Mathematical Model Formulation Aspects

The alternative pseudo-homogeneous model is a kind of mixture model (see sect 3.4.5) which is achieved by taking the sum of the individual equations for the two phases. 

The initial conditions used for d3mamics reactor simulations depend upon the start-up procedure adopted in industry for each particular chemical process. A possible set of initial conditions corresponds to uniform variable fields given by the inlet values as outlined in sect 1.2.6. The standard Danckwerts [39] boundary conditions are normally applied. 

The well-posedness of the two-fluid model has been a source of controversy reflected by the large number of papers on this issue that can be found in the literature. This issue is linked with analysis of the characteristics, stability and wavelength phenomena in multi -phase flow equation systems. The controversy originates primarily from the fact that with the present level of knowledge, there is no general way to determine whether the 3D multi-fluid model is well posed as an initial-boundary value problem. The mathematical theory of well posedness for systems of partial differential equations describing dispersed chemical reacting flows needs to be examined. 

Introductory work has however been made to analyze the mathematical properties of the equations. In their pioneer work Gidaspow [84] and Ly-ckowski et al [144] consider the ID, incompressible, in-viscid two-fluid flow equations with no added mass or lift effects given by  

If the characteristics of the system of equations are found to be complex, the initial-value problem is said to be ill-posed [178]. A physical interpretation of this mathematical statement can be found by analyzing the flow instabilities predicted by this set of model equations. The instabilities predicted by a well-posed model system has some realistic physical meaning, while the instability always present in an ill-posed system is a mathematical mode having no physical origin indicating that the model is not treating small-scale phenomena correctly. 

It was shown that this approach gives real characteristics only if (p/ A/ is lower than pi)vi-A value of is used by several researchers [145, 173]. However, Drew [58] also discusses this approach and argues that this formulation is questionable, as it may not be consistent with the interfacial drag parameterizations used. 

Sha and Soo [201] dealt with essentially the same problem from a more physical angle. Their discussion focuses on what form the term in the momen- 

In an attempt to close the non-constructive controversy, Boure [25] claims that both forms of the pressure terms are acceptable. The occurrence of complex roots in the momentum equations is not related to the correct use of the pressure terms, but is primarily a problem of mathematical form of the constitutive terms. 

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The mathematical formulation comprises of a number of mass balances and scheduling constraints. Due to the nature of the processes involved, the time aspect is prevalent in all the constraints in some form or another. A superstructure is used in the derivation of the mathematical model, as discussed in the following section. A description of the sets, variables and parameters can be found in the nomenclature list. 

One of the least well understood aspects of the whole field is the precise physical nature of the process whereby polymer chains of a different size are separated by passage through a gel column. On a qualitative level adec[uate explanations of the phenomenon exist but it has proved to be a more difficult task to formulate and solve anything other thcui the simplest of mathematical models of the chromatographic process. 

Aris (3) more formally defined a mathematical model thus a system of equations, S, is said to be a model of prototypical system, S, if it is formulated to express the laws of S and its solution is intended to represent some aspect of the behavior of S. Seinfeld and Lapidus (4) gave a more specific definition Mathematical model is taken to mean the formulation of mathematical relationships, which describe the behavior of actual systems such that the dependent and independent variables and parameters of the model are directly related to physical and chemical quantities in the real system. ... 

Usually, a mathematical model simulates a process behavior, in what can be termed a forward problem. The inverse problem is, given the experimental measurements of behavior, what is the structure A difficult problem, but an important one for the sciences. The inverse problem may be partitioned into the following stages hypothesis formulation, i.e., model specification, definition of the experiments, identifiability, parameter estimation, experiment, and analysis and model checking. Typically, from measured data, nonparametric indices are evaluated in order to reveal the basic features and mechanisms of the underlying processes. Then, based on this information, several structures are assayed for candidate parametric models. Nevertheless, in this book we look only into various aspects of the forward problem given the structure and the parameter values, how does the system behave ... 

One important aspect of this type of dissolution approach is that a good contact between the crystal surface and the sample is essential, not only for the imaging of the tablet or pharmaceutical formulation but also to ensure that water penetrates into the sample only from the side of the tablet. This is especially important when mathematical modeling of the dissolution process is compared with experimental data. When using the diamond ATR approach, the tablet is pressed onto the diamond, and hence the leakage of water between the diamond surface and the tablet is not expected [66]. 

However, the two-sink model as well as other existing adsorption (sink) models do not seem to be able to describe the strong asymmetry between the adsorption/desorption of VOCs on/from indoor surface materials (the desorption process is much slower than the adsorption process). Diffusion combined with internal adsorption is assumed to be capable of explaining the observed asymmetry. Diffusion mechanisms have been considered to play a role in interactions of VOCs with indoor sinks. Dunn and Chen (1993) proposed and tested three unified, diffusion-limited mathematical models to account for such interactions. The phrase unified relates to the ability of the model to predict both the ad/absorption and desorption phases. This is a very important aspect of modeling test chamber kinetics because in actual applications of chamber studies to indoor air quality (lAQ), we will never be able to predict when we will be in an accumulation or decay phase, so that the same model must apply to both. Development of such models is underway by different research groups. An excellent reference, in which the theoretical bases of most of the recently developed sorption models are reviewed, is the paper by Axley and Lorenzetti (1993). The authors proposed four generic families of models formulated as mass transport modules that can be combined with existing lAQ models. These models include processes such as equilibrium adsorption, boundary layer diffusion, porous adsorbent diffusion transport, and conveetion-diffusion transport. In their paper, the authors present applications of these models and propose criteria for selection of models that are based on the boundary layer/conduction heat transfer problem. 

The terms used in the CFD model equations are generally based on the continuum hypothesis. This means that individual molecules cannot be distinguished, and only a variety of molecules and average derived properties are relevant. Instead of positions and velocities of individual molecules, concentrations, average velocities, and temperatures are used. The differential balance for the border crossings (limiting processes) is formulated in such a way that they still move in the scope of the continuum hypothesis and do not lead to the description of single molecules. The balance equations need to be completed with boundary conditions that are discussed in mathematical and physical aspects. 

The system mass balance equations are often the most important elements of any modelling exercise, but are themselves rarely sufficient to completely formulate the model. Thus other relationships are needed to complete the model in terms of other important aspects of behaviour in order to satisfy the mathematical rigour of the modelling, such that the number of unknown variables must be equal to the number of defining equations. 

When the simulation of deep-well temperatures, pressures, and salinities is imposed as a condition, the number of codes that may be of value is reduced to a much smaller number. Nordstrom and Ball121 recommend six references as covering virtually all the mathematical, thermodynamic, and computational aspects of chemical-equilibrium formulations (see references 123-128). Recent references on modeling include references 45, 63, 70, 129, and 130. 

A standard continuous-time job-shop scheduling formulation [3] can be used to model the basic aspects of the production decisions, such as sequencing and assignment of jobs. Here, the key of the mathematical solution is to capture the durations of each processing step and to relate it to the amounts of material. Therefore, only a top-down approach will be presented to illustrate some main principles of the model. 

Theoretical bases of continuum models including their mathematical formulation and numerical implementation have already been discussed in the previous chapter of this book. We have therefore restricted our review to the environment effects on the NMR observables, without going into the theory of continuum models. This contribution is divided into five sections. After the Introduction, the definitions of the NMR parameters are recalled in the second section. The third section is focused on methodological aspects of the calculation of the NMR parameters in continuum models. The fourth section reviews calculations of the solvent effects on the nuclear magnetic shielding constants and spin-spin coupling constants by means of continuum models, and the final section presents a survey on the perspectives of this field. 

About 20 years ago, the theory of the be- and r-matrices, a global algebraic model of the logical structure of constitutional chemistry was formulated. This theory is the first direct mathematical approach to chemistry which also accentuates its dynamic aspect. The representation by mathematics comprises the individual objects of chemistry and also their relations, including their interconvertibility by chemical reactions. A decade later, the theory of the chemical identity groups was published in a monograph. It is a unified theory of stereochemistry that is primarily devoted to relations between molecular systems. 

A quantitative description of the diverse morphological features of PS requires the integration of the aspects discussed above as well as the fundamental reaction processes involved in silicon/electrolyte interface structure, anodic dissolution, and anodic oxide formation and dissolution as detailed in Chapters 2-5. Any mathematical formulation for the mechanisms of PS formation without such a global integration would be limited in the scope of its validity and in the power to explain details. In addition, a globally and microscopically accurate model would also require the full characterization of all of the morphological features of PS in relation to all of the... 

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