Solution-diffusion model

The above equation assumes that the swelling of the membrane separating layer is negligible. It is similar to the well-known equation presented by Wij- 

Bhanushali et al. [24] suggested that solvent viscosity and surface tension are dominant factors controlling solvent transport through NF membranes, and a 

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. 

Finally, Machado et al. [21] developed a resistances-in-series model and proposed that solvent transport through the MPF membrane consists of three main steps (1) transfer of the solvent into the top active layer, which is characterized by surface resistance (2) viscous flow through NF pores and (3) viscous flow through support layer pores, all expressed by viscous resistances, i.e. 

Where J , Rj and are the surface resistance and viscous resistances through NF active layer and support layers, respectively. The surface resistance is proportional to the surface-tension difference between the solvent and the OSN top layer, and viscous resistances are proportional to solvent viscosity. 

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Reverse osmosis models can be divided into three types irreversible thermodynamics models, such as Kedem-Katchalsky and Spiegler-Kedem models nonporous or homogeneous membrane models, such as the solution—diffusion (SD), solution—diffusion—imperfection, and extended solution—diffusion models and pore models, such as the finely porous, preferential sorption—capillary flow, and surface force—pore flow models. Charged RO membrane theories can be used to describe nanofiltration membranes, which are often negatively charged. Models such as Dorman exclusion and the... 

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. 

Extended-solution-diffusion model, 21 640 Extended X-ray absorption fine structure (EXAFS), 24 72 analysis, 22 564 Extender blacks, 21 776 Extender pigments, 19 410 Extenders paint, 18 59... 

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. 

Membrane deterioration may be merely caused by decrease of acetyl content(C ) in the active surface layer as a result of hydrolysis or oxidation, not by structure change. Analysis was carried out based on solution-diffusion model proposed by Lonsdale etal( ), using their measured values of solute diffusivity and partition coefficient in homogeneous membrnaes of various degree of acetyl content and also using those values of asymmetric membranes heat treated at various temperatures measured by Glueckauf(x) ... 

Concentration in the permeate is expressed by that in the feed as Equation (5), using solution-diffusion model assuming no concentration polarization. 

If g=l the solution-diffusion model is said to hold. By plotting R versus (AP-aAir) from Eq(12), the intercept is g and the slope is (P2RT/P v ). ... 

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. 

Parameters From the Solution-Diffusion Model And Spectroscopic Results of Aqueous Solutions... 

Feed water (Cp, Qp) is allowed to enter into the module from its left side. Permeated water at each membrane section is expressed simply by Solution Diffusion model (Eq (1), (2) in Table 3) Concentrated water (Cg, Qg) is discharged from the right side of the module. Summation of product water at each membrane section results in total product water (Cp, Qp). At each membrane section, material balance is taken on each component (Eq (7), (8) in Table 3). 

The solution-diffusion model (1 ) assumes that water and salt diffuse independently across the membrane and allows no convective salt transport. The reciprocal salt rejection, 1/r, is linearly related to the reciprocal volume flux, 1/q ... 

The results of a reverse osmosis study of radiation crossllnk-ed and heat treated polyvinyl alcohol(PVA) membranes are reported. In the framework of this study the permeability of water and salt through these membranes was investigated. In parallel, the diffusive transport of salt through PVA was also studied. The results suggest that the transport of salt and water through PVA is uncoupled, The salt transport data can be rationalized in terms of a modified solution-diffusion model. 

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. 

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 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. 

The solution-diffusion model applies to reverse osmosis, pervaporation and gas permeation in polymer films. At first glance these processes appear to be very... 

The second assumption concerns the pressure and concentration gradients in the membrane. The solution-diffusion model assumes that when pressure is applied across a dense membrane, the pressure throughout the membrane is constant at the highest value. This assumes, in effect, that solution-diffusion membranes transmit pressure in the same way as liquids. Consequently, the solution-diffusion model assumes that the pressure within a membrane is uniform and that the chemical potential gradient across the membrane is expressed only as a concentration gradient [5,10]. The consequences of these two assumptions are illustrated in Figure 2.5, which shows pressure-driven permeation of a one-component solution through a membrane by the solution-diffusion mechanism. 

By using osmosis as an example, concentration and pressure gradients according to the solution-diffusion model can be discussed in a somewhat more complex situation. The activity, pressure, and chemical potential gradients within this type of membrane are illustrated in Figure 2.6. 

Figure 2.6 Chemical potential, pressure, and solvent activity profiles through an osmotic membrane following the solution-diffusion model. The pressure in the membrane is uniform and equal to the high-pressure value, so the chemical potential gradient within the membrane is expressed as a concentration gradient...

Application of the Solution-diffusion Model to Specific Processes... 

In this section the solution-diffusion model is used to describe transport in dialysis, reverse osmosis, gas permeation and pervaporation membranes. The resulting equations, linking the driving forces of pressure and concentration with flow, are then shown to be consistent with experimental observations. 

The general approach is to use the first assumption of the solution-diffusion model, namely, that the chemical potential of the feed and permeate fluids are... 

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