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 heat diffusion), 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: Backward Euler is unconditionally stable for linear problems, but here every time step requires a Newton solve of a strongly non-linear system. During ignition, too large a time step can make Newton fail, so check convergence in every step. Also use a grid fine enough to resolve the steep concentration and temperature gradients near the surface. 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