PII: 0009-2509(95)00259-6 - DSpace at IIT Bombay: Homedspace.library.iitb.ac.in/jspui/bitstream/10054/886/1/...absorption of sulfur dioxide into aqueous slurries of magnesium hydroxide

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462 A. MEHRA

ISOLID (B)

Fig. 1. Schematic representation of mass transfer with in- stantaneous reaction near the gas-liquid interface (penetra-

tion element) in a slurry of “fine” particles.

dissolution itself could be enhanced in a manner sim- ilar to that for gas absorption accompanied by an instantaneous reaction. Sada and co-workers have subsequently reported a large number of studies; these include experimental investigations, mostly on the absorption of sulfur dioxide into suspensions of mag- nesium hydroxide (Sada et al., 1977b, 1979,1980), and theoretical extensions for the cases when the bulk aqueous phase is nor saturated with the dissolved solid material [finite slurries; Sada et al. (1979)] as well as when the reaction between the dissolved gas and solid is fast but not instantaneous [numerical solutions; Sada et al. (1977a)]. Some experimental data have also been reported by Uchida et al. (1978) on the absorption of lean sulfur dioxide into fine limestone slurries. All these studies have used the film model of mass transfer with chemical reaction to analyse and interpret the experimental data. A penetration theory analog of the model given by Ramachandran and Sharma (1969) has been proposed by Uchida et al. (1981) where they showed that for equal diffusivities of A and B, an analytical solution could be found for the problem of a moving reaction front in the presence of fine, sparingly soluble, solid reactant particles (using the concept of negative con- centrations). The prediction of rate values from this model have been demonstrated to be within a few percent of the film model computations. Yagi and Hikita (1987) have critiqued the work of Sada and co-workers as well as that of Uchida and co-workers by indicating that the claimed agreement between theoretical estimates of rates and the experimental values in some of their studies (Sada et al., 1977a; Uchida et al., 1978) is based on the use of unreason- able values of system parameters. Furthermore, Yagi and Hikita (1987) have proposed that an additional

parameter representing the average interparticle spac- ing is necessary to characterize the system and that the notion of a mass transfer coefficient for the par- ticle dissolution process is inappropriate. Sada et al. (1981, 1983) have reported some more work on the absorption of sulfur dioxide into aqueous slurries of magnesium hydroxide and limestone so that there are two reaction planes produced in the “film” zone - one due to the reaction between SO2 and OH-/SO:- and the other for the reaction of HSO; with CO:-. In these studies they proposed that there are “no particles suspending” in the zone enclosed by the first reaction plane and the gas-liquid interface since this zone thickness is of the order of the particle size itself. In another study (Sada et al., 1984), this concept of a thin, particle-free zone was extended to the interpretation of rate data for the absorption of carbon dioxide-sulfur dioxide into calcium hydroxide slurries, since, otherwise, the theory was found to yield rate values larger than the experimentally observed values.

It is indeed surprising that despite the large number of studies in the area of gas absorption into reactive slurries, there are none which focus on or account for the change in the size(s) of the dissolving particles and the concomitant efict of this phenomenon on the speci- fic rate itself. This point has been noted by Beenackers and van Swaaij (1993) in a recent review. These authors have cited their previous publication (Beenackers and van Swaaij, 1986) in which they have indicated that the situation of disappearing particles may actually occur in practical systems. The only other paper which makes a critical reference to the assumption of a fixed, unchanging particle size is that of Yagi and Hikita (1987) where the authors have provided an expression for the upper bound for the enhancement factor in the specific rate by assuming that all the particles are present in “completely dis- solved form” so that the extra reactant contained in the solid phase can be treated as being “immediately” available to the liquid phase. It will be shown later in the current work that even this upper limit is too high.

It is also clear from the above survey of the litera- ture that most of the studies deal with fast, in fact, instantaneous, reactions between the reactant derived from the solid phase and the dissolved solute gas. Since the reaction is fast, usually, only negligible amounts of the dissolved gas are ever able to reach the bulk slurry phase, implying thereby that all the reaction and particle dissolution occurs near the gas-liquid in- terface, exclusively. The above observations therefore imply that changes in particle size(s) and the conse- quence of this for the specific absorption rate have to be accounted for in the zone near the gas-liquid interface, i.e. the surface penetration element. As will be shown later, particles which are fine enough can dissolve to such an extent that these become significantly smaller (and may even dissolve completely!) in relation to their original size, during the time of contact of the surface element with the gas phase (at the gas-liquid

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Gas absorption in reactive slurries 463

interface). This dissolution may have drastic effects on Material and population balances in penetration the specific rate of gas absorption. elements

A related problem is the estimation of the particle size distribution as it evolves in the bulk slurry phase for a given type of slurry absorber. Consider, for instance, a batch slurry reactor. The evolution of the particle size distribution in the bulk slurry, on account of the particle dissolution, with batch time is impor- tant not only for determining the specific rate-batch time trajectory but also for calculating the overall extent of conversion of the solid reactant initially loaded into the slurry reactor with batch time. Track- ing this bulk particle size distribution involves particle accounting within the surface penetration elements since it is in this zone that dissolution may occur exclusively; the resulting distribution in the bulk phase then arises out of the convective mixing of these elements arriving from and departing to the gas-liquid interface.

Therefore, if the number concentration (number per unit volume of liquid phase) of solid reactant particles of size (radius) between R and R + dR, located at a distance x from the gas-liquid interface within a sur- face penetration element, at time t, is given by n(R, x, t) dR, the species balance for the diffusing, dis- solved gaseous solute, A, in the surface element (liquid) becomes

D a2cA Ax- = f$ + k2CACB (1)

while a similar balance for the liquid-phase reactant, B, which is derived from the dissolution of the solid particles, yields

This work was therefore carried out with two objec- tives in mind, namely (i) to assess the effect of particle dissolution and the consequent change in particle size(s) near the gas-liquid interface on the point speci- fic rate of absorption (“point”, here, signifies the rate at any given point on the rate-batch time curve), and (ii) to examine the nature of the evolution of the bulk-phase particle size distribution arising out of the phenomenon mentioned above in (i) and its effect on the specific rate as well as solid conversion trajectories with batch time for a typical batch slurry contactor. Some of the experimental data and obser- vations from the prior literature have also been reviewed in the light of the conclusions that emerge from this study.

DB a2cB acB - = at + zk2CACB a.9

s L. - ‘hDBShR(~B - c,) Rn(R, x, t) dR. (2)

0

The first and second terms in the above equations are the conventional diffusion and accumulation terms, respectively, while the third term in both the equa- tions represents a second-order, liquid-phase reaction where z is the stoichiometric factor for the reaction A + zB --t products. The last term in eq. (2) accounts for the amount of B that comes into the liquid phase by dissolution of the particles and may be derived as shown below.

The rate at which a single particle of size R supplies B to the liquid phase is given by

= k,, 4n R’(Cs, - C,) (3) THEORY

In order to account for the changes in particle size(s) of the slurry reactant with time, due to dissolu- tion near the gas-liquid interface, it is necessary to use an unsteady-state theory of mass transfer; the film model cannot be used since it is a steady- state description which allows only for the spatial variations in the concentration profiles of the relevant species. In this study, Higbie’s penetration theory has been used details of which may be found elsewhere (Danckwerts, 1970; Doraiswamy and Sharma, 1984).

where k,[ is the solid-liquid mass transfer coefficient. Therefore, the total volumetric rate at which B is transferred to the liquid phase from all the particles is given by

R,.. RLU =

I k,147cR2(Cb - C,) n(R, x, t) dR. (4)

0

Now, if the solid-liquid mass transfer coefficient for a single particle, in the above equation, is set to

Since, in general, the particles that constitute the slurry can have a size distribution it becomes impera- tive to use the population balance approach within the surface penetration element as well as the bulk slurry. The use of population balances within diffusion “films” is relatively novel and has been generally avoided on account of being highly computation in- tensive. Wachi and Jones (1991a, b) have used this approach for modelling precipitation near the gas-liquid interface and more recently Saraph and Mehra (1994) have reported the use of simplified population balances in their study on “near-the-inter- face” effects in microphase catalysis.

k,[ = Sh,Ds/R

the last term in eq. (2) results.

(5)

In writing the above equations, it has been assumed that the particles are spherical and that there are no “blockage” effects on the diffusion process due to the presence of solid particles. This is in consonance with the pseudo-homogeneous models, which have been postulated in the previous literature, for such “micro- heterogeneous” media (Bruining et al., 1986; Mehra, 1988). The overall change in the volume of the total slurry (i.e. solid + liquid) due to shrinkage/ dissolu- tion of the particles has been neglected as this change is small for low to moderately loaded slurries. In any

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464 A. MEHRA

case, this change in total volume cannot be accounted for by a pseudo-homogeneous approach and truly heterogeneous models for concentrated slurries need to be formulated for these situations as, for example, has been attempted by Yagi and Hikita (1987) or in the context of suspended catalysts, by Karve and Juvekar (1990). The particles have been taken to be stationary (non-diffusing) and hence a reasonable value of the radius-based Sherwood number, ShR, is 1.0, i.e. for an immobile particle suspended in a stag- nant fluid (this in fact represents the lowest possible rate of transfer). This also implies that the concentra- tion gradients of adjacent particles do not overlap which is consistent with a psuedo-homogeneous ap- proach that is rigorously valid at “low” hold-ups of the microdispersed phase; a more detailed discussion on the implications of pseudo-homogeneity may be found in Mehra (19881990). The Brownian diffusivity of particles of radius, say, 0.1 pm, as estimated from the Stokes-Einstein equation is about 1 x lo-r2 m”/s which is much less than the typical liquid-phase mo- lecular diffusion coefficients which are of the order of 1 x lo-’ m2/s. Hence, the assumption of stationary particles within a penetration element is entirely justi- fied and for this case the Sherwood number will have no dependence on the particle size R. Also, for typical particle sizes of l-10 pm, used to make fine slurries, eq. (5) (with ShR = 1) gives values of k,, which are of the order of 1 x 10-j to 1 x 10e4 m/s, which are in agreement with the typical values for this coefficient indicated by Doraiswamy and Sharma (1984). Inci- dentally, the values of ksl used by Ramachandran and Sharma (1969) in their numerical example are too low while the values used by Doraiswamy and Sharma (1984) in their solved example seem to be more realis- tic. Yagi and Hikita (1987) have criticized the use of a mass transfer coefficient for the solid dissolution effects and have suggested that, instead, a parameter representing the average spacing between particles be used. However, their critique is more applicable to concentrated slurries and in any case the model that these authors present predicts enhancement factor values which are as high as infinity! The modification of Uchida et al. (1975) for the enhancement of the dissolution process has been neglected in the current work in formulating eqs (1) and (2) as this effect is likely to have a significant impact only ‘when q = Cy (zC~) is << 1.

Now, eq. (3), after the left-hand side has been differ- entiated, gives the (negative) growth rate for a single particle and the resulting equation in rearranged form is

G(R,n,t)=$= - y+-CB) (R>O) P

(6)

the population balance being given by

an@, X, t) + a b(R. X9 t) G(R, X9 t)i at aP

= 0 (R > 0) (7)

the right-hand side of this equation being set to zero since there are no birth and death events within the particle size range 0 - R,,. (Death, i.e. exit of par- ticles from this size range occurs only at the boundary R = 0.)

The initial and boundary conditions on the primary set of equations, namely, eqs (l), (2) and (7) are given by the following.

ICs (t = 0, all x):

c* = C*b = 0 (8)

c, = cab = c; (9)

h(R, x, r) = hi,(R) (10)

BCs (t > 0):

x=0, C*=Cj, ac,=o ax (11)

x + co, C” = C,& = 0, Ce = CBb = cj, (12)

R = L,, n(R, x, t) = 0. (13)

The bulk concentration of A(CAa) is taken to be zero since the reaction is fast, whereas the bulk con- centration of B(Csb) is assumed to be that for a saturated solution of B [for fine particles this is a reasonable approximation; Sada et al. (1979)]. Reac- tant B is non-volatile as shown by the boundary condition setting the concentration gradient (hence, flux) at the gas-liquid interface (x = 0) to be zero. The condition on the particle size distribution function n(R, x, t) at R = R,,, implies that no particles are entering the particle size range from “above”, i.e. sizes greater than R,,,. (It may be noted that we have consistently shown the function arguments for n by writing it as n(R, x, t) . Rigorously speaking, the con- centrations of A and B should also be written as CA(x, t) and C,(x, t), respectively, but this has not been done following the usual literature conventions.)

A solution of the above equations, along with the relevant conditions, provides us with the main quanti- ty of interest, namely, the specific rate of absorption of A, which, from Higbie’s penetration theory, is given by

RA = f “RAi(t) dt s e 0

where, the instantaneous specific rate of absorption is

(15)

and t, = 4D,/(nki). The other variables of interest are the concentra-

tion profiles, CA and Cg, the particle size distribution function, n(R, x, t) and the moments thereof, such as the particle number concentration,

R,. N(x, t) = n(R, x, t) dR (16)

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Gas absorption in reactive slurries 465

the average particle size,

Rmca”(x, t) = St?-= Rn(R x, 4 dR N(x, 4 (17)

and, the local, volumetric ratio of solid to liquid,

Qx, t) = f I

&.. R3n(R, x, t) dR

0 (18)

which, for “low” values of 1, may also be taken to be the local, fractional volumetric hold-up of the solid reactant.

Bulk population balance Since the bulk concentrations of A and B have been

taken to be fixed [conditions (8) and (9)], in the formulation given above, the bulk material balances for these species are not required. In the case of the population balance, the bulk particle size distribution function n,(R) [rigorously, nb(R, 6)] needs to be com- puted as a function of batch time, 0 (which is likely to be much larger than the contact time represented by tJ. The appropriate population balance on a batch slurry reactor is

bn)(R, 0) = x, t,) - n*(R, @)I dx ” c

- & C(ndR, 0) Gb(R, 41 (19)

where the last term is on account of particle dissolu- tion in the bulk phase. Since, the bulk liquid phase has been taken to be saturated with E, the bulk (negative) growth rate, Gb(R, g), given by

Gb(R, 0) = ; = - ~~(c$-c,b) (R>O) * _

(20)

becomes zero for CBb = C$ and hence the last term in eq. (19) may be dropped. We thus have a situation of exclusive particle dissolution near the gas-liquid inter- face and none at all in the bulk. This is consistent with the case of fast reaction (no bulk reaction) and fine particles which keep the bulk phase saturated. The integral term in the bulk population balance, eq. (19X represents the exchange of material between the inter- face, composed of a mosaic of penetration elements, and the bulk. While the first term within the integral is the number (concentration) of particles in the differen- tial size range R to R + dR that are being carried awayfiom the interface by the departing penetration elements, the second term gives the equivalent number (concentration) of particles that are brought into the “near-interface” zone from rhe bulk. A more detailed discussion on this type of bulk balances may be found in Mehra et al. (1988). Equation (19) requires an initial condition which may be stated as

@ = 0, nb(R, f?) = n:(R) (21)

so that n:(R) describes the initial state of the particles, fed as a slurry, to the batch reactor at batch time, e = 0.

The bulk analogs of quantities defined in eqs (16), (17) and (18), and denoted by a subscript b, i.e. Nb(Q R mcnn,b(e) and /b(e) may be obtained by replacing n(R,x, t) in the above-mentioned equations by n,(R, @, which in turn is obtained from the solution of eq. (19) with condition (21).

Solution strategy A semi-numerical strategy has been developed to

solve the species balances for A [eq. (l)], B [eq. (2)] and the population balance [eq. (7)], along with the appropriate conditions, as described below.

Consider first the population balance given by eq. (7). Defining a new variable, o(R,x, t) = n(R, x, Q/R, this equation may be recast in the form

$+G(R,x,t)$=O (R>o) (22)

where G(R, x, t) is the (negative) growth rate, dR/dt, given by eq. (6). Now, from the method of character- istics, the left-hand side of eq. (22) may be taken to be the total derivative, do/dt, which set to zero implies that o(R, x, t) = constant = w(Ro, x, 0). Therefore,

n(R, = n(Ro, & 0) _ fib:01

R Ro t23J fR , oj

0

the last equality being obtained by the use of condi- tion (10). In order to eliminate the dummy variable, Ro, we may now integrate eq. (6) with the initial condition, R = R. at t = 0. This integration yields

Rij = RZ + 2ShRDsM, ’

PP I (Cf, - C,) dt’ (R > 0)

0

(24)

so that this result substituted into eq. (23) finally gives

n(R, x, 4

= n&/R’ + (2ShRD,N,/pp)j;(Cs, - CddflR JR” + (2Sh,&%4,/pp)J~(C~ - C,) dt’

(25)

provided, R. d R,,. (else n(R, x, t) = 0), and where, it should be recalled, that the dependence on x is through the concentration, CP Thus, the particle dis- tribution function, n(R, x, t), may be computed from the initial distribution alone, i.e. rib(R)) . The boundary condition (13) makes no contribution to the solution since n(R,&, x, t) is always zero.

If this final result, from the above eq. (25), is sub- stituted into eq. (2) i.e. the species balance for B, we have only the following equation:

D a2cs acB Bz ax = at + zk2CACB - 4nD,ShR(C; - C,)

(26)

s R-R2 n&h2 + (2&&&fw/P,)~&i - cd dt’l dR

0 ,/R2 + (2Sh,D,M&&,(Cs, - CL,) dt’

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466 A. MEHRA

Table 1. Data sets used in this study

Standard set Sada et al. (1984)’

Property Value Value Units

CZ 3.00x 1o-2 2.66 x lo-’ kmol/m” G 9.00 x lo-2 2.03 x lo-’ kmol/m3 DA 2.00 x 1o-9 2.00 x 1o-9 m’/s DB 2.00 x 1o-9 1.64 x 10-9 m2js kz 1 x 10s 1 x 10s m3/kmol s It:. 1.0 1.0 S

5.0 x 10-S 5.0x 1o-s m/s a 1 x lo2 m2/m3 hl 0.1 0.05n-aO.25 nd 1: 0.1 nd

ShR 1.0 Z nd LX 1.0 or 1 10.0 5.0 1 nd

MW 14 kg;:ol PP 1500 2GO kg/m3

Note: nd = non-dimensional; na = not applicable. t C02-Ca(OH)2 system.

and eq. (1) to be solved along with the initial condi- tions (B), (9) as well as boundary conditions (11) and (12).

The above-mentioned equations, in non-dimen- sionalform (see Appendix A), were solved numerically, simultaneously, since these are coupled [eqs (1) and (26)], by using an implicit, jinite difference discretiz- ation within an iterative procedure. This method involves solving eq. (1) for CA(x, t) and eq. (26) for CB(x, t) sequentially, using the last generated values of the other variable. For instance, when solv- ing eq. (1) for C,,, the last obtained solution for CB was used. This iterative process was carried out till convergence was achieved. The values of nb[ RZ + (2ShRDBM,/p,,) JA(C$ - CB) dt’] were ob- tained either by substitution into the expression for n,(R) [such as eqs (37) and (3811 or by a cubic spline interpolation on the discrete values of n*(R) stored in tabular form (this is done only when solving along batch time, 8, since only nb(R, 0) is known as an analytical expression; nb(R, 0) is known only at dis- crete values of R). All the integrations with respect to R were performed using Simpson’s rule; the integrals with respect to time, t, such as, for evaluating RA [eq. (14)] and that containing (C; - C,) as in eq. (26) were evaluated with the help of the Trap- ezoidal rule.

The bulk population balance [eq. (19)] was solved using the Euler’s method and the integrals in this equation, with respect to x, were obtained from Simpson’s rule. In this case, only nb at 0 = 0, i.e. n$! is available as an expression; the subsequent values of nb at 6 > 0 then become available only as a set of tabulated numbers at discrete values of R.

More details of the solution procedure may be found in Appendix A.

Table 1 lists the standard set of values for the input parameters (physico-chemical properties and oper- ating conditions) that were used for this study. The

proposed model was solved by varying one or two parameters in this set, while keeping the others fixed at the values shown.

RESULTS AND DEXXJSSION

In the first part of this study, the effects of changing particle size(s) near the gas-liquid interface, on the specific rate of absorption, have been considered. Here, the variation of the specific rate with batch time has not been examined and only the computation of the specific rate with respect to the conditions prevail- ing in a penetration element has been considered. For this purpose, a simple initial particle size distribution, nb(R), was chosen, namely, an impulse function

n,(R) = No&R - R,,,) (27)

which implies a monodisperse distribution, where all particles are of size R,., at a number concentration, No. This is thus the particle size distribution function when the penetration element arrives at the gas-liquid interface at time, t = 0. Substituting this into the last term of eq. (26) and integrating with respect to R, this dissolution term may now be written as

Rgd = 4nShR De(C; - C,)

*No R&x - (2ShRD&w/p,) J;(C;, - Cd dt’ . (281

Apart from the simplicity of this initial distribution function, the advantage of using eq. (28) is that numer- ical integration, in order to evaluate the integral in the last term of eq. (26) with respect to R, is avoided. It follows, then, from the definitions (16), (17) and (18) that for this situation,

N&t) = No, R,& Z (2Sh,D&,/p,)j;(CS - C,)dt’

(29)

N(x, t) = 0, R:,, < (2SW&Wp,) J!&; - ‘3 dt’ (30)

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Gas absorption in reactive slurries 467

x/(W.)‘”

Fig. 2. Variation of concentrations CA and Cs, average particle size R,,., and solid to liquid volumetric ratio I with distance from the gas-liquid interface x at t/tc = 0.1 and 1.0. Data used: standard set with R_ = 1.0 pm; Initial particle

size distribution: impulse function.

and

L,.(x, t) = JR:,, - (~SMM~,/P,) MC;, - CB) dt’

(31)

which is equal to the size of all the individual particles at that x and t, as well as

1(x, t) = y CR&,,

- (ZSh,D,M,/p,)I;(C$ - C,)dt’13”. (32)

The last two equations obviously hold only when N(x, t) # 0. The value of No may be determined by equating 1(x, 0), as evaluated from eq. (32), to the desired value of lo, after choosing the value of R,,.

The resulting, typical profiles of the concentrations of A and B, i.e. CA and Ce, respectively, the mean particle size, R,,,,, and the solid to liquid ratio, 1, as a function of the distance from the gas-liquid inter- face, x, at two different times, t, are shown in Fig. 2. While the profiles of CA and C, are as expected for the case of very fast/instantaneous reactions (A and B meet at a reaction plane, x = a), it may be observed that the particle size decreases rapidly as one moves towards the gas-liquid interface. In fact, very close to the interface there are no particles at all - all the particles in this zone have dissolved completely even in less than 0.1 s! At the end of the contact period lasting for time t = t, (= 1.0 s), the particle-free zone can be seen to have “penetrated” considerably into the ele- ment. This kind of a profile for the particle size is to be expected since the driving force, (C; - C,), is max- imum closest to the interface and decreases away from it as C, becomes significant (beyond the reaction plane at x = A).

Figure 3 shows R,,,, and I, vs x at the time of departure of the penetration element, i.e. at t = t,, for different initial sizes of the particles, R,,.. The vari- ation in these profiles with distance is steepest for the

- km _ _ _ ,

l:R,=l +rn 2:R,=5 pm

02- ; 3:R_=lO pm ___.I

x/m.) l/2

Fig. 3. Variation of average particle size R,,,, and solid to liquid volumetric ratio 1 with distance from the gas-liquid interface x at t = t, for different initial particle sizes R,,.. Data used: standard set; Initial particle size distribution:

impulse function.

smallest size of 1 pm-there is indeed a particle-free zone near the interface-but for the larger sizes (5 and 10 ,um) the particles can be seen to have only reduced in size and not dissolved completely. In fact, for the largest size of 10 w the maximum change in size (at x = 0) is of the order of 10%. For higher values of the contact time, tc, this change may be expected to be even more. Thus, for a given contact time, only those particles larger than a certain size, can be assumed not to undergo any significant change in size and exist in the “film” at a constant size. For the assumptions made in this study, the condition for a particle to undergo negligible change in size may be derived from eq. (31), wherein substituting t = t, and replacing R,,, by the initial size of any particle, R,,, we obtain

assuming the maximum driving force for dissolution by setting Ce = 0. Thus, for instance, for the standard data set listed in Table 1, particles with an initial radius, Ro>>4 pm (say, about 5 times this, i.e. 20 pm) will not undergo any significant change in size. Alter- natively, a particle of R,, = 1 m will simply vanish in about 0.06 s when subjected to the maximum driving force for dissolution. The points at which the profiles of R,,,, and 1 show a sudden change in slope (curves 2 and 3 of Fig. 3) correspond to the position of the reaction plane because in the zone that contains negli- gible amounts of reagent B, the driving force for dissolution is at its maximum and all particles in this zone dissolve to the same extent (having started from the same initial size)-hence the flat profile in this region close to the gas-liquid interface.

The behaviour of the instantaneous specific rate of absorption of A, RAi, defined by eq. (15), as a function of time, t, for “small” particles of 1 pm size is shown in Fig. 4. It is clear from this that the rates computed assuming a fixed, constant particle size are much higher that those obtained from the realistic scenario

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468 A. MEHRA

Fig. 4. Variation of instantaneous specific rate of absorption Fig. 5. Variation of instantaneous specific rate of absorption RAl with time t from constant and changing size models, and RAi with time t from constant and changing size models, and without particles. Also, variation of average particle size near without particles. Also, variation of average particle size near the gas-liquid interface R:..,, with t. Inset shows position of the gas-liquid interface R&,, with t. Inset shows position of reaction plane I with t for the three cases. Data used: stan- reaction plane I with t for the three cases. Data used: stan- dard set with R,, = 1 e; Initial particle size distribution: dard set with R,,, = 10 pm; Initial particle size distribution:

impulse function. impulse function.

where particles can change in size due to dissolution. The variation of the average particle size at the inter- face, R,..,,(O, t) , denoted by R&, is also depicted in this figure. The particles here dissolve completely within 0.06 s. The inset shows the position of the reaction front, 1, with respect to time, where A and B meet to react instantaneously. The effect of the solid reactant particles is to retard the rate of motion of this plane by continuously supplying B to the liquid phase. Therefore, this plane travels into the penetration ele- ment to the greatest extent when there are no particles present. If particles of a constant size are assumed then the extent of penetration is very small compared to the case when the particle size is allowed to change, since, in the former case, the particles continue to supply the reactant B inexhaustibly, even though these may have actually dissolved completely. Thus, the actual supply of B to the liquid phase is much less than what is assumed by models in the prior literature.

Figure 5 shows a similar set of plots for “large” particles of 10 pm. Here, the rates computed with either the constant or changing size assumptions leads to almost identical RAi vs t plots, since,the particles are large enough for their size to be affected only marginally in the given contact time. The maximum variation in the particle size, for those located at the gas-liquid interface, is about 10 %, as evidenced from the plot of RZ,,, vs t in this figure. A similar observa- tion holds for the reaction plane position, Iz, vs t shown in the inset to this figure.

The (surface averaged) specific rate of absorption of A, RA [eq. (14)], plotted as the enhancement factor, dA, vs the initial particle size, R,,, from the constant and changing particle size models is shown in Fig. 6. The enhancement factor is defined as

4A = R.,IRp

a.0 010

R- Cd

Fig. 6. Variation in enhancement factor c$” vs initial particle size R,,, from constant and changing size models at various stoichiometric ratios q. Inset shows the asymptotic and stoichiometric bounds on 4” vs 4. Data used: standard set;

Initial particle size distribution: impulse function.

where, R$., is the specific rate in the absence of the fine particles. In this figure, the concentration ratio, q = C&CA*), has been used as a parameter. For the case of instantaneous reactions, Ry is given by (Doraiswamy and Sharma, 1984)

/C$Cl = J(E+ &?). (35)

For the “small” particles, the difference between the corresponding enhancement factor values from the two models is very large. As the particle size increases, the predictions tend to become closer and finally coincide for “large” enough particles. An interesting observation to be noted from this figure is that while the enhancement factor curves for the constant par- ticle size cases are closely bunched together for all the

Page 9

Gas absorption in reactive slurries 469

I /

/

/ /

7.3 - / / 1 :ShR=O. 1 I

2 _ / 2:Sh.=l .O / / 3:Sh.=5.0 , ’

Fig. 7. Variation in enhancement factor 4A vs time of con- tact of penetration element t, from constant and changing size models for different Sherwood numbers ShR. Data used: standard set with R,,. = 1 pm; Initial particle size distribu-

tion: impulse function.

q values, these are clearly well separated for the changing particle size cases (in the smaller particle size range). This happens because for the greater q values (obtained by increasing the C; value) the rate of dissolution is faster and hence particles become smaller and/or disappear earlier resulting in the lar- gest q case to have the lowest enhancement value. Though the absolute value of RA increases with q the corresponding increase in Ry is relatively more [ac- cording to eq. (35), RA base increases linearly with q]. This causes a reduction in the 4A value. For the constant size runs, the increase in RA and RF, with q, is nearly the same - hence the bunching of these plots. The most significant observation from this fig- ure is that as the particle size decreases, the enhance- ments from the changing size model approach an asymptotic plateau. The variation of this asymptotic value with q is shown in the inset to Fig. 6. In other words, even smaller particles will not increase the rate of absorption, since these dissolve completely, any- way. This is in direct contrast to the predictions from the constant size models.

As has been stated earlier, for particles to be “small” enough for these to undergo significant change in size due to dissolution, what matters is not just their initial absolute size (and hence the dissolution rate) but also the total time available for dissolution, i.e. the contact time period, te. This is demonstrated in Fig. 7. For large rates of dissolution (as represented by larger values of the Sherwood number, Sh,J and/or for high values of the contact time, t,, the divergence between the enhancement factors predicted by the changing and constant particle size approaches, respectively, is substantial on account of the complete or partial (but significant) dissolution of the particles. Thus, it may be concluded that for higher values of t, (lower k,) even “large” particles may become “small” enough to dissolve substantially. Therefore, say, for the standard data set shown in Table 1, if tc is increased to a few seconds, even particles of 10 pm may be expected to

dissolve to an extent that will affect the rates signifi- cantly thereby lowering the enhancements vis-&is the constant size model predictions.

Interestingly, the enhancement factor values from the changing size model, in Fig. 7, for all the Sher- wood numbers, tend to approach an identical plateau at large contact times; the approach to this asymp- totic value is faster for higher ShR. Coupled with the discussion above, this implies that at this asymptotic extreme the rate (or alternatively, enhancement factor) becomes independent of the initial particle size as well as the Sherwood number and depends only on the initial solid loading, lW Also, if the modification of Uchida et al. (1975) were to be taken into account, by using an enhancement factor for the particle dissolution pro- cess, the overall benefit of this mechanism in further intensifying the specific absorption rate is likely to be moderated by quicker particle disappearance.

At this point it is also pertinent to look at the stoichiometric upper limit on the r$A values, as dis- cussed by Yagi and Hikita (1987), as this is especially relevant for “small” particles which tend to dissolve completely within the contact time, tc. If all the mater- ial contained in the solid phase is deemed to be avail- able “immediately” in dissolved form, a limiting value of the enhancement factor may be written as

These values of 4A are also plotted in the inset to Fig. 6. It may be seen that the values of e%,, predicted from the model of Ramachandran and Sharma (1969) can be made to exceed even this upper bound by choosing a particle size which is “small” enough. This is, however, stoichiometrically impossible. The 4” values computed from the proposed model are indeed smaller than the bounds given by eq. (36). The difference between the stoichiometric and asymptotic limits is on account of the fact that no matter how small the particles, there is always some point in the penetration element beyond which the driving force is insufficient to cause “immediate” dissolution, i.e. there will always be some B “bound and immobilized” in the form of particles. It should also be noted that rapid particle dissolution may cause small, local convective flows around a particle which may further enhance dissolution rates. We have not attempted to estimate these in our analysis in the present work, though these could be incorporated simply by increasing the value of ShR.

Figure 8 shows the effect of the initial (t = 0) solid loading, lo, on the enhancement factor, 4A, for differ- ent initial sizes of the solid reactant particles. While the difference in the enhancements from the two models, for the smaller particles is large, it becomes negligible for the largest particle size.

Uchida et al. (1978) have found that the absorption rate did not depend upon the particle size, R, for a fixed value of the solid-liquid interfacial area

Page 10

A. MEHRA

/ 15.0 / l:R,..=l.O +'m

/ 2:R,,=5.0 /on

125 / 3:R,,=lO.O pm

/

$ la0 /

I

Fig. 8. Variation in enhancement factor 4A vs initial solid loading of particles I, from constant and changing size models for different initial particle sizes R,,. Data used: standard set; Initial particle size distribution: impulse

function.

[A, = 3wt/@,R)] and deduced from this that kSi does not depend upon the size R. These workers used a constant value of the ratio of weight fraction of solid to particle size (wt/R) but varied both wt and R. An alternative explanantion is provided by the plots given in Fig. 8. The 44 values at lo = 0.1,0.05 and 0.01 read, respectively, from the curves for 10, 5 and 1 pm sizes give values between 2.1 and 2.4. This happens because for the large particle size the rates are low despite large loadings, whereas in the case of small particles, these dissolve away completely and the loadings are anyway low so that the rate may remain substantially the same. Therefore, it cannot be con- cluded from these experiments that kSl is a constant with respect to R. The above comparison is only to demonstrate the effect of dissolution and size change; the t&es of 4A mentioned above (read from Fig. 8) do not correspond to the system of Uchida et al. (1978).

The results discussed above have all been derived using the impulse function, i.e. an initial (t = 0), monodispersed particle size distribution. A logical corollary to the observations made above is that for a “wide” initial size distribution comprising “small” and “large” particles, the smaller ones may dissolve completely while the larger ones may remain substan- tially unchanged in size. This implies that the specific rate of absorption is likely to depend strongly on the width of the initial particle size distribution, for a fixed initial average size. In order to examine this hypothe- sis, the following initial particle size distribution func- tions are used.

Distribution I:

n,(R) = Ci exp C- G(R,.. - RI1 (R d LA. (37)

Distribution II:

rib(R)) = C1 exp (- Cz R) (R < R,,3. (38)

The values of the pre-exponential parameter Ci and the steepness parameter, C2 [a larger value of this implies a narrower distribution clustered around R = R,,, for eq. (37) and around R + 0 for eq. (3811 can be computed by fixing the maximum size, R,,,, and setting the initial solid loading expression as well as the average size expression, from eqs (18) [1(x, 0)] and (17) [R,,.Jx, 0)], respectively, equal to the de- sired values.

Table 2 shows the effect of the width of the initial particle distribution function on the enhancement fac- tor, 4A, for the two mean sizes, Rmcan,o = 1 and 10 pm, respectively, using the proposed theory. The enhance- ments (or alternatively, the specific absorption rate) may be seen to be substantially affected by the width; a broader distribution yields lower rates. Figure 9 shows the behaviour of the number concentration of particles, N* = N(0, t) and the solid to liquid ratio, I* = l(0, t), at the gas-liquid interface, as a function of time, t. Figure 10 shows the corresponding variations in the instantaneous specific rate, RAi, and the average particle size, R&., = R,,,&O, t). (These figures are for an initial, mean radius, R,,,cpn,o = 1 pm.) While the number and volume of particles decreases with time, the average size increases implying that the “fines” have dissolved away leaving behind larger particles which have not changed in size very significantly. For an initial distribution of greater width the average size increases by a larger amount, as can be noticed from Fig. 10 as well as Table 2. Therefore, the average size at any location x within the penetration element is likely to have a maxima with time, t. For the RL, curves shown in Fig. 10, this maxima is clearly visible for the two intermediate values of R,., = 3 and 5 pm. Initially, the smaller particles dissolve faster leaving the larger ones behind so that the average size goes up while later even these particles start reducing in size due to dissolution, thus decreasing the average size. For the impulse function there are no small and large particles; for the broadest distribution (curve 4 in Fig. 10) the reduction in size due to dissolution has not yet overtaken the increase due to dissolution of “fines”. From Table 2, it is interesting to observe that, for Rmean,,, = 10 pm, as R,,, is increased to make the distribution wider, the fraction of particles that dissolve completely near the gas-liquid interface in time t, increases (from 0 to 0.21) but the volume of the solid that has dissolved decreases (from 0.25 to 0.12). This indicates that, upon widening the distribu- tion, more “fines” are present in the system in order to keep the initial average fixed and these dissolve com- pletely. However, simultaneously, a larger volume of the solid, which is contained in the larger particles that are now present, now remains undissolved. This trend is reversed for the smaller size of R mean,0 = 1 pm. Now, all the particles for the case of narrower distributions disappear. As the distribution is widened, “coarser” particles which are present re- main undissolved.

It has been pointed out in some studies (Sada et al., 1981, 1984) that the specific rate of absorption

Page 11

Tabl

e 2.

E

ffect

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n. 0

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UI

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l. cl

max

ima

tltc

tltc

at

ALL

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ean,

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0.0

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2.16

D

istr

ibut

ion

I St

anda

rd d

ata

set

Impu

lse

fn

10.0

23

2 10

00

11.0

10

8 50

0 12

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19.5

10

0 16

.0

6.86

10

19

.5

599

1 20

.0

R m

un*o

= 1

.00

/ur&

gy-“

n’

= 21

.0

Impu

lse

fn

1.00

2,

322J

KQ

10

,000

1.

10

194,

500

1000

1.

60

71,4

90

100

1.93

239

2.20

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0.91

0.

75

2.32

21

8 1.

00

0.91

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75

216

2.11

0.

98

0.92

0.

77

1.55

1.

86

0.87

1.

02

0.84

1.

22

1.70

0.

80

1.10

0.

88

1.19

1.

68

0.79

1.

12

0.88

2387

23

20

1548

12

49

5.18

0.

00

5.18

0.

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5.14

5.

11

0.00

0.

00

0.00

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0.00

0.

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0.

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6 1

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00

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12

1.00

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06

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56

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Page 12

412 A. MEHRA

Fig. 9. Variation of number concentration of particles N* and solid to liquid volumetric ratio I+, near gas-liquid inter- face, vs time t for initial particle size distributions of different widths from changing size model. Data used: standard set;

R mem.0 = 1.0 lun.

II I2

0.0 I ,, , ,, , ,, 1, ,,,‘,, , , , , ,, 0.0 0.2

,I,1 ,o .o 0.4 0.1 01 IO

t/t

Fig. 10. Variation of instantaneous specific rate of absorp- tion RAI and average radius R&,, near the gas-liquid interface with time t for initial particle size distributions of different widths from changing size. model. Data used: stan-

dard set; Rmc.n.o = 1.0 pm.

obtained experimentally has been usually found to be lower than that computed from the models proposed by Ramachandran and Sharma (1969) or the equiva- lent unsteady one of Uchida et al. (1981). Sada et al.

(1984) have attempted to explain this by the use of an ad hoc hypothesis that a thin zone, of width equal to one particle diameter, near the gas-liquid interface has “no particles suspending” in it. This lowers the computed rate value since in the region closest to the interface there are no particles to provide the rate enhancing effect. In addition to the “fudge factor” approach implied by this hypothesis, there seems to be no physical reason as to why such a particle-free zone should exist per se. It is also clearly against the “spirit” of pseudo-homogeneous models, where the penetration element is viewed as a “pseudo-homo- geneous continuum”, to invoke a physical discontinu- ity at some arbitrary point. An alternative explanation, therefore, may be derived from the theory proposed in this study, in the light of the foregoing discussion; the process of dissolution (partial or complete) of “small” particles near the gas-liquid interface causes a lowering of the amount of solid present in this zone leaving behind larger particles, which in turn, is responsible for the reduced absorption rates vis-&vis the earlier model predictions. The results of calculations using the data of Sada et al. (1984), for the carbon dioxide-calcium hydroxide slurry system, are shown in Table 3. It can be seen that the rates obtained depend strongly upon the width of the initial particle size distribution for a fixed, initial, average size and solid loading. As the distribution is made wider, the enhancement factors decrease and approach the experimental values re- ported by Sada et al. (1984). A more precise compari- son than what is given in Table 3 is not possible since Sada et al. (1984) have not reported the size distribu- tion of the solids that they used. In fact, they have not even reported the mean particle size but merely sug- gested that it may be about 10 pm (diameter). For the computational results shown in Table 3, we have therefore used an initial radius of 5 pm. A slightly different initial, mean size or a different distribution may provide even lower calculated values for the enhancement factors. It is important to note that Sada et al. (1977a, 1984) have analysed their experimental data by using a lumped parameter, kSIA,, and back- calculating the value of this by fitting the data to the theory. The second study, mentioned above, attempts a two-parameter fit by introducing an additional

Table 3. Comparison of enhancement factors from changing size model, for different widths of the initial particle size distribution, with published experimental data of Sada et ~2. (1984); Rmcan,o = 5.00 pm

wt % + 5 10 1.5 20 25 10 + 0.024 0.051 0.080 0.114 0.152

Distribution R, (ka4

Constant+ 5.00 Impulse 5.00 I 8.75 I . 10.00 II 15.00

Experimental 5.00

4A N RL. I* Iv, R IIICP”, 0 I,

1.60 2.22 2.78 3.31 3.82 - - 1.73 2.32 2.85 3.35 3.83 1.00 0.95 0.87 1.44 1.85 2.27 2.59 2.94 0.89 1.06 0.94 1.37 1.74 2.08 2.41 2.73 0.85 1.09 0.94 1.23 1.49 1.74 1.99 2.24 0.78 1.18 0.97 1.1 1.3 1.5 1.6 1.9 - -

‘Constant size model predictions.

Page 13

Gas absorption in reactive slurries 473

3 ._----_---- r-f 0

-0d Q,

: e _ & I 1.h \

2 2.1, \

35-

50 100

0 (5)

Fig. 11. Variation in enhancement factor c#J,,, solid to liquid volumetric ratio Ii,, number concentration Nb and average particle size Rmcsn,,,, in the bulk, vs batch time 0. Data used: standard set; Imtial particle size distribution I with

R 0 mar = 1.1 pm and Rmean,& = 1.0 pm.

factor, namely, the thickness of the “particle-free zone” near the gas-liquid interface. This was probably done in order to obtain a higher fitted value for the lumped parameter k,,A, while still keeping the rates low. The proposed model essentially indicates that the “k&,” term cannot be treated as a constant and decreases with time so that its “effective” value is lower than that computed by using the initial values of solid loading and particle size. Unlike the analysis used by the previous workers, which is based on “fitting”, the theoretical values shown in Table 3 do not use any fitted parameter but require that the particle size distribution information to be known a priori for the rates to be truly predicted. The pub- lished data on sulfur dioxide absorption have not been analysed in the current work since this system has more than one instantaneous reaction and has somewhat more complex features.

Up to now it is the behaviour of the point rate that has been examined with respect to the initial particle size distribution that characterizes a penetration ele- ment when it joins the gas-liquid interface. In the second part of this study, the specific rate-batch time trajectories for a typical slurry batch reactor have been worked out (i.e. a sequence of point rates at different times along the batch time scale). Generating this sequence involves a solution of eq. (19) along with conditon (21). Figure 11 shows the specific rate of absorption, RA, and the bulk quantities-particle number concentration, Nr,, solid loading, Ib and the average size, Rmea+ as a function of batch time, 6, using the proposed theory as well as the unchanging particle size model for a starting average size, R,?,,,,J = 1 pm.

While the constant size model [see Appendix B for a simple Batch model based on Ramachandran and Sharma (1969)] predicts complete consumption of all particles in about 200 s, the changing size theory shows that about 64% of the solid loaded into the

2 h)

Fig. 12. Particle size distribution function Q(R) vs R at different values of batch time 0. Other details as in Fig. 11.

2

Fig. 13. Variation in enhancement factor 4A, solid to liquid volumetric ratio Ibr number concentration Nb and average particle size R,,..,,,,, in the bulk, vs batch time 0. Data used: standard set; Imtial particle size distribution I with

R mar = 11.0 pm and R&sn.b = 10.0 pm.

reactor is still unconsumed! This is consistent with the fact that the specific rates computed from the constant size models (plotted as 4” in Fig. 11) are much higher than those from the changing size model. The corres- ponding evolution of the bulk particle size distribution function is shown in Fig. 12 and the variation is as expected. The area under these curves, representing the particle number concentration, Nb, decreases as 0 increases while the mean radius, Rmea+ also de- creases. Figures 13 and 14 depict analogous plots for particles of a starting mean size of 10 pm. Here, the predictions of the two models nearly coincide for quite some time (approximately, 800 s) till the particle size(s) become “small” enough so that particle dissolution efiects near the gas-liquid interface cannot be ignored; the predictions now cease to match and the constant size assumption now overpredicts the specz$c rate value. Interestingly, this implies that even for slurry reactors which are initially loaded with “large” particles, there will exist a time (on the batch time scale) beyond

Page 14

474 A. MEHRA

R(w)

Fig. 14. Particle size distribution function n,(R) vs R at different values of batch time 0. Other details as in Fig. 13.

which effects related to the “fineness” of the particles cannot be ignored. Thus, the calculation of batch time for a specified conversion of the solid is likely to be grossly in error if particle size changes due to dissolu- tion near the gas-liquid interface are not taken into account.

CONCLUSIONS It has been shown in this study that sparingly

soluble, fine reactant particles are likely to dissolve significantly and undergo a substantial change in size, near the gas-liquid interface, when slurries con- stituted by these particles are used to absorb a react- ive gas.

This dissolution may occur, sometimes exclusively, near the gas-liquid interface, i.e. within the penetra- tion elements that make up this interface, provided the reaction between the absorbing dissolved gas and the sparingly soluble solid is fast enough to cause the depletion of the dissolving solid species near this interface.

The extent of particle dissolution depends on the (initial) particle size distribution, the Sherwood number for the dissolution process, the concentration driving force for the dissolution (which depends, in turn, on the depletion of the dissolved solid species) and the time of contact of the penetration element with the gas phase (which is inversely related to the liquid side mass transfer coefficient, i.e. the stirring intensity).

The specific rate of absorption into slurries of fine particles may be considerably overpredicted if the particle dissolution effects near the gas-liquid inter- face are ignored. Thus, the values of the rate com- puted from the theory proposed in this study are smaller than the corresponding values obtained from prior models in the literature, which assume a con- stant, unchanging particle size near the interface. However, for particles which are “large” enough (with respect to the physicochemical properties and the operating conditions) to allow for the neglect of these

dissolution effects the predictions from the proposed model match closely with those from the previous models. The current model therefore allows for estab- lishing the limits of validity of the models already available in the literature.

The models developed in this study have also been used to demonstrate that the width of the initial particle size distribution, that prevails within a pene- tration element upon its arrival at the gas-liquid interface, has a considerable influence on the absorp- tion rate because of the relatively faster dissolution of the smaller particles.

Thus, the phenomenon of particle dissolution near the gas-liquid interface may explain the commonly encountered experimental finding that the measured rates have usually been found to be lower than the theoretically calculated values.

The evolution of the bulk-phase particle size distributions for a batch slurry reactor have been generated and it has been shown that the solid reactant conversions computed from unchanging size models may result in drastic underestimates of the required batch times for specified conversion levels.

Therefore, particle dissolution effects near the gas-liquid interface are of tremendous importance and cannot be neglected for slurry reactor analysis when the particles in the slurry are in a fine state. These considerations will be of significance even for reactors starting out with “large” particles because the reactor must pass through a state of “fineness” on its way to complete conversion of the solid phase.

a -

CA

C Ab

CB

G CBb

Cl c2 DA DB G

Gb

NOTATION gas-liquid interfacial area per unit vol- ume of slurry, m2/m3 specific solid-liquid interfacial area C =3W(p,R)l, m2/m3 liquid-phase concentration of A, kmol/m3 solubility of A in liquid phase at gas- liquid interface, kmol/m3 concentration of A in bulk liquid phase, kmol/m3 liquid-phase concentration of B, kmol/m3 solubility of B in liquid phase, kmol/m3 concentration of B in bulk liquid phase, equal to CS, in this study, kmol/m3 constant in eqs (37) and (38), me4 constant in eqs (37) and (38) m-’ diffusivity of A in liquid phase, m’/s diffusivity of B in liquid phase, m2/s same as G(R, x, t); growth rate for single particle, m/s same as G,(R, 6); growth rate for single particle in bulk, m/s second-order rate constant in liquid phase for reaction between A and B, m3/kmol s

Page 15

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6.mIs zflnq uy snypm maw30 anIaa Ie!yu! w ‘(0 = 1) [(o ‘x)O’Umaw~ = ] 1uawaIa uoy

-ellauad u! smpm maw 30 an[m pzzyu! tu ‘AmIs

?Inq ul snfpw maux !(fj)q‘“namy SB sums Lu ‘CO ‘0) uBou121 = 1

a%~alu! pmb!I-se8 le smpw maw UI ‘uo!mqys!p B u! saID!ued 30

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-gs!p ET u! sapywd 30 smpm umugxetu s ~“/Iou=i

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u! uoye~iuamoc~ .xaqwnu 30 anIea Iwiy!

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E _ tu ‘AmIs yInq u! uoywuimo3 mqumu @)*N se aures

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SLP sapJn[s a$laeaJ u! uopdJosqe sef)

Page 16

476 A. MEHRA

Sada, E., Kumazawa, H. and Lee, C. H., 1984, Chemical absorption into concentrated slurry. Chem. Engng Sci. 39, 117-120.

Sada, E., Kumazawa, H. and Nishimura, H., 1983, Absorp- tion of SO, into aqueous double slurries containing lime- stone and Mg(OH),. A.1.Ch.E. J. 29, 60-65.

Sada, E., Kumazawa, H., Butt, M. A. and Sumi, T., 1977b, Removal of sulfur dioxide by aqueous slurries of magne- sium hvdroxide uarticles. Chem. Enana Sci. 32. 912-974.

Saraph, \;. S. and Mehra, A., 1994, M&ophase autocataly- sis: importance of near interface effects. Chem. Engng Sci. 49,949-956.

Uchida, S., Koide, K. and Shindo, M., 1975, Gas absorption with fast reaction into a slurry containing fine particles. Chem. Engng Sci. 30, 644-646..

Uchida. S.. Morinuchi. H.. Maeiima. H.. Koide. K. and Kageya&, S., ‘i978,’ Absorption of sulfur dioxide into limestone slurry in a stirred tank reactor. Can. J. Chem. Engng S6,690697.

Uchida, S., Miyachi, M. and Ariga, O., 1981, Penetration model of gas absorption into slurry accompanied by an instantaneous irreversible chemical reaction. Can. J. C/tern. Engng 59, 560-561.

Wachi, S. and Jones, A. G., 1991a, Mass transfer with chem- ical reaction and precipitation. Chem. Engng Sci. 46, 1027-1033.

Wachi, S. and Jones, A. G., 1991b, Effect of gas-liquid mass transfer on crystal size distribution during batch precipita- tion of CaCO,. Chem. Engng Sci. 46, 3289-3293.

Yagi, H. and Hikita, H., 1987, Gas absorption into a slurry accompanied by chemical reaction with solute from spar- ingly soluble particles. Chem. Engng J. 36, 169-174.

APPENDIX A: NOTES ON NUMERICAL PROCEDURE

The primary eqs (1) and (26) may be written in non- dimensional form as

-$=$+ K,ab 641)

and

Dz=-$+$ab-K,(l-b)

X s

‘- R’n,(,/R” + K&(1 - b)dT’) dR (A2)

cl ,/R2 + KS j;(l - b) dT’

respectively, where, a = C&x, 6 = C&i, X = .x/a, T = t/tc, Kt = k2C&, Kz = 4xD,Sh,,t,. K3 = 2ShRD&t,M,/p,, D = D,/D, and q = C;/(zCX).

The initial and boundary conditions (eqs (8), (9) and (1 l), (12), respectively) then become as follows.

(ICS, T = 0):

X20, a=ab=O (A3) b=b*=l (A4)

(BCs, T > 0)

ab X=0, a=l, -=0

ax VW

X+co, a=ab=O, b=br,=l. (A6) The rates given by eqs (14) and (15) can be rearranged to

aive

(A7)

G-W

The above equations can be solved by using an iterative, finite difference scheme which is implicit in time. Equations

(Al) and (A2) may be discretized by the use of a three-point central difference for the space derivative which is implicit in time. Each of these equations therefore yields a tridiagonal system of linear equations, in the X domain, where the non- linear terms such as, Klab, (K&J) ab and the last term of eq. (A2) can be split or computed using the last available values for the variable which is not being solved for from a given equation. Thus, eq. (Al) may be solved for a using the last computed values of b, followed by the solution of eq. (A2) for b using the latest available values of a. This solution sequence may be repeated till convergence, i.e. the values of a and b do not change anymore. This iterative procedure is required to be applied for obtaining a solution for a and b at T + dT knowing the values at T, i.e. for every time increment.

The output values (such as, RAi, RA, N, R,,,, and I) for every run were tested for their insensitivity to the value of kz used, in order to ensure that the rate constant was large enough for the reaction to be deemed instantaneous. The condition for the reaction to be treated as instantaneous, in the absence of reactant particles, is given by (Doraiswamy and Sharma, 1984)

- JKlW (A9)

whereas, the analogous condition, in the presence ofmonodis- persed particles, has been shown by Ramachandran and Sharma (1969) to be

(AlO)

Condition (A9) is essentially required to be satisfied so that depletion of species B occurs near the gas-liquid interface in order to provide a driving force for particle dissolution to occur in this zone. When particles are present, the reaction rate (consumption rate of B) needs to be even higher for depletion to occur since now the particles are also supplying (extra) B. This is embodied in condition (AlO) and this was satisfied for all the runs in this work. This condition is on the conseruatioe side since the Ramachandran and Sharma model assumes an inexhaustible supply of B.

A moving-boundary version of the model proposed in this study was also developed where the concept of a reaction plane, within the penetration element, moving with time is used explicitly [as proposed by Uchida et al. (1981) for the case of constant particle size]. This model yields results which are within a few percent of those reported in this work. However, this model is numerically not very stable since it requires numerical differentiation.

The results obtained in this study were also ensured to be free from any dependence on the various iteration tolerances as well as the numerical parameters such as the step sizes on X, T, 0 and the interval size of the table used to store values of nb(R, 0) (this is relevant only for the cases where the cubic spline interpolation is necessary).

It needs to be mentioned that R = 0 is strictly not an admissible value since it implies that a particle does not exist. Therefore, all integrals where R = 0 is given as one of the limits should more rigorously be written with the limit R = O+. Numerically, this problem was obviated by taking a “very small” but finite value instead of 0, typically, 10e3 pm and it was ensured that the solutions were inde- pendent of this small but finite value.

A material balance check was made for the batch runs by computing the amount of A absorbed till time 0 and compar- ing with the amount of solid reacted in as much time. This check was satisfied to within 10%.

APPENDIX B: BATCH MODEL BASED ON RAMACHANDRAN AND SHARMA (V&9)

The rate expressions from the model of Ramachandran and Sharma (1969) may be written as

Page 17

Gas absorption in reactive slurries 477

and conditions are

R~(Rmean.0) = 4 ,/4xI%h~Rmcan,oNo Z

x coth( 47rShRR~c.~,0N0@ - d))

+ 4~WWLR,,,.,oNo~

2 (B2)

where 6 = DA/kL. Here, RA may be obtained by first comput- ing I by equating the right-hand sides of eqs (Bl) and (B2) and substitution of this 1 value into either of the two equations.

An overall reactor balance, relating the rate of absorption of A to the rate of consumption of reactant B, is given by

> = zRA(Rms.n.& (B3)

For a monodisperse, initial particle size distribution, where all particles are of starting size R,,,, and assuming that all the A that has been absorbed till time 0 has consumed all the particles to an equal extent, the appropriate initial

0 = 0, &caa.b = ha. (R4)

where, Nz = 1:/(4/3)xRi.,. Integration of eq. (B3) gives

which provides the relationship of Rme.n.b vs 6’ (R’,,,...,b is a dummy variable). The values of RA at any 0 may be found by substituting the appropriate value of Rmc.n,b into eqs (Bl) or (B2). This formulation implies that

Nb(B) = Nb” (B6)

lb(e) = Nw3)nR:e,,,b. In the context of the constant size models, the predictions

using the above formulation and those obtained from the unsteady-state variant, as given by Uchida et al. (1981), are nearly the same and also match the values that may be computed from the model proposed in this study by deliber- ately holding the particle size constant.