Section 9.7 Heat Conduction in a Cooling Fin

Converting the DE to dimensionless form before we solve it.

>  restart;
>  A:=2*B*W; C:=2*W; Q:=z->-k*D(T)(z)*A;
>  eq := limit((Q(z)-Q(z+dz))/(k*A*dz),dz=0)=h*C*(T(z)-Ta)/(k*A); 
>  with(PDEtools,dchange); Now we change to dimensionless form for 
the equation.
>  trans:={z=zeta*L,T=Ta+Theta*(Tw-Ta)};
>  newde:=dchange(trans,eq,[zeta,Theta(zeta)]); Arguments:
1) the set of transformations
2) the differential equation
3) the new functional ralation

>  L:=N*sqrt(k*B/h); Replacing L with N as in eq. 9.7-8
>  de2:=simplify(N^2*k*B*newde/(h*(Tw-Ta))); eq. 9.7-9
>  s:=dsolve({de2,Theta(0)=1,D(Theta)(1)=0},Theta(zeta)); 
Using the BCs eq. 9.7-10 and 11
>  assign(s);Theta:=unapply(Theta(zeta),zeta);
>  simplify(Theta(zeta)-cosh(N*(1-zeta))/cosh(N)); Comparing our 
solution with eq. 9.7-13
>  expand(%,trig);
>  eta:=int(Theta(zeta),zeta=0...1); The effectiveness factor 
defined in eq. 9.7-15
>  eta:=unapply(simplify(convert(%,trig)),N); This agrees with 9.7-16
>  plot(eta(N),N=0.01...2); The effectiveness falls off as N increases

>