Porous diffusion models

Though a porous medium may be described adequately under non-reactive conditions by a smooth field type of diffusion model, such as one of the Feng and Stewart models, it does not necessarily follow that this will still be the case when a chemical reaction is catalysed at the solid surface. In these circumstances the smooth field assumption may not lead to appropriate expressions for concentration gradients, particularly in the smaller pores. Though the reason for this is quite simple, it appears to have been largely overlooked,... 

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... 

A fundamental difference exists between the assumptions of the homogeneous and porous membrane models. For the homogeneous models, it is assumed that the membrane is nonporous, that is, transport takes place between the interstitial spaces of the polymer chains or polymer nodules, usually by diffusion. For the porous models, it is assumed that transport takes place through pores that mn the length of the membrane barrier layer. As a result, transport can occur by both diffusion and convection through the pores. Whereas both conceptual models have had some success in predicting RO separations, the question of whether an RO membrane is truly homogeneous, ie, has no pores, or is porous, is still a point of debate. No available technique can definitively answer this question. Two models, one nonporous and diffusion-based, the other pore-based, are discussed herein. 

Diffusion within the largest cavities of a porous medium is assumed to be similar to ordinary or bulk diffusion except that it is hindered by the pore walls (see Eq. 5-236). The tortuosity T that expresses this hindrance has been estimated from geometric arguments. Unfortunately, measured values are often an order of magnitude greater than those estimates. Thus, the effective diffusivity D f (and hence t) is normally determined by comparing a diffusion model to experimental measurements. The normal range of tortuosities for sihca gel, alumina, and other porous solids is 2 < T < 6, but for activated carbon, 5 < T < 65. 

Figure 4 Essential features of a simple diffusion model consisting of a monolayer of cells grown on a porous support which separates fluid-filled donor and receiver compartments.

Feng, C., V. V. Kostrov and W. E. Stewart. 1974. Multicomponent diffusion of gases in porous solids. Models and experiments. Ind. Eng. Chem. Fundam. 13(1) 5-9. 

The next set of models treats the catalyst layers using the complete simple porous-electrode modeling approach described above. Thus, the catalyst layers have a finite thickness, and all of the variables are determined as per Table 1 with a length scale of the catalyst layer. While some of these models assume that the gas-phase reactant concentration is uniform in the catalyst layers,most allow for diffusion to occur in the gas phase. 

The final simple macrohomogeneous porous-electrode models are the ones that are more akin to thin-film models. In these models, the same approach is taken, but instead of gas diffusion in the catalyst layer, the reactant gas dissolves in the electrolyte and moves by diffusion and reaction. The... 

Tronconi et al. [46] developed a fully transient two-phase 1D + 1D mathematical model of an SCR honeycomb monolith reactor, where the intrinsic kinetics determined over the powdered SCR catalyst were incorporated, and which also accounts for intra-porous diffusion within the catalyst substrate. Accordingly, the model is able to simulate both coated and bulk extruded catalysts. The model was validated successfully against laboratory data obtained over SCR monolith catalyst samples during transients associated with start-up (ammonia injection), shut-down (ammonia... 

We have presented a general reaction-diffusion model for porous catalyst particles in stirred semibatch reactors applied to three-phase processes. The model was solved numerically for small and large catalyst particles to elucidate the role of internal and external mass transfer limitations. The case studies (citral and sugar hydrogenation) revealed that both internal and external resistances can considerably affect the rate and selectivity of the process. In order to obtain the best possible performance of industrial reactors, it is necessary to use this kind of simulation approach, which helps to optimize the process parameters, such as temperature, hydrogen pressure, catalyst particle size and the stirring conditions. 

Due to the fact that protein adsorption in fluidized beds is accomplished by binding of macromolecules to the internal surface of porous particles, the primary mass transport limitations found in packed beds of porous matrices remain valid. Protein transport takes place from the bulk fluid to the outer adsorbent surface commonly described by a film diffusion model, and within the pores to the internal surface known as pore diffusion. The diffusion coefficient D of proteins may be estimated by the semi-empirical correlation of Poison [65] from the absolute temperature T, the solution viscosity rj, and the molecular weight of the protein MA as denoted in Eq. (16). 

In laminar flows through porous media, the pressure is proportional to velocity and C2 can be taken as zero. Ignoring convective acceleration and diffusion, the porous media model can be changed into Darcy s Law ... 

The tortuosity term is intended to account for increases in diffusional path length due to windiness. Classical descriptions of the tortuosity predict a value of 1 to 3 for random porous media (.2.). Since the tortuosities inferred by these models are orders of magnitude greater than expected, other physical properties of the system must be important in determining release rates. Since continuum diffusion models provide an incomplete description of the release from these devices, the microscopic details of the system must be considered explicitly. 

Experiments have shown that the solution - diffusion imperfection model fits data better than the solution - diffusion model alone and better than all other porous flow models.6 However, the solution-diffusion model is most often cited due to its simplicity and the fact that it accurately models the performance of the perfect RO membrane. 

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