Linearization of the source term of the diffusion equation

M.L.QU · October 27, 2022, 02:43

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.

sbaffini · October 28, 2022, 05:22

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).

M.L.QU · October 31, 2022, 09:27

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

October 27, 2022, 02:43	Linearization of the source term of the diffusion equation	#1
M.L.QU New Member Zhejiang Join Date: Oct 2022 Posts: 2 Rep Power: 0	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.

October 28, 2022, 05:22		#2
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,120 Blog Entries: 29 Rep Power: 38	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). M.L.QU likes this.

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October 31, 2022, 09:27		#3
M.L.QU New Member Zhejiang Join Date: Oct 2022 Posts: 2 Rep Power: 0	Thank you for your reply, I will read the web you mentioned.