optmal bulk phase composition for a second order reactioin in a catalyst slab

Shorter Communications

Chemical Engineering Science, 197 1, Vol. 26, pp. 991-995. Pergamon Press. Printed in Great Britain.

Optimal bulk phase composition for a second order reaction in a catalyst slab

(First received 28 August; accepted 26 October 1970)

MAY& and Cunningham [ l] have examined the influence of intraparticle diffusion on the reaction

A, + A, bclct Products (1)

in a porous catalyst. The influences of varying ratios of Al and A2 effective diffusivities and of nonstoichiometric bulk phase concentrations of A, and A2 were discussed using effectiveness factor plots. This presentation did not, however, clearly show that there is an optimum ratio of bulk reactant concentrations for each value of the diffusivity ratio, and this fact was not mentioned there. As reactions of type (1) often involve two species of different diffusivities-as in a hydrogenation reaction, for example- this observation is of some practical interest.

Here, sensitivity information obtained by variational methods is employed in a gradient scheme to determine optimum bulk phase concentrations. The results clearly illustrate the strong dependence of the maximum overall rate of reaction and optimal bulk fluid composition on the diffusivity ratio.

MATHEMATICAL MODEL It is assumed that reaction (1) occurs in an isothermal slab

of porous catalyst and that bulk flow contributions to mass transport within the catalyst are negligible. In many cases intraparticle temperature gradients are insignificant com- pared to internal concentration gradients [2]. The mass transport model is justified if Knudsen diffusion prevails within the catalyst or if reaction (1) is equimolar. Under these conditions mass balances on A, and A2 may be written as follows:

DcY-kc,(z)c,(z) = 0, i= 1.2 (2)

with

c,(L) = ciO. i= 1.2 (3.1)

h(O) _ o -_

dz ’ i = 1.2. (3.2)

Assuming now that the total concentration of reactants in the fluid phase (co,) is fixed a priori. the control parameter 0 is introduced to relate cl,, and cZO with c,,?

Cl0 = %I, (4.1)

cXl= (I--B)c,, (4.2)

Since only nonnegative concentrations are allowed,

0<0<1. (5)

The mass balances may be cast in dimensionless form using the following set of dimensionless variables:

XI = c,.lc,, x2 = c,lco, E = z/L. (6.1)

The Thiele modulus h and diffusivity ratio d are defined by

(6.2)

A straightforward manipulation of (2)-(4) gives

@x,(e) -- h*x,(e)x,(e) = 0, de2

(7.1)

d%(e) -- ’ de2 h*x, (c)x,(e) = 0, (7.2)

with

x,(l) = 8. x,(l) = 1-e. (8.1)

br,(o)=O, i=12,

de

All of these equations may be combined into a boundary value problem in a single independent variable by introducing a dimensionless extent of reaction y which satisfies

where

y(1) =o, +$=o.

Then x, and X, satisfying conditions (7) and (8) are given by

X, = e-h*y (11.1)

x 2

= *_e-!?I. cb

(11.2)

Since the overall rate of reaction (1) is equal to A, con- sumption rate, the maximum rate is obtained when

(12)

is a maximum with respect to 8. The optimum 0 for any values of h and do may be easily determined using the method developed in the following section.

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SENSITIVITY ANALYSIS OF REACTIONS IN POROUS CATALYSTS

There are two methods available for computing the derivative of R with respect to 13. One involves direct differ- entiation of mass balance (9) to obtain a linear inhomogeneous differential equation for ay(e,@/af?. The appropriate boundary conditions are determined by dilIerentiating (10) with respect to 0. The desired result is then obtained as

Another approach is to introduce an adjoint variable A and utilize Eq. (9) to write (12) in the form

R =-y+(’ h(e)[w+r(y(e).e)]de (14) 0

where here

r(y,e) = (e-w (1 -e) -71 . [ (15)

A small change in 0 induces correspondingly small changes in y and R which to first order terms are related by

+ar(Y(w 68

ae 1 . de (16) Integrating the first term under the integral by parts and

utilizing boundary conditions (10) gives

6R= [A(l)-l]v+8y(O)y

I * + A(e) aw48) 6tic+ 1 w

0 ae I[ 0 de’

+*(e) a+(e),@) 8Y 1 Sy(e)de.

It follows that if the adjoint h is chosen to satisfy

d’A(e) T+*(e) ar(y(c),e) = o

8Y

and boundary conditions

dA(O) A(1) = 1, 7= 0

then

A(~) ar(Y(E)ye) de

de . (20)

The variational formulation of the sensitivity calculation has considerable advantage over the direct perturbation approach mentioned above. Because the adjoint Eq. (18) is linear and homogeneous, solution of the adjoint problem

(17)

(18)

(19)

can be effected simply by calculating a particular solution A * of Eq. ( 18) subject to the initial conditions


**(l-j) = 1, y=o.

The desired solution to the boundary value problem posed by (18) and (19) is now obtained by scaling. Thus a solution to the adjoint equations, namely

A(E) = A*(e)/A*(l), (22)

is obtained by integrating Eq. (18) once. This can be accom- plished very quickly using the Numerov integration scheme 131 once the steady-state extent of reaction profile has been determined. dR/d6 may then be calculated by a quadrature usina Ea. (20). Such a straightforward procedure is not ava&ble- for the perturbation-formulation which involves an inhomogeneous differential equation.

For the present problem,

(23)

which is nonpositive for nonnegative Al and Al concentrations. The suflicient condition given by Luss and Amundson for existence and utiiqueness[4] is therefore satisfied for both the extent of reaction mass balance and the adjoint problem. This insures a unique value for the derivative dRld0. Computed values of dR/dB were used in a gradient procedure to determine the optimum value of B. Convergence to the extremum of R was accelerated by using a Regula- Falsi procedure after each five gradient steps.

OPTIMIZATION OF BULK FLUID COMPOSITION

Physical intuition suggests that a nonstoichiometric bulk fluid composition may provide larger overall rates for reaction (1) when the diffusion coefficients of A, and As are different.. If, for example, A, diffuses faster than A*, -it is expected that an excess of A, in the bulk fluid may provide a larger rate than stoichiometric bulk concentrations. With an excess of A2 in the bulk phase, a larger concentration driving force for A2 is provided which will counteract to some extent the effect of a relatively low diision coefficient. This tends to equalize Al and A2 concentrations within the catalyst, resulting in a larger local rate of reaction and hence an increase in the overall rate.

Calculated results bear out these conjectures. In Fig. 1 the overall rate of reaction R is shown as a function of 0 for a Thiele modulus h = 3 and a diisivity ratio I#J = 0.1. It is evident that the magnitude of the rate depends strongly on 8. This plot also reveals that the bulk fluid should contain about l-5 as much A2 as A, to maximize the rate in this case.

In order to illustrate further the inlluence of diffusion on the best external conditions, the optimum 0 has been determined over a wide range of Q values for h = 3. The optimum value of 0 associated with each I#J is shown in Fig. 2. The general trend is as expected: As the ratio Dpe/Dle increases, a larger concentration of A, at the exterior catalyst surface is required to compensate for greater mass transfer resistance. Hence, the optimum B increases with 4.

If the size of the catalyst, total reactant concentration %f,

992


R 0.0.

0.c

02 04 0.6 06 I.0

Fig. 1. Dependence of overall reaction rate on fraction of reactant Al in the bulk fluid phase for Dz,/Dl, = 0.1 (h = 3).

and the rate constant are assumed fixed, holding h at a constant value corresponds to keeping D,, constant. Increasing $I then means that D,, is larger. Larger overall rates are expected as Q increases with h ftxed, then, since total mass transfer resistance is reduced. The plot in Fig. 3 of the maximum rate versus Q illustrates this result.

As D,, becomes much larger than D,,, an asymptotic region is expected where further increases in D,, have little effect on the desired boundary concentrations and the corresponding rate. This limiting case can be treated analytically since when #J is very large, Eq. (9) becomes approximately

W+(l-e)(e-h~y.(+=o, (24)

where y- denotes the extent of reaction function for 4 + a. Equation (24) is linear so that the solution to this equation satisfying boundary conditions (10) is easy to obtain. The resulting expression for the rate R, is

R =_dym(l) e- m -=-tanh(hfi). de h (25)

(A similar development is given in [I].) The optimum B for this case may be determined by tinding the root of the equation

(26)

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Fig. 2. Optimal bulk phase composition as a function of ditfusivity ratio (h = 3).

For h = 3 the resulting values of 0, and R, are

(em)optlmum = 0.627, (R,),, = 0.1213. (27)

These asymptotes are shown in Figs. 2 and 3 and agree well with the results of the exact calculation for sufficiently large 4.

The complementary limiting situation I#I + 0 is best treated by changing the dimensionless formulation of the problem somewhat. Small Q with fixed h corresponds to small D,, with fixed D,,. In the limit d + 0, the rate will approach zero. However, by holding D,, constant while increasing D,,, a nontrivial asymptotic case is obtained. The analysis of this situation is essentially complete already because of the symmetry of the problem.

That is, by defining

t3= i-e, (28)

the resulting mass balance in terms of the dimensionless extent of reaction will be identical to Eq. (7) except that the parameters 0, h, and $ will be replaced by 8, i;, and I$ respectively. Identical boundary conditions apply. Hence all results obtained for fixed h carry over directly to the case of fixed i;.

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CESVd.26No.6-P


IO

I

+ 1.

0

0.l

0

i

t

t

I I I I I 002 004 006 003 0.10

RIM

I ) Fig. 4. Optimal bulk fluid phase composition as a function of 014 Thiele modulus for the limiting case DJD,, + =.

Fig. 3. Maximum overall reaction rate as a function of dtisivity ratio (h = 3).

Because of relationship (28), it follows that &,, for fixed h is equal to one minus the optimum 13 for the same fixei h. This symmetry also resolves the case of small 4 since 4 is large when r#~ is small, and the analysis for large I$ has already been completed. I

The dependence of the optimal bulk phase composition on the Thiele modulus h will be explored only for the asymptotic regime I#J + m. In this case (&),,, is easily obtained by solving Eq. (26) for each h value of interest. The h- (&),,,, relationship is shown in Fig. 4. Stoichiometric bulk phase composition is called for when h is sufficiently small, while twice as much A, as A2 should be present in the bulk fluid in the diffusion limited regime. The existence of these two asymptotes is expected on physical grounds. The location of the small h asymptote in Fig. 4 follows from Eqs. (25), (26), and

lim tan h(hm) h

=X4-G. (29) h-r0

By noting that tan h(hV??) approaches unity for s&i- ciently large h, it is easy to see from (26) that the best 8, for large h is independent of h and is equal to 3.

h IC

01 I 1 I I 3 055 0.60 Q6! 07

(% b 066.

DISCUSSION The example case analyzed above shows that nonstoichio-

metric bulk phase composition may result in improved per- formance for a single catalyst pellet. This result suggests several interesting problems in integral reactor design. If a nonstoichiometric feed is introduced to a tubular or batch reactor, the ratio of the two reactants in the bulk phase will change with distance or time. One may then conjecture that the appropriate optimal feed composition must be chosen to compensate partially for this effect in order to maximize integral conversion. Moreover, the best feed composition in this case should depend strongly on residence time-close to the single particle result for a very short residence time and approaching stoichiometric for a sufficiently long residence time. In the latter situation distributed introduction of feed may be of interest.

Rice University Houston, Texas 77001 U.S.A.

JAMES E. BAILEYt

NOTATION A& chemical species

cl,cl concentrations of A, and A, respectively

‘lIVesent address: University of Houston, Houston, Texas 77004, U.S.A.

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CO? Dit?

h

7;

k L

R’ XIX2

Y 2

]ll MAYM6 J. 0. and CUNNINGHAM R. E.,J. Cafaf. 1967 6 186. PI CARBERRY J. J. and WHITE D., Ind. Engng Chem. 1969 6127. [31 FROBERG C. E., Introduction to NumericalAnalysis. Addison-Wesley, Reading, Mass. 1965. t41 LUSS D. and AMUNDSON N. R. Chem. Engng Sci. 1967 22 253.


total reactant concentration in the bulk fluid phase effective diffusion coefficient of species Al

Thiele modulus [=L@j

moditied Thiele modulus (= Lgg

rate constant catalyst half-thickness local reaction rate overall reaction rate dimensionless concentrations extent of reaction distance from catalyst centerline

Greek symbols 0 8

fraction of bulk fluid reactant which is AI fraction of bulk fluid reactant which is As

A adjoint variable A* particular solution to adjoint equations l dimensionless distance (=2/L) I$ ditfusivity ratio (=DpJDIe) #J ditfusivity ratio (=DJD&

Subscripts

0 evaluated in bulk fluid m limitingcaseoff$+ m

REFERENCES

Chemical Engineering Science, 197 1, Vol. 26, pp. 995-997. Pergamon Press. Printed in Great Britain,

A note on the initial motion of a fiuidiition bubble

(Received 16 October 1970)

INTRODUCTION MURRAY [ I] has analysed the initial motion of a cylindrical or spherical bubble in a fluidized bed and has shown that a cylindrical bubble accelerates from rest with an acceleration g, while a spherical bubble has an initial acceleration 2 R. He also studied the distortion of these idealized shaped into the characteristic kidney shape possessed by fluidization bubbles and he was able to demonstrate that the typical shape was achieved after the bubble had moved through a distance of the order of one bubble radius. Partridge and Lyall’s experi- ments[2] were not inconsistent with Murray’s theoretical findings. In this note similar results are produced from a rather simpler analysis.

THEORY 1. Initinl motion

Imagine a bubble of circular cross section constrained to be stationary in a bed of identical particles fluidized incipiently, as shown in Fig. I. In the three-dimensional situation the bubble is spherical. Rowe [3] has previously considered this situation and he showed that if the constraint is removed from the bubble, then, assuming that their spacing remains unaltered, the particles will begin to move relative to an observer fixed in space along streamlines defined by the irrotational dipole. The flow pattern associated with the initial motion is thus the same as would be produced by a solid cylindrical or spherical body moving through a perfect fluid.

The immediate implication of this result is that in the initial motion the bubble retains its form so that the initial

Fig. I. Forces on particles at the poles of a stationary bubble.

acceleration of its centre can be identified with the initial acceleration of any particle which subsequently maintains its position relative to the bubble. There are two such particles located at the poles of the bubble, which are stagnation points in the resulting particle flow relative to the bubble.

Consider the forces experienced by the particles at the stagnation points as shown in Fig. 1. Because there is an influx of fluid into the stationary bubble, the superficial gas

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optmal bulk phase composition for a second order reactioin in a catalyst slab

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