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
[Maple Plot]

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