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));