Model, mathematical phenomenological

The Fokker-Planck equation is essentially a diffusion equation in phase space. Sano and Mozumder (SM) s model is phenomenological in the sense that they identify the energy-loss mechanism of the subvibrational electron with that of the quasi-free electron slightly heated by the external field, without delineating the physical cause of either. Here, we will briefly describe the physical aspects of this model. The reader is referred to the original article for mathematical and other details. SM start with the Fokker-Planck equation for the probability density W of the electron in the phase space written as follows ... 

A sublattice phase can be envisaged as being composed of interlocking sublattices (Fig. 5.3) on which the various components can mix. It is usually crystalline in nature but the model can also be extended to consider ionic liquids where mixing on particular ionic sublattices is considered. The model is phenomenological in nature and does not define any crystal structure within its general mathematical formulation. It is possible to define internal parameter relationships which reflect structure with respect to different crystal types, but such conditions must be externally formulated and imposed on the model. Equally special relationships apply if the model is to be used to simulate order-disorder transformations. 

Empirical models are constituted by a mathematical equation that is able to describe the experimental trend under investigation by adopting proper values for its parameters (Grassi, 2007). Such models are phenomenological, because they are not a schematic mathematical representation of the observed phenomena and do not imply (or contain) any theoretical knowledge about the system of interest they are simply constructed from experimental data. The involved parameters have no physical meaning thus. 

Computer simulation of the reactor kinetic hydrodynamic and transport characteristics reduces dependence on phenomenological representations and idealized models and provides visual representations of reactor performance. Modem quantitative representations of laminar and turbulent flows are combined with finite difference algorithms and other advanced mathematical methods to solve coupled nonlinear differential equations. The speed and reduced cost of computation, and the increased cost of laboratory experimentation, make the former increasingly usehil. 

One of the reasons why a comprehensive CFSD code does not exist, is that the mathematical description of the thermochemical conversion of a packed fuel bed is extremely complex. In other words, the mathematical models we have today are far from able to describe the highly differentiated phenomenology of the conversion system. Today there exist comprehensive models describing the thermochemical conversion of single fuel particles, but from that step to the description of the thermochemical conversion of a packed bed is a great leap. Another reason for the slow progress of computer codes in this area is the limited computational capacity, that is, excessive computational time for these complicated models is required. 

The more complex the phenomenology for the conversion system becomes the more advanced and differentiated, the conceptual model, and in turn, the mathematical model must be to correctly describe the actual physical problem. 

Figure 56 above describes the phenomenology of the char combustion regime (III). The concept of the shrinking core or shrinking particle model is usually applied in mathematical modelling of char combustion in regime (III). 

An exact mathematical relationship is obtained between the salt rejection and total volume flux in reverse osmosis based on a purely phenomenological theory assuming constant salt permeability. This approach does not require a specific membrane model ... 

Warnock proposed a phenomenological model that mathematically captures the polishing process but that did not directly seek to incorporate the physical phenomena in CMP [63], The model has three parameters that can be used to fit measured data. The surface is divided into n discrete points each with x, y, and z coordinates. For each point i in the set of n points, the polish rate P,- is defined as... 

As the alternative, a phenomenological description of turbulent mixing gives good results for many situations. An apparent diffusivity is defined so that a diffusion-type equation may be used, and the magnitude of this parameter is then found from experiment. The dispersion models lend themselves to relatively simple mathematical formulations, analogous to the classical methods for heat conduction and diffusion. 

The first difficulty derives from the fact that given any values of the macroscopic expected values (restricted only by broad moment inequality conditions), a probability density always exists (mathematically) giving rise to these expected values. This means that as far as the mathematical framework of dynamics and probability goes, the macroscopic variables could have values violating the laws of phenomenological physics (e.g., the equation of state, Newton s law of heat conduction, Stokes law of viscosity, etc.). In other words, there is a macroscopic dependence of macroscopic variables which reflects nothing in the microscopic model. Clearly, there must exist a principle whereby nature restricts the class of probability density functions, SF, so as to ensure the observed phenomenological dependences. 

Following an idea of Zhabotinsky [4] that chemical oscillating systems could expediently be treated as a black box and that the relevant mathematical semi-phenomenological model has to focus on the basic reactions only neglecting those less important, Vasiliev, Romanovsky and Yakhno [5] suggested a concept of the basic model. These simplified models of an extended active medium could be obtained either by a reduction of pre-existing... 

Section IIA summarizes the physical assumptions and the resulting mathematical descriptions of the "concentration-dependent (5) and "dual-mode" ( 13) sorption and transport models which describe the behavior of "non-ideal" penetrant-polymer systems, systems which exhibit nonlinear, pressure-dependent sorption and transport. In Section IIB we elucidate the mechanism of the "non-ideal" diffusion in glassy polymers by correlating the phenomenological diffusion coefficient of CO2 in PVC with the cooperative main-chain motions of the polymer in the presence of the penetrant. We report carbon-13 relaxation measurements which demonstrate that CO2 alters the cooperative main-chain motions of PVC. These changes correlate with changes in the diffusion coefficient of CO2 in the polymer, thus providing experimental evidence that the diffusion coefficient is concentration dependent. 

This is the wish of most industrials "In general, industry would plead for less sophisticated mathematical models and more phenomenological models giving us more understanding of what s going on in the tank" (10). Among other recommendations (9, 10.>... 

The first section of this chapter is devoted to the presentation of some diffusion models constructed on the basis of phenomenological considerations. These so-called heuristic models are often cited in the literature and used for the interpretation of experimental results. A special emphasis is to discuss how the mathematical formulae of these models can correlate with experimetal data and moreover to predict diffusion coefficients beyond the ranges experimentally investigated. This latter aspect is of great interest not only from a fundamental point of view but also in many practical fields where the possibility to predict a diffusion process might be a more economic alternative to its experimental investigation. 

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