Example 3.11  Spherical Catalyst Pellet 

Concentration distribution inside a spherical catalyst pellet is governed by the following equation: 

 

Typesetting:-mrow(Typesetting:-mo(                                                                                                                                                                                                                                                                                                                                                                   (3.68) 

Typesetting:-mrow(Typesetting:-mo( 

 

This equation is solved below using Maple's 'dsolve' command: 

 

> restart:
 

> with(plots):
 

> eq:=diff(c(x),x$2)+2/x*diff(c(x),x)-phi^2*c(x);
 

`+`(diff(diff(c(x), x), x), `/`(`*`(2, `*`(diff(c(x), x))), `*`(x)), `-`(`*`(`^`(phi, 2), `*`(c(x))))) (1)
 

> BCs:=D(c)(0)=0,c(1)=1;
 

(D(c))(0) = 0, c(1) = 1 (2)
 

> sol:=dsolve({eq,BCs},c(x));
 

c(x) = `/`(`*`(sinh(`*`(phi, `*`(x)))), `*`(sinh(phi), `*`(x))) (3)
 

Maple is not able to solve this problem directly.  We can solve this problem without specifying the boundary conditions: 

> sol:=dsolve({eq},c(x));
 

{c(x) = `+`(`/`(`*`(_C1, `*`(sinh(`*`(phi, `*`(x))))), `*`(x)), `/`(`*`(_C2, `*`(cosh(`*`(phi, `*`(x))))), `*`(x)))} (4)
 

The solution obtained can be assigned as: 

> Ca:=rhs(sol[1]);
 

`+`(`/`(`*`(_C1, `*`(sinh(`*`(phi, `*`(x))))), `*`(x)), `/`(`*`(_C2, `*`(cosh(`*`(phi, `*`(x))))), `*`(x))) (5)
 

Now if ya has to be finite at x = 0, _C2 should be zero. 

> _C2:=0;
 

0 (6)
 

> Ca:=eval(Ca);
 

`/`(`*`(_C1, `*`(sinh(`*`(phi, `*`(x))))), `*`(x)) (7)
 

Next, the boundary condition at x = 1 is used to solve for _C1. 

> bc2:=subs(x=1,Ca)-1;
 

`+`(`-`(1), `*`(_C1, `*`(sinh(phi)))) (8)
 

> _C1:=solve(bc2,_C1);
 

`/`(1, `*`(sinh(phi))) (9)
 

> Ca:=eval(Ca);
 

`/`(`*`(sinh(`*`(phi, `*`(x)))), `*`(sinh(phi), `*`(x))) (10)
 

A three dimensional plot can be made as: 

> plot3d(Ca,x=0..1,phi=0..10,axes=boxed,orientation=[120,60],title="Figure Exp.3.1.15.",labels=[x,phi,"Ca"]);
 

Plot
 

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