Membrane Transport of Nonelectrolyte Solutions in Concentration Polarization Conditions: Hr Form of the Kedem–Katchalsky–Peusner Equations
In this paper, the Kedem–Katchalsky equations in matrix form for nonhomogeneous ternary nonelectrolyte solutions were applied for interpretation of transport through the membrane mounted in horizontal plane. Coefficients Hijr, Hijr, and Hdetr=detHr (for nonhomogeneous solutions), Hij and Hdet=detH (...
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Wiley
2019-01-01
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| Series: | International Journal of Chemical Engineering |
| Online Access: | http://dx.doi.org/10.1155/2019/5629259 |
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| author | Kornelia M. Batko Andrzej Ślęzak |
| author_facet | Kornelia M. Batko Andrzej Ślęzak |
| author_sort | Kornelia M. Batko |
| collection | DOAJ |
| description | In this paper, the Kedem–Katchalsky equations in matrix form for nonhomogeneous ternary nonelectrolyte solutions were applied for interpretation of transport through the membrane mounted in horizontal plane. Coefficients Hijr, Hijr, and Hdetr=detHr (for nonhomogeneous solutions), Hij and Hdet=detH (for homogeneous solutions) (i, j ∈ {1, 2, 3}, r = A, B), ψij=HijA−HijB/Hij, and ψdet=HdetA−HdetB/Hdet were calculated on the basis of experimentally determined coefficients (Lp, σ1, σ2ω11, ω22, ω21, ω12, ζ1r, and ζ2r) for glucose in aqueous ethanol solutions and two configurations of the membrane system. From the calculations, it results that the values of coefficients H12r, H13r, H22r, H23r, H32r, H33r, and Hdetr depend nonlinearly on solution concentration as well as on a configuration of membrane system. Besides, the values of coefficients H21r, H12, H21, H22, H33r, and Hdet depend linearly on solution concentration. The value of coefficients H13, H23, and H33 do not depend on solution concentration. The coefficients ψ12, ψ13, ψ22 = ψ23, ψ32 = ψ33, and ψdet depend nonlinearly on solution concentration and for C¯1 ≈ 9.24 mol m−3 are equal to zero. For C¯1 < 9.24 mol m−3, the values of coefficients ψ12 and ψ13 are negative and for C¯1 > 9.23 mol m−3, positive. In contrast, the values of coefficients ψ22 = ψ23, ψ32 = ψ33, and ψdet for C¯1 < 9.24 mol m−3 are positive and for C¯1 > 9.24 mol m−3, negative. For ψ = 0, we can observe nonconvective state, in which concentration Rayleigh number reaches the critical value RC = 1691.09, for ψ<0 is convective state with convection directed straight down and for ψ>0 is convective state with convection directed straight up. |
| format | Article |
| id | doaj-art-be61274a018a4444a83a4c3bcea60970 |
| institution | Kabale University |
| issn | 1687-806X 1687-8078 |
| language | English |
| publishDate | 2019-01-01 |
| publisher | Wiley |
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| series | International Journal of Chemical Engineering |
| spelling | doaj-art-be61274a018a4444a83a4c3bcea609702025-02-03T01:25:24ZengWileyInternational Journal of Chemical Engineering1687-806X1687-80782019-01-01201910.1155/2019/56292595629259Membrane Transport of Nonelectrolyte Solutions in Concentration Polarization Conditions: Hr Form of the Kedem–Katchalsky–Peusner EquationsKornelia M. Batko0Andrzej Ślęzak1Department of Business Informatics, University of Economics, B Bogucicka Str., 40287 Katowice, PolandDepartment of Biomedical Processes and Systems, Technical University of Czestochowa, 36b Armii Krajowej Al., 42200 Czestochowa, PolandIn this paper, the Kedem–Katchalsky equations in matrix form for nonhomogeneous ternary nonelectrolyte solutions were applied for interpretation of transport through the membrane mounted in horizontal plane. Coefficients Hijr, Hijr, and Hdetr=detHr (for nonhomogeneous solutions), Hij and Hdet=detH (for homogeneous solutions) (i, j ∈ {1, 2, 3}, r = A, B), ψij=HijA−HijB/Hij, and ψdet=HdetA−HdetB/Hdet were calculated on the basis of experimentally determined coefficients (Lp, σ1, σ2ω11, ω22, ω21, ω12, ζ1r, and ζ2r) for glucose in aqueous ethanol solutions and two configurations of the membrane system. From the calculations, it results that the values of coefficients H12r, H13r, H22r, H23r, H32r, H33r, and Hdetr depend nonlinearly on solution concentration as well as on a configuration of membrane system. Besides, the values of coefficients H21r, H12, H21, H22, H33r, and Hdet depend linearly on solution concentration. The value of coefficients H13, H23, and H33 do not depend on solution concentration. The coefficients ψ12, ψ13, ψ22 = ψ23, ψ32 = ψ33, and ψdet depend nonlinearly on solution concentration and for C¯1 ≈ 9.24 mol m−3 are equal to zero. For C¯1 < 9.24 mol m−3, the values of coefficients ψ12 and ψ13 are negative and for C¯1 > 9.23 mol m−3, positive. In contrast, the values of coefficients ψ22 = ψ23, ψ32 = ψ33, and ψdet for C¯1 < 9.24 mol m−3 are positive and for C¯1 > 9.24 mol m−3, negative. For ψ = 0, we can observe nonconvective state, in which concentration Rayleigh number reaches the critical value RC = 1691.09, for ψ<0 is convective state with convection directed straight down and for ψ>0 is convective state with convection directed straight up.http://dx.doi.org/10.1155/2019/5629259 |
| spellingShingle | Kornelia M. Batko Andrzej Ślęzak Membrane Transport of Nonelectrolyte Solutions in Concentration Polarization Conditions: Hr Form of the Kedem–Katchalsky–Peusner Equations International Journal of Chemical Engineering |
| title | Membrane Transport of Nonelectrolyte Solutions in Concentration Polarization Conditions: Hr Form of the Kedem–Katchalsky–Peusner Equations |
| title_full | Membrane Transport of Nonelectrolyte Solutions in Concentration Polarization Conditions: Hr Form of the Kedem–Katchalsky–Peusner Equations |
| title_fullStr | Membrane Transport of Nonelectrolyte Solutions in Concentration Polarization Conditions: Hr Form of the Kedem–Katchalsky–Peusner Equations |
| title_full_unstemmed | Membrane Transport of Nonelectrolyte Solutions in Concentration Polarization Conditions: Hr Form of the Kedem–Katchalsky–Peusner Equations |
| title_short | Membrane Transport of Nonelectrolyte Solutions in Concentration Polarization Conditions: Hr Form of the Kedem–Katchalsky–Peusner Equations |
| title_sort | membrane transport of nonelectrolyte solutions in concentration polarization conditions hr form of the kedem katchalsky peusner equations |
| url | http://dx.doi.org/10.1155/2019/5629259 |
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