Absorption with instantaneous reaction in a droplet with sparingly soluble fines

Absorption with Instantaneous Reaction in aDroplet with Sparingly Soluble Fines

Fabrizio ScalaIstituto di Ricerche sulla Combustione, C.N.R., 80125 Napoli, Italy

Michele D’AscenzoDipt. di Ingegneria Chimica, Universita degli Studi di Napoli Federico II, 80125 Napoli, Italy`

The problem of gas absorption followed by an instantaneous irre®ersible chemicalreaction in a rigid droplet containing sparingly soluble fine reactant particles is dis-cussed. A model was de®eloped for the general case when the reactant particle dissolu-tion near the gas � liquid interface is nonnegligible. Solids dissolution is treated takinginto account the a®erage spacing between the suspended particles. A simple analyticalexpression for the rate of gas absorption is deri®ed under the assumption of slow particledissolution, and conditions are gi®en for applicability of the model. A numerical exam-

( )ple, rele®ant for the spray-dry desulfurization process, for the system SO -Ca OH in2 2water is discussed.

Introduction

The absorption of a gas into a slurry, followed by a chemi-cal reaction between the dissolved gas and a solute dissolvingfrom the sparingly soluble fine particles is of great practicalrelevance in the chemical industry. One example is found ingas purification processes, where the suspended solid parti-cles provide a continuous supply of reactant in order to en-hance the gas absorption rate. Particularly important is thecase when the chemical reaction between the dissolved reac-tants can be considered instantaneous and irreversible. Theabsorption of sulfur dioxide into a lime slurry is an exampleof a widely used application of this case. A typical way ofbringing the gaseous stream into contact with the slurry is todisperse the liquid phase into droplets in order to increasethe gas�liquid interface area, as in spray-dry desulfurization.

Ž .Ramachandran and Sharma 1969 were the first to theo-retically discuss the problem of gas absorption with chemicalreaction and with sparingly soluble fine particles, and theyproposed two film models, depending on whether solid disso-lution in the liquid film was important or not. Analytical solu-tions were derived and results showed that the gas absorptionrate could be considerably enhanced by the presence of solids.

Ž .Uchida et al. 1975 extended the analysis to account for theincrease of solid dissolution in the liquid film due to the

Correspondence concerning this article should be addressed to F. Scala.

Ž .chemical reaction. Uchida et al. 1977 considered the casewhen the concentration of reactant dissolving from the parti-cles in the bulk liquid phase is lower than the saturation solu-

Ž .bility slow dissolution rate compared to gas absorption rate .Ž .Uchida and Wen 1977 classified the problem of gas absorp-

tion into a slurry into six categories according to the reactantconcentration profiles that are formed in the liquid phase.For each case a model was given based on film theory and ananalytical solution provided. Based on their own experimen-

Ž .tal results, Sada et al. 1984 further modified the theory inorder to account for the presence of an inert region without

Ž .particles near the gas�liquid interface. Yagi and Hikita 1987criticized the previous models, pointing out that a parameterrepresenting the average interparticle spacing is necessary tocharacterize the system and that it is inappropriate to use amass-transfer coefficient for particle dissolution. Recently

Ž .Mehra 1996 examined the process of particle dissolutionnear the gas�liquid interface and pointed out that the changein particle size should be accounted for when the particlesare small enough for significant dissolution. He also criticized

Ž .the approach of Sada et al. 1984 , indicating that there is nophysical reason why a particle-free zone should exist near thegas�liquid interface. In his article, he proposed a penetrationmodel based on a transient population balance near thegas�liquid interface that was solved numerically for somerepresentative cases.

August 2002 Vol. 48, No. 8AIChE Journal 1719

All the references reported so far refer to planar geometry.Ž .Partridge et al. 1990 presented a mechanistic model of sul-

fur dioxide absorption into a lime slurry in a spray dryer,Ž .where the solution of Ramachandran and Sharma 1969 for

the case of negligible solid dissolution in the liquid film wassimply combined with an existing mathematical description ofthe spray-dry operation. To simulate gas absorption into aslurry droplet, however, a more detailed model is needed.Few extensions of the problem to spherical geometry have

Ž .been reported in the literature. Newton et al. 1990 pre-sented a model of sulfur dioxide absorption into a lime slurrydroplet in a spray dryer, where solids circulation within thedroplet was assumed to be negligible. Mass-transfer pro-cesses within the droplet and around the suspended particleswere examined in detail and a complex trial-and-error proce-dure was used to solve the model equations. Recently,

Ž .Mungistein et al. 2001 developed a model for gas absorp-tion into a large slurry droplet, that took internal circulationinside the droplet into consideration. The model, relevant tospray towers, was solved numerically, and results indicatedthat for large droplets internal circulation enhances the masstransfer with respect to a stagnant droplet.

In this work the problem of gas absorption followed by aninstantaneous irreversible chemical reaction into a rigiddroplet containing sparingly soluble fine reactant particles isconsidered, assuming slow reactant particle dissolution. Thisassumption implies that particle shrinkage is negligible, sothat a constant particle size can be assumed. A model is de-veloped for the general case when the reactant particle disso-lution near the gas�liquid interface is nonnegligible. Solidsdissolution is treated in a rigorous way, and the average spac-ing between the suspended particles is taken into account. Asimple analytical expression is provided for the gas absorp-tion rate. A numerical example, relevant to the spray-dry

Ž .desulfurization process, for the SO -Ca OH system in wa-2 2ter is discussed.

TheoryThe problem can be represented by the following scheme

A g ™ A aqŽ . Ž .

B s ™B aqŽ . Ž .

A aq qB aq ™products 1Ž . Ž . Ž .

The theoretical treatment is based on the following simplify-ing assumptions:

1. Both gaseous and solid reactants have low solubility inthe liquid;

2. Reaction between the two species is instantaneous andirreversible;

3. The influence of product species on the process is negli-gible;

4. Solid particle dissolution is slow, so particle shrinkagecan be neglected;

5. Solid particles are spherical, uniform in size, uniformlydispersed in the liquid phase, and do not agglomerate;

6. There is no surface kinetic resistance to particle dissolu-tion, which is entirely controlled by mass transfer;

Ž .7. Solid particles are stationary nondiffusing and the liq-uid around the particles can be considered stagnant, as dis-

Ž .cussed by Mehra 1996 ;8. The droplet is spherical and rigid, and water and sor-

bent particles do not circulate within the droplet. This as-sumption is based on the consideration that typical dropletsizes encountered in the spray-dry process are of the order of10�100 �m. It has been reported that no internal circulation

Žoccurs for this small size Newton et al., 1990; Hill and Zank,.2000 .

9. Heats of reaction and dissolution are small and can beneglected, and the system is isothermal.

The system is shown in Figure 1. A sharp ‘‘macroscopic’’spherical reaction front concentric to the gas�liquid interfaceoccurs at the steady state at a distance � from the dropletcenter, where reactant concentration falls to zero. This reac-

Ž .tion front divides the droplet of radius R into two zones.DŽIn the zone between the interface and the reaction front �

.� R� R , only reactant A exists in the liquid phase. DueDto the presence of the dissolving solid particles, a small mi-croscopic spherical reaction front occurs around each parti-cle in this zone, as shown in Figure 2a. Beyond the macro-

Ž .scopic reaction front 0� R� � , only reactant B exists inthe liquid phase. Around each dissolving particle in this zonea gradient of reactant B occurs, as shown in Figure 2b.

If r is the solid particles radius and � is the solids vol-P Pume fraction in the slurry, the number of particles per unitvolume of slurry is

3�PNs 2Ž .34� rP

It is thought that the unit slurry volume can be divided intoŽ .N nonoverlapping cells assumed spherical , with a solid par-

ticle at their center. Each particle has a volume of slurry inits cell equal to 1rN. The radius of each cell is

rPr s 3Ž .C 3 �' P

Figure 1. Gas absorption with instantaneous reaction ina rigid slurry droplet.

August 2002 Vol. 48, No. 8 AIChE Journal1720

Figure 2. Reactants concentration profiles around onefine particle.Ž .a Particles between gas� liquid interface and the macro-

Ž .scopic reaction front. b Particles beyond the macroscopicreaction front.

Let us now consider the particles present in the zone be-tween the gas�liquid interface and the macroscopic reaction

Ž . Ž .front �� R� R Figure 2a . The concentration of B atDthe surface of the particles is at the equilibrium with the solidŽ .saturation solubility , namely B . On the other hand, at theSexternal cell surface, the concentration of reactant A is as inthe liquid phase, namely A . Somewhere between the parti-L

Ž .cle surface and the cell surface r � r � r , a microscopicP Creaction front occurs, at a distance r from the particle cen-�

ter, where reactant concentration falls to zero. This reactionfront divides the spherical cell into two zones: a zone near

Žthe particle surface where only reactant B is present r � rP.� r , and a zone near the external cell surface, where only�

Ž .reactant A is present r � r � r . The material balances in� Cthe two zones give

d dB rs r ´BsBP s2r s0 4Ž .ž / ½ rs r ´Bs0dr dr �

d dA rs r ´ As0�2r s0 5Ž .ž / ½ rs r ´ As Adr dr C L

The solutions of the two equations are, respectively

r rry1�BsB 6Ž .S ž /r rr y1� P

1y r rr�As A 7Ž .Lž /1y r rr� C

At the reaction front the fluxes of the two reactants mustequal each other

dB dAD syD 8Ž .B Adr drr r� �

Equation 8 allows the calculation of r�

D BB S1q

D AA Lr s r 9Ž .� P r D BP B S

1q ž /r D AC A L

The flux of reactant A per unit external surface of the cellis

dA D A qD BA L B SyD s 10Ž .A dr r r rr y1r Ž .C C C P

The consumption of reactant A per unit volume of slurryis given by

D A qD BA L B S 2 2G s3 s 1y� � D A qD B rRŽ . Ž .A P A L B S D2r r rr y1Ž .C C P

11Ž .

Ž .where using Eq. 3

R 3 R 3�D D P�s s( 3(r 1y� r rr y1 rŽ .Ž . 1y� 1y �'Ž .C P C P P Ž .P P

12Ž .

Let us now shift attention to the particles suspended in theŽ . Žzone inside the macroscopic reaction front 0� R� � Fig-

.ure 2b . No reactant A is present in this zone. At the surfaceof the particles the concentration of reactant B is B , whileSat the external cell surface, it is equal to that in the liquidphase, namely B . The material balance on reactant B in theLcell gives

d dB rs r ´BsBP S2r s0 13Ž .ž / ½ rs r ´BsBdr dr C L

The solution is

r rry1 1y r rrC PBsB qB 14Ž .S Lž / ž /r rr y1 1y r rrC P P C

The flux of reactant B per unit external surface of the cellis

dB D B yBŽ .B S LyD sy 15Ž .B dr r r rr y1r Ž .C C C P

The generation of reactant B per unit volume of slurry isgiven by

D B yBŽ .B S L 2 2G sy3 sy 1y� � D B yB rRŽ . Ž .B P B S L D2r r rr y1Ž .C C P

16Ž .


At this point, the material balances in the droplet liquidphase before and after the macroscopic reaction front can be

Ž .carried out with reference to Figure 1 using Eqs. 11 and 16 ,are

D d dAA L2 21y� R sG s 1y� �Ž . Ž .P A P2 ž /dR dRR

xs�´ A s0L2� D A qD B rR 17Ž .Ž . �A L B S D ½ xsR ´ A s AD L

D d dBB L2 21y� R sG sy 1y� � DŽ . Ž .P B P B2 ž /dR dRR

xs0´dB rdRs0L2� B yB rR 18Ž .Ž .S L D ½ xs�´B s0L

Ž . Ž .where the term 1y� on the lefthand side LHS accountsPfor the reduced volume for diffusion due to the presence ofthe solid particles. Let us make the following change of vari-ables

D AA L� s1q 19Ž .A D BB S

� s B yB rB 20Ž .Ž .B S L S

If we define �sRrR , Eqs. 17 and 18 becomeD

d d� �s� ´� s1A � A2 2 2� s� � � 21Ž .�A ½ž / �s1´� s�d� d� A

d d� �s0´d� rd�s0B B2 2 2� s� � � 22Ž .B ½ž / �s� ´� s1d� d� � B

where � s�rR and� D

D A�A�� s1q 23Ž .

D BB S

Noting that the boundary condition at the reaction front

dB dAL LD syD 24Ž .B AdR dR� �

implies the identity

d� d�A Bs 25Ž .d� d��

the problem can be simplified by solving the single equation

d d� �s0´d�rd�s02 2 2� s� � � 26Ž .�½ž / �s1´� s�d� d�

The solution of Eq. 26 reads

sinh ��Ž .�� s� 27Ž .

sinh � �Ž .

where � s� for 1�� and � s� for 0�� 1. Lo-A Bcation of the macroscopic reaction front along the droplet

radius can be found by imposing � s1 when �s� in Eq.�

27. The gas absorption rate per unit interface area is

dA L� �J sy 1y� D s D A qD BŽ . Ž .A P A A B SdR RRD D

D BB S�s D A 1q 28Ž .A �ž /R D AD A

where

�s 1y� y1 29Ž .Ž .P ž /tanh �Ž .

Equation 28 is very easy to handle; for example, if Henry’sŽ � �.law is assumed to hold at the interface p sHA , Eq. 28

can be combined with the gas-phase diffusion resistance togive

D p rHqD BA G B S�J s 30Ž .A R rqD r k HŽ .D A G

where p is the bulk-gas partial pressure of reactant A, andGk is the gas-side mass-transfer coefficient. The bulk gas isGsupposed to be perfectly mixed.

The treatment so far has been based on the assumptionthat a macroscopic reaction front occurs inside the droplet,somewhere between the droplet center and the gas�liquid

Ž .interface Figure 1 . Should the solids dissolution be very slowcompared to the gas absorption, the macroscopic reactionfront would be able to reach the droplet center. This wouldhappen if

sinh �Ž .�� G 31Ž .

�

In this case no macroscopic reaction front occurs at the steadystate, and only reactant A exists in the liquid phase in thewhole droplet. A microscopic reaction front is found around

Ž .each dissolving solid particle as shown in Figure 2a . AlsoEqs. 26�30 are able to handle this case, the only difference

Ž .being that in the solution Eq. 27 � s� �1 always wouldAbe along the droplet radius. Combining Eq. 31 with Eq. 28and considering the gas-phase diffusion resistance, the condi-tion for this case occurring results

H sinh � HŽ .p GD B q y 32Ž .G B S ž /D k R � DA G D A

It is interesting to study the asymptotic behaviors: if H™,or if the following condition is satisfied

p FD B rk R 33Ž .G B S G D

the surface concentration of reactant A goes to zero, the re-Ž .action front shifts to the gas�liquid interface �sR andD

the absorption rate is entirely controlled by gas-phase resis-Ž .tance reactant B is assumed to be nonvolatile , that is

J�sk p 34Ž .A G G

This also would be the case when r ™0, but in that case thePinitial assumption that solid-particle dissolution is slow would


not be valid and particle shrinkage could not be neglected.Ž .On the other hand, for a very dilute suspension � ™0 noP

steady-state solution can be found. The physical reason forthis is that solid particles are unable to provide a continuoussupply of reactant B to the liquid phase.

Conditions for applicability of the modelWe now discuss how limited the model is as regards to the

size of the suspended reactant particles. The condition is de-termined by the initial assumption that particle dissolution isslow, so that particle shrinkage can be neglected. One way toestimate the time interval in which this condition holds is asfollows. Using Eq. 10 to calculate the maximum rate of disso-lution of one particle, we get

�dA D A qD BA B S2G sy4� r D s4� 35Ž .P , max C A dr 1rr y1rrr Ž .C ,max P C

The particle volume change per unit time is given by

�V M M D A�qD BP A B Ss G s 4� 36Ž .p , max� t � � 1rr y1rrŽ .P P P C

where M is the molecular weight, and � is the particle den-Psity. Making the fractional volume change lower than a cer-

Ž .tain value: �V rV � . This implies that using Eq. 3P P

� r 3 1rr y1rrŽ .P P P C� t� �ž /3M D A qD BA B S

32r 1y �'� Ž .P PPs 37Ž .�ž /3M D A qD BA B S

Equation 37, thus, sets the time interval for the applicabilityof the model.

One last consideration should be added. From Yagi andŽ .Hikita’s 1987 discussion, one could point out that the con-

centration profiles around the dissolving solid particles andthose in the droplet liquid phase overlap, which causes a dis-tortion in the profiles. While for very low solids volume frac-tions, the solid particles have a very low effect on the pro-cess, Eq. 9 indicates that for large solids volume fractions themicroscopic reaction front nearly coincides with the particlesurface. This implies that the mathematical development re-ported in this work could be used to treat the whole range ofsolids volume fractions of interest.

Numerical ExampleAs a representative case, we chose the absorption of SO2

Ž . Ž . Žreactant A into an aqueous slurry droplet of Ca OH re-2.actant B at ambient conditions. This example is relevant to

spray-dry desulfurization processes. The ionic reaction in theliquid phase between the two dissolved reactants can be con-

Ž .sidered irreversible and instantaneous Babu et al., 1984 . Thephysical-parameter values for this system have been esti-

mated as follows

D s1.8�10y9 m2rsA

D s1.6�10y9 m2rsB

B s2.0�10y2 kmolrm3S

Hs7.3�10y1 atm�m3rkmol

Ms74 kgrkmol

� s2.2�103 kgrm3P

In order to evaluate the gas-side mass-transfer coefficient,the droplet is assumed to flow at its terminal velocity. Thiswould set the most conservative conditions. For 10�100-�m-size droplets, terminal velocities are always more than oneorder of magnitude lower than typical flue-gas velocities, so itis reasonable to assume that the droplet travels at the same

Ž .velocity as the flue gas no slip velocity . As a consequence, itŽ .is assumed that the droplet Sherwood number Sh is equal

to the limiting theoretical value of 2 relative to the stagnantboundary-layer condition. The gas-side mass-transfer coeffi-cient is then given by

D Sh DG Gk s s 38Ž .G 2 R �T R �TD D

where � is the gas-law constant, and T is the system temper-ature. The sulfur dioxide diffusion coefficient in the gas phaseis estimated to be D s1.4�10y5 m2rs.G

The first step is to establish the conditions for the applica-bility of the model. The typical range of particle sizes in theslurry used in practice is 1 �m� r �10 �m. According toP

ŽEq. 37 for this particle-size range a value of s0.1 was.used , the model would be applicable approximately for � t�

0.01y0.1 s. At this point, it is interesting to examine the spe-Ž .cific features of the system SO -Ca OH slurry. As will be2 2

shown later, for all the operating conditions of practical in-terest, the macroscopic reaction front is very close to thedroplet surface, as the solids dissolution rate is very fast withrespect to the gas absorption rate. As a consequence, the solidparticles near the gas�liquid interface are bound to be dis-solved much more rapidly than the others, so that after arelatively short time a particle-free zone would be creatednear the droplet surface. On the other hand, in a typicalspray-dry operation, water evaporates from the droplets,whose surface recedes as the droplet decreases in size. Thisprocess tends to compensate particle depletion near thedroplet surface, so the model developed here is likely to beapplicable to the whole droplet lifetime. Of course, for a de-tailed simulation of the process, the droplet-size decrease andthe particle-number depletion, as well as the solids volumefraction, increase with time should be taken into account bycombining the absorption model with a detailed spray-dry re-actor model. In any event, even simple considerations usingthe equations developed so far lead to quite interesting re-sults, as is shown in the following.

Figures 3 and 4 show the condition in Eq. 33 for the parti-cle- and droplet-size ranges of interest at three different solidsvolume fraction. The area below the curves represents therange of sulfur dioxide partial pressures in the gas phase forwhich the gas absorption is completely controlled by gas-side


Figure 3. Condition in Eq. 33 for the particle-size rangeof interest at three different slurry solids vol-ume fractions.

diffusion resistance, that is, Eq. 34 holds. It is clearly seenthat for most flue-gas purification processes, both the gas-and liquid-side resistances are relevant. In particular, for lowsulfur dioxide partial-pressure, gas-side resistance controls theabsorption process, while for high partial pressures, liquid-side resistance controls the process. Analysis of the order ofmagnitude of the two terms in the denominator in the RHSof Eq. 30 shows that for most practical operating conditions,liquid- and gas-side resistances have the same magnitude.

Ž .Decreasing the size of the suspended particles Figure 3 andŽincreasing the solids volume fraction or the droplet size Fig-

.ure 4 enhances the relevance of the gas-side resistance tothe absorption process.

On the other hand, the condition in Eq. 32 would be satis-fied only for sulfur dioxide partial pressures in a gas phasewell above one atmosphere for all operating conditions ofinterest. So in practical conditions the macroscopic reactionfront never reaches the droplet center, but always stays closeto the gas�liquid interface.

Figure 4. Condition in Eq. 33 for the droplet-size rangeof interest at three different slurry solids vol-ume fractions.

Figure 5. Reactant concentration profiles in the droplet( )a for three different particle sizes at R =50D

( )�m; b for three different droplet sizes at rP=5 �m: � =0.1; p =5,000 ppm.P G

Figures 5 and 6 report the reactant concentration profilesŽ .in the droplet for three particle sizes Figure 5a , for three

Ž .droplet sizes Figure 5b , for three solids volume fractionsŽ .Figure 6a , and for three gas sulfur dioxide partial pressuresŽ .Figure 6b . In the figures the reactant concentrations havebeen normalized, respectively, with the calcium hydroxidesaturation solubility and with the sulfur dioxide concentra-tion that would be in equilibrium with the actual gas partial

Ž .pressure p rH . Sulfur dioxide partial pressure values areGall above the curves reported in Figures 3 and 4. It can beclearly seen that, as a consequence of rapid particle dissolu-tion, the reaction front is always close to the droplet surface.The reaction front approaches the gas�liquid interface as theparticle size and the SO concentration decrease or as the2droplet size and solids volume fraction increase.

Figure 7 depicts the gas absorption rate per unit interfacearea as a function of the SO partial pressure for three dif-2ferent particle sizes. For comparison, the maximum gas ab-

Figure 6. Reactant concentration profiles in the droplet( )a for three different solids volume fractions

( )at p =5,000 ppm; b for three different SOG 2partial pressures at � =0.1: r =5 �m; R =p P D50 �m.


Figure 7. Sulfur dioxide absorption rate per unit inter-face area as a function of SO bulk gas partial2pressure for three different particle sizes: RD=50 �m; � =0.1.P

Ž .sorption rate that is, complete gas-side control; Eq. 34 isalso shown. Analysis of the figure shows that, while at lowSO partial pressures, the absorption rate is the maximum2possible, at relatively high partial pressures, the curves tendtoward lower values. In particular, the larger the particle size,the lower the gas absorption rate. Figure 8 shows the influ-ence of the droplet size and the solids volume fraction on thegas absorption rate per unit interface area, for a SO gas2partial pressure of p s5,000 ppm. Both these variables ex-Gert a limited influence, as the gas absorption rate is slightlylarger when the droplet size and solids volume fraction areincreasing.

On the whole, the model results suggest that at low SO2partial pressures the absorption rate is controlled by gas-sideresistance. In these conditions, the only way to enhance theabsorption rate is to increase the mass-transfer coefficient,for example, by reducing the droplet size. On the other hand,at high SO partial pressures liquid-side resistance controls2

Figure 8. Sulfur dioxide absorption rate per unit inter-face area as a function of droplet size for threedifferent solids volume fractions: r =5 �m;Pp =5,000 ppm.G

the absorption rate. In these conditions, the absorption ratecan be enhanced by decreasing the suspended particle sizeand to a lesser extent by increasing the droplet size and solidsvolume fraction. Solids dissolution is always very rapid andwould give an appreciable resistance to the absorption rateonly at SO partial pressures well above those typically en-2countered in flue-gas purification processes.

When analyzing the reported results, however, it must beborne in mind that optimization of the operating variables ina practical operation is much more complex and must takeinto account a number of other factors, such as droplet tra-jectory, residence time in the reactor, exposed interface area,heat transfer, evaporation, spray-nozzle characteristics, possi-ble solids recycle, and lime slaking.

ConclusionsThe problem of gas absorption followed by an instanta-

neous irreversible chemical reaction in a rigid droplet con-taining sparingly soluble fine reactant particles has been con-sidered. This problem has considerable industrial impor-tance, for example, in gas purification processes. A modelhas been developed for the general case when reactant parti-cle dissolution is nonnegligible next to the gas�liquid inter-face. Solids dissolution is treated in a rigorous way of ac-counting for the average spacing between the particles in theslurry. A simple analytical expression for the rate of gas ab-sorption is derived, assuming slow particle dissolution. Thetime interval for which the model is applicable is determined.

A numerical example for SO absorption into an aqueous2Ž .slurry droplet of Ca OH at ambient conditions is presented2

and discussed. This example is relevant for the spray-drydesulfurization process. Results show that for most practicaloperating conditions, the gas absorption rate is controlled bya combination of gas- and liquid-side diffusion resistances.Solids dissolution is always very rapid and exerts negligibleresistance to the absorption process. The gas absorption ratecan be enhanced by using a smaller particle size, and to alesser extent by increasing the solids volume fraction and thedroplet size. On the other hand, at low SO partial pressures2the only way to enhance SO absorption is to increase the2gas-side mass-transfer coefficient.

AcknowledgmentsUseful discussion with Prof. A. Lancia is gratefully acknowledged.

NotationAs liquid concentration of reactant A, kmolrm3

A�s interface liquid concentration of reactant A, kmolrm3

asgas�liquid interface area per unit volume of liquid, my1

Bs liquid concentration of reactant B, kmolrm3

Dsdiffusion coefficient, m2rsŽ .Gsconsumption generation rate per unit volume of slurry,

kmolrm3sG smaximum dissolution rate of one particle, kmolrsP ,max

HsHenry’s constant, atm m3rkmolJ�s gas absorption rate per unit interface area, kmolrm2 sk sgas-side mass-transfer coefficient, kmolrm2atm sGMsmolecular weight, kgrkmolNsnumber of particles per unit volume of slurry, my3

pspartial pressure, atm�sgas-law constant, atm m3rkmol KRsradial coordinate in the droplet, mrsradial coordinate around suspended particles, m


ShsSherwood numberTssystem temperature, Kts time, s

Vs volume, m3

Greek letters�scoefficient defined in Eq. 12scoefficient defined in Eq. 29�s volume fraction

Ž .�sdimensional radial coordinate in the droplet sRrRD�smacroscopic reaction front distance from the droplet cen-

ter, m�sdensity, kgrm3

�s variable defined in Eqs. 19 and 20� �s interface value of � defined in Eq. 23

scoefficient in Eq. 37

SubscriptsAsreactant ABsreactant BCscellDsdropletGsgas phaseLs liquid phasePssolid particleSssaturation�sreaction front

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Manuscript recei®ed Sept. 7, 2001, and re®ision recei®ed Jan. 28, 2002.


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Absorption with instantaneous reaction in a droplet with sparingly soluble fines