Solution diffusion model transport equation through membrane

Predictions of salt and water transport can be made from this application of the solution-diffusion model to reverse osmosis (first derived by Merten and coworkers) [12,13], According to Equation (2.43), the water flux through a reverse osmosis membrane remains small up to the osmotic pressure of the salt solution and then increases with applied pressure, whereas according to Equation (2.46), the salt flux is essentially independent of pressure. Some typical results are shown in Figure 2.9. Also shown in this figure is a term called the rejection coefficient, R, which is defined as... 

In the solution-diffusion model, the transport of solvent and solute are independent of each other, as seen in Equations 4.1 and 4.2. The flux of solvent through the membrane is linearly proportional to the effective pressure difference across the membrane (Equation 4.1) ... 

Water flux through the membrane is represented by Equation 4.3. This flux is based on the solutions - diffusion model with the added term to reflect transport due to the imperfections. 

Various models and mechanisms for the solvent and the solute transports through tight osmosis membrane have been developed [18-20]. The solution-diffusion model can be employed to calculate the pure water flux (Equation 21.2), and the solute flux (Equation 21.4),ys [21]. 

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. 

The differences between the surface force-pore flow model and the solution-diffusion model are (1) the microscopic structure of the membrane is incorporated explicitly into the transport equations as the pore radius in the surface force-pore flow model (2) the interaction force working between the permeant and the membrane is also incorporated into the transport equations as an interaction force parameter in the surface force-pore flow model (3) as mentioned in Chapter 5, the solution-diffusion model describes the transport of permeants through the membrane, as an uncoupled diffusive flow. Mass transfer by the diffusive flow is expressed by a set of transport parameters that are intrinsic to the polymeric material. Any flow other than the above intrinsic diffusive flow is... 

The permeate is continuously withdrawn through the membrane from the feed sueam. The fluid velocity, pressure and species concentrations on both sides of the membrane and permeate flux are made complex by the reaction and the suction of the permeate stream and all of them depend on the position, design configurations and operating conditions in the membrane reactor. In other words, the Navier-Stokes equations, the convective diffusion equations of species and the reaction kinetics equations are coupled. The transport equations are usually coupled through the concentration-dependent membrane flux and species concentration gradients at the membrane wall. As shown in Chapter 10, for all the available membrane reactor models, the hydrodynamics is assumed to follow prescribed velocity and sometimes pressure drop equations. This makes the species transport and kinetics equations decoupled and renders the solution of... 

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. 

Diffusion-type model. For diffusion-type membranes, the steady-state equations governing the transport of solvent and of solute are to a first approximation as follows (M2, M3). For the diffusion of the solvent through the membrane as shown in Fig. 13.9-2,... 

Equation 2 is an analytical statement of the solution-diffusion mqdel of penetrant transport in polymers, which is the most widely accepted explanation of the mechanism of gas permeation in nonporous polymers (75). According to this model, penetrants first dissolve into the upstream (i.e, high pressure) face of Ae film, diffuse through the film, and desorb at the downstream (Le. low pressure) face of the film. Diffusion, the second step, is the rate limiting process in penetrant permeation. As a result, much of the fundamental research related to the development of polymers with improved gas separation properties focuses on manipulation of penetrant diffusion coefficients via systematic modification of polymer chemical structure or superstructure and either chemical or thermal post-treatment of polymer membranes. Many of the fundamental studies recorded in this book describe the results of research projects to explore the linkage between polymer structure, processing history, and small molecule transport properties. 

There are a number of other models of transport of solvent and solute through a reverse osmosis membrane the Kedem-Katchalsky model, the Spiegler-Kedem model, the frictional model, the finely porous model, the preferential sorption-capUlary flow model, etc. Most of these models have heen reviewed and compared in great detail hy Soltanieh and GiU (1981). We will restrict ourselves in this hook to the solution-diffusion and solution-diffusion-imperfection flux expressions for a number of reasons. First, the form of the solution-diffusion equation is most commonly used and is also functionally equivalent to the preferential sorption-capiUary flow model. Secondly, the solution-diffusion-imperfection model is functionally representative of a number of more exact three-transport-coefficient models, even though the transport coefficients in this model are concentration-dependent... 

Peppas and Reinhart have also proposed a model to describe the transport of solutes through highly swollen nonporous polymer membranes [155], In highly swollen networks, one may assume that the diffusional jump length of a solute molecule in the membrane is approximately the same as that in pure solvent. Their model relates the diffusion coefficient in the membrane to solute size as well as to structural parameters such as the degree of swelling and the molecular weight between crosslinks. The final form of the equation by Peppas and Reinhart is... 

The theoretical description of the kinetics of transmembrane transport through a liquid membrane should be based on the principles of solvent extraction kinetics. It should be determined by the processes at both water/membrane interphases and should also involve the intermediate step of diffusion in the membrane. Thus the existence of all these three steps makes the membrane system and its description much more complicated than the relatively simple water/organic phase. However, even the kinetics mechanism in simpler extraction systems is often based on the models dealing only with some limiting situations. As it was pointed out in the beginning of this paper, the kinetics of transmembrane transport is a fimction both of the kinetics of various chemical reactions occurring in the system and of diffusion of various species that participate in the process. The problem is that the system is not homogeneous, and concentrations of the substances at any point of the system depend on the distance from the membrane surface and are determined by both diffusion and reactions. The solution of a system of differential equations in this case can be a serious problem. 

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