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Uptake of soluble gaseous pollutants by rain droplets in the atmospherewith nocturnal temperature profile

Tov Elperin, Andrew Fominykh, Boris Krasovitov⁎Department of Mechanical Engineering, The Pearlstone Center for Aeronautical Engineering Studies, Ben-Gurion University of the Negev, P. O. B. 653, 84105, Israel

a r t i c l e i n f o a b s t r a c t

Article history:Received 7 March 2010Received in revised form 16 September 2010Accepted 17 September 2010

We analyze the uptake of gaseous pollutants by the rain droplets falling in the atmosphere withnocturnal temperature inversion. The rate of uptake of soluble trace gases by falling raindroplets is determined by solving energy and mass conservation equations. In the analysis weaccounted for the accumulation of the soluble gas and energy in the bulk of the falling raindroplet. The problem is solved in the approximation of a thin concentration and temperatureboundary layers in the vicinity of the droplet surface. It is assumed that the bulk of a droplet,beyond the diffusion boundary layer, is completely mixed and distributions of concentration ofthe absorbate and temperature are homogeneous and time-dependent in the bulk. Theproblem is reduced to a system of linear-convolution Volterra integral equations of the secondkind which is solved numerically. Calculations are performed using available experimental dataon nocturnal temperature profiles in the atmosphere. It is shown than if the concentration ofgaseous pollutants in the atmosphere is homogeneous and the absolute temperature in theatmosphere increases with altitude, a droplet absorbs gas during all the period of its fall. In thecase when the temperature–altitude curve comprises the nocturnal inversion and temperaturefall segments, gas absorption by a falling rain droplet can be replaced by desorption and viceversa. Neglecting temperature inhomogeneity in the atmosphere caused by nocturnaltemperature profile leads to significant underestimation of the concentration of gaseouspollutants inside a droplet on the ground. The calculations performed using temperatureprofiles measured by Corsmeier et al. (1997) showed that the underestimation of theconcentration of gaseous pollutants in rain droplets at the ground can exceed 20%.

© 2010 Elsevier B.V. All rights reserved.

Keywords:AtmosphereGaseous pollutantsRain dropletsGas scavengingNocturnal temperature profile

1. Introduction

Gas absorption by the falling rain droplets is of relevancein meteorology and environmental engineering. Rains play animportant role in wet removal of gaseous pollutants from theatmosphere. Scavenging of atmospheric gaseous pollutantsby rain droplets is a result of a gas absorption mechanism(Pruppacher and Klett, 1997). Comprehensive study of masstransfer during gas absorption by falling rain droplets is alsorequired for predicting transport of hazardous gases in theatmosphere. Vertical transport of soluble gases in theatmospheric boundary layer (ABL) is an integral part of the

atmospheric transport of gases and is important for under-standing the global distribution pattern of soluble trace gases.An enhanced understanding of the cycle of soluble gases isalso important for the analysis of global climate change (seee.g., Aalto et al., 2006). Clouds and rains play a significant rolein vertical redistribution of SO2, NH3 and other soluble gasesin the atmosphere (see, e.g. Zhang et al., 2006). Scavenging ofsoluble gases, e.g., SO2, and NH3 by rain affects the evolutionof vertical distribution of these gases. At the same time thevertical gradients of the soluble gas concentration in theatmosphere affect the rate of gas absorption by rain droplets.Notably, the existing models of global transport in theatmosphere (see, e.g. de Arellano et al., 2004) do not takeinto account the influence of rains on biogeochemical cyclesof different gases.

Atmospheric Research 99 (2011) 112–119

⁎ Corresponding author. Tel.: +972 8 6477067; fax: +972 8 6472813.E-mail address: [email protected] (B. Krasovitov).

0169-8095/$ – see front matter © 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.atmosres.2010.09.012

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Vertical temperature distribution in the atmosphere wasdiscovered in 1749 by A. Wilson (see Wilson, 1826). InspiredbyWilson, numerous measurements andmodeling of verticaltemperature distribution in the atmosphere (see, e.g. Dines,1911; Taylor, 1960; Manabe and Wetherald, 1967) revealedthe existence of a 6.5K ⋅km−1 lapse rate. Evolution of thelapse rate during the last decades is discussed by Trenberthand Smith (2006).

Enhanced interest to a problem of nocturnal temperatureinversion during the last decades can be explained by theimportance of these meteorological conditions for thedispersion of air pollutants and for fog and frost formation.In his pioneering theoretical study Brunt (1934) showed thatinfrared radiative transfer behaves like a diffusive processleading to the exponential type temperature profile with astrong negative curvature. Brunt (1934) assumed that theheat flux at the ground is constant throughout the night. Laterstudies were devoted to forecasting of temperature profileevolution in the lower layers of the atmosphere, the height ofthe inversion layer and inversion strength (see, e.g. Anfossiet al., 1976; Surridge and Swanepoel, 1987; Anfossi, 1989;Gassmann and Mazzeo, 2001). Two latter parameters arerelated to the radiation heat flux from the ground to the nightsky and the rate at which heat propagates through theatmosphere to the ground. It was demonstrated (see, e.g.Anfossi, 1989) that the height of the inversion layer canexpand to hundreds of meters.

Influence of vertical distribution of the temperature in theatmosphere on the rate of gas scavenging by falling raindroplets is explained by a strong nonlinear dependence of thesolubility parameter (Henry's constant) for aqueous solutionsof different gases on the temperature (see, e.g. Reid et al.,1987). Accounting for vertical distributions of soluble gaseouspollutants and temperature in the atmosphere requiressolution of energy and mass conservation equations whichdescribe gas absorption by falling rain droplets.

Due to the differences in solubility of gases in liquids, masstransfer during absorption of a soluble gas by droplets in thepresence of an inert admixture can be continuous-phasecontrolled, liquid-phase controlled or conjugate. Continuous-phase controlled mass transfer by falling droplets wasdiscussed by Kaji et al. (1985), Altwicker and Lindhjem(1988), Waltrop et al. (1991) and Wurzler (1998). Liquid-phase controlledmass transfer was studied, e.g., by Amokraneand Caussade (1999) and Chen (2001). Mass transfercontrolled by both phases was analyzed by Walcek andPruppacher (1984), Chen (2004), Elperin and Fominykh(2005), Elperin et al. (2007, 2008, 2009).

Accumulation of the dissolved atmospheric gases in afallingwater droplet during gas absorption is determined by asystem of unsteady convective diffusion and energy conser-vation equations. An analytical solution of these equationsrequires application of rather sophisticated methods (see, e.g.Ruckenstein, 1967). Moreover, in the Earth atmosphere theproblem is complicated by the vertical gradients of theabsorbate concentration and temperature in the gaseousphase.

The effect of altitudinal distribution of the soluble gases inthe atmosphere on the rate of gas absorption by falling raindroplets was investigated by Elperin et al. (2009). Thesuggested approach includes applying the generalized similar-

ity transformation to a system of transient equations ofconvectivediffusion andDuhamel's theorem.Then theproblemreduces to the numerical solution of a linear-convolutionVolterra integral equation of the second kind. Note that in ourprevious study (Elperin et al., 2009) it was assumed that theatmospheric temperature distribution is uniform.

In the present study we investigate the effect of thenocturnal temperature profile in the atmosphere on the rateof gas absorption by falling droplets. In the case of theuniform vertical distribution of the trace gas and temperaturein the atmosphere the solution obtained in the present studyrecovers the expressions obtained in our previous study(Elperin and Fominykh, 2005). The latter approach wasvalidated by comparing the numerical solution with availableexperimental data for CO2 and SO2 absorption by a fallingdroplet (see Altwicker and Lindhjem, 1988; Amokrane andCaussade, 1999).

2. Description of the model

Consider absorption of a soluble gas from a mixturecontaining an inert gas by a moving droplet. At time t=0 thedroplet begins to absorb gas from the atmosphere. Distribu-tion of the concentration of the absorbate and temperature inthe gaseous phase in the vertical direction is assumed to beknown. Schematic view of a falling droplet with the attachedframe of coordinates is shown in Fig. 1.

In the analysis we account for the resistance to heat andmass transfer in both phases and use the following simplifyingassumptions: 1) we employ the approximation of the infinitedilution of the absorbate in the absorbent; 2) thicknesses of thediffusion and temperature boundary layers in both phases areassumed small compared with the droplet's size; 3) tangentialmolecular heat and mass transfer rates along the surface of aspherical droplet are assumed small compared with themolecular heat andmass transfer rates in the normal direction;

Fig. 1. Schematic view of a falling droplet.

113T. Elperin et al. / Atmospheric Research 99 (2011) 112–119


4) the bulk of a droplet, beyond the diffusion and temperatureboundary layers, is assumed to be completely mixed and theconcentration of the absorbate and the temperature arehomogeneous in the bulk; 5) the droplet has a sphericalshape; 6) internal circulation inside the droplet and the gasvelocity in the vicinity of the droplet have axial symmetry; and7) the solubility parameter (Henry's constant) depends uponthe temperature.

It must be emphasized that the timescale of gas absorptionby falling rain droplets is small in comparison with thetimescales of the macro processes occurring in the atmo-sphere. For example the timescale of rain droplet saturationby the absorbate for a given temperature varies in the rangefrom 10−1 to 1 s (depending on droplet radii) and the time ofa rain droplet fall is of the order of 102 s while a timescale forthe effects of turbulence on the whole Stable Boundary Layer(SBL) is of the order of 7 to 30 h (see e.g. Stull, 1988). Thecharacteristic vertical length scale of spatial stratification offalling droplets by turbulence is of the order of ℓ~DT /U,where DT is the coefficient of turbulent diffusion. In a stablystratified boundary layer ℓ~1÷10 m (see, e.g. Sofiev et al.,2009). Clearly,ℓbbL0, where L0=1 km is the initial height ofthe droplet. Consequently, the effects of large-scale turbulentmotion on spatial distribution of droplets can be neglected.

The assumptions about the circulation inside a droplet andthat the droplet has a spherical shape are valid in the followingranges of the falling in air water droplet radii, Reynoldsnumbers and velocities: 0.1mm≤R≤0.6mm, 10≤Re≤370and 0.7≤U≤4.6 (m s−1) (see, e.g. Pruppacher and Klett, 1997,Chapter 10). Circulations inside a falling droplet are caused byshear stresses at the droplet–gas interface. The existence ofinternal circulations allows us to assume that the liquid inside adroplet is fully mixed and distinguish between the two regionsin the droplet, adjacent to the droplet's surface diffusion ortemperature boundary layers and a bulk. A similar approachwas applied for the analysis of liquid extraction by thedispersed phase (see Uribe-Ramirez and Korchinsky, 2000).The analysis offluid flow around amoving droplet showed thatat different Reynolds numbers the tangential fluid velocitycomponent in the vicinity of a gas–liquid interface can beapproximated by the following equation (see, e.g. Pruppacherand Klett, 1997, p. 392):

vϑ = −kU sinϑ; ð1Þ

where the coefficient k is equal to 0.04 in the range of theexternal flow Reynolds numbers (Re=2URρ2/μ2) from 10 to103 (see, e.g., Pruppacher and Klett, 1997, p. 386). Thedependence of the terminal fall velocity of liquid droplets ontheir diameter was analyzed by Pruppacher and Klett (seePruppacher and Klett, 1997, Chapter 10). In this study weassume that gas absorption does not disturb temperaturedistribution in gaseous and liquid phases. At the same timeheattransfer between the atmosphere and a falling droplet affectsthe rate of gas absorption/desorption by a falling droplet. Thisdependence is explained by a very strong variation of thesolubility parameter (Henry's constant) with the temperature(see, e.g. Reid et al., 1987). Since the dependence of otherthermodynamic parameters on the temperature is by the orderof magnitude weaker, we assume them to be constant.Following the approach suggested by Ruckenstein (1967) we

arrive at the following system of transient equations ofconvective diffusion and energy conservation for the liquidand gaseous phases which account for convection in radial andtangential directions:

∂xi∂t + U⋅k − sinϑ

R ⋅∂xi∂ϑ +

2ycosϑR ⋅

∂xi∂y

� �= Di

∂2xi∂y2

; ð2Þ

∂Ti∂t + U⋅k − sinϑ

R∂Ti∂ϑ +

2ycosϑR

∂Ti∂y

� �= ai

∂2Ti∂y2

; ð3Þ

where i=1, 2. The radial fluid velocity component in Eqs. (2)and (3) is determined by Eq. (1) and the continuity equation:vr=2k cos ϑ Uy /R. Eqs. (2) and (3) are written in a frameattached to the falling droplet and valid for y≪R. Since thevelocity of the droplet fall is known and z=U⋅ t, the verticalcoordinate-dependent boundary conditions can be trans-formed into the time-dependent boundary conditions. Thevertical coordinate z is alignedwith the direction of the dropletfall. The initial and boundary conditions to Eqs. (2) read:

xi = xi0; at t = 0 ð4Þ

x2 = xb2 tð Þ as y→∞; ð5Þ

x1 = xb1 tð Þ as y→−∞; ð6Þ

x1 = mx2 at y = 0; ð7Þ

ND 1= ND 2

at y = 0: ð8Þ

The initial and boundary conditions to Eqs. (3) can bewritten as follows:

Ti = Ti0; at t = 0 ð9Þ

T2 = Tb2 tð Þ as y→∞; ð10Þ

T1 = Tb1 tð Þ as y→−∞; ð11Þ

T1 = T2 at y = 0; ð12Þ

NT1= NT2

at y = 0; ð13Þ

where NDi= −DiCi

∂xi∂y , NTi

= −λi∂Ti∂y , y=r−R, y≪R, coeffi-

cient m=HARgTC2 /C1 is a distribution coefficient thatcharacterizes the solubility of gases in liquids, HA is Henry'sconstant (see Seinfeld and Pandis, 2006, p. 288) and Rg is theuniversal gas constant. The equations of convective diffusionand energy conservation (2) and (3) describe variations ofconcentration and temperature near the gas–droplet interfaceinside and outside the droplet in a boundary-layer approxima-tion. The characteristic values of the diffusion Peclet number ofthe droplet in this study are of the order of 106 and 102 for theliquid and gaseous phases, respectively, and the characteristicvalues of the temperature Peclet number of the droplet are ofthe order of 104 for the liquid phase and 102 for the gaseousphase. Consequently, the use of the boundary-layer approxi-mation is justified. The solution of Eqs. (2) and (3) can beobtained using the similarity transformation method whichwas suggested first by Ruckenstein (1967).

114 T. Elperin et al. / Atmospheric Research 99 (2011) 112–119


3. Method of solution

Since the boundary conditions (5)–(8) and (10)–(13) toEqs. (2) and (3) are time-dependent, the solution can befound by combining the similarity transformation methodwith Duhamel's theorem. Let us introduce the following self-similar variables (for details see Ruckenstein, 1967):

ηTi=

yδTi t;ϑð Þ =

YΔTi

: ηDi=

yδDi

t;ϑð Þ =YΔDi

: ð14Þ

Since gas absorption (desorption) does not affect distri-bution of the temperature in both phases, heat transferequation (Eqs. (3) and (9)–(13)) can be solved independent-ly from the mass transfer equation. Variables ηTi allow us toobtain the solution of a system of partial differential Eq. (3) inthe following form:

Θi Y;ϑ;τð Þ= ∂∂τ∫

τ

0½Θbi ξð Þ+ −1ð Þi−1⋅ γTað Þi−1⋅ Θb2 ξð Þ−Θb1 ξð Þ½ �

1 + γTa

erfcY

ΔTiϑ;τ−ξð Þ

!�dξ: ð15Þ

The solution of Eq. (2) with boundary conditions (5)–(10)reads:

Xi Y ;ϑ; τð Þ = ∂∂τ∫

τ

0½Xbi ξð Þ + −1ð Þi Xb1 ξð Þ−m Θsð ÞXb2 ξð Þð Þ⋅ γDð Þi−1

1 + m Θsð ÞγD

erfcY

ΔD iϑ;τ−ξð Þ

!�dξ; ð16Þ

where γT=cp1C1/cp2C2, a =ffiffiffiffiffiffiffiffiffiffiffiffiffiffia1 = a2

p, Θs=Ts /T10,

Δ2Di

=4

PeD isin4 ϑð Þfcos ϑð Þ−1

3cos3 ϑð Þ−½ 1−f ϑ; τð Þ

1 + f ϑ; τð Þ

−13

1−f ϑ; τð Þ1 + f ϑ;τð Þ� �3�g; Δ2

Ti= Δ2

Di⋅ aiDi

;ð17Þ

f ϑ; τð Þ = tg2ϑ2

� �exp 2τð Þ, PeD i

= RkUDi

, PeTi =RkUai

, τ= tUk /R

is dimensionless time, Y=y /R. The expression for thetemperature at the surface of a droplet reads:

Θs τð Þ = Θb1 τð Þ + Θb2 τð Þ−Θb1 τð Þ1 + γTa

: ð18Þ

The variables Θb1(τ) and Xb1(τ) are the unknown func-tions of time which can be determined by means of anintegral energy andmaterial balances for the droplet (see, e.g.Elperin et al., 2009):

dΘb1

dτ=

32⋅PeT 1

∫π

0

∂Θ1

∂Y jY=0

sinϑ dϑ; ð19Þ

dXb1

dτ=

32⋅PeD 1

∫π

0

∂X1

∂Y jY=0

sinϑ dϑ: ð20Þ

Substituting expression for the temperature in the droplet(Eq. 15) into Eq. (19) yields:

Θb1 tð Þ = 1 +3ffiffiffi

πp

⋅PeT1 1 + γTað Þ∫τ

0Θb2 ξð Þ−Θb1 ξð Þ½ � ∫

π

0

sinϑ⋅dϑ⋅dξΔT1

ϑ; τ−ξð Þ ;

ð21Þ

where Θb1=Tb1/T10. Substituting expression (16) for theabsorbate concentration in the droplet into Eq. (20) yields:

Xb1 τð Þ = xb10m0xb20

+3ffiffiffi

πp

⋅PeD 1

∫τ

0

Xb1 ξð Þ−m Θs τð Þð ÞXb2 ξð Þ1 + m Θs τð Þð Þ⋅γD

� �∫π

0

sinϑdϑdξΔD 1

ϑ; τ−ξð Þ :

ð22Þ

Note that Henry's constant in Eq. (22) is a function of thetemperature. The magnitude of the temperature at a dropletinterface is determined by Eqs. (18) and (21). The functionaldependence of Henry's law constant vs. temperature reads(see, e.g. Seinfeld and Pandis, 2006):

lnHA T0ð ÞHA Tð Þ =

ΔH*Rg

1T− 1

T0

� �; ð23Þ

where ΔH* is the enthalpy change due to transfer of theabsorbate from the gaseous phase to liquid, Rg is the universalgas constant. Time dependence of solute concentration in thebulk of falling droplets is determined via solving Eqs. (18),(21)–(23).

If temperature distribution in the atmosphere is homoge-neous, the problem reduces to solving Eq. (22) with a constantparameter of solubilitym0.

Eqs. (21) and (22) are linear-convolution Volterra integralequations of the second kind and can be written in thefollowing form:

f tð Þ = ∫t

0f ξð ÞK t; ξð Þdξ + g tð Þ: ð24Þ

The method of the solution of the integral Eq. (24) isbased on approximating the integral in Eq. (24) using somequadrature formula:

∫b

aF ζð Þdζ = ∑

N

i=1αiF ζið Þ + RN F½ �;

where ζi∈ [a,b], i=1,2,...,N;αi — coefficients which areindependent of the function F; RN[F] — remainder of theseries after the N-th term. Using a uniform mesh with

an increment h(Ti=T0+ ih, h≡TN−T0N ) and applying the

trapezoidal integration rule yield:

f 0ð Þ = g 0ð Þ;

1−12h⋅Ki i

� �⋅fi = h

12Ki 0 f0 + ∑

i−1

j=1Ki j ⋅ fj

!+ gi;

ð25Þ

where i=1,…, N, fi= f(i ⋅h), gi=g(i ⋅h), Ki j=K(i ⋅h, j ⋅h).In order to solve the system of Eqs. (21) and (22) we can

represent Eq. (24) as a vector equation for the vector of M(M=1, 2) functions f(t). In this case thekernelK(t,ξ) is aM×M



matrix andEq. (24) canbe viewedasa vector equation. For eachiwe solved theM×M set of linear algebraic equations using theGaussian elimination method.

4. Results and discussion

Calculations of temperatures and concentrations of thedissolved gas inside water droplets were performed for therain droplets with diameter 1.2 mm falling in the non-homogeneous atmosphere containing a soluble gaseouspollutant. This diameter of rain droplets is equal to theaverage diameter of rain droplets of Feingold–Levin dropletsize distribution (Feingold and Levin, 1986) corresponding toa low rain intensity of 5–6 mm/h.

The suggestedmodelwas applied for the numerical analysisof uptakeof soluble ammonia (NH3) and sulfurdioxide (SO2) byrain droplets falling in the atmosphere with the nocturnaltemperature profile. The thermophysical properties of theammonia–water and sulfur dioxide–water systems are pre-sented in Table 1. In numerical calculations we used thetemperature profiles measured during the night from 23 to 24September 1994 by Corsmeier et al. (1997) in Essen andHannover, Germanywhile the concentration of soluble gaseouspollutants in the atmosphere was assumed to be uniform. Theknown concentration profile as well as the temperature profilein the external gaseous mixture appears explicitly in theobtained analytical solution (see Eqs. (18), (21) and (22)).Therefore numerical calculations require the knowledge of thetemperature and concentration profilesmeasured or calculatedusing a certainmathematicalmodel of heat andmass transfer inthe Atmospheric Boundary Layer (ABL). However, to the best ofour knowledge, there are no publications where nocturnaltemperature and concentration profiles were measured simul-taneously. Moreover, gas absorption by falling rain droplets isgoverned by various physical and chemical phenomena, e.g.droplet evaporation, internal circulation, effect of externalconcentration and temperature profiles, chemical reactions inthe atmosphere, dissociation of the dissolved molecules intoions, etc. Taking into account all these effects renders theproblem quite involved. Consequently, when a certain param-eter causes only the quantitative changes in the solution weprescribed a fixed value to this parameter while otherparameters vary. The nocturnal vertical profile of the atmo-spheric temperature measured by Corsmeier et al. (1997)shows an increase of the atmospheric temperature with heightfrom a ground up to approximately 500 m and a furthertemperaturedecreasewithheight forheights larger than500 m(see Fig. 2). At the same time the nocturnal profile in Hannoverreveals a strong temperature increase up to 1000 m altitude(see Fig. 3). The vertical profiles of the atmospheric tempera-ture were approximated using polynomial functions (the

coefficients of the polynomial function are showed in Figs. 2and 3). The droplet surface temperature was calculated usingEqs. (18) and (21).

In our previous study we showed that when distributionsof soluble trace gas and temperature in the atmosphere areuniform, the concentration of the dissolved gas in the dropletattains saturation after a certain time interval; at the finalstage of their fall, droplets do not absorb soluble trace gases(see Elperin et al., 2009). The presence of temperature andconcentration gradients in the atmosphere changes thescenario of gas absorption by falling rain droplets.

The effect of nocturnal temperature distribution in theatmosphere on soluble gas scavenging by falling rain droplet(see Figs. 4 and 5) is analyzed via the numerical solution ofthe system of Eqs. (21) and (22). The temperature of thedroplet surface is calculated using Eq. (18) and the depen-dence of Henry's constant on the temperature is taken intoaccount using Eq. (23). The initial temperature of a dropletwas assumed to be equal to an air temperature in the below-cloud atmosphere immediately adjacent to the cloud.Dependence of the interfacial temperature of falling raindroplet vs. altitude in the atmosphere is shown in Figs. 2 and3. The concentration of the trace soluble gas in theatmosphere was assumed to be uniform. Calculations of thedissolved average gas concentration were performed (seeFigs. 4 and 5) for rain droplets initially saturated with solublegas (solid curves) and for the rain droplet with a negligiblysmall initial concentration of the dissolved gas (dashedcurves). In the case when the soluble gas is distributeduniformly in the atmosphere and the atmospheric tempera-ture depends on the altitude (see Fig. 2) gas absorption by thefalling rain droplet is replaced by desorption and vice versa(see Fig. 4). In particular, in a region where the temperaturein the atmosphere decreases with height (as it is shown inFig. 2), the initially saturated falling droplet desorbs gas intothe atmosphere (see Fig. 4). In contrast, the droplet with aninitial negligibly small concentration of the dissolved gasabsorbs gas from the atmosphere in this region. Inspection ofFig. 4 shows that eventually absorption is replaced bydesorption. Up to the altitude of approximately 500 m thetemperature increases with height due to nocturnal inver-sion. In this region gas desorption is replaced by absorption.This complicated behavior can be explained by the depen-dence of Henry's constant and gas solubility in the liquid onthe temperature. The lower the temperature the higher thesolubility of the trace gas in the liquid and the higher theconcentration of saturation of a soluble gas in a droplet. In thecase of the inversed nocturnal atmospheric temperatureprofile the rain droplet absorbs the soluble gas during all theperiod of its fall (Fig. 5). The latter behavior does not dependon the initial concentration of the dissolved gas in a droplet.

Table 1Thermophysical properties.

Gas HA m D1 D2 ΔH*

10−2 mol m−3 Pa−1 10−9m2 s−1 10−5 m2 s−1 105 J mol−1

SO2 1.21 2.35 ⋅10−2 1.67 1.22 −2.615NH3 61.2 1.184 1.7 2.1 −3.421



The results of numerical calculations of gas absorption of SO2

by initially saturated rain droplets falling in the atmospherewith temperature profiles measured by Corsmeier et al.(1997) in Hannover and Essen are shown in Fig. 6. Thecomparison of Figs. 4–6 demonstrates that the dependenciesof the concentrations of the dissolved ammonia and sulfurdioxide in the bulk of a falling rain droplet on the altitude aresimilar, and the difference is only quantitative as can bereadily seen in Figs. 4–6.

In this study the numerical calculations were conductedfor homogeneous vertical distribution of soluble gases.Clearly, this assumption can be violated in some cases. Theconcentration of the soluble gases in the immediate vicinity ofthe ground or ocean surface decreases due to dry depositionor due to soluble gas absorption by water. This situation was

discussed in our previous study (see Elperin et al., 2009). Thedistribution of sulfur dioxide in the atmosphere is rarelyuniform since high stacks of power plants often emit it intothe troposphere. Consequently, feasibility of desorption ofSO2 which is indicated in this study, can be of relevance in theanalysis of gaseous pollutant scavenging by rain.

Neglecting temperature inhomogeneity in the atmospherecaused by the nocturnal temperature profile causes consider-able underestimation of the concentration of gaseous pollu-tants in a droplet on the ground. As it can be seen fromFigs. 4–6for the temperature profiles measured by Corsmeier et al.(1997) underestimation of the concentration of gaseouspollutants in rain droplets at the ground can exceed 20%.

Essen

X

, H

Fig. 4.Dependenceof the concentrationof thedissolvedNH3 gas in thebulk of afalling rain droplet vs. altitude (calculations are performed for the atmospherictemperature profile measured by Corsmeier et al. (1997) in Essen).

t

H

Essen

Fig. 2. Dependence of the atmospheric temperature tatm and droplet surfacetemperature ts vs. altitude (atmospheric temperature profile measured byCorsmeier et al. (1997) in Essen).

Hannover

t

H

Fig. 3. Dependence of the atmospheric temperature tatm and droplet surfacetemperature ts vs. altitude (atmospheric temperature profile measured byCorsmeier et al. (1997) in Hannover).

Hannover

X

H

Fig. 5. Dependence of the concentration of the dissolved NH3 gas in the bulkof a falling rain droplet vs. altitude (calculations are performed for theatmospheric temperature profile measured by Corsmeier et al. (1997) inHannover).



Inspection of Figs. 4–6 shows that in the case of homogeneoustemperature distribution in the atmosphere the dimensionlessconcentration of the absorbate in rain droplets Xb1 at theground is equal to 1.

5. Conclusions

In this study we considered coupled heat and masstransfer during soluble gas absorption by a falling dropletwith the internal circulation. The system of transient partialparabolic differential equations of convective diffusion andenergy conservation in liquid and gaseous phases with time-dependent boundary conditions has been solved by combin-ing the similarity transformation method with Duhamel'stheorem. The simple form of the obtained solutions allowsusing them in the analysis of the dependence of the rate ofheat and mass transfer on different parameters, e.g., upon theradius of the rain droplet, diffusion coefficient, concentrationgradient of the soluble gaseous pollutants and temperaturegradient in the atmosphere, etc.

The suggested model was applied for the analysis of theeffect of the nocturnal temperature profile in the atmosphereon the rate of gas absorption by falling rain droplets.

The results obtained in this study can be summarized asfollows:

1. The suggested model of gas absorption by a falling liquiddroplet in the presence of inert admixtures takes intoaccount a number of effects that were neglected in theprevious studies, such as the effect of dissolved gas andenergy accumulation inside a rain droplet and the effect oftemperature distribution in the atmosphere on the rate ofsoluble gas uptake.

2. It is shown that the dependence of the radius-averagedconcentration and temperature vs. time in a falling dropletis determined by a system of linear-convolution Volterraintegral equations of the second kind which is easier to

solve numerically than the original system of partialdifferential equations.

3. It is shown that in a region where the concentration of thesoluble species in the atmosphere is uniform and thetemperature increases with height, due to nocturnalinversion, the rain droplet absorbs gas from the atmosphereduring all the period of its fall. This behavior is explained bythe increase of trace gas solubility in a droplet with adecrease of the temperature. At the same time in our modeltheaverage temperature ina droplet at a givenaltitude is notequal to the atmospheric gas temperature at the samealtitude. Consequently, the instantaneous concentration ofthe dissolved gas in a droplet is not equal to theconcentration of saturation in a liquid corresponding to theconcentration of a trace soluble gas in an atmosphere at agivenheight. Therefore, theexact quantitative analysis of thesoluble trace gas concentration evolution in a droplet can beperformed only through the numerical solution of a systemof integral equations.

4. Neglecting temperature inhomogeneity in the atmospherecaused by the nocturnal temperature profile leads toconsiderable underestimation of the concentration ofgaseous pollutants in a droplet at the ground. Calculationsperformed using the available experimental data onnocturnal temperature profiles in the atmosphere showedthat the underestimation of the concentration of gaseouspollutants in a droplet at the ground can exceed 20%.

5. In the case when the soluble gas is distributed uniformly inthe atmosphere and the temperature–altitude curvecomprises nocturnal inversion and temperature fall seg-ments, gas absorption by the falling rain droplet can bereplaced by desorption and vice versa.

The developed model can be used for the analysis ofscavenging of hazardous gases in the atmosphere by raindroplets and can be used for the creation of various parame-terization and look-up tables that can be incorporated into theexisting computer codes.

Appendix A

Nomenclaturea square root of the temperature diffusivities ratio,ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a1 = a2p

cp specific heat, J mol−1 K−1

Ci molar density at the bulk of fluid, mol m−3

Di molecular diffusion coefficient, m2 s−1

D square root of the diffusivities ratio,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD1 =D2

pHA Henry's law constant, mol l−1 atm−1

k coefficient in Eq. (1)NDi = −DiCi

∂xi∂y molar flux density , mol m−2 s−1

NTi = −λi∂Ti∂y heat flux density , W m−2

m distribution coefficientm0 distribution coefficient at temperature T20PeDi

=kRU /Di Peclet number for a moving dropletPeTi=kRU /ai Peclet number for a moving dropletr radial coordinate, mR droplet radius, mRg universal gas constant, J mol−1 K−1

t time, sT temperature, K

X

H

Fig. 6. Dependence of the concentration of the dissolved SO2 gas in the bulkof a falling rain droplet vs. altitude (calculations are performed for theatmospheric temperature profiles measured by Corsmeier et al. (1997) inHannover and Essen).



U translational velocity of a droplet, m s−1

vr, vϑ velocity components, m s−1

x molar fraction of an absorbatexb10 initial value of molar fraction of an absorbate in a

dropletxb20 value of molar fraction of an absorbate in a gas phase

at height Hxb1(t) molar fraction of an absorbate in a bulk of a dropletxb2(t) molar fraction of an absorbate in a bulk of a gas phaseX1(t)=x1(t)/m0x20 relative molar fraction of an absorbate ina liquid phaseX2(t)=x2(t)/x20 relative molar fraction of an absorbate in agaseous phasey distance from the surface of a droplet, mY=y/R dimensionless distance from the surface of a dropletz coordinate in a vertical direction, m

Greek symbolsΘi=Ti /T10 dimensionless temperatureγ=C1 /C2 molar densities ratioδDi

thickness of a diffusion boundary layer, mδTi thickness of a thermal boundary layer, mΔi=δi /R dimensionless thickness of a boundary layerΔH* enthalpy change, J mol−1

λ thermal conductivity, W m−1K−1

ξ variableηi similarity variableϑ angleτ= tUk /R dimensionless time

Subscripts0 value at height H in the atmosphere1 liquid phase2 gaseous phaseb value in the bulkr radial directionϑ tangential directions value at the gas–liquid interface

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