Section 9.7 Heat Conduction in a Cooling Fin

p288-289 BSL

>  restart;
>  A:=2*B*W; C:=2*W;  The area for heat flow and the length of the 
fin where it contacts the gas.
It is possible to use Maple to solve the equation directly without 
performing a dimensionless analysis. But as the result is shown, 
it is obvious that the direct solution is very complicated.
Here is the heat conduction rate along the fin:
>  Q:=z->-k*D(T)(z)*A;
Using an energy balance and taking the limit as dz goes to 0.
>  eq := limit((Q(z)-Q(z+dz))/dz,dz=0)=h*C*(T(z)-Ta); 
 Leads to eq. 9.7-3
If we divide both sides by k*A we can see that it is exactly like 9.7-3
>  eq:=eq/(k*A);
>  sol := dsolve({eq,T(0)=Tw,D(T)(L)=0},T(z)):  Solve the 
differential equation with the BCs in eqs. 9.7-4 and 5: 
T at z=0 is Twall and the heat loss at the end of the fin is negligible.
>  assign(sol); T:=unapply(T(z),z);
>  L:=N*sqrt(k*B/h); Eq 9.7-8
>  Theta:=zeta->(T(zeta*L)-Ta)/(Tw-Ta); Eq 9.7-6 and Eq. 9.7-7
>  assume(h>0); assume(k>0); assume(B>0); Maple will need to know 
this.
>  simplify(Theta(zeta));
>  Thbook:=zeta->cosh(N*(1-zeta))/cosh(N); Eq. 9.7-13
>  difference:=simplify(Theta(zeta)-Thbook(zeta));
simplify(convert(difference,exp));