Phenomenological modeling variability

Because of all these minor components (e.g., catalysts and inhibitors, added to major ones) the cure of vinyl ester resins is very complex, involving many competitive reactions. There are some new variables to account for, such as the inhibitor and initiator concentrations and induction time. Several papers [81,96,200,201] use the mechanistic approach, claiming that the phenomenological models do not explicitly include these facts, resulting in a new parameter characterization after each change in resin formulation [96]. Despite these arguments, the phenomenological approach is the most widely used and is based on an autocatalytic model which has been successfully applied to epoxy resins. Many authors [30,34,74,199,202,203] proposed the Equation 2.30 to describe the cure kinetic of unsaturated polyesters ... 

The lattice model of mass transfer gives self-consistent expressions for sorption isotherms and permeability coefficients for microheterogeneous membranes of variable thickness at an arbitrary degree of filling. The parameters of the lattice model can be related to the molecular structure of a matrix and to the parameters of the interaction between diffusant and matrix [191]. The lattice model can serve as a basis to construct phenomenological models, which are capable of describing the features of a molecular system diffusant-membrane matrix . 

Phenomenological Model. The data reduction scheme developed for use with FTMA is based on a semi-empirical phenomenological model for polymeric materials with postulates corresponding to generally observed behavior. The constraints of current constitutive theory are satisfied and the model relates mechanical properties to both frequency and temperature with parameters that are material-dependent. It provides excellent interpolations of experimental results and also extrapolates to reasonable levels outside the ranges of the experimental variables. 

We may subsume all of the complexity of the full electromagnetic wave description of the Gaussian beam and its coupling to various elements of the resonator into two phenomenological constants the mutual inductances M, and M2 of Fig. 6. This procedure is equivalent to that used to model variable iris coupling into a waveguide cavity, for example. 

Phenomenological Modeling. Filtration Theory. Modeling of fines or particulate migration was first considered in deep bed filtration problems. The introduction of a more convenient time variable... 

The cavity model of solvation provides the basis for a number of additional models used to explain retention in reversed-phase chromatography. The main approaches are represented by solvophobic theory [282-286] and lattice theories based on statistical thermodynamics [287-291]. To a lesser extent classical thermodynamics combining partition and displacement models [292] and the phenomenological model of solvent effects [293] have also been used. Compared with the solvation parameter model all these models are mathematically complex, and often require the input of system variables that are either unknown or difficult to calculate, particularly for polar compounds. For this reason, and because of a failure to provide a simple conceptual picture of the retention process in familiar chromatographic terms, these models have largely remained the province of the physical chemist. 

The relationship between the molecular structure and the end-use properties of most polymer materials is poorly understood and relies heavily on empirical observation and testing [1,2]. This means that control of end-use properties caimot benefit completely from the fast development of phenomenological models of polymerization reactors, which provide detailed information about how operation variables affect the molecular structure of produced polymer materials. 

It seems clear that the fast development of computer resources and process instrumentation will make the implementation of on-line closed-loop control of polymerization processes much more frequent in the upcoming years. It also seems clear that the development of robust and sound phenomenological models will play a very important role in this scenario, as process models allow for estimation of several molecular properties that are difficult to measure otherwise. Besides, process models provide adequate decoupling of the very complex and non-linear relationships among the many process variables. 

The review of the most recent publications in the field indicates very clearly that the number of commimications of successful control implementations in laboratory and small-scale reaction processes has grown exponentially in the last few years. This certainly encourages the implementation of similar closed-loop strategies at plant site. However, measurement of end-use polymer properties in line and in real time remains an unsolved challenge in the field, especially because sound phenomenological models have yet to be developed for these variables. 

The present and the next sections address phenomenological models accounting for the effects of micro-mixing on the macro-scale averaged reaction rates. These models do not start from the one-point joint velocity-composition micro-PDF, f v, yf), and do not require micro-PDF moments other than first-order for the species concentration variables. They are in general less accurate, but computationally less demanding than micro-PDF or its moments based methods. 

Unified Plasticity Model The time-independent plastic deformation and fee time-dependent creep deformation arise from fee same fundamental mechanism of dislocation motion. Hence, a constitutive model which captures both of these deformation mechanisms is desirable. Such a constitutive model is referred to as a unified plasticity model. A commonly-used unified plasticity model is the Anand s model. This is a rate-dependent phenomenological model (Ref 17 and 18). There are two basic characteristics of fee Anand s model. First, no explicit yield criterion is specified, and second, a single internal state variable (ISV) s, the deformation resistance, represents the isotropic resistance to inelastic strain hardening. Anand s model can represent fee strain rate and temperature sensitivity, strain rate history effects, strain hardening, and fee restoration process of dynamic recovery. Equation 9 shows the functional form of fee flow equation that accommodates fee strain rate dependence on the stress ... 

Abstract In this paper, a new phenomenological model is developed to account variable coefficient of friction (COF) in space and time. The COF is no longer considered as a global value valid for the whole contact area. A local value is introduced instead, which evolution is governed by the local history of the contact and the amount of slip. The framework is inspired from elastoplasticity. The evolution of the COF depends on two variables an isotropic evolution related to cumulated slip and a kinematic component computed from the actual relative position of the bodies. 

If it cannot be guaranteed that the adsorbate remains in local equilibrium during its time evolution, then a set of macroscopic variables is not sufficient and an approach based on nonequihbrium statistical mechanics involving time-dependent distribution functions must be invoked. The kinetic lattice gas model is an example of such a theory [56]. It is derived from a Markovian master equation, but is not totally microscopic in that it is based on a phenomenological Hamiltonian. We demonstrate this approach... 

---