Porous Catalyst

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Prediction of the diffusion and reaction in a porous catalyst pellet is another important problem in chemical engineering. The reaction under consideration is A-> B which occurs inside the pellet. Mass balance and conservation of energy on the spherical pellet give:

- (6)
+

where is the concentration of A, is the effective diffusivity, the reaction rate expression as a function of concentration and temperature , the effective thermal conductivity and the heat of the reaction. Due to radial symmetry about the center of the pellet:

, , at . (7)

On the surface of the pellet the quantities are at their bulk values:

, , at . (8)

For a simple first-order irreversible reaction

;

the relationship between the reactant concentration and temperature at any point in the catalyst is given by [7]:

= = ( ) .

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Introducing dimensionless variables

, , ,

the reaction rate can be expressed as

.

Substituting these into (6) and the boundary conditions (7) and (8) leads to the two-point BVP ([7])

+ = ,

at

where and .

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When expressed as a first-order system, this problem is easily solved by shoot for specific values of , , and . For example:

> restart;

> with(Shoot):

Warning, the protected names norm and trace have been redefined and unprotected

Warning, the assigned name shoot now has a global binding

Note that there is a singularity at the initial point, . We appear to have been fortunate that Maple V, Releases 3 and 4, handled this potential problem without any adverse effects. In Release 5 and Maple 6, Maple detects this problem and requires that the user indicate how to handle this problem and not count on our good fortune. In this case, the ``solution'' to this problem is to move the initial point to a point with >0.

> ODE:={ diff(y(x),x) = z(x),

> diff(z(x),x) = -2/x*z(x)

> + phi^2*y(x)

> *exp(gamma*beta*(1-y(x))

> /(1+beta*(1-y(x))) ) }:

> FNS:={ y(x), z(x) }:

> IC:={ y(x0)=alpha, z(x0)=0 }:

> BC:=y(1)=1:

> COEF:=[ gamma=30, beta=2/5, phi=3/10 ]:

> x0 := 0.0001:

> 'IC' = IC;

> S1:=shoot( subs(COEF,ODE), IC, BC, FNS,

> alpha=1, value=array([0,x0,1]) );

This answer agrees with the one reported in the original paper to eight significant digits.

The effectiveness factor, , for the reaction is the ratio of the average reaction rate with diffusion to the average reaction rate when the reaction rate is evaluated at the bulkstream (or boundary) values ([3, pp. 58--62], [8, p. 83]). Here is one way in which this quantity can be computed from the results returned by shoot :

> eta:=3/phi^2*D(c)(1) = subs(COEF,3/phi^2*dcdr1);

> dcdr1:=S1[2,1][3,3];

> 'eta'=rhs(eta);

As is well-known [8, pp. 87-88], for a given value of , the effectiveness factor can be multiple-valued. The other values can be obtained by starting the shooting method with different values of the control parameter:

> Sh:=shoot( subs(COEF,ODE), IC, BC, FNS,

> alpha=1/2, value=array([0,x0,1]) );

> dcdr1:=Sh[2,1][3,3]:

> 'eta'=rhs(eta);

> S0:=shoot( subs(COEF,ODE), IC, BC, FNS,

> alpha=0, value=array([0,x0,1]) );

> dcdr1:=S0[2,1][3,3]:

> 'eta'=rhs(eta);

The effectiveness factor has been computed for a wide range of values for , , and . A sample of the results, for , are displayed in the following plot. The four datasets displayed in this plot correspond to (box -- top curve), (dotted line -- second curve from top), (solid line -- second curve from bottom), and (dashed line -- bottom curve). The multiple values for the exothermic catalyst ( ) were obtained using different initial guesses for the control parameter ( , , and ).

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Plot not feasible to include here. Please refer to the original journal article, or contact the first author for a copy of this plot.

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