658
Ind. Eng. Chem. Process Des. Dev. 1982, 21, 658-663 Sebastiin. H. M.; Simnick, J. J.; Lin, H. M.; Chao, K. C. J. Chem. Eng. Data 198Od.25,68. Sebastlan, H. M.; Simnick, J. J.; Lin. H. M.;Chao, K. C. J . Chem. Eng. Data 1980e,25, 138. Sebastian, H. M.; Simnick, J. J.; Lin. H. M.;Chao. K. C. J. Chem. Eng. Data 19801,25,246. Sebastlan, H . M.; Yao, J.; Lin. H. M.; Chao, K. C. J. Chem. €no. Data 1978c,23, 167. Simnick, J. J.: Lawson, C. C.; Lin, H. M.; Chao, K. C. AIChE J. 1977,23,
Kaminishi, G.; Arai. Y.; Salto, S.; Maeda, S.J . Chem. Eng. Jpn. 1968. 7, 109. Kiink. A. E.; Cheh, H. Y.; Amick, E. H. AIChE J. 1975,27, 1142. Kobayashi, R. “Phase and Volumetric Equilibria in Coal Hydrogenation Systems”; Ouarterly Report to DOE, Sept 1978. Krase, N. W.; Goodman, J. B. Ind. Eng. Chem. 1930,22, 13. Krlchevskli, I . R.; Sorina, G. A. Russ. J. Phys. Chem., 1960,34,679. Lee, B. I.; Kesier, M. G. AIChE J . 1975,27,510. Lee. T. J.; Lee, L. L.; Starling, K. E. Adv. Chem. Ser. 1979,No. 782,125. Leland, T. W.; Mueiler, W. H. Ind. Eng. Chem. 1959,51. 597. Leland, T. W.; Rowlinson, J. S.;Sather, G. A. Trans. faraday SOC. 1968,64, 1447. Lin, H. M.; .%bastlen, H. M.; Chao, K. C. J. Chem. Eng. Data 1980,25,252. Lin, H. M.; Sebastlan, H. M.; Simnick, J. J.; Chao, K. C. J. Chem. Eng. Data 1979,24, 146. Mraw, S.;Hwang, S.C.; Kobayashi, R. J. Chem. Eng. Data 1978,23, 135. Nasir, P.; Hwang, S.C.; Kobayashi, R. J. Chem. Eng. Data 1980,25,298. Ng, J. J.; Robinson, D. 8. J. Chem. Eng. Data 1978,23,325. Nichols, W. 8.; Reamer, H. H.; Sage. B. H. AIChE J. 1957,3,282. Olds, R. H.; Reamer, H. H.; Sage, B. H.; Lacey, W. N. Ind. Eng. Chem. 1949,47, 475. Olds, R. H.; Sage, B. H.; Lacey, W. N. Ind. Eng. Chem. 1942,34, 1008. Peter. S.;Rheinhartz, K. 2.W y s . Chem. 1960,24, 103. Rocker, V.; Knapp, H.; Prausnitz, J. M. Ind. Eng. Chem. Process D e s . D e v . 1978, 77,324. Price, A. R.; Kobayashi, R. J. Chem. Eng. Data 1959,4 ,40. Prodany, N. W.; WIHiams, B. J. Chem. Eng. Data 1971, 76,1. Reamer, H. H.; Sage, 8. H.; Lacey, W. N. Ind. Eng. Chem. 1950,42,535. R e a m , H. H.; Sage, B. H.; Lacey, W. N. J. Chem. Eng. Data 1958. 3,240. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. “The Properties of Gases and Liquids”, 3rd ed.;McGraw-HIII: New York, 1977. Sage, B. H.; Hicks, B. L.; Lacey, W. N. Ind. Eng. Chem. 1940,32, 1085. S a p , B. H.; Reamer, H. H.; OMS, R. H.; Lacey, W. N. Ind. Eng. Chem. 1942,34, 1108. Sebastian. H. M.; Lin, H. M.; Chao, K. C. J. Chem. Eng. Data 1980a,25, 381. Sebastlan, H. M.; Lin. H. M.; Chao, K. C., AIChE J. 198la,27, 138. Sebastian, H. M.; Lin, H. M.; Chao, K. C. Ind. Eng. Chem. fundam. IgElb, 20,348. Sebastian, J. M.; Lin, H. M.; Chao, K. C. Ind. Eng. Chem. Process Des. D e v . 1981c,20, 508. Sebastiin, H. M.; Nageshwar, G. D.; Lin. H. M.; Chao, K. C. J. Chem. Eng. Data 1980b,25, 145. Sebastian, H. M.; Nageshwar, G. D.; Lin, H. M.; Chao, K. C. Fluid Phase Equiib. 1980~. 4, 257. Sebastian, H. M.; Simnick. J. J.; Lin, H. M.; Chao, K. C. J. Chem. €no. Data 1978a,23. 305. .%bastian, H. M.; Simnick, J. J.; Lin, H. M.; Chao, K. C. Can. J. Chem. Eng. 1978b,56,743. Sebastian, H. M.; Simnick, J. J.; Lin. H. M.; Chao, K. C. J. Chem. Eng. Data 1979,24, 149.
469.
Simnick, J. J.; Liu, K. D.; Lin, H. M.; Chao, K. C. Ind. Eng. Chem. Process D e s . D e v . 1978a, 77, 204. Simnick, J. J.; Sebastian, H. M.;Lin, H. M.;Chao, K. C. J . Chem. Eng. Data 1978b,23,339. Simnick, J. J.; Sebastian, H. M.;Lin, H. M.; Chao, K. C. J. Chem. Thermodyn. 1979a. 77. 331. Simnick, J. J.; Sebastian, H. M.; Lin, H. M.; Chao. K. C. Fiuid Phase Eouilib 1979b.3, 145. Simnick, J. J.; Sebastiin. H. M.;Lin, H. M.; Chao, K. C. J . Chem. Enq. Data 1979c,24,239. Smith. W. R. Mol. Phys. 1971,27, 105. Smith, W. R. Can. J. Chem. Eng. 1972,50, 271. Soave, G. Chem. Eng. Sci. 1972,27, 1197. Spano. J. 0.; Heck, C. K.; Barrick, P. L. J. Chem. Eng. Data 1968, 73,168. Stryjek, R.; Chappelear, P. S.;Kobayashi, R. J. Chem. Eng. Data 1974, 79, 334. Stuii, D. R. Ind. Eng. Chem. 1947,39,517. Timmermans, J. “PhysicoChemicai Constants of Pure Organic Compounds”, 2nd ed.; Elsevier: Amsterdam, 1950. Wichterie, I.; Kobayashi, R. J. Chem. Eng. Data 1972, 77,4. Wieczorek, S.A.; Kobayashi, R. J. Chem. Eng. Data 1980,25,302. Wieczorek, S.A.; Kobayashi, R. J. Chem. Eng. Data 1981,26,8. Wiihoit, R. C.; Zwoiinski, B. J. “Handbook of Vapor Pressures and Heats of Vaporization of Hydrocarbons and Related Compounds”; API-4dTRC 101, Texas A&M University, 1971. Wilson, G. M.; Johnston, R. H.; Hwang, S. C.; Tsonopoulos, C. Ind. Eng. Chem. Process D e s . Dev. 1981,20, 94. Yao, J.; Sebastian, H. M.; Lin, H. M.; Chao, K. C. Fluid Phase Equilib. 19771 1978, 7, 293 (see also 1980,4, 321 for the data at 456.9 “C). Yorizane, M.; Yoshimura, S.; Masuoka, H. Kagaku Kogaku 1970, 34,953. Zwoiinski, B. J. “Thermodynamic Properties of Chemical Compounds of Importance to Processing Lignite”; Report No. 2, Cresols, Texas A&M University, 1977.
Received for reuiew July 16, 1981 Accepted April 26, 1982
Funds for this research were provided by the Electric Power Research Institute through Grant RP-367.
Effect of Bulk Flow Due to Volume Change in the Gas Phase on Gas-Solid Reactions: Initially Porous Solids H. Y. Sohn” and Osvaldo A. Bascur Department of Metallurgy and Metallurgical Engineering, University of Utah, Sa# Lake City#Utah 84 7 12
The effect of bulk flow due to volume change in the gas phase on the rate of noncatalytic gas-soli reactions in porous sollds has been studied for systems in which diffusion is in the molecular regime. The model has been formulated in general terms so as to allow the incorporation of specific details of an actual system. The computed results show that the effect of bulk flow can be quite large. This effect increases with the importance of pore diffusion through the product layer in determining the overall rate of reaction. The law of additive reaction times previously proposed for reactions without volume change has been applied and found to yield a useful approximate solution also for this type of reaction systems. As a result, an approximate analytical equation for the conversion vs. time relationship incorporating chemical reaction, product layer diffusion, and external mass transfer has been obtained.
Introduction Because of the important role played by heterogeneous noncatalytic gas-solid reactions in many chemical and metallurgical processes, much research has been devoted to the subject. Recent advances due to the development 0198-4305/82/1121-0658$01.25/0
of more sophisticated mathematical models and experimental techniques have contributed to our understanding of reaction mechanisms and reactor design problems. The reader is referred to a recent monograph (Szekely et al., 1976) and review articles (Sohn, 1976a,b, 1978,1979) for 0
1982 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 4, 1982 659
comprehensive reviews on this topic. The analysis of the heterogeneous reaction system must start from the recognition that mass and heat must be transported between phases. It follows that the structure of the solid, pertinent to heat and mass transfer, before, during, and after the reaction, plays an important role in determining the overall rate of reaction. Thus, the previously proposed models describing this type of fluid-olid reactions can be largely divided into the following two groups: (1) for an initially nonporous solid producing a porous product layer: the shrinking core model; and (2) for an initially porous solid producing a porous product: the grain model. However, most of the work has been restricted to the systems in which bulk flow within the solid due to the volume change of gas is negligible. Among heterogeneous noncatalytic gas-solid reactions, the roasting of sulfide minerals is a typical example in which the volume of the reactant gas is different from that of the product gas. This group of reactions can in general be written as: MeS, 1 . 5 ~= 0 MeO, ~ xS02. Other examples of this type of reactions are the Boudouard reaction
+
+
the carbonylation of nickel Nib) + 4CO(g) = Ni(C0)Jg) and the reduction of halides FeC12(s)+ H2(g) = Fe(s) + 2HCl(g) Recently, Sohn and Sohn (1980) presented a systematic analysis of the bulk flow effects on the reaction of an initially nonporous solid according to the shrinking core scheme. They showed that the effect of bulk flow on the rate of a gas-solid reaction which is controlled in general by both intrinsic kinetics and mass transfer processes can be quite large, increasing with the importance of the product-layer diffusion. They obtained an exact analytical solution when the overall rate is completely controlled by diffusion and an approximate closed-form equation for conversion vs. time was derived which includes the effect of chemical reactions, product-layer diffusion, effect of bulk flow, and external mass transfer. They also described examples of reaction systems in which the bulk flow effect may be important and reviewed previous work dealing with the subject. Following this previous investigation, in this paper, we present the results for the reaction of an initially porous solid. Mathematical Formulation Let us consider a gas-solid reaction which can in general be expressed by A(g) b.B(s) &(g) + d-D(s) (1)
+
-
In the following formulation we shall consider an isothermal, first-order irreversible reaction of a porous solid. Because of the initial porosity of the solid, the reaction occurs while the fluid reactant diffuses into the pores. Thus, the reaction occurs in general in a diffuse zone within the pellet. We shall also consider only the case where the product layer is sufficiently permeable so that the total pressure is uniform throughout the system, and viscous flow effects are negligible. The conditions in which viscous flow becomes important have been examined by Evans (1972). I t was shown that in many cases of practical interest the effect of viscous flow resistance is small. Furthermore, most pellets of practical interest can structurally withstand only moderate internal pressure. Any large
pressure generated internally will tend to be relieved preferentially through cracks formed due to the pressure. Thus, the bulk flow effect will be important mainly in pellets with sufficiently large permeability in which total pressure is uniform. The grain model assumes the individual grains to be nonporous and each grain to react according to the shrinking-core scheme, but the nucleation and growth kinetics and fiit-order kinetics with respect to the amount of solid reactant remaining unreacted can be incorporated. In this case the nucleation and growth kinetics will be used since in the reaction of very fine solid particles nucleation is very important and may control the reaction over the entire duration. Many porous pellets are made by pelletizing very fine particles and thus it is of interest to consider the reaction of porous pellets in which the reaction of grains follows the nucleation and growth kinetics. There are various types of nucleation and growth kinetics most of which give sigmoidal curves for conversion as a function of time. There are comprehensive reviews of these kinetics in the literature (Young, 1966; Pannetier and Souchay, 1967; Delmon, 1969). We shall formulate the problem in general by writing the conversion vs. time relationship under constant fluid reactant concentration as mt = f ( w ) (2) where o is the local value of the fractional conversion of the solid reactant and m is a parameter containing reaction-rate constant and other quantities. In keeping with our consideration of kinetics that are of first order with respect to the fluid reactant concentration, we shall let m = bkCA (3) In general k could depend on grain size and initial molar density of the solid reactant, but it shall be independent of fluid-reactant concentration and fractional conversion of the solid. Reversible first-order reactions could be incorporated following the procedure described elsewhere (Szekely et al., 1976). The following rate expression for the nucleation and growth attributed to Erofeev (Young, 1966, p 35) will be used f ( o ) = g N ( o )=
[-ln (1-
(4)
where n is a positive number. This rate expression has an advantage of including, as a special case for n = 1, a first-order kinetics with respect to the amount of solid reactant remaining unreacted. The governing equations are obtained by considering the conservation of the fluid reactant in the porous solid and the mass balance of the solid reactant. It will further be assumed that the pseudo-steady-state approximation for gas-phase-mass balance is valid, and the pellet retains its original overall shape and size during the course of reaction. For the three basic geometries, slabs, long cylinders, and spheres, we have the following mass balance for the fluid reactant A (5)
where uA is the local rate of consumption of the fluid reactant A per unit volume and the shape factor Fp has the value of 1, 2, or 3 for each of the three geometries, respectively. The molar flux N A is given by N A
=
(NA
+NC)XA-DeCTVxA
and from stoichiometry
(6)
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Ind. Eng. Chem. Process Des. Dev., Vol.
21, No. 4, 1982
Nc = - U N A (7) The expression for flux in eq 6 was written for diffusion in the molecular regime, not including the Knudsen or surface diffusion. When only the ordinary molecular diffusion is important, the viscous effect can be neglected (Evans, 1972). Substituting eq 7 in eq 6 and rearranging, we get -DeeT dxA/dr NA = (8) 1- (1 - U)xA Combining eq 5 and 8 and rewriting the resulting equation, we obtain
In this derivation the effective diffusivity is assumed to remain constant. This would be a reasonable assumption for a system in which the initial porosity is substantial or the porosity does not change substantially upon reaction. The change in effective diffusivity with the extent of reaction could be incorporated. We solve the problem only for the case of relatively constant effective diffusivity, because our major objective is to assess the effect of bulk flow. Furthermore, as will be seen later, in the diffusioncontrolled regime the results obtained will depend on the effective diffusivity in the product layer only and not in the solid reactant. The mass balance for the solid reactant can be written from eq 2 and 3 as (10) where the prime designates the derivative with respect to w. The expression for UA can be obtained by considering the stoichiometry and the apparent density of the porous solid as
Equations 9 and 10 may be written in dimensionless form as
and
where
sufficient to note that this modulus represents the ratio of time for a molecule to diffuse the characteristic distance to the characteristic time for chemical reaction. The boundary and initial conditions for eq 12 and 13 are dy/dq = 0 at q = 0 (19) y = l
- 1)XAb
(18) We see that the effect of bulk flow is represented by a single parameter 6. The particular choice of the definition of fluid-solid reaction modulus (dN) will be explained shortly. It is
(20)
w =0 at tN* = 0 (21) The boundary condition (eq 20) is written for the case of negligible resistance due to external mass transfer. The effect of external mass transfer can be incorporated by writing an appropriate equation in lieu of eq 20. A separate discussion on the effect of external mass transfer will be presented in a subsequent section. The system of eq 12, 13, and 19 to 21 is applicable to pellets of various geometries and solid kinetics of the form given by eq 10. The solution yields y and w as functions of q and tN*. The overall conversion, which is of greater practical interest, may be obtained from the following relationship
Solution Procedure. The procedure for numerical solution of the system of eq 12 with boundary conditions 19 and 20 and eq 13 with initial condition 21 is as follows. At t N * = 0, w = 0 and eq 12, a second-order nonlinear ordinary differential equation is solved for y as function of q using boundary conditions eq 19 and 20. This twopoint boundary value problem is solved by rewriting eq 12 as a set of two fiist-order nonlinear ordinary differential equations which can be solved using a modified fourthorder Runge-Kutta method together with a suitable shooting technique. With the concentration profile (y vs. q) thus obtained at t N * = 0, w at t N * = AtN* is computed from eq 13 using a fourth-order Rung-Kutta method. A new value of w is obtained which can be used to solve eq 12 to give y vs q at tN* = AtN*. This procedure is repeated for increasing time steps. Since eq 12 presents a singularity at q = 0 a value of q = 1 X was used. This procedure can be avoided by using the SincGalerkin method using Whittaker’s cardinal function (Stenger, 1979). (This method is much more involved and was used in this work only to check the complete numerical solution.) The total conversion of the solid is calculated using eq 22 using Simpson’s numerical integration. Results and Discussion Asymptotic Behavior. (i) When CN approaches zero, Le., when the overall reaction is controlled by chemical reaction, the reactant concentration is uniform throughout the pellet and is equal to that in the bulk (y = 1). Therefore, w is independent of q. It has been shown (Sohn, 1978) that the following results applies tN*
6 = (u
atq=1
= f(w)
(23)
In this case volume change has no effect on the overall rate. (ii) When 6 = -1, which is the case when u = 0 and ZAb = 1, again y = 1. This result can also be expected from physical reasoning that, when there is no gaseous product and the bulk gas is pure reactant, only the reactant gas exists in the system. Under the conditions of a uniform total pressure considered in this work, the gaseous reactant concentration will be uniform throughout the pellet and equal to the bulk value.
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 4, 1982 881
, 0
IO
20
I
I
I
30
40
50
0
30
20
IO
40
,
I
50
60
+;
+; Figure 1. The effect of B on the conversion vs. time relationship for a s m d value of bN2.
Figure 3. The effect of B on the conversion vs. time relationship for bN2
= 3.
101
x
08r " U
0
I O
2 0
30
40
5 0
6 0
70
80
9 0
",
+; Figure 2. The effect of B on the conversion m. time relationship for ENz = 1.
Figure 4. The effect of B on the conversion vs. time relationship for a large value of bN2.
(iii) When dN approaches infinity, i.e., when chemical reaction is very fast compared with diffusion, the overall rate of reaction is controlled by product layer difbion and the concentration of gas A will be zero at the reaction interface. This situation is identical with the diffusioncontrolled shrinking unreacted-core system of a nonporous solid, and the results presented by Sohn and Sohn (1980) hold, namely
expected that regardless of pellet geometry the regime for chemical reaction control will be defined by &N < 0.3 and that for diffusion control by dN > 3 (Sohn, 1978). T h e General Case. In the general case, eq 12 and 13 must be integrated with the aid of boundary conditions 19 and 21 using the numerical procedure outlined previously. Figures 1-4 show conversion vs. time for spheres for various 0 and 8N2. For a small value of dN2 (say less than 0.1) in which case chemical reaction is predominately rate-controlling, the effect of volume change is seen to be negligible as shown in Figure 1. As dN2 increases, however, the effect becomes quite large as can be seen in Figures 2-4. It is noted that when the volume of the product gas is greater than that of the reactant gas, i.e. 0 > 0, the outward bulk flow makes it more difficult for the reactant to diffuse into the interior. This results in a lower rate of reaction. When the volume of the product gas is smaller (e < 0), the inward bulk flow speeds up the diffusion, thus increasing the rate.
where
The conversion function PF ( X )in eq 24 can be rewritten for different values of F p in {he following familiar forms: for Fp = 1 (slabs) In (1 + 0 ) t+ = Pl(X) x2
e
f
for Fp= 2 (long cylinders), by applying the L'Hospital's rule In (1 + e) t+ = P2(X)= X (1- X ) In (1- X) (27)
+
e
for Fp = 3 (spheres) ~n ( 1 + e)
e
+
t+ = p3(X) = 1 - 3(1 - X)2/3 2(1 - X) (28)
The form of eq 25 actually determines the definition of dW With this definition the criteria for the asymptotic
regimes of chemical reaction or diffusion control can be given by the same numerial values of the fluid-solid reaction modulus for various geometries. Thus, it can be
Application of the Law of Additive Reaction Times One of the authors (Sohn, 1978) proposed the law of additive reaction times for gas-solid reactions which can be applied in either an integrated or a differential form. This law states that the time required to attain a certain conversion can be approximated by the sum of the time required to reach the same conversion in the absence of resistance due to the intrapellet diffusion of fluid reactant and the time required to reach the same conversion under the control of intrapellet diffusion. It would be extremely useful to have a closed form solution for the system under consideration in which the effect of volume change is significant. Therefore, the law of additive reaction times was tested. The mathematical representation of the law is given by (Sohn, 1978) t(X) = [ t ( X ) l b p a + [t(X)lap.. (29)
882
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 4, 1982 L" 11 e x
02
2 0
I C
I
I
,
3 0
41
51
3
IC
I
1
1
1
I
20
3.0
4 0
50
60
70
1.N
't Figure 5. Comparison between the exact and the approximate solutions for cN2= 1.
Figure 7. Comparison between the exact and the approximate solutions for Sh* = 10.
Bird et al. (1960) recommended the following equation for external mass transfer accompanied by bulk flow
? 11 A 1
I O
2 5
4 3
30
5 0
6 0
70
Nh - X A b ( N h Nc,) = k z o ( X h-XAb) (32) The mass transfer coefficient defined by this equation is reported to be less dependent on concentration and mass transfer rate than that defined otherwise (Bird et al., 1960). Again from stoichiometry Ncs = -UNA, (33) From eq 32 and 33, we get
+; Figur 0 6. Comparison between the exact and the approximate solutions for Sh* = 3.
(34)
Using eq 8 at r = R we have
Using eq 23, 24, and 29 the approximate solution can be written as Equating eq 34 and 35 and arranging we obtain the necessary boundary condition when external mass transfer is considered. Figure 5 shows the comparison between the exact numerical solution and the approximate solution given by eq 31. It is seen that the approximate solution gives a reasonable representation of the exact solution (maximum 10% error). The comparison was made for %N = 1 for which the difference between the exact and the approximate solutions is largest. As BN becomes smaller or larger, the agreement becomes better (Sohn and Szekely, 1972; Sohn and Sohn, 1980). In fact, as gN approaches either zero or infinity, eq 31 becomes asymptotically exact. It has thus been verified that the law of additive reaction times is valid even for the reaction of a porous solid in which there is volume change in the gas phase. The advantages of such a closed-form solution, especially in the analysis and design of multiparticle systems in which a particle size distribution and/or bulk concentration and temperature gradients may exist, have been discussed elsewhere (Sohn and Szekely, 1972; Sohn, 1978). Effect of External Mass Transfer In the preceding discussion we assumed that external mass transfer does not influence the overall rate. In laboratory scale experimental studies such conditions are readily realized by using sufficiently high gas velocities. In practical systems involving packed or fluidized beds of solid particles, however, external mass transfer may be fully or partially rate-controlling. When the effect of external mass transfer cannot be neglected, the governing eq 12 and 13 and the conditions 19 and 21 still apply, but the boundary condition 20 must be replaced.
Rearranging 1 i + e dy Sh* 1 + Oy dq where
+y = 1
at v = 1
(37)
The solution procedure employed to solve this system is the same. A relationship analogous to eq 31 has been derived for the case when the external mass transfer has an influence on the overall rate of reaction
(39) The reader is referred to Sohn and Sohn (1980) for the derivation of eq 39. Figures 6 and 7 present the effect of bulk flow for the case where Sh* = 3 and 10, respectively, with gN2= 1.0 and Fp= 3. Also shown is the result of applying the approximate solution given by eq 39. It is seen that eq 39 gives a satisfactory representation of the exact solution. Concluding Remarks The effect of bulk flow on the reaction of a porous solid with a gas has been examined for systems in which diffu-
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 4, 1982 863
sion in the pores is in the ordinary molecular regime. It has been shown that this effect can be quite large, increasing with the importance of the produd-layer diffusion. For a wide range of aN2,the law of additive reaction times has been shown to provide a useful approximate closedform solution. Thus, an equation for conversion vs. time has been presented which includes the effect of chemical reaction, product-layer diffusion, bulk flow, and external mass transfer. The results presented in this paper were obtained for constant temperature and bulk fluid concentrations. It has been shown, for a gas-solid reaction without volume change, that the differential form of the approximate solution holds valid even when the temperature of the system or the bulk concentration varies with time, as long as the solid is isothermal at any given instant (Sohn, 1978). The same can be extended to the present case. The approximate closed-form solution is very useful in analyzing multiparticle systems because it obviates the necessity to numerically solve the governing differential equations such as eq 12 and 13 at every time step and for different particle sizes.
Nomenclature A, = external surface area of the particle b = stoichiometry coefficient CT = total molar concentration of gas D, = molecular diffusivity De = effective diffusivity Fp= particle shape factor (equal to 1,2, and 3 for slabs, long cylinders, and spheres, respectively) .g&) = conversion function defined by eq 4 k = reaction rate constant k,’ = mass transfer coefficient given in eq 32 n = constant in eq 4 N = flux of species PF,(X) = conversion function defined by eq 26, 27, and 28 r = distance coordinate R = radius of a particle Sh = Sherwood number Sh* = modified Sherwood number defined by eq 38 t = time tN* = dimensionless time for nucleation growth model defined by eq 16 t+ = dimensionless time defined by eq 25 U A = local rate of consumption of fluid reactant A
V , = volume of the pellet = mole fraction of gas A X = fractional conversion of solid y = dimensionless concentration defined by eq 14 XA
Greek Letters
fraction of pellet volume initially occupied by solid reactant B 9 = dimensionless distance coordinate defined by eq 15 B = volume change parameter defined by eq 18 v = number of moles of gaseous product produced from one mole of gaseous reactant p~ = number of moles of B per unit of volume of the solid d~ = gas-solid reaction modulus for nucleation and growth kinetics defined by eq 17 (J = local value of fractional conversion of solid reactant ~ r g=
Subscripts A = gas A b = bulk phase B = solid B C = gas C D=gasD s = value at external surface Literature Cited Bird, R. B.; Stewart, W. E., Lightfoot, E. N. ”Transpm Phenomena”; Wiley: New York, 1960; Chapter 21. Deimon, B. "Introduction a la Cinetique Heterogene”, 1st ed.;Editions Technip: Paris, 1969. Evans, J. W. Can. J . Chem. Eng. W72, 50, 811. Pannetler, G.; Souchay, P. “Chemical Kinetics”, translated by Gesser, H. D.; Emond, H. H., Elsevier: Amterdam, 1967; Chapter 9, pp 393-408. Sohn, H. Y. Hwahak Konghak 1976a. 14, 3. Sohn. H. Y. Hwatrak Konghak 1076b, 14, 65. Sohn. H. Y. Met. Trans. 8 , 1978, 98, 89. Sohn. H. Y. I n “Rate Processes of Extractive Metallurgy”; Sohn, H. Y.; Wadsworth, M. E., Ed.; Plenum Press: New York, 1979; Chapter 1, pp 1-51. Sohn, H. Y.; Sohn. H. J. I d . €ng. Chem. process Des. D e v . 1980, 19, 237. Sohn, H. Y.; Srekely, J. Chem. Eng. Sci., 1972, 27, 763. Stenger, F. Mem. Comp. 1979, 33, 85. Szekely, J.; Evans, J. W., Sohn, H. Y., “Gas-Solid Reaction”; Academic Press: New York, 1976. Young, D. A. ”DecomposMn of Solids”;The International Encyclopedia of Physical Chemlstry and Chemical Physics, Topic 21 Solid and Surface Klnetlcs. Voi. 1, Tompklns, F. C.. Ed.; Pergemon Press: 1966 Chapter 2.
Received for review July 20, 1981 Accepted March 22, 1982
This work was supported in part by a research grant from the University of Utah Research Committee.
