Modeling Membrane Hydrodynamics

In modeling an RO unit, two aspects should be considered membrane transport equations and hydrodynamic modeling of the RO module. The membrane transport equations represent the phenomena (water permeation, solute flux, etc.) taking place at the membrane surface. On the other hand, the hydrodynamic model deals with the macroscopic transport of the various species along with the momentum and energy associated with them. In recent years, a number of mathematical... 

An alternative description of membrane stability has been based on hydrodynamic models, originally developed for liquid films in various environments [54-56]. Rupture of the film was rationalized by the instability of symmetrical squeezing modes (SQM) related to the thickness fluctuations. In the simplest form it can be described by a condition [54] d Vdis/dh < where is the interaction contribution related to the dis-... 

One of the primary compounds that the cell membrane is composed of is DMPC, because of its surface activity caused by a hydrophobic zwitterion. A hydrodynamic model of the DMPC membrane can evaluate the intrinsic viscosity (r]i) of the surfactant monolayer, eliminating the contribution of the viscosity of dodecane and aqueous phases. The T]i values listed in Table 10.1 are about 2-4 times higher than the apparent i. The maximum t]i value, 0.75 Pa s, is comparable to that of a common viscous liquid such as glycerin (0.945 Pa s). This study demonstrated that a single molecule probing method could successfully measure the hydrodynamic properties of the interface. 

The permeate is continuously withdrawn through the membrane from the feed sueam. The fluid velocity, pressure and species concentrations on both sides of the membrane and permeate flux are made complex by the reaction and the suction of the permeate stream and all of them depend on the position, design configurations and operating conditions in the membrane reactor. In other words, the Navier-Stokes equations, the convective diffusion equations of species and the reaction kinetics equations are coupled. The transport equations are usually coupled through the concentration-dependent membrane flux and species concentration gradients at the membrane wall. As shown in Chapter 10, for all the available membrane reactor models, the hydrodynamics is assumed to follow prescribed velocity and sometimes pressure drop equations. This makes the species transport and kinetics equations decoupled and renders the solution of... 

A theoretical prediction for the receptor translational diffusion coefficient Dr based on viscous interactions of proteins with membrane lipids was derived by Saffman and Delbruck (1975). In their hydrodynamic model, a solitary cylindrical protein (i.e., a protein at infinite dilution) of radius s embedded in a membrane of thickness h and viscosity 17 m,... 

A.G. Winger, R. Ferguson and R. Kunin, The electroosmotic transport of water across permselective membranes, J. Phys. Chem., 1956, 60, 556 B.R. Breslau and I.F. Miller, A hydrodynamic model for electroosmosis, Ind. Eng. Chem., Fundam., 1971, 10, 554-565 T. Okada, G. Xie, O. Gorseth, S. Kjelstrup, N. Nakamura and T. Arimura, Ion and water transport characteristics of Nafion membranes as electrolytes, Electro-chim. Acta, 1998, 43, 3741-3747. 

The EOD coefficient ( drag) is defined as the ratio of the flux of water through the membrane to the flux of protons in the absence of a concentration gradient of water [224]. EOD increases with current density and often exceeds the ability of the membrane to redistribute water by back diffusion. A hydrodynamic model for electroosmosis has been developed [225] which treats ions as spherical particles moving in a continuous viscous medium. The model describes the variation of the EOD within polystyrene-based model membranes quite well. Several methods have been utilized to measure EOD coefficients including streaming potential measurements [226,227], the use of concentration cells [224,228], water flux measurements [191], DMFC experiments [229,230], and NMR spectroscopy [231]. Plots of n rag values for selected membranes are shown in Fig. 30. 

When the membranes are macroporous and the fluid flow is laminar, a simple hydrodynamic theory can be applied to describe the transfer across the membrane. Three models are nsnally nsed in this case. The first is the Darcy s law developed for the flux through porous media [5] ... 

The second hydrodynamic model is the Hagen-PoiseuiUe model. It is used to macroporous membranes with cylindrical straight pores of same diameter ... 

Membrane Pore diameter (nm) Pore density (cm ) Thickness (pm) Enhancement over Knudsen model Enhancement over hydrodynamic model Minimum slip length... 

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. 

Most theoretical studies of osmosis and reverse osmosis have been carried out using macroscopic continuum hydrodynamics [5,8-13]. The models used include those that treat the wall as either nonporous or porous. In the nonporous models the membrane surface is assumed homogeneous and nonporous. Transport occurs by the molecules dissolving in the membrane phase and then diffusing through the membrane. Mass transfer across the membrane in these models is usually described using the solution-diffusion... 

Similar instability is caused by the electrostatic attraction due to the applied voltage [56]. Subsequently the hydrodynamic approach was extended to viscoelastic films apparently designed to imitate membranes (see Refs. 58-60, and references therein). A number of studies [58, 61-64] concluded that the SQM could be unstable in such models at small voltages with low associated thinning, consistent with the experimental results. However, as has been shown [60, 65-67], the viscoelastic models leading to instability of the SQM did not account for the elastic force normal to the membrane plane which opposes thickness... 

A significant size exclusion phenomenon was observed for the IPN membranes. Theophylline R =1.3 A), proxyphylline = 2.3 A), ox-prenolol HCl R = 2.6 A), and FITC-Dextran R = 49 A) were used as model drugs in the diffusion study where R denotes the hydrodynamic radius of the solute. The solute size, membrane mesh size, pH, temperature, and the affinity of the solute with the membrane can affect the permeation of the solute. 

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,... 

The permeability Ps is a measure of the transport of a molecule by diffusion. The reflection coefficient a of a given component is the maximal possible rejection for that component (at infinite solvent flux). Various models have been proposed for the reflection coefficient [75-77]. In the lognormal model [78], a lognormal distribution is assumed for the pore size. No steric hindrance in the pores or hydrodynamic lag is taken into account, but it is assumed that a molecule permeates through every pore that is larger than the diameter of the molecule. Moreover, the diffusion contribution to the transport through the membrane is considered to be negligible. Therefore, the reflection curve can be expressed as ... 

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... 

The basic hydrodynamic equations are the Navier-Stokes equations [51]. These equations are listed in their general form in Appendix C. The combination of these equations, for example, with Darcy s law, the fluid flow in crossflow filtration in tubular or capillary membranes can be described [52]. In most cases of enzyme or microbial membrane reactors where enzymes are immobilized within the membrane matrix or in a thin layer at the matrix/shell interface or the live cells are inoculated into the shell, a cake layer is not formed on the membrane surface. The concentration-polarization layer can exist but this layer does not alter the value of the convective velocity. Several studies have modeled the convective-flow profiles in a hollow-fiber and/or flat-sheet membranes [11, 35, 44, 53-56]. Bruining [44] gives a general description of flows and pressures for enzyme membrane reactor. Three main modes... 

If in stationary operation conditions, membrane clogging does not occur or is negligible, then the modelling case becomes banal. Nevertheless, vhen surface clogging cannot be eliminated by the tangential flow rate, we must introduce a continuous increase in the hydrodynamic resistance of the membrane [3.16-3.18],... 

For ease of fabrication and modular construction, tubular reactors are widely used in continuous processes in the chemical processing industry. Therefore, shell-and-tube membrane reactors will be adopted as the basic model geometry in this chapter. In real production situations, however, more complex geometries and flow configurations are encountered which may require three-dimensional numerical simulation of the complicated physicochemical hydrodynamics. With the advent of more powerful computers and more efficient computational fluid dynamics (CFD) codes, the solution to these complicated problems starts to become feasible. This is particularly true in view of the ongoing intensified interest in parallel computing as applied to CFD. 

None of the above studies, however, deals with the detailed hydrodynamics in a membrane reactor. It can be appreciated that detailed information on the hydrodynamics in a membrane enhances the understanding and prediction of the separation as well as reaction performances in a membrane reactor. All the reactor models presented in Chapter 10 assume very simple flow patterns in both the tube and annular regions. In almost all cases either plug flow or perfect mixing is used to represent the hydrodynamics in each reactor zone. No studies have yet been published linking detailed hydrodynamics inside a membrane reactor to reactor models. With the advent of CFD, this more complete rigorous description of a membrane reactor should become feasible in the near future. 

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