Method to determine the critical concentration Rayleigh number in isothermal passive membrane transport processes

Bajdur

2020

Entropy

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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.

“…The process of transition from diffusion to convective concentration field can be controlled by the Rayleigh concentration number ( R C ) [ 25 ]. Assuming that

this number for ternary solutions can be described by the equation [ 26 , 27 ]

where

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.…”

Section: Electrochemical Membrane Cellmentioning

confidence: 99%

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

Bajdur

2020

Entropy

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“…Its value depends on both the concentration of solutions separated by the membrane (C k ) and the configuration of the membrane system (r A, B). More speci cally 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 coe cient of concentration polarization (ζ r k ) exceeds its critical value (ζ r k ) crit suitably [34][35][36]. For diluted ternary nonelectrolyte solutions, the concentration polarization factor (ζ r ks ) and the thickness of concentration the boundary layers (δ r h and δ r l ) can be described by the expression…”

Section: Membrane Systemmentioning

confidence: 95%

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

International Journal of Chemical Engineering

2019

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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.

“…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

.…”

Section: Theorymentioning

confidence: 99%

“…The following conditions can be saved for these densities:

> ρ l or

and

> ρ h or

< ρ h . If the solution with lower density is under the membrane, the system

/M/

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 (

).…”

Section: Theorymentioning

confidence: 99%

“…It should be noted that solutions which are vigorously mechanically stirred are considered as homogeneous solutions [ 29 , 30 ]. In turn, for heterogeneous solutions (solutions in which concentration polarization occurs), consisting in the formation of concentration boundary layers (CBLs) on both sides of the membrane separating solutions [ 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 ]. These layers serve as additional kinetic barriers for rapidly penetrating substances through membranes in artificial and biological systems [ 37 , 38 , 39 , 40 , 41 ].…”

Section: Introductionmentioning

confidence: 99%

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

Grzegorczyn

et al. 2020

Entropy

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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).