Model homogeneous diffusion

The homogeneous diffusion model is slightly more complex in cyUndrical coordinates relative to the model described above in rectangular coordinates. Additional complexity arises because the radial term of the Laplacian operator (V V = V ) accounts for the fact that the surface area across which radial diffusion occurs increases linearly with dimensionless coordinate r/ as one moves radially outward. Basic information for = f(t]) is obtained by integrating the dimensionless mass balance with radial diffusion and chemical reaction ... 

Bilicki, Z. Kestin, J. "Physical Aspects of the Relaxation Model in Two-Phase Flow," submitted to Proceedings of the Royal Society, London. Bilicki, Z., Kestin, J. 1983. "Two-phase flow in a vertical pipe and the phenomenon of choking homogeneous diffusion model-I. 

The overall observed kinetics of exchange is of course the result of all the above-described mechanisms working in concert [45,89]. To cope with the complexities of the system, a simple approach one may adopt is the homogeneous diffusion model, which assumes that the behaviour of each distinct diffusing species within the macroparticle may be described in terms of a single solid-phase effective diffusivity [89]. More sophisticated approaches include the heterogeneous diffusion models, where the macropore and micropore diffusion processes are described separately and are then assiuned in different mathematical treatments either to occur in series or in parallel [45,89]. 

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 ... 

Let us examine three examples of how these times are used in model selection. If << and t << t, there is rapid chemical change before any movement occurs. It t >> t and << t, there is little chemical change and diffusion spreads the pollutant rapidly so that the mixture is homogeneous. If t t, all processes act simultaneously. Taking these cases in order, we see that the first case is trivial requiring no model (except possibly a reacting plume in the near field). The second case is approximated by a nonreactive box model and the third, by a full reactive diffusion model. 

Fisher, J.F. and Cho, A.K., Chemical release of dopamine from striatal homogenates evidence for an exchange diffusion model, J. Pharmacol. Exp. Ther., 208, 203, 1979. 

The diffusion model assumes that (1) Fick s law is valid as modified by reactions (2) reactant concentrations are interpreted as probability densities (3) specific rates correspond to otherwise homogeneous reactions—Monchick et al. (1957) show that this implies neglect of interparticle correlation, which is... 

Great simplification is achieved by introducing the hypothesis of independent reaction times (IRT) that the pairwise reaction times evolve independendy of any other reactions. While the fundamental justification of IRT may not be immediately obvious, one notices its similarity with the molecular pair model of homogeneous diffusion-mediated reactions (Noyes, 1961 Green, 1984). The usefulness of the IRT model depends on the availability of a suitable reaction probability function W(r, a t). For a pair of neutral particles undergoing fully diffusion-con-trolled reactions, Wis given by (a/r) erfc[(r - a)/2(D t)1/2] where If is the mutual diffusion coefficient and erfc is the complement of the error function. 

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... 

The diffusive models treat the membrane system as a single phase. They correspond more-or-less to the vapor-equilibrated membrane (panel c of Figure 6). Because the collapsed channels fluctuate and there are no true pores, it is easiest to treat the system as a single, homogeneous phase in which water and protons dissolve and move by diffusion. Many membrane models, including some of the earliest ones, treat the system in such a manner. 

This difference is largely accounted for by the difference in thickness of the homogeneous membranes involved, such that the product of water permeation constant and membrane thickness is about the same for both membranes. The constant arises from the diffusion model of permeation in which ... 

Membrane deterioration may be merely caused by decrease of acetyl content(C ) in the active surface layer as a result of hydrolysis or oxidation, not by structure change. Analysis was carried out based on solution-diffusion model proposed by Lonsdale etal( ), using their measured values of solute diffusivity and partition coefficient in homogeneous membrnaes of various degree of acetyl content and also using those values of asymmetric membranes heat treated at various temperatures measured by Glueckauf(x) ... 

Jaberi, F.A., and P. Givi. 1995. Inter-layer diffusion model of scalar mixing in homogeneous turbulence. Combustion Science Technology 104(4-6) 249-72. 

Meshko et al. (2001) used a homogeneous solid model taking into account both internal and external diffusion. They found that the adsorption of the dye had not been significantly affected by the agitation speed, which indicated that the process was solid diffusion-controlled. Furthermore, for the specified conditions, they found that kf = 6.66 X 10 s m/s and /), =10 12 m2/s. 

Choy and McKay used a homogeneous surface diffusion model (HSDM) taking into account both external and internal transport, and found that the mean value of the solid diffusion coefficient is 3.72 X 10-9 cm2/s while kf = 6.06 X 10- 4 cm/s. 

Application of the dual mode sorption and diffusion models to homogeneous polymer blend-gas systems 26,65) and filled polymers 66) has also been reported. 

Smith, E.H. 1991. Modified solution of homogeneous surface diffusion model for adsorption. J Environ. Eng-ASCE 117(3) 320-338. 

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. 

Historically most of the microscopic diffusion models were formulated for amorphous polymer structures and are based on concepts derived from diffusion in simple liquids. The amorphous polymers can often be regarded with good approximation as homogeneous and isotropic structures. The crystalline regions of the polymers are considered as impenetrable obstacles in the path of the diffusion process and sources of heterogeneous properties for the penetrant polymer system. The effect of crystallites on the mechanism of substance transport and diffusion in a semicrystalline polymer has often been analysed from the point of view of barrier property enhancement in polymer films (35,36). 

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