> restart;

> Area:=r->4*Pi*r^2; Area of a sphere

> WAr:=r->Area(r)*NAr(r); WAr is the molar flow of species A across spherical surface. This molar flow is obtained from multiplying the area of A by the molar flux of A. (eqn 17.2-21 along with eqn 17.0-1)

> mass_balance:=WAr(r)-WAr(r+dr); This mass balance is taken for species A over the spherical shell. This balance illustrates the law of conservation of mass. {rate of mass in - rate of mass out +rate of production} (eqn 17.1-1)

> dmass_bal:=limit(mass_balance/(dr*Area(r)),dr=0); take the limit of the mass balance above

> s:=dsolve({dmass_bal,NAr(r1)=NAr1},NAr(r)); Solve with the boundary condition that NAr(r1) = NAr1

> assign(s);NAr:=unapply(NAr(r),r); Help on assign tells us that it assigns s to NAr. Unapply returns a functional operator.

> deqn:=NAr(r)=-c*DAB*D(xA)(r)+xA(r)*NAr(r); eqn 17.0-1 with NBr =0. Equation 17.0-1 is derived from equation 16.2-2 by relating the molar flux to the concentration gradient.

> s:=dsolve({deqn,xA(r1)=xA1},xA(r)); Dsolve solves the ODE deqn

> assign(s);

> xA:=unapply(xA(r),r);

> eqn:=xA(r2)=xA2;

> NAr1:=solve(eqn,NAr1);

> xA1:=1-xB1;xA2:=1-xB2;

The expressions above relate xA and xB at r1 and r2 respectively. Now solve for the molar flow of A across a spherical surface.

> WA_q:=4*Pi*c*DAB*ln(xB2/xB1)/((1/r1)-(1/r2));

The "questionable" expression (WA_q) is above. This looks very similar, perhaps exactly, like equation 17.2-21 given in BSL. But, we must check. Let's check by the taking the difference of the equation we derived, WAr(r1), and the expression above, WA_q.

> difference:=WAr(r1)-WA_q;

> difference:=simplify(difference);

Well, look here. The two expressions are the same!!! Thus, we have solved for expressions of diffusion through a spherical shell that are analogous to eqn 17.2-10 (concentration profile) and eqn 17.2-14 (molar flux).

> assume(xB1>0,xB2>0); from above answer, xB1 and xB2 were automatically assumed to be zero