Section 18.4 Diffusion with Homogeneous Chemical Reaction > restart; The area for transport is constant: call it S as BS&L do. > WA:=z->NAz(z)*S; Mass flow at z > eq:=WA(z)-WA(z+dz)+R(z)*S*dz; Mass balance where R is the rate of production of A by chemical reaction: mols/(time*volume) > R:=z->-kppp*cA(z); the reaction rate. kppp is the reaction rate constant per volume. > deq:=-limit(eq/(S*dz),dz=0)=0; eq. 18.4-2 > NAz:=z->-DAB*D(cA)(z); eq. 18.4-3 for c constant and NBz=0. > -deq; eq. 18.4-4 > with(PDEtools,dchange); > tr:={cA=Gam*CA0,z=zeta*L};kppp:=phi^2/L^2*DAB; > deq:=simplify(deq,assume=positive); > newde:=-simplify(L^2*dchange(tr,deq,[Gam(zeta),zeta])/DAB/CA0); eq.18.4-7
> s:=dsolve({newde,Gam(0)=1,D(Gam)(1)=0},Gam(zeta)): > assign(s); > Gam:=unapply(Gam(zeta),zeta); eq. 18.4-9 > Gbook:=zeta->cosh(phi*(1-zeta))/cosh(phi); > simplify(Gam(zeta)-Gbook(zeta)); > expand(%); The right hand form in eq. 18.4-9 is also correct.
> AvgG:=int(Gam(zeta),zeta=0...1); > simplify(convert(AvgG,trig)); > %-tanh(phi)/phi; Subtracting off 18.4-11 > simplify(%); The average value of cA/cA0 is tanh(phi)/phi.
> cA:=cA0*Gam(z/L):cA:=unapply(cA,z); phi=sqrt(kppp*L^2/DAB) > D(cA)(0); > simplify(NAz(0)); This agrees with eq. 18.4-12
> plot(AvgG,phi=0...2,title="Average value of cA/cA0 vs sqrt(kppp*L^2/DAB)");
> plot(NAz(0)*L/DAB/cA0,phi=0...2,title="L*NAz(0)/(CA0*DAB) vs sqrt(kppp*L^2/DAB)");