Facilitated transport via carrier—mediated diffusion in membranes: Part II. Mathematical aspects and analyses

Abstract

Part I1 of this review is concerned with the mathematical analysis of facilitated transport. An exposition is given of the most generally useful techniques for obtaining asymptotic or approximate solutions to one-dimensional carrier-mediated diffusion in membranes, involving multiple permeant and carrier species which undergo one or more chemical reactions. Primary emphasis is devoted to the limiting regimes of weakly-perturbed membranes (small driving forces) and slow or fast reactions (small or large Damkohler numbers). Many of the results appearing in the literature are unified and extended, and a systematic procedure for using these to estimate membrane performance is put forth. Finally, some areas for further work are identified.In the absence of convection and electric-field effects, it appears possible now to estimate mathematically, with some confidence, the steady state performance characteristics of highly complex carrier-mediated membranes, given the requisite kinetic, equilibrium, and diffusion constants. In many cases, predictions can be obtained by relatively straightforward and rapid analytical methods, based on asymptotic or approximate formulae.The most useful and generally applicable results appear to be, roughly in the order of complexity and applicability, 1. The classical type of reaction-equilibrium approximation for investigating nonlinearities in the driving force or concentration gradients and the effects of transport parameters and binding constants, 2. The linearized-kinetic formulae for investigating reaction-rate limitations and nonequilibrium departures from l,.provided the linear approximation is valid in the equilibrium regime, that is, provided it gives reasonably accurate prediction of equilibrium fluxes under the im-posed driving force, and, if not, 3. A strong boundary-layer analysis, based on extensions proposed here of an approximate method due to Kreuzer and Hoofd (1972) and Smith et al. (1973), to replace 2, together with 4. Near-diffusion or slow-reaction, perturbation formulae to determine the approximate lower limits of validity of 3.At the very least, it appears that such formulae can provide useful information about the likely operational regimes and some valuable guidelines for the application of more difficult, numerical methods or the development of special approximate methods.In the summary, suggestions are made for some further theoretical work, including extensions to other interesting geometric configurations and to unsteady operation, as well as to systems with electric-field and convection effects.Only that notation from Part I which is most directly relevant to Part 11, is included here. Some of the mathematical symbols of Part 11, defined only in isolated contexts, are not listed here. No notational distinction is generally made between dimensional quantities x, D, k, . . . and their dimensionless counterparts r / L ,

A review on olefin/paraffin separation using reversible chemical complexation technology

“…Yung and Probstien [138] used a similarity transform to simplify the differential reactions and obtain a numerical solution. Schultz et al [91] mentioned that it is generally difficult to decide which solution results in the accurate facilitation factors. Smith and Quinn [139] developed an approximate solution under the assumption of large excess of carrier.…”

“…Ward [132] and Smith et al [131] have presented numerical solutions, but their algorithms were not given in detail. Smith et al [131], Suchdeo et al [143], Goddard et al [91], on the semi-infinite interval 0 x < 1 and subject to the exact boundary conditions at x = 0 solved the system. Jain and Schultz [144] solved the system by the method of successive substitutions or Picard iteration.…”

“…The lattice energy less than 1000 kJ/mol is suitable for a transition metal salt to be used as a carrier in the facilitated transport membrane. Thus, according to the table the tendency of the anion to form a strong ion pair with the cation decreases in the order of decrease in the lattice energy [90,91] as the following:…”

Separation in Mass-Exchange Devices with Reactive Membranes

“…The diffusive flux of the permeate through the membrane can be increased by introducing a carrier species into the membrane. The augmentation of the flux of a solute by a mobile carrier species, which reacts reversibly with the solute, is known as carrier-facilitated transport (25). The use of carrier-facilitated transport in industrial membrane separation processes is of considerable interest because of the increased mass transfer rates for the solute of interest and the improved selectivity over other solutes (26).…”

Liquid Membrane Technology

“…These fields include: There have been previous review articles that cover various aspects of liquid membrane separations. The reader is referred to them for further information (11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23).…”