10.5 Heat Conduction with Chemical Heat Source
Derivation of eq. 10.5-5
>  restart;
>  A:=Pi*R^2;V:=A*dz;  The bed area is constant and the shell volume is just A*dz.
>  eq1:=A*(ez(z)-ez(z+dz))+V*Sc(z); eq. 10.5-1
>  eq2:=z->limit(eq1/dz/A,dz=0);
>  eq2(z); eq. 10.5-2
>  ez:=z->rho*(vz(z)^2/2+Hhat(z)+tauzz(z))*vz(z)+qz(z); eq. 9.8-6 using the fact that vz is the only component of the velocity vector and we need the z component of e
>  eq2(z);
>  tauzz:=-2*mu*D(vz)(z); eq. 1.2-6 for incompressible fluid and i=j
>  tauzz:=unapply(tauzz,z);
>  qz:=-keff*D(T)(z); Fourier's Law (approximate for a packed bed)
>  qz:=unapply(qz,z);
>  Hhat:=z->Cphat*(T(z)-Tref); eq. 9.8-8 for Cp constant and incompressible fluid.
>  eq2(z); eq. 10.5-4
>  vz:=z->vz0;
>  eqgen:=unapply(eq2(z),z,T,Sc); eq. 10.5-5
>  eqI:=eqgen(z,TI,0); Reactor inlet before the reaction starts
>  eqII:=eqgen(z,TII,Sc1*F(Theta)); The reaction zone
>  eqIII:=eqgen(z,TIII,0); After the reaction zone
>  with(PDEtools,dchange);  Used in DEs to change variables. New in Maple Vr5.
>  transf:={z=Z*L,T=To+Theta*(T1-To)};  A set that gives the transformation to get  old  variables from the  new  ones. Note the use of {} to make a set.
>  Newde:=dchange(transf,eqgen(z,T,Sc),[Theta(Z),Z]); Arguments in dchange:
1) the set of transformations
2) the differential equation
3) the new functional relation
>  vz0:=B*keff/(rho*Cphat*L);Sc:=(Z,L)->N*Theta(Z)*rho*vz0*Cphat*(T1-To)/L;
>  simplify(Newde/(T1-To)/keff*L^2,assume=positive);

> 
Theta is (T-To)/(T1-To) and Z is z/L
B is vz0*rho*Cphat*L/keff and N is L/(rho*vz0*Cphat*(T1-To))