Transport, phenomenological models

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

The broadest class of models, phenomenological models, account explicitly for individual phenomena such as swelling, diffusion, and degradation by incorporation of the requisite transport, continuity, and reaction equations. This class of models is useful only if it can be accurately parameterized. As phenomena are added to the model, the number of parameters increases, hopefully improving the model s accuracy, but also requiring additional experiments to determine the additional parameters. These models are also typically characterized by implicit mean-field approximations in most cases, and model equations are usually formulated such that explicit solutions may be obtained. Examples from the literature are briefly outlined below. 

Analysis of mass transfer in ternary media, until now, has mainly involved experimental studies of model and real food. Phenomenological models could be applied to obtain a more detailed description of the mechanisms involved. However, this would require an understanding of factors such as mass transport properties and transfer dynamics of different active compounds in concentrated solutions, which have yet to be characterized. 

However, a distinction should be made in that Eq. (12) is purely phenomenological and does not require any transport mechanism model while the Nermst-Planck equation used in the previous finely-porous membrane model requires a specific pore model. Another difference is that the salt concentration in Eq. (12) is that in the membrane while the quantity appearing in the Nernst-Planck equation refers to the salt concentration in the membrane pores. 

Chapter 8 addresses the treatment of contaminated air streams using photocatalysis. Special attention is given to the distinction between reaction kinetics and mass transport processes. The reviewed studies show the evolution from the early days of Ti02 photocatalysis, where the aim was to understand the basic process parameters, to today s development of phenomenological models assisting in the scaling-up of units. 

Cation, anion, and water transport in ion-exchange membranes have been described by several phenomenological solution-diffusion models and electrokinetic pore-flow theories. Phenomenological models based on irreversible thermodynamics have been applied to cation-exchange membranes, including DuPont s Nafion perfluorosulfonic acid membranes [147, 148]. These models view the membrane as a black box and membrane properties such as ionic fluxes, water transport, and electric potential are related to one another without specifying the membrane structure and molecular-level mechanism for ion and solvent permeation. For a four-component system (one mobile cation, one mobile anion, water, and membrane fixed-charge sites), there are three independent flux equations (for cations, anions, and solvent species) of the form... 

A substantial number and variety of models of gas transport in polymers have been proposed during the last 20-30 years, in view of the great practical and scientific importance of this process. Molecular-type models are potentially most useful, since they relate diffusion coefficients to fundamental physicochemical properties of the polymers and penetrant molecules, in conjunction with the pertinent molecular interactions. However, the molecular models proposed up to now are overly simplified and contain one or more adjustable parameters. Phenomenological models, such as the dual-mode sorption model and some free-volume models, are very useful for the correlation and comparison of experimental data. 

It is a vector-matrix equation in s space and an integral equation in time. The unknown is the vector of functions This equation is called continuous time random walk (CTRW) and was used in phenomenological modeling of transport [17]. Equation 13.1 is closed and can be solved provided that A. /x) is known. Our contribution is to show how detailed microscopic dynamics is used to compute or its moments (see below). 

A phenomenological model of epithelial transport in the conductive form Eqn. 20... 

Thus, this interspace model has been cast in the form of the general phenomenological model of Section 2b (Eqns. 33 and 35), and the relevant parameters of coupled water transport may be calculated as outlined there. 

Linear diffusion satisfactorily describes the transport mechanism for a single population. For interacting populations, linear diffusion terms imply that the populations are able to mix completely, with the movement of one cell type unaffected by the presence of cells of the other type. The reality is exactly the opposite. Cell movement is typically halted by contact with another cell. This phenomenon is known as contact inhibition and is very well documented for many types of cells. Sherratt introduced a phenomenological model to account for contact inhibition [402]. Consider the interaction between normal and tumor cells with concentrations pm(xj) and Pt(x, t), respectively. The overall cell flux of both populations is given by x(Pn + Pt)- a fraction Pn/(Pn + Pt) of this flux corresponds to normal cells, so that the flux of normal cells is - [pn/(Pn + Pr)] x(Pn + Pr)> a similar expression for the flux of tumor cells. These expressions indicate that the movement of one population is inhibited by the presence of the other. The system of dimensionless reaction-diffusion equations reads [402]... 

G. J. M. Janssen, A Phenomenological Model of Water Transport in a Proton Exchange Membrane Fuel Cell, Journal of the Electrochemical Society, 148, A1313 (2001). 

The nature of the bottlenecks for proton conductance in the dry membrane state or on the way to it is, however, still the subject of debates. This wiU only be resolved after more detailed experimental studies (of macroscopic transport parameters such as proton conductance and electro-osmotic coefficients as a function of water content, or gas and liquid permeability before and after operation, and of microscopic structural probes such as small-angle neutron and X-ray scattering) will have discriminated between competing models. By and large, the direction of effects that go with dehydration is obvious enough to be introduced into phenomenological models of overall cell performance. 

Janssen GJM (2001) A phenomenological model of water transport in a proton-exchange-membrane fuel cell. J Electrochem Soc 148 A1313-A1323... 

In Section 3.11.2 we discussed the phenomenological model of electron transport in the conduction band of tetramethylsilane modulated by trapping by biphenyl. Such a model can be generalized for the electron transport in low mobility hydrocarbons, as, for instance, n-hexane. Localized electrons have been detected by their optical absorption. The traps in hydrocarbons are assumed to be structural voids which upon occupation by an electron increase further in depth. The electrons are only partially localized in these traps (Schiller et al., 1973). By thermal activation they... 

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