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Linearization of the source term of the diffusion equation

Hi guys,

I want to solve a convection-diffusion equation with an absorption source term (Liner Driving Force model and Langmuir-Freundlich isothermals), but I don't know how to linearise this source term. Please see the following equations:

$\frac{\partial c}{\partial t}+u \frac{\partial c}{\partial z} -D_{z}\frac{\partial^{2} c}{\partial z^{2}}+\rho_{p} \frac{\partial q}{\partial t} =0$

$\frac{\partial q}{\partial t} =k^{*}_{LDF} (q^{*}-q)$

$q^{*}=\frac{q_{max} (bcRT)^{\frac{1}{n}}}{1+(bcRT)^{\frac{1}{n}}}$

where c is the variable, and others are constant. I wonder how to linearise this $q^{*}$ . Please give me some advice, and I really appreciate your time and patience.

One typically writes the source term as a linearization around a 0 state (i.e., previous iteration value) $S\left(\phi\right) = S\left(\phi_0\right) + \partial S / \partial \phi|_0 \left(\phi - \phi_0\right)$ , but then the actual implementation depends from the specific method you are using (i.e., solving in delta form or not). See also https://www.cfd-online.com/Wiki/Sour..._linearization.

Hopefully, you can do $\frac{\partial S}{\partial c}$ by yourself, but the possibly problematic point I see here is that the derivative is not defined at c=0 (but the source term is). Honestly, I never dealed with such problem, but a possible way out is to simply saturate the derivative at a minimum value of c (but, again, this might depend from the implemetation).

Thank you for your reply, I will read the web you mentioned.