Unsteady mass transfer in a long composite cylinder with interfacial resistances

Abstract

A solution is given for unsteady state concentration profiles in an infinitely long composite cylinder resulting from a step-change in concentration in a large reservoir surrounding the cylinder. The composite cylinder consists of an inner cylinder with diffusivity D1 surrounded by a permeable tube with aliffusivity Dz and has interfacial mass transfer resistapces at the cylinder-tube and tube-reservoir interfaces. Numerical values are given for the first eight roots and various coefficients of the solution for physical properties typical of tubular (hollow fiber) membranes. These results can be used in the analysis of data from unsteady state mass transfer experiments to determine the permeability of homogeneous tubular membranes. A simple, approximate method for data analysis is suggested for the rapid estimation of the permeability of both homogeneous and asymmetric (skinned) tubular membranes. SCOPEAlthough permeability measurements are routinely made for flat membaanes, no convenient and accurate method is available at present for making these measurements for small tubular (hollow fiber) membranes. I n this paper a method based on an unsteady state diffusion experiment is proposed for measuring the permeability of tubular membranes. The objective of this paper is to present the mathematical analysis which is necessary for the interpretation of the experimental results. A detailed description of the elrperimental method and experimental data will be presented in a subsequent paper.Small tubular membranes are finding increasing use in mass transfer devices such as the artificial kidney and are now being considered for novel applications such as reaction vessels for enzyme-catalyzed reactions. A convenient and accurate method for measuring the permeability of these membranes will be useful for modeling membrane behavior, evaluating candidate membranes for specific applications, optimizing membrane composition and processing conditions, and maintaining quality control during manufacturing. The analysis presented here can easily be applied to the analogous heat transfer problem

Evaluation of dialysis membranes using a tortuous pore model

“…Changes in solute concentration over time were found by measuring a transmittance inside the hollow fibers. From these data, an intramembrane diffusion coefficient was calculated by the method for analyzing nonsteady state diffusion in a single hollow-fiber membrane (Stevenson, 1974).…”

An unsteady state method for measuring the permeability of small tubular membranes

“…This solution can also be obtained from a more general analysis given by a$er (1968) when a misprint noted by Stevenson (1974) is corrected. For the case of no mass transfer resistance at the inner radius of the cylinder, the solution by Stevenson (1974) is a special case of a general analysis by Shperling (1972). ' A numerical procedure for analyzing a similar problem, unsteady state diffusion with nonlinear adsorption in a composite cylinder, is given in a paper by Crank and Godson (1947).…”

“…The mathematical solution of the unsteady state diffusion problem in a long composite cylinder with interfacial resistances and a computational procedure for evaluating the solution are given in a previous paper (Stevenson, 1974). This solution can also be obtained from a more general analysis given by a$er (1968) when a misprint noted by Stevenson (1974) is corrected.…”

Structural analysis of hollow fiber dialysis membranes for clinical use.

“…Surface porosity was also calculated by Eq. (10). Calculated values for x and Ak are summarized in Table 4.…”

“…T =JH/Axl(AJxlAx) (9) A^y/HAxiAJx/Ax) (10) In the tortuous pore model presented here, dialysis The experimental apparatus shown in Fig. 1 was utilized to measure solute permeability at a dialysate flow rate (Fig.…”