Ind. Eng. Chem. Fundam. 1883, 22, 55-61
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Self-Inhibited Rate in Gas-Solid Noncatalytic Reactions: The “Rotten Apple” Phenomenon and Multiple Reaction Pathways Henry F. Elk’ and M. P. Dudukovle’ Chemlcal Reaction Enginmrlng Laboratory, Washington Universlty, St. Louis, Missouri 63 130
The self-inhibited rate form (bimolecular Langmuk-Hinshelwoodrate) is Incorporated in the diffusion with simultaneous reaction model (DSRM) to describe the progress of a noncatalytlc gas-solid reaction in a spherical solid particle. The model wRh this rate form, solved numerically by orthogonal collocation, provides a plausible explanation for the “rotten apple” phenomenon (internal burning) which occurs when the reaction begins in the interior of the solid particle and progresses outwards. It is shown that the position of the highest reaction rate in the solid particle depends on model parameters. It is also shown that at given gas reactant concentration, solid particle size, and initial soli reactant concentration, muttiple solki conversion histories are possible for a certain range of parameters.
Introduction Gas-solid noncatalytic reactions are frequently used in current chemical processing technologies. The reduction of metal oxides, the oxidation of coked catalyst, and desulfurization in coal combustion are a few samples of important gas-solid reactions. To design and operate gassolid reactors efficiently, all the processes that affect reaction progress must be described and understood. Various popular models for gas-solid reactions include the diffusion with simultaneous reaction model, the shrinking core model (Levenspiel, 1972; Lacey et al., 1965), the two-stage zone model (Ishida and Wen, 1968),and the structural models (Szekely et al., 1976). These models describe what is normally observed experimentally: the conversion of solid reactant to product which starts either on the outside and moves inward or starts everywhere in the particle but preferentially on the outside. In these models reaction rate is assumed to be nth order (n 1 0) with respect to the gaseous reactant and mth order (0 I m I 1)with respect to the solid reactant (Szekely et al., 1976). It has been reported, however, that with some reactions the exact opposite phenomenon occurs. The solid is converted to product with the reaction progressing from the interior and moving outwards. Examples of this apparent anomaly include the disproportionation of potassium benzoate to terephthalate (Gokhale et al., 1975),reactions of certain calcinated dolomites in the desulfurization of fuels (Hubble, 1978), and the so-called “internal burning” of graphite in a carbon monoxide/carbon dioxide environment (Turkdogan et al., 1968). Perhaps this phenomenon, which we call the “rotten apple” phenomenon because of the similarity to an apple rotting from within, has occurred in other systems, but experimenters considered the observation as “bad data”! Some observations of this type remain unpublished (Hubble, 1978). The “rotten apple” phenomenon may be unnoticed if partially reacted pellets are not analyzed for the solid reactant concentration profile and if overall conversion-time curves resemble those predicted by other models. Kulkarni and Doraiswamy (1980) presented a two-step consecutive reaction scheme along with the concept of delayed diffusion to provide an explanation for the appearance of the ”rotten apple” phenomenon in the disproportionation of potassium benzoate to terephthalate. It is not known whether such a specific reaction scheme can be applied to other reactions ‘Monsanto Company,
P.O.Box 8, St. Peters, MO 63376. 0196-4313/83/1022-0OSS~01 .SO10
that display this phenomenon. On the other hand, it is known that certain catalytic reactions, e.g., carbon monoxide oxidation on supported platinum catalyst, display a self-inhibited rate form represented by klCA/(1 + k 2 C ~ )(Cavandish 2 and Oh, 1979; Pereira and Varma, 1978). This rate form can lead to a high reaction rate at low gas concentrations, CA,and a low reaction rate at high gas concentrations. For example, under certain conditions carbon monoxide oxidizes faster in the interior of a uniformly distributed platinum catalyst than at the outer surface of the catalyst since the interior has the lowest concentration of carbon monoxide. If a parallel catalyst deactivation reaction were to occur simultaneously with the carbon monoxide oxidation, it would be possible to deactivate the catalyst from the center outwards, which would be an example of the “rotten apple” phenomenon. Extensive studies have been made of the self-inhibited rate, or bimolecular Langmuir-Hinshelwood rate, particularly with regard to effectiveness factors (Pereira and Varma, 1978; Becker and Wei, 1976; Elnashaie and Mahfouz, 1978). Roberts and Satterfield (1966) first reported the existence of three steady states in the catalyst. Later, Pereira and Varma (1978) and Elnashaie and Mahfouz (1978) independently found five steady states. At least two of the states are stable. The objective of this study is to examine what happens in a solid particle when reaction rate follows the self-inhibited rate form. The diffusion and simultaneous reaction model is selected to describe the phenomena occurring within the particle. The goal is to examine whether this rate form may provide a possible model for the “rotten apple” phenomenon. At the same time, since the model leads to multiple steady states in catalytic reactions, it is of interest to determine whether multiple reaction pathways are possible in gas-solid noncatalytic reactions and how much they affect the conversion-time profile. Model Equations Consider a noncatalytic gas-solid reaction with a general stoichiometry aA(g) + bB(s) = gG(g) + sS(s) (1) The following assumptions are made: (i) The solid particle and the surrounding gas are at constant temperature. (ii) The particle size, shape, pore structure remain constant during the reaction. (iii) The gaseous reactant is dilute and diffusion in the pellet obeys Ficks law with a constant diffusion coefficient. (iv) Concentration profiles in the 0 1983 American Chemical Society
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Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983
pellet are symmetric. (v) Extemal mass transfer resistance is negligible. (vi) The reaction rate of consumption of gaseous reactant A is described by the self-inhibited rate form with pth order solid-reactant concentration dependence, as given by eq 2.
The order of reaction with respect to the solid reactant, p , is a measure of the availability of the solid reactant sites on a microscopic level (subparticle dimensions) during the course of reaction. The zeroth order (p = 0) implies that solid reactant sites a t any point of the pellet are equally accessible to the gas for reaction at all stages of reaction, i.e., at all local conversion levels of the solid reactant, until complete conversion occurs. The order of one (p = 1) implies that with the progress of reaction and with increased local solid conversion, solid reactant sites become less accessible to the gas, thus reducing (slowing down) the local rate of reaction. Mass balances on the gaseous reactant A and solid reactant B yield the equations
~ C =B - -b ~ICACB' -
a (1 + k&A)'
(4)
cumulation parameter, $; the self-inhibition constant, K; and the Thiele modulus, 4. The role of the reaction order, p, has already been discussed. The accumulation parameter, +, determines the effect, if any, of the gas accumulation term in the mass balance equation. For gas-solid reactions this term is generally very small and the pseudo-steady-state assumption (PSSA) can be safely used (Bischoff, 1965; King and Jones, 1979). According to the PSSA, eq 8 becomes
The use of PSSA implies that the time required to establish a steady-state profile of the gas reactant concentration in the particle at any given level of solid conversion is so short that during that time local and global solid conversion do not change; i.e., they change only infinitesimally. The self-inhibition constant, K,measures the importance of self-inhibition. When K 0 the rate becomes first order with respect to gas reactant. When K >> 1 a negative order is observed. The Thiele modulus squared, 4', is the ratio of the particle diffusion time and characteristic reaction time (in the limit of zero self-inhibition constant, K 0). The actual Thiele modulus squared (the kinetic rate at surface conditions divided by the maximum diffusional transport rate) is @/(1 IT)'.
-
-
+
t = 0 ; CA=o
(5b)
t = 0 ; CA=C&
(5c)
Rate Form and Multiplicity of Solutions The "rotten apple" phenomenon occurs when the reaction rate has a maximum somewhere within the interior of the solid pellet. For zeroth reaction order, p = 0, it is known from the previous studies (Matsuura and Kato, 1967) that the rate has a maximum when dimensionless gas concentration is
(6)
y = y m = 1/K
at
The initial and boundary conditions are
t
=o;
CB
= CBo
r = R ; C,=C,
(54
r = 0; aCA/ar = 0 (7) These equations can be cast in dimensionless form using the variables described in the Nomenclature section. The equations for a spherical pellet become
8=0; z = 1
(loa)
o=o; y = o
(lob) 8=0; y = l (10c) 5=1; y = l (11) t = 0; ay/aE = o (12) Solid conversion, which measures the fraction of solid reactant in the particle that has reacted, is given by
X = 1 - 3S1E2z 0 dE
(13)
The time derivative of solid conversion measures the volume averaged rate of reaction and is given by
There are four parameters in the above model: the reaction order (with respect to solid reactant), p ; the ac-
(15)
For all K > 1the maximum rate occurs some place within the particle in the presence of any amount of diffusional resistance, 4 > 0. Some combinations of the degree of self-inhibition,K,and of the ratio of diffusional and kinetic resistance, 4, lead to the maximum rate at the center of the pellet when y c = ym. For some values of parameters one may find y c < ym and the rate has a maximum somewhere between the outer surface and the center. For other values of parameters, y c > ymand the rate decreases monotonically from the center outwards, still having the highest value at the center. For reaction order of one, p = 1, additional complications arise since the solid is depleted faster at points of high rage, which tends to reduce the rate, and a simple analysis does not seem possible. At small times solid concentration is uniform, z = 1, and eq 8a reduces to the form that describes steady-state diffusion with catalytic, self-inhibited reaction rate, e.g., carbon monoxide oxidation on supported platinum. Multiple solutions are known to exist for certain values of 4 and (Becker and Wei, 1976; Pereira and Varma, 1978; Elnashaie and Mahfouz, 1978). Table I summarizes the findings of these investigators on the range of Thiele modulus and self-inhibition constant within which multiple solutions are possible. Three or five multiple solutions have been reported. Of interest in this work are only the two stable solutions (Aris, 1975). These two solutions correspond to the steepest and flattest gaseous reactant concentration profiles or, equivalently, to the highest and lowest value of the effectivenessfactor. Pereira and Varma (1979) have shown that the steepest gas concentration profile is established by starting the reaction with zero gas concentration in the solid (initial condition given by eq
Ind. Eng.
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Table I. Bifurcation Points for Isothermal Bimolecular Langmuir-Hinshelwood Kinetics and Spherical Geometry
Parameter: @
-
self-
inhibition parameter, R Thiele Modulia 20 3 3 < @< 33.6 50 69
