Axial Convection with Radial Dispersion - Multiphase Reactor Modeling using PyMRM

Implement species transport in a tube with reaction at the tube wall. The governing equation is:

v \frac{\partial c}{\partial x} = \frac{1}{r} \frac{\partial}{\partial r} \left( r D_{rad} \frac{\partial c}{\partial r} \right) = \mathrm{div}\left( D_{rad} \, \mathrm{grad}(c) \right).

(1)

The inlet profile is uniform. Assume an infinitely fast surface reaction at the tube wall, meaning $c(R) = 0$ . Choose your own parameters or make the equations dimensionless.

Questions:

What is the proper boundary condition at $r = 0$ ?
Use the method of lines: Perform a spatial discretization in the radial direction to obtain a set of ODEs:
$\frac{d\mathbf{c}}{dx} = \ldots$
(2)
Use a standard SciPy ODE solver, solve_ivp, to solve the set of equations.
Compute the flux at the wall as a function of $x$ .
Compute the Sherwood number (based on $\langle c \rangle$ ) as a function of $x$ .

Note that all numerical implementations should be consistent with the provided equations and boundary conditions.