Concentration polarization boundary layer model

When concentration polarization effects are present an estimated value of j8 can be used to make an approximate correction for this effect. This is used in Eq. (13.10-1) to obtain a value of Am for use in Eqs. (13.9-2) and (13.9-9). Also, Eq. (13.10-2) will replace Eq. (13.9-5). A more detailed analysis of this complete mixing model is given by others (HI, Kl) in which the mass transfer coefficient in the concentration polarization boundary layer is used. 

According to this model the permeate flux at steady-state is independent of the imposed pressure drop and it is controlled by the concentration polarization boundary layer. The increasing pressure drop results in a thicker solute layer until convection and difliision in the boundary layer will... 

Concentration polarization can dominate the transmembrane flux in UF, and this can be described by boundary-layer models. Because the fluxes through nonporous barriers are lower than in UF, polarization effects are less important in reverse osmosis (RO), nanofiltration (NF), pervaporation (PV), electrodialysis (ED) or carrier-mediated separation. Interactions between substances in the feed and the membrane surface (adsorption, fouling) may also significantly influence the separation performance fouling is especially strong with aqueous feeds. 

The phenomenon of concentration polarization, which is observed frequently in membrane separation processes, can be described in mathematical terms, as shown in Figure 30 (71). The usual model, which is weU founded in fluid hydrodynamics, assumes the bulk solution to be turbulent, but adjacent to the membrane surface there exists a stagnant laminar boundary layer of thickness (5) typically 50—200 p.m, in which there is no turbulent mixing. The concentration of the macromolecules in the bulk solution concentration is c,. and the concentration of macromolecules at the membrane surface is c. 

Using this model and the assumption that concentration polarization occurs only on the feed side of the membrane, the flux across the combined resistances of the feed side boundary layer and the membrane can be written as... 

The second approach to concentration polarization, and the one used in this chapter, is to model the phenomenon by assuming that a thin layer of unmixed fluid, thickness S, exists between the membrane surface and the well-mixed bulk solution. The concentration gradients that control concentration polarization form in this layer. This boundary layer film model oversimplifies the fluid hydrodynamics occurring in membrane modules and still contains one adjustable parameter,... 

Figure 4.2 Fluid flow velocity through the channel of a membrane module is nonuniform, being fastest in the middle and essentially zero adjacent to the membrane. In the film model of concentration polarization, concentration gradients formed due to transport through the membrane are assumed to be confined to the laminar boundary layer...

Figure 4.4 Salt concentration gradients adjacent to a reverse osmosis desalination membrane. The mass balance equation for solute flux across the boundary layer is the basis of the film model description of concentration polarization...

The formation of concentration gradients caused by the flow of ions through a single cationic membrane is shown in Figure 10.8. As in the treatment of concentration polarization in other membrane processes, the resistance of the aqueous solution is modeled as a thin boundary layer of unstirred solution separating the... 

The concentration polarization occurring in electrodialysis, that is, the concentration profiles at the membrane surface can be calculated by a mass balance taking into account all fluxes in the boundary layer and the hydrodynamic conditions in the flow channel between the membranes. To a first approximation the salt concentration at the membrane surface can be calculated and related to the current density by applying the so-called Nernst film model, which assumes that the bulk solution between the laminar boundary layers has a uniform concentration, whereas the concentration in the boundary layers changes over the thickness of the boundary layer. However, the concentration at the membrane surface and the boundary layer thickness are constant along the flow channel from the cell entrance to the exit. In a practical electrodialysis stack there will be entrance and exit effects and concentration... 

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. 

In discussing the Bartlett model, we assumed there was no concentration polarization in the solution at the film surface. This effect can be incorporated into the model for species B if the boundary condition at X = L is altered so that fluxes at the film surface match the transport across the diffusion layer. This is done by Bartlett and coworkers in their papers. " In the context of the RDE, we find that at high rotation speeds, the current response decreases considerably. This experimental observation can be accounted for quite adequately by using the modified boundary condition at x = L. The steady-state current response decreases as rotation speed is increased because at higher values of the... 

---