Pore-flow model

Stable pores are assumed to be present inside the membrane and the driving force for transport is the pressure gradient across the membrane. Assuming a system at constant temperature where there are no external forces except pressure, one can derive, from the Stefan-MaxweU equations [34], the following equations describing the total volumetric flux through a membrane  

Hagen-PoiseuiUe equation - if the membrane is assumed to be composed of more or less cyhndrical pores 

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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... 

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. 

Transport equations, for the surface force-pore flow model, 21 640—641 Transport gasifier, 6 798 Transport models, reverse osmosis, 21 638-639... 

Calculation of Solute Separation and Product Rate. Once the pore size distribution parameters R, ou R >,2, 02, and h2 are known for a membrane and the interfacial interaction force parameters B and D are known for a given system of membrane material-solute, solute separation f can be calculated by eq 6 for any combination of these parameters. Furthermore, because the PR-to-PWP ratio (PR/PWP) can also be calculated by the surface force-pore flow model (9), PR is obtained by multiplying experimental PWP data by this ratio. 

The interfacial force constants available in the literature for many organic solutes that constitute potential pollutants in water enable one to calculate the separation of such solutes at various operating conditions by a membrane of a given average pore size and pore size distribution on the basis of the surface force-pore flow model. The product rate of the permeate solution can also be calculated. Such data further allow us to calculate the processing capacity of a membrane to achieve a preset ratio of concentration in the concentrate to concentration of the initial feed solution. 

Pressure-driven convective flow, the basis of the pore flow model, is most commonly used to describe flow in a capillary or porous medium. The basic equation covering this type of transport is Darcy s law, which can be written as... 

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. 

For relatively porous nanofiltration membranes, simple pore flow models based on convective flow will be adapted to incorporate the influence of the parameters mentioned above. The Hagen-Poiseuille model and the Jonsson and Boesen model, which are commonly used for aqueous systems permeating through porous media, such as microfiltration and ultrafiltration membranes, take no interaction parameters into account, and the viscosity as the only solvent parameter. It is expected that these equations will be insufficient to describe the performance of solvent resistant nanofiltration membranes. Machado et al. [62] developed a resistance-in-series model based on convective transport of the solvent for the permeation of pure solvents and solvent mixtures ... 

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. 

These two quantities may be described on the basis of the pore flow model presented above. 

Adopting the assumptions of the laminar pore flow model described above, it is possible to compute these two sources of dissipation and minimize the total dissipation. This leads to the following relation ... 

In the earlier work (1 ) transport equations were developed on the basis of surface force-pore flow model in which a surface potential function and a frictional function are incorporated. The results can be briefly summarized as follows ... 

Thus, in a "pore-flow" model, R would not be expected to increase with pressure since the solute and solvent flux are "coupled". As the pressure is increased, both fluxes increase. 

The Pore Flow Model uses the Hagen-Poiseuille Equation to describe solvent flow through cylindrical pores of the membrane. No membrane characteristics other than pore size or pore density are accounted for, and neither limitation of flux due to friction nor diffusion is considered. Flux occurs due to convection under an applied pressure. The equation is derived from the balance between the driving force pressure and the fluid viscosity, which resists flow (Braghetta (1995), Staude (1992)). Solvent flux ( ) is described by equation (3.26) and solute flux (Js) by equation (3.27), where rp is the pore radius, np the number of pores, T the tortuosity factor. Ax the membrane thickness and ct the reflection coefficient. 

Transport in OSN membranes occurs by mechanisms similar to those in membranes used for aqueous separations. Most theoretical analyses rely on either irreversible thermodynamics, the pore-flow model and the extended Nemst-Planck equation, or the solution-diffusion model [135]. To account for coupling between solute and solvent transport (i.e., convective mass transfer effects), the Stefan-Maxwell equations commonly are used. The solution-diffusion model appears to provide a better description of mixed-solvent transport and allow prediction of mixture transport rates from pure component measurements [136]. Experimental transport measurements may depend significantly on membrane preconditioning due to strong solvent-membrane interactions that lead to swelling or solvent phase separation in the membrane pore structure [137]. 

Both solution-diffusion and pore-flow models have been used to analyze the experimental solvent flux data. 

The data for binary systems of mixed solvents can be described well using a two-parameter model, when the two parameters are obtained from the fluxes of each of the pure solvents. The solution-diffusion model provides a slightly better fit, but there is not much difference between this and the pore-flow model. 

Shao and Huang (2007) and Feng and Huang (1997) wrote reviews on polymeric membrane PV, which are focused on the fundamental understanding of the membrane. There are principally two approaches to describing mass transport in PV (i) The solution-diffusion model and (ii) The pore flow model. 

FIGURE 9.3 Schematic representation of the pervaporation transport mechanism (a) solution-diffusion model and (b) pore flow model. 

The distinguishing feature of the pore flow model is that it assumes a liquid-vapor phase boundary inside the membrane, and PV is considered to be a combination of liquid transport and vapor transport in series. 

The knowledge of the influence of these operating conditions on Ljn is of crucial interest to develop accurate and practical models of transport processes. Unfortunately, this influence is until now not very well known. We presented previously a first attempt in this direction, by minimizing the dissipation of energy related to the liquid flow creation (Crine (1978)). Adopting the assumptions of the laminar pore flow model (see figure 8), we obtained the following expression for... 

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