Section 10.7 Heat Conduction in a Cooling Fin

Rectangular fin of length L, width W and thickness 2*B.
     |   |___________
     |   /          /|
   TW| W/          / |
     | /          /  /   Ta
     |/__________/  /
     |   2*B     | /
     |___________|/
     |<----L---->|
     | z-->
At top and bottom of the fin: q=h*(T(z)-Ta)
Neglect heat loss at the end and on the sides of the fin.
Assume T in the fin varies only in the z direction
>  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. 10.7-3
If we divide both sides by k*A we can see that it is exactly like 10.7-3
>  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. 10.7-4 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 10.7-8
>  Theta:=zeta->(T(zeta*L)-Ta)/(Tw-Ta); Eq 10.7-6 and Eq. 10.7-7 used to put the equation in dimensionless form
>  Theta1:=zeta->simplify(Theta(zeta),assume=positive);
>  Thbook:=zeta->cosh(N*(1-zeta))/cosh(N); Eq. 9.7-13
>  difference:=simplify(Theta1(zeta)-Thbook(zeta));


>  simplify(convert(difference,exp));

>  Qactual:=simplify(Q(0),assume=positive);
>  Qmax:=simplify(C*L*h*(Tw-Ta),assume=positive);
>  eta1:=simplify(convert(Qactual/Qmax,exp),assume=positive); The definition of effectiveness for the fin.

>  eta:=int(Theta1(zeta),zeta=0...1); The effectiveness of the fin as calculated in the text.

>  etabook:=tanh(N)/N; A simple form for the effectiveness.

>  simplify(convert(eta-etabook,exp));