Theoretical and experimental determination of a boundary contition at a thin membrane for diffusion

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Katarzyna Lewandowska (Medical University of Gdansk)


We present a method of deriving a boundary condition for diffusion at a thin membrane from experimental data. Within this method the Laplace transform of a boundary condition is assumed to be in the form $ $ $ \hat{C}_2 (0^+,p)=\hat{\Phi}(p)\hat{C}_1(0^-,p)\;, $ $ $ where $\hat{\Phi}(p)$ is a function to be determined. Next, we find the Laplace transform of some theoretical function containing $\Phi$, which is a relatively easy to measure experimentally. Then, this function is also determined by means of a numerical calculation of the Laplace transform of the experimental data obtained for normal diffusion of ethanol in water in a system with a nephrophan membrane. Finally, comparing both Laplace transforms mentioned above, we find the function $\Phi$. The derived boundary condition at a membrane contains a term with a Riemann-Liouville fractional time derivative $ $ $ \alpha C_2(0^+,t)+\beta\frac{\partial^{1/2}}{\partial t^{1/2}}C_2(0^+,t)=C_1(0^-,t)\;. $ $ $ Such a form of the boundary condition shows that particles transfer through a thin membrane is a “long-memory process.” The presented method is an example that an important part of the mathematical model of physical processes may be derived directly from experimental data. $\;$ This work was partially supported by the Polish National Science Centre under Grant No. 2014/13/D/ST2/03608. $\;$ [1] T. Kosztołowicz, S. Wąsik, K.D. Lewandowska, Phys. Rev E 96, 010101(R) (2017). [2] T. Kosztołowicz, Int. J. Heat Mass Transf. 111, 1322 (2017). [3] T. Kosztołowicz, J. Chem. Phys. 146, 084114 (2017).

Primary author

Tadeusz Kosztołowicz (Institute of Physics, Jan Kochanowski University in Kielce)


Katarzyna Lewandowska (Medical University of Gdansk) Sławomi Wąsik (Institute of Physics, Jan Kochanowski University in Kielce)

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