Experimental Systems and CharacteristicCarrier-mediated transport in membranes as a globally nonreactive process is distinguished from film theory with chemical reaction and other facilitated diffusion phenomena. With the concept of stoichiometric and system invariants, an approach is developed for the analysis of carriermediated transport with multiple permeants involving multiple reactions in the membrane. Approximate solutions of the requisite differential equations according to the relative importance of diffusion and reaction rates are reviewed, as well as typical experimental studies. Criteria for evaluating whether a membrane is in the diffusion or equilibrium regime are given, and, in the latter case, the effects of some system parameters are given, for example, binding constants, competitive permeants. Advances in membrane technology have made it possible in recent years to manufacture membranes in diverse forms such as sheets, tubes, and hollow fibers. In most current applications to separations processes, the membrane functions as a physical diffusion barrier or simple (micro-) sieve. However, &rough recent studies on models of biological membranes, it has become evident that artificial membranes, often in the form of liquid films, can be made functionally very specific in their properties by incorporating mobile or partially mobile compounds within the membrane structure which selectively react with a restricted class of permeants, for example, in ionspecific electrodes. These compounds serve effectively as carriers which not only can render the membrane very specific in its transport properties but also can enhance the relative rate at which the preferred permeants diffuse across the barrier.While various mechanistic models have been proposed to describe carrier-mediated transport in membranes, those models based on diffusion accompanied by chemical reaction have received the most theoretical and experimental study and are the main subject of this review. Recent studies of carriepmediated membrane transport of this type has led to a basic and more precise understanding of the effect of the major parameters involved, including the reaction kinetics, equilibrium (binding) constant, membrane diffusivities, concentration gradients, membrane thickness, and solubilities of the permeants in the membrane phase.Currently available numerical methods, together with various asymptotic and approximate analytic methods, based on the respective concepts of weak gradients or fast and slow reactions, allow one to estimate with a good deal of confidence the response of a particular carriermediated membrane to a variety of operational conditions. Apart from their potential for direct applications to processes such as drug transport into cells, the concepts and methods developed in these studies, relating to reaction boundary-layer analysis, global nonreactivity, and competitive interactions with carriers, suggest similar theoretical treatments in a diversity of related phenomena, such as facilitated heat transfer, ion-...

A theoretical and experimental analysis of facilitated transport of CO2 across membranes containing NaHCO3 and the enzyme carbonic anhydrase (carbonate hydro-lyase EC 4.2.1.1) is presented. The necessary diffusion reaction equations are derived and the system constraints defined. For the CO2-HCO3-system, mathematical simplifications based on the magnitude of various reaction and concentration terms are made to make the equations tractable to solution. The resultant equations are solved by a number of analytical and numerical techniques, each having a limited, though useful, range of validity. The experimental arrangement consists of a liquid membrane (created by soaking a porous filter paper in the test solution), a diffusion chamber, and gas metering and analysis equipment. Conditions were selected to cover the range from diffusionto reaction-dominated behavior. The flux of CO2 across a membrane containing ! M NaHCO3 was measured at various partial pressures of CO2 (2-28 in Hg) and with membrane thicknesses of 0.02, 0.06 and 0.10 cm. The extent of facilitation (defined as the ratio of reactionrelated flux to the expected Fick's Law flux in the absence of reaction) ranged from near zero to nearly 5 in these experiments. The agreement between model calculations and experimental observation was found to be excellent over the entire range of neardiffusion to near-equilibrium behavior. In the presence of enzyme carbonic anhydrase (0.10 mg/ml, activity approx. 80%) and 1 M NaHCO3, the CO2 flux across a 0.02 cm membrane was 3-10-fold higher than the corresponding flux in the absence of enzyme. From experiments at various enzyme concentrations and membrane thicknesses, it appeared that the apparent CO2 reaction rate was directly proportional to the enzyme concentration. The model calculations for the enzyme-catalyzed reactions agreed with the experimental observations to within ± 10 %. * Carbonic anhydrase isolated from bovine red blood cells. ** See text: Physiochemical parameters.

The transport of a gas across a stationary hquld film containing reactive species 1s investigated for the purpose of determining gas permeablhtles or mass transfer coefficients m reacting solutions Under hnutmg condltlons when the reaction time constant far exceeds the dtiuslonal time constant, the flux of the transported gas follows Flck's law of dlffuslon Analytical series solution for the contnbutlon of the chemical reaction to the transport process 1s obtained usmg the technique of perturbation analysis, criteria for the validity of various terms m the series solution are presented The permeablhty of carbon dioxide m water and m 1N NaHCOrNa2C0, solution 1s estimated It 1s shown that a hleh degree of accuracy m the data 1s necessary for obtammg separate estimates of dlffuslvlty and sol&htyby this technique

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 ,

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It is shown how an asymptotic, boundary-layer method, developed specifically for application to facilitated mass transport, can also be applied to the corresponding problem of facilitated enthalpy transport. The method, which merely entails an implicit algebraic computation, is applied to the (hydrodynamic) film-theory model for heat transfer between a hot or cold wall and a reversibly dissociating gas mixture, 2N0 &2NO + 02, which has previously been treated by Brian and Bodman[l]. Good agreement is found with their approximate analytical formula over the parameter range of their computations. However, for more extreme states of dissociation and larger overall temperature differences, the present results differ from those given by their approximate formula, especially for the cooling (cold wall) problem.