Estimation of thickness of concentration boundary layers by osmotic volume flux determination

Abstract

Abstract. The estimation method of the concentration boundary layers thicknesses (δ) in a singlemembrane system containing non-electrolytic binary or ternary solutions was devised using the Kedem-Katchalsky formalism. A square equation used in this method contains membrane transport (L p , σ, ω) and solution (D, C) parameters as well as a volume osmotic flux (J v ). These values can be determined in a series of independent experiments. Calculated values δ are nonlinearly dependent on the concentrations of investigated solutions and the membrane system configuration. These nonlinearities are the effect of a competition between spontaneously occurring diffusion and natural convection. The mathematical model based on Kedem-Katchalsky equations and a concentration Rayleigh number (R C ) was presented. On the basis of this model we introduce the dimensionless parameter, called by us a Katchalsky number (Ka), modifies R C of membrane transport. The critical value of this number well describes a moment of transition from the state of diffusion into convective diffusion membrane transport.

Network Hybrid Form of the Kedem–Katchalsky Equations for Non-homogenous Binary Non-electrolyte Solutions: Evaluation of $$P_{ij}^{*}$$ P i j ∗ Peusner’s Tensor Coefficients

“…Under conditions of concentration polarization, i.e., when solutions separated by the membrane are not stirred mechanically, at the both sides of the membrane the CBLs are created limiting the volume and diffusive flows of the solution (Abu-Rjal et al 2014; Barry and Diamond 1984;Jasik-Ślęzak et al 2011;Kargol 1999Kargol , 2000Ślęzak 1989;Wang et al 2014). The CBLs are created as a result of molecular diffusion of a dissolved substance from the solution with the higher concentration into the solution with the lower concentration (Dworecki et al 2003;Nikonenko et al 2010;Wang et al 2014).…”

$$H^{*}$$ H ∗ Peusner’s Form of the Kedem–Katchalsky Equations for Non-homogenous Non-electrolyte Binary Solutions

“…For the R C > (R C ) ct the hydrodynamic instability leads to the natural convection which reduces the thickness of CBL and increases the value of concentration gradient of the mem-brane and consequently volume and solute fluxes (Ślęzak et al 2010). The existence of such regulatory mechanism at the presence of a gravitational field justifies amplification and rectification of the volume and solute fluxes (Kargol 1992;Ślęzak 1989;Jasik-Ślęzak et al 2011;Ślęzak et al 2012a, b). These effects occur in single-and double-membrane systems containing binary or ternary solutions in conditions of concentration polarization when the density gradient in CBL formed in surroundings of horizontally mounted membrane is antiparallel to the vector of gravity (Kargol 1992;Ślęzak 1989;Ślęzak et al 2012a, b).…”

“…We assume that the mass density (ρ) of the solutions of concentrations C l , C e , C i and C h fulfills the condition ρ l < ρ e < ρ i < ρ h . The thickness (δ) is a basic parameter of CBL (Schlichting and Gersten 2000) that can be evaluated by the optical (Dworecki 1995;Nikonenko et al 2010) and the volume or solute fluxes methods (Barry and Diamond 1984;Ślęzak et al 2010;Jasik-Ślęzak et al 2011). In certain hydrodynamic conditions CBLs can be partially destroyed by natural convection Nield and Bejan 2006).…”

“…The value of R C depends on the gravity acceleration, concentration, viscosity and density of the transported solutions Jasik-Ślęzak et al 2011;Ślęzak et al 2010). For configuration B, when the value of R C reaches a critical value (R C ) cr , transition from non-convective (stable) to the convective (unstable) state is observed.…”

Resistance Coefficients of Polymer Membrane with Concentration Polarization

“…This process limits the increase of thickness of CBL and accelerates diffusion of substances outside the layers (Śle˛zak 1989;Kargol 2000). Natural convection occurs when the hydrodynamic conditions cause the CBL thicknesses δ h and δ l to reach their critical values (δ h ) crit and (δ l ) crit , and when the concentration Rayleigh number (R C ) that control the natural convection process also reaches critical value Jasik-Śle˛zak et al 2011). Then, the process of natural convection appears and in certain conditions can even lead to the liquid-type structure called 'plum structure' Puthenveettil and Arakeri 2008).…”