Osmotic flow equations for leaky porous membranes

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$\pi$ and dP 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 $\begin{align*}J_v = L_p dP + \sigma_sL_p d\pi \\ J_s = c^*_s(1 - \sigma_f) J_v + P_s d\pi/RT \\ \sigma_s = \theta(1 - D_s V_s/D_w V_w - D_s/D^o_s) \\ \sigma_f = 1-\thetaD_s V_s/D_w V_w - D_s/D^o_s\end{align*$ 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 $\sigma_s$ and $\sigma_f$ for osmosis and ultrafiltration is then given by $\sigma_s = \sigma_f - (1 - \theta) (1 - D_s/D^o_s),$ where $\theta$ 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 $\sigma_s = \sigma_f.$ Symmetry holds in the limits where either the pore is small, when $\sigma_s = \sigma_f = 1,$ or where the pore is large when $\sigma_s = \sigma_f = 0.$