A method for af®nity membrane column design, based on the analytical solution of the Thomas model for frontal analysis in membrane column adsorption, was developed. The method permits to ®nd the operating conditions to reach a 93.5% of the column capacity as operating capacity, using a sharpness restriction for the system breakthrough curve.The solution of the model is presented in a graphic form and can be used in a wide range of operational conditions, provided that four design restrictions are ful®lled. The application of the method was illustrated using experimental data and a simple procedure. The implications of the results on the design and optimiztion of af®nity membrane chromatographic columns are discussed.
List of symbolsA membrane column cross-sectional area, cm 2 c solute concentration in the bulk phase, M c 0 solute concentration in the bulk phase at column inlet, M c à solute concentration in the bulk phase at equilibrium, M I 0 modi®ed zero-order Bessel function of the ®rst kind d p average pore diameter, cm D diffusion coef®cient, cm 2 s À1 F volumetric¯ow-rate, ml min À1 k 1 forward adsorption rate constant, M À1 s À1 k À1 reverse adsorption rate constant, s À1 K d dissociation constant, M L length of membrane column, lm L m membrane thickness, lm n dimensionless number of transfer units for overall process P molecule of protein PS complex between protein and ligand adsorbent q average protein concentration, M q à adsorbate concentration at equilibrium, M q m maximum binding capacity of the membrane, based on the solid volume, M r dimensionless separation factor S ligand adsorption site t time, s z axial distance along the membrane column Greek letters vtY L dimensionless solute concentration e void fraction C dimensionless ef¯uent volume s dimensionless time v linear velocity, cm s À1
IntroductionAf®nity membrane chromatography is a novel puri®ca-tion method that exploits the biospeci®c interactions between a protein and a ligand to economically purify proteins present at very low concentrations in complex solutions [1]. Af®nity membranes were developed to overcome the limitations encountered in conventional commercial processes using beads [2±5]. Af®nity membranes operate in convective mode, which can signi®-cantly reduce diffusion limitations commonly encountered in column chromatography. As a result, higher throughput and faster processing times are achieved in membrane systems [6]. Axial and radial diffusion, sorption kinetics, and nonuniformities in membrane porosity and thickness have been shown to affect af®nity membrane performance key factors such as breakthrough curve (BTC) sharpness and residence time. Degradation of membrane performance can be minimized working with axial Peclet number greater than 40, and radial Peclet numbers smaller than 0.04. Stacking more than 30 membranes averages outmembrane porosity and thickness nonuniformities [7]. Under these conditions, the membrane system performance can be predicted using the analytic solution of the