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Add linearization to Mobile Phase Modulator Langmuir Binding#261

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jbreue16 merged 2 commits intomasterfrom
feature/linearize_mobile_phase_modulator
Aug 6, 2024
Merged

Add linearization to Mobile Phase Modulator Langmuir Binding#261
jbreue16 merged 2 commits intomasterfrom
feature/linearize_mobile_phase_modulator

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@ronald-jaepel
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@ronald-jaepel ronald-jaepel commented Aug 2, 2024

This PR stabilizes the Mobile Phase Modulator Langmuir Binding for $c_{p,0} <= 0$.

The isotherm equation:

$$$ \begin{aligned} \frac{\mathrm{d} q_i}{\mathrm{d} t} = k_{a,i} e^{\gamma_i c_{p,0}} c_{p,i} q_{\text{max},i} \left( 1 - \sum_{j=1}^{N_{\text{comp}} - 1} \frac{q_j}{q_{\text{max},j}} \right) - k_{d,i} c_{p,0}^{\beta_i} q_i && i = 1, \dots, N_{\text{comp}} - 1. \end{aligned} $$$

can be undefined for $c_{p,0} <= 0$ for two reasons:

  1. The flux implementation is not defined for $c_{p,0} < 0$ due to $c_{p,0}^{\beta_i}$ if $\beta_i$ is a fraction with an even denominator.
  2. For $c_{p,0} == 0$ and $\beta_i < 1.0$ the first derivative is also not defined.

Therefore a Taylor series linearization with respect to $c_{p,0}$ has been introduced around $c_{p,0} <= \epsilon$ where $\epsilon$ is a small number.

The CADET-core tests failed with the old formulation for $c_{p,0} <= 0$ and now pass. A full simulation where $c_{p,0}$ dips below zero due to oscillations has been performed to confirm that the linarization works.

Once this is merged we can merge fau-advanced-separations/CADET-Process#163.

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Looks good to me, but I think we should add a separate test case for the linearization

@jbreue16 jbreue16 merged commit 1127925 into master Aug 6, 2024
@jbreue16 jbreue16 deleted the feature/linearize_mobile_phase_modulator branch August 6, 2024 09:28
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