International Journal of Chemical Engineering

Volume 2019, Article ID 5629259, 10 pages

https://doi.org/10.1155/2019/5629259

## Membrane Transport of Nonelectrolyte Solutions in Concentration Polarization Conditions: Form of the Kedem–Katchalsky–Peusner Equations

^{1}Department of Business Informatics, University of Economics, B Bogucicka Str., 40287 Katowice, Poland^{2}Department of Biomedical Processes and Systems, Technical University of Czestochowa, 36b Armii Krajowej Al., 42200 Czestochowa, Poland

Correspondence should be addressed to Kornelia M. Batko; lp.eciwotak.eu@oktab.ailenrok

Received 26 November 2018; Accepted 21 February 2019; Published 1 April 2019

Academic Editor: Michael Harris

Copyright © 2019 Kornelia M. Batko and Andrzej Ślęzak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 , , and (for nonhomogeneous solutions), *H*_{ij} and (for homogeneous solutions) (*i*, *j* ∈ {1, 2, 3}, *r* = A, B), , and were calculated on the basis of experimentally determined coefficients (*L*_{p}, *σ*_{1}, *σ*_{2}*ω*_{11}, *ω*_{22}, *ω*_{21}, *ω*_{12}, , and ) for glucose in aqueous ethanol solutions and two configurations of the membrane system. From the calculations, it results that the values of coefficients , , , , , , and depend nonlinearly on solution concentration as well as on a configuration of membrane system. Besides, the values of coefficients , , , , , and depend linearly on solution concentration. The value of coefficients *H*_{13}, *H*_{23}, and *H*_{33} do not depend on solution concentration. The coefficients *ψ*_{12}, *ψ*_{13}, *ψ*_{22} = *ψ*_{23}, *ψ*_{32} = *ψ*_{33}, and *ψ*_{det} depend nonlinearly on solution concentration and for ≈ 9.24 mol m^{−3} are equal to zero. For < 9.24 mol m^{−3}, the values of coefficients *ψ*_{12} and *ψ*_{13} are negative and for > 9.23 mol m^{−3}, positive. In contrast, the values of coefficients *ψ*_{22} = *ψ*_{23}, *ψ*_{32} = *ψ*_{33}, and *ψ*_{det} for < 9.24 mol m^{−3} are positive and for > 9.24 mol m^{−3}, negative. For = 0, we can observe nonconvective state, in which concentration Rayleigh number reaches the critical value *R*_{C} = 1691.09, for is convective state with convection directed straight down and for is convective state with convection directed straight up.

#### 1. Introduction

The membrane transport study is still a scientific challenge fulfilling the cognitive and application criteria in many processes occurring in living organisms (in membranes and biological cells, kidneys, etc.) observed in laboratory conditions and used in industry (hemodialysis, desalination of water, concentration of juices, etc.) [1, 2]. Moreover, in recent years, experimental techniques using microfluidic methods and nanoscale interface engineering to study interalia colloidal flows have been developed [3, 4]. The exchange of fluids and substances dissolved in it by biological membranes facilitates the transport of substances needed by living organisms to maintain their metabolic activity and regulation of pressure equilibrium by membranes in order to maintain the structural integrity of biological systems [5]. Passive and active transport processes control permeation of these substances. Passive transport mechanisms allow the flows of water and/or solutes to reduce their concentration gradients without energy using. In turn, active transport mechanisms allow penetration of dissolved substances against their concentration gradients, at the expense of energy supplied from metabolic reactions. The interaction between these mechanisms determines the hydrostatic fluid pressure and osmotic pressure differences through biological membranes, which are important features of biological systems [6, 7].

To describe transport phenomena in biological and technical systems, models developed in the framework of nonequilibrium thermodynamics [8–12], diffusion models [13, 14], and friction [15–17]; models developed in the framework of statistical physics [18–20]; and models developed as part of the network thermodynamics [21–26] are used. In order to characterize the relationship between generalized streams of liquid and solutes and their generalized driving forces, which are derived from the electrochemical and/or chemical affinity gradient, the thermodynamics of Onsager are used [9, 11, 27, 28]. This model is based on the existence of the dissipation function, which describes the total change in the entropy of the system. In near-equilibrium systems, i.e., those in which the dissipation rate of free energy is small, linear dependencies between the flow and the driving forces (thermodynamic stimuli) can be assumed [10, 11]. The classic Kedem–Katchalsky model equations were developed in accordance with these principles [9]. This model takes into account the interaction between solvents and nonionic solutes. This gives a set of phenomenological membrane coefficients that can be easily determined experimentally in a series of independent experiments. Moreover, this formalism provides the theoretical basis for the analysis of the volume and solute fluxes in various membrane systems [10, 11]. Therefore, the KK equations are one of the basic research tools for membrane transport in both biological and artificial systems. Many versions of these equations are used: classical [7], Kargol’s [29], Chang and Pinsky [6], and network thermodynamics [22–26]. These versions of KK equations for nonelectrolytes show the relationship between volume flow () and dissolved matter (*J*_{s}) and thermodynamic forces: osmotic (Δ*π*) and/or hydrostatic (Δ*P*). The network form of KK equations is obtained by symmetrical and/or hybrid transformation of classical KK equations using Peusner network thermodynamics (Peusner’s Network Thermodynamics, Peusner’s NT) [22]. For homogeneous and nonhomogeneous binary solutions of nonelectrolytes, two symmetrical and two hybrid forms of KK equations are known. The symmetrical form of these equations contain Peusner coefficients: and (for homogeneous solutions) and hybrid forms- and (for inhomogeneous solutions) (*i*, *j* ∈ {1, 2}, *r* = A, B) [22, 24].

Therefore, the concepts of *L*, *R*, *H*, and *P* have been introduced in the form of Kedem–Katchalsky network equations for homogeneity conditions and , , , and in the form of Kedem–Katchalsky network equations for polarization concentration conditions of solutions separated by the membrane [25]. For the homogeneity conditions of nonelectrolytic ternary solutions, there are two symmetrical and six hybrid forms of network KK equations. The symmetric forms of these equations, similarly to homogeneous ternary solutions, contain and coefficients and are derived directly from Onsager’s thermodynamics and hybrid forms- , , , , , and (*i*, *j* ∈ {1, 2, 3}), which are a consequence of the application of network thermodynamics techniques [30]. For concentration polarization conditions, these coefficients should be written in the form , *, *, *, **, *, , and (*i*, *j* ∈ {1, 2, 3}, *r* = A, B). Therefore, the concept of the form *L*, *R, H*, *K, P, N*, *S*, and *W* can be introduced in the network equations KK for the conditions of homogeneity and the form , , , , , , , and of the KK equations for the concentration polarization conditions of ternary solutions separated by the membrane.

The aim of the next series of papers is to present the form , , , , , , , and of the KK equations for concentration polarization conditions of ternary solutions. The aim of this work is to develop the form of of the KK equations, containing the Peusner coefficients (*i*, *j* ∈ {1, 2, 3}, *r* = A, B). Besides, we compare and coefficients and matrix coefficients = det [] and = det []. We will present the results of calculations of coefficients and matrix coefficients = det [] and = det [] and the quotients _{ij} *=* ()/ and *=* ( − )/.

#### 2. Materials and Methods

##### 2.1. Membrane System

Similarly as in previous papers [31, 32], we will consider transport of nonhomogeneous ternary nonelectrolyte solutions with concentrations at the initial moment (*t* = 0) *C*_{kh} and *C*_{kl} (*C*_{kh} > *C*_{kl}, *k* = 1, 2) through the membrane (M) in the single-membrane system (Figure 1). This membrane separates compartments *l* and *h* and is isotropic, symmetric, electroneutral, and selective for solvent and nonionized dissolved substances. In the case of membrane located in horizontal plane that is perpendicularly to the gravity vector, there are configurations A or B of arrangement of solutions in relation to the membrane (*r* *=* A or B). In configuration A, the solution with concentration *C*_{kl} is in the compartment over the membrane while the solution with concentration *C*_{kh} is in compartment under the membrane. In configuration B of the membrane system, location of solutions is reversed. We will consider only isothermal and stationary processes of membrane transport, for which the measure is the volume fluxes () and solutes fluxes () (*k* = 1, 2 and *r* = A, B). These fluxes can be described by the KK equations for nonhomogeneous ternary nonelectrolyte solutions [33].