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... 

In accordance with the Spiegler-Kedem model (Krishna and Wesselingh, 1997), Avd is proportional to the net pressure difference across the membranes. Since the pressure difference (AP) may be regarded as negligible, Ava is mainly controlled by the corresponding instantaneous osmotic pressure difference (Are), this being proportional to the difference in solute concentration across the membranes for a great number of solutes (Lo Presti and Moresi, 2000) ... 

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 ... 

Note Here Cj, tt and P correspond to infinitely thin solutions in equilibrium with the local section of the membrane therefore C, is the molar concentration of solute i in a solution of osmotic pressure w.) There are three parameters here Qjj (the intrinsic hydraulic permeability), P (the local solute permeahility coefficient) and <7,- (the local solute reflection coefficient). When these two equations are integrated across a membrane of thickness assuming Qsi, P and <7 to be essentially constant across the membrane thickness, one obtains, for the whole membrane, two equations for the Spiegler-Kedem model (based on the Kedem-Katchalsky model) ... 

One would like to know now what the relation is between the solute rejection, / , and these parameters in the Spiegler-Kedem model. Focus on the ratio (4/4) as given by the relation (6.3.153a). If, instead, we consider the ratio... 

Thus, the Spiegler-Kedem model predicts correctly that, at very high AP, i ,- reaches a limiting value which is less than 1 (o , can have a maximum value of 1) for a given solute. SimpUstically, a value of o = 1 means that the solute i is completely rejected by the membrane it cannot enter the membrane. 

This is identical to the Spiegler-Kedem relationship, Eq. (2), and that of finely-porous membrane model, Eq. (3), with a = r. However, it should be noted that Eq. (28) is derived phenomenologically without any assumptions on the transport mechanism. 

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 the Fig. 7, we have reported the flux as function of the transmembrane pressure (AP) for a NaCI solution at a concentration of 10-1molL-1 (6gL 1) which is typical of a synthetic brackish water. The linearity observed suggests that this salty solution follows the Kedem-Katchalscky model (i.e. Spiegler-Ke-dem model, with pressure and osmotic linear gradients). For linear gradients, equation (1) amounts to... 

In addition to the approach using phenomenological equations for modelling ion transport in soils, the theory of irreversible thermodynamics may be adapted to soils [26], as for the case of ion-exchange membranes. Spiegler [251 and Kedem and Katchalsky [27,28] are the prime examples of this approach to transport models. The detailed review by Verbrugge and Pintauro contains a number of other references to mathematical approaches for modelling the fundamental electrokinetic phenomena. 

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