Membrane transport of the non-homogeneous ternary solutions: mathematical model of thicknesses of the concentration boundary layers.

We have presented the mathematical model, which can be the basis of determination of a numerical algorithm for the thickness of concentration boundary layers calculation for ternary nonelectrolyte solutions. This model is based on Rayleigh equation and second Kedem-Katchalsky equation for ternary solutions. In proposed model, the thickness of concentration boundary layers is controlled by concentration Rayleigh number.

[Computer modeling the hydrostatic pressure characteristics of the membrane potential for polymeric membrane, separated non-homogeneous electrolyte solutions].

On the basis of model equation depending the membrane potential deltapsis, on mechanical pressure difference (deltaP), concentration polarization coefficient (zetas), concentration Rayleigh number (RC) and ratio concentration of solutions separated by membrane (Ch/Cl), the characteristics deltapsis = f(deltaP)zetas,RC,Ch/Cl for steady values of zetas, RC and Ch/Cl in single-membrane system were calculated. In this system neutral and isotropic polymeric membrane oriented in horizontal plane, the non-homogeneous binary electrolytic solutions of various concentrations were separated. Nonhomogeneity of solutions is results from creations of the concentration boundary layers on both sides of the membrane.

[Computer modeling the concentration characteristics of the membrane potential for polymeric membrane, separated non-homogeneous electrolyte solutions].

The influence of the concentration boundary layers on membrane potential (deltapsis) in a single-membrane system on basis of the Kedem-Katchalsky equations was described in cases of horizontally mounted neutral polymeric membrane separates non-homogeneous (mechanically unstirred) binary electrolytic solutions at different concentrations. Results of calculations of deltapsis as a function of ratio solution concentrations (Ch/Cl) at constant values of: concentration Rayleigh number (Rc), concentration polarization coefficient (zetas) and hydrostatic pressure (deltaP) were presented. Calculations were made for the case where on a one side of the membrane aqueous solution of NaCl at steady concentration 10(-3) mol x l(-1) (Cl) was placed and on the other aqueous solutions of NaCl at concentrations from 10(-3) mol x l(-1) to 2 x 10(-2) mol x l(-1) (Ch).

[Computer modeling the dependences of the membrane potential for polymeric membrane separated non-homogeneous electrolyte solutions on concentration Rayleigh number].

On the basis of model equation describing the membrane potential delta psi(s) on concentration Rayleigh number (R(C)), mechanical pressure difference (deltaP), concentration polarization coefficient (zeta s) and ratio concentration of solutions separated by membrane (Ch/Cl), the characteristics delta psi(s) = f(Rc)(delta P, zeta s, Ch/Cl) for steady values of zeta s, R(C) and Ch/Cl in single-membrane system were calculated. In this system neutral and isotropic polymeric membrane oriented in horizontal plane, the non-homogeneous binary electrolytic solutions of various concentrations were separated. Nonhomogeneity of solutions is results from creations of the concentration boundary layers on both sides of the membrane.

Mathematical model describing thickness changes of concentration boundary layers controlled by polymeric membrane parameters and concentration Rayleigh number.

In the paper, by applicating the classic definition of concentration Rayleigh number and the second Kedem-Katchalsky equation, there was deriven the equation of the fourth degree, which makes thicknesses (deltah and deltal) dependent on the concentration difference (Ch-Cl), concentration Rayleigh number (Rc), membrane permeability parameters (omega, xi s) and solutions (Dl, Dh), physico-chemical parameters of solutions (v(l), v(h), rho l, rho h, delta rho/deltaC), temperature (T) and gravitational acceleration (g). On the basis of the obtained formula for isothermal conditions (T = const) and constant gravitational field (g = const), there were calculated non-linear dependencies delta h = f(Ch-Cl)(Rc, zeta s), delta h = f (Rc)((Ch-Cl),zeta s) and delta h = f(delta s)((Ch-Cl),Rc).