Osmotic flow equations for leaky porous membranes

Abstract

A basic set of equations describing the flows of volume ( J v ) and solute ( J s ) across a leaky porous membrane, coupled to the differences of osmotic and hydrostatic pressures d π and d P has been derived by using general frictional theory. Denoting the mean pore concentration of solute by c * s and the hydraulic and diffusive conductances by L p and P s / RT the equations take the form J v = L p d P + σ s L p d π J s = c * s (1 – σ f ) J v + P s d π / RT σ s = θ (1 – D s V s / D w V w – D s / D o s σ f = 1 – θ D s V s / D w V w – D s / D o s in which D w and D s are the diffusion coefficients for water and solute in the pore and D o s that for free solution. The relation between the reflection coefficients σ s and σ f for osmosis and ultrafiltration is then given by (σ s = σ f – (1 – θ) (1 – D s / D o s ), where θ is the diffusive-driven: pressure-driven flow ratio. These equations follow from the fact that in leaky pores osmosis occurs by diffusion alone and that there cannot be any Onsager symmetry leading to σ s = σ f . Symmetry holds in the limits where either the pore is small, when σ s = σ f = 1, or where the pore is large when σ s = σ f = 0. Symbols used in the text c * s , c * w arithmetic mean solute and water concentration across a pore c s , c w solute and water concentration within a pore D o s diffusion coefficient in free solution D s diffusion coefficient within the pore core f ij partial molar frictional coefficient between the components i and j f sw , f sm molar frictional coefficients between solute and water, solute and membrane f d wm molar water–membrane frictional coefficient during diffusive flow f p wm molar water–membrane frictional coefficient during pressure-driven flow f π wm molar water–membrane frictional coefficient during osmotic flow J d exchange flow between solute and water J v volume flow K i partition coefficient, c i / c * i L d , L pd osmotic conductivity for volume and exchange flows L p , L dp hydraulic conductivity for volume and exchange flows P hydrostatic pressure P s , P w solute and water permeability r s , r p radius of solute molecule and pore S ( r i ) restricted diffusion series for species of radius r i in a pore V i partial molar volume of component i v i velocity of component i β ratio of the two frictional coefficients f p wm : f π wm δ x pore length ɳ viscosity of water θ diffusive-driven: pressure-driven flow ratio μ i chemical potential of the component i π osmotic pressure σ s , σ f osmotic and hydraulic reflection coefficients ϕ Rayleigh dissipation function