Reverse osmosis Spiegler-Kedem 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... 

S. Jain, S.K. Gupta, Analysis of modified surface pore flow model with concentration polarization and comparison with Spiegler-Kedem model in reverse osmosis system, J. Membr. Sci. 232 (2004) 45-61. 

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... 

Another three-parameter model of reverse osmosis membrane transport of some importance is the Spiegler-Kedem model (Spiegler and Kedem, 1966). In its differential form (similar to equations (3.4.47) and (3.4.48)), the model proposes the following local flux expressions at any location z in the membrane ... 

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