The effect of reversibility on the selectivity of parallel reactions in a porous catalyst

Pergamon

Chemical

Engrneerirq Science, Vol. 51, No. 3, pp. 44-448, 1996 Copyright 0 1996 Ekvier Science Ltd Printed in Great Britain. All rights reserved OOOV2509/96 $15.00 + 0.00

0009-2509(95)00266-9

THE EFFECT OF REVERSIBILITY ON THE SELECTIVITY OF PARALLEL REACTIONS IN A POROUS CATALYST GEORGE

W. ROBERTS

and H. HENRY LAMB

Department of Chemical Engineering, North Carolina State University, Box 7905, Raleigh, NC 27695-7905, U.S.A. (First received 20 March 1995; revised manuscript received and accepted 19 July 1995)

Abstract-The selectivity of two first-order, parallel reactions, one reversible and one irreversible, can be altered substantially by the presence of an intraparticle diffusion resistance. At very large resistances, the relative selectivity approaches an asymptotic limit which depends on the kinetic, transport and thermodynamic parameters of the reaction system. It is possible for the selectivity to the product of the reversible reaction to change sign as the pore diffusion resistance increases, i.e. there can be a net formation of that product at low resistances and a net consumption at high resistances. In some cases, the rate of consumption of the product of the reversible reaction can be so high that the ratio of product concentration to reactant concentration in the bulk fluid moves away from the equilibrium ratio. This behavior contrasts markedly with that of two irreversible, first-order reactions in parallel, where the selectivity is independent of the iniraparticle diffusion resistance.

INTRODUCTION

are several industrially-important reaction networks where a desired product is produced from a reactant by a reversible reaction and an undesired by-product is produced from the same reactant by a parallel reaction, which is essentially irreversible. This type of network can be represented as:

There

A

$ \

B (Scheme A) C

where B is the desired product and C is the undesired by-product. Important examples of this reaction network are as follows: (1) The synthesis of methanol from carbon monoxide and hydrogen accompanied by a parallel methanation reaction to form methane. Methane formation is more significant with the older, highpressure, Zn/Cr catalyst than it is with the low-pressure Cu/ZnO catalyst (Pasquon and Dente, 1962; Satterfield, 1991; Stiles, 1977). However, a low-pressure catalyst that has become contaminated by iron and/or nickel via the decomposition of their respective carbonyls may have substantial methanation activity (Chinchen et al., 1988, Satterf-ield, 1991; Strelzoff, 1970; Supp, 1973); (2) The isomerization of n-paraffins to branched paraffins accompanied by cracking to lower-molecular-weight paraffins (Greenough and Rolfe, 1986; Otten et al., 1994; Rosati, 1986; Weiszmann, 1986);

(3) The reaction of naphtha with steam to form carbon monoxide and hydrogen accompanied by

‘Corresponding

author.

pyrolysis of the naphtha to form “coke” (Bridger and Chinchen, 1970; Rostrup-Nielsen, 1975); (4) The dehydrogenation of paraffins to olefins accompanied by cracking to lower-molecular-weight paraffins (Craig and Spence, 1986; Hutson and McCarthy, 1986). Another important example of the reaction network of Scheme A is the irreversible Fischer-Tropsch reaction taking place in parallel with the reversible water-gas shift reaction. These reactions both occur at substantial rates on iron-based Fischer-Tropsch catalysts (Riiper, 1982). In contrast to the previous examples, the primary hydrocarbon product, the object of the process, is formed via the irreversible reaction. The water-gas shift reaction may or may not be desirable, depending on the H&O ratio of the synthesis gas fed to the process. There is evidence in the literature (Otten et al., 1994; Chinchen et al., 1988; Rostrup-Nielsen, 1975; Stiles, 1977; Skrzypek et al., 1985; Pasquon and Dente, 1962) that the primary, reversible reaction in at least the first three of these examples may operate under conditions where intraparticle diffusion (pore diffusion, internal diffusion) exerts a significant influence on the overall reaction rate. This raises the question of what influence intraparticle diffusion may have on the reaction selectivity. Wheeler (1951, 1955) was the first to explore the effect of pore diffusion on reaction selectivity. However, his analysis of parallel reactions involving a common reactant was qualitative, and was limited to the case where both B and C are produced by irreversible reactions. Wheeler pointed out that intraparticle diffusion will not influence the apparent selectivity of irreversible parallel reactions in an 441

G. W. ROBERTS and H. H. LAMB

442

isothermal catalyst particle unless the orders of the reactions A + B and A + C are different. Several quantitative analyses of the effect of intraparticle heat and mass transport have been carried out for parallel, irreversible reactions. Ostergaard (1964) and McGreavy and Thornton (1970) investigated the behavior of two first-order reactions in a non-isothermal catalyst particle. Pawlowski (1961) performed an asymptotic analysis for the case where the catalyst particle is isothermal, the reaction A + B is first-order in A, and the reaction A -+C has an order in A between l/3 and 5. Finally, Skrzypek et al. (1984), using the dusty gas model of mass transport, carried out a numerical analysis of the effect of pore diffusion on the behavior of two irreversible, firstorder, parallel reactions. Roberts (1972) developed a general asymptotic solution for an isothermal catalyst particle and any arbitrary combination of reaction orders. He also provided exact solutions for three pairs of orders: (0, l), (0,2) and (1,2). For any combination of reaction orders, the reaction selectivity approaches an asymptotic limit as the intraparticle diffusion resistance increases. This limit is given by

WA) -= &,(B/A)

r+l 2q - r + 1’

In eq. (la), q is the order of the higher-order reaction, arbitrarily taken to be A -+ B, and r is the order of the lower-order reaction, A + C. The selectivity of B relative to A, S(B/A), is defined as the ratio of the actual rate of formation of B to the actual rate of disappearance of A. The intrinsic selectivity, &,(B/A), is the selectivity that is observed in the absence of any resistance to intraparticle heat and mass transport. The ratio S(B/A)/&(B/A) is referred to as the relative selectivity. Equation (la) has two important implications: (1) For “normal” values of q and r, the asymptotic value of the relative selectivity is reasonably high. For example, if q = 2 and r = 1, the actual selectivity will never be less than one-half the intrinsic value. (2) When the pore diffusion resistance is high, temperature has no effect on the relative selectivity, unless one or both of the reaction orders are functions of temperature. One purpose of this paper is to show that neither of these inferences is valid if one of the parallel reactions is reversible. The selectivity loss caused by a resistance to pore diffusion can be far greater than suggested by eq. (la) when reversibility is considered. Moreover, the influence of temperature on the relative selectivity can be significant. Wei, in his remarks on the paper by Ostergaard (1964), pointed out that pore diffusion can influence the selectivity of reversible reactions in an isothermal pellet, even when all reactions are of the same order. Wei (1962) further provided a mathematical framework for analyzing the effects of intraparticle diffusion

on complex systems of reversible first-order reactions, although the operational consequences of reversibility were not explored. THEORETICALANALYSIS Consider the reactions of Scheme A taking place at steady state in a porous catalyst particle. All three reactions are assumed to be first-order in their respective reactants. The particle is assumed to be a semi-infinite slab of thickness 2L, the centerline is at x = 0 and the external surfaces that are in contact with the bulk fluid are at k L. The catalyst is assumed to be isothermal and the external resistances to heat and mass transfer are not considered. The effective diffusivities of all three species inside the catalyst are assumed to be constant. This model is a very simplified representation of diffusion and reaction inside a real catalyst particle. It is unlikely that three reactions would all follow firstorder kinetics over a wide range of concentrations and temperature. In addition, there are relatively few situations where the rate of mass transport is equal to the product of the concentration gradient and an effective diffusivity that is independent of concentration and position. More realistic kinetic and transport equations are usually required to provide an accurate description of diffusion/reaction interactions inside porous catalysts. The virtues of the present set of assumptions are the mathematical simplicity that results and the widespread prior use of this model in other pore diffusion problems. The latter is especially important because it helps to isolate the effects caused by reversibility in the reaction network from effects that might arise primarily from more complex kinetic or transport equations. The differential equations and associated boundary conditions that describe the simultaneous diffusion and reaction of A, B and C inside the catalyst particle are: d2C,

DA-=

dx2

d2Cs dx2

Dg-=

d2Cc

DC-=

dx2

k,C., -;

Cg + k&.,

-k&.,+;C,

- k&A

dCi = 0, dx Ci=Ci,s,

x = 0; x=L;

i = A,B,C

i=A,B,C.

In these equations, Ci is the concentration of Species i, Di is the effective diffusivity of Species i, x is the particle coordinate, measured from the centerline of the slab, kl and k3 are the rate constants for the reactions A -+ B and A + C, respectively, and K is the equilibrium constant of the reaction A o B. The subscript “s” denotes a condition at the surface of the catalyst particle, i.e. at x = L.

Effect of reversibility

on the selectivity

of parallel

443

reactions

Evaluating the above derivative from eq. (1) and combining the result with eqs (5) and (6) leads to:

The solutions of these equations are:

CA -= CAd

(1)

-=G3

-

KC,,

-

Similarly,

4: 4: - 4: Gosh(41x/L) (4: - 4:) 441 + Y)- 4: Cosh4t 4: - 422 Cod b#w/L) 4: - Ml + Y) Cosh &

1 (2) (8) and

t

(3) where where

(10)

E = C,sIKG,,. The intrinsic selectivity is rB(cA.s,cd

S&VA) =

- rA(cA.stcd

ci+,s >

k,CA,s- ; =

kl +

ks 1cA.s

-

-

K

1

1-E (11)

=l+K-E

Ce,s

and the actual selectivity is given by

b=$(l+..$ ’

(W

K

W)

7‘5

IC= k3/kl

(4i)

6 = D,/D,.

(4)

In eq. (4f), the plus sign is arbitrarily assigned to al. The effectiveness factor, q, is defined as actual reaction q=.

Intrinsic

rate

reaction

rate

= -

rAtiA??

cd

@)

where - rA is the rate of disappearance of reactant A per unit volume of catalyst particle. The intrinsic rate of disappearance of A is -

rA(cA.st

cd

=

(k,

i-

kd’&

and the actual rate of disappearance

-

rA

=

DA

dCA

t

dx

x=L‘

kx

- 7;;; Cs+

is given by

(6)

S(B/A) = rg/-

rA

=

bhc-

rA(cA.,?

cB,s)l.

(12)

A. Limiting cases 1. One reversible reaction (k3 = 0). If there is no irreversible side reaction, k3 = c = as = 42 = 0 and

eq. (4a) becomes 4, = L (k,/D,) + (kl/KDB). Equation (7) reduces to 9 = tanh +J$i, which is the result of Smith and Amundson (1951) as presented by Satterfield (1970). 2. Both reactions irreversible (K + a). For this situation, c = ai = 4~~= & = 0 and 4, = Ldw. Once again, eq. (7) reduces to rl = tanh4,/4,. Equations (6)-(8) and (lo)-(12) reduce to S(B/A)/S,(B/A) = 1, confirming the reasoning of Wheeler (1951, 1955) and the calculations of Skrzypek et al. (1984). This result is independent of the value of $i, i.e. it does not depend on the magnitude of the resistance to intraparticle diffusion. 3. Fast reversible reaction, slow irreversible reaction (k, 9 k,). If k, is sufficiently small compared to kt, a Maclaurin series expansion of the square root term in eq. (4f) can be truncated after the first-order term to yield

444

G.

W. ROBERTSand H. H. LAMB

shows that the relative selectivity for product B can be negative, i.e. B can be consumed rather than formed, even if the concentrations in the bulk fluid favor the formation of B, i.e. CA,, > Ce,JK(E < 1). This “selectivity reversal” results from the fact that the irreversible conversion of A to C can reduce the conLet the rate constants, the equilibrium constant and the diffusion coefficients be specified. The values of CI~ centration of A to less than (CJK) throughout a large and u2 are therefore fixed and the value of L will portion of the catalyst particle. The asymptotic value of the relative selectivity will be positive when: determine the values of I#J~and &. Consider a case where 4i is large, such that tanh 4i/$i is very small, yet & is still sufficiently small that tanh&/& z 1. C.&s> (1 +&%}. Clearly, for any value of L, the values of kI and k3 can be selected such that this condition is satisfied. For This equation shows that when the resistance to pore this situation, the above equation reduces to diffusion is very high, the concentration of A in the bulk stream must be more than marginally higher (DACA.~ + DBCLA = kJCA,eq. rc z k3C..,,sT = k3 than its equilibrium value, (CB,,/K), in order to have (DA + KDB) 41 a net production of B. (13) The fact that pore diffusion can cause a net conThe quantity (DACAps+ D&B,+)/(DA + KDJ, denoted sumption of Product B under conditions where E < 1 CA,eq,is the concentration of A that would exist inside raises the question of whether the “selectivity reverthe catalyst if the reaction A o B came to equilibrium sal” effect can be strong enough to cause the ratio of throughout the particle, in the absence of the reaction B to A in the bulk fluid to decrease, i.e. to move away A+C. from the equilibrium ratio. Assuming that E < 1, the Equation (13) can be used to calculate approximate ratio C&CA,, will decline if values of the selectivities, i.e. Moreover, & will be negligible compared to & and &, so that eq. (9) becomes:

(2 1

S&/A)

= k&,s+:/&V S.,(B/A)

=

[-

1 -

rA(cA,st

CB,~)]

Se&IA).

(14) (15)

B. Asymptotic analysis

As the characteristic dimension of the catalyst pellet increases to the point that tanh b1 z tanh & g 1, the expressions for q [eq. (7)] and rB [Eq. (8)] approach limiting forms, i.e. f V = (41 + 42)

(16)

(17) f=

C(1 + K)K + &Q

- bm

[(I + K)K - (I’/~)]

’

(18)

The superscript co denotes the limiting value as both +i and & become very large. If a generalized Thiele modulus, 4, is defined as (Froment and Bischoff, 1990): 4 = (41 +

42)/f

S(B/A) < -

(&l,s/cA,s)~

(20)

This criterion requires that the ratio of the consumption rate of B to that of A in the catalyst be greater than the ratio of their concentrations in the bulk fluid. Obviously, the inequality can be satisfied only if S(B/A) < 0. Since the asymptotic selectivity, Sm(B/A) is the lowest selectivity that can result for a given set of parameter values, a comparison of S” with - (C,,/CA,,) will provide some understanding of the circumstances under which the criterion of eq. (20) can be satisfied. Equations (ll), (17) and (18) can be combined to give an expression for the asymptotic selectivity. Sm(A/B) =

(K6 - EKh[l

+ @I)

(K&l + ic) + ,/%

. c21j

- EK6)

For fixed values of K, E and 6, S” goes through a minimum with K. The position of this minimum is

(19)

then eq. (16) becomes q” = l/4. Equation (17) provides important insight into the selectivity of the catalyst. For example, it shows that the asymptotic relative selectivity is a function of E and the various kinetic, thermodynamic and transport parameters. Since these parameters depend on temperature, (S/S,)m also will be a function of temperature. This dependence contrasts with the behavior of irreversible reactions in parallel, where the asymptotic relative selectivity is only a function of the two reaction orders, as shown by eq. (1). Equation (17) also

and the value of S” at the minimum

is

(23) If 6 G 1, Sq, cannot be less than - (C,,/C’A,s) according to eq. (23). However, Szi, can be less than - (C,,/C,+) when 6 > 1 and the value of E is sufficiently high.

Effect of reversibility on the selectivity of parallel reactions

445

SELECl-EDNUMERICALBESULTG

Figures l-4 illustrate some of the more important features of the behavior of the reactions of Scheme A when the intraparticle dilI’usion resistance is significant. The calculations for the first three figures were performed by specifying K, K, 6, and E. The value of 4 was then varied by changing the value of Lm.

0.9 ill z 2 > 0e

0.8 o.7 0.8 0.5 0.4 1 10 Equilibrium Constant,K

Fig. 4. The critical value of E as a function of the equilibrium constant for selected values of the diffusion coefficient ratio.

s iii d

I

-1.50 ’ 0.1

1

10

100

9 Fig. 1. The effect of the generalized Thiele modulus I#Jon relative selectivity for E = 0.75 and 6 = 1; Curve A: K = 1, K = 1; Curve B: K = 10, K = 0.01.

EffecUveness Factor Fig. 2. The relationship between relative selectivity and the effectiveness factor for E = 0.75 and S = 1: Curve A: K = 1, K = 1; Curve B: K = 10, K = 0.01; Curve C: K = 1, K = 0.01.

0

10

20

30

40

50

0 Fig. 3. Comparison of the actual selectivity with the equilibrium and asymptotic selectivities: K = 1, K = 0.01, d = 1, E = 0.75.

In Fig. 1, the concentration of B in the bulk stream is 75% of its equilibrium concentration, i.e. CBS8= 0.75 KC”,, (E = 0.75). Equation (17) shows that the asymptotic relative selectivity decreases as E increases, for fixed kinetic, thermodynamic and transport properties. Therefore, plots for lower values of E would show higher values of S/S, and vice versa for larger values of E. For Curve A, K = 1. This corresponds to a situation where both the reversible and irreversible reactions occur at comparable rates, such as in the earlier example of Fischer-Tropsch hydrocarbon synthesis in parallel with the water-gas shift reaction. For Curve B, K = 0.01, representing the situation where the irreversible “side” reaction is slow compared to the main, reversible reaction. Figure 1 shows that intraparticle diffusion can have a very significant effect on the apparent selectivity of the reaction network when CB,sis close to its equilibrium value. For Curve A, the relative selectivity decreases from 1 at very small values of 4 to an asymptotic value of - 1.1 when 4 is greater than about 5. There is a net formation of B when the resistance to intraparticle diffusion is small, i.e. for 4 less than about 1. However, when this resistance is large, all of the A that reacts goes to C and there is also a significant net conversion of B through A to C. For Curve B, negative values of S(B/A)/&,(B/A) never occur, as the asymptotic value of the relative selectivity is 0.040. Nevertheless, given the low ratio of k3/kl for this curve, it is noteworthy that the selectivity to B is essentially zero when the resistance to pore diffusion is substantial. Figure 2 shows the relative selectivity as a function of the effectiveness factor for the two cases in Fig. 1 (Curves A and B) and for a third case (Curve C) where the rate constant ratio is the same as for Curve B and the equilibrium constant is the same as for Curve A. For Curve A, the value of the relative selectivity does not begin to increase from the asymptotic limit until the effectiveness factor has reached a value of about 0.2. From that point, S/S0 increases rapidly with q. However, S/S0 is only about 0.80 when q 10.90. In contrast, S/S0 for Curves B and C begins to increase at

446

G. W. ROBERTS and H. H. LAMB

0.02, and has reached a value of about 0.90 when

before it is even possible for pore diffusion to cause C&Z,,= to move away from the equilibrium ratio.

q z 0.2. These curves show that the relative selectivity does not track the effectiveness factor directly. In particular, the increase in S/S0 is delayed relative to the increase in q when the value of K is high. Figure 3 shows a comparison of the actual selectivity, S(B/A), with the “equilibrium.” selectivity, S,,(B/A), calculated from eqs (6), (7), (14) and (15). The actual and “equilibrium” selectivities are very close (AS < 0.02) for values of 4 less than about 8, which corresponds to an effectiveness factor of about 0.12. The agreement at low values of 4 is somewhat surprising since one of the assumptions in the derivation of eq. (14) is that cbl is large. However, it can be shown that S,,(B/A) will be close to &(B/A) as 4 + 0, provided that K is small, a condition that was previously imposed in the development of eq. (14). The two selectivities, S and SC,,,diverge as I$ increases beyond about 8 because the assumption that tanh &/& z 1 breaks down. Physically, this corresponds to a situation where Ca 6 CA,sqthroughout a substantial portion of the catalyst particle because A cannot diffuse into the interior rapidly enough to maintain the equilibrium between A and B. Figure 3 suggests a method for approximating S(B/A) that may be useful for situations where KC< 1. The path formed by Sm(B/A), from high values of 9 until S”(B/A) intersects S,,(B/A), and by S,,(B/A) thereafter, provides a lower bound to S(B/A) except at low values of 4. However, when K 4 1, the difference between S,(B/A) and S,,(B/A) is immaterial at low values of I$. Figure 4 defines the conditions under which an intraparticle diffusion resistance can cause the reversible reaction to appear to move away from equilibrium. Let a “critical” value of E, denoted E*, be defined as the value at which Szi” = - (C&C,,,) for specified values of 6 and K. If the value of E for a given system is less than E*, then the asymptotic selectivity, S”, can never be low enough to cause the ratio C,,JC,,s to decrease, i.e. to move away from the equilibrium ratio. If the actual value of E is greater than E*, S” may be low enough to cause C,,/C.+ to decrease, depending on whether the actual value of K is close to K,in. Figure 4 shows the behavior of E* for values of K from 0.1 to 100 and for 6 = 2,4 and 10. These ranges probably are reasonably representative of systems that may be of practical interest. For the simple chemistry of Scheme A, where all three species have the same molecular weight, 6 should be close to 1 in most catalysts. The value of 6 = 10 might be representative of diffusion in a zeolite if there were size and/or shape differences between the various species, such as in the skeletal isomerization of n-paraffins cited earlier. The values of E* in Fig. 4 were calculated from eq. (23) with - Szi” set equal to EK. For a given value of K, the critical value of E* can be quite low if 6 is high. However, for 6 < 2 and K > 0.1, the ratio of B to A in the bulk fluid must be greater than about 90% of the equilibrium value

The selectivity of parallel, first-order reactions in the presence of a resistance to pore diffusion depends rather dramatically on whether one of the reactions is reversible. When both reactions are irreversible, the relative selectivity, S(B/A)/Se(B/A), always is unity, independent of the magnitude of the intraparticle diffusion resistance. However, when one of the reactions is reversible, S/S,, falls below unity when the intrapartitle diffusion resistance becomes significant and continues to decrease to an asymptotic limit as this resistance increases. Depending on the system parameters, the asymptotic limit for S/S, can be much lower than the asymptotic limits for parallel, irreversible reactions of different orders. A very important result of this analysis is the phenomenon of “selectivity reversal”. When there is a significant concentration of the product of the reversible reaction (B) in the bulk fluid, a large pore diffusion resistance can give rise to a net consumption of B under certain circumstances, even when the net concentration driving force in the bulk fluid favors the production of B. In other words, when the value of CB,J is sufficiently high, the relative selectivity can decrease from + 1 at low values of the Thiele modulus, 4, to a negative value as 4 --) co. In fact, a large pore diffusion resistance can cause the selectivity to B to be so negative that the composition of the bulk fluid tends to move away from the equilibrium ratio of B to A, rather than towards it. The fact that intraparticle diffusion can cause the relative selectivity to be well below unity, and even be negative under certain circumstances, may have important implications for the design and operation of industrial reactors. With some reversible reactions, it is common practice to try to achieve a close approach to equilibrium at the reactor outlet, i.e., to convert A to the point that C,,, z CB,JK. If an undesired, irreversible side reaction occurs in parallel with the main, reversible reaction, this strategy can have very negative consequences. A significant net loss of the product of the reversible reaction can occur under some circumstances if there is even a modest resistance to intraparticle diffusion. Moreover, an attempt to achieve a very close approach to equilibrium can have exactly the opposite effect if the pore diffusion resistance is high. The parameter E, a measure of the approach to equilibrium, will increase only when S(B/A) > - (C&C,,,). If this criterion is not satisfied (C&C,,,) will decrease, i.e. move away from the equilibrium ratio. A significant intraparticle diffusion resistance may have the overall effect of causing the reversible reaction to appear to “stall” well before it reaches equilibrium. Finally, for two irreversible parallel reactions with different orders, the asymptotic value of S/S, is a function only of the orders of the two reactions. If these

relatively low values of the effectiveness factor, about

DI!SCIJSSION OF RFSULTS

Effect of reversibility on the selectivity of parallel reactions orders

do not change

with temperature,

the asymp

totic relative selectivity will be independent of temperature. This is not the case when one of the parallel reactions is reversible. For this case, the relative selectivity generally will be strongly temperature dependent because the asymptotic limit depends on the rate constants and on the equilibrium constant. As acknowledged earlier, the present analysis is based on highly simplified kinetic and transport equations. However, as long as one of the parallel reactions is reversible, the new behavioral features that have been revealed should be present in systems that follow different rate equations and require more sophisticated descriptions of mass transport in the porous structure of the catalyst particle. The “selectivity reversal” phenomenon, the potential for the ratio of products to reactants in the bulk fluid to move away from the equilibrium ratio, and the equilibrium selectivity approximation all arise from the basic structure of the reactions of Scheme A. Such features should be present in analyses based on more sophisticated kinetic and transport equations.

k3 K L

-

rA

rll, rc S(X/A) X

Intraparticle diffusion can have a pronounced effect on the apparent selectivity of parallel, first-order reactions when one of the reactions is reversible. This behavior contrasts markedly with that of irreversible parallel reactions. If the concentration of the product of the reversible reaction in the bulk fluid stream is significant, the selectivity to this product can be positive at low intraparticle diffusion resistances and negative at high resistances. The magnitude of the effect depends on the values of the kinetic, thermodynamic and transport properties of the system. In the extreme, the selectivity to the product of the reversible reaction can be so negative that the product/reactant ratio in the bulk fluid moves away from the equilibrium ratio. An approximation to the reaction selectivity in the presence of an intraparticle diffusion resistance can be derived by assuming that the reversible reaction reaches equilibrium over a major portion of the catalyst interior. When the irreversible reaction is slow relative to the reversible reaction, this equilibrium approximation is valid over a reasonable range of diffusion resistances. The path formed by the asymptotic selectivity and the “equilibrium” selectivity appears to provide a lower bound to the actual selectivity, except at very low diffusion resistances.

NOTATION

parameter defined by eq. (4g) m-’ parameter defined by eq. (4h), rnd4 concentration of Species i, moles/m3 fluid diffusion coefficient of Species i, m’/s C&KC_,, [eq. (lo)], dimensionless parameter defined by eq. (18), dimensionless rate constant for A 4 B, m3 fluid/m3 catalyst, s

rate constant for A -+ C, m3 fluid/m3 catalyst, s equilibrium constant for A o B, dimensionless characteristic dimension of catalyst particle (half-width of catalyst slab), m rate of disappearance of reactant A, moles/m3 catalyst, s rate of formation of B and C respectively, moles/m3 catalyst, s selectivity of product X with respect to reactant A, moles X formed/moles A consumed distance from centerline of catalyst particle, m

Greek Letters: Ulr a2

6 Y ‘I

; CONCLUSIONS

447

di

parameters defined by eq. (4f), m- ’ D,/DA [eq. (4j)], dimensionless parameter defined by eq. (4e), dimensionless effectiveness factor, defined by eq. (5), dimensionless k3/kl [(eq. (4i)], dimensionless generalized Thiele modulus, defined by eq. 19, dimensionless moduli defined by eqs (4a)-(4d), dimensionless

Subscripts: eq

min s 0

refers to equilibrium conditions refers to minimum value of S” refers to conditions in bulk stream, i.e. at catalyst surface intrinsic

Superscript m refers to limit as $i and & + co

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G. W. ROBERTSand H. H. LAMB

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