Extending the finite difference method

We next extend the finite difference method to treat BVPs of greater complexify, with non-Cartesian coordinates and nommiform grids, von Neumann-type boundary conditions, multiple fields, time dependence, and PDFs in more than two spatial dimensions. We do so through the examples in the following sections. 

Chemical reaction and diffusion in a spherical catalyst pellet 

Similarly, if A, is the effective thermal conductivity of the pellet, the temperature profile r(r) is governed by the enthalpy balance 

Neglecting external heat or mass transfer resistance, we have known values of the concentration and temperature at the surface, r = i . At the pellet center, we use the symmetry conditions dcA/dr = dT/dr = 0. Thus, we solve (6.32) and (6.33) subject to the boundary conditions 

This BVP introduces several new issues (1) nonCartesian (spherical) coordinates, (2) more than one coupled PDE, and (3) a BC at r = 0 that specifies the local value of the gradient (a von Neumann-type boundary condition). Also, experience tells us that when internal mass transfer resistance is strong, reaction only occurs within a thin layer near the surface over which the local concenUation of A drops rapidly to zero. Thus, we use a computational 

---