Example 3.11 Spherical Catalyst Pellet

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

(3.68)

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);

| (1)

> | BCs:=D(c)(0)=0,c(1)=1;

| (2)

> | sol:=dsolve({eq,BCs},c(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));

| (4)

The solution obtained can be assigned as:

> | Ca:=rhs(sol[1]);

| (5)

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

> | _C2:=0;

| (6)

> | Ca:=eval(Ca);

| (7)

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

> | bc2:=subs(x=1,Ca)-1;

| (8)

> | _C1:=solve(bc2,_C1);

| (9)

> | Ca:=eval(Ca);

| (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"]);

> |