Simple solution diffusion model

This simple equation gives the maximum possible enrichment of one gas of a two-component mixture when separated by a membrane with a selectivity of a. The equation becomes more complex when the permeate pressure cannot be neglected [320]. Following the simple solution-diffusion model the gas fluxes for gas 1 and 2 through the membrane are... 

White [25] investigated the transport properties of a series of asymmetric poly-imide OSN membranes with normal and branched alkanes, and aromatic compounds. His experimental results were consistent with the solution-diffusion model presented in [35]. Since polyimides are reported to swell by less than 15%, and usually considerably less, in common solvents this simple solution-diffusion model is appropriate. However, the solution-diffusion model assumes a discontinuity in pressure profile at the downstream side of the separating layer. When the separating layer is not a rubbery polymer coated onto a support material, but is a dense top layer formed by phase inversion, as in the polyi-mide membranes reported by White, it is not clear where this discontinuity is located, or whether it wiU actually exist The fact that the model is based on an abstract representation of the membrane that may not correspond well to the physical reality should be borne in mind when using either modelling approach. 

If the simple solution-diffusion model applies, the penetrant solubility can be calculated from P and D from the simple equation ... 

The driving force of a membrane for gas separation is the pressure difference across the membrane. The yield of the separated gas can be expressed in terms of membrane permeance, which can be characterised by the amount of permeated gas that passes through a certain membrane area in a given time at a definite pressure difference. The values of permeability are often quoted in Barter (1 Barter=10 cm s cm cm Hg = 3.35 x 10 mol m m s Pa STP, standard temperature and pressure). Gas permeation phenomena can be described by a simple solution diffusion model, which involves (l) sorption or dissolution of the permeating gas in the membrane at the higher pressure side, (2) diffusion through the membrane and (3) desorption or dissolution at the lower pressure side. Thus, the permeability coefficient P can be determined by the product of the solubility coefficient S and the mutual diffusion coefficient D [eqn (5.1)] ... 

The simple solution-diffusion model was often used to explain the gas permeation of a polymer membrane. The permeability coefficient (P) is the product of the solubility coefficient (S) and the diffusion coefficient (D). For a given gas molecule, the main factor for the diffusion coefficient of a certain... 

Diffusion, the basis of the solution-diffusion model, is the process by which matter is transported from one part of a system to another by a concentration gradient. The individual molecules in the membrane medium are in constant random molecular motion, but in an isotropic medium, individual molecules have no preferred direction of motion. Although the average displacement of an individual molecule from its starting point can be calculated, after a period of time nothing can be said about the direction in which any individual molecule will move. However, if a concentration gradient of permeate molecules is formed in the medium, simple statistics show that a net transport of matter will occur... 

The last part of.Ais chapter will be devoted to a comparison of meiribr c processes v where transport occurs through nonporous membranes. A solution-diffusion model will be used where each component dissolves into the membrane and diffuses through the membrane independently [41]. A similar approach was recently followed by Wijmans[43]. As a result, simple equations will be obtained for the component fluxes involved in the various processes which allows to compare the processes in terms of transport parameters. 

Fig. 8. Dependence of (A) corrected diffusion coefficient (D), (B) steady-state fluorescence intensity, and (C) corrected number of particles in the observation volume (N) of Alexa488-coupled IFABP with urea concentration. The diffusion coefficient and number of particles data shown here are corrected for the effect of viscosity and refractive indices of the urea solutions as described in text. For steady-state fluorescence data the protein was excited at 488 nm using a PTI Alphascan fluorometer (Photon Technology International, South Brunswick, New Jersey). Emission spectra at different urea concentrations were recorded between 500 and 600 nm. A baseline control containing only buffer was subtracted from each spectrum. The area of the corrected spectrum was then plotted against denaturant concentrations to obtain the unfolding transition of the protein. Urea data monitored by steady-state fluorescence were fitted to a simple two-state model. Other experimental conditions are the same as in Figure 6.

A solution of the diffusion equation for an electrode reaction for repetitive stepwise changes in potential can be obtained by numerical integration [44]. For a stationary planar diffusion model of a simple, fast, and reversible electrode reaction (1.1), the following differential equations and boundary conditions can be formulated ... 

Ham [508] considered that the growth of a random array of precipitating particles could be approximated to a simple cubic lattice of spherical sinks of radius R to which more material diffused from the supersaturated solution. A model of the type is very similar to those models discussed by Reek and Prager [507] and Lebenhaft and Kapral [492], The analysis which Ham introduced highlights a similarity between the competitive effect and the Wigner—Seitz model of metals. 

Test of Uptake Model Based on a Slow Surface Reaction Combined with Diffusion within the Particle. Since the simple diffusion model is inadequate to describe the uptake behavior of the molybdenum and tellurium oxide vapors by the clay loam particles, a more complex model is required, in which the effects of a slow surface reaction and of diffusion of the condensed vapor into the particle are combined. Consider the condensation of a vapor at the surface of a substrate (of any geometry) and the passage by diffusion of the condensed vapor through a thin surface layer into the body of the substrate. The change in concentration of solute per unit volume in the surface layer caused by vapor condensa-... 

Considerable progress has been made in going beyond the simple Debye continuum model. Non-Debye relaxation solvents have been considered. Solvents with nonuniform dielectric properties, and translational diffusion have been analyzed. This is discussed in Section II. Furthermore, models which mimic microscopic solute/solvent structure (such as the linearized mean spherical approximation), but still allow for analytical evaluation have been extensively explored [38, 41-43], Finally, detailed molecular dynamics calculations have been made on the solvation of water [57, 58, 71]. 

The importance of adsorbent non-isothermality during the measurement of sorption kinetics has been recognized in recent years. Several mathematical models to describe the non-isothermal sorption kinetics have been formulated [1-9]. Of particular interest are the models describing the uptake during a differential sorption test because they provide relatively simple analytical solutions for data analysis [6-9]. These models assume that mass transfer can be described by the Fickian diffusion model and heat transfer from the solid is controlled by a film resistance outside the adsorbent particle. Diffusion of adsorbed molecules inside the adsorbent and gas diffusion in the interparticle voids have been considered as the controlling mechanism for mass transfer. 

One possible solution to this problem is to develop microscopic diffusion models for glassy polymers, similar to those already presented for rubbery polymers. Ref. (90) combines some of the results obtained with the statistical model of penetrant diffusion in rubbery polymers, presented in the first part of Section 5.1.1, with simple statistical mechanical arguments to devise a model for sorption of simple penetrants into glassy polymers. This new statistical model is claimed to be applicable at temperatures both above and below Tg. The model encompasses dual sorption modes for the glassy polymer and it has been assumed that hole"-filling is an important sorption mode above as well as below Tg. The sites of the holes are assumed to be fixed within the matrix... 

Sedimentary denitrification rates have been estimated from measured pore-water solute profiles using diagenetic models, determined direcdy via sediment incubation both on deck and in situ, and determined from N-incubation techniques. Sedimentary diagenetic process can be thought of as a simple reaction—diffusion-transport system (Berner, 1980 Boudreau, 1997). In a simple fine-grained sediment system, transport is via molecular diffusion and the diagenetic equation describing this system can be expressed as ... 

Batch kinetic data on the removal of reactive dye from solution using thermally chaired dolomite have been well described by empirical external mass transfer and intra-particle diffusion models. It was found that external mass transfer and intra-particle diffusion had rate limiting effects on the removal process which were attributed to the relatively simple... 

This classification has been discussed extensively within the context of a one-dimensional advection-diffusion model, along with simple solutions to the relevant equations (Craig, 1969). It should be noted, however, that specific tracers may fall into different categories depending on the nature of the specific application. For example, radiocarbon is a transient tracer in the surface waters of the ocean because its natural inventory (due to cosmic ray production) has been affected... 

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