The Role of Gravity in the Evolution of the Concentration Field in the Electrochemical Membrane Cell

The subject of the study was the osmotic volume transport of aqueous CuSO4 and/or ethanol solutions through a selective cellulose acetate membrane (Nephrophan). The effect of concentration of solution components, concentration polarization of solutions and configuration of the membrane system on the value of the volume osmotic flux (Jvir) in a single-membrane system in which the polymer membrane located in the horizontal plane was examined. The investigations were carried out under mechanical stirring conditions of the solutions and after it was turned off. Based on the obtained measurement results Jvir, the effects of concentration polarization, convection polarization, asymmetry and amplification of the volume osmotic flux and the thickness of the concentration boundary layers were calculated. Osmotic entropy production was also calculated for solution homogeneity and concentration polarization conditions. Using the thickness of the concentration boundary layers, critical values of the Rayleigh concentration number (RCr), i.e., the switch, were estimated between two states: convective (with higher Jvir) and non-convective (with lower Jvir). The operation of this switch indicates the regulatory role of earthly gravity in relation to membrane transport.


Introduction
The membrane is a selective barrier separating the interior of the cell from its surroundings and plays a key role in the biological cell [1]. Attempts have been made to apply some features of cell membranes in membrane technologies used in various fields of science, technology and medicine as well as in various industries for a long time. Therefore, studies on membrane transport processes are carried out in order to learn, among others, mechanisms of transport across cell membranes or the development of membrane technologies and techniques useful in medicine (hemodializer) and industrial technologies (bioreactors, biorefineries, modules for food processing and water treatment, wastewater treatment, etc.) [2]. Polymers constitute the majority of film-forming materials: polymers highly stable (e.g., polybenzimidazole, polyamide, polytriazole, cellulose acetate, cellulose triacetate, etc.) and biodegradable polymers (e.g., poly/lactic acid, cellulose, bacterial cellulose, chitozan, etc.) [3]. They provide membrane materials for osmotic-based membrane system [4,5].
The membrane diffusion processes occurring spontaneously in real conditions are accompanied by the phenomenon of concentration polarization [6][7][8][9]. It consists in changing the concentration field or density of solutions in the areas on both sides of the membrane caused by the creation of concentration boundary layers. These layers significantly reduce membrane transport, which leads to a reduction in the efficiency of membrane processes in industrial technologies [2]. In biological systems as well as microchip systems of artificial membranes, concentration creation can have positive impact due to the spontaneous regulatory properties of the value of flux through the membrane, which in turn translates into slowing the source of entropy, and thus, slowing down the aging of the system [10]. S-entropy is the only general physical quantity that indicates irreversible and one-way flow of processes, including biological processes [11]. This means that entropy is produced in any non-equilibrium thermodynamic system, including membrane systems. Local entropy production is the sum of four contributions: thermal, diffusion, viscous and chemical [12]. Under isothermal, non-viscous conditions and without chemical reactions, the diffusion contribution plays a major role. It also applies to membrane transport processes.
All "earthly phenomena" occur in the resultant gravitational field, whose main source is the Earth, the Moon and the Sun. The research into the impact of gravity on the concentration (density) field, generated in the environment of the separation membrane of non-mechanically mixed solutions, began in the 1970s. In 1972, the pioneering paper of S. Przestalski and M. Kargol about the discovery of the phenomenon of graviosmosis was published [8]. These studies were undertaken and continued by researchers directly or indirectly associated with the scientists. So far, several hundred papers on this issue have been published [7,[13][14][15][16][17][18][19][20].
In previous papers, the results of experimental studies on the volume osmotic flux (J r vi , r = α, β, i = 1, 2) and solute flux (J r i , r = α, β, i = 1, 2) were presented. The solutions separated by the membrane contained aqueous solutions of glucose and/or ethanol [15,16], potassium chloride and/or ammonia [19]. The first of these substances causes an increase in and the second decreases the density of solutions. The characteristics of J r vi = f (∆C i , r = α, β, i = 1, 2) and J r i = f (∆C i , r = α, β, i = 1, 2) presented in these papers are non-linear and show typical transitions from convective to non-convective state and inversely. However, for the same membrane, they differ in terms of details that are related to the physico-chemical properties of the solutions. These papers also showed that the value of the volume osmotic flux depends on the membrane transport properties, the configuration of the membrane system as well as the physicochemical properties and composition of the solutions separated by the membrane. The common feature of these transports is that the value of this flux is higher in convective than non-convective conditions.
The purpose of the present paper was to investigate the effect of earthly gravity on concentration fields in the membrane areas. To achieve this goal, the authors will determine volume osmotic fluxes (J r vi ) in a single-membrane system, in which a Nephrophan membrane (used in plate hemodialyzers) located in a horizontal plane, separates water and a ternary solution consisting of water, CuSO 4 and/or ethanol. In addition, the authors will examine the effect of the concentration of individual solution components and the configuration of the membrane system on the value of J r vi . The study will be carried out under conditions of mechanical mixing of the solutions and after it has been turned off. Based on the obtained measurement results J r vi , the authors will calculate the effects of: concentration polarization, natural convection, asymmetry and amplification of the volume osmotic flux, as well as the thickness of concentration boundary layers. The authors will also calculate the osmotic entropy production for solution homogeneity and concentration polarization conditions as well as interpret the results obtained using the osmotic concentration polarization factor (ζ r i ). This factor, through the concentration permeability coefficient of the boundary layer (ω r o ), treated as a liquid membrane with a reflection coefficient equal to zero, will be related to the thickness of the concentration boundary layers. The thickness of these layers will be used to estimate the Rayleigh concentration number (R r C ), i.e., the parameter controlling the transition from non-convective to convective state. The Rayleigh concentration number acts as a switch between two states: convective (with higher J r vi ) and non-convective (with lower J r vi ). The operation of this switch indicates the regulatory role of earthly gravity in relation to membrane transport.

Electrochemical Membrane Cell
Let us consider membrane transport in a physicochemical cell, shown in Figure 1. In this cell, the membrane (M), arranged in a horizontal plane, at the initial moment (t 0 = 0), separated two homogeneous solutions of the same non-electrolytic substance with concentrations C ui i C di (C ui > C di ). If the membrane in question is isotropic, symmetrical, electro-neutral and selective for water and solute, its transport properties are characterized only by the coefficients: hydraulic permeability (L p ), reflection (σ i ) and permeability of solute (ω i ) [21]. For times satisfying the condition t > t 0 , on both sides of the membrane, the creation of concentration boundary layers begins, which change the concentration field in the areas around the membrane, generating concentration polarization [6,21].

Electrochemical Membrane Cell
Let us consider membrane transport in a physicochemical cell, shown in Figure 1. In this cell, the membrane (M), arranged in a horizontal plane, at the initial moment (t0 = 0), separated two homogeneous solutions of the same non-electrolytic substance with concentrations Cui i Cdi (Cui > Cdi). If the membrane in question is isotropic, symmetrical, electro-neutral and selective for water and solute, its transport properties are characterized only by the coefficients: hydraulic permeability (Lp), reflection (σi) and permeability of solute (ωi) [21]. For times satisfying the condition t > t0, on both sides of the membrane, the creation of concentration boundary layers begins, which change the concentration field in the areas around the membrane, generating concentration polarization [6,21]. The nature of the concentration field in the areas around the membrane is determined by the density of the solutions separated by the membrane. If the density of the solution with Cui concentration reaches a critical value in relation to the density of the solution with Cdi concentration, then the concentration field changes its nature from diffusive to diffusion -convective. Under the conditions of the diffusion field of concentration, the concentration of the solution, which initially was Cui, decreases to the value  or  , and the concentration of the solution, which initially was Cd, increases to the value of  or  (  >  ,  >  ). In turn, under the conditions of diffusion-convective concentration field, the concentration of the solution that initially amounted to Cui decreases to the value  or  , and the concentration of the solution that initially amounted to Cdi increases to the value of  or Therefore, under the conditions of the diffusion field of concentration, on both sides of the membranes there are concentration boundary layers l  , l  , l  and l  under conditions of the diffusion-convective field of concentration; concentration boundary layers l  , l  , l  and l  . The thickness of the layers l  , l  , l  and l  is much smaller than the layers l  , l  , l  i l  . The thicknesses of layers are denoted by   ,   ,   i   respectively. The concentration boundary layers are treated as pseudomembranes, whose transport properties are determined by the coefficients  =  =  =  = 0 and   ,   ,   and   . The volume flux through the complexes l  /M/l  and l  /M/l  will be denoted by  and  . respectively. Membrane volume transport processes occurring under the conditions of concentration polarization of areas on both sides of the membrane can be described using the first Kedem-Katchalsky equation (for volume flux) [21]. For the homogeneity conditions of diluted electrolyte solutions, this equation can be written as follows. The nature of the concentration field in the areas around the membrane is determined by the density of the solutions separated by the membrane. If the density of the solution with C ui concentration reaches a critical value in relation to the density of the solution with C di concentration, then the concentration field changes its nature from diffusive to diffusion -convective. Under the conditions of the diffusion field of concentration, the concentration of the solution, which initially was C ui , decreases to the value C αd ui or C βd ui , and the concentration of the solution, which initially was C d , increases to the value of C αd . In turn, under the conditions of diffusion-convective concentration field, the concentration of the solution that initially amounted to C ui decreases to the value C αk ui or C βk ui , and the concentration of the solution that initially amounted to C di increases to the value of C αk  [21]. For the homogeneity conditions of diluted electrolyte solutions, this equation can be written as follows.
In turn, for concentration polarization conditions, this equation will take the form [19] In the above equation, the coefficients of the hydrostatic permeability of the solvent and the reflection of the solute are respectively denoted by L p and σ i . In turn, ζ r p and ζ r i are the coefficients of pressure and osmotic concentration polarization, respectively. The symbol f i (1 ≤ f i ≤ 2) means the Vant Hoff coefficient. Expressions (P h; P l ) = ∆P and RT(C h ; C l ) = ∆π refer to the difference of respectively hydrostatic pressures (P h , P l ) and osmotic pressures on both sides of the membrane (RT is the product of gas constant and absolute temperature and C h and C l ; concentration of solutions). The coefficients ω α ui , ω α di , ω is the appropriate diffusion coefficient. The coefficients ζ r i , δ r u , δ r d , ω mi , D r ui and D r di are related by the equation [22] where: r = α or β and i = 1 or 2. This equation shows that the value of the coefficient ζ r i depends on the thickness of the concentration boundary layers δ r u i δ r d . The process of creating these layers can be followed using a Mach-Zehnder interferometer [7,22,23]. It is also possible, based on interferograms, to determine the time-spatial evolution of the concentration field and to determine the time dependence of the concentration thicknesses of boundary layers [24]. The process of transition from diffusion to convective concentration field can be controlled by the Rayleigh concentration number (R C ) [25]. Assuming that δ r u = δ r d = δ r 0 , D r ui = D r di = D i this number for ternary solutions can be described by the equation [26,27] where g is the gravitational acceleration; ρ i is the mass density, ν i is the kinematic viscosity of fluid, is the variation of density with the concentration.
Entropy is produced in every membrane system, including the biological one. In the case where the driving forces in the membrane system are the differences in hydrostatic pressure (∆p) and osmotic pressure (∆π k ), entropy production (P r S ) can be described by the equation [10,11] where: J r i is the flux of i-th solute, C i = (C ui − C di ) ln(C ui C di −1 ] −1 is the average solution concentration.

Methodology for Measuring the Volume Flux
The study of the volume osmotic flux (J r vi ) was carried out using the measuring set described in the previous paper [18]. This set consisted of two cylindrical measuring vessels (U, D) made of Plexiglas with a volume of 200 cm 3 each. Vessel U contained the tested binary or ternary solution, while vessel D had pure water. As binary solutions, aqueous CuSO 4 solutions or aqueous ethanol solutions were used. The ternary solutions were ethanol solutions in an aqueous CuSO 4 solution or CuSO 4 solutions in an aqueous ethanol solution. It should be noted that the density of aqueous ethanol solutions is less than the density of water, and the density of the aqueous solution of CuSO 4 is greater than the density of water. In turn, the density of ethanol solutions in aqueous CuSO 4 and the density of CuSO 4 solutions in aqueous ethanol may be less than, equal to or greater than the density of water.
The U and D vessels were separated by a cellulose acetate membrane called Nephrophan situated in a horizontal plane with an area of S = 3.36 cm 2 and transport properties determined, in accordance with Kedem and Katchalsky formalism, by the factors: hydraulic permeability (L p ), reflection (σ i ) and diffusion permeability (ω i ). The Nephrophan membrane is the microporous, highly hydrophilic polymeric filter used in medicine (VEB Filmfabrik, Wolfen, Germany). This membrane is made of cellulose acetate (cello-triacetate (OCO-CH 3 ) n ) [28,29]. The electron microscope image of surface and cross-section of these membrane it was presented in ref. [18]. The values of these coefficients for CuSO4 (index 1) and ethanol (index 2), determined in a series of independent experiments, are: The U vessel was connected to a graduated pipette (K) positioned in a plane parallel to the membrane plane, which was used to measure the volume increase of the solution (∆V) filling the vessel. In turn, the vessel D was connected to the water reservoir (N) with adjustable height relative to the pipette K, which served to compensate for the hydrostatic pressure (∆p = 0) present in the measuring set.
Each experiment was performed for the α and β configuration of the membrane system. In the α configuration, the test solution was in the vessel above the membrane, and the water, in the vessel under the membrane. In the β configuration, the order in which the solution and water were positioned relative to the membrane was reversed. The flow tests consisted of measuring the volume increase (∆V) of the solution in the pipette K at 10 min intervals (∆t). For each configuration, the tests were carried out according to a two-step procedure [15]. In the first stage, the volume flux was determined under mechanical mixing conditions of the solutions separated through the membrane at a speed of 500 rpm. until steady state was achieved. The second stage began with switching off the mechanical stirring of the solutions and consisted in testing the flux until the second steady state was obtained. All the investigations of volume osmotic flows were carried out under isothermal conditions for T = (295 ± 0.5) K. The volume osmotic flux, which is a measure of the volume osmotic flows, was calculated on the basis of the measurement of the change in volume (∆V) in the pipette K occurring during ∆t, through the membrane surface area S, using the formula J r vi = (∆V r i )S −1 (∆t) −1 for conditions ∆p = 0. The volume osmotic fluxes always occurred from the solution with a lower concentration to the solution with a higher concentration. Investigations of volume osmotic flux in both configurations consisted in determining the for different concentrations and composition of solutions. Each measurement series was repeated three times. The relative error made in determining J r vi was not greater than 3%. Based on these characteristics, for the steady state, the characteristics

Results and Discussion
The results of the volume osmotic flux study for the conditions of homogeneity of solutions and conditions of concentration polarization of solutions separated by the membrane are presented in Figures 2 and 3. Figure 2 shows the experimental dependences J α v1 = f (∆C 1 , ∆C 2 = constant) and J for CuSO4 solutions in aqueous ethanol and the α and β configurations of the membrane system. Graphs 1, 3α and 3β were obtained for ΔC2 = 0, graphs 2, 4α and 4β; for ΔC2 = 750 mol m −3 . for ethanol solutions in aqueous CuSO4 and the α and β configurations of the membrane system. Graphs 1, 3α and 3β were obtained for ΔC1 = 0, graphs 2, 4α and 4β; for ΔC1 = 50 mol m −3 .
Adding a fixed amount of ethanol to aqueous CuSO4 solutions or a fixed amount of CuSO4 to aqueous ethanol solutions causes a parallel shift of the line (1) by a constant and positive volume flux.
The concentration characteristics of the volume flux for concentration polarization conditions look completely different (after switching off the mechanical stirring of solutions). Graphs 3α and 3β presented in Figure 3 show that an increase in ΔC1 value in binary solutions (water solutions of CuSO4) for ΔC2 = 0, except for the segment 0 < ΔC1 ≤ 50 mol m −3 , causes a linear increase in the fluxes and ( > ). In turn, graphs 4α and 4β show that, unlike binary solutions, an increase in ΔC1 in ternary solutions (ΔC2 = 750 mol m −3 ), causes a non-linear increase in the value of the flux for the α configuration and an initial increase followed by a non-linear decrease in value flux for the β configuration of the membrane system. In the case of the 4α curve shown in  for CuSO4 solutions in aqueous ethanol and the α and β configurations of the membrane system. Graphs 1, 3α and 3β were obtained for ΔC2 = 0, graphs 2, 4α and 4β; for ΔC2 = 750 mol m −3 . for ethanol solutions in aqueous CuSO4 and the α and β configurations of the membrane system. Graphs 1, 3α and 3β were obtained for ΔC1 = 0, graphs 2, 4α and 4β; for ΔC1 = 50 mol m −3 .
Adding a fixed amount of ethanol to aqueous CuSO4 solutions or a fixed amount of CuSO4 to aqueous ethanol solutions causes a parallel shift of the line (1) by a constant and positive volume flux.
The concentration characteristics of the volume flux for concentration polarization conditions look completely different (after switching off the mechanical stirring of solutions). Graphs 3α and 3β presented in Figure 3 show that an increase in ΔC1 value in binary solutions (water solutions of CuSO4) for ΔC2 = 0, except for the segment 0 < ΔC1 ≤ 50 mol m −3 , causes a linear increase in the fluxes and ( > ). In turn, graphs 4α and 4β show that, unlike binary solutions, an increase in ΔC1 in ternary solutions (ΔC2 = 750 mol m −3 ), causes a non-linear increase in the value of the flux for the α configuration and an initial increase followed by a non-linear decrease in value flux for the β configuration of the membrane system. In the case of the 4α curve shown in  The concentration characteristics of the volume flux for concentration polarization conditions look completely different (after switching off the mechanical stirring of solutions). Graphs 3α and 3β presented in Figure 3 show that an increase in ∆C 1 value in binary solutions (water solutions of CuSO 4 ) for ∆C 2 = 0, except for the segment 0 < ∆C 1 ≤ 50 mol m −3 , causes a linear increase in the fluxes J α

The Effect of Concentration Polarization
The measure of the concentration polarization effect (∆J r vk ) is the equation where J vk is the volume osmotic flux determined for mechanical stirring conditions of solutions, J r vk is the volume osmotic flux determined for concentration polarization conditions, k = 1 or 2 and r = α or β. Figure 4 shows the dependence ∆J r v1 = f (∆C 1 , ∆C 2 = constant). This graph shows that for binary solutions ∆J

The Effect of Concentration Polarization
The measure of the concentration polarization effect (Δ ) is the equation where is the volume osmotic flux determined for mechanical stirring conditions of solutions, is the volume osmotic flux determined for concentration polarization conditions, k = 1 or 2 and r = α or β. Figure 4 shows the dependence Δ = (∆ , ∆ = constant). This graph shows that for binary solutions Δ  > Δ  in the whole range of ΔC1. In the case of ternary solutions Δ  > Δ  , for ΔC1 < 47 mol m −3 and Δ  < Δ  , for ΔC1 > 47 mol m −3 .

Convection Effect
The measure of convective effect (Δ ) is an equation where  is the volume flux determined for concentration polarization conditions of solutions and α configuration of the membrane system,  is the volume flux determined for the conditions of concentration polarization of solutions and configuration of the membrane system, k = 1 or 2. Figure 6 shows the dependence Δ = (∆ , ∆ = constant). This graph shows that for binary solutions (ΔC2 = 0) Δ > 0 in the whole range of ΔC1. For ternary solutions (ΔC2 = 750 mol m −3 ), Δ < 0 for ΔC1 < 47 mol m −3 and Δ > 0, for ΔC1 > 47 mol m −3 .

Convection Effect
The measure of convective effect (∆J vk ) is an equation where J α vk is the volume flux determined for concentration polarization conditions of solutions and α configuration of the membrane system, J β vk is the volume flux determined for the conditions of concentration polarization of solutions and configuration of the membrane system, k = 1 or 2. Figure 6 shows the dependence ∆J v1 = f (∆C 1 , ∆C 2 = constant). This graph shows that for binary solutions (∆C 2 = 0) ∆J vk > 0 in the whole range of ∆C 1 . For ternary solutions (∆C 2 = 750 mol m −3 ), ∆J vk < 0 for ∆C 1 < 47 mol m −3 and ∆J vk > 0, for ∆C 1 > 47 mol m −3 .
Entropy 2020, 22, 680 9 of 18 concentration polarization of solutions and configuration of the membrane system, k = 1 or 2. Figure 6 shows the dependence Δ = (∆ , ∆ = constant). This graph shows that for binary solutions (ΔC2 = 0) Δ > 0 in the whole range of ΔC1. For ternary solutions (ΔC2 = 750 mol m −3 ), Δ < 0 for ΔC1 < 47 mol m −3 and Δ > 0, for ΔC1 > 47 mol m −3 .   Figure 7 shows the dependence ∆J v2 = f (∆C 2 , ∆C 1 = constant). This graph shows that for binary solutions (∆C 1 = 0), ∆J v2 < 0 in the whole range of ∆C 2 . For ternary solutions (∆C 1 = 50 mol m −3 ), ∆J v2 > 0, for ∆C 2 < 750 mol m −3 , and ∆J v2 < 0, for ∆C 2 > 750 mol m −3 . It should be noted that the test results presented in Figures 6 and 7 are similar to the results of studies on the gravity-osmotic flux measured in a two-membrane system [14,15]. The membranes in this system were horizontally oriented and separated aqueous solutions of glucose and/or ethanol. The concentrations of these solutions met the condition C ui = C di < C mi (C ui , C di ; solution concentrations in the external compartments, C mi ; solution concentration in the inter-membrane compartment). The equivalent of such a membrane system is two single-membrane systems connected in parallel. It should be noted that the test results presented in Figures 6 and 7 are similar to the results of studies on the gravity-osmotic flux measured in a two-membrane system [14,15]. The membranes in this system were horizontally oriented and separated aqueous solutions of glucose and/or ethanol. The concentrations of these solutions met the condition Cui = Cdi < Cmi (Cui, Cdi; solution concentrations in the external compartments, Cmi; solution concentration in the inter-membrane compartment). The equivalent of such a membrane system is two single-membrane systems connected in parallel.

The Effect of Asymmetry of the Volume Osmotic Flux
The comparison of the 3α and 3β and 4α and 4β plots presented in Figures 2 and 3 shows the asymmetry of the volume osmotic fluxes, which is the evidence of the osmotic rectifying properties of the membrane system. The measure of this asymmetry is the asymmetry coefficients k1= / and k2 = / . The curves in Figures 8 and 9 show the characteristics of k1 = f(ΔC1, ΔC2 = constant) and k2 = f(ΔC2, ΔC1 = constant). Graphs 1 in Figures 8 and 9 illustrate the dependences k1 = f(ΔC1, ΔC2 = 0) and k2 = f(ΔC2, ΔC1 = 0). respectively. In turn, graphs 2 presented in these graphs illustrate the k1

The Effect of Asymmetry of the Volume Osmotic Flux
The comparison of the 3α and 3β and 4α and 4β plots presented in Figures 2 and 3 shows the asymmetry of the volume osmotic fluxes, which is the evidence of the osmotic rectifying properties of the membrane system. The measure of this asymmetry is the asymmetry coefficients k 1 = J α v1 /J β v1 and k 2 The curves in Figures 8 and 9 show the characteristics of k 1 = f (∆C 1 , ∆C 2 = constant) and k 2 = f (∆C 2 , ∆C 1 = constant). Graphs 1 in Figures 8 and 9 illustrate the dependences k 1 = f (∆C 1 , ∆C 2 = 0) and k 2 = f (∆C 2 , ∆C 1 = 0). respectively. In turn, graphs 2 presented in these graphs illustrate the k 1 = f (∆C 1 , ∆C 2 = 750 mol m −3 ) and k 2 = f (∆C 2 , ∆C 1 = 50 mol m −3 ). The values of k 1 and k 2 coefficients, different from unity, indicate that the tested membrane system has rectifying properties, which are manifested as the asymmetry of the volume osmotic flux.

The Effect of Amplification the Volume Osmotic Flux
The measure of the amplification effect of the osmotic volume flux is the amplification coefficient, the definition of which is the equation where (Δ ) is the volume flux increase for ternary solutions, (Δ ) is the volume flux increase for ternary solutions, k = 1 or 2 and r = α or β.

The Effect of Amplification the Volume Osmotic Flux
The measure of the amplification effect of the osmotic volume flux is the amplification coefficient, the definition of which is the equation where (Δ ) is the volume flux increase for ternary solutions, (Δ ) is the volume flux increase for ternary solutions, k = 1 or 2 and r = α or β. Figures 10 and 11 show the dependencies = ( ̅ , ∆ = constant), where ̅ = 0.5(Cj + Cj+1), j = 1, 2, …). Figure 10 shows that for binary solutions (ΔC2 = 0) > 0 in the whole range ̅ and takes values from = 2.1 to = 3.3. In the case of ternary solutions (ΔC2 = 750 mol m −3 ), the

The Effect of Amplification the Volume Osmotic Flux
The measure of the amplification effect of the osmotic volume flux is the amplification coefficient, the definition of which is the equation where (∆J r vk ) ternary is the volume flux increase for ternary solutions, (∆J r vk ) binary is the volume flux increase for ternary solutions, k = 1 or 2 and r = α or β. Figures 10 and 11 show the dependencies a r vk = f C 1 , ∆C 2 = constant , where C 1 = 0.5(C j + C j+1 ), j = 1, 2, . . . ). Figure 10 shows that for binary solutions (∆C 2 = 0) a r v1 > 0 in the whole range C 1 and takes values from a r v1 = 2.1 to a r v1 = 3.3. In the case of ternary solutions (∆C 2 = 750 mol m −3 ), the dependence a r vk = f C 1 , ∆C 2 = constant is nonlinear, with a clearly marked minimum, and the coefficient a r v1 is negative. The minimum of this dependence has the coordinates C 1 = 43.75 mol m −3 and a r v1 = −54. .  In turn, Figure 11 shows that for binary solutions (ΔC1 = 0), > 0 in the whole range ̅ and takes values from = 0.5 to = 1.4. In the case of ternary solutions (ΔC1 = 50 mol m −3 ), the dependence = ( ̅ , ∆ = constant) is non-linear, with the maximum clearly indicated, and the .  In turn, Figure 11 shows that for binary solutions (ΔC1 = 0), > 0 in the whole range ̅ and takes values from = 0.5 to = 1.4. In the case of ternary solutions (ΔC1 = 50 mol m −3 ), the dependence = ( ̅ , ∆ = constant) is non-linear, with the maximum clearly indicated, and the In turn, Figure 11 shows that for binary solutions (∆C 1 = 0), a r v2 > 0 in the whole range C 2 and takes values from a r v2 = 0.5 to a r v2 = 1.4. In the case of ternary solutions (∆C 1 = 50 mol m −3 ), the dependence a r v2 = f C 2 , ∆C 1 = constant is non-linear, with the maximum clearly indicated, and the coefficient a r v2 Entropy 2020, 22, 680 12 of 18 assumes positive values for C 2 < 760 mol m −3 and negative for C 2 < 760 mol m −3 . The maximum of this dependence has the coordinates C 2 = 515.75 mol m −3 and a r v2 = 36.7. Rectifying properties along with amplification properties and oscillation generation belong to the group of regulatory phenomena [19].

Evaluation of Osmotic Entropy Production
The osmotic entropy production (P r S ) will be calculated using Equation (5), omitting the term i J r i ∆π i C i −1 and assuming that ∆p = 0 and i = 1, 2. With such assumptions the Equation (5) will take the form This equation shows that P r S is directly proportional to, among others, J r vi . Taking into account the results of J r vi presented in Figures 2 and 3 in the above equation, the relationships P r S1 = f (∆C 1 , ∆C 2 = constant) and P r S2 = f (∆C 1 , ∆C 2 = constant), (r = α, β). The results of the calculations are presented in Figures 12 and 13. These figures show that for the same values ∆C 1 i ∆C 2 , both P r S1 and P r S2 follow the changes in the values of J r v1 or J r v2 . Under the conditions of homogeneity of the solutions P r S1 and P r S2 they increase with the increase of the values of J r v1 and J r v2 , respectively. On the other hand, under the conditions of concentration polarization, the values P r S1 and P r S2 increase when free convection appears in the membrane system and decreases when convection disappears. Due to the fact that concentration polarization reduces J r v1 and J r v2 , it also reduces P r S1 and P r S2 .
Entropy 2020, 20, x 12 of 18 coefficient assumes positive values for ̅ < 760 mol m −3 and negative for ̅ < 760 mol m −3 . The maximum of this dependence has the coordinates ̅ = 515.75 mol m −3 and = 36.7. Rectifying properties along with amplification properties and oscillation generation belong to the group of regulatory phenomena [19].

Evaluation of Osmotic Entropy Production
The osmotic entropy production ( ) will be calculated using Equation (5), omitting the term ∑ ∆ ̅ and assuming that Δp = 0 and i = 1, 2. With such assumptions the Equation (5) will take the form This equation shows that is directly proportional to, among others, . Taking into account the results of presented in Figures 2 and 3   solutions in aqueous ethanol and the α and β configurations of the membrane system. Graphs 1, 3α and 3β were obtained for ΔC2 = 0, graphs 2, 4α and 4β; for ΔC2 = 750 mol m −3 . Figure 12. Graphic illustration of the dependencies P r S1 = f (∆C 1 , ∆C 2 = constant), (r = α, β) for CuSO 4 solutions in aqueous ethanol and the α and β configurations of the membrane system. Graphs 1, 3α and 3β were obtained for ∆C 2 = 0, graphs 2, 4α and 4β; for ∆C 2 = 750 mol m −3 . Equations (2)-(4) will be used to interpret the results of osmotic volume flux tests for concentration polarization conditions and presented in Figures 2 and 3. For this purpose, Equation (2), for pu; pd = 0, will be transformed into the form Having Equation (3) in the above equation, we get Assuming that = = , = = and f2 = 1, the equation can be written in a simplified form, namely Based on Equation (10)  Equations (2)-(4) will be used to interpret the results of osmotic volume flux tests for concentration polarization conditions and presented in Figures 2 and 3. For this purpose, Equation (2), for p u ; p d = 0, will be transformed into the form Having Equation (3) in the above equation, we get Assuming that δ r u = δ r u = δ r i , D r ui = D r di = D i and f 2 = 1, the equation can be written in a simplified form, namely Based on Equation (10), the dependencies δ α   The curves 1α and 1β presented in Figure 14   The curves 1α and 1β presented in Figure 14 illustrate the dependencies δ α 1 = f (∆C 1 , ∆C 2 = 0) and δ For the 2α and 2β curves in this figure, the values of δ α 1 initially increase linearly and then, after reaching the maximum value δ α 1 = 9.9 × 10 −3 m for ∆C 1 = 6.25 mol m −3 decrease non-linearly. In turn, the values of δ β 1 increase non-linearly. For ∆C 1 = 50 mol m −3 δ α 1 = δ β 1 = 1.02 × 10 −3 m, which means that the value of δ r 1 is independent of the configuration of the membrane system and thus also of the dependence between the gravity vector and the density gradient of ternary solutions separated through the membrane. Comparing graphs 2α and 2β, it can be seen that for ∆C 1 < 50 mol m −3 , δ α 1 < δ β 1 while for ∆C 1 > 50 mol m −3 , δ α 1 > δ β 1 . This means that for ∆C 1 > 50 mol m −3 and the β configuration of the membrane system (curve 2β), and for ∆C 1 < 50 mol m −3 and the configuration of the membrane system (curve 2α), the convection fluxes generated in the membrane areas cause concentration destruction of boundary layers, increasing the volume flow through the membrane.
The curves 1α and 1β presented in Figure 15 illustrate the dependencies δ α 2 = f (∆C 2 , ∆C 1 =0) and δ = 0.94 × 10 −3 m, which means that the value of δ r 2 is independent of the configuration of the membrane system and thus also of the dependence between the gravity vector and the density gradient of binary solutions separated through the membrane. For ∆C 2 ≥ 375 mol m −3 δ α 2 = 6.8 × 10 −3 m = const. and for ∆C 2 ≥ 375 mol m −3 δ β 2 = 0.2 × 10 −3 m = const., and therefore δ α 2 > δ β 2 . This means that for ∆C 2 ≥ 375 mol m −3 in the β configuration of the membrane system, convection fluxes generated in the membrane regions destroy the concentration boundary layers, increasing the volume flow through the membrane.
In the case of the 2α and 2β curves in this figure, the values of δ β 2 initially increase and then, after reaching the maximum value δ β 2 = 5.1 × 10 −3 m for ∆C 2 = 250 mol m −3 decrease non-linearly. In turn, the values of δ α 2 change non-linearly. For ∆C 2 = 850 mol m −3 δ α 2 = δ β 2 = 0.92 ×10 −3 m, which means that the value of δ r 2 is independent of the configuration of the membrane system and thus also of the dependence between the gravity vector and the density gradient of ternary solutions separated through the membrane. Comparing graphs 2α and 2β, it can be seen that for ∆C 2 < 840 mol m −3 δ α 2 < δ β 2 , while for ∆C 2 > 840 mol m −3 , δ α 2 > δ β 2 . This means that for ∆C 1 > 840 mol m −3 and the β configuration of the membrane system (graph 2β), and for ∆C 1 < 840 mol m −3 and the α configuration of the membrane system (graph 2 α), the convection fluxes generated in the membrane areas cause concentration destruction of the boundary layers, increasing the volume flux through the membrane.
Graphs 1α and 1β show that for R C1 < (R C1 ) kryt. and R C2 > (R C2 ) kryt. non-convective state in both configurations of the membrane system is being dealt with. R C1 > (R C1 ) kryt. in the α configuration (graphs 1α and 2α) a convective state is obtained and in the β configuration (graphs 1β and 2β)-the non-convective state. On the other hand, for R C2 < (R C2 ) kryt. in the α configuration (graphs 1α and 2α) a non-convective state is obtained, and for the β configuration (graphs 1β and 2β); the convective state. Therefore, the authors have shown that the concentration Rayleigh number (R r C ) is a parameter controlling the transition from non-convective to convective state. This number also acts as a switch between two convective states (with a higher J r vi value) and non-convective states (with a lower J r vi value). The operation of this switch indicates the regulatory role of earthly gravity in relation to membrane transport. Investigations on membrane transport are one of the most forward-looking directions in biotechnology, biomedical engineering and environmental protection and engineering, especially in water treatment and purification. Moreover, in recent years the research on integrated membrane processes has also been carried out [30]. The research results presented in the paper may also be relevant for nature-inspired chemical engineering (NICE) [31].

Conclusions
In this article, the authors presented the results of studies on the impact of the concentration of individual solution components and the configuration of the membrane system on the value of the volume osmotic flux (J r vi ) in a single-membrane system, in which the polymer membrane was positioned in a horizontal plane and separated water and a ternary solution consisting of water, ethanol and/or CuSO 4 . From the studies it results, that for conditions of concentration polarization and binary solutions J r vi is a linear and for ternary solutions a non-linear function of the solution concentration differences. In addition, J r vi depends on the configuration of the membrane system. For mechanically stirred solutions, J r vi is independent of the membrane system configuration and is a linear function of the difference in solution concentrations. The effects of concentration polarization, convective polarization, asymmetry and amplification of the volume osmotic flux calculated on the basis of J r vi measurements are a consequence of the concentration polarization of solutions adjacent to the membrane. The effects of concentration polarization and convective polarization for binary solutions are linear and for ternary ones a non-linear function of the concentration difference. The measures of asymmetry and amplification of the volume osmotic flux (which are a consequence of concentration polarization) are the corresponding asymmetry coefficients k 1 and k 2 and the amplification coefficients a v1 and a v2 . The k 1 coefficient for both binary and ternary solutions is a non-linear function of the difference in concentration of CuSO 4 . In turn, the value of the coefficient k 2 for binary solutions is independent of the concentration and for ternary solutions; it is a non-linear function of the difference in ethanol concentration. For binary solutions, the values of a v1 and a v2 coefficients are constant and positive. In turn, for ternary solutions, these coefficients are a non-linear function of the respective concentration differences and assume both positive and negative values. It has been shown that entropy production occurs in the single-membrane system study, which is a consequence of two thermodynamic forces (one variable and the other constant) and the generation of an osmotic flux. It has been shown, that the factor ζ r i , by the thickness of the concentration boundary layer (δ r i ), can be associated with the Rayleigh concentration number (R r C ), i.e., the parameter controlling the transition from non-convection (diffusion) to convective concentration field. Four different concentration Rayleigh number, which differ in values and signs were obtained.
The R r C signs is conditioned by the relationship between the gravity vector and the solution density gradient. It has been shown that this number also acts as a switch between two states of the concentration field: convective (with a higher J r vi value) and non-convective (with a lower J r vi value). The operation of this switch indicates the regulatory role of earthly gravity in relation to membrane transport. Funding: This research received no external funding.