Solution-diffusion approach

Because of the phase change associated with the process and the non-ideal liquid-phase solutions (i.e., organic/water), the modeling of pervaporation cannot be accomplished using a solution-diffusion approach. Wijmans and Baker [14] express the driving force for permeation in terms of a vapor partial pressure difference. Because pressures on the both sides of the membrane are low, the gas phase follows the ideal gas law. The liquid on the feed side of the membrane is generally non-ideal. 

There seems to be a historical reason for the difference in these two approaches. The solution-diffusion approach was established for the permeation of liquid and gas through the membrane before reverse osmosis membranes of practical usefulness were developed by the phase-inversion technique. Dense membranes without asymmetricity were prepared from polymeric materials, and their transport properties were measured, assuming that the membranes were defect-free and the transport parameters so produced were the values intrinsic to the material. The membrane with the highest separation capacity for a given polymer was believed to be that which could exhibit the transport properties intrinsic to the polymer. The goal of membrane production engineering was to ensure the membrane intrinsic transport properties of the polymeric material. This approach is still popular in the membrane manufacturing industry. 

Although transport processes in aqueous NF systems have been studied for several years and much knowledge has been gained, OSN systems are not yet well understood. While some studies support the use of pore-flow models, others suggest using a solution-diffusion approach. The basic equations of these models are outhned below along with the major simplifying assumptions. 

The water flux from cathode to anode is driven by either diffusion or convection. In the case of solution diffusion approach, the diffusional flux of water according to Pick s law is... 

Low-PressureAlulticomponent Mixtures These methods are outlined in Table 5-17. Stefan-MaxweU equations were discussed earlier. Smith-Taylor compared various methods for predicting multi-component diffusion rates and found that Eq. (5-204) was superior among the effective diffusivity approaches, though none is very good. They so found that hnearized and exact solutions are roughly equivalent and accurate. 

To cross the filter, the solute must diffuse through the filter pores. Therefore, pore size, density, and tortuosity must be taken into consideration. Many filter configurations are commercially available and have been employed for these types of studies. In general, the greater the pore size and porosity, the less the potential for the filter to act as a significant diffusion barrier. For small solutes, the filter will probably not present much of a problem. However, as the molecular size of the solute increases and approaches the dimension of the pore, these considerations become more important. In principle, solute diffusion through the... 

In an early attempt, Mozumder (1968) used a prescribed diffusion approach to obtain the e-ion geminate recombination kinetics in the pure solvent. At any time t, the electron distribution function was assumed to be a gaussian corresponding to free diffusion, weighted by another function of t only. The latter function was found by substituting the entire distribution function in the Smoluchowski equation, for which an analytical solution was possible. The result may be expressed by... 

Even when using LDPE and PP, the diffusive losses of most nonpolar solvents may be unacceptably large for A designs that exceeded 1 cm . This is especially true at higher exposure temperatures because both solute diffusion and polymer free volume increase with temperature (Comyn, 1985). Even when solvent losses were not excessive, uptake of HOCs by membrane-enclosed solvents appeared to become curvilinear well before thermodynamic equilibrium was approached. This phenomenon is likely due to the outward flux of sampler solvent with elevated HOC levels (relative to water), which appears to facilitate residue... 

In order to compare data obtained with otherwise similar chromatographic systems in which only the particle size of the column packing and solute diffusivity may vary, Eq. (21) should be written in dimensionless form. Using an approach taken from chemical engineering, Knox (7, 34) has shown that a corresponding reduced plate height equation is given by Eq. (22)... 

If the diffusion process is coupled with other influences (chemical reactions, adsorption at an interface, convection in solution, etc.), additional concentration dependences will be added to the right side of Equation 2.11, often making it analytically insoluble. In such cases it is profitable to retreat to the finite difference representation and model the experiment on a digital computer. Modeling of this type, when done properly, is not unlike carrying out the experiment itself (provided that the discretization error is equal to or smaller than the accessible experimental error). The method is known as digital simulation, and the result obtained is the finite difference solution. This approach is described in more detail in Chapter 20. 

The general approach is to use the first assumption of the solution-diffusion model, namely, that the chemical potential of the feed and permeate fluids are... 

Single-stage simulations reveal that intermolecular friction forces do not lead to reverse diffusion effects, and thus the molar fluxes calculated with the effective diffusion approach differ only slightly from those obtained via the Maxwell-Stefan equations without the consideration of generalized driving forces. This result is as expected for dilute solutions and allows one to reduce model complexity for the process studied (143). 

Similar to the approach for solvents, both diffusive and convective transport of solutes can be modeled separately. For dense membranes, a solution-diffusion model can be used [14], where the flux / of a solute is calculated as ... 

Good quality RO membranes can reject >95-99% of the NaCl from aqueous feed streams (Baker, Cussler, Eykamp et al., 1991 Scott, 1981). The morphologies of these membranes are typically asymmetric with a thin highly selective polymer layer on top of an open support structure. Two rather different approaches have been used to describe the transport processes in such membranes the solution-diffusion (Merten, 1966) and surface force capillary flow model (Matsuura and Sourirajan, 1981). In the solution-diffusion model, the solute moves within the essentially homogeneously solvent swollen polymer matrix. The solute has a mobility that is dependent upon the free volume of the solvent, solute, and polymer. In the capillary pore diffusion model, it is assumed that separation occurs due to surface and fluid transport phenomena within an actual nanopore. The pore surface is seen as promoting preferential sorption of the solvent and repulsion of the solutes. The model envisions a more or less pure solvent layer on the pore walls that is forced through the membrane capillary pores under pressure. 

Electron spin resonance (ESR) studies of radical probe species also suggest complexity. Evans et al. [250] study the temperature dependence of IL viscosity and the diffusion of probe molecules in a series of dissimilar IL solvents. The results indicate that, at least over the temperature range studied, the activation energy for viscous flow of the liquid correlates well with the activation energies for both translational and rotational diffusion, indicative of Stoke-Einstein and Debye-Stokes-Einstein diffusion, respectively. Where exceptions to these trends are noted, they appear to be associated with structural inhomogeneity in the solvent. However, Strehmel and co-workers [251] take a different approach, and use ESR to study the behavior of spin probes in a homologous series of ILs. In these studies, comparisons of viscosity and probe dynamics across different (but structurally similar) ILs do not lead to a Stokes-Einstein correlation between viscosity and solute diffusion. Since the capacities for specific interactions are... 

In the case of selective neutrality—this means that all variants have the same selective values—evolution can be modeled successfully by diffusion models. This approach is based on the analysis of partial differential equations that describe free diffusion in a continuous model of the sequence space. The results obtained thereby and their consequences for molecular evolution were recently reviewed by Kimura [2]. Differences in selective values were found to be prohibitive, at least until now, for an exact solution of the diffusion approach. Needless to say, no exact results are available for value landscapes as complicated as those discussed in Section IV.3. Approximations are available for special cases only. In particular, the assumption of rare mutations has to be made almost in every case, and this contradicts the strategy basic to the quasi-species model. 

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