Equilibrium flux models, membranes

As with the equilibrium solvation models introduced earlier, it is also possible to incorporate quantum mechanical effects into the non-equilibrium transport model. Our motivation is to account for non-equilibrium ion fluxes and induced response in the electronic structure of the solute or membrane protein. To this end, we combine our DG-based DFT model with our DG-based PNP model as illustrated in Fig. 12.4 to develop a free energy functional and derive the associated governing equations. 

Solute Flux Solute partitioning between the upstream polarization layer and the solvent-filled membrane pores can be modeled by considering a spherical solute and a cylindrical pore. The equilibrium partition coefficient 0 (pore/bulk concentration ratio) for steric exclusion (no long-range ionic or other interactions) can be written as... 

The results in sections 2 and 3 describe the adsorption isotherms and diffusivities of Xe in A1P04-31 based on atomistic descriptions of the adsorbates and pores. The final step in our modeling effort is to combine these results with the macroscopic formulation of the steady state flux through an A1P04-31 crystal, Eq. (1). We make the standard assumption that the pore concentrations at the crystal s boundaries are in equilibrium with the bulk gas phase [2-4]. This assumption cannot be exactly correct when there is a net flux through the membrane [18], but no accurate models exist for the barriers to mass transfer at the crystal boundaries. We are currently developing techniques to account for these so-called surface barriers using atomistic simulations. 

Carrier facilitated transport involves a combination of chemical reaction and diffusion. One way to model the process is to calculate the equilibrium between the various species in the membrane phase and to link them by the appropriate rate expressions to the species in adjacent feed and permeate solutions. An expression for the concentration gradient of each species across the membrane is then calculated and can be solved to give the membrane flux in terms of the diffusion coefficients, the distribution coefficients, and the rate constants for all the species involved in the process [41,42], Unfortunately, the resulting expressions are too complex to be widely used. 

An alternative approach is to make the simplification that the rate of chemical reaction is fast compared to the rate of diffusion that is, the membrane diffusion is rate controlling. This approximation is a good one for most coupled transport processes and can be easily verified by showing that flux is inversely proportional to membrane thickness. If interfacial reaction rates were rate controlling, the flux would be constant and independent of membrane thickness. Making the assumption that chemical equilibrium is reached at the membrane interfaces allows the coupled transport process to be modeled easily [9], The process is... 

Experiments were carried out with Ionac MC 3470 to determine the self-diffusion coefficient values for H+ and Al + in the coupled transport. Data points were used from the experiment involving 2N acid sweep solution in Figure 34.24b, presented later. These values formed the basis for aluminum transport rate or flux (7ai) calculation at different time intervals. The equilibrium data generated in Figure 34.20b were used in conjunction with Equation 34.25 to determine the interdiffusion coefficient values. Local equilibrium was assumed at the membrane-water interface. Eigure 34.24a shows computed Dai,h values for this membrane. When compared with Dai,h values for Nafion 117, it was noticed that the drop in interdiffusion coefficient values was not so steep, indicative of slow kinetics. The model discussed earlier was applied to determine the self-diffusion coefficient values of aluminum and hydrogen ions in Ionac MC 3470 membrane. A notable point was that the osmosis effect was not taken into account in this case, as no significant osmosis was observed in a separate experiment. 

A mathematical model to be solved numerically has been developed and used to predict the separation effects caused by nonstationary conditions for a bulk liquid membrane transport. Numerical calculations compute such pertraction" characteristics as input and output membrane selectivity (ratio of respective fluxes), concentration profiles for cations bound by a carrier in a liquid membrane phase, and the overall separation factors all being dependent on time. The computations of fluxes and separation factors as dependent on time have revealed high separation efficiency of unsteady-state pertraction as compared with steady or near-steady-state process (with reactions near equilibrium). 

The result was negative separation in RO application. The solution-diffusion model in its existing form proved to be insufficient to quantitatively describe the negative separation. Therefore, the authors suggested the development of a coupled flow model (l.e. the solute flux is coupled to the water flux). Subsequent attempts to develop such a model met with only moderate success ( ). Anderson et al. O) conducted studies that paralleled the work of Lonsdale. The emphasis was placed on separately measuring equilibrium sorption and diffusion coefficients for the solute in the membrane material. They attributed the negative separation to the strong interaction between the phenol and the cellulose acetate membrane material. Pusch et al. (O also studied the separation of phenol and water. 

Concentration Polarisation is the accumulation of solute due to solvent convection through the membrane and was first documented by Sherwood (1965). It appears in every pressure dri en membrane process, but depending on the rejected species, to a very different extent. It reduces permeate flux, either via an increased osmotic pressure on the feed side, or the formation of a cake or gel layer on the membrane surface. Concentration polarisation creates a high solute concentration at the membrane surface compared to the bulk solution. This creates a back diffusion of solute from the membrane which is assumed to be in equilibrium with the convective transport. At the membrane, a laminar boundary layer exists (Nernst type layer), with mass conservation through this layer described by the Film Theory Model in equation (3.7) (Staude (1992)). cf is the feed concentration, Ds the solute diffusivity, cbj, the solute concentration in the boundary layer and x die distance from the membrane. 

Molecular dynamics (MD) simulations have been used to simulate non-equilibrium binary diffusion in zeolites. Highly anisotropic diffusion in boggsite provides evidence in support of molecular traffic control. For mixtures in faujasite, Fickian, or transport, diffusivities have been obtained from equilibrium MD through appropriate correlation functions and used in macroscopic models to predict fluxes through zeolite membranes under co- and counterdiffusion conditions. For some systems, MD cannot access the relevant time scales for diffusion, and more appropriate simulation techniques are being developed. 

Hydrogen sulfide and methane fluxes were measured at ambient conditions for 200 um perfluorosulfonic acid cation exchange membranes containing monoposltlve EDA counterions as carriers. Facilitation factors up to 26.4 and separation factors for H2S/CH up to 1200 were observed. The HjS transport Is diffusion limited. The data are well represented by a simplified reaction equilibrium model. Model predictions Indicate that H S facilitated transport would be diffusion limited even at a membrane thickness of 1 um. 

In this study, monopositive ethylene diamine (EDA) Ions were exchanged Into perfluorosulfonlo acid (PFSA) lonomer films to prepare facilitated transport membranes. The flux of HjS was measured with and without carrier present at ambient conditions as a function of HjS mole fraction In the feed gas stream. The selectivity of these membranes was determined by simultaneous measurements of HgS and CH fluxes from binary mixtures as a function of composition. Reaction equilibrium models were derived to predict the observed experimental data. 

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