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

Authors: Kornelia M. Batko; Andrzej Ślęzak
Journal: International Journal of Chemical Engineering
 (2019)

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## 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. CoefficientsHijr, 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, σ2 ω11, 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.


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## Body

## 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 (Jv) and dissolved matter (Js) and thermodynamic forces: osmotic (Δπ) and/or hydrostatic (ΔP). 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: Lij and Rij (for homogeneous solutions) and hybrid forms- Lijr and Rijr (for inhomogeneous solutions) (i, j ∈ {1, 2}, r = A, B) [22, 24].Therefore, the concepts ofL, R, H, and P have been introduced in the form of Kedem–Katchalsky network equations for homogeneity conditions and Lr, Rr, Hr, and Pr 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 Lij and Rij coefficients and are derived directly from Onsager’s thermodynamics and hybrid forms- Pij, Hij, Kij, Nij, Sij, and Wij (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 Lijr, Rijr, Hijr, Kijr, Pijr, Nijr, Sijr, and Wijr (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 Lr, Rr, Hr, Kr, Pr, Nr, Sr, and Wr 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 formLr, Rr, Hr, Kr, Pr, Nr, Sr, and Wr of the KK equations for concentration polarization conditions of ternary solutions. The aim of this work is to develop the form of Hr of the KK equations, containing the Peusner coefficients Hijr (i, j ∈ {1, 2, 3}, r = A, B). Besides, we compare Hijr and Hijcoefficients and matrix coefficients Hdetr = det [Hr] and Hdet = det [H]. We will present the results of calculations of coefficients Hijr and Hij matrix coefficients Hdetr = det [Hr] and Hdet = det [H] and the quotients ψij = (HijA−HijB)/Hij and ψdet = (HdetA − HdetB)/Hdet.


## 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) Ckh and Ckl (Ckh > Ckl, 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 Ckl is in the compartment over the membrane while the solution with concentration Ckh 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 (Jvr) and solutes fluxes (Jkr) (k = 1, 2 and r = A, B). These fluxes can be described by the KK equations for nonhomogeneous ternary nonelectrolyte solutions [33].Figure 1
The model of single-membrane system: M, membrane;llA and lhA, the concentration boundary layers in configuration A; llB and lhB, the concentration boundary layers in configuration B; Ph and Pl, mechanical pressures; Ckh and Ckl, concentrations of solutions; JkA and JvA, solute and volume fluxes in configuration A; and JkB and JvB, solute and volume fluxes in configuration B.Under such conditions, solutes, which diffuse through the membrane, create the concentration boundary layers (CBLs) on both sides of the membrane signed bylhr and llr [32, 33]. The CBL lhr has a thickness marked as δhr and CBL llr has a thickness marked as δlr.The mean concentrations of solutes “1” and “2” in membrane (C¯1, C¯2) can be calculated using expressions C¯k = (Ckh−Ckl) [ln (CkhCkl−1)]−1 (k = 1, 2). Appearance of CBLs causes those concentrations at the interfaces of the membrane and solutions, respectively, to decrease from Ckh to Ckir and increase from Ckl to Cker. For steady state, the following relation is fulfilled: Ckir > Cker, Cker > Ckl, Ckh > Ckir (k = 1, 2) [31, 34]. In addition, ρer and ρir denote the densities at interfaces llr/M and M/lhr, respectively, while ρl and ρh (ρl < ρh or ρl > ρh) denote the densities of solutions outside the layers. Moreover, ρer > ρl or ρer < ρir, ρer > ρir or ρer < ρir, and ρir > ρh or ρir < ρh [34]. If the solution with lower density is under the membrane the system, lhr/M/llr loses its hydrodynamic stability, and convective instabilities in the near membrane area are observed [33, 34].The measure of the concentration polarization is the concentration polarization factor (ζkr). Its value depends on both the concentration of solutions separated by the membrane (C¯k) and the configuration of the membrane system (r = A, B). More specifically for this case, the thicknesses of CBLs δhr and δlr exceed values (δhr)crit and (δlr)crit, and coefficient of concentration polarization (ζkr) exceeds its critical value (ζkr)crit suitably [34–36]. For diluted ternary nonelectrolyte solutions, the concentration polarization factor (ζksr) and the thickness of concentration the boundary layers (δhr and δlr) can be described by the expression ζksrC¯k,C¯s=1+RTωksδlrC¯k,C¯s+δhrC¯k,C¯sDks−1−1 [31].


### 2.2. Matrix Form of the Kedem–Katchalsky Equations

According to the Kedem–Katchalsky formalism [7, 10, 33], transport properties of the membrane are determined for solutions containing a solvent and dissolved two substances (ternary solution) by practical coefficients: hydraulic permeability (Lp), reflection (σk, k = 1, 2), and permeability of solute (ωkf, k, f ∈ {1, 2}). In turn, the transport properties of the complex llr/M/lhr are characterized by hydraulic permeability (Lpr), reflection (σkr,σar), and permeability coefficients of solute (ωkfr). The coefficients of hydraulic, osmotic, advective, and diffusive concentration polarization are defined by expressions ζpr=Lpr/Lp, ζvkr=σkr/σk, ζar=σar/σk, and ζsr=ωkfr/ωkf [23]. The decrease of the value of volume and solute fluxes from Jv and Jk (in conditions of homogeneous solutions) to Jvr and Jkr (in condition of concentration polarization) is caused by formation of the concentration boundary layers llr and lhr, respectively [33].The classical form of Kedem–Katchalsky equations for concentration polarization conditions can be written as(1)Jvr=ζprLpΔP−ζv1rσ1Δπ1−ζv2rσ2Δπ2,(2)J1r=ζs11rω11Δπ1+ζs12rω12Δπ2+C¯11−ζa1rσ1Jvr,(3)J2r=ζs21rω21Δπ1+ζs22rω22Δπ2+C¯11−ζa2rσ2Jvr,where Jvr, J1r, and J2r are volume and solutes “1” and “2” fluxes, respectively; Lp is the hydraulic permeability coefficient; σ1 and σ2 are reflection coefficients suitably for solutes “1” and “2”; ω11 and ω22 are the solute permeability coefficients for solutes “1” and “2” generated by forces with indexes “1” and “2” and ω12 and ω21 are the cross coefficients of permeability for substances “1” and “2” generated by forces with indexes “2” and “1” respectively. ΔP=Ph−Pl is the hydrostatic pressure difference (Ph, Pl are higher and lower values of hydrostatic pressure suitably). Δπk = RT(Ckh − Ckl) is the difference of osmotic pressure (RT is the product of gas constant and thermodynamic temperature whereas Ckh and Ckl are solutes concentrations, k = 1, 2). C¯k is the mean solute concentration in membrane and is expressed by C¯k = (Ckh − Ckl) [ln (CkhCkl−1)]−1 (k = 1, 2).Relatively simple algebraic transformations allow transforming equations (1)–(3) into the form(4)ΔP−Δπ1−Δπ2=Jvrζpr−C¯11−ζv1rσ1Δπ1C¯1−C¯21−ζv2rσ2Δπ2C¯2,(5)J1r=C¯11−ζa1rσ1Jvr+C¯1ζs11rω11Δπ1C¯1+C¯2ζs12rω12Δπ2C¯2,(6)J2r=C¯21−ζa2rσ2Jvr+C¯1ζs21rω21Δπ1C¯1+C¯2ζs22rω22Δπ2C¯2,The above equations are one of the forms of Kedem–Katchalsky equations obtained by the hybrid transformation of Peusner’s thermodynamic networks.Equations (4) and (5) can be transformed by simple algebraic transformations to the matrix form:(7)ΔP−Δπ1−Δπ2J1rJ2r=H11rH12rH13rH21rH22rH23rH31rH32rH33rJvrΔπ1C¯1Δπ2C¯2=HrJvrΔπ1C¯1Δπ2C¯2,where H11r=ζprLp−1, H12r=−C¯11−ζv1rσ1, H13r=−C¯21−ζv2rσ2, H21r=C¯11−ζa1rσ1, H22r=C¯1ζs11rω11, H23r=C¯2ζs12rω12, H31r=C¯21−ζa2rσ2, H32r=C¯1ζs21rω21, H33r=C¯2ζs22rω22, and [Hr] is the matrix of Peusner’s coefficients Hijr (i, j ∈ {1, 2, 3}) for ternary nonelectrolyte solutions in conditions of concentration polarization.It results from equation (7) for the nondiagonal coefficients H12r ≠ H21r, H13r ≠ H31r, and H23r ≠ H32r. On the basis of equation (7), we get H12r=−H21r1−ζv1rσ11−ζa1rσ1−1, H31r=−H31r1−ζv2rσ21−ζa2rσ2−1, and H23r=H32rC¯2ω12ζs12rC¯1ω21ζs21r−1. Moreover, the determinant of the matrix Hr is equal to(8)detHr=Hdetr=C¯1C¯2ζpr−1Lp−1γ1+C¯1C¯221−ζa2rσ2γ2+C¯12C¯21−ζa1rσ1γ3,where γ1=ω11ζs11rω22ζs22r−ω12ζs12rω21ζs21r, γ2=ω11ζs11r1−ζv2rσ2−ω12ζs12r1−ζv1rσ1, γ3=ω22ζs22r1−ζv1rσ1−ω21ζs21r1−ζv2rσ2.Index “r” in equations (3)–(10) indicates that the fluxes Jvr, J1r, and J2r, coefficients Hijr (i, j ∈ {1, 2, 3} and r = A, B) and matrix [Hr] of these coefficients, depend on configuration of the membrane system. For homogeneous conditions (ζpr = ζv1r = ζv2r = ζa1r = ζa2r = ζs11r = ζs12r = ζs22r = ζs21r = 1), we get(9)ΔP−Δπ1−Δπ2J1J2=H11H12H13H21H22H23H31H32H33JvΔπ1C¯1Δπ2C¯2=HrJvΔπ1C¯1Δπ2C¯2,where H11=Lp−1, H12=−C¯11−σ1=−H21, H13=−C¯21−σ2=−H31, H22r=C¯1ζs11rω11, H23=C¯2ω12, H32=C¯1ω21, H33=C¯2ω22, and [H] is the matrix of Peusner’s coefficients Hij (i, j∈ {1, 2, 3}) for ternary nonelectrolyte homogeneity solutions.On the basis of equation (9), we get H23=H32C¯2ω12C¯1−1ω21−1. The determinant of the matrix [H] is equal to(10)detH=Hdet=C¯1C¯2Lp−1γ1+C¯1C¯221−σ2γ2+C¯12C¯21−σ1γ3,where γ1=ω11ω22−ω12ω21, γ2=ω111−σ2−ω121−σ1, and γ3=ω221−σ1−ω211−σ2.In order to show the relationship between coefficientsHijr and Hij and between Hdetr and Hdet for A and B configurations of the membrane system (r = A, B), we can calculate using equations (6), (10), (11), and (12), the coefficients ψij = (HijA − HijB)/Hij and ψdet = (HdetA − HdetB)/Hdet. The expressions for the coefficients ψij and ψdet are given as follows:(11)ψ11=H11A−H11BH11=ζpB−ζpAζpAζpB,(12)ψ12=H12A−H12BH12=σ1ζv1B−ζv1A1−σ1,(13)ψ13=H13A−H13BH13=σ2ζv2B−ζv2A1−σ2,(14)ψ21=H21A−H21BH21=σ1ζa1B−ζa1A1−σ1,(15)ψ22=H22A−H22BH22=ζs11A−ζs11B,(16)ψ23=H23A−H23BH23=ζs12A−ζs12B,(17)ψ31=H31A−H31BH31=σ2ζa2B−ζa2A1−σ2,(18)ψ32=H32A−H32BH32=ζs21A−ζs21B,(19)ψ33=H33A−H33BH33=ζs22A−ζs22B,(20)ψdet=HdetA−HdetBHdet=C¯1C¯2γ1AζpB−γ1BζpA+LpζpAζpAC¯1C¯22γ2A−γ2B+C¯12C¯2γ3A−γ3BζpAζpBC¯1C¯2γ1+Lpγ2+γ3,where γ1A=ω11ζs11Aω22ζs22A−ω12ζs12Aω21ζs21A, γ1B=ω11ζs11Bω22ζs22B−ω12ζs12Bω21ζs21B, γ2A=1−ζa2Aσ2ω11ζs11A1−ζv2Aσ2−ω12ζs12A1−ζv1Aσ1, γ2B=1−ζa2Bσ2ω11ζs11B1−ζv2Bσ2−ω12ζs12B1−ζv1Bσ1, γ3A=1−ζa1Aσ1ω22ζs22A1−ζv1Aσ1−ω21ζs21A1−ζv2Aσ2, γ3B=1−ζa1Bσ1ω22ζs22B1−ζv1Bσ1−ω21ζs21B1−ζv2Bσ2, γ1=ω11ω22−ω12ω21, γ2=C¯21−σ2ω111−σ2−ω121−σ1, and γ3=C¯11−σ1ω221−σ1−ω211−σ2.The values of coefficientsψij and ψdet show the influence of concentration polarization and natural convection on the membrane transport.


## 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) Ckh and Ckl (Ckh > Ckl, 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 Ckl is in the compartment over the membrane while the solution with concentration Ckh 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 (Jvr) and solutes fluxes (Jkr) (k = 1, 2 and r = A, B). These fluxes can be described by the KK equations for nonhomogeneous ternary nonelectrolyte solutions [33].Figure 1
The model of single-membrane system: M, membrane;llA and lhA, the concentration boundary layers in configuration A; llB and lhB, the concentration boundary layers in configuration B; Ph and Pl, mechanical pressures; Ckh and Ckl, concentrations of solutions; JkA and JvA, solute and volume fluxes in configuration A; and JkB and JvB, solute and volume fluxes in configuration B.Under such conditions, solutes, which diffuse through the membrane, create the concentration boundary layers (CBLs) on both sides of the membrane signed bylhr and llr [32, 33]. The CBL lhr has a thickness marked as δhr and CBL llr has a thickness marked as δlr.The mean concentrations of solutes “1” and “2” in membrane (C¯1, C¯2) can be calculated using expressions C¯k = (Ckh−Ckl) [ln (CkhCkl−1)]−1 (k = 1, 2). Appearance of CBLs causes those concentrations at the interfaces of the membrane and solutions, respectively, to decrease from Ckh to Ckir and increase from Ckl to Cker. For steady state, the following relation is fulfilled: Ckir > Cker, Cker > Ckl, Ckh > Ckir (k = 1, 2) [31, 34]. In addition, ρer and ρir denote the densities at interfaces llr/M and M/lhr, respectively, while ρl and ρh (ρl < ρh or ρl > ρh) denote the densities of solutions outside the layers. Moreover, ρer > ρl or ρer < ρir, ρer > ρir or ρer < ρir, and ρir > ρh or ρir < ρh [34]. If the solution with lower density is under the membrane the system, lhr/M/llr loses its hydrodynamic stability, and convective instabilities in the near membrane area are observed [33, 34].The measure of the concentration polarization is the concentration polarization factor (ζkr). Its value depends on both the concentration of solutions separated by the membrane (C¯k) and the configuration of the membrane system (r = A, B). More specifically for this case, the thicknesses of CBLs δhr and δlr exceed values (δhr)crit and (δlr)crit, and coefficient of concentration polarization (ζkr) exceeds its critical value (ζkr)crit suitably [34–36]. For diluted ternary nonelectrolyte solutions, the concentration polarization factor (ζksr) and the thickness of concentration the boundary layers (δhr and δlr) can be described by the expression ζksrC¯k,C¯s=1+RTωksδlrC¯k,C¯s+δhrC¯k,C¯sDks−1−1 [31].


## 2.2. Matrix Form of the Kedem–Katchalsky Equations

According to the Kedem–Katchalsky formalism [7, 10, 33], transport properties of the membrane are determined for solutions containing a solvent and dissolved two substances (ternary solution) by practical coefficients: hydraulic permeability (Lp), reflection (σk, k = 1, 2), and permeability of solute (ωkf, k, f ∈ {1, 2}). In turn, the transport properties of the complex llr/M/lhr are characterized by hydraulic permeability (Lpr), reflection (σkr,σar), and permeability coefficients of solute (ωkfr). The coefficients of hydraulic, osmotic, advective, and diffusive concentration polarization are defined by expressions ζpr=Lpr/Lp, ζvkr=σkr/σk, ζar=σar/σk, and ζsr=ωkfr/ωkf [23]. The decrease of the value of volume and solute fluxes from Jv and Jk (in conditions of homogeneous solutions) to Jvr and Jkr (in condition of concentration polarization) is caused by formation of the concentration boundary layers llr and lhr, respectively [33].The classical form of Kedem–Katchalsky equations for concentration polarization conditions can be written as(1)Jvr=ζprLpΔP−ζv1rσ1Δπ1−ζv2rσ2Δπ2,(2)J1r=ζs11rω11Δπ1+ζs12rω12Δπ2+C¯11−ζa1rσ1Jvr,(3)J2r=ζs21rω21Δπ1+ζs22rω22Δπ2+C¯11−ζa2rσ2Jvr,where Jvr, J1r, and J2r are volume and solutes “1” and “2” fluxes, respectively; Lp is the hydraulic permeability coefficient; σ1 and σ2 are reflection coefficients suitably for solutes “1” and “2”; ω11 and ω22 are the solute permeability coefficients for solutes “1” and “2” generated by forces with indexes “1” and “2” and ω12 and ω21 are the cross coefficients of permeability for substances “1” and “2” generated by forces with indexes “2” and “1” respectively. ΔP=Ph−Pl is the hydrostatic pressure difference (Ph, Pl are higher and lower values of hydrostatic pressure suitably). Δπk = RT(Ckh − Ckl) is the difference of osmotic pressure (RT is the product of gas constant and thermodynamic temperature whereas Ckh and Ckl are solutes concentrations, k = 1, 2). C¯k is the mean solute concentration in membrane and is expressed by C¯k = (Ckh − Ckl) [ln (CkhCkl−1)]−1 (k = 1, 2).Relatively simple algebraic transformations allow transforming equations (1)–(3) into the form(4)ΔP−Δπ1−Δπ2=Jvrζpr−C¯11−ζv1rσ1Δπ1C¯1−C¯21−ζv2rσ2Δπ2C¯2,(5)J1r=C¯11−ζa1rσ1Jvr+C¯1ζs11rω11Δπ1C¯1+C¯2ζs12rω12Δπ2C¯2,(6)J2r=C¯21−ζa2rσ2Jvr+C¯1ζs21rω21Δπ1C¯1+C¯2ζs22rω22Δπ2C¯2,The above equations are one of the forms of Kedem–Katchalsky equations obtained by the hybrid transformation of Peusner’s thermodynamic networks.Equations (4) and (5) can be transformed by simple algebraic transformations to the matrix form:(7)ΔP−Δπ1−Δπ2J1rJ2r=H11rH12rH13rH21rH22rH23rH31rH32rH33rJvrΔπ1C¯1Δπ2C¯2=HrJvrΔπ1C¯1Δπ2C¯2,where H11r=ζprLp−1, H12r=−C¯11−ζv1rσ1, H13r=−C¯21−ζv2rσ2, H21r=C¯11−ζa1rσ1, H22r=C¯1ζs11rω11, H23r=C¯2ζs12rω12, H31r=C¯21−ζa2rσ2, H32r=C¯1ζs21rω21, H33r=C¯2ζs22rω22, and [Hr] is the matrix of Peusner’s coefficients Hijr (i, j ∈ {1, 2, 3}) for ternary nonelectrolyte solutions in conditions of concentration polarization.It results from equation (7) for the nondiagonal coefficients H12r ≠ H21r, H13r ≠ H31r, and H23r ≠ H32r. On the basis of equation (7), we get H12r=−H21r1−ζv1rσ11−ζa1rσ1−1, H31r=−H31r1−ζv2rσ21−ζa2rσ2−1, and H23r=H32rC¯2ω12ζs12rC¯1ω21ζs21r−1. Moreover, the determinant of the matrix Hr is equal to(8)detHr=Hdetr=C¯1C¯2ζpr−1Lp−1γ1+C¯1C¯221−ζa2rσ2γ2+C¯12C¯21−ζa1rσ1γ3,where γ1=ω11ζs11rω22ζs22r−ω12ζs12rω21ζs21r, γ2=ω11ζs11r1−ζv2rσ2−ω12ζs12r1−ζv1rσ1, γ3=ω22ζs22r1−ζv1rσ1−ω21ζs21r1−ζv2rσ2.Index “r” in equations (3)–(10) indicates that the fluxes Jvr, J1r, and J2r, coefficients Hijr (i, j ∈ {1, 2, 3} and r = A, B) and matrix [Hr] of these coefficients, depend on configuration of the membrane system. For homogeneous conditions (ζpr = ζv1r = ζv2r = ζa1r = ζa2r = ζs11r = ζs12r = ζs22r = ζs21r = 1), we get(9)ΔP−Δπ1−Δπ2J1J2=H11H12H13H21H22H23H31H32H33JvΔπ1C¯1Δπ2C¯2=HrJvΔπ1C¯1Δπ2C¯2,where H11=Lp−1, H12=−C¯11−σ1=−H21, H13=−C¯21−σ2=−H31, H22r=C¯1ζs11rω11, H23=C¯2ω12, H32=C¯1ω21, H33=C¯2ω22, and [H] is the matrix of Peusner’s coefficients Hij (i, j∈ {1, 2, 3}) for ternary nonelectrolyte homogeneity solutions.On the basis of equation (9), we get H23=H32C¯2ω12C¯1−1ω21−1. The determinant of the matrix [H] is equal to(10)detH=Hdet=C¯1C¯2Lp−1γ1+C¯1C¯221−σ2γ2+C¯12C¯21−σ1γ3,where γ1=ω11ω22−ω12ω21, γ2=ω111−σ2−ω121−σ1, and γ3=ω221−σ1−ω211−σ2.In order to show the relationship between coefficientsHijr and Hij and between Hdetr and Hdet for A and B configurations of the membrane system (r = A, B), we can calculate using equations (6), (10), (11), and (12), the coefficients ψij = (HijA − HijB)/Hij and ψdet = (HdetA − HdetB)/Hdet. The expressions for the coefficients ψij and ψdet are given as follows:(11)ψ11=H11A−H11BH11=ζpB−ζpAζpAζpB,(12)ψ12=H12A−H12BH12=σ1ζv1B−ζv1A1−σ1,(13)ψ13=H13A−H13BH13=σ2ζv2B−ζv2A1−σ2,(14)ψ21=H21A−H21BH21=σ1ζa1B−ζa1A1−σ1,(15)ψ22=H22A−H22BH22=ζs11A−ζs11B,(16)ψ23=H23A−H23BH23=ζs12A−ζs12B,(17)ψ31=H31A−H31BH31=σ2ζa2B−ζa2A1−σ2,(18)ψ32=H32A−H32BH32=ζs21A−ζs21B,(19)ψ33=H33A−H33BH33=ζs22A−ζs22B,(20)ψdet=HdetA−HdetBHdet=C¯1C¯2γ1AζpB−γ1BζpA+LpζpAζpAC¯1C¯22γ2A−γ2B+C¯12C¯2γ3A−γ3BζpAζpBC¯1C¯2γ1+Lpγ2+γ3,where γ1A=ω11ζs11Aω22ζs22A−ω12ζs12Aω21ζs21A, γ1B=ω11ζs11Bω22ζs22B−ω12ζs12Bω21ζs21B, γ2A=1−ζa2Aσ2ω11ζs11A1−ζv2Aσ2−ω12ζs12A1−ζv1Aσ1, γ2B=1−ζa2Bσ2ω11ζs11B1−ζv2Bσ2−ω12ζs12B1−ζv1Bσ1, γ3A=1−ζa1Aσ1ω22ζs22A1−ζv1Aσ1−ω21ζs21A1−ζv2Aσ2, γ3B=1−ζa1Bσ1ω22ζs22B1−ζv1Bσ1−ω21ζs21B1−ζv2Bσ2, γ1=ω11ω22−ω12ω21, γ2=C¯21−σ2ω111−σ2−ω121−σ1, and γ3=C¯11−σ1ω221−σ1−ω211−σ2.The values of coefficientsψij and ψdet show the influence of concentration polarization and natural convection on the membrane transport.


## 3. Results and Discussion

The values of coefficientsHij, Hijr, Hdetr, ψij = (HijA − HijB)/Hij, and ψdet = (HdetA − HdetB)/Hdet, (i, j ∈ {1, 2, 3}, r = A, B), which describe equations (7)–(20), are calculated for polymer membrane Nephrophan and glucose solutions in aqueous solution of ethanol. The glucose concentration was marked by index “1” and the ethanol concentration by index “2”. The concentration of substance “1” in the compartment (h) take values from C1h = 1 mol·m−3 to C1h = 101 mol·m−3. In turn, concentration of a substance “2” in the compartment (h) was constant and amounted to C2h = 201 mol·m−3. The concentrations of both components in the compartment (l) were established and amounted to C1l = C2l = 1 mol·m−3. In expressions under equations (7)–(20), there are coefficients that describe transport properties of membrane (Lp, σ1, σ2 ω11, ω22, ω21 and ω12), average concentrations of solutions “1” and “2” in the membrane (C¯1, C¯2), and coefficients of concentration polarization (ζpr, ζa1r, ζa2r, ζv1r, ζs11r, ζs12r, ζv2r, ζs22r, and ζs21r). The values of these coefficients are calculated, using conditions ζpr = ζa1r = ζa2r = 1, ζv1r = ζs11r = ζs12r = ζ1r, and ζv2r = ζs22r = ζs21r = ζ2r [33, 37].In order to calculateHij, Hijr, Hdetr, ψij = (HijA − HijB)/Hij, and ψdet = (HdetA − HdetB)/Hdet, (i, j ∈ {1, 2, 3}, r = A, B) on the basis of equations (7)–(20), we used the characteristics ζ1r = f(C¯1, C¯2=const.) and ζ2r = f(C¯1, C¯2=const.) presented in Figure 2 and following data: Lp = 4.9 × 10−12 m3·N−1·s−1, σ1 = 0.068, σ2 = 0.025, ω11 = 0.8 × 10−9 mol·N−1·s−1, ω12 = 0.81 × 10−13 mol·N−1·s−1, ω22 = 1.43 × 10−9 mol·N−1·s−1, ω21 = 1.63 × 10−12 mol·N−1·s−1, C¯1 = 2.79 ÷ 21.67 mol·m−3 and C¯2 = 37.71mol·m−3.Figure 2
Dependencies of concentration polarization coefficient (ζkr, k = 1, 2) on glucose concentration in 201 mol m−3 aqueous ethanol solution for configuration A (r = A and B (r = B)) of the single-membrane system [31, 32].The calculations based on equations (7) and (9) show that H11A = H11B = H11 = 0.204 × 1012 N·s·m−3, and H31A = H31B = H31 = 36.77 mol·m−3 are independent of both concentration of solution and configuration of the membrane system.Dependencies of coefficientsH12A = f(C¯1, C¯2 = const.), H12B = f(C¯1, C¯2 = const.), H12 = f(C¯1, C¯2 = const.), and H21A = H21B = H21 = f(C¯1, C¯2 = const.) are shown in Figure 3. Graphs 1A, 1B, and 1 show that values of coefficients H12A, H12B, and H12 are decrease almost linearly with the increase in C¯1 for C¯2  = 37.71 mol·m−3. These coefficients fulfill the following conditions: for C¯1 = 9.24 mol·m−3, H12A = H12B = −8.98 mol·m−3, for C¯1 < 9.24 mol·m−3, H12 > H12A > H12B < 0, and for C¯1 > 9.24 mol·m−3, H12 > H12B > H12A < 0.Figure 3
The graphic illustration of the dependenceHijr = f(C¯1, C¯2 = 37.71 mol·m−3), (i ≠ j, r = A, B) for the glucose in aqueous ethanol solution in conditions of concentration polarization for configurations A and B of the membrane system, respectively: H12A = f(C¯1, C¯2  = const.)—curve 1A; H12B = f(C¯1, C¯2 = const.)—curve 1B; and H21A = H21B = f(C¯1, C¯2 = const.)—line 2. The lines illustrate the dependence H12 = f(C¯1, C¯2 = const.)—line 1 and H21 = f(C¯1, C¯2 = const.) (line 2) in conditions of homogeneity of solutions.Graph 2 shows that values of coefficientsH21A, H21B, and H21 increase linearly with the increase in C¯1 for C¯2 = 37.71 mol·m−3. These coefficients fulfill the condition H21A = H21B = H21 > 0. Dependencies of coefficients H13A = f(C¯1, C¯2 = const.), H13B = f(C¯1, C¯2 = const.), and H13 = f(C¯1, C¯2 = const.) are presented in Figure 4. Graphs 1A and 1B show that values of coefficients H13A and H13B decrease and increase nonlinearly with the increase in C¯1 for C¯2 = 37.71 mol·m−3, respectively. The value of coefficient is equal to H13 = −36.77 mol·m−3 and is independent of both concentration of solution and configuration of the membrane system. These coefficients fulfill the following conditions: for C¯1 = 9.24 mol m−3, H13A = H13B = −37.49 mol·m−3, for C¯1 < 9.24 mol·m−3, H13 < H13A < H13B < 0, and for C¯1 > 9.24 mol·m−3, H13 < H13B < H13A < 0.Figure 4
The graphic illustration of the dependenceH13r = f(C¯1, C¯2  = 37.71 mol·m−3), (r = A, B) for the glucose in aqueous ethanol solution in conditions of concentration polarization for configurations A (H13A = f(C¯1,C¯2 = const.), curve 1A) and B (H13B = f(C¯1, C¯2 = const.), curve 1B) of the membrane system. Line 1 illustrates the dependence H13 = f(C¯1, C¯2 = const.) for conditions of homogeneity of solutions.The graphs 1A, 1B, and 1 shown in Figure5 illustrate the dependencies H22A = f(C¯1, C¯2 = const.), H22B = f(C¯1, C¯2 = const.), and H22 = f(C¯1, C¯2 = const.). Graph 1A shows that the values of coefficient H22A first increase in a nonlinear way and then decrease in a nonlinear manner with the increase of C¯1value and graph 1B shows that the H22Bvalues increase nonlinearly with the increase of the of C¯1value (for C¯2 = 37.71 mol·m−3). In turn, the course of graph 1 shows that H22 is a linear function of C¯1(for C¯2 = 37.71 mol·m−3). These coefficients fulfill the following conditions: for C¯1 = 9.24 mol·m−3, H22A = H22B = 1.81 × 10−9 mol2N−1·s−1·m−3, for C¯1 < 9.24 mol·m−3, H22 > H22A > H22B > 0, and C¯1 > 9.24 mol·m−3, H22 > H22B > H22A > 0.Figure 5
The graphic illustration of the dependenceH22r = f(C¯1, C¯2  = 37.71 mol·m−3) (r = A, B) for the glucose in aqueous ethanol solution in conditions of concentration polarization for configurations A (H22A = f(C¯1, C¯2 = const.), curve 1A) and B (H22B = f(C¯1, C¯2 = const.), curve 1B) of the membrane system. Line 1 illustrates the dependence H22 = f(C¯1, C¯2 = const.) in conditions of homogeneity of solutions.Dependencies of coefficientsH23A = f(C¯1, C¯2 = const.), H23B = f(C¯1, C¯2 = const.), and H23 = f(C¯1, C¯2 = const.) are presented in Figure 6. Graphs 1A and 1B shows that values of coefficients H23A and H23B decrease and increase almost nonlinearly with the increase in C¯1 for C¯2 = 37.71 mol·m−3, respectively. The value of coefficient is equal to H23 = 30.54 mol·m−3 and is independent of both concentration of solution and configuration of the membrane system. These coefficients fulfill the following conditions: for C¯1 = 9.24 mol·m−3, H23A = H23B = 6.98 × 10−13 mol2·N−1·s−1·m−3, for C¯1 < 9.24 mol·m−3, H23 > H23A > H23B > 0, and C¯1 > 9.24 mol·m−3, H23 > H23B > H23A > 0.Figure 6
The graphic illustration of the dependenceH23r = f(C¯1, C¯2 = 37.71 mol·m−3) (r = A, B) for the glucose in aqueous ethanol solution in conditions of concentration polarization for configuration A (H23A = f(C¯1,C¯2 = 37.71 mol·m−3), curve 1A) and B (H23B = f(C¯1,C¯2  = 37.71 mol·m−3), curve 1B), of the membrane system. Line 1 illustrates the dependence H23 = f(C¯1,C¯2 = const.) in conditions of homogeneity of solutions.The graphs 1A, 1B, and 1 shown in Figure7 illustrate the dependencies H32A = f(C¯1, C¯2 = const.), H32B = f(C¯1, C¯2 = const.), and H32 = f(C¯1, C¯2 = const.). Graph 1A shows that the values of coefficient H32A first increase in a nonlinear way and then decrease in a nonlinear manner with the increase of C¯1 value, and graph 1B shows that the H32B values increase nonlinearly with the increase of the of C¯1 value (for C¯2 = 37.71 mol·m−3). In turn, the course of graph 1 shows that H32 is a linear function of C¯1 (for C¯2 = 37.71 mol·m−3). These coefficients fulfill the following conditions: for C¯1 = 9.24 mol·m−3, H32A = H22B = 3.35 × 10−12 mol2·N−1·s−1·m−3, for C¯1 < 9.24 mol·m−3, H32 > H32A > H32B > 0, and for C¯1 > 9.24 mol·m−3, H32 > H32B > H32A > 0.Figure 7
The graphic illustration of the dependenceH32r = f(C¯1, C¯2 = 37.71 mol·m−3) (r = A, B) for the glucose in aqueous ethanol solution in conditions of concentration polarization for configurations A (H32A = f(C¯1, C¯2 = const.), curve 1A) and B (H32B = f(C¯1, C¯2 = const.), curve 1B) of the membrane system. Line 1 illustrates the dependence H32 = f(C¯1, C¯2 = const.) in conditions of homogeneity of solutions.Dependencies of coefficientsH33A = f(C¯1, C¯2 = const.), H33B = f(C¯1, C¯2 = const.), and H33 = f(C¯1, C¯2  = const.) are presented in Figure 8. Graphs 1A and 1B show that values of coefficients H33A and H33B decrease and increase almost nonlinearly with the increase in C¯1 for C¯2 = 37.71 mol·m−3, respectively. The value of coefficient is equal to H33 = 53.92 × 10−9 mol2·N−1·s−1·m−3 and is independent of both concentration of solution and configuration of the membrane system. These coefficients fulfill the following conditions: for C¯1 = 9.24 mol m−3, H33A  = H33B  = 12.64 × 10−9 mol2·N−1·s−1·m−3, for C¯1 < 9.24 mol·m−3, H33 < H33A < H33B > 0, and for C¯1 > 9.24 mol·m−3, H33 < H33B < H33A > 0.Figure 8
The graphic illustration of the dependenceH33r = f(C¯1,C¯2 = 37.71 mol·m−3) (r = A, B) for the glucose in aqueous ethanol solution in conditions of concentration polarization for configuration A (H33A = f(C¯1,C¯2  = 37.71 mol·m−3), curve 1A) and B (H33B = f(C¯1, C¯2 = 37.71 mol·m−3), curve 1B), of the membrane system. Line 1 illustrates the dependence H23 = f(C¯1, C¯2 = const.) in conditions of homogeneity of solutions.The graphs 1A, 1B, and 1 shown in Figure9 illustrate the dependencies HdetA = f(C¯1, C¯2 = const.), HdetB = f(C¯1, C¯2 = const.), and Hdet = f(C¯1, C¯2 = const.). Graph 1A shows that the values of coefficient HdetA first increase in a nonlinear way and then decrease in a nonlinear manner with the increase of C¯1 value, and graph 1B shows that the HdetB values increase nonlinearly with the increase of the of C¯1 value (for C¯2 = 37.71 mol·m−3). In turn, the course of graph 1 shows that det [H] is a linear function of C¯1 (for C¯2 = 37.71 mol·m−3). These coefficients fulfill the following conditions: for C¯1 = 9.24 mol·m−3, HdetA = HdetB = 6.74 × 10−6 mol2·N−1·s−1·m−3, for C¯1 < 9.24 mol·m−3, Hdet > HdetA > HdetB > 0, and for C¯1 > 9.24 mol·m−3, Hdet > HdetB > HdetA > 0.Figure 9
The graphic illustration of the dependenceHdetr = f(C¯1, C¯2  = 37.71 mol·m−3) (r = A, B) for the glucose in aqueous ethanol solution in conditions of concentration polarization for configurations A (HdetA = f(C¯1, C¯2 = const.), curve 1A) and B (HdetB = f(C¯1, C¯2 = const.), curve 1B) of the membrane system. Line 1 illustrates the dependence Hdet = f(C¯1, C¯2 = const.) in conditions of homogeneity of solutions.Figures3–9 show that there are three groups of characteristics Hijr = f(C¯1, C¯2 = const.), Hij = f(C¯1, C¯2 = const.), and Hdetr = f(C¯1, C¯2 = const.), (i, j ∈ {1, 2, 3} and r = A, B). The first group includes the characteristics presented in Figure 3, the second Figures 4, 6, and 8, and the third the characteristics shown in Figures 5, 7, and 9. In the case of group 1, which includes concentration characteristics of coefficients H12r, H12, H21r, and H21 (r = A, B), these coefficients are expressed in the same units, and their values are in the range −21.62 mol·m−3 ÷ 20.19 mol·m−3.In the case of the second group of characteristics, the shape of the concentration characteristics of the coefficientsH13r, H13, H23r, H23, H33r, and H33 (r = A, B) is very similar. However, the values of these coefficients differ by up to several orders of magnitude: −37.68 mol·m−3 ≤ H13r, H13 ≤ −37.23 mol·m−3, 0.95 × 10−13 mol2·N−1·s−1·m−3 ≤ H23r, H23 ≤ 30.54 × 10−13 mol2·N−1·s−1·m−3 and 1.51 × 10−9 mol2·N−1·s−1·m−3 ≤ H23r, H23 ≤ 53.92 × 10−9 mol2·N−1·s−1·m−3.Moreover, the values of coefficientsH13r and H13 are expressed in different units than coefficients H23r, H23, H33r, and H33.The third group of characteristics includes the concentration characteristics of coefficientsH22r, H22, H32r, H32, Hdetr, and Hdet. Values of coefficients Hdetr and Hdet are expressed in units other than H22r, H22, H32r, and H32. In addition, the values of these coefficients differ from each other by several orders of magnitude: 0.02 × 10−9 mol2·N−1·s−1·m−3 ≤ H22r, H22 ≤ 17.33 × 10−9 mol2·N−1·s−1·m−3, 0.03 × 10−12 mol2·N−1·s−1·m−3 ≤ H32r, H32 ≤ 35.32 × 10−12 mol2·N−1·s−1·m−3, and 0.03 × 10−6 mol4·N−1·s−1·m−9 ≤ Hdetr, Hdet ≤ 236.09 × 10−6 mol4·N−1·s−1·m−9.The graphs 1–5 shown in Figure10 illustrate the dependencies ψ22 = ψ23= f(C¯1, C¯2 = const.), ψ32 = ψ33= f(C¯1, C¯2 = const.), ψdet = f(C¯1, C¯2 = const.), ψ13 = f(C¯1, C¯2 = const.), and ψ12 = f(C¯1, C¯2 = const.), respectively. These coefficients fulfill the following conditions: for C¯1 = 9.24 mol·m−3, ψ22 = ψ23 = ψ32= ψdet = ψ13 = ψ12 = 0, for C¯1 < 9.24 mol·m−3, ψ22 = ψ23 ≈ ψ32 = ψ33 > ψdet > 0 and ψ12 > ψ13 < 0, and for C¯1 > 9.24 mol·m−3, ψ22 = ψ23 ≈ ψ32 = ψ33 < ψdet < 0 and, ψ12 > ψ13 > 0. Besides, throughout the range of solution concentrations ψ11 = ψ21 = ψ31 = 0.Figure 10
The graphic illustration of the dependenceψij = f(C¯1,C¯2 = 37.71 mol·m−3) (i ≠ j, r = A, B) for the glucose in aqueous ethanol solution in conditions of concentration polarization of the membrane system, respectively: ψ22 = ψ23 = f(C¯1, C¯2 = const.)—curve 1; ψ32 = ψ33 = f(C¯1, C¯2  = const.)—curve 2; ψdet = f(C¯1, C¯2 = const.)—curve 3; ψ13 = f(C¯1, C¯2 = const.)—curve 4; and ψ12 = f(C¯1, C¯2 = const.)—curve 5.For example, we will consider equationψ22=ζ1A−ζ1B and dependencies ζ1r = f(C¯1, C¯2=const.) (r = A, B) presented in Figure 2. It is drawn from the equation and Figure 2 that if ψ22 = 0, then ζ1A = ζ1B = 0.234. Moreover, by taking expressions ζ1A = D1 (D1 + 2RTω11δA)−1 and ζ1B = D1 (D1 + 2RTω11δB)−1 into account in equation (15), one can show that ψ22=2RTω11δA−δBD11−1. From the equation, it becomes apparent that if ψ22 = 0, then δA = δB. Moreover, using equations δr=Dks1−ζir2RTωksζir−1and ρh−ρl = (∂ρ/∂C1)(C1h−C1l) + (∂ρ/∂C2)(C2h−C2l), where (∂ρ/∂C1) = 0.06 kg·mol−1, (∂ρ/∂C2) = −0.0095 kg·mol−1, we can show that, if ψ22 = 0, it is for C¯1  = 9.24 mol·m−3 (C1h = 33.44 mol·m−3 and C1l = 1 mol·m−3) and C¯2  = 37.71 m−3 (C2h = 201 mol·m−3 and C2l = 1 mol·m−3) we get ρh−ρl = 0.046 kg·m−3 and δA = δB ≈ 1.4 × 10−3 m.Taking these data into consideration as well asD11 = 0.69 × 10−9 m2·s−1, g = 9.81 m·s−2, ω11 = 0.8 × 10−9 mol·N−1 s−1, ν = 1.063 × 10−6 m2·s−1, ρl = 998.3 kg·m−3, ζ1A = ζ1B = ζ = 0.234, and δA = δB = δ = 1.4 × 10−3 m in the expression for the concentration Rayleigh number RC = [gρh−ρlδ3](ρhνhD11)−1 = [gρh−ρl1−ζD1123] [8ρhvhD14ζRTω113]−1 [31, 34], we get RC = 1691.09. The value is similar to the classical Rayleigh number critical value for Bernard problem RC = 1707.76 [38].


## 4. Conclusions

(1)
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. CoefficientsHijr and Hdetr = det [Hr] (for nonhomogeneous solutions), Hij and Hdet = det [H] (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, σ2 ω11, ω22, ω21, ω12, ζ1r, and ζ2r) for glucose in aqueous ethanol solutions and two configurations of the membrane system.
(2)
We can conclude that Peusner’s Network Thermodynamics (PNT) is an alternative manner of description of membrane transport both for homogeneity of solutions separated by a membrane and in conditions of concentration polarization.
(3)
The values of coefficientsH12r, H13r, H22r, H23r, H32r, H33r, and Hdetrdepend nonlinearly on solution concentration as well as on a configuration of membrane system. The values of these coefficients in the convective state are greater then their values in the nonconvective state.
(4)
The values of coefficientsH21r, H12, H21, H22, H33r, and Hdet depend linearly on solution concentration. The value of coefficients H13, H23, and H33 does not depend on solution concentration.
(5)
We can distinguish three groups of characteristicsHijr = f(C¯1, C¯2 = const.), Hij = f(C¯1, C¯2 = const.), and Hdetr = f(C¯1, C¯2 = const.), (i, j ∈ {1, 2, 3} and r = A, B). In the case of group 1, which includes concentration characteristics of coefficients H12r, H12, H21r, and H21 (r = A, B), these coefficients are expressed in the same units and their values are in the range −21.62 mol·m−3 ÷ 20.19 mol·m−3. In the case of the second group of characteristics, the shape of the concentration characteristics of the coefficients H13r, H13, H23r, H23, H33r, and H33 (r = A, B) is very similar. The values of coefficients H13r and H13 are expressed in different units than coefficients H23r, H23, H33r, and H33. The third group of characteristics includes the concentration characteristics of coefficients H22r, H22, H32r, H32, Hdetr, and Hdet. Values of coefficients Hdetr and Hdet are expressed in units other than H22r, H22, H32r, and H32.
(6)
There is a threshold value of concentrationC¯1 ≈ 9.24 mol·m−3 above which the values of coefficients ψ12, ψ13, ψ22 = ψ23, ψ32 = ψ33, and ψdet are equal to zero. For C¯1 < 9.24 mol·m−3, the values of coefficients ψ12 and ψ13 are negative and fulfill the conditions ψ22 = ψ23 ≈ ψ32 = ψ33 > ψdet > 0 and ψ12 > ψ13 < 0. For C¯1 > 9.23 mol·m−3 values of coefficients ψ12 and ψ13 are positive and fulfill the conditions ψ22 = ψ23 ≈ ψ32 = ψ33 < ψdet < 0 and, ψ12 > ψ13 > 0. 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. Besides, throughout the range of solution concentrations, ψ11 = ψ21 = ψ31 = 0.
(7)
Ifψ22 = 0, then ζ1A = ζ1B = 0.234 and it is conjugated with critical value of concentration Rayleigh number RC = [gρh−ρlδ3](ρhνhD11)−1 = [gρh−ρl1−ζD1123] [8ρhνhD14(ζRTω11)3]−1 = 1691.09. The value is similar to the classical Rayleigh number critical value for Bernard problem RC = 1707.76 [38].



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*Source: 5629259-2019-04-01.xml*