What Is Mathematical Modeling

A mathematical model is an abstract representation of reality, in mathematical language, used to find solutions to different types of problems. 

For example, we would like to determine the number of poles (Fig. 9.1) needed to separate two adjacent sites. According to experts, we should put the poles d meters apart (including poles at both extremes). Further, the distance between poles could be less than d meters but no greater. The experts also suggest that this fence utilize poles that are e meters in diameter e d). Then, if the fence has a total length of a meters a d), how many poles are needed  

The variables in the problem are as follows a Total length of fence in meters n Number of poles 

Equation (9.2) is a very simple but practical mathematical model that allows us to calculate the number of poles as a function of the fence length, the distance between poles, and the pole diameter. For example, if you decide that 25 poles is too many, and your budget will allow for just 20 poles, then, using (9.2), you can simulate different scenarios and determine the distance between poles if you are employing 20 poles or any other number of poles on this fence. 

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Parameter estimation and identification are an essential step in the development of mathematical models that describe the behavior of physical processes (Seinfeld and Lapidus, 1974 Aris, 1994). The reader is strongly advised to consult the above references for discussions on what is a model, types of models, model formulation and evaluation. The paper by Plackett that presents the history on the discovery of the least squares method is also recommended (Plackett, 1972). 

Without definite examples to focus our thinking, it is easy to get entangled in a quasi-philosophical discussion of just what a mathematical model might be. To avoid this, I present a dogmatic statement on modeling and proceed to consider an elementary example, returning later to the philosophical caveats and more general considerations. 

Chemistry, and in particular physical and analytical chemistry, often requires a numerical or statistical approach. Not only is mathematical modelling an important aid to understanding, but computations are often needed to turn raw data into meaningful information or to compare them with other data sets. Moreover, calculations are part of laboratory routine, perhaps required for making up solutions of known concentration (see p. 170 and below) or for the calibration of an analytical instrument (see p. 171). In research, trial calculations can reveal what input data are required and where errors in their measurement might be amplified in the final result, e.g. flame atomic absorption spectrometer (see Chapter 27). Table 39.7 Sets of numbers and operations ... 

In the introductory paragraphs earlier we nicjfed that often the physical equipment of the chemical process wp wap) j control have not been constructed. Consequently, we cannot experiment to determine how the process reacts to various inputs and therefore we cannot design the appropriate control system. But even if the process equipment is available for experimentation, the procedure is usually very costly. Therefore, we need a simple description of how the Rfflfcess reacts to various inputs, and this is what the mathematical models can provide to the control designer. 

Keep records that document when and what was invented. It is important that accurate records are kept showing your original sketches with a disclosure statement describing what and how your invention works. It is useful to have someone witness this disclosure document and verify the date that this invention took place. It is often during this step that the invention concept is either modeled (mathematical or physical or both) and tested. Thus, accurate records of these analyzes or test results should also be kept. In the U.S. it is the first to invent that will obtain a patent in the event of two individuals inventing the same thing. Keep the disclosure document secret until the patent application is submitted to the patent office. 

The first step is to be certain of the basis of the published data and consider in what ways this will be affected by different conditions. Revised figures can then usually be determined. For extensive interpretation work, simple mathematical models of performance can be constructed [69]. 

Mathematical models are the link between what is observed experimentally and what is thought to occur at the molecular level. In physical sciences, such as chemistry, there is a direct correspondence between the experimental observation and the molecular world (i.e., a nuclear magnetic resonance spectrum directly reflects the interaction of hydrogen atoms on a molecule). In pharmacology the observations are much more indirect, leaving a much wider gap between the physical chemistry involved in drug-receptor interaction and what the cell does in response to those interactions (through the cellular veil ). Hence, models become uniquely important. 

Obviously, construction of a mathematical model of this process, with our present limited knowledge about some of the critical details of the process, requires good insight and many qualitative judgments to pose a solvable mathematical problem with some claim to realism. For example what dictates the point of phase separation does equilibrium or rate of diffusion govern the monomer partitioning between phase if it is the former what are the partition coefficients for each monomer which polymeric species go to each phase and so on. 

What is a mathematical model The group of unknown physical quantities which interest us and the group of available data are closely interconnected. This link may be embodied in algebraic or differential equations. A proper choice of the mathematical model facilitates solving these equations and providing the subsidiary information on the coefficients of equations as well as on the initial and boundary data. 

What is commonly understood by a fundamental approach is applying theoretically based mathematical models of necessary equipment items. Intrinsic (not falsified by processes other than a chemical transformation) kinetics of all processes are investigated, transport phenomena are studied, flow patterns are identified, and relevant microscopic phenomena are studied. It is intended to separately study as many intrinsic stages as possible and to combine results of these investigations into a mathematical model. Such a model contains only a limited amount of theory (grey models, gross models, or tendency models). Obviously, the extrapolation power of these models strongly depends on the content of theory. The model... 

The formulation of the parameter estimation problem is equally important to the actual solution of the problem (i.e., the determination of the unknown parameters). In the formulation of the parameter estimation problem we must answer two questions (a) what type of mathematical model do we have and (b) what type of objective function should we minimize In this chapter we address both these questions. Although the primary focus of this book is the treatment of mathematical models that are nonlinear with respect to the parameters nonlinear regression) consideration to linear models linear regression) will also be given. 

Denormalization of data is needed when a statistical procedure requires that the information to be analyzed must be on the same observation. Procedures in SAS that perform data modeling are often the ones that require denormalized data, as they require that the dependent variable be present on the same observation as the independent variables. For example, imagine that you are trying to determine a mathematical model that predicts under what conditions a therapy is successful. That model might look like this ... 

What are some of the mathematical tools that we use In classical control, our analysis is based on linear ordinary differential equations with constant coefficients—what is called linear time invariant (LTI). Our models are also called lumped-parameter models, meaning that variations in space or location are not considered. Time is the only independent variable. 

The derivation and development of a mathematical model which is as general as possible and incorporates detailed knowledge from phenomena operative in emulsion polymerization reactors, its testing phase and its application to latex reactor design, simulation, optimization and control are the objectives of this paper and will be described in what follows. 

Scanning electron microscopy and other experimental methods indicate that the void spaces in a typical catalyst particle are not uniform in size, shape, or length. Moreover, they are often highly interconnected. Because of the complexities of most common pore structures, detailed mathematical descriptions of the void structure are not available. Moreover, because of other uncertainties involved in the design of catalytic reactors, the use of elaborate quantitative models of catalyst pore structures is not warranted. What is required, however, is a model that allows one to take into account the rates of diffusion of reactant and product species through the void spaces. Many of the models in common use simulate the void regions as cylindrical pores for such models a knowledge of the distribution of pore radii and the volumes associated therewith is required. 

One must understand the physical mechanisms by which mass transfer takes place in catalyst pores to comprehend the development of mathematical models that can be used in engineering design calculations to estimate what fraction of the catalyst surface is effective in promoting reaction. There are several factors that complicate efforts to analyze mass transfer within such systems. They include the facts that (1) the pore geometry is extremely complex, and not subject to realistic modeling in terms of a small number of parameters, and that (2) different molecular phenomena are responsible for the mass transfer. Consequently, it is often useful to characterize the mass transfer process in terms of an effective diffusivity, i.e., a transport coefficient that pertains to a porous material in which the calculations are based on total area (void plus solid) normal to the direction of transport. For example, in a spherical catalyst pellet, the appropriate area to use in characterizing diffusion in the radial direction is 47ir2. 

We examine these models not only as mathematical entities but also as a means of determining what the mathematical properties of schemes tell us regarding programming problems and languages. In studying alternative models an important point to consider is their relative power. 

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