Solution-diffusion model for

D.R. Paul, The Solution-diffusion Model for Swollen Membranes, Sep. Purif. Meth. 5, 33 (1976). 

Understand the solution-diffusion model for separation of liquid and gas mixtures using dense, nonporous membranes. 

Solution-Diffusion Model for the Transport of Binary Gas Mixtures 9... 

In practical reverse osmosis with a positive (AP — Atc), there is considerable flow of solvent from the feed to the permeate. However, the membrane is designed to reject the solute species. Thus, from the feed solution next to the membrane, solvent is continuously withdrawn through the membrane, whereas the solute species is not. This leads to a build-up of solute concentration near the membrane-feed solution interface (Figure 3.4.7) in excess of the bulk feed solute concentration C,y This phenomenon is called concentration polarization. The feed-membrane interface is now exposed to a solute concentration instead of Cif < Cfji). Consequently, the solvent and solute flux expressions in the solution-diffusion model for one solute in a solvent system are changed to... 

Wang CY, Beckermann C (1993) A multiphase solute diffusion model for dendritic alloy solidification. Metall Trans A 24 2787—2802... 

Solution—Diffusion Model. In the solution—diffusion model, it is assumed that (/) the RO membrane has a homogeneous, nonporous surface layer (2) both the solute and solvent dissolve in this layer and then each diffuses across it (J) solute and solvent diffusion is uncoupled and each is the result of the particular material s chemical potential gradient across the membrane and (4) the gradients are the result of concentration and pressure differences across the membrane (26,30). The driving force for water transport is primarily a result of the net transmembrane pressure difference and can be represented by equation 5 ... 

When it was recognized (31) that the SD model does not explain the negative solute rejections found for some organics, the extended solution—diffusion model was formulated. The SD model does not take into account possible pressure dependence of the solute chemical potential which, although negligible for inorganic salt solutions, can be important for organic solutes (28,29). 

When a polymer film is exposed to a gas or vapour at one side and to vacuum or low pressure at the other, the mechanism generally accepted for the penetrant transport is an activated solution-diffusion model. The gas dissolved in the film surface diffuses through the film by a series of activated steps and evaporates at the lower pressure side. It is clear that both solubility and diffusivity are involved and that the polymer molecular and morphological features will affect the penetrant transport behaviour. Some of the chemical and morphological modification that have been observed for some epoxy-water systems to induce changes of the solubility and diffusivity will be briefly reviewed. 

Farmer (6) reviewed the various diffusion models for soil and developed solutions for several of these models. An appropriate model for field studies is a nonsteady state model that assumes that material is mixed into the soil to a depth L and then allowed to diffuse both to the surface and more deeply into the soil. Material diffusing to the surface is immediately removed by diffusion and convection in the air above the soil. The effect of this assumption is to make the concentration of a diffusing compound zero at the soil surface. With these boundary conditions the solution to Equation 8 can be converted to the useful form ... 

Dense membranes are used for pervaporation, as for reverse osmosis, and the process can be described by a solution-diffusion model. That is, in an ideal case there is equilibrium at the membrane interfaces and diffusional transport of components through the bulk of the membrane. The activity of a component on the feed side of the membrane is proportional to the composition of that component in the feed solution. 

Wang TF, Kasting GB, Nitsche JM (2006) A multiphase microscopic diffusion model for stratum comeum permeability. 1. Formulation, solution, and illustrative results for representative compounds. J Pharm Sci 95 620-647. 

Reaction rates of nonconservative chemicals in marine sediments can be estimated from porewater concentration profiles using a mathematical model similar to the onedimensional advection-diffusion model for the water column presented in Section 4.3.4. As with the water column, horizontal concentration gradients are assumed to be negligible as compared to the vertical gradients. In contrast to the water column, solute transport in the pore waters is controlled by molecular diffusion and advection, with the effects of turbulent mixing being negligible. 

In Fig.la a lower slope ((j)=2.25) and higher Intercept (g=1.32) Is obtained for the low pressure region as compared to the slope (((>=2.60) and Intercept (g=1.09) for high pressure region. Since from Eq(13), a value of g close to 1 implies adherence to the solution-diffusion model. It appears that at low pressures where g = 1.32>1.0, a combined model of viscous and diffusive flow Is operative. This correlates with previous SEM studies In our laboratory (unpublished), where mlcro-pln holes were postulated to exist In the skin. The presence of such m-LcAO-p-Ln hoZ 6 In the surface can be used to explain the high g-value. Above 10 atm, the DDS-990 membrane Is compressed or compacted and the mlcro-pln holes filled. Thus g = l.O l.O implies adherence to the solution-diffusion model. 

The finite solution volume model for solid-diffusion control (Patterson s model) will be used (eq. (4.52)). Following the procedure presented in the section Design of a batch reactor system for adsorption and ion exchange (eqs. (4.119)-(4.125)), we obtain the results shown in Table 4.23. 

In the performance data of various polyamide and related membranes published to date there should be valuable information for molecular design of more excellent barrier materials. But at present a means for their evaluation and optimization is still not clear. One of the reasons may at least come from the competitive flood of proposals for the detailed mechanisms of reverse osmosis, e.g. the solution-diffusion model, the sieve model, the preferential sorption model and so on. 109)... 

The transport of gas in polymers has been studied for over 150 years (1). Many of the concepts developed in 1866 by Graham (2) are still accepted today. Graham postulated that the mechanism of the permeation process involves the solution of the gas in the upstream surface of the membrane, diffusion through the membrane followed by evaporation from the downstream membrane surface. This is the basis for the "solution-diffusion model which is used even today in analyzing gas transport phenomena in polymeric membranes. 

Smith, E.H. 1991. Modified solution of homogeneous surface diffusion model for adsorption. J Environ. Eng-ASCE 117(3) 320-338. 

The difference between the solution-diffusion and pore-flow mechanisms lies in the relative size and permanence of the pores. For membranes in which transport is best described by the solution-diffusion model and Fick s law, the free-volume elements (pores) in the membrane are tiny spaces between polymer chains caused by thermal motion of the polymer molecules. These volume elements appear and disappear on about the same timescale as the motions of the permeants traversing the membrane. On the other hand, for a membrane in which transport is best described by a pore-flow model and Darcy s law, the free-volume elements (pores) are relatively large and fixed, do not fluctuate in position or volume on the timescale of permeant motion, and are connected to one another. The larger the individual free volume elements (pores), the more likely they are to be present long enough to produce pore-flow characteristics in the membrane. As a rough rule of thumb, the transition between transient (solution-diffusion) and permanent (pore-flow) pores is in the range 5-10 A diameter. 

Reverse osmosis, pervaporation and polymeric gas separation membranes have a dense polymer layer with no visible pores, in which the separation occurs. These membranes show different transport rates for molecules as small as 2-5 A in diameter. The fluxes of permeants through these membranes are also much lower than through the microporous membranes. Transport is best described by the solution-diffusion model. The spaces between the polymer chains in these membranes are less than 5 A in diameter and so are within the normal range of thermal motion of the polymer chains that make up the membrane matrix. Molecules permeate the membrane through free volume elements between the polymer chains that are transient on the timescale of the diffusion processes occurring. 

Sorption data were used to obtain values for A" L. As pointed out by Paul and Paciotti, the data in Figure 2.17 show that reverse osmosis and pervaporation obey one unique transport equation—Fick s law. In other words, transport follows the solution-diffusion model. The slope of the curve decreases at the higher concentration differences, that is, at smaller values for c,eimi because of decreases in the diffusion coefficient, as the swelling of the membrane decreases. 

In the 1940s to 1950s, Barrer [2], van Amerongen [3], Stem [4], Meares [5] and others laid the foundation of the modem theories of gas permeation. The solution-diffusion model of gas permeation developed then is still the accepted model for gas transport through membranes. However, despite the availability of interesting polymer materials, membrane fabrication technology was not sufficiently advanced at that time to make useful gas separation membrane systems from these polymers. 

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