Irreversible thermodynamic model equations of the transport across a horizontally mounted membrane

Diffusion and hydrodynamic instabilities in membrane systems with water solutions of NaCl and ethanol

The characteristic manifestations of instability were observed in the form of voltage pulsations measured between electrodes immersed directly in solutions of membrane system chambers, in different configurations of membrane systems. The reason for this type of voltage pulsations is Rayleigh-Benard type instabilities of near-membrane layers caused by density gradients of solutions in these layers. The time of build-up of the concentration boundary layer, after which hydrodynamic instability appears is one of important parameters of these phenomena. The concentration characteristics of these times, measured for one- and two-membrane systems, are nonlinear. With increasing differences in the density of solutions on the membrane at the initial moment, the times of build-up of concentration boundary layers were reduced. In two-membrane systems containing ternary solutions (water, NaCl, ethanol), ethanol was used to control the initial differences in the density of solutions on the membrane. The times of hydrodynamic instabilities in two-membrane system were symmetrical due to the concentration of ethanol, for which the densities of solutions on both sides of the membrane were the same at the initial moment. This dependence is similar for both configurations of the membrane system and is characterized by two nonlinear curves converging to the concentration of ethanol at which, at the initial moment, the densities of the solutions in the chambers of the two-membrane system are the same. In turn, the steady-state voltages of the two-membrane system as a function of the initial concentration of ethanol in the middle chamber with the same initial NaCl concentration in the middle chamber, are a complex function depending on the membrane arrangement. These voltages are characterized by a transition in the ethanol concentration range, for which, at the initial moment, the densities of the solutions in the chambers of the two-membrane system are comparable.

Evaluation of Transport Properties and Energy Conversion of Bacterial Cellulose Membrane Using Peusner Network Thermodynamics

We evaluated the transport properties of a bacterial cellulose (BC) membrane for aqueous ethanol solutions. Using the R^r version of the Kedem-Katchalsky-Peusner formalism (KKP) for the concentration polarization (CP) conditions of solutions, the osmotic and diffusion fluxes as well as the membrane transport parameters were determined, such as the hydraulic permeability (L_p), reflection (σ), and solute permeability (ω). We used these parameters and the Peusner (Rijr) coefficients resulting from the KKP equations to assess the transport properties of the membrane based on the calculated dependence of the concentration coefficients: the resistance, coupling, and energy conversion efficiency for aqueous ethanol solutions. The transport properties of the membrane depended on the hydrodynamic conditions of the osmotic diffusion transport. The resistance coefficients R11r, R22r, and Rdetr were positive and higher, and the R12r coefficient was negative and lower under CP conditions (higher in convective than nonconvective states). The energy conversion was evaluated and fluxes were calculated for the U-, F-, and S-energy. It was found that the energy conversion was greater and the S-energy and F-energy were lower under CP conditions. The convection effect was negative, which means that convection movements were directed vertically upwards. Understanding the membrane transport properties and mechanisms could help to develop and improve the membrane technologies and techniques used in medicine and in water and wastewater treatment processes.

Management of Energy Conversion Processes in Membrane Systems

The internal energy (U-energy) conversion to free energy (F-energy) and energy dissipation (S-energy) is a basic process that enables the continuity of life on Earth. Here, we present a novel method of evaluating F-energy in a membrane system containing ternary solutions of non-electrolytes based on the Kr version of the Kedem–Katchalsky–Peusner (K–K–P) formalism for concentration polarization conditions. The use of this formalism allows the determination of F-energy based on the production of S-energy and coefficient of the energy conversion efficiency. The K–K–P formalism requires the calculation of the Peusner coefficients Kijr and Kdetr (i, j ∈ {1, 2, 3}, r = A, B), which are necessary to calculate S-energy, the degree of coupling and coefficients of energy conversion efficiency. In turn, the equations for S-energy and coefficients of energy conversion efficiency are used in the F-energy calculations. The Kr form of the Kedem–Katchalsky–Peusner model equations, containing the Peusner coefficients Kijr and Kdetr, enables the analysis of energy conversion in membrane systems and is a useful tool for studying the transport properties of membranes. We showed that osmotic pressure dependences of indicated Peusner coefficients, energy conversion efficiency coefficient, entropy and energy production are nonlinear. These nonlinearities were caused by pseudophase transitions from non-convective to convective states or vice versa. The method presented in the paper can be used to assess F-energy resources. The results can be adapted to various membrane systems used in chemical engineering, environmental engineering or medical applications. It can be used in designing new technologies as a part of process management.

Modelling of the Electrical Membrane Potential for Concentration Polarization Conditions

Based on Kedem–Katchalsky formalism, the model equation of the membrane potential (Δψs) generated in a membrane system was derived for the conditions of concentration polarization. In this system, a horizontally oriented electro-neutral biomembrane separates solutions of the same electrolytes at different concentrations. The consequence of concentration polarization is the creation, on both sides of the membrane, of concentration boundary layers. The basic equation of this model includes the unknown ratio of solution concentrations (Ci/Ce) at the membrane/concentration boundary layers. We present the calculation procedure (Ci/Ce) based on novel equations derived in the paper containing the transport parameters of the membrane (Lp, σ, and ω), solutions (ρ, ν), concentration boundary layer thicknesses (δl, δh), concentration Raileigh number (RC), concentration polarization factor (ζs), volume flux (Jv), mechanical pressure difference (ΔP), and ratio of known solution concentrations (Ch/Cl). From the resulting equation, Δψs was calculated for various combinations of the solution concentration ratio (Ch/Cl), the Rayleigh concentration number (RC), the concentration polarization coefficient (ζs), and the hydrostatic pressure difference (ΔP). Calculations were performed for a case where an aqueous NaCl solution with a fixed concentration of 1 mol m⁻³ (Cl) was on one side of the membrane and on the other side an aqueous NaCl solution with a concentration between 1 and 15 mol m⁻³ (Ch). It is shown that (Δψs) depends on the value of one of the factors (i.e., ΔP, Ch/Cl, RC and ζs) at a fixed value of the other three.

A Novel Mass Transfer Model to Describe the Separation Process in Reverse Osmosis of Glucose

Basic and theoretical research on processes such as reverse osmosis (RO) is essential in the fermentation industry to improve production efficiency and reduce cost. Here, we focus on the RO concentration of glucose solutions. We constructed a mathematic model that incorporates various membrane and experimental parameters to characterize the mass transfer process of RO membrane and validated the model output with experimental data. Calculation results were highly consistent with the experimental data, demonstrating that this model can be useful for predicting the RO concentration process.

Evaluation of the Global S-Entropy Production in Membrane Transport of Aqueous Solutions of Hydrochloric Acid and Ammonia

The results of experimental studies of volume osmotic fluxes (Jvkr) and fluxes of dissolved substances (Jkr) in a system containing a synthetic Nephrophan® membrane (Orwo VEB Filmfabrik, Wolfen, Germany) set in a horizontal plane are presented. The membrane separated water and aqueous HCl or ammonia solutions or aqueous ammonia and HCl solutions. It was found that for the homogeneity conditions of the solutions Jvk and Jk depend only on the concentration and composition of the solutions. For concentration polarization conditions (where concentration boundary layers are created on both sides), Jvkr and Jkr depend on both the concentration and composition of the solutions and the configuration of the membrane system. The obtained results of the Jvk and Jk flux studies were used to assess the global production of entropy for the conditions of homogeneity of solutions (ΦSk), while Jvkr and Jkr-to assess the global production of entropy for concentration polarization conditions (ΦSkr). In addition, the diffusion-convective effects and the convection effect in the global source of entropy were calculated. The concentration polarization coefficient ζir was related to modified concentration Rayleigh number, e.g., the parameter controlling the transition from non-convective (diffusive) to convective state. This number acts as a switch between two states of the concentration field: convective (with a higher entropy source value) and non-convective (with a lower entropy source value). The operation of this switch indicates the regulatory role of earthly gravity in relation to membrane transport.

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).

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 ( J v i r ) 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 J v i r , 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 ( R C r ), i.e., the switch, were estimated between two states: convective (with higher J v i r ) and non-convective (with lower J v i r ). The operation of this switch indicates the regulatory role of earthly gravity in relation to membrane transport.

A Review on Reverse Osmosis and Nanofiltration Membranes for Water Purification

Sustainable and affordable supply of clean, safe, and adequate water is one of the most challenging issues facing the world. Membrane separation technology is one of the most cost-effective and widely applied technologies for water purification. Polymeric membranes such as cellulose-based (CA) membranes and thin-film composite (TFC) membranes have dominated the industry since 1980. Although further development of polymeric membranes for better performance is laborious, the research findings and sustained progress in inorganic membrane development have grown fast and solve some remaining problems. In addition to conventional ceramic metal oxide membranes, membranes prepared by graphene oxide (GO), carbon nanotubes (CNTs), and mixed matrix materials (MMMs) have attracted enormous attention due to their desirable properties such as tunable pore structure, excellent chemical, mechanical, and thermal tolerance, good salt rejection and/or high water permeability. This review provides insight into synthesis approaches and structural properties of recent reverse osmosis (RO) and nanofiltration (NF) membranes which are used to retain dissolved species such as heavy metals, electrolytes, and inorganic salts in various aqueous solutions. A specific focus has been placed on introducing and comparing water purification performance of different classes of polymeric and ceramic membranes in related water treatment industries. Furthermore, the development challenges and research opportunities of organic and inorganic membranes are discussed and the further perspectives are analyzed.

A Mathematical Modeling of the Reverse Osmosis Concentration Process of a Glucose Solution

A mathematical modeling of glucose–water separation through a reverse osmosis (RO) membrane was developed to research the membrane’s performance during the mass transfer process. The model was developed by coupling the concentration–polarization (CP) model, which uses one-dimensional flow assumption, with the irreversible thermodynamic Spiegler–Kedem model. A nonlinear parameter estimation technique was used to determine the model parameters Lp (hydraulic permeability constant), σ (reflection coefficient), and Bs (solute transport coefficient). Experimental data were obtained from the treatment of a pre-treated glucose solution using a laboratory-scale RO system, and studies on the validation of the model using experimental results are presented. The calculated results are consistent with the experimental data. The proposed model describes the RO membrane concentration process and deduces the expression of k (mass transfer coefficient in the CP layer). The verification shows that the expression of k well-describes the reverse osmosis mass transfer of a glucose solution.

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

In this paper, the Kedem–Katchalsky equations in matrix form for nonhomogeneous ternary nonelectrolyte solutions were applied for interpretation of transport through the membrane mounted in horizontal plane. Coefficients Hijr, 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, σ2ω11, 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.

The mathematical model of concentration polarization coefficient in membrane transport and volume flows

In this paper, the authors investigate the membrane transport of aqueous non-electrolyte solutions in a single-membrane system with the membrane mounted horizontally. The purpose of the research is to analyze the influence of volume flows on the process of forming concentration boundary layers (CBLs). A mathematical model is provided to calculate dependences of a concentration polarization coefficient (ζ _s ) on a volume flux (J _vm ), an osmotic force (Δπ) and a hydrostatic force (ΔP) of different values. Property ζ _s  = f(J _vm ) for J _vm  > 0 and for J _vm  ≈ 0 and property ζ _s  = f(ΔC ₁) are calculated. Moreover, results of a simultaneous influence of ΔP and Δπ on a value of coefficient ζ _s when J _vm  = 0 and J _vm  ≠ 0 are investigated and a graphical representation of the dependences obtained in the research is provided. Also, mathematical relationships between the coefficient ζ _s and a concentration Rayleigh number (R _C ) were studied providing a relevant graphical representation. In an experimental test, aqueous solutions of glucose and ethanol were used.

The role of mechanical pressure difference in the generation of membrane voltage under conditions of concentration polarization

The mechanical pressure difference across the bacterial cellulose membrane located in a horizontal plane causes asymmetry of voltage measured between electrodes immersed in KCl solutions symmetrically on both sides of the membrane. For all measurements, KCl solution with lower concentration was above the membrane. In configuration of the analyzed membrane system, the concentration boundary layers (CBLs) are created only by molecular diffusion. The voltages measured in the membrane system in concentration polarization conditions were compared with suitable voltages obtained from the model of diffusion through CBLs and ion transport through the membrane. An increase of difference of mechanical pressure across the membrane directed as a difference of osmotic pressure always causes a decrease of voltage between the electrodes in the membrane system. In turn, for mechanical pressure difference across the membrane directed in an opposite direction to the difference of osmotic pressure, a peak in the voltage as a function of mechanical pressure difference is observed. An increase of osmotic pressure difference across the membrane at the initial moment causes an increase of the maximal value of the observed peak and a shift of this peak position in the direction of higher values of the mechanical pressure differences across the membrane.

Effect of hydroxyapatite nanoparticles on osmotic responses of pig iliac endothelial cells

In order to fully explore the potential applications of nanoparticles in biopreservation, it is necessary to study the effect of nanoparticles on cell membrane permeabilities. The aim of this study is therefore to comparatively evaluate the osmotic responses of pig iliac endothelial cells in the absence and presence of commercially available hydroxyapatite nanoparticles. The results indicate that, after the introduction of 0.0 1 wt% hydroxyapatite nanoparticles, the dependence of cell membrane hydraulic conductivity (Lp) on temperature still obeys the Arrhenius relationship, while the reference value of the hydraulic conductivity of the cell membrane at 273.15K (Lpg) and the activation energy for water transport across cell membrane (ELp) change from 0.77 × 10(-14)m/Pa/s and 15.65 kJ/mol to 0.65 × 10(-14)m/Pa/s and 26.14 kJ/mol. That is to say, the reference value of the hydraulic conductivity of the cell membrane has been slightly decreased while the activation energy for water transport across cell membrane has been greatly enhanced, and thus it implies that the hydraulic conductivity of cell membrane are more sensitive to temperature in the presence of nanoparticles. These findings are of potential significance to the optimization of nanoparticles-aided cryopreservation.

Evaluation of the efficiency of venous leg ulcer treatment with a membrane dressing

A quantitative method for the evaluation of ulcer healing has been developed. The method consists of calculating an ulcer healing efficiency ratio (zetaz) interpreted as the second derivative of the regenerated ulcer area over time. The zetaz value is in direct proportion to the acceleration of healing, and in inverse proportion to the initial ulcer area. In order to test this method, we measured the surface area that was delimited by the ulcer edge and the initial depth of the ulcer in three patient groups. The first and third groups were treated with a cellulose membrane and the second group-with Unna boot. The initial ulcer area was the same in groups 1 and 2, but in group 3, it was six times smaller. The greatest zetaz value was obtained in group 3, the smallest-in group two.

Study of the solute flows of multicomponent and heterogeneous non-ionic solutions in double-membrane system

The solute flows were studied in a double-membrane osmotic-diffusive cell, in which two membranes mounted in horizontal planes separate three compartments (l,m,r) containing the non-homogeneous, non-electrolytic binary and ternary solutions. The volume of inter-membrane compartment (m), which is the infinitesimally layer of solution, and volume of external compartments (l and r) fulfill the conditions V(m)→ 0 and V(l) =V(r)→ ∞, respectively. In an initial moment, the solution concentrations satisfy the condition (C(o) (s))(l) < (C(o) (s))(m) >(C(o) (s))(r). The double-membrane osmotic-diffusive cell is composed of two complexes: boundary layer/membrane/boundary layer, mounted in horizontal planes. In the cell, solute flux was measured as a function of concentration and gravitational configuration. The linear dependencies of the solute flux on concentration difference in binary solutions and nonlinear - in ternary solutions were obtained. It was shown that the double-membrane osmotic-diffusive cell has rectifying and amplifying properties of solute flows.

Gravitational effects in a passive transmembrane transport: the flux graviosmotic and gravidiffusive effects in non-electrolytes

In this paper the classification ofthe gravitational effects in a passive transmembranetransport is presented. Among these effects there arethe flux and force gravitational effects (fluxgraviosmotic effect, osmotic pressure graviosmoticeffect, flux gravidiffusive effect, osmotic pressuregravidiffusive effect, voltage gravielectric effectand current gravielectric effect). The volume fluxgraviosmotic and solute flux gravidiffusive effectsmodel equations for a single-membrane system areelaborated. These models for binary and ternarynon-electrolyte solutions have been verified using anexperimental data volume and solute fluxes forosmotic-diffusion cell with horizontally mountedmembrane. In the experimental set-up, water was placedon one side of the membrane. The opposite side of themembrane was exposed to binary or ternary solutions ofdensities greater than that of water (aqueous glucoseor glucose-0.2 mole/l aqueous ethanol) and binary andternary solutions of densities larger than that ofwater (aqueous ethanol or ethanol-0.05 mole/l aqueousglucose). These experimental results are interpretedin terms of the convective instability that increasesthe diffusive permeability coefficient of junction:boundary layer/membrane/boundary layer.

Thermodynamic model equations for heterogeneous multicomponent non-ionic solution transport in a multimembrane system

Non-equilibrium thermodynamic model equations for non-ionic and heterogeneous n-component solution transport in a m-membrane system are presented. This model is based on two equations. The first one describes the volume transport of the solution and the second the transport of the solute. Definitions of the hydraulic permeability, reflection and diffusive permeability coefficients of the m-membrane system and relations between the coefficients of the m-membrane system and the respective membranes of the system are also given. The validity of this model for binary and ternary solutions was verified, using a double-membrane cell with a horizontally mounted membrane. In the cell, volume and solute fluxes were measured as a function of concentration and gravitational configuration.

A model equations of the volume transport of multicomponent and heterogeneous non-ionic solutions in double-membrane system

The volume flows model equation for a double-membrane system, in which two membranes separate three compartments (l,m,r) containing the heterogeneous, non-ionic n-component solutions is elaborated. In this system the solution concentrations fulfill the condition Clk > Cmk > Crk. The inter-membrane compartment (m) consists of the infinitesimal layer of solution. The volume of compartment m and external compartments (l and r) fulfill the conditions Vm→ 0 and Vl =Vr→∞ respectively. The linear dependences of the volume flux on concentration differences in binary solutions and nonlinear - in ternary solutions, were obtained. This model for binary and ternary non-electrolyte solutions is discussed. It is shown, that the double-membrane system has rectifying and amplifying properties for osmotic transport and mechanical pressure.

A model equation for the gravielectric effect in electrochemical cells

The gravielectric effect model equation for a single-membrane system was elaborated. This model for binary and ternary ionic solutions was verified using a cell with a horizontally mounted membrane. In this cell, the membrane and transition potentials were measured as a function of gravitational configuration. In these experiments, a 0.001 M aqueous solution of sodium chloride was placed on one side of the membrane. The opposite side of the membrane was exposed to either aqueous sodium chloride solutions, with densities greater than that of 0.001 M aqueous NaCl, or ethanol/NaCl/water solutions. On the basis of the experimental results, the influence of constrained release and the gravielectric effect were established. These experimental findings are interpreted in terms of a convective gravitational instability that reduces boundary layer dimensions and increases the permeability coefficient of the complex system: boundary layer/membrane/boundary layer. A concentration-gradient Rayleigh number is used in a mathematical model for gravitationally sensitive membrane potential.