Solution-diffusion theory

Mathematical treatments (34) of various models suggest that pore defects combined with predictions from solution-diffusion theory yield the best agreement between theory and experiment. 

Transport in dense discriminating layers is most commonly described using the well developed solution-diffusion theory [36]. The theory is based on the assumptions that 1) the driving force for transport is a gradient in chemical potential, 2) at a fluid-membrane interface the chemical potential in the two phases are equal (i.e., equilibrium exists), and 3) the pressure within the membrane is constant and equal to the highest value at either interface. Baker [37] summarizes the application of the theory to a variety of membrane separation processes ranging from dialysis to gas separation. 

For gas separations, solution-diffusion theory leads to the conclusion that gas permeation flux (J) is proportional to the difference in gas partial pressure across the membrane (Ap) J=(P//)Ap. The proportionality constant is equal to the intrinsic permeability (P) for the membrane material divided by the effective membrane thickness (/). In turn, the permeability is equal to the product of a solubility (S) and diffusivity (D) P=D S. The ability to separate two... 

D. R. Paul, Reformulation of the solution-diffusion theory of reverse osmosis, J. Memhr. Sci. 241 (2004) 371-386. 

In the FO processes, the ICP can be modeled based on the solution-diffusion theory for the rejection layer coupled with the convection and diffusion of solute in the porous support layer. The modeling equations in AL-DS and AL-FS orientations can be expressed by Equations (14.1) and (14.2) (Tang etal, 2010) ... 

The solution-diffusion theory models the performance of the perfect membrane. In reality, industrial membranes are plagued with imperfections that some argue must be considered when developing a complete theory that models performance. The basis of the Diffusion Imperfection Model is the assumption that slight imperfections in the membrane occur... 

In general, the foUowing steps can occur in an overall Hquid—soHd extraction process solvent transfer from the bulk of the solution to the surface of the soHd penetration or diffusion of the solvent into the pores of the soHd dissolution of the solvent into the solute solute diffusion to the surface of the particle and solute transfer to the bulk of the solution. The various fundamental mechanisms and processes involved in these steps make it impracticable or impossible to describe leaching by any rigorous theory. 

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... 

Analytical information taken from a chromatogram has almost exclusively involved either retention data (retention times, capacity factors, etc.) for peak identification or peak heights and peak areas for quantitative assessment. The width of the peak has been rarely used for analytical purposes, except occasionally to obtain approximate values for peak areas. Nevertheless, as seen from the Rate Theory, the peak width is inversely proportional to the solute diffusivity which, in turn, is a function of the solute molecular weight. It follows that for high molecular weight materials, particularly those that cannot be volatalized in the ionization source of a mass spectrometer, peak width measurement offers an approximate source of molecular weight data for very intractable solutes. 

As this kind of verification of classical J-diffusion theory is crucial, the remarkable agreement obtained sounds rather convincing. From this point of view any additional experimental treatment of nitrogen is very important. A vast bulk of data was recently obtained by Jameson et al. [270] for pure nitrogen and several buffer solutions. This study repeats the gas measurements of [81] with improved experimental accuracy. Although in [270] Ti was measured, instead of T2 in [81], at 150 amagat and 300 K and at high densities both times coincide within the limits of experimental accuracy. 

Combining hindered diffusion theory with the diffusion/convection problem in the model pore, Trinh et al. [399] showed how the effective transport coefficients depend upon the ratio of the solute to pore size. Figure 28 shows that as the ratio of solute to pore size approaches unity, the effective mobility function becomes very steep, thus indicating that the resolution in the separation will be enhanced for molecules with size close to the size of the pore. Similar results were found for the effective dispersion, and the implications for the separation of various sizes of molecules were discussed by Trinh et al. [399]. 

The development of the theory of solute diffusion in soils was largely due to the work of Nye and his coworkers in the late sixties and early seventies, culminating in their essential reference work (5). They adapted the Fickian diffusion equations to describe diffusion in a heterogeneous porous medium. Pick s law describes the relationship between the flux of a solute (mass per unit surface area per unit time, Ji) and the concentration gradient driving the flux. In vector terms. 

The solubility-diffusion theory assumes that solute partitioning from water into and diffusion through the membrane lipid region resembles that which would occur within a homogeneous bulk solvent. Thus, the permeability coefficient, P, can be expressed as... 

Since the prediction of the solute diffusion coefficient in a swollen matrix is complex and no quantitative theory is yet possible, Lustig and Peppas [74] made use of the scaling concept, arriving at a functional dependence of the solute diffusion coefficient on structural characteristics of the network. The resulting scaling law thus avoids a detailed description of the polymer structure and yet provides a dependence on the parameters involved. The final form of the scaling law for description of the solute diffusion in gels is... 

NA Peppas, CT Reinhart. Solute diffusion in swollen membranes. Part I. A new theory. J Membrane Sci 15 275-287, 1983. 

Although the hybrid theory is the most correct theory to use in the prediction of unattached fractions, the error in using the kinetic-diffusion theory in place of the hybrid is small. The kinetic-diffusion theory has the advantage that the solution is in analytical form and thus is more convenient to use than the hybrid theory, which must be solved numerically. 

Employing experimental supersaturated solution diffusion coefficient data and the cluster di sion theory of Cussler (22), Myerson and Lo (27 attempted to estimate the average cluster size in supersaturated glycine solutions. They estimated an average cluster size on the order of two molecules. Their calculations indicated that while the average cluster size was small, large clusters of hundreds of molecules existed, only there were very few of them. Most of the molecular association was in the form of dimers and trimers. 

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