Weisz and Hicks model - Multiphase Reactor Modeling using PyMRM

This exercise considers diffusion and reaction in a spherical particle for first-order kinetics. The reaction rate is temperature-dependent via an Arrhenius dependence.

\begin{align*} \frac{\partial c}{\partial t} &= \frac{1}{r^{2}}\frac{\partial}{\partial r}\left( r^{2}D_{e}\frac{\partial c}{\partial r} \right) + r = \mathrm{div}\left( D_{e}\ \mathrm{grad}(c) \right) + r, \\ r &= - k_{0}\exp\left[ \frac{E_{a}}{R_{G}}\left( \frac{1}{T_{f}} - \frac{1}{T} \right) \right] c, \\ \rho C_{p}\frac{\partial T}{\partial t} &= \frac{1}{r^{2}}\frac{\partial}{\partial r}\left( r^{2}\lambda_{S}\frac{\partial T}{\partial r} \right) + r\ \Delta H_{r} = \mathrm{div}\left( \lambda_{S}\ \mathrm{grad}(T) \right) + r\ \Delta H_{r}. \end{align*}

(1)

Assume Dirichlet boundary conditions for both concentration and temperature on the particle surface.

Questions:

Formulate the model in a dimensionless way by introducing characteristic numbers $\varphi$ , $\beta$ , and $\gamma$ . The Thiele modulus is defined as $\varphi = R\sqrt{\frac{k_{0}}{D_{e}}}$ , $\beta = \frac{c_{f}(-\Delta H_{r})D_{e}}{T_{f}\lambda_{S}}$ (ratio of heat generation to transport), and $\gamma = \frac{E}{R_{G}T_{f}}$ (ratio of activation to thermal energy).
Provide a backward Euler Python implementation of both the diffusion and temperature equations.
Can you reproduce the case of effectiveness, $\eta > 1$ , for a specific choice of $\beta$ , $\gamma$ , and $\varphi$ ?

Hint: Use a small enough time step to avoid stability issues. Based on the profiles you find, explain what causes the efficiency to be larger than 1.

Effectiveness factor as a function of Thiele modulus (Weisz & Hicks, Chem. Eng. Sci. 1962).

Source: Chem. Engin. Sc., 1962, Vol. 17, pp. 265-275