Finely Porous Model

H. Mehdizadeh, J. M. Dickson. Theoretical modifications of the finely porous model for reverse osmosis. J Appl Polym Sci 42 1143, 1991. 

The finely-porous model is based on a balance of applied and frictional forces in a 1-dimentional pore.7 The model considers friction between the solute and solvent, and between the solute and the membrane material. The model also includes the membrane thickness and the fractional pore area of the membrane surface. 

As with the finely-porous model, (Chapter 4.1.3), the mathematical representation of solvent and solute fluxes for the irreversible thermodynamic model is quite complex and beyond the scope of this work. However, it is recommended that readers consider references1 and8 for details on this transport model. 

The same evolution of the rejection coefficient with volume flow and indirectly with transmembrane pressure was predicted by Tremblay [54] using the finely porous model proposed by Merten [55] and modified by Mehdizadeh... 

A.Y. Tremblay, Finely porous models and radially averaged friction factor, /. Appl. Polym. Sci., 45 (1) (1992) 159. 

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. 

Finely Porous Model. In this model, solute and solvent permeate the membrane via pores which connect the high pressure and low pressure faces of the membrane. The finely porous model, which combines a viscous flow model eind a friction model (7, ), has been developed in detail and applied to RO data by Jonsson (9-12). The most recent work of Jonsson (12) treated several organic solutes including phenol and octanol, both of which exhibit solute preferential sorption. In his paper, Jonsson compared several models including that developed by Spiegler eind Kedem (13) (which is essentially an irreversible thermodynamics treatment), the finely porous model, the solution-diffusion Imperfection model (14), and a model developed by Pusch (15). Jonsson illustrated that the finely porous model is similar in form to the Spiegler-Kedem relationship. Both models fit the data equally well, although not with total accuracy. The Pusch model has a similar form and proves to be less accurate, while the solution-diffusion imperfection model is even less accurate. 

In all models, the largest discrepeincy between the predicted performance and the experimental data occurred when negative separation was observed. Jonsson concluded that the finely porous model is preferred over the alternatives, although the Pusch relationship is easier to use and yields reasonable results in most cases. 

The advantage of the preferential sorption-capillary flow approach to reverse osmosis lies in its emphasis on the mechanism of separation at a molecular level. This knowledge is useful when it becomes necessary to predict membrane performance for unknown systems. Also, the approach is not restricted to the so-called "perfect", defect-free membranes, but encompasses the whole range of membrane pore size. Until recently, the application of a quantitative model to the case of solute preferential sorption has been missing. Attempts to change this situation have been made by Matsuura and Sourirajan (21) by using a modified finely porous model. In addition to the usual features of this model (9-12), a Lennard-Jones type of potential function is Incorporated to describe the membrane-solute interaction. This model is discussed elsewhere in this book. 

The Fine Porous Model as presented by Xu and Spencer (1997), describes equilibrium and non-equilibrium factors of rejection. Only coupling between solvent and solute is taken into account, and no solute-solute coupling is permitted. Equilibrium parameters dominated separation, and these are described by the reflection coefficient (J in equation (3.28), where kii is the solute mass transfer coefficient in the membrane. 

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 absence of any concentration polarization, and Cfi are equal to Cg and respectively. The extent of concentration polarization and its effects on the solvent flux and solute transport for porous membranes and macrosolutes/proteins can be quite severe (see Section 6.3.3). This model is often termed the combined diffusion-viscous flow model (Merten, 1966), and it can be used in ultrafiltration (see Sections 6.3.3.2 and 7.2.1.3). The relations between this and other models, such as the finely porous model, are considered in Soltanieh and Gill (1981). 

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