The isothermal selectivity of porous catalysts: competitive—consecutive reactions

Chmicol

Engineering

Science. 1975. Voi

30. pp. M745l.

Pergamon Press.

Printed in Great Britain

THE ISOTHERMAL SELECTIVITY OF POROUS CATALYSTS: COMPETITIVE-CONSECUTIVE REACTIONS H.-P. WIRGES and W. RAHSE Institut fiir Technische Chemie, Technische Universitlt Berlin, Berlin, Germany (Received 23 July 1974;accepted 9 October 1974) Abstract-The influence of mass transfer on the selectivity of three reaction types has been studied for an isothermal catalyst particle: Type 1is a consecutive reaction, Types 2 and 3 are competitive-consecutive reactions. Type 1and Type 2 are treated analytically for the case that the diffusivities are not equal. Equations and diagrams are presented that show the importance of different parameters to maximize the concentration of an intermediate in a plug-flow reactor. Type 3 has been dealt with on an analog computer. It can be proved that the selectivity loss for Type 3 is less than for Type 1.

Type 3 is a more complicated competitive-consecutive reaction:

INTRODlJCTlON

The influence of intraparticle mass transfer on the consecutive reaction AA&-%S

S

or

a

Type 3

Type 1

A’

of

A+B’I,R B+Rt’-S

has been extensively studied by Wheeler[ 11 who developed some equations for the selectivity of two first-order reactions in an isothermal pellet. Butt[2] extended the investigations by dealing with nonisothermal particles, yet McGreavy and Cresswell [3] showed that the assumption of negligible temperature gradients within the pellet is often justified and nonisothermal behaviour is primarily to be found across the gas film and not within the particle. Carberry [4] amplified Wheeler’s isothermal treatment by assuming the macromicropore model for the pellet and Butt [5] studied the selectivity in variable diffusivity systems. A sequence of n consecutive first-order reactions has been treated by (dstergaard[6] who gave analytical expressions for the influence of intraparticle diffusion on the reaction kinetics. The research of Type 1 was extended to zero and second-order reactions by van de Vusse[7] and to kinetical equations which follow the Langmuir-Hinshelwood types of rate expressions[8]. Weisz and Swegler [9], further Komiyama and Inoue [lo] published experimental results and took the theory proposed by Wheeler as a basis. The purpose of this paper is to investigate two additional reaction types in the isothermal case, Types 2 and 3, and to compare the different types with one another. Type 2 represents a reaction with consecutive and parallel steps:

A-p

case

Type 2 TT

This reaction type can often be found in the practice of chemical engineering. The problem and its solution: Types 1 and 2

A lot of hydrogenation, halogenation and partial oxidation reactions conform to these two schemes and can be assumed to be pseudo-first order if one reactant is used in large excess. As an example of Type 2 the gas phase air oxidation of o-xylene into phthalic anhydride on V205catalysts may be considered where A represents the key component o-xylene, R phthalic anhydride and S the oxidation products COZ and Hz0[12]. The differential equations for the simultaneous diffusion and reaction of A and R are, if no volume change occurs:

d’cn DRY=

kzcn - k,c.+

(2)

The analytical solution of this system of differential equations depends on the ratio of the effective diffusion coefficients q = DR/Dn. This ratio is nearly independent of pressure and temperature. For one reaction system q has a fixed value, which in a lot of cases is not far from one. The omission of the third reaction step, k3 = 0, changes Type 2 into Type 1. Aris [13] proved that the treatment of (1) and (2) may be rendered independent of particle shape by using the proper measure of length for the pellet. The analysis presented below assumes the slab-model and the boundary conditions

Roberts [ 111only dealt with the parallel steps of Type 2 and gave equations which allow to calculate the effect of pore diffusion on selectivity for three chosen pairs of reaction orders. 647

x=0 x=L

CA= CA,

dcA,o dx

CR= CRs dCR z=O.

(3) (4)

H.-P. WIRGESand W. RAISE

648

It follows that the ratio of the concentration gradients of A and R at the pore mouth is given by

_ --- dens dca,

_ (dcn/dxLo

_

(dc,,/dxh

Setting

S,

qSw - 1

yields: tanh h tanh(h4(qSwN

(‘)

where h is a specially defined Thiele modulus and S1 and SWare the selectivity factors: slv = s13

for

k,iO

for

ks = 0.

Equation (13) states a criterion for the appearance of a maximum in the concentration of R:

and SW=

s,

1* (12) *<

(6)

CAE

As it is interesting to calculate the rates of reaction of A and production of R, further to know the composition of the reaction mixture in an integral reactor, the model of a steady state plug-flow-reactor will be employed:

St x4qSv)-cp

q&-l

(p

In order to obtain a better impression of the intluence of the various parameters, Fig. 1 shows

(7) (8)

for different values of S, and SWon the condition that cp= 1, that is to say, strong resistance to pore diffusion and c& = 0. It is to be seen that the maximum of

The reactor coordinate is represented by z and the velocity in the flow reactor by u. Division of Eq. (7) by Eq. (8) yields (dcn/dX)x=o dCRS dca, = ’ (dc,Jdx),-0.

(9)

A comparison of Eq. (5) and Eq. (9) shows that the pore and the reactor give different results for the ratio of the concentration gradients. The following considerations apply to the reactor terminology because it is more important to evaluate the changes of cAs and cRswith the position in the reactor and to realize which are the optimum conditions of finding the maximum concentration of the intermediate R, being desired. The ratio of the gradients of R and A can be obtained from Eq. (5) and substituted into Eq. (9). Integrating Eq. (9) with the boundary condition CA,= CA,

for

cn, =

CRB

vs q is situated near q = 1; so in a lot of cases, where q approaches one, the maximum concentration of the intermediate is obtained. This statement can be maintained even for cp9 1, though the curves have a different structure in this case. Moreover Fig. 1 clearly demonstrates the influence of the selectivity factors: if S, equals SW,Type 2 turns into Type 1; if S, decreases and SW remains constant,

(10)

the following result is received:

with tanh h ’ = tanh hV/(qS,)

I

and qsIW(qSw)- cpl I= l~(qSlV) - qrpl(qSw- 1)’ The best operating conditions of competitive-consecutive of R.

reactions are verified at the maximum concentration

O,l

0,s

l,o

5

10

Fig. 1. Maximum yield of R, (cRS/cAJm., vs the ratio of dilkivities, q, for cp= 1 and cne = 0.

Theisothermalselectivity of porouscatalysts will diminish, because the parallel step of Type 2 now gains more importance. When these findings are taken into account, it is sufficient to refer to the case q = 1. Figure 2 illustrates the effect of various initial concentrations of R in the feed for cp= 1, q = 1 and constant selectivity factors. The basis of this plot is Eq. (11) that gives a relationship between the yield of R

OoI

0.02

649

0.05

0)

42

2 0,s

1,o

5

10

I

h

FE3 Fig.4. Maximumyieldof R, (c~~/c~~)~~, vs the Thielemodulush forq=landcn,=O.

and conversion of A

(

l-2.

I?>

If the initial concentration cREis increased, the maximum yield of R decreases, so that it will be best to set CR,= 0. Figure 3 also relates to Eq. (11) and indicates that this equation applies to the general case in so far as it can be used for systems with small and large values of the Thiele modulus h. In the case of strong resistance to pore diffusion (h > 3; h . v(qS,) > 3) cpequals one and gives rise to a considerable yield loss of the intermediate R in

comparison with the case of negligible pore diffusion (h-CO.1; hV(qS,)
(14) 0

1-3

0;s

It follows from Eqs. (7) and (8) for

‘k

Fig.2. Yield of R, (cRs- C&L~ vs conversion of A, l(c&~,), for cp= 1and q = 1. 7% = kzr0p,=

h+q V(qS,) tanh (hV/(q&N- q tad h .,,m 9’Q.Y

(15)

where

Fig. 3. Yield of R, (c,&,,), vs conversion of A, forq=landcRe=O.

1-

(c.&&

It will be best to control the reaction in such a way that the maximum of R is reached at the end LX of the plug flow reactor. Figure 5 shows for q = 1 and c& = 0 that the dimensionless residence time T&, increases with increas-

H.-P. WIRGESand W.

650

411 0.05

RAHSE

Figs. 6(a, b). Yield of R, (cns/cee), vs conversion of B, 1- (cB,/ce,), for systems with small Thiele moduli (Fig. 6a) and with high Thiele moduli (Fig. 6b); cnEis assumed to be zero.

0,l

h

Fig.5. Optimumresidence time & as a function of the Thiele modulus h.

ing Thiele modulus h. High values of SWgive rise to a low value of &. The problem and its solution: Type 3

Several heterogeneous gas phase chlorination reactions are reported to be second order with the rates proportional to the concentrations of both the chlorine and the hydrocarbon. Some hydrogenation and oxidation reactions are good examples of Type 3, if the adsorption constants appearing in the Langmuir-Hinshelwood expressions have small values. So it is assumed for the treatment of this problem that the partial reaction order with regard to A, B and R equals one. The differential equations describing the diffusion-reaction-problem for systems with small and large values of Thiele moduli cannot be treated analytically in the general case. The differential equation system was solved on an analog computer for the region of strong pore diffusion, and for negligible volume change. It is known from chemical kinetics applicable to the case of very small Thiele moduli that a certain minimum concentration ratio (cBE/cAE),+at the reactor entrance must be exceeded to guarantee the maximum in the yield of R [14,15]. If the concentration of R in the feed equals zero, these observations also prove true for porous catalysts in so far as the concentration ratio (c&c& plays an important role and prevails against the influence of the selectivity factor. The decrease of (cB,/c,,) reduces the difference between a nonporous catalyst treated by chemical kinetics only (Fig. 6a) and a porous catalyst which can be described by the slab model including strong resistance to pore diffusion and reaction (Fig. 6b). The Thiele moduli hi and hi are given values high enough so that the curves do not change if hl and hi are increased. The diffusivities are assumed to be equal. Provided that there is some R in the feed, (c,,/cB,) = 0.5, Figures 7a and 7b display that a maximum in the concentration of the intermediate only occurs, if the selectivity factor is greater than one. The Figs. 6a, 6b, 7a and 7b give plots of

cn, %

vs

( > 1-z

E

,

Figs. 7(a, b). (cRS/cBE)vs conversion of B, 1-(cBs/cBE), for systems with small Thiele moduli (Fig. 7a) and with high Thiele moduli (Fig. 7b).

that can be changed into plots of vs

by multiplication with the concentration ratio (cB,/c,,). If a plot of vs is desired and a system with small Thiele moduli is considered, there is only one curve for the different values of (cB~/c~~). This finding can easily be understood by writing down the first-order differential equations for chemical kinetics with regard to A and R and dividing them by each other. A system with high values of the Thiele moduli is to be described by differential equations of the second order which must be dealt with in a more complicated way. Thus a plot of CRs C&

vs

(l-2 > E

gives a different curve for each value of (C&C,+). Figure 8 compares the reaction types presented here and illustrates the relationship between the selectivity loss a and the selectivity factor SI, referring to Types 1 and 3 or SW,referring to Type 2. The value of (r indicates the ratio

651

The isothermal selectivity of porous catalysts a

0,7. 96.

ki q=l Q =l

reaction-rate constant

VOl”_’ for n th’ Moles”-’ . time

order

cue c’2E

L length of the pore, length LR length of the reactor, length

O>4.

4=

2

43.

A

ratio of diffusivities of R and A,

dimensionless s, =

O? 91.

s,, =

01 O,l Sl

45

1

5

It1

,sw

u=

k, I;; k, + kj

selectivity factors, dimensionless

-I kz

2

velocity, length/time

J

UL superficial velocity, length/time

Fig.8. Selectivityloss, (I,as a function of the selectivityfactor S, or SWfor Types 1-3. and displays how the maximum value (c~Jc,+)~~~ is intluenced by pore diffusion. The upper curve is evaluated on the basis of Eq. (12) for C,Q= 0 and q = 1 and relates to Types 1 and 2. The lower curve is obtained from analog computations and is valid for Type 3. Both curves show that the maximum selectivity loss results if SI or SWequals one. It is evident that high values of (c&c,,) turn the reaction Type 3 to Type 1. Very low values of (c,,/c,,) considerably reduce the selectivity loss (Y.In any case it is found that a increases if the reaction order diminishes, in agreement with van de Vusse’s results[7]. Acknowledgement-The authors wish to express their appreciation of helpful discussions with Professor Dr. Peter Hugo. NOTATION CA, CB,CR

D Da, 4p

concentration of A, I3 and R, moles/vol. diffusivity of any substance, if assumed equal, area/time diffusivity of A and R, area/time

h,= h,s =

LJ(F)

h=

h dcss,)

hi=

LJ(+s)

h;=

hl

VW

=&

* Thiele moduli, dimensionless

1

X Z

length coordinate in the pore, length length coordinate of the reactor, length

Greek symbols a

selectivity loss, dimensionless e porosity, dimensionless cp,=y,I’ abbreviations, see Eqs. (11) and (15), dimensionless roPl optimum residence time, time r&r optimum residence time, dimensionless, see Eq. (15)

Subscripts S surface

E entrance of plug flow reactor REFERENCES

[l] Wheeler A., Adu. Carol. 1951III 249. [2] Butt J. B., Chem. Engng Sci. 1!%621 275. [3] McGreavy C. and Cresswell D. L., Chem. EngngSci 196924 608. [4] Carberry J. J., Chem. Engng Sci. 196217 675. [5] Butt J. B., Can. J. Chem. Engng 196442 211. [6] Ostergaard K., Acta Chem. Stand. l%l I5 2037. [7j van de Vusse J. G., Chem. Engng Sci. 196621 645. [t?] Komiyarna H. and Inoue H., J. Chem. Engng Japan 19703 117. [9] Weisz P. B. and Swegler E. W., J. Phys. Cbem. 195559 823. [lo] Komiyama H. and Inoue H., J. Chem. Engng Japan 19681 142. [ll] Roberts G. W., Chem. Engng Sci. 197227 1409. [12] Froment G. F., Ind. Engng Chem. 196759 18. [13] Ark R., Cfiem. Engng Sci. 19576 262. [I41 Kerber R. andGestrich W., Chem. Ing. Techn. 196638536. [IS] van de Vusse J. G., Chem. Engng Sci. 1%6 21 1239.