GENERAL MEMBRANE EQUATION

It is not possible at present to provide an equation, or set of equations, that allows the prediction from first principles of the membrane permeation rate and solute rejection for a given real separation. Research aimed at providing such a prediction for model systems is under way, although the physical properties of real systems, both the membrane and the solute, are complex. An analogous situation exists for conventional filtration processes. The general membrane equation is an attempt to state the factors which may be important in determining the membrane permeation rate for pressure driven processes. This takes the form  

The solids-liquid separation of slurries containing particles below 10 xm is difficult by conventional filtration techniques. A conventional approach would be to use a slurry thickener in which the formation of a filter cake is restricted and the product is discharged continuously as a concentrated slurry. Such filters use filter cloths as the filtration medium 

All of the membrane processes listed in Table 8.1 are operated with such a cross-flow of the process feed. The advantages of cross-flow filtration over conventional filtration are  

Ideally, cross-flow microfiltration would be the pressure-driven removal of the process liquid through a porous medium without the deposition of particulate material. The flux decrease occurring during cross-flow microfiltration shows that this is not the case. If the decrease is due to particle deposition resulting from incomplete removal by the cross-flow liquid, then a description analogous to that of generalised cake filtration theory, discussed in Chapter 7, should apply. Equation 8.2 may then be written as  

Schneider and Klein(5) have pointed out that the early stages of cross-flow microfiltration often follow such a pattern although the growth of the cake is limited by the cross-flow of the process liquid. There are a number of ways of accounting for the control of cake growth. A useful method is to rewrite the resistance model to allow for the dynamics of polarisation in the film layer as discussed by Fane 6. Equation 8.3 is then written as  

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For gas and vapor systems, by combining the laws of sorption and diffusion in the sequence (l)-(3), general permeation equations are obtained. For sheet membrane samples of polymers above Tg, if the definition is made that permeation coefficient Q = Ds,... 

The concept of permeability. Pm, described first in Section 4.3.2.2 also applies to membranes. Equation (4.77) relates the permeability to the diffusion coefficient and solubility. Some representative values of permeabilities for common gases in common polymer films are given in Table 4.17. The units of permeability in Table 4.17 are obtained when diffusivity is in units of m /s, and gas solubility is in units of m gas m /(m soUd-N). Note that carbon dioxide permeabilities are generally 3-4 times... 

The permeability coefficient depends on the characteristics of the membrane and solute, and can vary considerably for various solutes. For example,/) = 10-21 m/s for sucrose and 10 4 m/s for water in the human red blood cell membrane. Equation (1) may be generalized by including the effect of pressure gradient APm = P(0) - P(L), and we have... 

Generalized Formalism A generalized membrane transport model, in the form of the black box models discussed earlier, can be considered in order to compare alternative mechanisms of water backflow in gradients of chemical potential, activity or concentration of water. Each of these gradients can be expressed by a gradient in w. The equation of net water flow is, thus,... 

The physical model bridges the gap between the two types of mathematical models in the literature. Furthermore, it does so with a physically based description of the structure of the membrane. However, to put it to use in simulations a mathematical model and approach is required that describes the governing phenomena discussed above. In this section, the general governing equations based on the physical model are developed using concentrated-solution theory and the approach of having two transport modes is introduced. 

The above relationship allows the simplification of the driving forces in the membrane system. Using the above formulation, substituting the resultant driving forces into the generalized transport equation, Eq. (5.1), and inverting the resultant equations yields the two independent transport equations... 

In this approach, the smaller scale models are used to determine the closure equations to be used in larger scale models. The final aim is to obtain better and more general closure equations for heat, momentum and mass transfer that can be applied in phenomenological models and account for the presence of and permeation of gas through membranes in membrane assisted fluidized bed reactors instead of the previous described (empirical) closure equations obtained for reactors without membranes. 

Countercurrent and Cocurrent Plug Flows, The model equations for these flow patterns cannot be solved analytically. Oishi and coworkers first derived the general model equations for a binary-component system with porous media. Walawender and Stem, Blaisdell and Karnmermeyer, and Pan and Habgood later reported solutions for similar membrane separators. The cocutrent-countercurrem combination flow pattern also has been studied by Pan and Habgo. ... 

The general membrane stresses in the nozzle are calculated using the basic equation... 

This is a general-purpose equation applicable to both RO and UF which can be used for design (membrane area lurks in J) and predictive purpose. It is also a nonlinear equation in which must be solved for numerically. We can use this equation to estimate water flux drawing on values given in Table 8.8 or those for the Peclet number listed in Table 8.9. Let us choose an average value for Pe = 0.4, a water permeability of 3 x 10" mol/m s Pa (see Table 8.4), and an applied hydrostatic pressure of AP = 1000 psig = 6.8 X 10 Pa. The particle molarity C n for seawater is approximately 10 mol/m . We obtain from Equation 8.13f,... 

This equation shows that the separation achieved in pervaporation is proportional to the product of the separation achieved by evaporation of the Hquid and the separation achieved by permeation of the components through a membrane. To achieve good separations both terms should be large. It follows that, in general, pervaporation is most suited to the removal of volatile components from relatively involatile components, because will then be large. However, if the membrane is sufficientiy selective and P g is large, nonvolatile components can be made to permeate the membrane preferentially (88). 

In membrane separations, the product S Dj is referred to as the per-meabihty, p (kmoL/m s Pa). The rate of passage of material through a membrane is referred to as flux, with symbol J . Jj is equal to Ni in the equations given above. Generally Jj has the dimensions of velocity, m/s (more conveniently, Im/s), or conventionally as /m hr, gal/ft day, or ftVft day. For most apphcations, throughput is expressed in volumes instead of moles or mass. 

Basic Principles of Operation RO and NF are pressure-driven processes where the solvent is forced through the membrane by pressure, and the undesired coproducts frequently pass through the membrane by diffusion. The major processes are rate processes, and the relative rates of solvent and sohite passage determine the quality of the product. The general consensus is that the solution-diffusion mechanism describes the fundamental mechanism of RO membranes, but a minority disagrees. Fortunately, the equations presented below describe the obseiwed phenomena and predict experimental outcomes regardless of mechanism. 

For the usual case when R = (total retention of the solute), L petm = 0 3.nd combining these equations gives a general expression for flux in a turbulent-flow membrane system. For any given solute concentration ... 

The simplifying assumptions that make Tick s law useful for other processes are not vaHd for pei vaporation. The activity gradient across the membrane is far more important than the pressure gradient. Equation (22-110) is generally used to describe the pei vaporation process ... 

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