an eulerian model for scavenging of pollutants by raindrops

Almo~ic Etllii-nr Vol. 19, No. 5. pp. x9-:x 198.5. oocd-6981 8s s3.w + 0.00 Rioted in Great Britain. sg 1985. Pcrgamoo Prrs Ltd.

AN EULERIAN MODEL FOR SCAVENGING OF P~LL~TA~S BY RAINDROPS

SUDARSHAN KUMAR Environmental Science Department, General Motors Research Laboratories, Warren, Michigan 48090-9055,

U.S.A.

(First receiued 8 June 1984 and receiuedfor pubhtion 10 October 1984)

Abstract-An Eulerian model for simulating the coupled processes of gas-phase depletion and aqueous- phase accumulation of the pollutant spaies during a rain event has been formulated. The model is capable of taking into account any realistic vertical profile of pollutant species concentrations and time-dependent initial aqueous-phase concentrations at the cloud base. The model considers the processes of single species absorption and dissociation in the aqueous phase. The coupled partial differential equations constituting the model are discretized into a set of ordinary differential equations by using the Galerkin method with chapeau functions as the basis functions. These equations are solved to obtain the pollutant concentrations of the gas phase and raindrops as well as the pH of raindrops as a function of time and distance below cloud-base.

Simulations are performed for scavenging of gaseous HNOs, H202, SO2, formaldehyde and NH3. For the case of highly soluble HNO, and H202, raindrops are far from equilibrium with the gas phase and their capacity for absorption of these gases is und~nish~ even as they reach ground level. The gas-phase concentrations for these species decrease exponentially with time and the washout is determined primarily by the rain intensity and mass-transfer coefficient of the gaseous species to the raindrops. The pollutant species concentrations in raindrops are an almost linear function of the distance below the cloud base. For the simulation conditions considered in this study, the half-life periods of these gases for removal from the atmosphere range from 15 to 40 min.

For SO1 and formaldehyde, the aqueous-phase concentrations approach equilibrium as the drops fall to ground level and the gas-phase concentrations show large gradients in the vertical. Half-life periods for SO, range from 1.3 to 13 h depending on the initial raindrop pH and rain intensity. For formaldehyde, the half-life ranges from 19 to 63 min.

Solubiiity of NH3 is a strong function of the raindrop pH. As NH3 is absorbed, the raindrop pH increases and NHs solubility decreases. For pre-acidified drops (pH = 4.6). ammonia soIubility is very high and the drops are far from equilibrium with the gas phase throughout the falling period. The half-life for ammonia ranges from 11 min to over 3 h in our simulations.

Key word index: Precipitation scavenging, scavenging of pollutants, scavenging model, washout model. scavenging by rain.

The chemical composition of rain is influenced by trace gas absorption and aerosol scavenging occurring both in-cloud and below-cloud. Although the relative contri- butions of these processes in influencing the chemical composition of rain can vary widely, absorption of atmospheric trace gases by raindrops plays an important rote in influencing the chemical composition of rain and in cleansing the atmosphere. Adamowicz (1979) and Durham et nl. (1981) have treated the problem of trace gas absorption using a Lagrangian approach by following raindrops descending in the atmosphere. The drops absorb trace gases as they fall through the polluted layer and these absorbed gases undergo dissociation and chemical reaction in the aqueous phase. Levine and Schwartz (1982) have considered the problem of irreversibIe scavengjng for I-INO, vapor and calculated washout coefficients for removal of HNOJ from the atmosphere. Adewuyi and Carmichael (1982) considered the absorption of a number of trace gases but did not include any chemical

reactions. In most of these models, the spatial distribution of trace gases in the atmosphere cannot be taken into account.

However, as pointed out by Hates (1972), the process of trace gas scavenging is reversible in nature and the phenomenon of absorption and desorption may cause a redistribution of pollutants in the atmosphere. Therefore, it is desirable to develop a model for determining washout rates of trace gases and chemical composition of rain that can take into account the spatial distribution of pollutants. Furthermore, the pollutant species concentrations in the gas phase directly influence the aqueous-phase concentrations; therefore, it is necessary to consider simultaneousiy the coupled processes of gas-phase depletion and aqueous- phase accumulation of the pollutant species. With these objectives in mind, in this work we formukte an Eulerian model to describe the simultaneous processes of trace gas removal from the atmosphere and absorption of these gases in raindrops. The solution of the resulting equations gives us space-time profilesof both the gas-phase and aqueous-phase concentrations. We

769

770 SUDARSHAN Kutd~R

consider the absorption of HN03, HrOr, SOz, formaldehyde and NH,. For each case. we take into account the processes of absorption and dissociation in the aqueous phase. The case of simultaneous absorption of various gases and chemical reactions in the aqueous phase will be treated in a separate paper.

The approach used in this work allows us to follow the space-time profiles of gas-phase concentrations and aqueous-phase concentrations of various species as the rain event progresses. From the gas-phase profiles, we can calculate the washout coefficients of removal of various gases. In addition, it is possible to observe the change in pH of raindrops as a function of time and distance below the cloud layer. One major advantage of this approach is that any reaiistic initial vertical profile of gas-phase concentration can be employed in the model simulations. In addition, it is possible to incorporate time-dependent initial aqueous-phase concentrations at the cloud base. Because of these capabilities, this model can also be used to consider precipitation scavenging of a plume containing higher concentrations of S or N com- pounds than the surrounding atmosphere. Fisher (1982) and Mofenkamp (1983) have attempted to couple the dynamics of air motion and precipitation production with the uptake of pollutanis by precipitation. However. the emphasis in this work is to develop a model with the capability of including aqueous-phase chemistry associated with scavenging of a number of pollutant species.

In the next section, we present the formulation of our scavenging model. Relevant solubility and chemical equilibria are discussed in Section 3 together with a brief description of the numerical solution technique. The results of a number of model simulations are presented in Section 4. Finally, we present the conciu- sions in the last section.

2. MODEL FORMUtATiON

In this work, we are primarily interested in the precipitation scavenging of gases present in the polluted layer below the cloud base. We wish to calculate the concentrations of pollutant species in both the gas and aqueous phases as a function of time and distance below the cloud base. We assume the atmosphere to be divided into two regions-a cloud and a polluted layer beneath it,as shown schematically in Fig. 1. Raindrops with known concent~tions of various species fall vertically through the polluted layer at their terminal velocities. The terminal velocity u of a drop depends on its radius r and is determined simply by a balance between the buoyancycorrected gravitational force and the drag force acting on the drop. Mass transfer of the trace species to the drops takes place in this polluted layer. We denote the pollutant concentration in the gas phase as p(x, I) and the pollutant concentration referred to a unit volume of aqueous phase in drops of radii r as c(x, C; r). Here x is the vertical distance from the cloud base and t is the time after the

x = o a _ _ ,_ _ _ _ _ _--; -,- - Cloud-Base

x 1 I ;I 1: I/’

Fig. 1. A schematic diagram of raindrops falling through the polluted layer.

start of the rain event. To account for transport of material from the gas-phase to the aqueous-phase, ordinarily we need to consider both the gas-phase and the aqueous-phase resistance to mass transfer, However, Garland (1978) points out that the shear on the surface of a falling drop creates significant velocities inside the drop resulting in rapid mixing inside the drop and a negligible Iiquid-phase resistance to mass transfer. Therefore, we assume that the Iiquid- phase resistance to mass-transfer is negligible in com- parison to the gas-phase resistance. The gas-phase mass-transfer coefficient k, for a drop in motion can be calculated from the Friissling equation (Frossling, 1938).

Sh = 1 f0.3 Re”Z Sc’!3, (1)

where Sh, the Sherwood number = k, r/D,; Re, the Reynolds number = 2 w/v and SC, the Schmidt number = vi/i$. In these definitions, u is the drop terminal velocity, r is the drop radius, Y is the kinematic viscosity of air and I), is the moiecuiar diffusivity of the gaseous species in air. For drops of radii r, the equation for the concentration of the absorbed species can be derived by mass-balance over a horizontal plane of infinitesimal thickness dx. The resulting equation is

t?c(x, t; r) dc(x, f; r) 3k (r) -+u(r)ax =-A----

& rRT

x P(X, @z$L) [ 1 . (2) Here H is the Henry’s law solubiiity constant of the pollutant species, R is the universal gas constant and T

is the absolute temperature. The terminal velocity II and the mass-transfer coefficient k, are functions of drop radius r. The terminal velocity generally increases with drop size while the mass-transfer coefficient decreases. The exact value of the terminal velocity for a given drop was obtained from the measurements of Beard and Pruppacher (1969) and Gunn and Kinzer (1949). As mentioned above k, was obtained from the Friissiing equation.

The equation for the gas-phase concentration of the species must take into account the total mass trans- ferred to the drops over the entire size spectrum of

An Eulerian model for scavenging of pollutants by raindrops 771

raindrops. We assume that there is no advection in the Some of the other assumptions inherent in our model gas-phase and that turbulent mixing in the gas-phase are that the variations in drop size distribution (due to can be described by a constant eddy-delusion coef- eva~ration and coalescence) and temperature in the ficient D. In the absence of such turbulent mixing, below-cloud layer do not affect the scavenging process considerable gradients in the gas-phase concentration significantly and that the drops fall vertically to the can develop. Thus, the equation for the gas-phase ground. concentration p(x, t) is

dP(X* 4 _ D d’p(x, r) r p --

Zr 2x2 ./o 4nr*n(r)k,(r) 3. DISSOLUTIOS ASD DISSOCIATION OF HSO,, H,OZ,

SO2 AND NH,

x p(x, ,)-w 1 dr, (3) Some of the most important gases which influence the composition and acidity of precipitation are

where n(r) is the number density function and n(r) dr is HNOX, H202, SO2 and NHJ. SO2 is absorbed into

the number ofdrops between radii r and r + dr per unit raindrops, undergoes dissociation and is oxidized to

volume of air. The cumulative number distribution produce sulfate. HNOs vapor directly contributes to

N(r) = Sin(r) dr of raindrops has been measured by the rain acidity through absorption and dissociation.

various authors including Best (1950), iMarshal and NH3 is highly basic and serves to neutralize some of

Palmer (194X), and Sekhon and Srivastava ( 197 1). The the rain acidity. H*O,, though itself not a significant

gas-phase concentration p(x, t), though itsetf in- source of H + through dissociation, is considered to be

dependent of the size of a particular drop r, involves the most important oxidizing agent for dissolved S(IV)

knowledge of c(x, r; r) for drops over the entire size species in producing SO:-. Formaldehyde is import-

spectrum r. Thus, in order to solve for p(x, t), one must ant because it can form a complex with HSO; and

solve for c(x, t; r) over a discretized spectrum of drop reduce the HSO; amount available for oxidation.

sizes making the calculations very cumbersome. In Formaldehyde is quite soluble in water but does not

order to simplify the calculations, we assume that all dissociate, In this paper, we will not take into account

the drops are of a uniform size r,corresponding to the any chemical reactions. We will simply assume that the

maximum in the fractional volume dist~bution (4nj3) gases are absorbed in the raindrops and then undergo

r3n(r) of the drops. The number of drops per unit dissociation in the aqueous-phase. The case of multiple

voiume of air N, is then calculated by enforcing the species absorption and chemical reaction in the

constraint that ct: the liquid water content per unit aqueous phase will be considered in a separate paper.

volume of air during rainfall, obeys the empirical Table 1 shows the equilibrium processes and equilib-

relationship observed between Wand rainfall intensity rium constants (Ki) at 15°C for absorption and

I. The equation for p(x, c) is thus simplified and dissociation of HNOs, H202, SO*, NH3 and HCHO.

given by The square brackets [ ] indicate the molar concen-

apex, t) _ D EZP(& r) tration of an aqueous-phase species. For these pol-

ar ------4nr;N,kg(rp) ax=

lutant species, we define the effective solubility constants (indicating total solubihty of a pollutant species including the dissociated forms) as follows

(4) c [HNO,] + [NO; I = where c(x, r) is the aqueous-phase concentration in a

H,=--Ili_. PHNO, PHNO,

drop of radius rp, and the governing equation for c(x, I) (7) becomes

SC@, 0 Zc(x, 0 3k,(r,) C,, _ W,W + D-WI

dttu(r,f- = - H,,=--

ZX r,,RT PH:02 P&o%

(5) = &(I+&)

with initial condition c(x, 0) = +e (x) H = c, = [SOz. I-W] + [HSO;] + [SO; -1

and boundary condition ~(0, t) = #e(r). s Psoa Pso,

To specify the boundary conditions for p(x, r) we assume that the flux of pollutant species in the gas- (9)

phase at the cloud base (x = 0) as well as at ground level (x = L) is zero. Therefore, the gas-phase concen- (neglecting SO:- in aqueous solution)

tration has zero spatial gradient at x = 0 and L. Thus, the initial and boundary conditions for p(x, t) are

H c, _ [NH,.H,Ol+W;l A - PNH, PNH,

P(X, 0) = PO b)

and iaP/wj, -0 = f&vW~, P L = 0. @I = Kg(l+&)

712 SL’DARSHAN KUUAR

Table 1. Equilibrium processes and values of equilibrium constants at 1YC

HNO, &HNOJ (I) K =EHNo3’ 5.8 (+j)*Matm“ I ---= PHNO,

HNO,(&H’+NO; K2 = W’l WXI CHNW

= 15.4 M

K WA1 3 = ~ = 1.65 (+S)Marm-’

PH;Oz

H20&H+ +HOf K 4 = [H+l~Ho~l = I.50 (-11)M D-W,1

SO2 (8) 2 SOz. Hz0 (I) K 5 = [S01~H@l = 1,795 bf atm-’ PSO,

SOL.H,O?H’ + HSO;

HSO; %-I+ +SO:- I( 7 zz ‘H+3cso’-1 = 6.87 (-8) M EHSO; 1

HCHO (g) %H2(OH)2 (I) K

8

= CCH,W%l = I.49 (+4) IM atm-’

PHCHO

NH3 (g) 2 NH,. Hz0 (I) K 9

= [NH3.H201 = 9j.l Matm-1

PNH,

Kt.3 NH,. Hz0 (I) *NH: +OH- K = [NH’l’“H-l = 1.61 (_ j)M

lo [NH,,H20]

(SW). t

tsw

WD)

*The notation 5.8 (+ 5) denotes 5.8 x 10’. tSW, Schwartz and White, 1981; IMD, Martin and Damschen, 1981; CW, Cotton and Wilkinson, 1972;

SM, Siflen and Martell, 1964; LB, Ledbury and Blair, 1925.

HF=K8. (11)

The effective solubility constants H,, H,,, H, and H, as defined above are functions of raindrop acidity, and the solubility of HNOJ, H202 and SO2 decreases as [H+] increases. NH3 solubility on the other hand increases as [H’] increases. In order to calculate the effective solubility constants, we need a relationship between [H’] and the dissolved concentration of dissociated and undissociated species. Such a relationship can be easily obtained by utilizing the equilibrium relationships and the electroneutrality condition. For example, for the case of SO* dissolution, let us assume that the droplets contain an initial concentration [X-l0 of univalent ions from a strong acid and have an initial pH of pHo. The electroneutraiity condition is

[X-Jo = [H’],-[OH-], = IO-P”o--K,. lOPHo.

(12)

In the presence of S(IV) species, the electroneutrality condition is

[H’] = [HSO;] + [X-l, + [OH-] (13)

neglecting the contribution of [SO: -1. Furthermore, from the equilibrium relationships in Table 1, we obtain

[HSO;] = K, (K, + [H+])- ' C,. (14

The two relationships above can be combined to obtain a cubic equation relating [H’] and C,

EH+]‘~~~~-~X-1,)~H+1’-~~~EX-1~

+K,Cs+Kw)[H+]--K,K, =o, (15)

where A w = [H’] [OH-]. Similar equations relating [H+] and C, or C,, can be easily obtained. In the case of NH3 dissolution, we get a similar equation relating [OH-] and C,. These equations in conjunction with the definitions of eff’ective solubility constants can be used to update the effective soiubifity constants as the concentration of dissoIved species and droplet pH change.

Solution of the coupled partial differential equations with appropriate boundary conditions would give us the space-time profiles of pollutant concentration in the gas-phase and in the predominant size drops. We note here that for the limiting case of infinite solubility (H -* 30) it is possible to calculate the gas- phase and liquid-phase concentration profiles analytically. These analytical solutions will be presented in the next section, For the general case, however, one has to resort to a numerical solution of the coupled partial diflerential equations. We used the Galerkin method (Lapidus and Pinder, 1982) with chapeau functions as the basis functions to formulate discretized equations.


These discretized equations were then solved by an implicit Crank-Nicolson method.

4. .MODEL SKMULATIONS AND RESULTS

In this section we report the results for scavenging of gaseous HNOs, H202, SOr, formaldehyde and NHJ. In these simulations we assume the cloud base height to be 200 m. Calcufations are performed for rain intensities of 5 mm h - i and 20 mm h- ‘. The initial droplet pH is assumed to be 5.6 representing the acidity due solely to CO, absorption and dissociation, or 4.6 representing a pre-acidified condition. For the Marshall-Palmer size distribution (Marshall and Palmer, 1948), the predominant drop radius rp and rainwater concentration W in the atmosphere are given by (Mason, 1971)

r,(mm) = 0.3659 IO,” (16)

W(mg m-‘) = 72 1°.88, (17)

where 1, the rain intensity is in mm h- i. Table 2 shows the rainfall parameters for rain intensities of 5 and 20 mm h-‘. The gaseous diffusivities D, and mass- transfer coefficients k, used in the simulations in this paper are shown in Table 3. For both cases,- we employed an eddy diffusivity D of 100 m2 s-i.

4.1. Nitric acid and hydrogen peroxide

Gaseous HN03 and H202 play a major role in infiuencing the acidity and chemical composition of rain. Both of these gases are highly soluble in water and behave in a similar fashion in terms of gas-phase and aqueous-phase concentrations. Let us consider the absorption and scavenging of HNOs by rain. Figures 2(a)-(b) show the gas-phase and aqueous-phase concentrations of HNO, at various times after the start of

Table 2. Rainfall parameters for two rain intensities

rP W

NP u

Rain intensity I 5 mm h-‘ 20mm h-’

0.5130 mm 0.6864 mm 296.8 mg m-’ 1005.2 mg m-3 525 m-’ 742 mm3 4.11 m se1 3.10 m s-’

Table 3. Gaseous diffusivities and mass-transfer coetficients to predominant size drops

ic,(cms-t)

D+cm* s-* E I=Smmh-’ i=20mmh-’

HNO, 0.124 15.4 14.3 Hz% 0.166 19.0 17.6 SGZ 0.130 15.9 14.8 HCHO 0.172 19.5 18.1 NH, 0.218 23.2 21.4

*Calculated from the Wilke-Lee modification of the H~r~hfelder-Bird-Spot method (Treybal, 1980).

z 0.0 0.4 0.8 1.2

is Concentration lppbl

15min

z ibl : ; 2000 , I

\\ I I I

i5 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 Concentration in Raindrops LMI

;i 2

$ 0 F 2 1 500 f

G 1 :loooL 5 ! 3

4 1500’- m

s c 2000 r , I c- z 4.0 4.2 4.4 4.6 4 8 5 0 5 2 5 4 5 6

a pH in Raindrops

Fig. 2(a) Gas-phase concent~tion profiles for HNO, at various times. Rain intensity, I = 5 mm h- ‘, initial raindrop pH = 5.6, and initial HNOJ gas-phase concentration 2 ppb. (b) Concentration of dissolved nitric acid ([HNO,(I)] + [NO;]) in raindrops at various times. Same conditions as in (a). (c) pH in raindrops at various times. Same conditions as in la).

the rain event. In these simulations, the initial concentration of HN03 was assumed to be 2 ppb throughout the polluted layer and the initial pH of the raindrops was assumed to !x 5.6. As the rain event progresses, the gas-phase concentration is depleted and the aqueous- phase concentration in turn decreases. The gas-phase concentration does not show any appreciable vertical gradient while the aqueous-phase concentration increases almost linearly as a function of fall distance. This is because the raindrops are far from equilibrium with the gas phase throughout the failing period and their capacity for absorption of HN03 is undimin- ished. It is noteworthy that in the absence of turbulent diffusion (D = 0), the gas-phase and aqueous-phase

774 SUDaRsHAN KUMAR

concentration profiles (not shown here) remain unchanged. The gas-phase concentration profile does not show any vertical gradients in the absence of turbulent diffusion because of the extremely high solubility of HNO, in water. Since turbulent diffusion serves only to reduce the gradients, the profiles are unchanged for the case of non-zero diffusion. The vertical profile of the pH in the raindrops at various times is shown in Fig. Z(c). The pH of raindrops at the ground level IS min after the rain event starts is approximately 4.5 and increases with time because of the depletion of gaseous HNOj in the atmosphere.

Similar calculations were done for H102 which is also highly soluble in water. However, HZOz dis- sociates only slightly in water and therefore its solubility does not depend on the pH of raindrops to any significant degree. Just as in the case of HN03, there is a negligible vertical gradient in the gas-phase concentration and the aqueous-phase concentration is a linear function of the distance below the cloud base. For the case of uniform concentration in the polluted layer and complete absorption in raindrops, scavenging is an exponential decay process (Engelmann, 1968) with the gas-phaseconcentration varying with timeasp = p0 exp (- 21). This equation defines the washout coefficient i.. The washout coefficients d and.half-lives r,,,2 (tli2 = ln2/i.) for removal of gaseous pollutants were calculated from the gas-phase concentration profiles. Table 4 shows the washout coefficients and half-life periods for HNO, and H202 for rain intensities of 5 and 20 mm h- ‘. For both of these gases, washout coeficients were also calculated for the case of pre- acidified raindrops at an initial pH of 4.6. There were no significant changes in gas-phase and aqueous-phase concentration profiles and washout coeficients. This is because even at a lower pH, the solubilities of HN03 and H,Ot are very high and the transfer of HN03 and H20Z to raindrops is mass-transfer controlled as before.

For highly soluble gases, the equation for gas-phase concentration p(x, t) can be simplified by assuming

p B c/H, and neglecting c H. The resulting equation

for p(x, t) is

SP(X, [) -=

?r D (18)

I.C. p(x. 0) = po (xf.

This equation is a linear equation for p(x. t) and is independent of c(x, 1). It can be solved analytically (Carslaw and Jaeger, 1959) by the standard separation- of-variables technique. The solution in an infinite- series form is

p(x, t) = (l/L)exp(-fit)

+i,exp[--(F+B)r]cosy

X p. (x’) ~0s F dx’.

where B = 4713: P$,k,. For the case of a uniform initial gas-phase concentration profile, the solution is considerably simplified and we get

P(X 0 = p. exp ( -PO. (20)

The gas-phase concentration is thus not a function of the fall distance and decreases exponentially in time with the washout coefficient /? = -Ixr~N,k,. For highly soluble gases like HN03 and H202, the numerical solution and the analytical solution for the gas-phase concentration are in excellent agreement. Since the predominant drop size rp and the droplet number density N,are functions of rain intensity, one can express the washout coefficient in terms of the rain intensity I and the mass-transfer coefficient k, for the gaseous species in question. For the case of a uniform initial gas-phase profile, one can substitute the exponential expression for p(x, I) in the equation for the

Table 4. Washout coefficients E. (min- t) and half-life periods I( z (mm) of the gases for various combinations of rain intensity f and initial pH values of raindrops

I=jmmh-’ f=20mmh-’

pHo = 5.6 pHo = 4.6

HNOz i. 1.67 ( - 2). 1.67 (-2) fI;2 4.14 (e 1) 4.14 (+ 1)

H20L I 1.94 (-2) 1.94 (-2) ‘t:r 3.56 (cl) 3.56 (+ 1)

SOL 1 1.62 (-3) 9.02 (- 4) fl..Z 4.28 ( + 2) 7.68 ( + 2)

HCHO L 1.10 (-2) 1.10 (-2) [1,.2 6.30 (i. I) 6.30 (i- 1)

NH3 A: 3.50 ( - 3) 2.67 (- 2) t1,1 1.98 (+2) 2.60 (-k 1)

*The notation 1.67 (-2) indicates 1.67 x !O-‘.

pHo = 5.6 pH, = 4.6

3.93 (- 2) 3.93 ( - 2) 1.76 (+ I) 1.76 (+ 1)

4.68 ( -2) 4.68 (- 2) 1.48 (+ 1) 1.48 (+I)

3.84 (-3) 4.03 ( - 3) 7.85 (+ 1) 1.72 (+2)

3.60 (- 2) 3.60 ( - 2) 1.90 (+ 1) 1.90 (tl)

3.96 ( - 2) 6.26 ( - 2) 1.75 (+ 1) 1.11 (tl)


liquid-phase concentration c(x, r). If we neglect the term c/H in this expression, it is possible to solve the resulting first-order partial differential equation by application of the Laplace transform technique. The resulting solution is

c(x, t) = sexp(-Pr)[exp(F)-I]. (21)

For the cases considered above, p is very small, and U

therefore,

c(x, t) c 3QcJ r nRT exp(-Br), (22)

P

thus showing a linear dependence on the fall distance x. The numerical and analytical results agree quite well. Although it was possible to obtain an analytical solution for the case mentioned above, it is clear that for the general case one has to resort to numerical methods.

4.2. Sulfur dioxide and formaldehyde

SO2 is perhaps the most important gaseous pollutant from a precipitation acidity point of view. It can be absorbed by water drops and oxidized in the tiquid- phase by various oxidants. It has a moderately high effective solubility constant H,. For S02, there is a substantial change in H, as the droplet pH changes; therefore, calculation of H, as a function of [H’] is necessary and useful in this case.

Figure 3 shows the gas-phase concentration profile for SO*, the aqueous-phase concentration profile for C,(C, = [SO* .HzO] + [HSO;] + [SO:-]) and the pH for raindrops at various times. The initial concentration of SO* is assumed to be 10 ppb throughout the mixed layer. The rain intensity I is 20 mm h- ’ and the drops are assumed to be at an initial pH of 5.6. The pH of raindrops decreases slowly with distance below the cloud-base, from an initial pH of 5.6 to a final value of approximately 4.8. Over the 2-h period, the pH of drops reaching ground level does not change ap- preciably. As the drops fall, they get concentrated in S species, the capacity of raindrops for SO2 absorption is thus reduced and the gas-phase concentration is not depleted as fast as it is at the top of the polluted layer. The gas-phase SO2 concentration, therefore, shows a significant vertical profile. In the aqueous-phase, the C, concentration increases fast at first but slowly later on as the drops approach equilibrium near the ground level. For the case of pre-acidified drops (pH, = 4.6), the effective solubility of SO2 is reduced and the C, concentration in drops reaches the equilibrium value at a shorter distance below the cloud-base. The pH of raindrops reaching ground level is approximately 4.5, and the gas-phase concentrations show appreciable vertical gradients as before. Figure 4 shows the results for this case.

Since the gas-phase concentrations for SO2 do not show a uniform vertical profile, the washout coefficient

0.0 2.0 4.0 6.0 8.0 10.0

Concentration (ppb)

(b)

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0

Concentration in Raindrops ItiM)

z 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6

a pH in Raindrops

Fig. 3. (a) Gas-phase concentration profiles for SOa at various times. Rain intensity I = 20 mm h- r, initial raindrop pH = 5.6 and initial gas-phase concentration = 10 ppb. (b) Concentration of S(W) species ([SO,. H,O] + [HSO;] + [SO:-] in raindrops at various times. Same conditions as in (a). (c) pH in raindrops at various times. Same conditions as in (a).

must be approximated. In order to calculate a washout coefficient, we calculated vertically averaged gas-phase concentrations at 15 min intervals and obtained a washout coefficient by a least square fit of In [p,,,(t)/po] vs time. The washout coefficients thus obtained are shown in Table 4 for various cases.

Similar calculations were also performed for formaldehyde scavenging. Formaldehyde gas, though not as soluble as the gases considered in the previous section, is an order of magnitude more soluble than S02. Furthermore, the solubility of formaldehyde is not pH dependent, therefore, the dissociation constant in our equations was assumed to be zero. The gas- phase concentration for formaldehyde shows a non- uniform vertical profile. The washout coefficient was

776 SUDARSHAN KUMAR

P -2 soo-

$’ $

a c

Ij lOOO-

is

f BOO-

-zoo0 t,. 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Concentration (ppb)

8 5 2000 I I I t I I : 0.0 2.0 4.0 6.0 8.0 10 0

Concentration in Raindrops (uM)

F 2; 5 z O_ 9 500 -

3 5 1000 -

1 -0 41

m t500 -

:: 5 2000 , I 1 .)? 4.30 4.35 4.40 4.45 4.50 4.55 4.60 0 pH in Raindrops

Fig. 4. (a) Gas-phase concentration profiles for S02. Same conditions as in Fig. 3(a) except initial raindrop pH = 4.6. (b) Same as Fig. 3(b) except initial raindrop pH = 4.6. (c)Same as Fig. 3(c) except initial raindrop pH

= 4.6.

calculated in a fashion similar to that of SC&; the washout coefficients thus obtained are shown in Table 4 for various cases.

4.3. Ammonia

From the viewpoint of rain acidity, NH3 is an important trace gas species because it is very soluble in water and is highly basic in nature. Thus, it serves to neutralize some of the rain acidity. As NH3 dissolves in raindrops, the raindrop pH increases and the effective soiubility ofNHJ decreases. Solubitity of ammonia is. thus, a self-limiting process. Figure 5 shows the gas- phase and aqueous-phase concentrations. and pH of raindrops for an initial ammonia concentration of 1 ppb, pHo = 5.6, and I = 5 mm h- I. The gas-phase concentration profiles show a significant vertical

~::/ , , I , / ~~~~i ,

$ 2000

6 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Concenrration in Raindrops IuM)

3 -0 1500-

m”

~2000 f

1 1 1 1 t 1 t ! 1 ; 4.0 5.0 6.0 7.0 8.0

a pH in Raindrops

Fig. 5. (a) Gas-phase concentrafion profiles for NH3 at various times. I = 5 mm h- ‘, initial raindrop pH = 5.6, and initial gas-phase concentration = 1 ppb. (b) Concentration of dissolved ammonia ({NH, .H,O] + [NH:]) in raindrops. Same conditions as in (a). (c) pH in raindrops at various times. Same conditions as

in (a).

gradient. In the aqueous-phase, the NH3 concentration approaches its equilibrium value at a distance of 400-700m and the pH of raindrops increases to approximately 8.0. The pH of raindrops reaching ground level does not change over the 2-h time period considered.

Simulations were also performed for a reduced value of initial pH (pHo = 4.4) while all the other parameters were kept at the same values as before. The results in this case are strikingly different and are shown in Fig. 6. For raindrops with initial pH of 4.6, the solubility of NH, is very high and the gas-phase and aqueous-phase concentration profiles show charac- teristics similar to that of absorption of HNOs or

;i i 0

d r (8)

8 500 d 20 min. 60 min. 30 min. 15 min.

ir

2 ij 1000

z

f I 3 tsooy

c 2,oo; 0.0 0.2 0.4 0.6 0.8 1 .o considered.

Concentration (ppbl

species in raindrops are far from equilibrium throughout the falling period. For uniform initial gas-phase concentrations, the gas-phase concentrations for HNOJ and H202 show uniform vertical profiles as the rain event progresses. The aqueous-phase concentrations increase almost linearly with distance below cloud-base. For pre-acidified raindrops, the gas-phase and aqueous-phase concentration profiles remain unchanged. The half-life periods of these gases range from about 1540 min under the simulation conditions

For SO, and HCHO, the gas-phase concentrations have a substantial gradient in the vertical. The

ib) aqueous-phase concentrations increase rapidly during the initial stages of fall but slowly later on as the drops approach equilibrium. Under pre-acidified conditions, SO2 solubility decreases resulting in a substantial increase in the half-life of SO,; the half-life periods range from about 1.3 to 13 h. The solubility of formaldehyde is not affected by the acidity of raindrops and its half-life ranges from 19 to 63 min.

NH, solubility is a strong function of the raindrop I

; 0.0 4.0 8.0 12.0 16.0 20.0 24.0 acidity. For pre-acidified conditions, ammonia profiles ._ a Concentration in Raindrops (r;Ml in the gas-phase and the aqueous-phase are similar to that of HNOs. NH, is depleted quite fast (half-life of approximately 15 min) because of its high solubility and high mass-transfer coefficient. For initial raindrop pH of 5.6, the gas-phase concentration profiles show gradients in the vertical and aqueous-phase concentrations approach equilibrium because of iowered solubility as drop pH increases. The half-life increases considerably compared to the case of pre-acidified drops.

0 pE rn Ramdrops

Fig. 6. (a) Gas-phase concentration profiles for NHJ. Same conditions as in Fig, S(a) except initial raindrop pH = 4.6. (b) Same as Fig. 5(b) except initial raindrop pH = 4.6. (c) Same as Fig. 5(c) except initial raindrop

pH = 4.6.

H202. The pH of raindrops reaching ground level at 15 min is about 5.4 and decreases slowly with time as Adamowin R. F. (1979) A model for the reversible washout of

ammonia in the atmosphere is depleted. The washout sulfur dioxide, ammonia and carbon dioxide from a

coefficients and half-life periods are shown in Table 4 polluted atmosphere and the production of sulfates in

for various cases. raindrops. Atmospheric Etmironmenr 13, 105-I 2 1.

Adewuyi Y. G. and Carmichael G. R. (1982) A theoretical investigation of gaseous absorption by water droplets from SO,-HNOs-NH,-C02-HCI mixtures. Armospheric

5. COBCLUSIONS Enoironntenr 16, 7 19-729. Beard K. V. and Pruppacher H. R. (1969) A determination of

An Eulerian model for simultaneous simulation of the terminal velocity and drag of small water drops by

gas-phase and aqueous-phase concentrations of va- means of a wind tunnel. J. armos. Sci. 26, 1066.

rious pollutant species during a rain event has been Best A. C. (1950) The size distribution of raindrops. Q. Jl R.

met. Sot. 76, 16. developed. Simulations were performed for single- CarsIaw H. S. and Jaeger J. C. (1959) Co~~rio~ of Hear in

species scavenging of HN03, H202, S02, HCHO and Solids (2nd Edition). University Press, Oxford.

NH, under various conditions. Gaseous HN03 and Cotton F. A. and Wilkinson G. (1972) &fvanced Inorgnnic

HzOz are highly soluble in water and under typical Chemistry (3rd Edition). Wiley-Interscience, New York.

Durham J. L., Overton J. H. and Aneja V. P. (198 I) Influence atmospheric conditions, the concentrations of these of easeous nitric acid on sulfate nroduction and acidity in _


The model at this stage has been applied to single- species scavenging from the atmosphere by rain. It is currentIy being extended to include multiple-species absorption and chemical reaction in the aqueous phase.

Acknowledgement--The author thanks Peter Berzins for help in the computations.

REFERENCES

778 SUDARSHAN KUMAR

rain. Atmospheric Encironmenr 15, 1059-1068. Engelmann R. J. (1968) The calculation of precipitation

scavenging. In Mrteoro/oyp and Aromic Energy (edited by Slade D. H.), pp. 208-221. U.S. Atomic Energy Commission.

Fisher B. E. A. (1982) The transport and removal of sulfur dioxide in a rain system. Atmospheric Encironmenr 16, 775-783.

Frossling N. (1938) The evaporation of falling drops. Gerlands Beirr. Geophys. 52, 170.

Garland J. A. (1978) Dry and wet removal of sulfur from the atmosphere. Atmospheric Entiironmenr 12, 349-362.

Gunn R. and Kinzer G. D. (1949) The terminal velocity of fail for water droplets in stagnant air. J. Mer. 6, 243.

Hales J. M. (1972) Fundamentals of the theory of gas scavenging by rain. Atmospheric Encironmenr 6, 635-659.

Lapidus L. and Pinder G. F. (1982) Numerical Solurion of Partial Differenrial Equarions in Science and Engineering. Wiley, New York.

Ledbury W. and Blair E. W. (1925) The partial formaldehyde vapour pressure of aqueous solutions of formaldehyde. J. Chem. Sot. 127. 2832-2839.

Levine S. 2. and Schwartz S. E. (1982) In-cloud and below-

cloud scavenging of nitric acid vapor. Atmospheric Encironmenr 16, 172j_1734.

Marshall J. S. and Palmer W. McK. (1948) The distribution of raindrops with size. J. .Vrt. 5. 165.

Martin L. R. and Damschen D. E. (1981) Aqueous oxidation of sulfur dioxide by hydrogen peroxide at low pH. Armospheric Enkonmenr 15. 161s162 I.

Mason B. J. (1971) The Physics of Clouds. Clarendon Press, Oxford.

Molenkamp C. R. (1983) A scavenging model for stratified precipitation. In Precipirarion Scatienging, Dry Deposirion, and Resuspension (edited by Pruppacher H. R.. Semonin R. G. and Slinn W. G. N.), pp. 597-608 Elsevier, New York.

Schwartz S. E. and White W. H. (198 1) Solubility equilibria of the nitrogen oxides and oxyacids in dilute aqueous solution. Ada. Encir. Sci. Engny 4. l-45.

Sekhon R. S. and Srivastava R. C. (1971) Doppler radar observations of drop-size distributions in a thunderstorm. J. ormos. Sci. 28, 983-994.

Sillen L. G. and Martell A. E. (1964) Stability constants of metal-ion complexes. Spec. Pub/. I7 Chem. Sot. Land.

Treybal R. E. (1980) Mass-Transfer Operarions. McGraw-Hill, New York.

an eulerian model for scavenging of pollutants by raindrops

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