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Network Hybrid Form of the Kedem–Katchalsky Equations for Non-homogenous Binary Non-electrolyte Solutions: Evaluation of $$P_{ij}^{*}$$Pij∗ Peusner’s Tensor Coefficients

Kornelia M. Batko,I. Ślęzak-Prochazka,A. Ślęzak

Published 2015 in Transport in Porous Media

ABSTRACT

Methods of Peusner’s network of thermodynamics enable the symmetric or hybrid transformation of classic Kedem–Katchalsky (K–K) equations into a network form. In the case of binary non-electrolyte solutions (homogenous and non-homogenous ones), two symmetric and two hybrid forms of the K–K equations may be obtained, containing relatively symmetric ($$R_{ij}^{*},\,R_{ij},\,L_{ij}^{*}$$Rij∗,Rij,Lij∗ or $$L_{ij}),$$Lij), or hybrid ($$P_{ij}^{*},\,P_{ij},\,H_{ij}^{*}$$Pij∗,Pij,Hij∗ or $$H_{ij})$$Hij) Peusner’s coefficients. In the following paper, the network form of the K–K equations was obtained, containing the Peusner’s coefficients $$P_{ij}^{*}\,(i,\, j\in \{1,\,2\}),$$Pij∗(i,j∈{1,2}), and creating matrix of the second row of the Peusner’s coefficients $$[P^{*}].$$[P∗]. The equations were used to study transport of aqueous glucose solutions through a Nephrophan membrane oriented horizontally as well as configurations A and B of a membrane system. The configuration A involves a solution with a higher concentration placed under the membrane, whereas a solution with a lower concentration is placed above the membrane. In the configuration B, the solutions are swapped with places. Dependences of the Peusner’s coefficients $$P_{ij}^{*}$$Pij∗ and $$P_{ij}\,(i,\, j \in \{1,\,2\})$$Pij(i,j∈{1,2}) for non-homogenous ($$P_{ij}^{*})$$Pij∗) and homogenous ($$P_{ij})$$Pij) solutions upon the average concentration of glucose in the membrane ($$\overline{{C}})$$C¯) were calculated. The transport properties of membrane are characterized by coefficients determined experimentally: the coefficient of reflection ($$\sigma $$σ), hydraulic permeability ($$L_\mathrm{p})$$Lp), and solution permeability ($$\omega $$ω) for aqueous glucose or ethanol solutions. The calculations show that values of coefficients $$P_{11}^{*},\,P_{12}^{*},\,P_{21}^{*}$$P11∗,P12∗,P21∗, and $$P_{22}^{*}$$P22∗ depend non-linearly on both the membrane $$\overline{{C}}$$C¯ and the configuration of the membrane system. The values of the coefficients are different from the values of the coefficients $$P_{11},\,P_{12},\,P_{21}$$P11,P12,P21 and $$P_{22}.$$P22. Moreover, the coefficients $$P_{11},\,P_{12},\,P_{21}$$P11,P12,P21 and $$P_{22}$$P22 do not depend on the configuration of the membrane system. It was shown that there is a threshold value of concentration above which relations $$P_{11}^{*}/P_{11},\,P_{12}^{*}/P_{12}$$P11∗/P11,P12∗/P12 and $$P_{22}^{*}/P_{22}$$P22∗/P22 depend on the configuration of the membrane system.

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