration of carbon dioxidein the ambient gaseous mixture while in the case ofSO2 and NH3 dissolution the maximum of tempera-ture arises even for small concentration of absorbatein the gaseous phase. In contrast to droplet evapo-

ration during droplet condensation vapor flux andthe hydrodynamic flow (Stefan flow) are directedtoward the droplet surface. It is shown that vanish-ing of the surface temperature maximum during thetransient period of condensation is a result of thesephenomena.

3) The results obtained using the suggested model forCO2 absorption by water droplets agree with theexperimental data by Taniguchi and Asano (1992)for CO2 absorption by falling noncirculating waterdroplets and with analytical solution obtained forthe case of aqueous-phase controlled mass transferinside a stagnant nonevaporating droplet.

4) It is demonstrated that the time of saturation duringgas absorption by liquid droplets only weakly de-pends on the value of the RH.

5) It is shown that the average concentration of thedissolved species strongly depends on RH for highlysoluble and for slightly soluble gaseous atmosphericpollutants. Therewith the difference between the av-erage concentrations of the dissolved species in wa-ter droplets attains tens of percent for different val-ues of RH.

The performed analysis of gas absorption by liquiddroplets accompanied by droplets evaporation and va-por condensation on the surface of liquid droplets canbe used in calculations of scavenging of hazardous gasesin atmosphere by rain and atmospheric clouds evolu-tion.

APPENDIX A

Symbol Definitions

cp Specific heat at a constant pressure,J kg�1 K�1

Dj Diffusion coefficient of jth species inthe gaseous phase, m2 s�1

Djk Binary diffusion coefficient in the gas-eous phase, m2 s�1

Fo � DLt/4R2 Fourier numberHA Henry’s law constant, mole m�3 Pa�1

k Thermal conductivity, W m�1 K�1

L Latent heat of evaporation, J kg�1

La Heat of dissolution, J kg�1

Mj Molar mass of jth species, kg mole�1

mL Mass of the droplet, kgp Pressure, Par Radial coordinate, mR Radius of the droplet, mRg Universal gas constant, J mole�1 K�1

RG Gas constant, J kg�1 K�1

RH Relative humidity


T Temperature, Kt Time, s , u Velocity, m s�1

Vd Droplet volume, m3

Y Mass fraction

Greek symbols

� Thermal diffusivity, m2 s�1

Density, kg m�3

Subscripts and superscripts

0 Initial value1 Volatile speciesA Absorbatee Value outside a dropletj Number of a speciesL, (L) Liquids Value at the droplet surface� Value at infinity

APPENDIX B

Transformed System of Eqs. (17)–(5) andAppropriate Initial and Boundary Conditions

After the transformation of the spatial coordinatesystem with the aid of dimensionless independent vari-ables (27)–(29) and the following dimensionless vari-ables:

� �R�t�

R0, �e �

Te

Te,�, � �L� �

T �L�

Te,�, �B1�

the Eqs. (1)–(5) can be rewritten in the following di-mensionless form:

DL�2

�L

�� L�

� �

�� L�

�x � 2

�1 � x��

DL�

�L�1 � x��

�2� �L�

�x2 ,

�B2�

�2�YA

�L�

� �

�YA�L�

�x � 2

�1 � x�� 1 � x��

�2YA�L�

�x2 ,

�B3�

�2�2��e

� �

��e

�w��uR0�

DL� ��w � 1� �

2ke

��w � 1��

1�

�

�w �ke

��e

�w �, �B4�

�2�2�Yj

� �

�Yj

�w��uR0�

DL� ��w � 1� �

2�Dj

��w � 1��

1�

�

�w ��Dj

�Yj

�w �, �B5�

where � � cpDL, � � DL, and u � u(w, �) is thefunction of dimensionless time � and dimensionless co-ordinate w.

Equations (1)–(5) must be supplemented with twoequations for determining gas velocity and for the rateof change of droplet’s radius

u ��sus

��w � 1�2 , �B6�

where

us ��LDL

�R0��1 � Y1�

�YA�L�

�x�

x�0�

D1

�R0��1 � Y1�

�Y1

�w�

w�0

and

�� 1

�1 � Y1�

�YA�L�

�x�

x�0�

�D1

�LDL��1 � Y1�

�Y1

�w�

w�0.

�B7�

Dimensionless boundary conditions for x � 0, w � 0read

�Yj

�w�w�0�

��R0

DjusYj, s , j � 2, . . . K � 1, j � A,

�B8�

Y1, s�0, � � Y1, s��s� ��1, s

��

p1, s��s�M1

p�M, �B9�

��L� � �e , �B10�

�YA

�w�

w�0�

��R0us

DAYA, s �

DL�L�

DA�

�YA�L�

�x�

x�0, �B11�

��L

DL

�� L�

�x�

x�0�

La

cLT�

�YA�L�

�x�

x�0

�ke

�LDLcL�

��e

�w�w�0�

L�

cLT�

�d�

d , �B12�

and YA,s can be found using Eq. (22).Dimensionless boundary conditions for x � 1 and

w � 1 read

�Yj

�w� 0, �B13�

��e

�w� 0, �B14�

�� L�

�x� 0, �B15�

�YA�L�

�x� 0. �B16�


Initial conditions for � � 0, and 0 � x � 1, 0 � w � 1,are as follows:

� �L� � �0�L�, YA

�L� � YA,0�L� , �B17�

�e � �e,0�w�, Yj � Yj,0�w�. �B18�

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