Example 7.3 Mass Transfer in a Spherical Pellet
| > |
| > | with(plots): |
| > | eq:=diff(u(x,t),t)=diff(u(x,t),x$2)+2/x*diff(u(x,t),x); |
| (1) |
| > | IC:=u(x,0)=1; |
| (2) |
| > | bc1:=diff(u(x,t),x)=0; |
| (3) |
| > | bc2:=u(x,t)=0; |
| (4) |
| > | Eq:=subs(u(x,t)=X(x)*T(t),eq): |
| > | Eq:=expand(Eq/X(x)/T(t)): |
| > | Eq_T:=lhs(Eq)=-lambda^2; |
| (5) |
| > | T(t):=rhs(dsolve({Eq_T,T(0)=T0},T(t))); |
| (6) |
| > | Eq_X:=rhs(Eq)=-lambda^2: |
| > | Eq_X:=expand(Eq_X*X(x)); |
| (7) |
| > | dsolve({Eq_X},X(x)); |
| (8) |
The solution for X(x) can be taken as:
| > | X(x):=c[1]*sin(lambda*x)/x+c[2]*cos(lambda*x)/x; |
| (9) |
| > | Bc1:=diff(X(x),x)=0; |
| (10) |
| > | Bc2:=X(x)=0; |
| (11) |
x = 0 cannot be substituted. Hence, the limit at x = 0 is obtained:
| > | Eq_Bc1:=eval(subs(x=0,Bc1)); |
| Error, numeric exception: division by zero |
| > | limit(Bc1,x=0); |
| (12) |
So the constant c2 should be zero.
| > | c[2]:=0; |
| (13) |
| > | Eq_Bc2:=eval(subs(x=1,Bc2)): |
| > | Eq_Eig:=sin(lambda)=0; |
| (14) |
| > | _EnvAllSolutions := true: |
| > | solve(Eq_Eig,lambda); |
| (15) |
| > | U:=eval(X(x)*T(t)): |
| > | Un:=subs(c[1]=A[n]/T0,lambda=lambda[n],U): |
| > | u(x,t):=Sum(Un,n=1..infinity): |
| > | u(x,t):=subs(lambda[n]=n*Pi,u(x,t)); |
| (16) |
| > | eq_An:=eval(subs(t=0,u(x,t)))=rhs(IC); |
| (17) |
The eigenfunction is:
| > | phi[n]:=sin(n*Pi*x)/x; |
| (18) |
The weighting function is:
| > | r(x):=x^2; |
| (19) |
Next, the integrals are calculated to obtain An:
| > | I1:=int(phi[n]^2*r(x),x=0..1): |
| > | IC; |
| (20) |
| > | I2:=int(rhs(IC)*phi[n]*r(x),x=0..1): |
| > | vars:={sin(n*Pi)=0}: |
| > | I1:=subs(vars,I1): |
| > | I2:=subs(vars,I2): |
| > | A[n]:=I2/I1; |
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(21) |
The dimensionless concentration profile is given by:
| > | u(x,t):=eval(u(x,t)): |
| > | u(x,t):=subs(infinity=N,u(x,t)): |
| > | ua:=subs(N=20,u(x,t)): |
| > | uu:=piecewise(t=0,rhs(IC),t>0,ua); |
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(22) |
| > | plot3d(uu,x=1..0,t=0.3..0,axes=boxed,title="Figure Exp. 7.7.",labels=[x,t,"u"],orientation=[60,60]); |
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| > | plot([subs(t=0,uu),subs(t=0.01,uu),subs(t=0.05,uu),subs(t=0.1,uu),subs(t=0.2,uu)],x=0..1,axes=boxed,title="Figure Exp. 7.8.",thickness=5,labels=[x,"u"],legend=["t=0","t=0.01","t=0.05","t=0.1","t=0.2"]); |
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| > |