386
Ind. Eng. Chem. Process Des. Dev. 1986, 25, 386-394
= reaction rate constants for free-radical initiator influenced reactions k , = second-order rate constant for the forward reaction, 2C6H6 e CI2Hl0+ HP,K mol m-3 s-' kPa-2 K = equilibrium constant for the reaction (eq 1) L = length of the reactor, m R = universal gas constant, kJ mol-' K-' P = total pressure, kPa S = cross section of the reactor tube, m2 T = temperature, K T R = reference temperature, K VE = equivalent reactor volume, m3 x = fractional conversion of benzene f ( x ) = In ([((4K- 1)x - 4K - 2K112)(-2K+ K*/2)]/[((4K - 1)x - 4K + 2K11')(-2K - K'")] Registry No. C6Hs,71-43-2; Cl2Hlo,92-52-4;CI8Hl2,217-59-4; CH,, 74-82-8; C2H4, 74-85-1; C3H6, 115-07-1;Hz, 1333-74-0. Literature Cited klLk6'
Asaba. T.; Fujii, N. 13th International Symposium on Combustion, Salt Lake City, UT, 1971, p 155. Bauer, S.H.: Aten, C. F. J . Chem. Phys. 1963, 39 (5), 1253. Brooks, C. T.; Peacock. S. J.; Reuben, B. G. J . Chem. Soc.. Faraday Trans. 11979, 75, 652. Brown, J. L.; Bell, J. T.. Monsanto Chemical Co. US. Patent, 3 876 719, April 8 1975 - . -. -1
Cipau, R.: Moraru, M. Bull. Stint, Tech. Znst. Politech. Timisoura , 1966, 13
(2),327
Froment, G. F.; Picke. H.; Goethals, G. Chem. Eng. Sci. 1961, 1 3 , 173. HOU,K. C.; Palmer, H. B. J . Phys. Chem. 1965, 69, 863. Hougen, 0.A.; Watson, K. M. "Chemical Process Principles"; Wiley: New York, 1947; Part 3. Indyukov, N. M.; Gasanova, R. I. Khlm. Prom. st. (Moscow) 1975, 7, 555. Indyukov, N. M.; Gasanova, R. I.; Abilov, A. G.; Imamverdiev, A. A,; Orudzheva, N. I . Neff. Khim. (Sofia) 1976, 16(4), 555. Kinney, C. R.; Delbel, E. Znd. Eng. Chem. 1954, 46 (3), 548. Korzum, N. V.; Magaril, R. 2.; Ioanidis, N. V. Khim. Tekhnol, Tr. Tyumen, Ind. Znst. 1972, 196. Louw, R.: Lucas, H. J. Recl. Trav. Chim. 1973, 92,55. Mariyasin, I. L.; Nabutovskii, 2 . A. Kinet. Catal. 1969, 10, 983. Mead, F. C., Jr.; Burk, R. E. Znd. Eng. Chem. 1935, 27, 229. Murphy, G. B.; Lamb, G. C.; Watson, K. M. Trans. Am. Znst. Chem. Eng. 1936, 34, 42. Ostroff, N.; Miller, I. F. Chem. Eng. Prog., Symp. Ser. 1971, No. 67(112), 109. Pease, R. N.; Morton, J. M. J . Am. Chem. SOC. 1933, 55, 3190. Walters, W. D. J . Am. Chem. SOC. 1941, 63, 1701. Rice, F. 0.; Sakai, T.; Wada, S.:Kunugi. T. Znd. Eng. Chem. Process Des. Dev. 1971, 10 (3), 305. Saunders, J. H.; Anniston, A.; Solocombe, R. J., Monsanto Chemical Co. US. Patent 2 702 307, 1964. Slysh, R. S.;Kinney, C. R. J . Phys. Chem. 1961, 6 5 , 1044. VanDamme, P. S.;Narayanan, S.; Froment, G. F. AZChE J . 1975, 21 (S), 1065. Virk, P. S.; Chambers, L. E.; Woebeke, H. N. Adv. Chem. Ser. 1974, No.
131.
Received for review May 23, 1983 Revised manuscript received March 19, 1985 Accepted July 16, 1985
Effect of Nonuniform Distribution of Solid Reactant on Fluid-Solid Reactions. 1. Initially Nonporous Solids Hong Yong Sohn" and Yong-Nian Xla Department of Metallurgy and Metallurgical Engineering, University of Utah, Salt Lake City, Utah 84 112- 1183
This paper describes the results of a study on the effect of a nonuniform distribution of the solid reactant within the pellet on the overall rate of a fluid-solid reaction. Part 1 considers nonporous pellets. The case of porous pellets will be described in part 2. The shrinking-core model is used to describe the reaction of a nonporous solid. I t is shown that for an arbitrary distribution of the solid reactant, the critical factors affecting the overall rate of the conversion processes are the effective surface area of the solid reactant at the reaction interface and the porosity of the reacted layer of the pellets, both of which are determined by the form of the distribution function. Wlthin the frame of the model, a solid reactant concentration which increases monotonically with the radius will cause the time for complete conversion to be shorter than for a uniform concentration which is identical with the volumetric average concentration of the nonuniform distribution. For a monotonically decreasing concentration, the effect is opposite. This is the case when pore diffusion affects the overall rate. When chemical reaction controls the overall rate, the time for complete reaction is independent of the distribution function regardless of the geometry of the pellet.
There has been a great deal of work treating the subject of fluid-solid reactions of pure or uniformly distributed solid reactants (Lu, 1963; Seth and Ross, 1965 and 1966; Shen and Smith, 1965; Sohn, 1979 and 1981; Sohn and Szekely, 1972; St. Clair, 1965; Szekely et al., 1976). However, in many practical chemical and metallurgical systems, solid particles with a nonuniform distribution of solid reactant are commonly encountered. For example, in the leaching of ore particles, the formation of coke in catalyst particles, and the adsorption by solid, it is not always the case in real systems that the solid particles are actually pure or the solid reactant is uniformly distributed within a particle. Only in certain circumstances can we treat those problems with the theory for pure or uniformly distributed solid reactant with a reasonable approximation. Then, it is important to establish the criterion for the assumption
of uniform distribution and to estimate the degree of the approximation. In addition, there are certainly other circumstances under which the approximation cannot be used, because of a large unacceptable error. Therefore, a systematic analysis of the reaction of a solid with nonuniform distribution of solid reactant is necessary in order to obtain more accurate information. Because there are numerous possible forms of distribution of the solid reactant, we will treat the problem in a more general way and emphasize those aspects such as model establishment and general results for an arbitrary form of distribution, as well as some explicit results for certain specific distributions. Since we started looking a t this problem, Dudukovic (1984) has published a paper also on the reactions of particles with nonuniform distribution of solid reactant
0196-4305/86/1125-0386$0 1.50/0 0 1986 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986 387
and the effective diffusivity of fluid A through the reacted layer is
where DiC is the molecular diffusivity for the mixture of A and C (combined with Knudsen diffusivity, if its effect is significant), and T is the tortuosity of the product layer. Approximating T with l / t (Wakao and Smith, 1962),the effective diffusivity of A through the reacted layer can be expressed as
or Figure 1. Sketch of the system in which the solid reactant forms the continuous phase.
following the shrinking-core model. We are in disagreement with certain aspects of his formulation of the model and thus his results. We will discuss the differences between the two models at appropriate points in the sections of Model Formulation and Results and Discussion below.
Model Formulation For a nonporous pellet, we base our analysis on the following basic physical properties and postulates: (i) The pellet consists of reactant and inert solids. The concentration of the solid reactant is a function only of the distance from the center of symmetry. The reactant solid constitutes a continuous matrix of the pellet, which also contains an inert solid phase, as shown in Figure 1. (ii) The reacted layer of the pellet is porous. (iii) The diffusivity of the fluid reactant and product through the solid phase can be neglected. (iv) The overall pellet size and shape do not change with the reaction, and the unreacted core maintains the original shape as it shrinks toward the center of symmetry. (v) The system is isothermal. (vi) The pseudo-steady-state approximation is appropriate for describing the concentration of fluid reactant within the pellet. The fluid-solid reaction under consideration is A(f) + bB(s) = C(f) + dD(s) (1)
where
i.e., the effective diffusivity of fluid A through pure product D generated from B according to reaction 1, and is a constant for any particular system. In a real system, another relationship may exist between 7 and 6. This may change the actual magnitude of the effect of nonuniform distribution of the solid reactant, but it is thought that the qualitative nature of the effect will remain substantially the same as that which will be presented subsequently. Equation 7 expresses the fact that under nonuniform distribution of solid reactant, the effective diffusivity of the fluid through the product layer is not constant but related to the distribution by a factor of f ( R ) ;physically, this means that existing inert has the effect of decreasing the cross-sectional area available for diffusion and increasing the tortuosity. Now we set up the mass balance equations for the fluid and solid species under the condition of simultaneous chemical reaction and diffusional control for infinite slabs, long cylinders, and spheres:
with intrinsic kinetics
RA = ~ ( C -A CC/KE)
(2)
We further propose that the distribution of the solid reactant B within the pellet can be described as a function of the distance from the center of symmetry of the pellet
PB(R)= _PB~(R) subject to the constraint 0 6 f ( R )6 1, for 0 4 R 4 R,
(3)
CA = CAO and Cc = Cco at R = R, dCA
at R = R, D;fl(R)dR = k f ( R ) ( C A- Cc/KE)
dR
R(Fp-l)p j ( R ) - = b-const dt
R=R, (4)
where -pB is the molar density of pure solid reactant B, and pB(R)is the local concentration of B at any distance R. It is observed that f(R) denotes the volume fraction of solid reactant in the pellet and also the fraction of crosssectional area of the solid reactant per unit cross-sectional area of the pellet at any distance R. On the basis of f(R), the void fraction per unit volume of product layer is (5)
att=O
(11) (12) (13)
(14)
where R, is the radius of the unreacted core, and F, is the shape factor of the pellet ( = l , 2, and 3 for infinite slabs, long cylinders, and spheres, respectively). Now we rewrite (9)-(14) in dimensionless form. First, the expression of (3), which is based on the molar density of pure B, is changed to its dimensionless form, (15); which is based on the average concentration of B over the pellet PB(7) = pBh(7) (15) where 7 RIR, is the dimensionless distance from the center of symmetry of the pellet, h(7)is the distribution function of the solid reactant in terms of the dimensionless
388
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986
distance p , and P B is the volumetric average density of the solid reactant defined as LRPR(Fp-1)pBf(R)
dR for Fp = 1-3
pB I
(16)
d~
LRP~(Fp-l)
We further define a dimensionless parameter 6 as PB
6I-
_PB
(17)
which measures the relative magnitude of the average density of the solid reactant to its molar density. Thus, the relation between f ( R )and h(p) is f ( R )= 6h(d (18) in which the h(p) is subjected to the following two constraints: 1 0 < h(p) < - for 0 S p < 1 (19) 6
Substituting the above-mentioned expressions into (9)-(14), we obtain the following:
pc=l
i.e., the conversion function in terms of 7 for diffusional control (Sohn, 1979 and 1981; Szekely et al., 1976). Equation 26 together with eq 27-30 is the general relationship between the dimensionless reaction time and distance coordinate for the arbitrary distribution function and different geometries; its explicit form is dependent on the form of h(q)and the value of Fp. In terms of the dimensionless variables, the expression of overall extent of conversion is
att=O
Solving (20)-(25), we obtain the final result
where
i.e., the dimensionless reaction time based on the average concentration of the solid reactant,
Comparing the dimensionless governing equations of the model, (20)-(25), with the corresponding equations of Dudukovic's model, (5)-(8) in Dudukovic (1984), the basic differences can be seen. Firstly, throughout the formulation, Dudukovic's model assumed the effective diffusivity of gas species through the reacted layer to be constant. The shrinking-core model is valid for an initially nonporous solid, producing a porous product layer, and thus in our work, it is treated as a function of the position through the distribution function h(q). Secondly, in the boundary condition at the reaction interface, Dudukovic's model assumed the chemical reaction rate per unit cross-sectional area to be constant regardless of the distributions of solid reactant. In our model, the rate is expressed to be proportional to the cross-sectional area of solid reactant, which in turn is proportional to the solid reactant concentration at p = pc. In Dudukovic's model, the effect of nonuniform distribution of the solid reactant was reflected only in the equation of mass balance for unreacted solid reactant. This leads to a very different effect of a nonuniform distribution on the overall reaction rate as will be seen later.
Results and Discussion Because there are too many possible forms of the distribution function to show each of the results individually, we will just discuss bhe results for a general linear distribution function. These results at least can qualitatively reflect the general features and trends of the results of any other monotonic distribution function which cover a large part of the practically encountered distribution functions. The discussion will focus on comparing the results with those for the case of a uniform solid reactant concentration which is identical with the volumetric average of the nonuniform distribution under consideration. We have also obtained the analytical results for a general quadratic distribution function. Some of the results for slabs are shown in Table I. For the cylinders and spheres, the analytical results are too long and cumbersome to present here, and it is thought that in practice, numerical computation might be more convenient. We express the general linear distribution function as f ( R ) = PR + Y (32)
i.e., the shrinking-core modulus (Sohn, 1979 and 1981; Szekely et al., 1976), G(pc) I 1 - 7, (29) i.e., the conversion function in terms of p for chemical reaction control (Sohn, 1979 and 1981; Szekely et al., 1976),
in which p and y are constant coefficients; then, its corresponding dimensionless form is
sF,(pc) E
where
h(p) = (up + (1 - -a)FP
Fp+ 1
PRP C y I -
6
Fp = 1-3
(33)
(34)
and is subject to the following constraints: When a 2 0,
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986
Fp+ 1
,
O I a I -
forOC6I-
(: )
O I a 5 ( F p + l )- - 1
for-
1c
FP
Fp+ 1
(354
56 I 1
(35b)
FP
389
Of
FP
Fp+l
W
and when a 5 0, 0 1 a 1 -(Fp
+ 1)
for 0
1
