The Rr Form of the Kedem–Katchalsky–Peusner Model Equations for Description of the Membrane Transport in Concentration Polarization Conditions

The paper presents the Rr matrix form of Kedem–Katchalsky–Peusner equations for membrane transport of the non-homogeneous ternary non-electrolyte solutions. Peusner’s coefficients Rijr and det [Rr] (i, j ∈ {1, 2, 3}, r = A, B) occurring in these equations, were calculated for Nephrophan biomembrane, glucose in aqueous ethanol solutions and two different settings of the solutions relative to the horizontally oriented membrane for concentration polarization conditions or homogeneity of solutions. Kedem–Katchalsky coefficients, measured for homogeneous and non-homogeneous solutions, were used for the calculations. The calculated Peusner’s coefficients for homogeneous solutions depend linearly, and for non-homogeneous solutions non-linearly on the concentrations of solutes. The concentration dependences of the coefficients Rijr and det [Rr] indicate a characteristic glucose concentration of 9.24 mol/m3 (at a fixed ethanol concentration) in which the obtained curves for Configurations A and B intersect. At this point, the density of solutions in the upper and lower membrane chamber are the same. Peusner’s coefficients were used to assess the effect of concentration polarization and free convection on membrane transport (the ξij coefficient), determine the degree of coupling (the rijr coefficient) and coupling parameter (the QRr coefficient) and energy conversion efficiency (the (eijr)r coefficient).


Introduction
Membrane transport belongs to the group of processes described by thermodynamics of irreversible processes, now called modern thermodynamics. This theory was created and described by Lars Onsager, Theophile De Donder, Ilya Prigogine and others [1]. This field of knowledge has provided many research tools for transport mechanisms, including membrane transport, which is used in many areas of science (physics, biology, chemistry) and technology (biotechnology, biomedical engineering, water and sewage engineering, bioenergetics) [2][3][4][5][6][7][8][9]. One of the basic research tools for membrane transport are the Kedem-Katchalsky Equations (K-K Equations) derived from Onsager thermodynamics. The K-K Equations show the relationship between volume (J v ), solute (J s ) fluxes and thermodynamic forces (osmotic ∆π and/or hydrostatic ∆P) [10,11]. Currently, several versions of these equations classical form [12] and forms presented by Kargol and Kargol [13,14], Peusner [15], Elmoazzen et al. [16], Cheng and Pinsky [17] and Cardoso and Cartwright [18]. L r 32 . The aim of this paper is to develop the form of R r of the K-K Equations, containing the Peusner coefficients R r ij (i, j ∈ {1, 2, 3}, r = A, B). We will present the results of calculations of coefficients R r ij and R ij matrix coefficients R r det = det [R r ] and R det = det [R] and the quotients ξ ij = (R A ij − R B ij )/R ij and ξ det = (R A det − R B det )/R det which were obtained on the basis of experimentally determined coefficients (L p , σ 1 , σ 2 , ω 11 , ω 22 , ω 21 , ω 12 , ζ r 1 and ζ r 2 ) for glucose in aqueous ethanol solutions and Configurations A and B of the membrane system. These coefficients were calculated on the basis of experimentally measured volume (J r v ) and solute fluxes (J r k ) (k = 1, 2 and r = A, B) using the procedure described in [11,30,34]. Besides, we will present the results of calculations of the degree of coupling r ij = −R ij R ii R jj −0.5 (for homogeneous ternary nonelectrolyte solutions), r r ij = R r ij R r ii R r jj −0.5 (for non-homogeneous ternary nonelectrolyte solutions), coupling parameter Q R = r ij r ji 2 − r ij r ji

Theory
Similarly, as in previous papers (e.g., [42,43]), let us consider the membrane system presented in Figure 1. In this system the membrane (M) is located in horizontal plane and separates compartments (l) and (h) filled with non-homogeneous ternary non-electrolyte solutions with concentrations at the initial moment (t = 0) C kh and C kl (C kh > C kl , k = 1, 2). This membrane treated as a "black box" type is isotropic, symmetrical, electroneutral and selective for solvent and non-ionized dissolved substances. For a membrane located in a horizontal plane that is perpendicular to the gravity vector, two configurations of the membrane system are possible. These configurations are denoted by A and B. In Configuration A, the C kl solution is in the chamber above the membrane, and the C kh solution is in the chamber under the membrane. In Configuration B, the arrangement of the solutions relative to the membrane is reversed. described in [11,30,34]. Besides, we will present the results of calculations of the degree of coupling (for non-homogeneous ternary nonelectrolyte solutions) in which (i, j ∈ {1, 2, 3}, r = A, B).

Theory
Similarly, as in previous papers (e.g., [42,43]), let us consider the membrane system presented in Figure 1. In this system the membrane (M) is located in horizontal plane and separates compartments (l) and (h) filled with non-homogeneous ternary non-electrolyte solutions with concentrations at the initial moment (t = 0) Ckh and Ckl (Ckh > Ckl, k = 1, 2). This membrane treated as a "black box" type is isotropic, symmetrical, electroneutral and selective for solvent and non-ionized dissolved substances. For a membrane located in a horizontal plane that is perpendicular to the gravity vector, two configurations of the membrane system are possible. These configurations are denoted by A and B. In Configuration A, the Ckl solution is in the chamber above the membrane, and the Ckh solution is in the chamber under the membrane. In Configuration B, the arrangement of the solutions relative to the membrane 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 K-K Equations for ternary non-electrolyte solutions [42,43]. Under such conditions water and solutes which diffuse through the membrane create concentration boundary layers (CBLs), and , on both sides of the membrane [35][36][37]. The thicknesses of and are equal suitably to and . The mean concentrations of solutes "1" and "2" in membrane ( ̅ , ̅ ) can be calculated using expressions ̅ = (Ckh -Ckl)[ln(CkhCkl −1 )] −1 (k = 1, 2). Appearance of CBLs causes that concentrations at the interfaces of the membrane and solutions respectively decreases from Ckh to and increases from Ckl to ( > , > Ckl, Ckh > . k = 1, 2). We will consider only isothermal and stationary processes of membrane transport, for which the measure is the volume fluxes (J r v ) and solutes fluxes (J r k ) (k = 1, 2 and r = A, B). These fluxes can be described by the K-K Equations for ternary non-electrolyte solutions [42,43]. Under such conditions water and solutes which diffuse through the membrane create concentration boundary layers (CBLs), l r h and l r l on both sides of the membrane [35][36][37]. The thicknesses of l r h and l r l are equal suitably to δ r h and δ r l . The mean concentrations of solutes "1" and "2" in membrane (C 1 , C 2 ) can be calculated using expressions C k = (C kh -C kl )[ln(C kh C kl −1 )] −1 (k = 1, 2). Appearance of CBLs causes that concentrations at the interfaces of the membrane and solutions respectively decreases from C kh to C r kh and increases from C kl to C r kl (C r kh > C r kl , C r kl > C kl , C kh > C r kh . k = 1, 2). Let us denote by ρ r l and ρ r h the densities of solutions in the interfaces l r l /M and M/l r h while by ρ l and ρ h (ρ l < ρ h or ρ l > ρ h ) the density of solutions outside the CBLs. The following conditions can be saved for these densities: ρ r l > ρ l or ρ r l < ρ r h , ρ r l > ρ r h or ρ r l < ρ r h and ρ r h > ρ h or ρ r h < ρ h . If the solution with lower density is under the membrane, the system l r h /M/l r l loses its hydrodynamic stability and convective instabilities in near membrane area are observed [35][36][37]. The measure of the concentration polarization (CP) is the CP coefficient (ζ r k ). Using this coefficient, we can write the relation: C r kh − C r kl = ζ r k (C kh − C kl ). The value of coefficient ζ r k 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 δ r h and δ r l exceed values (δ r h ) crit and (δ r l ) crit and CP coefficient (ζ r k ) exceed its critical value (ζ r k ) crit suitably [42,43]. The dependency between the CP coefficient (ζ r k ) and the thickness of CBLs (δ r h and δ r l ) can be described by the following expression [37].
where (i, j ∈ {1, 2} and r = A, B = D ks . Besides, we can also assume that δ r h = δ r l = δ r . According to the Kedem-Katchalsky formalism [11] transport properties of the membrane are determined for solutions containing a solvent and two dissolved substances (ternary solution) by practical coefficients: hydraulic permeability (L p ), reflection (σ k , k = 1, 2) and permeability of solute (ω kf , k, f ∈ {1, 2}). In turn, the transport properties of the complex l r h /M/l r l are characterized by coefficients of hydraulic permeability (L r p ), reflection (σ r sk , σ r ak ) and permeability of solute (ω r k f ). The coefficients of hydraulic, osmotic, advective and diffusive concentration polarization are defined by expressions: ζ r p = L r p /L p , ζ r v = σ r sk /σ k , ζ r a = σ r ak /σ k and ζ r k = ω r k f /ω kf [26]. For osmotic volume and diffusive fluxes of homogeneous (evenly stirred) solutions, the values of volume (J v ) and solute (J k ) fluxes does not depend on the configuration of the membrane system. Besides, the dependencies J v = f (C kh − C kl ) and J k = f (C kh − C kl ) are linear, while J r v = f (C kh − C kl ) and J r k = f (C kh − C kl ) are nonlinear [33,43]. The formation of the layers l r l and l r h reduce the value of volume and solute fluxes from J v and J k (in conditions of homogeneous solutions) to J r v and J r k (in condition of CP), respectively. The Kedem-Katchalsky Equations for CP conditions can be written as: where J r v , J r 1 and J r 2 -volume and solutes "1" and "2" fluxes respectively, L p -hydraulic permeability coefficient, σ 1 and σ 2 -reflection coefficients suitably for solutes "1" or "2", ω 11 and ω 22 -solute permeability coefficients for solutes "1" or "2" generated by forces with indexes "1" or "2" and ω 12 and ω 21 -cross coefficients of permeability for substances "1" or "2" generated by forces with indexes "2" or "1" respectively. ∆P = P h − P l is the hydrostatic pressure difference (P h , P l are higher and lower values of hydrostatic pressure suitably). ∆π k = RT (C kh − C kl ) is the difference of osmotic pressure (RT is the product of gas constant and thermodynamic temperature whereas C kh and C kl are solutes concentrations, k = 1, 2). C k is the mean solute concentration in membrane and is expressed by C k = (C kh − C kl )[ln(C kh C kl −1 )] −1 (k = 1, 2). By means of this expression one can show that ∆π k /C k = ln (C kh C kl −1 ). Equations (2)-(4) are modified Kedem-Katchalsky Equations for ternary solutions [33]. The Equations (2)-(4) can be transformed by simple algebraic transformations to the matrix form of the Kedem-Katchalsky-Peusner equations for non-homogenous non-electrolyte ternary solutions: Index "r" in Equations (2)- (6) indicate that the fluxes J r v , J r 1 , J r 2 , Coefficients R r ij (i, j ∈ {1, 2, 3} and matrix [R r ] of these coefficients (R r form of the matrix of Peusner's coefficients), depend on configuration of the membrane system (r = A, B). From a formal point of view, the case of R r det = 0 is excluded, because in order for the denominator of Equation (6) to be different from zero, the condition ω 11 ζ r s11 ω 22 ζ r s22 ω 12 ζ r s12 ω 21 ζ r s21 must be satisfied. If ω 11 ζ r s11 ω 22 ζ r s22 > ω 12 ζ r s12 ω 21 ζ r s21 then R r det > 0, and if ω 11 ζ r s11 ω 22 ζ r s22 < ω 12 ζ r s12 ω 21 ζ r s21 then R r det < 0. In order to write Equations (5) and (6) for the conditions of homogeneity of solutions, the superscript "r" should be removed and assumption that the condition ζ r p = ζ r v1 = ζ r v2 = ζ r a1 = ζ r a2 = ζ r s11 = ζ r s12 = ζ r s22 = ζ r s21 = 1 is fulfilled. Then Equations (5) and (6) are taking the following form: where R r Besides the determinant of matrix [R] is given by the relationship: As in the case of Equation (6), the case of R det = 0 is excluded, because in order for the denominator of Equation (8) to be different from zero, the condition ω 11 ω 22 ω 12 ω 21 must be fulfilled. If ω 11 ω 22 > ω 12 ω 21 then R r det > 0, and if ω 11 ω 22 < ω 12 ω 21 then R det < 0.  [15] in the above equation, symmetry of non-diagonal coefficients (R ij = R ji i j) is not required. In the case considered above for non-diagonal coefficients, we have R 12 = R 21 , R 13 = R 31 only when ω 12 = ω 21 . Besides, from Equation (7) it results that R 23 = R 32 only when ω 12 C 1 = ω 21 C 2 .
In order to show the relations between coefficients R r ij and R ij and between determinants of matrixes [R r ] and [R] for A and B configurations of the membrane system (r = A, B) we calculate using Equations (4)-(7) the expressions: The values of coefficients ξ ij and ξ det show the influence of CP and natural convection (NC) on the membrane transport. These coefficients are a measure of the distance of convective processes from the critical state (non-convection). Assuming that the coefficients R A ij , R B ij , R ij , R A det , R B det , ξ ij and ξ det have the same sign, on the basis of Equations (9) and (10), we can write the criteria listed in Table 1.
In order to show the relationship between coefficients R ij , R ji , R ii and R jj and coefficients R r ij , R r ji , R r ii and R r jj for A and B configurations of membrane system we will calculate the Kedem-Caplan-Peusner (KCP) degree of coupling r ij and r r ij in which i, j ∈ {1, 2, 3}, superscript r = A, B, using Equations (5), (7), (11) and (12) [19,20]. The expressions for these coefficients take the following forms: The second law of thermodynamics imposes the conditions R r ii R r jj ≥ R r ij 2 and R r ii R r jj ≥ R r ji 2 which means that r r ij and r r ji is limited by the relation −1 ≤ r r ij , r r ji ≤ +1. For ternary solutions, taking into consideration Equations (5) and (11) and (7) and (12) Peusner proposed the "super Q R "-coupling parameter, defined by the following expression [15,21,22]:
In addition, it can be seen from the Figures 2 and 3 that for C 1 < 9.24 mol/m 3 and ∆ρ ≤ 0.046 kg/m −3 in Configuration A, the complex of CBLs is hydrodynamically unstable and in Configuration B-hydrodynamically stable, because the solutions of ethanol prevailing over glucose are under the membrane, and for that case the solution density under the membrane is lower than the solution density over the membrane. In Configuration B, the complex of CBLs is stable because density of the solution under the membrane is greater than the solution above the membrane. In turn for C 1 > 9.24 mol/m 3 and ∆ρ > 0.046 kg/m −3 in Configuration A, the complex of CBLs is hydrodynamically stable, and in Configuration B-hydrodynamically unstable due to the fact that in solutions separated by the membrane, glucose concentration is greater than ethanol and density of solution under the membrane is greater than the solution over the membrane. In Configuration B, the complex of CBLs is unstable because density of the solution under the membrane is smaller than the solution above the membrane. This causes the convection movements vertically downward. For C 1 = 9.24 mol/m 3 and ∆ρ = 0.046 kg/m −3 the CBL S complex is independent of the membrane system configuration and therefore ζ A In Configuration A, a non-convective state occurs, when the density of the solution in the compartment above the membrane is higher than density of the solution in the compartment under the membrane. In Configuration A natural convection occurs when ρ l > ρ A e , ρ A i > ρ h and ρ A e > ρ A i and is directed vertically upwards. On the other hand, in Configuration B, a natural convection occurs when ρ l < ρ B e , ρ B i < ρ h and ρ B e < ρ B ei and is directed vertically downwards [38]. Natural convection allows it to increase the value fluxes of J r vk and J r k .
The Graphs 1A, 1B, 2A and 2B present dependencies Figure  5, were obtained suitably for Configurations A and B of the membrane system, respectively. In the The Graphs 1A, 1B, 2A and 2B illustrating dependencies Figure 5, were obtained suitably for Configurations A and B of the membrane system, respectively. In the case of Configuration A, the value of coefficients  (Lines 1 and 2).
The Graphs 1A, 1B, 2A and 2B present dependencies R A 12 = f (C 1 , C 2 = 37.71 mol/m 3 ), R B 12 = f (C 1 , C 2 = 37.71 mol/m 3 ), R A 21 = f (C 1 , C 2 = 37.71 mol/m 3 ) and R B 21 = f (C 1 , C 2 = 37.71 mol/m 3 ) presented in Figure 5, were obtained suitably for Configurations A and B of the membrane system, respectively. In the case of Configuration A, the value of coefficients  (Lines 1 and 2).   Figure  6, were obtained suitably for Configurations A and B of the membrane system, respectively. The dependencies shown in this figure are similar to the dependencies shown in Figure 5.   Figure 6, were obtained suitably for Configurations A and B of the membrane system, respectively. The dependencies shown in this figure are similar to the dependencies shown in Figure 5.

for
, Curve 2B-for , Line 1-for R21 and Line 2-for R31.  Figure  6, were obtained suitably for Configurations A and B of the membrane system, respectively. The dependencies shown in this figure are similar to the dependencies shown in Figure 5.      To calculate coefficients and , the Equations (6) and (8) were used, respectively. The Graphs 1A and 1B presented in Figure 10 (9) and (10) were used, respectively. The graph presented in Figure 11 illustrating the dependencies = f( ̅ , ̅ = 37.71 mol/m 3 ) was calculated on the basis of Equation (9). In that case the value of coefficient initially decreases to = −4.8 (for ̅ = 1.44 mol/m 3 ) and next increases nonlinearly to = 5.41 (for ̅ = 13.66 mol/m 3 ) and then increases linearly to = 6.39 (for ̅ =   Figure 9, Graphs 1A and 1B illustrating the dependencies R A 33 = f (C 1 , C 2 = 37.71 mol/m 3 ) and R B 33 = f (C 1 , C 2 = 37.71 mol/m 3 ) were obtained for Configurations A and B of the membrane system. The value of coefficient R A 33 increases nonlinearly from R A 33 = 0.37 × 10 9 m 3 Ns/mol 2 (for C 1 = 1.44 mol/m 3 ) to R A 33 = 6.51 × 10 9 m 3 Ns/mol 2 (for C 1 = 13.66 mol/m 3 ). For C 1 > 13.66 mol/m 3 R A 33 = 6.61 × 10 9 m 3 Ns/mol 2 and is constant. The value of coefficient R B 33 in Configuration B of the membrane system initially is constant and for C 1 > 1.44 mol/m 3 increases nonlinearly from R B 33 = 6.61 × 10 9 m 3 Ns/mol 2 (for C 1 = 0.69 mol/m 3 ) to R B 33 = 0.38 × 10 9 m 3 Ns/mol 2 (for C 1 = 13.66 mol/m 3 ) and next achieves constant value R B 33 = 0.37 × 10 9 m 3 Ns/mol 2 (for C 1 > 13.66 mol/m 3 ). Besides R A 33 = R B 33 = 0.82 × 10 9 m 3 Ns/mol 2 for C 1 = 9.24 mol/m 3 and C 2 = 37.71 mol/m 3 . For homogeneous solutions R A 33 = R B

33
= R 33 , R 33 = 0.18 × 10 9 m 3 Ns/mol 2 (for C 1 = 0.69 mol/m 3 ). Besides, it follows from this figure that for C 1 < 9.24 mol/m 3 R A 33 < R B 33 and for C 1 > 9.24 mol/m 3 R A 33 > R B 33 . The curves presented in Figures 2-9, marked with a number and letters A or B, show that there are transition points from a linear wave to a non-linear wave or vice versa. It is related to the change of the nature of membrane transport from osmotic-diffusion to osmotic-diffusion-convective, or-inversely. The mechanism of this process is as follows. As the concentration of glucose increases at a given concentration of ethanol, the density of the solution, filling the compartment under the membrane in Configuration B, increases. If the density of this solution is lower than the density of the solution filling the compartment above the membrane, natural convection occurs in Configuration B, which causes destruction of CBLs, increasing driving forces and increasing the value of the coefficient. The addition of glucose stabilizes the layers and finally eliminates natural convection and changes the nature of transport from osmotic-diffusion-convective to osmotic-diffusion. In Configuration A, the process of creating gravitational convection is in the reverse order. This means that in Configuration A we have a transition from non-convective to convective, and in Configuration B-from convective to non-convective states. These transitions have a pseudo-phase transition character.
In all cases of the dependencies, R r ij = f (C 1 , C 2 = 37.71 mol/m 3 ) (i, j ∈ {1, 2, 3}, r = A or B) and R r det = f (C 1 , C 2 = 37.71 mol/m 3 ), (r = A or B) shown in Figures 5-10 show clearly that their values are determined by the hydrodynamic conditions in solutions near membrane which separates ternary non-electrolytes with different concentrations. It means that values of these coefficients in concentration polarization conditions are strongly connected with concentrations C 1 and C 2 and configuration of the membrane system. In turn, in the case of mechanical stirring of solutions, the values of these coefficients depend only on concentrations C 1 and C 2 . Therefore, for interpretation of calculation results, the combinations of coefficients R A ij , R B ij and R ij (i, j ∈ {1, 2, 3) of the same indicators and R A det , R B det and R det were used. These combinations are presented by Equations (5) Table 2.
From the results presented in Figures 4-10, it also appears that the R r ij and R r det (i, j ∈ {1, 2, 3}, r = A, B), have different physical significance. The unit of coefficients R r 11 , R r 21 and R r 31 is Ns/m 3 . Therefore, they have the character of flow resistance coefficients (hydraulic resistance). In turn, the unit of coefficients R r 12 i R r 13 is Ns/mol, what makes them coefficients of flow resistance of dissolved substances (diffusion resistance). The unit of coefficients R r 22 , R r 23 , R r 32 and R r 33 is m 3 Ns/mol 2 . This unit is a measure of the ratio of diffusion resistance to concentration. The unit of the coefficient R r det is m 3 N 3 s 3 /mol 4 . It corresponds to the ratio of diffusion resistance raised to the power of third and concentration.

Concentration Dependencies of r r
ij , r ij , e r ij , e ij , Q r R and Q R Figures 14-16 show the dependences r r ij = f (C 1 , C 2 = 37.71 mol/m 3 ) and r ij = f (C 1 , C 2 = 37.71 mol/m 3 ), (i, j ∈ {1, 2, 3} and r = A, B) calculated on the basis of Equations (11) and (12) and data presented in Figures 4-9. Figure 14 shows that Curves 1A and 1B intersect at a point with coordinates: r A 12 = r B 12 = 0.36 and C 1 = 9.15 mol/m 3 , and the Curves 2A and 2B-at a point with coordinates: r A 21 = r B 21 = 0.35 and C 1 = 9.33 mol/m 3 . The course of Curves 1A, 1B and 1 shows that for C 1 < 9.15 mol/m 3 , r B 12 > r A 12 > r 12 and for C 1 > 9.15 mol/m 3 , r A 12 > r B 12 >r 12 . Similarly, Curves 2A, 2B and 2 show that for C 1 < 9.33 mol/m 3 , r B 21 > r A 21 > r 21 and for C 1 > 9.33 mol/m 3 , r A 21 > r B 21 > r 21 . Curves 1B and 2B have maxima. The coordinates of the maximum of Curve 1B are r B 12 = 0.48 and C 1 = 6.53 mol/m 3 . In turn, the coordinates of maximum of the 2B curve are r B 21 = 0.46 and C 1 = 7.14 mol/m 3 . This means that the maximum of Curve 1B is shifted relative to the maximum of Curve 2B vertically by (r B 12 − r B 21 ) = 0.02 and horizontally by ∆C 1 = 0.61 mol/m 3 . In addition, Curves 1A and 2A and Curves 1B and 2B are shifted relative to each other, except for the point with coordinates r A 12 = r A 21 = 0.14 and C 1 = 2.15 mol/m 3 . This means that for C 1 < 2.15 mol/m 3 r A 21 = r A 12 while for C 1 > 2.15 mol/m 3 r A 12 = r A 21 . Curves 1B and 2B coincide on the section with coordinates r B 12 = r B 21 = 0.48 and C 1 = 8.03 mol/m 3 and r B 12 = r B

21
= 0.33 and C 1 = 9.47 mol/m 3 . For C 1 < 8.03 mol/m 3 and C 1 > 9.47 mol/m 3 the condition r B 12 > r B 21 is fulfilled. Curves 1 and 2 show that the condition r 12 = r 21 is fulfilled.        From the course of curves shown in Figure 16, it follows that r A 23 = r B 23 = r 23 and r A 32 = r B 32 = r 32 . Curves 1 and 2 and 1A and 2B intersect at a point with coordinates r 23 = r 32 = r A 23 = r B 32 = 0.11 × 10 −3 and C 1 = 2.33 mol/m 3 , while Curves 1B and 2A-at a point with coordinates r B 23 = r A 32 = 0.12 and C 1 = 2.6 mol/m 3 . For C 1 < 2.33 mol/m 3 , r 23 = r A 23 = r B 23 > r 32 = r A 32 = r B 23 = r B 32 and for C 1 > 2.33 mol/m 3 , r 23 = r A 23 = r B 23 > r 32 = r A 32 = r B 32 . As can be seen, the values of the coefficients r ij , r r ij , r ji and r r ji (i, j ∈ {2, 3} and r = A, B) ( Figure 16) are three orders of magnitude smaller than the values of the coefficients r ij , r r ij , r ji and r r ji (i, j ∈ {1, 2} and r = A, B) and r ij , r r i j , r ji and r r ji (i, j ∈ {1, 3} and r = A, B) (Figures 14 and 15).     Figure 8a,b and Figure 9a fulfilled the conditions 0 ≤ r ij ≤ 1, 0 ≤ r r ij ≤ 1, 0 ≤ r ji ≤ 1 and 0 ≤ r r ji ≤ 1 determined by Roy Caplan [20]. Graphs in Figures 14 and 15 have characteristic shapes, depending on the configuration of the membrane system and the properties of the solutions. In the case of homogeneous solutions (mechanically stirred solutions-Graphs 1 and 2), the coefficients do not depend on the configuration of the membrane system and are approximately linearly dependent on the concentration of glucose. This means that mechanical stirring of solutions at a sufficiently high speed eliminates CBL creation and causes maximization of fluxes and forces on the membrane. In the case of heterogeneous solutions (without mechanical stirring of the solutions in the chambers), the appearance of CBL near the membrane, reduces the value of the respective fluxes and increases the value of coupling factors for the same concentrations of solute in relation to homogeneous conditions. In addition, coupling coefficients for heterogeneous conditions strongly depend on the membrane configuration.
In Configuration A, the increase in glucose at a constant ethanol concentration at the beginning causes an increase in the coupling coefficients. In Configuration B, an increase in glucose causes a decrease in the value of coupling coefficients. The range of glucose concentrations for which the change in coupling coefficients in Configuration B is maximum is within the range similar to Configuration A of the membrane system. Analyzing the characteristics of coupling coefficients in heterogeneous conditions, we observed that for the respective characteristics in the A and B configurations of the membrane system, the respective graphs pairs (1A and 1B, 2A and 2B) intersect at a concentration of about 9.2 mol m −3 . At this glucose concentration, the densities of the ternary solutions in the upper and lower chambers at the initial moment are the same. In this case, we observed the appearance of hydrodynamic instabilities that cause a disturbance of CBL diffusion reconstruction. Despite the fact that the solution densities were the same at the initial moment, the diffusion of glucose and ethanol through the membrane caused the appearance of sufficiently large and concentration gradients (and density gradients) in opposite direction to the gravitational field in the CBL areas. These gradients can cause hydrodynamic instabilities in the membrane system. Figure 17 show that in the case of heterogeneous solutions (solutions not mechanically mixed-Graphs 1A and 1B, 2A and 2B), the coupling factors do not show their dependence on the configuration of the membrane system. Perhaps, because their value is very small.  (13) and (14) and data presented in Figures 14-16. Figure 17 shows that Curves 1A and 1B intersect at a point with coordinates: e A          From the course of Curves 1A, 1B and 1 presented in Figure 19 it follows that e A     The results of experimental research indicate that ω 11 >> ω 12 , ω 22 >> ω 21 , ζ r p = ζ r a1 = ζ r a2 = =1, ζ r v1 = ζ r s11 = ζ r s12 = ζ r 1 and ζ r v2 = ζ r s22 = ζ r s21 = ζ r 2 (r = A, B). By accepting the above conditions and that ζ r 1 ≈ ζ r 2 = ζ r . Given this condition, and Equations (5), (9) and (10), we can write:

Graphs in
Equations (17)- (20) contain the factor ζ A and Equation (22)-the factor This factor, using Equation (1) can be written in a form containing the thickness of CBLs. To simplify the accounts, using the conditions D r ij l = D r ij h = D ij and δ r h = δ r l = δ r , we write the Equation (1) in the form: Using Equation (22) we can write: From all the foregoing considerations, it is clear that coefficients ξ ij (i, j ∈ {1, 2, 3} and ξ det are measures of the natural convection effect. If the conditions ξ ij < 0 and ξ det < 0 are fulfilled, fluxes of natural convection in single-membrane system are directed vertically upwards. In turn, for coefficients ξ ij > 0 and ξ det > 0, the fluxes are directed vertically downwards. Zeroing of the coefficients (ξ ij = 0 and ξ det = 0) means that the system is in the critical point where the flux turns its direction from vertically upwards to vertically downwards. In this point, the structure of layers lose its stability, but natural convection does not have precise turn yet, what means that the membrane system is not sensitive to changes in the gravitational field. This is shown by dependencies ξ ij = f (C 1 , C 2 = 37.71 mol/m 3 ), (i, j ∈ {1, 2, 3} and ξ det = f (C 1 , C 2 = 37.71 mol/m 3 ), presented in Figures 11-13 as well as interferograms presented in the previous publication [37,38]. Hydrodynamic stability in the membrane system is controlled by the concentration Rayleigh number [34][35][36][37][38]. The Rayleigh number value depends on the concentration of solutions separated by the membrane [34,35]. For the points where ξ ij = 0 and ξ det = 0, (i, j ∈ {1, 2, 3}) the critical value of concentration Rayleigh number (R C ) can be specified.

Conclusions
From the above presented studies, the following results are obtained:

1.
In order to describe transport processes of ternary solutions of nonelectrolytes through horizontally oriented membrane, nine Peusner's coefficients should be calculated R r ij (i, j ∈ {1, 2, 3}, r = A, B) and the determinant of the matrix of these coefficients is det [R r ] = R r det . For the Nephrophan membrane and aqueous solutions of glucose and ethanol, the values of coefficients R r ij (i, j ∈ {1, 2, 3}, r = A, B) and R r det are dependent on concentration solutions and configuration of the membrane system. For i j these coefficients fulfill the relations R r ij R r ji .

2.
Concentration dependencies of coefficients ξ ij = (R A ij − R B ij )/R ij = f (C 1 , C 2 = 37.71 mol/m 3 ) and ξ det = (det [R A ] -det [R B ])/det [R]) = f (C 1 , C 2 = 37.71 mol/m 3 ) facilitate estimation of natural convection direction: for ξ ij < 0, natural convection is directed vertically upwards and for ξ ij > 0-vertically downwards. The value of coefficients ξ ij and ξ det (ξ ij < 0, ξ det < 0, ξ ij = 0, ξ det = 0, ξ ij > 0 or ξ det > 0) shows the influence of concentration polarization and natural convection on the membrane transport. For ξ ij = 0 the critical value of the concentration Rayleigh number (R C ) can be estimated, for the point where convective stream changes its direction from vertical upwards into vertical downwards. The R C value estimated in this paper for the considered case amounts to (R C ) crit. = 1353.1.

3.
For Curves marked with a number and the letters A or B are evidence that there are transition points associated with the change in the nature of membrane transport from osmotic-diffusion to osmotic-diffusion-convective or vice versa. This means that in Configuration A, we have a transition from convective to convective, and in Configuration B-from convective to non-convective. These transitions are a pseudo-phase transition. 5.
The presented equations are a new research tool for membrane transport and the influence of gravity field on this transport. Funding: This research received no external funding.