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))