Section 18.7 Diffusion and Chemical Reaction inside a porous Catalyst: 
"The Effectiveness Factor"
>  restart;
>  A:=r->4*Pi*r^2;
>  WA:=r->A(r)*NAr(r);
>  eq:=WA(r)-WA(r+dr)+RA(r)*A(r)*dr;
>  de:=limit(eq/(A(r)*dr),dr=0); Compare with eq. 18.7-3
>  NAr:=r->-DA*D(cA)(r); eq. 18.7-4: note we must assume constant c and neglect convection.
>  de;  Compare to 18.7-5
>  RA:=r->-kpp*a*cA(r); Volumetric rate of production of cA for 1st order rxn. a is the surface area in the catalyst per volume of catalyst.
>  s:=dsolve({de,cA(R)=cAs,cA(0)=finite},cA(r));
>  assign(s);
>  cA:=unapply(cA(r),r); A little complicated, but we can deal with it.
>  cA:=simplify(cA(r),assume=positive); This agrees with eq. 18.7-9

>  cA:=unapply(cA,r);
>  WAs:=WA(R); Eq. 18.7-10
>  combine(simplify(%,assume=positive)); Oh well, it was worth a try.
>  rt:=sqrt(kpp*a/DA);dif:=WAs-4*Pi*R*DA*cAs*(1-rt*R*coth(rt*R)): Subtracting off 18.7-11
>  simplify(dif,assume=positive); 18.7-11 is OK.

>  VP:=4*Pi*R^3/3;SP:=4*Pi*R^2; volume and surface area of the pellet
>  WA0:=VP*a*(-kpp*cAs);
>  etaA:=WAs/WA0;
>  kpp:=(Lambda*SP/VP)^2*DA/a; Lambda=sqrt(kpp*a/DA)*VP/SP
>  etaA;
>  etaA:=simplify(%,assume=positive);

>  simplify(etaA-(3*Lambda*coth(3*Lambda)-1)/(3*Lambda^2));

>  plot(etaA,Lambda=0.4....5,title="Effectiveness vs sqrt(kpp*a/DA)*VP/SP");