Phenomenological equations models

The phenomenology of model B, where (j) is conserved, can also be outlined simply. Since (j) is conserved, it obeys a conservation law (continuity equation) ... 

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

The above phenomenological equations are assumed to hold in our system as well (after appropriate averaging). Below we derive formulas for P[Aq B, t), which start from a microscopic model and therefore makes it possible to compare the same quantity with the above phenomenological equa tioii. We also note that the formulas below are, in principle, exact. Therefore tests of the existence of a rate constant and the validity of the above model can be made. We rewrite the state conditional probability with the help of a step function - Hb(X). Hb X) is zero when X is in A and is one when X is ill B. 

The phenomenological equations proposed by Felix Bloch in 19462 have had a profound effect on the development of magnetic resonance, both ESR and NMR, on the ways in which the experiments are described (particularly in NMR), and on the analysis of line widths and saturation behavior. Here we will describe the phenomenological model, derive the Bloch equations and solve them for steady-state conditions. We will also show how the Bloch equations can be extended to treat inter- and intramolecular exchange phenomena and give examples of applications. 

In the literature, there are many transport theories describing both salt and water movement across a reverse osmosis membrane. Many theories require specific models but only a few deal with phenomenological equations. Here a brief summary of various theories will be presented showing the relationships between the salt rejection and the volume flux. 

In order to quantify diffiisional effects on curing reactions, kinetic models are proposed in the literature [7,54,88,95,99,127-133]. Special techniques, such as dielectric permittivity, dielectric loss factor, ionic conductivity, and dipole relaxation time, are employed because spectroscopic techniques (e.g., FT i.r. or n.m.r.) are ineffective because of the insolubility of the reaction mixture at high conversions. A simple model, Equation 2.23, is presented by Chem and Poehlein [3], where a diffiisional factor,//, is introduced in the phenomenological equation, Equation 2.1. 

A simple model is the diode circuit with two metals with known work functions, as in fig. 16 in VI.9. The whole is in thermal equilibrium at temperature T ( Alkemade s diode ) ). The charge Q on the condenser obeys the phenomenological equation... 

The above example gives us an idea of the difficulties in stating a rigorous kinetic model for the free-radical polymerization of formulations containing polyfunctional monomers. An example of efforts to introduce a mechanistic analysis for this kind of reaction, is the case of (meth)acrylate polymerizations, where Bowman and Peppas (1991) coupled free-volume derived expressions for diffusion-controlled kp and kt values to expressions describing the time-dependent evolution of the free volume. Further work expanded this initial analysis to take into account different possible elemental steps of the kinetic scheme (Anseth and Bowman, 1992/93 Kurdikar and Peppas, 1994 Scott and Peppas, 1999). The analysis of these mechanistic models is beyond our scope. Instead, one example of models that capture the main concepts of a rigorous description, but include phenomenological equations to account for the variation of specific rate constants with conversion, will be discussed. 

Certainly, Monod s formula has been used extensively in phenomenological (unstructured) models, although the literature presents other equations for one limiting substrate systems (Equations 17 and 18). In Moser s formulation it was necessary to introduce a third parameter ( n in Equation 17) to represent experimental data. 

In any chemical or electrochemical process, the application of the conservation principles (specifically to the mass, energy or momentum) provides the outline for building phenomenological mathematical models. These procedures could be made over the entire system, or they could be applied to smaller portions of the system, and later integrated from these small portions to the whole system. In the former case, they give an overall description of the process (with few details but simpler from the mathematical viewpoint) while in the later case they result in a more detailed description (more equations, and consequently more features described). 

These symmetry relationships do not depend on the specific features of any given model but follow quite generally from the linear phenomenological equations of nonequilibrium thermodynamics. Therefore, any linear model that does not predict these relations is likely to be incorrect. 

The rate constant (/ ,) was expressed in terms of the results of the computer simulations, for which a non-adiabatic transition-state theory (TST) model was used. Since the experimental results were analyzed in terms of a phenomenological Arrhenius model [158], we relate experiment (left-hand side) and theory (right-hand side) in terms of the following two equations. For the weakly temperature-dependent prefactor we have ... 

Almost all physical models use simple pore geometries. Practical pore systems are, however, very complicated and contain parameters which are difficult to measure or which have a wide distribution of their characteristic parameters. The applicability of a rigorous treatment and of very refined models and physical expressions is therefore doubtful. The treatment in this chapter will make use mainly of phenomenological equations which allow description of data, data reduction and some extrapolation and which rely on experimentally determined parameter values. Gas kinetic theory and expressions based on the microscopic (atomic) level will be used only to estimate some parameter values and to predict trends. 

Future efforts should be directed to develop more advanced hydration theory models than the simple form examined here. Ion pairing should be treated in an explicit manner, even for the 1 1 electrolytes. Other possible changes that might be explored include the use of a more sophisticated electrostatic model and the usage of virial coefficient terms in the phenomenological equations. These models must be tested against not only the ability to fit data, but also the abilities to satisfy additivity relationships and to predict the properties of mixtures of aqueous electrolytes. 

In addition to the approach using phenomenological equations for modelling ion transport in soils, the theory of irreversible thermodynamics may be adapted to soils [26], as for the case of ion-exchange membranes. Spiegler [251 and Kedem and Katchalsky [27,28] are the prime examples of this approach to transport models. The detailed review by Verbrugge and Pintauro contains a number of other references to mathematical approaches for modelling the fundamental electrokinetic phenomena. 

The model should be a model - not a collection of phenomenological equations based on curve fits. 

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