Construction of Mathematical Models

First of all are determined physicochemical processes enabling the solution of the set problem and are selected those laws of thermodynamics and kinetics or known empiric correlations, which formalize them and set sought for final data in correlation with available initial data in the studied system. Chemical processes in the geological medium are mutually associated by strict restrictions of charge neutrality, mass and energy conservation. Structurization of models in these conditions boils down to identification of cause and effect associations between various physicochemical processes and in the selection or derivation of equations describing them. Such structurization is intended for the creation of a system of interrelated equations, which characterize the physicochemical state of groimd water and may be considered as an independent physicochemical submodel. 

This submodel deals with microprocesses within ground water and at its bounds. It determines a minimum set of the necessary physicochemical 

Among the outside factors the main role belongs to geological ones. They include the composition, properties and mutual position of deposits and rocks, their hydrogeological stratification. They affect ground water hydrodynamics but practically do not depend on it. In connection with this the geologic structure and properties of the host rocks may be considered an independent stationary submodel of the geologic structure 

The geological submodel is a structural one as it determines spatial position of boundaries most important for modeling. They separate deposits and rock from each other, from the atmosphere, surface water. Such boxmdaries have rigid space coordinates and actually do not change their position in time. Parameters of this submodel characterize the composition of host deposits and rocks, porosity and permeability properties. A geological submodel may be constructed in coordinates x, y and 2 and graphically imaged in the forms of maps and vertical cross-sections. 

In the final analysis in the structure of most hydrogeochemical models are identified the following groups of mutually associated parameters, which form three submodels, inserted into one another as in Russian nesting dolls  

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The Supplement B (reference) contains a description of the process to render an automatic construction of mathematical models with the application of electronic computer. The research work of the Institute of the applied mathematics of The Academy of Sciences ( Ukraine) was assumed as a basis for the Supplement. The prepared mathematical model provides the possibility to spare strength and to save money, usually spent for the development of the mathematical models of each separate enterprise. The model provides the possibility to execute the works standard forms and records for the non-destructive inspection in complete correspondence with the requirements of the Standard. 

In any environmental management plan, it is essential to understand the pertinent quality criteria and their relationship to the variables that can be manipulated directly. The construction of mathematical models imposes the requirement of a logical framework that connects causes and effects, identifies all pertinent variables, and defines their interrelationships. Even if a particular technique never becomes practical, the discipline imposed by the logical structure is valuable in highlighting important relationships. 

Fundamentals of the various methods of plastisol technology have been considered in a number of papers, nevertheless the theoretical analysis and the construction of mathematical models have not yet been completed so far. The dipping process has been studied in more detail cf.2 6 10). 

Apart from its usefulness in the construction of mathematical models, the shortsightedness principle packs notable predictive power. As an example, an olefinic double bond can exist in only six configurations ... 

Since the most powerful way to summarize our quantitative understanding is through the construction of mathematical models, the sections which follow will, in addition to giving a qualitative description, present key features of mathematical models which have been proposed to describe these microscale and macroscale phenomena. Furthermore, once constructed, such models can be used to explore system behavior, including making predictions about... 

The pursuit of operations research consists of (a) the judgment phase (what are the problems ), (b) the research phase (how to solve these problems), and (c) the decision phase (how to act on the finding and eliminate the problems). These phases require the evaluation of objectives, analysis of an operation and the collection of evidence and resources to be committed to the study, the (mathematical) formulation of problems, the construction of theoretical models and selection of measures of effectiveness to test the models in practice, the making and testing of hypotheses as to how well a model represents the problem, prediction, refinement of the model, and the interpretation of results (usually as possible alternatives) with their respective values (payoff). The decision-maker generally combines the findings of the... 

The next step in formulating the problem is to construct a mathematical model of the process by considering the fundamental chemical and physical phenomena and physical limitations that influence the process behavior. For the case of the blast furnace, typical features are... 

These differences probably contribute to the fact that mathematical modeling is, as yet, not seen as a mainstream research tool in many areas of molecular biology. However, as will be described in the remainder of this chapter, many obstacles in the construction of kinetic models of cellular metabolism can be addressed using a combination of novel and established experimental and computational techniques, enabling the construction of metabolic models of increasing complexity and size. 

Much research has been applied toward understanding the ways that various animals use odor plumes. It is not within the scope of this book to exhaustively survey the resulting literature. One aspect of this research has direct bearing on the use of plumes to locate sources of chemical molecules such as explosives. As they are studying their animal subjects, these researchers construct two kinds of mathematical models to assist them. 

Successful construction and use of mathematical models for individual processes soon leads to more ambitious projects. It has already been mentioned, for example, that in petroleum refining most processes cannot be regarded as separate entities because they consume products of other processes and supply raw materials for still other operations. If one tries, however, to construct a mathematical model which will directly represent the operations of an entire refinery, the task can become overwhelming. The size of the model, when looked at in its entirety, is simply too large for a proper perspective. 

Once the ions have entered the skimmer tip, it is necessary to extract and focus them into the analyser by subjecting the charged ions to constant electric fields. In order to construct an elfective ion optical array, it is necessary to calculate the path followed by the ions in the electrostatic fields. We can resort to a number of mathematical models, such as SIMION, for a better understanding and optimisation of the ion-optical design for ICP-MS and the processes involved [12]. 

It is interesting to note that a number of researchers have attempted, more or less successfully, to construct reliable mathematical models which combine structural elements and experimental data to predict the toxicities of new substances not yet experimentally tested for humans and other species.110152 Furthermore pharmacology, toxicity, metabolism and enzyme-inhibiting effects of fluorine-containing aromatic systems (e.g., anilines, ben-zothiadiazines, butyrophenones, corticoids, phenothiazines, steroids, uracils) have been discussed in depth in the literature.153-156... 

A mathematical model is a representation, in mathematical terms, of certain aspects of a nonmathematical system. The arts and crafts of mathematical modeling are exhibited in the construction of models that not only are consistent in themselves and mirror the behavior of their prototype, but also serve some exterior purpose. 

Values of both parameters, p and Mn, are generalized characteristics of any polymerization process therefore, it is reasonable to construct a mathematical model of the kinetics for the parameter P and then calculate Mn fromp with Eq. (2.3). 

Toxicokinetics studies are designed to measure the amount and rate of the absorption, distribution, metabolism, and excretion of a xenobiotic. These data are used to construct predictive mathematical models so that the distribution and excretion of other doses can be simulated. Such studies are carried out using radiolabeled compounds to facilitate measurement and total recovery of the administered dose. This can be done entirely in vivo by measuring levels in blood, expired air, feces, and urine these procedures can be done relatively noninvasively and continuously in the same animal. Tissue levels can be measured by sequential killing and analysis of organ levels. It is important to measure not only the compound administered but also its metabolites, because simple radioactivity counting does not differentiate among them. 

Both branches of kinetics, i.e. applied and detailed, are connected with mathematical kinetics dealing with the construction and examination of mathematical models of chemical kinetics. Thus applied mathematical kinetics is useful for both applied and detailed kinetics. 

Constructing a mathematical model involves developing a set of equations for each control region. This set consists of the following ... 

Figure ll-18a shows a flow diagram for constructing the mathematical model of a fermentation process. 

Figure 11-18a. The scheme of constructing the mathematical model of a fermentation process. (Source B. Volesky and J. Votruba [4].)...

In other work cyclic operation of a metal hydride refrigerator [3] was analyzed. With the help of mathematical model for calculation of transfer processes to HHP the algorithm of definition of optimum switching of heat flows in system has been developed at given cycle time. The algorithm is constructed from a condition of achievement of the maximal refrigerating effect (real COP) on unit of brought heat. 

After these preliminary remarks we now proceed to construct the mathematical model of the system. 

Although under certain experimental condition, step 2 or step 3 may be ignored due to its non-conspicuous influence to the flnal pyrolysis product distribution, however, a good mathematical model for biomass pyrolysis should be versatile applicable to other pyrolysis conditions, and thus it should be involved the above-mentioned three steps of process, of course heat and mass transfer equations should be included also. This paper presents this kind of mathematical model. Although the model is constructed based on sawdust pyrolysis, it is quite straightforward to apply the same approach to other cases such as straw and municipal solid waste pyrolysis even if to biomass or coal gasification or metal ore reduction. 

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