lable at ScienceDirect

Journal of Environmental Radioactivity 143 (2015) 29e39

Contents lists avai

Journal of Environmental Radioactivity

journal homepage: www.elsevier .com/locate/ jenvrad

Scavenging of radioactive soluble gases from inhomogeneousatmosphere by evaporating rain droplets

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

a r t i c l e i n f o

Article history:Received 4 November 2014Received in revised form29 January 2015Accepted 1 February 2015Available online

Keywords:Radioactive gasesScavengingGas absorptionAtmosphereDroplet evaporationPrecipitation

* Corresponding author. Tel.: þ972 8 6477078; fax:E-mail addresses: elperin@bgu.ac.il (T. El

(A. Fominykh), borisk@bgu.ac.il (B. Krasovitov).

http://dx.doi.org/10.1016/j.jenvrad.2015.02.0010265-931X/© 2015 Elsevier Ltd. All rights reserved.

a b s t r a c t

We analyze effects of inhomogeneous concentration and temperature distributions in the atmosphere,rain droplet evaporation and radioactive decay of soluble gases on the rate of trace gas scavenging byrain. We employ a one-dimensional model of precipitation scavenging of radioactive soluble gaseouspollutants that is valid for small gradients and non-uniform initial altitudinal distributions of temper-ature and concentration in the atmosphere. We assume that conditions of equilibrium evaporation ofrain droplets are fulfilled. It is demonstrated that transient altitudinal distribution of concentration underthe influence of rain is determined by the linear wave equation that describes propagation of a scav-enging wave front. The obtained equation is solved by the method of characteristics. Scavenging co-efficients are calculated for wet removal of gaseous iodine-131 and tritiated water vapor (HTO) for theexponential initial distribution of trace gases concentration in the atmosphere and linear temperaturedistribution. Theoretical predictions of the dependence of the magnitude of the scavenging coefficient onrain intensity for tritiated water vapor are in good agreement with the available atmosphericmeasurements.

© 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Scavenging of the radioactive atmospheric soluble gaseouspollutants by rain droplets is the result of gas absorption mecha-nism (see, e.g. Pruppacher and Klett, 1997). Variation of altitudinaldistribution of concentration of a soluble radioactive gas in theatmosphere due to rain scavenging changes also radioactivity dis-tribution in the atmosphere (see, e.g. Chamberlain, 1991). In thepresent study we analyze dynamics of soluble radioactive gasscavenging by rain taking into account the effects which wereneglected in the previous studies, e.g. droplet evaporation, inho-mogeneous altitudinal temperature and concentration distribu-tions in the atmosphere and radioactive decay of soluble gases. Thegoal of this study is to determine evolution of concentration dis-tribution of radioactive soluble trace gases in the atmospherebelow the cloud under the influence of gas scavenging by fallingrain droplets. The analysis is focused on three radioactive soluble in

þ972 8 6472813.perin), fominykh@bgu.ac.il

water atmospheric trace gases which can be appreciably scavengedby rain of moderate intensity and duration, namely radon (Rn-222),iodine-131 and tritiated water vapor (HTO).

Radon-222 (radon) is a naturally occurring radioactive noble gasof terrestrial origin that has a half-life of 3.8 days. Radon-222 isformed as progeny of uranium and thorium in rocks and soil. Rn-222 is emitted from the ground into the atmosphere where it de-cays and forms daughter products, isotopes of polonium, bismuthand lead. These products either remain airborne till they decay, orare deposited by rain and by diffusion at the ground. Radon con-centration in air always decreases with height (see, e.g.Chamberlain, 1991; Williams et al., 2011).

Iodine-131 is a radioactive isotope formed in nuclear fission,either directly or by decay of a tellurium precursor, and has a half-life period of 8.02 days. Iodine-131 was a significant contributor tothe health hazards from open-air atomic bomb testing in the 1950s,and from the Chernobyl disaster, as well as being a large fraction ofthe contamination hazard in the Fukushima nuclear crisis (see, e.g.Pham et al., 2012; Steinhauser et al., 2014). The gaseous releasefraction of Iodine-131 is typically as high as the particulate fraction.In the Fukushima accident emissions about 70% of the released 131-I was gaseous (Kristiansen et al., 2012). In the Chernobyl accident,

Nomenclature

a raindrop radius, mc total concentration of soluble trace gas in gaseous and

liquid phases, mole l�1

cðGÞ concentration of soluble trace gas in gaseous phase,mole l�1

cðGÞc concentration of soluble gas at cloud bottom, mole l�1

cðGÞgr concentration of soluble gas at the ground, mole l�1

cðLÞ concentration of dissolved gas in droplet, mole l�1

d raindrop diameter, mHA Henry's law constant, mole l�1 atm�1

k1 growth constant, m�1

k2 coefficient in a linear dependence of temperature oncoordinate, K m�1

k3 coefficient in linear dependence of solubilityparameter on temperature, K�1

k4 ¼ k2$k3coefficient in a linear dependence of solubility oncoordinate, m�1

Kv evaporation coefficient, m2s�1

L distance between the ground and cloud bottom, mLv latent heat of evaporation, kcal g�1

m ¼ HARgT dimensionless Henry's law coefficientR rainfall rate, ms�1

Rg universal gas constant, atm$l$mole�1$K�1

qc mass flux density of dissolved gas transferred by raindroplets, mole$m�2$s�1

t time, sT temperature, Ku terminal velocity of droplet, ms�1

U wash-down front velocity, ms�1

U0 wash-down front velocity for non-evaporatingdroplets, ms�1

xðGÞ mole fraction of a soluble trace gas in gaseous phasez vertical coordinate, m

Greek symbolsb coefficient of mass transfer, ms�1

4 volume fraction of droplets in airl radioactive decay constant, s�1

L scavenging coefficient, s�1

Subscripts and superscripts0 initial valuec value at the cloud bottomgr value at the groundG gaseous phaseL liquid phasev vapor

Table 1Scavenging of soluble radioactive gases by a single droplets and rain.

Problem Experimental ortheoretical

References

Single droplet Experimental Booker, 1965; Belovodsky et al., 1997.Single droplet Theoretical Booker, 1965.Precipitation Experimental Dana et al., 1978; K€oll€o et al., 2011;

Abrol, 1990; Nankara et al., 2012;Matsumoto et al., 2013;Piskunov et al., 2012; Gautam et al., 2013

Precipitation Theoretical Chamberlain and Eggleton 1964;Dana et al., 1978; Atanassov andGaleriu 2011; Patry et al., 2011

T. Elperin et al. / Journal of Environmental Radioactivity 143 (2015) 29e3930

about 25% of the total reactor inventory of 131-I was released toatmosphere as vapor or particulate aerosol (see, e.g. Steinhauseret al., 2014).

Tritium was discovered in water as hydrogen tritium oxide(HTO) by Grosse et al. (1951). Naturally occurring tritium is pro-duced by cosmic radiation. However, the distribution of naturaltritium in the atmosphere and hydrosphere was severely disturbedsince the testing of thermonuclear weapons began in 1954 (see, e.g.Junge, 1963). Most of the tritium from the atmospheric nuclearweapons tests was formed as HTO. Tritium is also formed as aproduct of ternary fission in nuclear power reactors. Althoughmostof the tritium is retained in the reactor fuel somemay be released toatmosphere as HTO during fuel reprocessing. Concentration mea-surements in the atmosphere revealed the decrease of HTO con-centration with height (see, e.g. Ehhalt, 1971).

Inspired by the studies of Chamberlain and Eggleton (1964) andBooker (1965), a number of theoretical and experimental in-vestigations were carried out to determine the rate of solubleradioactive gas scavenging by single droplets and precipitation.Radioactive gas absorption by a single falling drop was investigatedexperimentally by Belovodsky et al. (1997). Scavenging of solubleradioactive gas by rain was studied theoretically by Dana et al.(1978), Atanassov and Galeriu (2011) and experimentally by Abrol(1990), K€oll€o et al. (2011) and Piskunov et al. (2012). Sorption ofradioactive Iodine-131 by aerosols was investigated experimentallyby Noguchi et al. (1988, 1990). The state-of-the art in the field ofsoluble radioactive gas absorption by single droplets and by rainwas overviewed quite comprehensively (see e.g. by Piskunov et al.,2012; Nankara et al., 2012). These previous investigations aresummarized in Table 1. Note that most published investigations ofradioactive gas absorption by falling liquid droplets and by pre-cipitation are devoted to HTO absorption by water. Since the life-time of radioactive HTO is by orders of magnitudes larger than thetotal time of soluble gas scavenging, these studies do not take intoaccount radioactive decay in the equation of mass balance.

Theoretical models which allow us to take into account a radioac-tive decay during gas absorption by a single stagnant and fallingdroplet are presented in the Appendixes A and B, respectively.Taking into account radioactive decay is important e.g. in theanalysis of absorption of radioactive gaseous iodine-131 becausethe lifetime of radioactive I-131 is of the same order of magnitudeas the time of complete gas absorption by a droplet.

2. Description of the model

2.1. Scavenging of radioactive gases with low solubility by non-evaporating droplets

Consider absorption of radioactive gas having a low solubilityfrom a mixture containing inert gas by rain droplets falling in theatmosphere with the known initial non-uniform concentration andtemperature distributions. Since the velocity of scavenging frontpropagation is proportional to the solubility of scavenged gases(see, e.g. Elperin et al., 2013b), the velocity of temperature frontpropagation is by orders of magnitude larger than the velocity of ascavenging front propagation for radioactive gases having a low

T. Elperin et al. / Journal of Environmental Radioactivity 143 (2015) 29e39 31

solubility, e.g. radon-222 or iodine-131. Scavenging front is amoving plane parallel to the bottom of a cloud and to the groundthat separates the region in the atmosphere between the cloudbottom and the ground into two sub-domains. In the upper sub-domain, that is located between the cloud bottom and a scav-enging front, the soluble trace gas in the atmosphere is washeddown by rain to the concentration in the under cloud region, andthe spatial concentration distribution of the trace gas in this sub-domain is uniform. In the lower sub-domain, that is located be-tween the ground and the scavenging front, the soluble trace gas isnot yet washed down by rain, and its concentration distribution isaltitude and time dependent. Scavenging velocity is the velocity ofa scavenging front propagation. Similarly, temperature front is amoving plane parallel to the bottom of a cloud and to the groundthat separates the region in the atmosphere between the cloudbottom and the ground into two sub-domains. In the upper sub-domain, that is located between the cloud bottom and the tem-perature front, the temperature in the atmosphere is uniform, i.e.homogenized by rain to the temperature in the region adjacent tothe cloud. In the lower sub-domain, that is located between theground and the temperature front, the temperature in the atmo-sphere is not uniform, i.e. it is not homogenized by rain yet, andtemperature distribution is altitude and time dependent. Therefore,during the period of time when temperature front propagates froma bottom of a cloud to the ground and temperature distributionbecomes homogeneous, the changes of the concentration profileare negligibly small. Consequently, scavenging of radioactive gaseshaving a low solubility by precipitation in the inhomogeneous at-mosphere can be analyzed assuming a homogeneous temperaturedistribution (see, e.g. Elperin et al., 2011b, 2013b). In the case ofgases having a low solubility the approximation of small gradient ofconcentration in the atmosphere is always valid, i.e.

ðamu=cðGÞgr bÞ��ðdcðGÞ=dzÞ��< <1. Consequently, it can be assumed thatthe instantaneous concentration of the dissolved radioactive gas ina droplet is equal to the concentration of saturation in the liquidcorresponding to the concentration of a trace soluble radioactivegas in an atmosphere at a given height, i.e. cðLÞðzÞ ¼ mcðGÞðzÞ. HerecðLÞ is concentration of the dissolved gas in a droplet (in [mole l�1]),m ¼ HARgT is the dimensionless Henry's law coefficient, and HA isHenry's law coefficient (see Seinfeld and Pandis, 2006, Chapter 7).

When the concentration gradient is small, the total concentra-tion of soluble compound in the gaseous and liquid phases reads(see, e.g. Elperin et al., 2013b):

c ¼ cðGÞ½ð1� 4Þ þm4�: (1)

The total mass flux density of the dissolved gas transferred by raindroplets is determined by the following expression:

qc ¼ m4ucðGÞ; (2)

where u is the terminal fall velocity of droplets. It must be notedthat the effect of droplet size distribution (DSD) on the rate of tracegas scavenging by precipitation was investigated in our previousstudies using the Monte Carlo method whereby the trace gasconcentration profile was calculated by solving the initialboundary-value problem for the droplet diameter that wasrandomly sampled from the appropriate probability density func-tion and averaged over a large number of sampled diameters (see,e.g. Elperin et al., 2011b; Baklanov et al., 2013). These studiesshowed that accounting for the droplet size distribution does notlead to significant changes in the rate of gas scavenging in com-parison with that calculated using the average radius approxima-tion. Therefore, in the present study we performed calculations for

the mono size droplets having the average rain droplet radius.Equation of mass balance for radioactive soluble trace gas in thegaseous and liquid phases reads:

vcvt

¼ �vqcvz

� lc; (3)

where z axis is directed from cloud bottom to the ground and l is aradioactive decay constant. Combining Eqs. (1)e(3) we obtain thefollowing equation:

vxðGÞ

vtþ U0

vxðGÞ

vzþ lxðGÞ ¼ 0; (4)

where t is time, xðGÞ ¼ cðGÞ=CðGÞ, xðGÞ is mole fraction of soluble tracegas in the gaseous phase, CðGÞ is a molar density of gas in the bulk ofa gaseous phase,U0 ¼ m4u=½ð1� 4Þ þm4�. Since volume fraction ofdroplets in air during precipitation is of the order of 10�7÷10�6, itcan be assumed that U0 ¼ mR, where R is a rainfall rate. The initialand boundary conditions for Eq. (4) read:

xðGÞ ¼ f ðzÞ for t ¼ 0; (5)

xðGÞ ¼ xðGÞc e�lt for z ¼ 0; (6)

where xðGÞc is mole fraction of soluble gas at the cloud bottom att ¼ 0. The boundary condition (6) takes into account a radioactivedecay of gas. Introducing the new variable wðGÞ ¼ xðGÞelt , Eqs.(4)e(6) can be rewritten as follows:

vwðGÞ

vtþ U0

vwðGÞ

vz¼ 0; (7)

wðGÞ ¼ f ðzÞ for t ¼ 0; (8)

wðGÞ ¼ xðGÞc for z ¼ 0: (9)

Eq. (7) describes propagation of a linear wave in one directionwithout change of shape. Since U0 >0 these “scavenging waves”travel in the direction of increasing z. Eqs. (7)e(9) can be solved bythe method of characteristics (see e.g. Pinchover and Rubinstein,2005). The solution is given by the following formula:

xðGÞ ¼"f ðz� U0$tÞe�lt z>U0$txðGÞc e�lt z<U0$t

(10)

In the case of the exponential initial distribution of soluble gas inthe atmosphere, i.e. when

t ¼ 0; xðGÞ ¼ xðGÞc $expðk1$zÞ; (11)

where z is measured from the cloud to the ground, the solution ofEq. (7) with the initial and boundary conditions (9) and (11) reads:

xðGÞðzÞ ¼"xðGÞc $expðk1$zÞexpð � ðU0$k1 þ lÞ$tÞ z>U0$t

xðGÞc e�lt z<U0$t:

(12)

The obtained solution allows us to calculate the scavengingcoefficient L for the soluble radioactive trace gas absorption fromthe atmosphere:

T. Elperin et al. / Journal of Environmental Radioactivity 143 (2015) 29e3932

L ¼ � 1xðGÞ

vxðGÞ

vt: (13)

Eqs. (12) and (13) yield the following expression for the scavengingcoefficient:

L ¼�k1$U0 þ l z>U0$tl z<U0$t

: (14)

Eq. (14) implies that if the initial altitudinal concentration dis-tribution of a soluble trace gas in the atmosphere is exponential,scavenging coefficient in the region between a ground and ascavenging front is the sum of the radioactive decay constant and aproduct of the rain intensity, solubility parameter and the growthconstant in the initial profile of concentration distribution in agaseous phase. Scavenging coefficient in a region between thescavenging front and the cloud bottom is equal to the radioactivedecay constant.

In the casewhen the initial altitudinal distribution of the solublegas in the atmosphere can be approximated by the polynomials:

t ¼ 0; xðGÞ0 ðzÞ ¼Xki¼0

aizi; (15)

where a0 ¼ xðGÞc , the solution of Eq. (7) with the initial andboundary conditions (9) and (15) reads:

xðGÞðzÞ ¼

264Pki¼0

aiðz� U0$tÞie�lt z>U0$t

xðGÞc ¼ a0e�lt z<U0$t

: (16)

Eqs. (13) and (16) yield the following formula for the scavengingcoefficient:

L ¼

266666664lþ

U0$Xki¼1

aiiðz� U0$tÞi�1

Pki¼0 aiðz� U0$tÞi

z>U0$t

l z<U0$t

(17)

2.2. Scavenging of highly soluble radioactive gases by evaporatingdroplets in the atmosphere with non-uniform altitudinalconcentration and temperature distributions

For highly soluble gases the velocity of scavenging front propa-gation is by orders of magnitude larger than velocity of temperaturefront propagation. Therefore, it can be assumed that during the timerequired for complete scavenging of radioactive gases by precipita-tion, the changes of temperature profile can be neglected. Conse-quently, analysis of scavenging of highly soluble gases in theinhomogeneous atmosphere in this case reduces to solving equationsof mass transfer in the atmosphere with the stationary inhomoge-neous temperature distribution. Since in the considered problem thegradients of concentration and temperature are small, it can beassumed that the instantaneous concentration of the dissolvedradioactive gas in a droplet is equal to the concentration of saturationin liquid corresponding to the concentration of a trace solubleradioactive gas in the atmosphere at a given height (see, e.g. Elperinet al., 2013b). Similarly, the instantaneous temperature of thedroplet is equal to the local atmospheric temperature at a givenheight. Consider gas absorption by slowly evaporating falling raindroplets with diameter much larger than the mean free path of gas

molecules at normal conditions. In this case the kinetic effects can beneglected.We assume also that the humidity of the ambient air doesnot change strongly during time required for complete scavenging ofradioactive trace gas. The times of thermal tT ¼ a2=c (where c isthermal diffusivity) and diffusion relaxation tD ¼ a2=Dv in a gaseousphase in the vicinity of the droplet are much smaller than dropletevaporation time. For example, for the droplet having 1mmdiameterevaporating in air with temperature 294 K at 80% relative humidity,evaporation time is of the order of 10 minwhile the relaxation timestT � tD areof theorderof 0.1 s. Therefore, evaporationof raindropletsproceeds in a steady-state regime (see Fuchs, 1959). Elperin et al.(2013b) showed that under these assumptions and in the casewhen relative temperature differences in the vicinity of the dropletare small, i.e., jTa � T∞j=T∞ < <1 (where Ta is the droplet surfacetemperature and T∞ is the ambient temperature at a given height farfrom the droplet) the dependence of the evaporating droplet radiusvs. time for the falling droplet can be determined as follows:

a ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia20 � Kvt

q; (18)

where Kv is the evaporation constant of a falling droplet. In theatmosphere where lapse rate is of the order of 7 K=km thedependence of air temperature on altitude reads TðzÞ ¼ Tc þ VT$z(where Tc is the air temperature at the cloud bottom andVTzconst). During droplet fall the radius of the droplet does notchange significantly, i.e. , the condition ða0 � agrÞ=a0 < <1 is satis-fied, where a0 is the initial radius of a rain droplet at the bottom of acloud and agr is radius of evaporating droplet at the ground. Takinginto account small temperature gradient in the atmosphere (VT~10�3 K/m) the evaporation constant of a droplet falling in the at-mosphere with nonuniform temperature depending on altitudecan be determined as follows:

KvðzÞ ¼ Kv0 þ KT$z (19)

where Kv0 ¼ ðMDvPv;satðTcÞð1 � fÞ=rwRgTcÞ 1 �

Pv;satðTcÞ=

kaR2

gT3c

L2vM2Dvþ Pv;satðTcÞ

!!!f vðaÞ is evaporation constant for the

droplet falling in homogeneous atmosphere (see Elperin et al.,

2010), M e molar mass of water, f v is mean ventilation coeffi-cient, Dv is coefficient of diffusion of water vapor in air, rw is waterdensity, Pv;sat is a pressure of saturated water vapor at the droplet

surface, and KT ¼ �ðKv0k2=TcÞ þ ð3a1b1Tck2=ðb1T3c þ 1Þ2Þ,

a1 ¼ ððMDvPv;satðTcÞð1 � fÞf vðaÞÞ=ðrwRgÞÞ, b1 ¼ ððkaR2gÞ=ðL2vM2DvPv;satðTcÞÞÞ, k2 ¼ ðTgr � TcÞ=L, and ka is thermal conduc-tivity of air. The dependence of droplet fall velocity on dropletradius can be approximated by the following correlation for theterminal fall velocity:

u ¼ c1½aðzÞ�a; (20)

where for droplets having diameter d<10�4 m, c1 ¼ 3:075�107 m�1s�1 and a ¼ 2; for droplets having diameter in the range10�4 m<d<10�3 m, c1 ¼ 3:8� 103 s�1, a ¼ 1 and for larger drop-lets when 10�3 m< d, c1 ¼ 133:046 m1=2s�1, a ¼ 0:5 (see, e.g.Pruppacher andKlett,1997).Note that in correlation (20) the terminalfall velocity ismeasured inm s�1. Eqs. (18)e(20) yield formula for thedependence of radius of a falling evaporating droplet vs. coordinate:

aðzÞ ¼ a0

1� ðaþ 2Þ$ðKv0 þ KTzÞ

2c1aaþ20

!1=ðaþ2Þ: (21)

T. Elperin et al. / Journal of Environmental Radioactivity 143 (2015) 29e39 33

Formula for volume fraction of evaporating droplets vs. dropletdiameter reads:

4ðzÞ ¼ ½aðzÞ�340

.a30: (22)

Taking into account that UðzÞ ¼ mðTðzÞÞ$4$u,m ¼ m0 � k3$ðT � T0Þ and TðGÞ ¼ TðGÞ

0 þ k2$z,m ¼ m0 � k4$z, wherek4 ¼ k2$k3, we obtain the following expression for the dependenceof the scavenging front velocity vs. coordinate:

UðzÞ ¼ 40c1aa0ðm0 � k4zÞ

1� ðaþ 2ÞzðKv0 þ KTzÞ

2c1aaþ20

!ðaþ3Þ=ðaþ2Þ:

(23)

Since ðððaþ 2ÞLðKv0 þ KTLÞÞ=ð2c1aaþ20 ÞÞ< <1, Eq. (23) yields:

UðzÞ ¼ U0$�1� k4m

�10 z�

1� ðaþ 3ÞzðKv0 þ KTzÞ2c1aaþ2

0

!;

U0 ¼ 40c1aa0m0:

(24)

Note that m0 ¼ 1:19� 105, for tritiated water vapor (HTO) atT ¼ 283:15 K, k3 ¼ 8565 K�1 in the temperature range from273:15 K to 293:15 K. The dependence of the velocity of scav-enging front propagation vs. distance from a cloud bottom is shownin Fig. 1. Inspection of Fig. 1 reveals that UðzÞ decreases with coor-dinate z if the initial temperature distribution in the atmosphere isdetermined by the environmental temperature lapse. This behavioris explained by the dependence of the solubility of trace gas inwater on temperature. Indeed, when the initial temperature dis-tribution is determined by environmental lapse, falling rain dropletmoves from regions with lower temperature to regions with highertemperature. Therefore, gas solubility in a droplet decreases withincrease of the distance from the cloud bottom because solubility isa decreasing function of temperature. Since the scavenging velocity

Fig. 1. Dependence of scavenging velocity for absorption of HTO vapor by evaporatingdroplets vs. distance from cloud bottom for various values of humidity,R0 ¼ 14 mm h�1. Temperature distribution is determined by environmental lapse,k2 ¼ 7� 10�3 K m�1.

decreases when solubility decreases (see, e.g. Elperin et al., 2013b),UðzÞ is a decreasing function of z. Evaporation of rain droplets alsocauses the decrease of the scavenging front velocity. Indeed, in theresult of evaporation of rain droplets their size, velocity and volu-metric fraction in the atmosphere decrease. Therefore scavengingfront velocity, U, that is proportional to droplet velocity and dropletvolumetric fraction, decreases. Hence, when the initial altitudinaltemperature distribution in the atmosphere is determined by theenvironmental lapse, droplet evaporation as well as altitudinaltemperature dependence cause a decrease of the scavenging frontvelocity. On the contrary, when the initial temperature distributionin the atmosphere is determined by nocturnal inversion, altitudinaltemperature dependence increases scavenging velocity whiledroplet evaporation decreases scavenging front velocity. Competi-tion of these two mechanisms e nocturnal inversion and dropletevaporation e leads to the increase of the scavenging front velocityat smaller distances from the cloud and to the decrease of thescavenging front velocity at larger distances from the cloud (seeFig. 2). The latter behavior occurs for small values of humidity whenthe influence of droplet evaporation on the dynamics of radioactivesoluble gas scavenging is essential. At larger values of humidity, theinfluence of nocturnal inversion on the dynamics of scavenging isdominant, and scavenging front velocity increases with the dis-tance from the cloud bottom during all the time of rain droplets fall.

Using equation of mass balance for the soluble radioactive tracegas in the gaseous and liquid phases and applying the approachoutlined in the previous Section (see also Elperin et al., 2011b) wearrive at the following equation for soluble trace gas distribution inthe atmosphere:

vxðGÞ

vtþ UðzÞ vx

ðGÞ

vzþ lxðGÞ ¼ 0; (25)

where U(z) is determined by Eq. (24). The initial and boundaryconditions to Eq. (25) are described by Eqs. (6) and (11). Introducingthe new variables wðGÞ ¼ xðGÞelt and

Fig. 2. Dependence of scavenging velocity for absorption of HTO vapor by evaporatingdroplets vs. distance from cloud bottom for various values of relative humidity RH,R0 ¼ 14 mm h�1. Temperature distribution is determined by nocturnal inversion,k2 ¼ �10�2 K m�1.

Fig. 3. Dependence of scavenging coefficient vs. rain intensity for iodine-131 wash outby non-evaporating droplets for different values of growth constant k1 (Eq. (14)). Theinitial distribution of soluble trace gas in the atmosphere is exponential.

T. Elperin et al. / Journal of Environmental Radioactivity 143 (2015) 29e3934

h¼ 1U0

$1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

b22�4a2q $ln

8><>:2a2zþb2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib22�4a2

q2a2zþb2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib22�4a2

q $b2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib22�4a2

qb2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib22�4a

q2

9>=>;;

(26)

wherea2 ¼ �ðððaþ 3ÞKTÞ=ð2c1aaþ2

0 ÞÞ þ ðk4ðaþ 3ÞKv0 Þ=ðm02c1aaþ20 Þ,

b2 ¼ �ðððaþ 3ÞKv0 Þ=ð2c1aaþ20 Þ þ ðk4=m0ÞÞ.

Eq. (25) can be rewritten as follows:

vwðGÞ

vtþ vwðGÞ

vh¼ 0: (27)

The initial and boundary conditions to Eq. (27) read:

xðGÞ ¼ f ðhÞ for t ¼ 0; (28)

xðGÞ ¼ xðGÞc for h ¼ 0: (29)

Solution of Eq. (27) with the initial and boundary conditions(28) and (29) is given by the following formula:

xðGÞ ¼"f ðh� tÞ$e�lt h> txðGÞc $e�lt h< t

: (30)

The coordinate of a scavenging front is determined by equationh ¼ t. If the initial altitudinal distribution of the radioactive solublegas in the atmosphere is exponential, Eq. (30) yields:

xðGÞðzÞ ¼

26666664xðGÞc e�lt$exp

0BB@�b2 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib22 � 4a2

q �F�1 exp

��h� t

�$U0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib22 � 4a2

q �� b2 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib22 � 4a2

q

2a2k1

�1� F�1$exp

��h� t

�U0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib22 � 4a

q2

�1CCA h> t

xðGÞc e�lt h< t

; (31)

where F ¼ ðb2 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib22 � 4a2

qÞ=ðb2 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib22 � 4a2

qÞ. Expression for the

scavenging coefficient is determined by Eqs. (13) and (31).

3. Results and discussion

The above model of radioactive atmospheric trace gases scav-enging by rain was applied to study evolution of soluble trace gasconcentration in the atmosphere during rain. In the calculations weused exponential altitudinal distribution for the initial concentra-tions of soluble trace gases whereby the growth constants in theexponential dependence were taken from the measured altitudinaldistributions of a radioactive trace gases in the atmosphere (see,e.g. Ehhalt, 1971). Calculations were performed for scavenging ofgaseous iodine-131 having a low solubility and highly solubletritiated water vapor (HTO) by rain. Data on solubility of iodine andHTO in water can be found in Sanemasa et al. (1984), Jones (1968),Van Hook (1968) and Ogram (1985).

The results of solution of Eqs. (4)e(6) with the exponentialinitial distribution of gaseous Iodine-131 in the atmosphere areshown in Fig. 3. Eq. (4) and Fig. 3 show that neglecting radioactivedecay in the equation of mass balance leads to large underesti-mation of scavenging coefficient for Iodine-131.

The altitudinal and temporal evolution of HTO vapor concen-tration calculated using Eq. (31) is shown at Fig. 4a. Correctness ofmodeling of HTO vapor uptake by water as soluble gas absorptionprocess was shown by Hales (1972). Inspection of Fig. 4a shows thattritiated water vapor is scavenged by non-evaporating rain dropletsto the concentration in the interstitial air in a cloud after 40 minwhen the initial temperature distribution in the atmosphere isdetermined by the environmental lapse. Time required for scav-enging to the concentration in the interstitial air in a cloud for HTOabsorption by evaporating rain droplets is larger. Comparison ofFig. 4a and b shows that time of scavenging of tritiated water vapor(HTO) by evaporating rain drops is larger for the environmentallapse temperature distribution than for the temperature distribu-tion in the nocturnal inversion. This behavior is explained bydependence of gas solubility on temperature. The dependence ofscavenging coefficient for tritiated water vapor on rain intensity forthe initial exponential distribution of radioactive trace gas con-centration in a gaseous phase is calculated using Eqs. (13) and (31)and is shown in Fig. 5a and b. Calculations presented in Fig. 5a and bcorrespond to different altitudinal dependencies of atmospherictemperature. Calculations are performed for RH equal to 70%.Whentemperature inversion determines altitudinal atmospheric tem-perature distribution, the value of the scavenging coefficient de-creases with time for given rain intensity. On the contrary, when

T. Elperin et al. / Journal of Environmental Radioactivity 143 (2015) 29e39 35

environmental lapse rate determines the altitudinal atmospherictemperature distribution, the value of scavenging coefficient in-creases with time for given rain intensity. Clearly, scavenging co-efficient always increases when rain intensity increases.

The dependence of the scavenging coefficient vs. altitude in thecase of tritiated water vapor wash out by evaporating rain dropletsfor the exponential initial profile of soluble gas in the atmosphere is

Fig. 4. a. Evolution of HTO vapor distribution in the atmosphere caused by rainscavenging (Eqs. (16) and (40)). The initial distribution of soluble trace gas in the at-mosphere is exponential and is taken from Ehhalt (1971), R0 ¼ 14 mm h�1, 1 e

droplets evaporation is taken into account (RH ¼ 70%); 2 e evaporation of droplets isneglected. Temperature distribution is determined by environmental lapse. b. Evolu-tion of HTO vapor distribution in the atmosphere caused by rain scavenging (Eq. (31)).The initial distribution of soluble trace gas in the atmosphere is exponential and istaken from Ehhalt (1971), R0 ¼ 14 mm h�1, 1 e droplets evaporation is taken intoaccount (RH ¼ 70%); 2 e evaporation of droplets is neglected. Temperature distributionis determined by nocturnal inversion.

calculated using Eqs. (13) and (31) and is shown in Fig. 6a and b.Inspection of Fig. 6a shows that when the environmental lapse ratedetermines altitudinal atmospheric temperature distribution, scav-enging coefficient increases with height in a region between thescavenging front and the ground. On the contrary, when temperatureinversion determines the altitudinal atmospheric temperature dis-tribution, scavenging coefficient decreases with height in a regionbetween the scavenging front and the ground (see Fig. 6b). At theground, the magnitude of scavenging coefficient increases with timewhen the environmental lapse rate determines the altitudinal at-mospheric temperature distribution, and decreases with time fortemperature inversion. Dependence of scavenging coefficient onaltitude at different times is described by converging lines, when

Fig. 5. a. Dependence of scavenging coefficient vs. rain intensity for HTO vapor washout by evaporating droplets. Temperature distribution is determined by environmentallapse. b. Dependence of scavenging coefficient vs. rain intensity for HTO vapor washout by evaporating droplets. Temperature distribution is determined by nocturnalinversion.

T. Elperin et al. / Journal of Environmental Radioactivity 143 (2015) 29e3936

altitudinal atmospheric temperature distribution is determined byenvironmental lapse rate (see Fig. 6a). On the contrary, when thealtitudinal atmospheric temperature distribution is determined bytemperature inversion, altitudinal dependence of scavenging coef-ficient at different times is described by the diverging lines. Notably,the altitudinal dependence of scavenging coefficient at differenttimes during soluble gas scavenging by non-evaporating rain drop-lets in the atmosphere with inhomogeneous temperature distribu-tion is described by parallel lines.

Fig. 6. a. Dependence of scavenging coefficient vs. altitude for HTO vapor wash out byevaporating rain droplets (Eqs. (13) and (31)). The initial distribution of soluble tracegas in the atmosphere is exponential and is taken from Ehhalt (1971), R0 ¼ 14 mm h�1,air humidity is 70%. Temperature distribution is determined by environmental lapse. b.Dependence of scavenging coefficient vs. altitude for HTO vapor wash out by evapo-rating rain droplets (Eqs. (13) and (31)). The initial distribution of soluble trace gas inthe atmosphere is exponential and is taken from Ehhalt (1971), R0 ¼ 14 mm h�1, airhumidity is 70%. Temperature distribution is determined by nocturnal inversion.

Scavenging coefficients in the atmosphere for tritium-oxidewashout by rain was measured by Piskunov et al. (2012). Sincethe latter study did not include information about soluble trace gasconcentration and temperature distributions in the atmospherebefore rain, in order to compare our results with the experimentalestimates of the scavenging coefficient we conducted calculationsfor the uniform atmospheric temperature typical for rain season atthe measurement sites. We assumed that vertical distribution ofsoluble trace gas concentration before rain can be approximated bythe exponential dependence and calculated scavenging coefficientfor different values of the growth constants based on the atmo-spheric measurements (see, e.g. Ehhalt (1971, 1974)). In the case ofuniform temperature distribution in the atmosphere Eq. (14) de-termines the rain intensity dependence of scavenging coefficient.Inspection of Fig. 7 shows that Eq. (14) yields the same estimate ofthe scavenging coefficient during tritium-oxide washout by rain asthe experimental estimates obtained by of scavenging coefficienton rain intensity as the experimental results.

4. Conclusions

We suggested a model for scavenging of radioactive solubletrace gases in inhomogeneous atmosphere by evaporating raindroplets. It is shown that gas scavenging in the case of low gradi-ents of radioactive soluble trace gas concentration and temperaturein the atmosphere is determined by linear wave equation thatdescribes propagation of a linear wave in one direction withoutchanging shape. The obtained equation was solved by the methodof characteristics. The simple form of the obtained solution allowsanalyzing the dependence of the rate of soluble gas scavenging ondifferent parameters, e.g. value of radioactive decay constant, rainintensity, radioactive trace gas solubility, gradients of the trace gasconcentration and temperature in a gaseous phase, humidity etc.

Using the developedmodel we calculated scavenging coefficientand the rates of scavenging of different radioactive trace gases(iodine-131 and tritiated water vapor (HTO)). The obtained resultscan be summarized as follows:

Fig. 7. Comparison of theoretical predictions with atmospheric measurements ofPiskunov et al. (2012) for HTO vapor scavenging by rain.

T. Elperin et al. / Journal of Environmental Radioactivity 143 (2015) 29e39 37

1. It is demonstrated that if the initial altitudinal concentrationdistribution of a radioactive trace gas having low solubility inthe atmosphere is exponential, scavenging coefficient in theregion between a ground and a scavenging front is the sum ofradioactive decay constant and a product of rain intensity, sol-ubility parameter and the growth constant in the initial profileof concentration in a gaseous phase. Scavenging coefficient in aregion between a scavenging front and the bottom of a cloud isequal to radioactive decay constant. Neglecting radioactivedecay in equation of mass balance for radioactive soluble tracegas in the gaseous and liquid phases leads to large underesti-mation of the value of scavenging coefficient for Iodine-131washout by rain.

2. It is demonstrated that when initial temperature distribution inthe atmosphere is determined by the environmental lapse,scavenging velocity is smaller than for the isothermal temper-ature distribution (see Fig. 1). Droplet evaporation causesfurther reduction of scavenging velocity. When the initial tem-perature distribution in the atmosphere is determined bynocturnal inversion, the competition of two factors e nocturnalinversion and droplet evaporation e leads to increase of scav-enging velocity with coordinate at smaller distances from thecloud and to decrease of scavenging velocity at larger distancesfrom the cloud. The latter behavior is observed for smallervalues of humidity when the influence of droplet evaporation onthe dynamics of radioactive soluble gas scavenging is essential.At larger values of humidity the influence of nocturnal inversionon the dynamics of scavenging is dominant, and scavengingvelocity increases with the distance from the cloud bottomduring all the time of rain droplets fall (see Fig. 2).

3. It is demonstrated that altitudinal dependence of scavengingcoefficient for radioactive soluble gas scavenging by evaporatingrain droplets at different times is described either by convergingor diverging lines depending on whether the altitudinal atmo-spheric temperature distribution is determined by the envi-ronmental lapse rate or temperature inversion (see Fig. 6a andb). It should be noted that altitudinal dependence of the scav-enging coefficient at different times during soluble gas scav-enging by non-evaporating rain droplets in the atmospherewithinhomogeneous temperature is described by parallel lines.

4. The suggestedmodel yields the same estimate of the scavengingcoefficient during tritiated water vapor (HTO) washout by rainand the same dependence of the scavenging coefficient on rainintensity as the atmospheric measurements conducted byPiskunov et al. (2012) (see Fig. 7).

The developed model can be used for the analysis of precipita-tion scavenging of radioactive hazardous gases in the atmosphereand for validating the advancedmodels for predicting scavenging ofradioactive soluble gases by rain.

Acknowledgments

This study was performed in the framework of COST ES1004“European Framework for Online Integrated Air Quality andMeteorology Modeling”.

Appendix A. Radioactive gas absorption by a stagnant liquiddroplet

Conjugate mass transfer during soluble radioactive gas absorp-tion from a mixture with inert gas by a stagnant liquid droplet isdescribed by the following system of equations of diffusion withcorresponding initial and boundary conditions (see, e.g. Elperinet al., 2007, 2008, 2013a):

vxðLÞ

vt¼ DL

v2xðLÞ

vr2þ 2

rvxðLÞ

vr

!� lxðLÞ 0< r< a (A1)

vxðGÞ

vt¼ DG

v2xðGÞ

vr2þ 2

rvxðGÞ

vr

!� lxðGÞ r> a (A2)

xðLÞ ¼ xðLÞ0 ; xðGÞ ¼ xðGÞ0 at t ¼ 0 (A3)

vxðLÞ

vr¼ 0 at r ¼ 0; xðGÞ ¼ xðGÞð∞Þe�lt as r/∞ (A4)

xðLÞðtÞ ¼ mxðGÞðtÞ; DLvxðLÞ

vr¼ DG

vxðGÞ

vrat r ¼ a; (A5)

where a is a droplet radius. After introducing new variables,wðLÞ ¼ xðLÞelt , wðGÞ ¼ xðGÞelt , Eqs. (A1)e(A5) can be rewritten asfollows:

vwðLÞ

vt¼ DL

v2wðLÞ

vr2þ 2

rvwðLÞ

vr

!0< r< a (A6)

vwðGÞ

vt¼ DG

v2wðGÞ

vr2þ 2

rvwðGÞ

vr

!r> a (A7)

wðLÞ ¼ xðLÞ0 ; wðGÞ ¼ xðGÞ0 at t ¼ 0 (A8)

vwðLÞ

vr¼ 0 at r ¼ 0; wðGÞ ¼ xðGÞð∞Þ as r/∞ (A9)

wðLÞðtÞ ¼ mwðGÞðtÞ; DLvwðLÞ

vr¼ DG

vwðGÞ

vrat r ¼ a (A10)

Following the approach suggested by Brown (1965) we obtainsolution of Eqs. (A6)e(A10) in the following form:

xðLÞðr;tÞelt¼mxðGÞ0 þ2�xðLÞ0 �mxðGÞ0

�$a$D

prm

�Z ∞

0

ðsinu�ucosuÞ$sinður=aÞexp�DLu2t�a2�

ðucosuþLsinuÞ2þD$m�1usinu�2 du

(A11)

xðGÞðr; tÞelt ¼ mxðGÞ0 þ2�xðLÞ0 �mxðGÞ0

�$a

prm

$

Z ∞

0

ðsin u� u cos uÞ$FðuÞexp� DLu2t�a2�

ðu cos uþ L sin uÞ2 þ D$m�1u sin u�2 $duu ;

(A12)

where

FðuÞ ¼ ðu cos uþ L sin uÞsinfuðr � aÞ=Dagþ D$m�1u sin u cosfuðr � aÞ=Dag;

D ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiDG=DL

p; L ¼ D2

.m� 1; Q ¼ D=m:

T. Elperin et al. / Journal of Environmental Radioactivity 143 (2015) 29e3938

Appendix B. Radioactive gas absorption by a falling liquiddroplet

Consider absorption of a soluble radioactive gas from a mixturecontaining an inert gas by a moving droplet. At time t ¼ 0 thedroplet begins to absorb gas from the atmosphere. Initial altitudinaldistribution of the concentration of the absorbate in the gaseousphase in is assumed to be known. Following the approach sug-gested by Elperin et al. (2009, 2011a) we arrive at the followingsystem of transient equations of convective diffusion for the liquidand gaseous phases which accounts for convection in radial andtangential directions and a process of radioactive decay:

vxðGÞ

vt� ku

asin q

vxðGÞ

vqþ 2yku

acos q

vxðGÞ

vy¼ DðGÞv

2xðGÞ

vy2� l$xðGÞ;

(B1)

vxðLÞ

vt� ku

asin q

vxðLÞ

vqþ 2yku

acos q

vxðLÞ

vy¼ DðLÞv

2xðGÞ

vy2� l$xðLÞ;

(B2)

where coefficient k is equal to 0.04 in the range of the external flowReynolds numbers from 10 to 300 (see, e.g., Pruppacher and Klett,1997, p. 386), y is a distance from the surface of a droplet. Eqs. (B1)and (B2) are written in a frame attached to the falling droplet andare valid for y< < a. Since the velocity of the droplet fall is knownand z ¼ u$t, the vertical coordinate-dependent boundary condi-tions can be transformed into the time-dependent boundary con-ditions. The vertical coordinate z is aligned with the direction of thedroplet fall. The initial and boundary conditions to Eqs. (B1) and(B2) read:

xðLÞ ¼ xðLÞ0 ; xðGÞ ¼ f ðzÞ at t ¼ 0 (B3)

xðGÞ ¼ xðGÞb ðtÞ$e�lt as y/∞; (B4)

xðLÞ ¼ xðLÞb ðtÞ as y/�∞; (B5)

xðLÞ ¼ mxðGÞ at y ¼ 0; (B6)

DðGÞCðGÞvxðGÞ

vy¼ DðLÞCðLÞvxðLÞ

vyat y ¼ 0; (B7)

where C is a molar density in the bulk of fluid. After introducingnew variables, L ¼ ðla=kuÞ, Y ¼ y=a, T ¼ ðtukÞ=a,PeðGÞ ¼ ðkauÞ=DðGÞ, PeðLÞ ¼ ðkauÞ=DðLÞ, Eqs. (B1)e(B7) can berewritten as follows:

uðLÞðY; q; TÞ ¼ xðLÞðY; q; TÞe�Ll ¼ v

vT

Z T

0

"xðLÞb ðlÞeLl � xðLÞb ðlÞeLl �mxðb

1þmgD

uðGÞðY ; q; TÞ ¼ xðGÞðY ; q; TÞe�Ll ¼ v

vT

Z T

0

24xðGÞb ðlÞ �

�xðLÞb ðlÞeLl �mxðGb

1þmgD

vxðGÞ

vT� sin q

vxðGÞ

vqþ 2Y cos q

vxðGÞ

vY¼ 1

PeðGÞv2xðGÞ

vY2 �L$xðGÞ;

(B8)

vxðLÞ

vT� sin q

vxðLÞ

vqþ 2Y cos q

vxðLÞ

vY¼ 1

PeðLÞv2xðLÞ

vY2 �L$xðLÞ; (B9)

The initial and boundary conditions to Eqs. (B8) and (B9) read:

xðGÞ ¼ xðGÞb ðTÞ$e�LT as Y/∞; (B10)

xðLÞ ¼ xðLÞb ðTÞ as Y/�∞; (B11)

xðLÞ ¼ mxðGÞ for Y ¼ 0; (B12)

DðGÞCðGÞvxðGÞ

vY¼ DðLÞCðLÞvxðLÞ

vYfor Y ¼ 0; (B13)

xðLÞ ¼ xðLÞ0 ; xðGÞ ¼ f ðzÞ for T ¼ 0: (B14)

Introducing new variables, xðGÞ ¼ uðGÞðTÞe�LT , xðLÞ ¼ uðLÞðTÞe�LT ,Eqs. (B8)e(B9) can be rewritten as follows:

vuðGÞ

vT� sin q

vuðGÞ

vqþ 2Y cos q

vuðGÞ

vY¼ 1

PeðGÞv2uðGÞ

vY2 : (B15)

vuðLÞ

vT� sin q

vuðLÞ

vqþ 2Y cos q

vuðLÞ

vY¼ 1

PeðLÞv2uðLÞ

vY2 : (B16)

The initial and boundary conditions to Eqs. (B15) and (B16) read:

uðLÞ ¼ xðLÞ0 ; uðGÞ ¼ f ðzÞ for T ¼ 0 (B17)

uðGÞ ¼ xðGÞb ðTÞ as Y/∞; (B18)

uðLÞ ¼ xðLÞb ðTÞeLT as Y/�∞; (B19)

uðLÞ ¼ muðGÞ for Y ¼ 0; (B20)

DðGÞCðGÞvuðGÞ

vY¼ DðLÞCðLÞvuðLÞ

vYfor Y ¼ 0: (B21)

Solution of Eqs. (B15)e(B16) with initial and boundary conditions(B17)e(B21) reads (for details see Elperin et al., 2009):

GÞðlÞerfc

�Y

DLðq; T � lÞ�#

dl (B22)

ÞðlÞÞgDerfc

�Y

DLðq; T � lÞ�35dl; (B23)

T. Elperin et al. / Journal of Environmental Radioactivity 143 (2015) 29e39 39

where

D2L ¼ 4

PeðLÞ sin4ðqÞ

(cosðqÞ � 1

3cos3ðqÞ

�"1� f ðq; TÞ1þ f ðq; TÞ �

13

�1� f ðq; TÞ1þ f ðq; TÞ

�3#)

;

(B24)

and f ðq; TÞ ¼ tg2�

q2

�expð2TÞ.

The variable xðLÞb is the unknown function of time that can bedetermined by means of an integral material balance over thedroplet (see Ruckenstein et al., 1971, Uribe-Ramirez and Korchinsky,2000):

�VCðLÞdxðLÞb

dt¼ 2pa2

Z p

0DðLÞCðLÞvxðLÞ

vy

����y¼0

sin q$dq� VCðLÞlxðLÞb ;

(B25)

where V is a volume of a droplet. Substituting expression for thesoluble trace gas concentration in the droplet (Eq. (B21)) into Eq.(B24) yields the following integro-differential equation:

dXðLÞb ðTÞdT

¼ 3PeðLÞ

ffiffiffip

p ð1þmgDÞv

vT

�8<:Z T

0

hXðLÞb ðlÞeLl �mXðGÞ

b ðlÞiZ p

0

sin q dqDLðq; T � lÞdl

9=;

�LXðLÞb ;

(B26)

where XðLÞb ðTÞ ¼ ðxðLÞb ðTÞ �mxðGÞb0 Þ=ðxðLÞb0 �mxðGÞb0 Þ is the dimension-

lessmolar fraction of soluble trace gas in the bulk of a droplet, xðGÞb0 isthe value of soluble trace gas molar fraction in a gaseous phase atthe bottom of a cloud. Neglecting radioactive decay, i.e. in the casewhen L ¼ 0, Eq. (B25) recovers the equation obtained by Elperinet al. (2009, 2011a).

References

Abrol, V., 1990. Estimation of wash out of tritiated water (HTO) effluent by rain-drops. Bull. Radiat. Prot. 13, 23e26.

Atanassov, D., Galeriu, D., 2011. Rain scavenging of tritiated water vapour: a nu-merical Eulerian stationary model. J. Environ. Radioact. 102, 43e52.

Baklanov, A., Elperin, T., Fominykh, A., Krasovitov, B., 2013. Cell model of in-cloudscavenging of highly soluble gases. J. Atmos. Sol-Terr. Phys. 97, 135e142.

Belovodsky, L.F., Gaevoy, V.K., Golubev, A.V., Kosheleva, T.A., 1997. Tritium oxidewashout by drops. J. Environ. Radioact. 36, 129e139.

Booker, D.V., 1965. Exchange between water droplets and tritiated water vapor. Q. J.R. Meteorol. Soc. 91, 73e79.

Brown, A., 1965. Diffusion of heat from a sphere to a surrounding medium. Aust. J.Phys. 18, 483e489.

Chamberlain, A.C., 1991. Radioactive Aerosols. Cambridge University Press, UK, 255p.Chamberlain, A.C., Eggleton, A.E.J., 1964. Washout of tritiated water vapour by rain.

Int. J. Air Water Pollut. 8, 135e149.Dana, M.T., Wogman, N.A., Wolf, M.A., 1978. Rain scavenging of tritiated water

(HTO): a field experiment and theoretical consideration. Atmos. Environ. 12,1523e1529.

Ehhalt, D.H., 1971. Vertical profiles and transport of HTO in the troposphere.J. Geophys. Res. 76, 7351e7367.

Ehhalt, D.H., 1974. Vertical Profiles of HTO, HDO, and H 20 in the Troposphere.NCAR-TN/STR-100, National Center for Atmospheric Research, AtmosphericQuality and Modification Division, Boulder, Colorado, November 1974, 129pp.

Elperin, T., Fominykh, A., Krasovitov, B., 2007. Evaporation and condensation of largedroplets in the presence of inert admixtures containing soluble gas. J. Atmos.Sci. 64, 983e995.

Elperin, T., Fominykh, A., Krasovitov, B., 2008. Scavenging of soluble gases byevaporating and growing cloud droplets in the presence of aqueous-phasedissociation reaction. Atmos. Environ. 42, 3076e3086.

Elperin, T., Fominykh, A., Krasovitov, B., 2009. Effect of altitude concentrationgradient of soluble gaseous pollutants on their scavenging by falling raindroplets. J. Atmos. Sci. 66, 2349e2358.

Elperin, T., Fominykh, A., Krasovitov, B., 2010. Scavenging of soluble trace gases byfalling rain droplets in inhomogeneous atmosphere. Atmos. Environ. 44,2133e2139.

Elperin, T., Fominykh, A., Krasovitov, B., 2011a. Uptake of soluble gaseous pollutantsby rain droplets in the atmosphere with nocturnal temperature profile. Atmos.Res. 99, 112e119.

Elperin, T., Fominykh, A., Krasovitov, B., Vikhansky, A., 2011b. Effect of rain scav-enging on altitudinal distribution of soluble gaseous pollutants in the atmo-sphere. Atmos. Environ. 45, 2427e2433.

Elperin, T., Fominykh, A., Krasovitov, B., Lushnikov, A., 2013a. Isothermal absorptionof soluble gases by atmospheric nanoaerosols. Phys. Rev. E 87, 012807 (1e8).

Elperin, T., Fominykh, A., Krasovitov, B., 2013b. Rain scavenging of soluble gases bynon-evaporating and evaporating droplets from inhomogeneous atmosphere.Meteorol. Atmos. Phys. 122, 215e226.

Fuchs, N.A., 1959. Evaporation and Droplet Growth in Gaseous Media. PergamonPress, NY.

Gautam, Y.P., Sharma, S., Sharma, A.K., Kumar, A., Ravi, P.M., Sarkar, P.K., 2013.Studies on the rain scavenging process of tritium in a tropical site at Narora inIndia. J. Nucl. Chem. Article ID 849732, 6 pages.

Grosse, A.V., Johnston, W.M., Wolfgang, R.L., Libby, W.F., 1951. Tritium in nature.Science 113, 1e2.

Hales, J.M., 1972. Scavenging of Gaseous Tritium Compounds by Rain AEC Researchand Development Report. Pacific Northwest Laboratories, Atmospheric AnalysisSection, Atmospheric Sciences Department. BNWL-1659 UC33, Richland,Washington 99352 April 1972.

Jones, W.M., 1968. Vapour pressures of tritium oxide and deuterium oxide. Inter-pretation of the isotope effects. J. Chem. Phys. 48, 207e214.

Junge, C.E., 1963. Air Chemistry and Radioactivity. Academic Press, New York,p. 382p.

K€oll€o, Z., Palcsu, L., Major, Z., Papp, L., Moln�ar, M., Ranga, T., Domb�ov�ari, P., Manga, L.,2011. Experimental investigation and modelling of tritium washout by precip-itation in the area of the nuclear power plant of Paks. Hung. J. Environ. Radioact.102, 53e59.

Kristiansen, N.I., Stohl, A., Wotawa, G., 2012. Atmospheric removal times of theaerosol-bound radionuclides 137Cs and 131I measured after the FukushimaDai-ichi nuclear accident e a constraint for air quality and climate models.Atmos. Chem. Phys. 12, 10759e10769.

Matsumoto, T., Maruoka, T., Shimoda, G., Obata, H., et al., 2013. Tritium in Japaneseprecipitation following the March 2011 Fukushima Daiichi Nuclear Plant acci-dent. Sci. Total Environ. 445, 365e370.

Nankara, D.P., Patraa, A.K., Ravib, P.M., Joshia, C.P., Hegdeb, A.G., Sarkarb, P.K., 2012.Studies on the rain scavenging process of tritium in a tropical site in India.J. Environ. Radioact. 104, 7e13.

Noguchi, H., Murata, M., Suzuki, K., 1988. Adsorption of radioactive iodine gas ontofly ash aerosol. Jpn. J. Health Phys. 23, 19e26.

Noguchi, H., Murata, M., Suzuki, K., 1990. Adsorption of radioactive iodine gas ontoatmospheric aerosol. Jpn. J. Health Phys. 25, 209e219.

Ogram, G.L., 1985. Precipitation Scavenging of Tritiated Water Vapour (HTO). ReportNo 85-233-K. Ontario Hydro Research Division, Atmospheric Research Section,Chemical Research Department, 32 pp.

Patry, L., Galeriu, D., Armand, P., 2011. Sensitivity analysis of rain characteristics onHTO concentration in drops. Fusion Sci. Technol. 60, 1228e1231.

Pham, M.K., Eriksson, M., Levy, I., Nies, H., Osvath, I., Betti, M., 2012. Detection ofFukushima Daiichi nuclear power plant accident radioactive Traces in Monaco.J. Environ. Radioact. 114, 131e137.

Pinchover, Y., Rubinstein, J., 2005. An Introduction to Partial Differential Equations.Cambridge University Press, Cambridge.

Piskunov, V.N., Golubev, A.V., Balashov, Y.S., Mavrin, S.V., et al., 2012. The effect ofrain characteristics on scavenging rate of tritium-oxide from the atmosphere.Atmos. Environ. 62, 573e583.

Pruppacher, H.R., Klett, J.D., 1997. Microphysics of Clouds and Precipitation. KluwerAcademic Publishers, Dordrecht.

Ruckenstein, E., Dang, V.D., Gill, W., 1971. Mass transfer from chemical reaction fromone or two component bubbles or drops. Chem. Eng. Sci. 26, 647e668.

Sanemasa, I., Kobayashi, T., Piao, C.Y., Deguchi, T., 1984. Equilibrium solubilities ofiodine vapor in water. Bull. Chem. Soc. Jpn. 57, 1352e1357.

Seinfeld, J.H., Pandis, S.N., 2006. Atmospheric Chemistry and Physics. From AirPollution to Climate Change, second ed. John Wiley & Sons, NY.

Steinhauser, G., Brandl, A., Johnson, T.E., 2014. Comparison of the Chernobyl andFukushima nuclear accidents: a review of the environmental impacts. Sci. TotalEnviron. 470e471, 800e817.

Uribe-Ramirez, A.R., Korchinsky, W.J., 2000. Fundamental theory for prediction ofsingle-component mass transfer in liquid drops at intermediate Reynoldsnumbers (10 � Re �250). Chem. Eng. Sci. 55, 3305e3318.

Van Hook, W.A., 1968. Vapour pressures of the isotopic waters and ices. J. Phys.Chem. 72, 1234e1244.

Williams, A.G., Zahorowski, W., Chambers, S., Griffiths, A., Hacker, J.M., Element, A.,Werczynski, S., 2011. The vertical distribution of Radon in clear and cloudydaytime terrestrial boundary layers. J. Atmos. Sci. 68, 155e174.