Forced diffusion model

Kramers H A 1940 Brownian motion in a field of force and the diffusion model of chemical reactions Physica 7 284-304... 

Fig. 9. Uptake curves for N2 in two samples of carbon molecular sieve showing conformity with diffusion model (eq. 24) for sample 1 (A), and with surface resistance model (eq. 26) for example 2 (0)j LDF = linear driving force. Data from ref. 18.

Fig. 15. Theoretical breakthrough curves for a nonlinear (Langmuir) system showing the comparison between the linear driving force (—), pore diffusion (--------------------), and intracrystalline diffusion (-) models based on the Glueckauf approximation (eqs. 40—45). From Ref. 7.

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... 

Solution—Diffusion Model. In the solution—diffusion model, it is assumed that (/) the RO membrane has a homogeneous, nonporous surface layer (2) both the solute and solvent dissolve in this layer and then each diffuses across it (J) solute and solvent diffusion is uncoupled and each is the result of the particular material s chemical potential gradient across the membrane and (4) the gradients are the result of concentration and pressure differences across the membrane (26,30). The driving force for water transport is primarily a result of the net transmembrane pressure difference and can be represented by equation 5 ... 

This gives a model for Eq. (4,305b), but not a model for force While force gives the flow force caused by the material, it is normal to represent this fact so that gives the pure diffusion resistance force that is not caused by the material. This requires treating independently from the material or porosity. For (f) = 1 or E = °°, where 0 EQ- (4.306) gives... 

The important fact is that the number of collisions Zr increases with temperature. It may be attributed to the effect of attraction forces. They accelerate the molecule motion along the classical trajectories favouring more effective R-T relaxation. This effect becomes relatively weaker with increase of temperature. As a result the effective cross-section decreases monotonically [199], as was predicted for the quantum J-diffusion model in [186] (solid line) but by classical trajectory calculations (dotted and broken lines) as well. At temperatures above 300 K both theoretical approaches are in satisfactory mutual agreement whereas some other approaches used in [224, 225] as well as SCS with attraction forces neglected [191] were shown to have the opposite temperature dependence for Zr [191]. Thus SCS results with a... 

The ideas of Overton are reflected in the classical solubility-diffusion model for transmembrane transport. In this model [125,126], the cell membrane and other membranes within the cell are considered as homogeneous phases with sharp boundaries. Transport phenomena are described by Fick s first law of diffusion, or, in the case of ion transport and a finite membrane potential, by the Nernst-Planck equation (see Chapter 3 of this volume). The driving force of the flux is the gradient of the (electro)chemical potential across the membrane. In the absence of electric fields, the chemical potential gradient is reduced to a concentration gradient. Since the membrane is assumed to be homogeneous, the... 

As we have seen, the electric state of a surface depends on the spatial distribution of free (electronic or ionic) charges in its neighborhood. The distribution is usually idealized as an electric double layer one layer is envisaged as a fixed charge or surface charge attached to the particle or solid surface while the other is distributed more or less diffusively in the liquid in contact (Gouy-Chapman diffuse model, Fig. 3.2). A balance between electrostatic and thermal forces is attained. 

The surface diffusion model (Eq. 60) is usually approximated by the linear-driving-force relation [124] ... 

The diffusion model and the hydraulic permeation model differ decisively in their predictions of water content profiles and critical current densities. The origin of this discrepancy is the difference in the functions D (T) and /Cp (T). This point was illustrated in Eikerling et al., where both flux terms occurring in Equation (6.46) were converted into flux terms with gradients in water content (i.e., VA) as the driving force and effective transport coefficients for diffusion, A), and hydraulic permeation,... 

In this section the solution-diffusion model is used to describe transport in dialysis, reverse osmosis, gas permeation and pervaporation membranes. The resulting equations, linking the driving forces of pressure and concentration with flow, are then shown to be consistent with experimental observations. 

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. 

First, with no exception, in all classical diffusion models one or more adjustable parameters enter in the formula of D. To calculate the magnitude of this/these param-eter/s a number of diffusion experiments must be performed with the very penetrant polymer system which one intends to simulate theoretically. In practice such experiments most often require quite sophisticated equipment to obtain the experimental data, and often non-trivial theoretical schemes to evaluate them. The attempt to save experimental work by using the adjustable parameters determined for a certain penetrant polymer system in order to estimate/predict Ds in a related system is generally not recomendable. Hence, in a first step, in order to use one or other of the classical diffusion models, one is forced to replace migration experiments with diffusion ones. 

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