Reverse osmosis solution-diffusion model

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... 

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. 

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)... 

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... 

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. 

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... 

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. 

A number of models have been developed over the years to describe reverse osmosis. These models Include the solution-diffusion model, the finely porous model, and the preferential sorption - capillary flow model. In each case, the model was originally developed based on the separation of aqueous,salt solutions. The application of each of these models to systems which exhibit anomalous behavior will be discussed in this section. 

Reverse osmosis is simply the application of pressure on a solution in excess of the osmotic pressure to create a driving force that reverses the direction of osmotic transfer of the solvent, usually water. The transport behavior can be analyzed elegantly by using general theories of irreversible thermodynamics however, a simplified solution-diffusion model accounts quite well for the actual details and mechanism in most reverse osmosis systems. Most successful membranes for this purpose sorb approximately 5 to 15% water at equilibrium. A thermodynamic analysis shows that the application of a pressure difference, Ap, to the water on the two sides of the membrane induces a differential concentration of water within the membrane at its two faces in accordance with the following (31) ... 

The solution-diffusion model seems to represent the performance of a reverse osmosis membrane. Figure 4.3 shows the salt rejection and flux of a low pressure polyamide membrane as a function of applied pressure. The membrane was operated on a 5,000 mg/ aqueous solution of sodium chloride at 25°C. As can be seen, there was no product water flow until the applied pressure exceeded the osmotic pressure (50 psi). After this, the flux increased linearly as would be predicted by the above water flux equation. Rejection is poor at lower pressures and increases rapidly until it reaches an asymptote at an applied pressure of about 150 psig. This can be attributed to a near constant flow of salt with a rapidly increasing product water flow which results in a more dilute product or in increased rejection. These data tend to substantiate the assertion of the solution-diffusion model that flow is uncoupled. 

The solution-diffusion model is currently being used by the majority of the membrane community. An excellent summary of the applicability of this model for various membrane transport phenomena is given in the classic book by Crank and Park (100]. Lonsdale applied this model for reverse osmosis transport [ 101) when the latter process emerged as an important desalination process. 

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... 

Develop systematically an explicit solvent flux expression in the reverse osmosis separation of two noninteracting solutes / = 2, 3 from water in the manner of equation (3.4.66a). Assume that (3.4.65c) is valid for each solute. Further, i y ktf and Ctp for i = 2, 3. Write down the solute flux expressions as welL Use the solution-diffusion model assume dilute solution of each solute. 

In Section 3.4.2.1, the phenomenon of reverse osmosis (RO) through a nonporous membrane was introduced. If the hydraulic pressure of a solution containing a microsolute, e.g. common salt, on one side of a nonporous membrane exceeds that of another solution on the other side of the same membrane by an amount more than the difference of the osmotic pressures of the same two solutions, then, according to the solution-diffusion model, the solvent will flow from the solution at higher pressure to the one at a lower pressure (equation (3.4.54)) at the following rate ... 

Ultrafiltration separations range from ca 1 to 100 nm. Above ca 50 nm, the process is often known as microfiltration. Transport through ultrafiltration and microfiltration membranes is described by pore-flow models. Below ca 2 nm, interactions between the membrane material and the solute and solvent become significant. That process, called reverse osmosis or hyperfiltration, is best described by solution—diffusion mechanisms. 

Most theoretical studies of osmosis and reverse osmosis have been carried out using macroscopic continuum hydrodynamics [5,8-13]. The models used include those that treat the wall as either nonporous or porous. In the nonporous models the membrane surface is assumed homogeneous and nonporous. Transport occurs by the molecules dissolving in the membrane phase and then diffusing through the membrane. Mass transfer across the membrane in these models is usually described using the solution-diffusion... 

An effect not considered in the above models is the added resistance, caused by fouling, to solute back-diffusion from the boundary layer. Fouling thus increases concentration polarization effects and raises the osmotic pressure of the feed adjacent to the membrane surface, so reducing the driving force for permeation. This factor was explored experimentally by Sheppard and Thomas (31) by covering reverse osmosis membranes with uniform, permeable plastic films. These authors also developed a predictive model to correlate their results. Carter et al. (32) have studied the concentration polarization caused by the build-up of rust fouling layers on reverse osmosis membranes but assumed (and confirmed by experiment) that the rust layer had negligible hydraulic resistance. 

Both 7 and 7 are diffusive fluxes through the membrane. If there are large pores in the membrane, there will be convection through such pores or defects. The total flux of any species will no longer be completely diffusive and therefore should be expressed in terms of Ni (Soltanieh and GiU, 1981). A simple model for the fluxes through a reverse osmosis membrane having large pores (or defects or imperfections) has been provided by Sherwood et al. (1967) it is called the solution-diffusion-imperfection model ... 

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... 

In the article on electro-osmosis (q.v.) a similar formula, but with 4 in place of the factor 6, is derived. Since electrophoresis is the reverse of electro-osmosis, the same expression should apply in both cases to the potential at the surface of shear between the two phases. The explanation of the apparent discrepancy is that instead of applying Stokes s law to a small sphere, the derivation of the electro-osmotic effect is based on the model of a parallel plate capacitor, i.e. on a large solid surface whose radius of curvature is negligible (compared with the thickness of the diffuse double layer). Closer analysis of the problem by Henry and Booth has shown that 4 is the correct factor for large particles, independent of their size and shape, but that for most systems, e.g. stable colloidal solutions, the factor varies between 4 and 6, depending on the size of the particle and the thickness of its atmosphere. 

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