On One Mathematical Model of the Extraction Process of Polydisperse Porous Material

We consider a mathematical model which represents the extraction process of a target component from the polydispersed porous material. The suggested model is demonstrated by the example of a flat solid material with bidispersed pores of different size in the form of a system of channels of macropores with micropores facing their walls. The macropores and the micropores in the material have homogeneous size. We model a case when micropores of the solid material (dispersed medium) are initially filled with an oil (dispersion phase), which is our target component. The macropores are filled in with a pure solvent. In the process of extraction the oil diffuses from the micropore to the macropore, and then from the micropores to the external solvent volume, wherein the ratio of concentrations in the macropore and the micropore is taken in accordance with the linear law of adsorption. The well-posedness of the formulated mathematical model has been justified.

Keywords

processes of the extraction; polydisperse porous materials; target component; density of sources; inverse problem; diffusion equation.

References

Литература:
1. Sabitov K.B., Martem'yanova N.V. An Inverse Problem for an Equation of Elliptic-Hyperbolic Type with a Nonlocal Boundary Condition. Siberian Mathematical Journal, 2012, vol. 53, no. 3, pp. 507-519. DOI: 10.1134/S0037446612020310
2. Kirane M., Malik S.A. Determination of an Unknown Source Term and the Temperature Distribution for the Linear Heat Equation Involving Fractional Derivative in Time. Applied Mathematics and Computation, 2011, vol. 218, no. 1, pp. 163-170. DOI: 10.1016/j.amc.2011.05.084
3. Kozhanov A.I. Parabolic Equations with an Unknown Absorption Coefficient. Doklady Mathematics, 2006, vol. 74, no. 1, pp. 573-576. DOI: 10.1134/S1064562406040272
4. Kostin A.B. Counterexamples in Inverse Problems for Parabolic, Elliptic, and Hyperbolic Equations. Computational Mathematics and Mathematical Physics, 2014, vol. 54, no. 5, pp. 797-810. DOI: 10.1134/S0965542514020092
5. Ashyralyev A., Sarsenbi A. Well-Posedness of a Parabolic Equation with Nonlocal Boundary Condition. Boundary Value Problems, 2015, vol. 2015: 32, 11 p. DOI: 10.1186/s13661-015-0297-5
6. Ashyralyev A., Hanalyev A. Well-Posedness of Nonlocal Parabolic Differential Problems with Dependent Operators. The Scientific World Journal, 2014, vol. 2014, no. 519814. DOI: 10.1155/2014/519814
7. Ashyralyev A., Sharifov Y.A. Counterexamples in Inverse Problems for Parabolic, Elliptic, and Hyperbolic Equations. Advances in Difference Equations, 2013, vol. 2013, no. 173, pp. 797-810.
8. Kal'menov T.Sh., Tokmagambetov N.E. On a Nonlocal Boundary Value Problem for the Multidimensional Heat Equation in a Noncylindrical Domain. Siberian Mathematical Journal, 2013, vol. 54, no. 6, pp. 1023-1028. DOI: 10.1134/S0037446613060086
9. Kalmenov T.Sh., Shaldanbaev A.S. On a Criterion of Solvability of the Inverse Problem of Heat Conduction. Journal of Inverse and Ill-Posed Problems, 2010, vol. 18, no. 5, pp. 471-492. DOI: 10.1515/jiip.2010.022
10. Orazov I., Sadybekov M.A. One Nonlocal Problem of Determination of the Temperature and Density of Heat Sources. Russian Mathematics, 2012, vol. 56, no. 2, pp. 60-64. DOI: 10.3103/S1066369X12020089
11. Orazov I., Sadybekov M.A. On a Class of Problems of Determining the Temperature and Density of Heat Sources Given Initial and Final Temperature. Siberian Mathematical Journal, 2012, vol. 53, no. 1, pp. 146-151. DOI: 10.1134/S0037446612010120
12. Orazov I., Makhatova A. An Inverse Problem of Mathematical Modeling of the Extraction Process of Polydisperse Porous Materials. AIP Conference Proceedings, 2014, vol. 1611, pp. 225-230. DOI: 10.1063/1.4893837
13. Orazov I., Sadybekov M.A. One-Dimensional Diffusion Problem with Not Strengthened Regular Boundary Conditions. AIP Conference Proceedings, 2015, vol. 1690, pp. 040007. DOI: 10.1063/1.4936714
14. Pyatkov S.G., Shergin S.N. On Some Mathematical Models of Filtration Theory. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 2, pp. 105-116.
15. Pyatkov S.G., Safonov E.I. Some Inverse Problems for Convection-Diffusion Equations. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 4, pp. 36-50.
16. Pyatkov S.G., Borichevskaya A.G. Some Inverse Problems for Mathematical Models of Heat and Mass Transfer. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2013, vol. 6, no. 4, pp. 63-72. (in Russian)
17. Mihailov V.P. On Riesz Bases in L_{2}(0,1). Doklady Akademii Nauk SSSR, 1962, vol. 144, no. 5, pp. 981-984. (in Russian)
18. Kesel'man G.M. On the Unconditional Convergence of Eigenfunction Expansions of Certain Differential Operators Izvestiya VUZ. Matematika, 1964, no. 2, pp. 82-93. (in Russian)
19. Dunford N., Schwartz J.T. Linear Operators, Part III. New York, John Wiley & Sons, 1971.