Example 3.2.6 Diffusion with Second Order Reaction
Example 3.2.1 is solved here using Maple's 'dsolve' command. The boundary condition at the surface, x = 1 is taken as:
Example 3.2.1 is solved below in Maple with a modified boundary condition:
Enter the governing equation:
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Eq:=diff(c(x),x$2)=Phi^2*c(x)^2; |
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(1) |
The value of the parameter is substituted here:
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(2) |
The boundary conditions are entered here:
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BCs:=(D(c)(0),D(c)(1)=100*(1-c(1))); |
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(3) |
The numerical solution is obtained here:
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sol:=dsolve({eq,BCs},{c(x)},numeric); |
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(4) |
The concentration profile obtained is plotted here:
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odeplot(sol,[x,c(x)],0..1,thickness=4,title="Figure Exp. 3.2.10.",axes=boxed,color=gold); |
Next, the problem is solved for a higher value of Φ:
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(5) |
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BCs:=(D(c)(0),D(c)(1)=100*(1-c(1))); |
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(6) |
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sol:=dsolve({eq,BCs},{c(x)},numeric); |
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(7) |
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odeplot(sol,[x,c(x)],0..1,thickness=4,title="Figure Exp. 3.2.11.",axes=boxed,color=brown); |
We observe that as Φ increases, the profile becomes steeper and the time taken to solve the problem also increases.