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Particle model: first order reaction

Numerically solve the stationary spherically symmetric problem

1r2∂∂r(r2D∂c∂r)−kc=div(D grad(c))−kc=0\frac{1}{r^{2}}\frac{\partial}{\partial r}\left( r^{2}D\frac{\partial c}{\partial r} \right) - kc = \mathrm{div}\left( D\ \mathrm{grad}(c) \right) - kc = 0
(1)

with the boundary condition c(R)=1c(R) = 1.

Questions:

  1. Perform the spatial discretization (start with a uniform grid) and implement it in Python. Note that the spherical geometry is accounted for by the proper definition of the divergence operator.

  2. Construct, implement, and solve the matrix-vector equation.

  3. Compute the apparent reaction rate from the concentration gradient at the surface of the particle.

  4. Investigate the effectiveness as a function of the Thiele modulus.

  5. Does the result correspond to the analytical solution?

  6. Consider the high Thiele modulus case and improve the solution by using a spatial discretization that is refined near the wall.

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