WY-6981 82 Ol~liJI-09 M3.000

t 1981 Pergmon Press Ltd.

NON-ISOTHERMAL EFFECTS ON SO2 ABSORPTION BY WATER DROPLETS-II. RESULTS AND DISCUSSION

MAHMOUD REDA and GREGORY R. CARMICHAEL

Chemical and Materials Engineering Program. L’niversitk- of Iowa. Iowa City. 1~1 5X10. U.S.4.

Abstract-Presented results indicate that non-isothermal effects on SO2 absorption by water droplets can be important in gas-absorption calculations and interpretation ofqround level rain samples. For example, small droplets can desorb SO, before reaching ground level and m some cases these droplets can evaporate completely. The desorption of SO, can lead to a modest increase in droplet pH and. in effect. redistribute ambient SO2 from higher to lower elevations

ISTRODLX3’IOS Spec@d condirions

During recent years appreciable effort has been devoted to the analysis of gas scaven_tig by precipitation. This effort, which has resulted in a significant advancement of rain scavenging theory, has produced principally two types of models: physical models, exemplified by the work of Hales and his colleagues (e.g. Dana et al., 1973; Hales et al.. 1973), and chemical models, stemming primarily from the work of Scott and Hobbs (1967). These studies, as we11 as those more recent works which have combined these two model approaches (e.g. Hill and Adamowin, 1977), have treated only the non-isothermal no-growth drop- let case. However, non-isothermal phenomena are present to some extent in all gas absorption processes, especially during absorption or desorption in a droplet undergoing evaporation or condensation, as is the case for a drop falling in a rain event.

Unless stated otherwise, nomenclature is the same as in RC-I.

(a) Simulations were carried out for fall distances of 2 km;

(b) The relative humidity was assumed to be 80 7,: (c) The droplet temperature at cloud-base was

assumed to be 2’ C; (d) The termina1 velocity for the drop was calcu-

lated from that given by Johnson (1979).

6; = 9.62 - 10.30 exp [ - 12r]

for r < 0.03 cm and

Vz = 86.2r - 0.155 (2,

for r > 0.03 cm: (e) The ambient temperature was assumed to be

In this paper, simulation results obtained from a model for the washout of sulfur dioxide which includes

non-isothermal effects are presented and discussed. (The model is discussed in detail in the companion paper (Reda and Carmichael, 1981) referred to as RC- I). Results indicate that non-isothermal effects can be important in calculating SO? washout and in analyzing ground level rain samples.

T, = 276 + lo- ‘Z, (3)

where Z is the distance traveled by the droplet in meters and Z = 0 at the cloud base;

(f) The pressure was assumed to change with height according to the hydrostatic equilibrium equation

p = pO exp[-$‘I, V

SIML’LATION CONDITIONS

The developed model was used to simulate SO, absorption below cloud by raindrops falling through a layer in the atmosphere near the surface of the earth which is 2 km in height and in which sulfur dioxide is distributed in some specified manner. Rain is formed above this layer and is comprised of droplets of an assumed size. The raindrops are assumed to be spheri- cal and to fall vertically at their terminal velocities. Before entering this layer. the raindrops attain an initial pH designated as pH,.

(1)

(4)

where c is the average virtual temperature change in height AZ;

(g) The virtual temperature was calculated from the following

(5)

where m is the liquid water mixing ratio (g water vapor;g dry air)

e, w = (0.8)(0.62+-- p-e,’

(6)

and e, is the saturated vapor pressure of water vapor

151

152 MAHMOUD REDA and GREGORY R. CARMICHAEL

(.in millibars) and is given by and ‘i’ indicates the transferred species and

17.269T - 47 17.3

(T - 35.86) I ’ (7)

(h) The background concentration of SO, was assumed to be 10 ppb 2: lo- ’ atm.

Calculations

Equations (39), (4O), (41) and (29) from RC-I comprise the system of equations which describe the non-isothermal absorption of SO,. These equations were solved simultaneously to yield the droplet radius, temperature. pH, and bulk and interfacial S(W) con- centrations profiles as a function of fall distance. Equations (39)-(41) were solved using the Gear method and Equation (29) was solved using the Kotor- vach or so called Newton-Raphson method.

Due to the temperature changes of the droplet during the fall, physical properties of the system have to be reevaluated continuously. The diffusivities, viscosity, etc. were predicted using various relation- ships as given by Reid et al. (1977). The pressure of H,O at the interface was derived from a curve-fit to steam table values as given by Johnson (1979), the Henry’s constant and first dissociation constant were estimated from values cited in Carmichael and Peters (1979). and the gas phase mass transfer coefficients were estimated using the Frassling equation

where

Nf, = 2+x’Re’:ZSc’3, (8)

Nf, - 2k9r D;C/ (9)

(10)

SC = v/D: (11)

293 t

Z’ = 0.6 for SO, _a

I’ = 0.95 for water vapor.

This expression has been shown to describe the physical behavior with reasonable accuracy over all conditions of practical interest to the present in- vestigation (Skelland, 1974)

RESULTS A.s;D DISCUSSION

Predicted droplet temperatures as a function of fall distance are plotted in Fig. 1 for daerent initial drop sizes. Droplet temperatures are seen to be below the ambient values. The ground level temperature for a OScm droplet is _ 8’C as opposed to the ambient value of 23’ C, whereas the 0.1 cm droplet has a ground level temperature of _ 20’ C. The larger droplets have a larger mass and lower residence times due to their large terminal velocities and thus have insufficient time to heat up during descent. The smaller droplets have a lower-than-ambient temperature due principally to the evaporation of the droplet.

The evaporation rates under conditions simulated were sufficient to maintain the droplets at a tempera- ture about 3°C below the ambient value. These evaporation rates were insufficient to cause any ap- preciable change in droplet size during descent for droplets larger than 0.08cm. However, for smaller droplets the change in droplet radius was significant. The radius of a 0.04 cm droplet was found to change by 15 % during the 2 km fall distance and a 0.01 cm droplet decreased to 10 4; of the original size after only 170m fall distance (see Fig. 2).

Bulk and interfacial total sulfur (S(IV)) concen- trations as a function of fall distance are presented in Figs 3-10. Concentration profiles for droplets smaller than 0.1 cm for pH,, = 4 are similar in that desorption

273 I I I I I I I !

0 200 400 600 800 1000 1200 ICC0 i6-c )8CC ix,;

FALL DISTANCE (m)

Fig. 1. Temperature of a drop as a function of fall d&an=.

Non-isothermal effects on SO, absorption by water droplets-Ii

,001 1 0 10 20 30 40 50 60 70 a0 90 IO0 110 120 I40 161:

FALL DISTANCE (“1

Fig. 2. The radius of a drop as a function of fall distance for a drop of radius of0.01 cm.

i

INTERFACE

1 /

/

r BL’iK

I / I I I

200 400 600 800 1000 1230 1430 1600 1800 2x3

F4LL DISTANCE (m)

Fig. 3. Concentration of total sulfur S(W) for a drop of radius 0.5 cm and pH, = 4 as a function of fall distance.

I , ! I I / / !

2cc 400 500 800 1000 1200 14c3 1600 1830 2coz

FALL DISTANCE (m)

Fig. 4. Concentration of total sulfur S(W) for a drop of radius 0.1 cm and pH, = 4 as a function of fall distance.

,UAHYOCD REDA and GREGORY R. CARMICHAEL

0 2% 439 533 800 1000 1 ZOO l4GO 1600 1800 2200 FA.LL DIS’1&UCE cm!

Fig. 5, Concentration of total sulfur SW) for a drop of radius 0.04 cm and pH, = 4 as a function of fall distance.

INTERFACE

1 / I ! : / I / / I

0 13 23 30 43 50 60 70 80 90 100 :2c 143 15:

FALL DlSTAU’E (m: ”

Fig. 6. Concentration of total sulfur S(N) for a drop of radius 0.01 cm and pH, = 3 as a function of fall distance.

3 2x 4x 600 800 1000 1200 1400 1600 lBO0 2oi3

FALL DISTANCE (m)

Fig. 7. Concentration of total sulfur S(N) as function of fall distance for a drop of radius 0.1 cm and pH, = 7.

Non-isothermal effects on SO: absorption by water droplets--II

200 5~0 600 600 1000 1200 1430 1600 1800 20c2 FALL DISTANCE fmi

Fig. 8. Concentration of total sulfur S(N) as a function of fall distance for a drop of radius 0.1 cm and pH, = 10.

aULK

8-Y / / I I I I I

0 200 a0 600 800 1000 1200 1400 1600 i8CO 2000

FALL DISTANCE (ml

Fig. 9. Concentration of total sulfur S(N) as a function of fall distance for a drop of radius 0.5 cm and pH, = 10.

‘, 70 Y

g 60

i c 50 .-

” 1 40

2 30

20

10

0 200 400 600 800 1000 1200 1400 1600 I a30 2OW FALL DISTANCE (ml

Fig. 10. Concentration of total sulfur S(IVJ as a function of height for a drop of radius 0.04cm and pH, = IO.

156 M~HMOUD REDA and GREGORY R. CARMICHAEL

occurs before reaching ground level. This desorption can be seen in the profiles of Figs 5 and 6 and is due to both the increase in temperature of the drop (and the subsequent decrease in solubility) and to evaporation. The smaller the droplet the earlier that desorption occurs, the higher the maximum concentration of S(W) and the lower the ground level concentra- tion. The maximum values ranged from 3.9 to 6 p moles !’ - 1 and the ground level values varied from 2.5 to 2.1 ,U molest - ‘. Note also that the smaller the droplet the closer the bulk and interfacial concen- trations are to being equal. Desorption does not occur for the 0.5 cm droplet because a saturation condition is never approached (see Fig. 3). In all these cases, the droplet pH decreased only very slightly.

The effect of initial pH on the droplet S(W) concentration profiles can be seen by comparing Figs 4, 7 and S. The bulk concentration increases as the initial pH increases, varying from 2.5 p mole / - ’ (at ground level) for pH,, = 4, to 5.7p mole! -I for pH, = 7, to 6.3 p mole / - * for pHO = 10. This effect of initial pH is even more dramatic for the smaller droplets (e.g. ground level concentration of a 0.04cm droplet is 2p molet’ - ’ for pH, = 4 and 103umolet-’ for a pH, = 10).

The S(IV) concentration profiles for pH, = 10 for droplets greater than 0.06cm are similar in that these drops never reach equilibrium (see Figs S and 9). Due to the presence of a strong base, these droplets absorb significant quantities of SO,, with ground level bulk concentrations reaching 24 and 45 p moles f - ’ for the O.Og and 0.06 cm droplets. respectively. The pH at these levels are 9.99 and 9.7, respectively.

For droplets less than z 0.06ctn and pH, = 10 sutlicient SO1 is absorbed to reach a breakthrough point and the pH of these droplets decreases dramatically. The S( IV) concentrations and droplet pH of a 0.04cm drop are presented in Figs 10 and 11. The breakthrough point in this case occurred at a fall distance of _ 15OOm with the pH falling to 5.6. For larger fall distances the values remained constant. The smaller the droplet, the sooner breakthrough occurs (e.g. for a 0.01 cm drop, breakthrough occurs at a fall distance of 40 m).

.A few simulations were also conducted for different ambient relative humidities. The S(N) concentrations for the case when RH = 407, and pH, = 10 for a 0. I cm drop are presented in Fig. 12. The ground level concentration for this case is _ 2.5 times the value for RH = 80:“. This is due principally to the increased

a-

7- I

t”6- Y P 5-

E cl 4-

3-

2-

1 I ! I I I

0 200 430 630 800 1000 1200 1400 1600 1800 2000 FALL DISTA?iCE (m)

Fig. 11. The pH of a drop of radius 0.04 cm and pH, = 10 as a function of height.

Fig. 12. Concentration of total sulfur S(W) for a drop of radius 0.1 cm and pH, = 10 as a function of height. The ambient relative humidity is 407;

Non-isothermal effects on SO, absorption by water droplets-11 15:

ev-aporation for the RH = 40”, case which keeps this

droplet at a cooler temperature (droplet temperature at ground level is ZSZK compared to 293 K for

RH = SO”& and the radius smaller (droplet radius is

O.G%cm at ground level compared to O.W7cm for RH = 80”~ These factors cause an increase in SO, solubility and an increase in the mass transfer rate, respectively. The ground level droplet pH values were also slightly lower for RH =40”,, than for RH =SO”,.

The cases presented so far have been for a droplet

ialling through a uniform distribution ofambient SO?.

Simulations were also performed for different con-

centration profiles. The cases for a 0.1 cm droplet with

pH, = 10 falling through a step-change in ambient

SO2 concentrations are presented in Figs 13 and 14. In Fig. 13. the SO, concentrationchanged from 1Oppb to 1 ppm at a fall distance of 16OOm. This simulation indicates that a droplet responds quickly to changes in ambient concentrations. Of particular interest is the simulation where the ambient concentration decreased

by a factor of 100, 400m from ground level (see Fig.

14). This is similar to a raindrop falling through an

0 200 400 600 800 1000 1200 1400 1600 1800 2000

FALL DISTANCE (m)

Fip. 13. Concentration of total sulfur S(W) for a drop ofradius 0.1 cm and pH, = IOas a function of height. The ambient sulfur dioxide concentration is given by:

C!02 = 10 PPb. Z < 16OOm;

CEO2 = 1 PPm, Z > 1600m.

260 c

240

220

200

200 400 600 800 1000 1200 1400 1600 1800 2300

FALL DISTANCE (m)

Fig. 14. Concentration of total sulfur S(N) as a function of height for a drop of radius 0.1 cm and pH, = 10. The ambient sulfur dioxide concentration is given by:

Cto2 = 1 PPm, Z < 1600m;

c:02 = 10 PPb, Z 2 16OOm.

I% ~fanuour, REDA and G~ioou~ R. ~ARHICHAEL

eIevated non-dispersed plume. These results indicate predicted ground level concentrations are highly de- that a significant amount of absorbed SO2 is desorbed pendent on drop size. with drop size conrroliing the before reaching the surface. This leads to a modest rate at which saturation is approached. The effects of increase in pH (from 4.3 to 4.7), but more importantly multicomponent diffusion appear IO slow down the to a redistribution of ambient S02, with SO1 being in rate at which saturation is approached as compared to etTect transported from higher to lower elevations. the isothermal model of Hit1 and Adamowicz.

Finally, a comparison between the ground level bulk S(N) concentrations and droplet temperatures for a 0.1 cm droplet resulting from a fall through an adia- batic atmosphere and those resulting from a fall through isothermal atmospheres is provided in Table 1, Even in an isothermal atmosphere, the dropiet temperature never reaches the ambient value. For this droplet the temperature reaches an equilibrium value after c MOm, but due to evaporation effects this tem- perature remains below ambient. The lower the temperature, the higher the ground level S(W) con- centrations for both pH, = 4 and pH, = 10. The adiabatic atmosphere results in higher S(W) con- centrations than those of an isothermal atmosphere with the same ground level temperature (i.e. + 20” C).

Results obtained from a SO.. sbsorption*model which includes non-jso~hermal &rs indirate that these effects can be important in gas absorption calculations and in the interpretation of ground level samples. For example. initially acidic small droplets desorb SO1 before reaching the surface and small basic droplets absorb suflifisnt SO, ta reach a break- through point with the droplet PI-I falling rapidly to

__ 4 3.3.

The isothermal results presented in Table 1 can also be compared with those results predicted by models which do not consider multicomponent effects. For example, Hill and Adamowicz (1977) have developed an isothermal model for SO, absorption by water droplets and presented results which can be compared to those in Table 1. For a 0.1 cm droplet with T= 298 K and C,, = 10 ppb. their model predicts ground level

suifur [Iti concentrations after a 2 km fall distance of _ l.j~moles!-‘and - 5~~moles~-~forapH~of 3 and 10, respectively. Their model predicts that under these conditions the droplet is saturated for pH, = 4 but not for pH, = 10. Their results for pH, = 4 are in excellent agreement with those predicted by the non- isothermal effects model.

Resultsdemonstrate that the absorption behavior of a raindrop is strongly dependent on the initial pH of the droplet, the ambient RH, and rhc ambient profiles of temperature and SO:. In order ic analyze ground Level samples in terms of the cozrribution of gas scavenging to the measu:ed pH, i; is necessary that field studies obtain information on rhese parameters. In addition. simulation results indiare that initial drop size and fall distance are also imporr;int. For example. for the same set of initial and a&ient conditions. ground level samplers located at di%rsnt geographical locations {e.g. mountain and sal!e:;: Gil record dif- ferent chemical compsirion cc the collected precipitation. For initial!! acidic rains with mostly small droplets. the sampler at the higher elevation would record the higher S(W) concentrations.

However. in I-M and Adamowicz’s model under these conditions saturation is reached after approxi- mately 2OOm, whereas in the non-isothermal effects model saturation requires a 1800 m fall. For pH, = 10 the predicted ground level concentrations differ by nearly a factor of ten. However, for a 0.01 cm droplet, under the same conditions. both models predict ground level concentrations to be + 100~ moles / - ’ (which is the saturation value). In both models satu- ration is reached after a short fall distance.

In addition, small droplets have significant evap- oration during descent and their volume contribution to a bulk sample can be quite diifsrem from that based on the volume at cloud base. Veiy small drops. although they absorb quickly, a;: evaporate com- pletely and never contribute to ground level samples.

These results indicate that for high initial pH values

Simulations also indicate thar drops respond quickty to changes in ambient vaiz-s. Thus droplets passing through elevated plumes Lxn absorb sig- nificant amounts of SO, in-plume and then desorb SO2 blow-plume. This can lead to a xdest increa’se in pH at ground level, but more importantly. to a

Table 1. Comparison of ground tevel bulk [S(W)] and droplet temperature for a Qi cm dro+t for isothermal versus adiabatic atmospheres

Ambient temperature profile Ground level [S(W)]

(I( moles I-‘) Ground level drop&

temperatures (‘C!

Isothermal pH,=4 pH, = 10 3YZ 4.3 6.6 -2

1O’C 3.3 6.5 8.0 1S”C 2.7 6.2 12.5 70% 2.0 6.0 16.5 WC 1.5 5.7 - ‘2

Adiabatic (given by Equation 31) 2.5 6.3 * 20

Non-isothermal effects on SO, absorption b) water droplets-II 159

redistribution ofambient SO,. w-ith SO, being in effect transported from higher to lower elevations.

The model is currently being used to calculate

ground level bulk St IV) concentrations and pH based

on given cloud-base raindrop distributions. Finally. it should be emphasized that these obser-

vations are based on the simulations described and

that in a real rain situation the background quantities

(temperature protile. RH. gaseous SO: profiles. etc.)

will change with time. thus complicating the system. The results presented in this paper indicate that non- isothermal effects can be significant, but more u-ork is needed in order to assess their importance in real rain

events.

REFERESCES

Carmichael G. R. and Peters L. K. ( 1979) Some aspects of SO, absorption by water-generalized treatment. Atmos. Ewir. 13. 1505-1513.

Dana If. T Hales J. \f _ Siin U G. N. and Woii Xl. .A. t 19~IJ Satural pxcipitarion *xas.hou~ o!‘ sulfur compounds from plumrs. En~ironmsntal Proixtion hgencv r2port ?;a. R j- 7 ;-‘-: -

Hales J. Xi Throp J 11. and LL’olf !vl. .A. I 1971j Fteld investigation of suiiur dioxide washout and rainout from ths plum2 of a larg: coal-tired po*er pidn: 0) natural prectpitartw. En,wn.menta ! ?xtection .Agencb An Poi- lution Crntral O&2. Contras: 50. 2?-69-150: ’

Hill F. B. and .Adamorta R. F. 119771 .A mod21 for rain composition and it: Hashout pi sulfur diouidr. .4trna.~. Etnir. I I. 917-9’1.

Johnson P. B. ( 1931 T'nr role 01 ;salesence nuclst in aarm rain tmt:atton. P5.D. dissertation. The L’niversity of Chicaga.

Rsda Xl. and Carmicnacl G. R. rlYS11 Non-isothsrmal2tT2cts on SO; absorption 51. wattr droplets. .-t:moi. Ewir. 15.

145-l 50. Reid R. C.. Prausnitz J. \I. and Sherwood T. K. (1977) ntr

Proprri:<i 0/ Gaws 3id LiyuiA. pp. 566595. &t&raw- Hill. S2yr- York,

Scott H’. D. and Hobbs P. V. (196-i The formation of sulfate in watzr drops. J. .xn:. Sci. 24. 5+57.

Skelland A. H. P. (13-41 Dijkiocal Mass Tran.y/>r, p. 275, John N’iley. New 1.ork.