Example 14.5-1 Radiation Shields

>  restart;
>  E1:=e1*sigma*T1^4; Emitted from surface 1.
>  Abs1:=e2*E1; Absorbed on surface 2 from the emitted ray E1.
>  Refl1:=E1-Abs1; Reflected from surface 2.
>  Reabs:=e1*Refl1; Reabsorbed out of the reflected ray.
>  E2:=Refl1-Reabs; Sent on its way back to surface 2.
>  rat:=simplify(E2/E1); Note that this ratio is exactly as given 
in the text: (1-e1)*(1-e2). Each time a new beam gets to surface 2, it
 has been diminished by the factor rat. The same thing works for rays 
from both surfaces.
>  Q12:=A*sigma*e1*e2*(T1^4-T2t4)*sum(rat^i,i=0...infinity); 
Eq. 14.5-6. Note that Maple does the sum for us. I used T2t4 instead 
of T2^4 to simplify the process of eliminating it later. This checks 
with eq. 14.5-8.
>  Q23:=A*sigma*e2*e3*(T2t4-T3^4)*sum(((1-e3)*(1-e2))^i,
i=0....infinity);
>  eq:=Q12=Q23; The net flux to the shield should be zero.
>  T2t4:=solve(eq,T2t4); Here is T2^4.
>  Q12; After eliminating T2^4 from Q12.
>  Q12:=simplify(%); This should check with eq. 14.5-10
>  Q12maybe:=A*sigma*(T1^4-T3^4)/((1/e1)+(1/e2)-1+(1/e2)+
(1/e3)-1); Maybe it does.
>  simplify(Q12-Q12maybe); Looks OK.

>  Q13:=A*sigma*e1*e3*(T1^4-T3^4)*sum(((1-e3)*(1-e1))^i,
i=0...infinity); The flux if the shield were not there.
>  rat:=simplify(Q12/Q13); The ratio of the flux with the shield 
to that without a shield.

>  e1:=e2; e3:=e2; If all the emissivities are the same:
>  rat; This should be 1/2.
>  simplify(%);