7

n + I

P, pT r

Re

R, SC t T

f .i

NON-ISOTHERMAL EFFECTS ON SO2 ABSORPTION BY WATER DROPLETS-I. MODEL DEVELOPMENT

MAHYOUD RED.& and GREGORY R. CARMICHAEL Chemical and Materials Engineering Program. The Universit? of IoNa. Iowa City, IA 52240, U.S.A.

Abstract-4 theory for treating the simultaneous absorption and chemical reaction of SO, into (~5:s~ droplets falling in a non-uniform temperature field is presented. The analysis includes non-isothermal effects on droplet growth. reaction rates and physical parameters, and treats multicomponent diffusion. The model can predict droplet radius. temperature. pH and S02.H20. HSO; and S(W) interfacial and buik concentrations. Simulation results are presented and indicate that non-isothermal effects can be important in gas-scavenging calculations.

SO\lENCLATCRE Grruk luiwrs 6 tilm thickness. cm

I - D&;D&, as defined. in Eiuation (42) back_nround concentration of hydrogen ion before entermg mixed layer, g mole ! - ’ total gas concentration, gmole/ _ ’ heat capacity. cal g-’ ‘K-’ total molar liquid phase concentration diffusivity of i in the mixture. cm’s_ ’ “cl Ix Ja ,I,

flux of component i at the interface, gmole cm / _ ’ s- ’ Henry constant. atm gmoie- ’ f - ’ enthalphy. cal g-’ first dissociation constant. gmolei-’ thermal conductivity. cal cm _ ’ s - ’ K - ’

1 1;: i heat of vaporization for water, cal g _ ’ \ kinematic viscosity. cm2 s-’ 5 distance at a point in the bulk gas mixture from the

center of drop

$ density, gem-j [SOLHI - [HSO;] = total sulfur SIV concen- trations, g mole t- ’

cc, as defined in Equation (371.

Subscrips B bulk gas mixture condition * inrcrfacial condition 9 gas phase L liquid phase.

mass transfer coefficient in gas. ems-’

D;o;. d = mass transfer coefticient of SO2 gas in the ISTRODUCTION

liquid phase. cm s _ ’ dissociation constant for water, gmole2 /-’

The absorption and desorption of gaseous pollutants

flux of component i, gmole cm! -’ s- ’ by water droplets is a key step in the removal process

flux ofcomponent iat the interface,gmolecm/ -’ s-’ for many trace gases in the atmosphere. The removal of

.V’ s02H20+&$SO;. gmolecm(-‘s-’ SO, by u-ater droplets in clouds and fog and by falling

Sherwood Number = 3 raindrops is an important example. The absorption of D tm SO, by water droplets is a complex mechanism

involving the transport to the air-water interface, the total number of components in the gas mixture pressure of component i

absorption of gaseous SO, into the liquid while the

total pressure liquid is undergoing either evaporation or conden-

radius of the drop. cm sation. and the simultaneous partial hydrolysis of SO2

Reynolds Number = 2 in water.

Y Although the absorption of SO, is an important

rate of disappearance of i atmospheric process, most generalized analyses treat

Schmidt Number = c;Df only the isothermal, no-growth droplet case (e.g.

time. 5 Carmichael and Peters, 1979). However. noniso-

temperature, K thermal phenomena are present to some extent in all

ambient temperature, K drop interfacial temperature. K

gas absorption processes, especially during absorption

terminal velocity of the drop. m s- ‘ and desorption in a drop undergoing evaporation and

distance from the interface into liquid. cm condensation. Under certain conditions these effects

molz fractions of i at interface can be very important. For example. Bogaevskii (1969)

mole fractions of i in the bulk gas mixture showed that gas absorption is enhanced when the drop mole fractions of i at distance 5 from center of drop. is undergoing condensation as compared with a drop

145 -,

146 MAHMOUD REDA and GREGORY R. CARWCHAEL

in the no-growth condition. lie found that droplets growing by water vapor condensation in a mine absorbed six times more sulfur dioxide than that predicted by the steady state absorption. In addition, Matterson and Oliver (1978) showed that absorption of oxygen and nitrogen dioxide was enhanced during condensation of a humid gas stream on cold suspended droplets.

These studies point out that in a condensed water film a supersaturated amount of absorbed gas can be accumulated during the growth phase. When the droplet begins to evaporate, this excess gas can be desorbed. These non-isothermal effects can be par- ticularly important in chemical absorption processes, affecting both the reaction rates and the selectivities.

In this paper, SO, absorption in a drop undergoing evaporation or condensation is analyzed and modeled. The analysis includes non-isothermal effects on droplet growth, physical parameters and reaction rates, and treats multicomponent diffusion. Results demonstrate that non-isothermal effects can be important in gas- scavenging calculations.

DERIVATION OF THE DROPLET EQUATIONS

The derivation of ordinary differential equations which describe sulfur dioxide transport to a singie droplet undergoing evaporation (or condensation) in a non-uniform temperature is carried out in two steps. First, the transport equations for gas-phase multicom- ponent diffusion are solved to generate expressions for the SO, ffux and water vapor flux to the droplet. A similar analysis including chemical dissolution and electral-neutrality considerations are presented and an expression for the total sulfur [S(IV)] flux in the liquid phase is obtained. In this analysis, the gas-liquid interface is assumed to be at equilibrium. This implies that interfacial resistance is negligible compared with the diffusive resistance of the gas, and should be valid except at quite low pressures. In addition, the effect of surface tension on vapor pressure is ignored. Finally, these equations are coupled to droplet radius and energy equations.

The transport equation describing ordinary diffu- sion in multicomponent gases at low density is the Stefan-Maxwell equation (Bird et al., 1960):

where tvi is the flux of component i, C, the total gas concentration, Di, the diffusivity of i in the mixture and yi the mole fraction of component i. This equation applies to a drop of radius r surrounded by a gas composed on n components where the n f 1 com- ponent is the inert and < is the distance from the center of the droplet.

The following boundary conditions apply:

.i’i = .vt and f = P at < = r (‘I

and

yi =x; and T=T* at f= X. (3)

where T is the temperature and superscripts ***” and “B” denote the interfacial and bulk quantities, respectively.

Now Equation (1) must be modified to account for the motion of the droplet in the atmosphere (see Newbold and Amundson. 1973):

where N,, is the Sherwood Number. Now, if pseudosteady-state is assumed, then the rate

of transfer of component i through any sphere 4 = constant will be constant. Thus, if Ji is the flux at c = r, then

and Equation (4) becomes

- .V:,, C,D, +dfi Ji=--_I--7-,

?i; n+ ’ __ _rr4,.vi c Jj.

i r UC i i=t

Substituting

I- 1

%--

‘I

and

into Equation (6) yields

Integrating the above equation subject to the bound- ary conditions (2) and (3) results in

where

“+t

J= c Jr. k=t

Simplificarions for SO, absorption

Equation (10) represents the general expression for multicomponent diffusion of n + I species. In the process of SO, absorption by water droplets in the atmosphere the major gaseous species are SOz, Ii,0 and air (assuming SO1 is the only trace gas being absorbed).

Non-isothermal et%cts on SO2 absorption by water droplrrs--I 147

Since air is inert

J4,, = 0.

and from Equation (10)

Or

where

.r;,r = f - Gtio -Go:. (16)

The expressions for the flux of SO2 and H,O from the gas phase to the interface are

and

The expressions derived above describe gas-phase mass transfer. However, in solution SOI exists as physically dissolved S02, designed as SOz. H 20. and in dissociated forms such as bisulfite (HSO;) and sulfite (SO:-). (The discussion that follows will be restricted to a SO,.H,O and HSO; system, although it can be extended to include SOi- and other forms).

The expressions for the flux of SOz.H20 and HSO; in solution are

,v* [SO,.H,O]’ SO,-HI0 - CT

>I (1%

x = 0

and

Equations ( 19) and 120) can be combined to yield

.I’,; = v* v* SO, H,O T. HSO;’ (23

If tiim theory assumptions are used. then the following boundary conditions apply:

at .x = 0, 05 = (P* 1731

and

where

Q = [S02.H,0] -i [HSO;].

Equation (21) can be integrated, giving

(25)

with

-k$.-t([HSO;]*-[HSO;]% (26)

The equilibrium relationships at the interface are given by

and

.Vz,20 = - JRIO. (30)

Solurion composition ryuarions

Equation (29) couples the liquid phase and gas phase mass transfer processes by stating that the flux of SO2 from the gas phase must equal the flux of absorbed SO, (and dissociated forms) in the liquid phase. The compositions on the liquid side of the interface are coupled to the gas-side interfacial SO2 concentrations through the equilibrium expression

[SO,(g)]* = H[S02.H20]*, (31)

where H is Henry’s law constant. Now, before SO2 absorption occurs, the droplet may

contain a strong base or acid. Under these conditions, the electral-neutrality condition dictates that

[I-I’], = C,lt[OH-],. (32)

where

K, = [H+],[OH-I,, (33)

148 MAHMOUDREDA and GREGORY R. CARMICHAEL

and C, is positive For a strong acid. negative for 3 strong base. and zero at pH = 7.

The eiectral-neutrality condition after SO, ab- sorption is

[H’] = [HSO;]+C,,&

Litilizine the equilibrium expression.

(34)

Et-r”l[HsO;l = K

[SO,.H,O] * (35)

Equation (35) can be solved for [HSO;], i.e.

[HSO;] = [ - li/ + +’ + SK#];Z (36)

where

Ij/ = ‘K + C,, + ,I( [HSO;] + C&f’ + 4K, (37)

and

[SOI-Hz01 = 4--[HSO;]. (38)

Iterative solution of these equations enables the calcu- lation of droplet pH. (HSO;] and [SO,.H,O].

The equations described up to now have dealt with the steady-state multicomponent mass transfer process and constitute a system which for a droplet of fixed radius and temperature, and specified initial con- ditions can be solved for .I&,, J&o, [H-l, [HSO;], [S02.H20]. etc. With this information the change in the bulk concentration of absorbed SO, with time can be calculated from

However, in non-isothermal applications the tem- perature as well as the radius of the droplet change with time. Since the mass transfer equations are coupled to the droplet radius (e.g. see Equations 39 and 17) and to the temperature of the droplet ~principaIly through the evaluation ofthe physical properties ofthe system), it is necessary also to include the droplet radius and temperature equations in the analysis.

The change of the droplet radius with time can be expressed as

dr

z= -aJ”H ,,

i ’

where Z. = 180/p,, and the temperature of the drop can be determined from (Newbold and Amundson, 1973):

+zJ”H .i.+ IO-‘J&&$ *

-C$ ) _ *

x 0-s - Lop) -i 10-j J"so W

1 $02

+- lo-‘3’ cq (T-,-T,,,,) 50: P, t

+ 0.95oiTJ3 -P&J 1 J’

(41)

where a is given by

+ J”H oC”p .10-3, : . I (42)

k is the thermal conductivity in cal cm- ’ S- ’ K- ’

given by

LI = 549 x 10Y5 + 1.68 x 10-*(T- 273.16), (43)

and i: is the heat o~vaporization for the water given by

;. = 753.68 - 0.5725T cal g - ‘. Fw

DISCUSSION

Equations (391, (40), (41) and (29) comprise the basic model which describes the non-isothermal absorption of gaseous SO?. These equations can be solved simul- taneously to yield the droplet radius, temperature, pH, [SO,*H,O], [HSO;] and [S(IV)], etc.

This model is being used to analyze SOI absorption by rain droplets falling from cloud base. A description of the simulations and a detailed presentation and discussion of the results are presented in a companion paper. However, typical results are presented in Figs i-3. These results are For a 0.08cm droplet with initial pH of 4 falling 2 km through a uniform fietd of 1Oppb SO? in an adiabatic atmosphere with a relative humidity of SO*, and a ground level temperature of 23’ c.

The droplet temperature as a function of fall distance is shown in Fig. 1. The droplet temperature increases with the ambient temperature but is always below the ambient value. This is due principally to the fact that the dropfet is evaporating. This can be seen by noting that the droplet radius decreases during fall (see Fig. 2).

The interracial and bulk [S{iV)f in the droplet as a function of fait distance are presented in Fig. 3. The concentrations are seen to pass through a maximum. This phenomena is due to the coupled effects of heat and mass transfer. The fact that the solubility of SO2 decreases as the temperature increases is also an important factor.

The phenomena shown in Fig. 3 has also been demonstrated experimentally. The results of Flack er al. (1979). where a suspended cold water droplet was exposed to a humid nitrogen stream containing re- latively high concentrations of SO? ( - 1OOOppm) at different supersaturation levels, are presented in Fig. 4. Their results also showed that the droplet temperature was below the ambient value.

The effects of multicomponent diffusion are aIso illustrated in Fig. 3. Recall that at the interface Equa- tion (29) is assumed to apply, i.c.

i”; = - JgSol.

Son-isothermal effects on SO, absorption by water droplets-1

273 ’ I 1 I I I , I I I

0 200 400 600 800 1000 1200 1400 1600 1300 2000

~CLOUO 3465 FALL DISTANCE (m) GROUND LEVEL!

Fig. I. Temperature of a droplet with pH, = 4 and r0 = 0.08 as a function of fall distance from cloud base. Also presented is the inputted atmospheric temperature profile.

3 IOCQ 2000

CLOLC SASE GRC)UND

FALL DISTANCE (m) LEVEL

Fig. 2. Droplet radius as a function of fall distance from cloud base.

V I I I I I I I I

0 200 400 600 800 1000 1200 1400 is00 i33c 2000

FALL DISTANCE (m)

119

Fig. 3. Interfacial and bulk S(IV) concentrations as a function of fall distance from cloud base.

150 MAHMOCD REDA and GREWRY R. C~RWICHAEL

TIME. S

Fig. 3. Measured S(IV) concentrations as a function of time from the study by Flack et ai. t 1979).

Now after a certain amount of time equilibrium must be attained and

X; = J”so = 0. 1 (4%

and from Equation ( 17)

Go, exp Tr

- - P;o, = 0. v9,02 -

w

Therefore, at equilibrium when absorption is equal to desorption we have from Equation (46)

Cf40, P$. $= _A In p*. r so,

(47)

Substituting Equation (14) for Jyields

(48)

or

Now under the assumption that Diir = D9,,. the multicomponent equilibrium criteria is

Equilibrium is approached in multidiffusion when Equation (50) is satisfied. This condition is in contrast to that which arises when the effect of other fluxes is neglected, i.e.

p;o> = Co,- (51)

The importance of temperature on the equilibrium criteria can be seen by rewriting Equation (50) as

fYob? II1 - G*o(=)I. (52)

[SO2 HzOl& = [I_ pBH o, H(T)

The location where JgoI equals zero is located in

Fig. 3. From this point on Jg,,changes sign and SO, is desorbed from the solution. it can also be seen that J to, = 0 occurs at a different location from that which sat&es Equation (51).

Sf.W#AR’r

A theory for treating the simultaneous absorption and chemical reaction of SO2 into a water droplet in a non-uniform temperature field has been presented. The analysis includes non-isothermal effects on drop- let growth (such as evaporation or condensation) on the physical parameters of the system including solubility, diffusivities. etc., and reaction rates. and treats multicomponent diffusion. The model can be used to predict the droplet temperature, radius, pH, and S02.H20, HSO; and S(W) interfacial and bulk concentrations, as a function of time or fall distance. Model simulations indicate that non-isothermal effects can be important in gas-scavenging calculations.

Ackna~~~edgemmr-This research was supported in part by the National .Aeronautics and Space Administration under Research Grant NAG l-36.

REFERESCES

Bird R. B.. Steward W. E. and Lightfoot E. N. (1960) Transport Phenomena. Wiley. New York.

Bopevskii 0. .A. 11969) Absorption of a gas on a growing arop. Zh. fi:. Khim. 43, 719.

Carmichael G. R. and Peters L. K. (1979) Some aspects of SOI absorption by water-generalized treatment. Almospheric Encironmmt. 13. 1505-l 5 13.

Flack W. W. and Matteson M. J. (1980) Mass transfer of gases to growing water droplets. In Pollurrd Rain (Edited hy T. Y. Toribarn). pp. 61-83. Plenum Press. New York.

Matteson M. J. and Olivers M. J. (1978) The absorption of oxygen by condensing and evaporating aster droplets. itm. id. Hjq ;Iss. J. 39, 783.

Sewbold F. R. and Amundson 5. R. (19731 A model for evaporation ofa multicomponent droplet. .4./.C.H.E. J. 19. 112- 30.