On equations for combined convective and diffusive transport of neutral solute across porous membranes

Abstract

The process of combined convection and diffusion of solute across membranes of finite thickness is discussed. An exact solution is available for an open (nonsieving) homoporous membrane. This solution is nonlinear in the volume flux (Jv) for any nonzero volume flow and concentration difference. Extension to the more general case of a partially sieving membrane does not change the nonlinear form of this equation. A linearization of this transport equation about small Jv/Ps yields an approximate equation (the arithmetic mean) that is useful over a reasonable range of conditions near equilibrium. The application of linear nonequilibrium thermodynamics to this process has led to the derivation of a third transport equation, linear in Jv but logarithmic in the concentrations. This finite difference equation is shown to give a less accurate approximation to the exact equation than does the arithmetic mean equation, even in regions near equilibrium. Use of approximate equations may lead to error when the solute reflection coefficient is determined from ultrafiltration experiments or when applied to the individual elements of a membrane array. The early origins of the concept of reflection and sieving coefficients and their relation to one another are discussed. The importance of structural detail in membranes, even at a fine-grained level, and the distinction between the terms "homogeneous" or "black box" and "homoporous" is emphasized. Although structural complexity creates problems in any attempt to write a transport equation without detailed knowledge of the membrane stucture, proper consideration of the local equation and its subsequent integration makes this fact explicit and allows for an assessment of the magnitude of these effects.