Mathematical model computational results

Statistical analysis can range from relatively simple regression analysis to complex input/output and mathematical models. The advent of the computer and its accessibiUty in most companies has broadened the tools a researcher has to manipulate data. However, the results are only as good as the inputs. Most veteran market researchers accept the statistical tools available to them but use the results to implement their judgment rather than uncritically accepting the machine output. 

Considerable work has been done on mathematic models of the extmsion process, with particular emphasis on screw design. Good results are claimed for extmsion of styrene-based resins using these mathematical methods (229,232). With the advent of low cost computers, closed-loop control of... 

Theoretically based correlations (or semitheoretical extensions of them), rooted in thermodynamics or other fundamentals are ordinarily preferred. However, rigorous theoretical understanding of real systems is far from complete, and purely empirical correlations typically have strict limits on apphcabihty. Many correlations result from curve-fitting the desired parameter to an appropriate independent variable. Some fitting exercises are rooted in theory, eg, Antoine s equation for vapor pressure others can be described as being semitheoretical. These distinctions usually do not refer to adherence to the observations of natural systems, but rather to the agreement in form to mathematical models of idealized systems. The advent of readily available computers has revolutionized the development and use of correlation techniques (see Chemometrics Computer technology Dimensional analysis). 

As with any constitutive theory, the particular forms of the constitutive functions must be constructed, and their parameters (material properties) must be evaluated for the particular materials whose response is to be predicted. In principle, they are to be evaluated from experimental data. Even when experimental data are available, it is often difficult to determine the functional forms of the constitutive functions, because data may be sparse or unavailable in important portions of the parameter space of interest. Micromechanical models of material deformation may be helpful in suggesting functional forms. Internal state variables are particularly useful in this regard, since they may often be connected directly to averages of micromechanical quantities. Often, forms of the constitutive functions are chosen for their mathematical or computational simplicity. When deformations are large, extrapolation of functions borrowed from small deformation theories can produce surprising and sometimes unfortunate results, due to the strong nonlinearities inherent in the kinematics of large deformations. The construction of adequate constitutive functions and their evaluation for particular... 

Catalytic crackings operations have been simulated by mathematical models, with the aid of computers. The computer programs are the end result of a very extensive research effort in pilot and bench scale units. Many sets of calculations are carried out to optimize design of new units, operation of existing plants, choice of feedstocks, and other variables subject to control. A background knowledge of the correlations used in the "black box" helps to make such studies more effective. 

The introduction of computers to many companies allows proprietary software to be used for layout design. Spreadsheet, mathematical modeling and computer-aided design (CAD) techniques are available and greatly assist the design process, and have added to the resources available to planners. However, the traditional scale models described above will still be useful to present the result to management and shop floor personnel. 

Computer tools can contribute significantly to the optimization of processes. Computer data acquisition allows data to be more readily collected, and easy-to-implement control systems can also be achieved. Mathematical modeling can save personnel time, laboratory time and materials, and the tools for solving differential equations, parameter estimation, and optimization problems can be easy to use and result in great productivity gains. Optimizing the control system resulted in faster startup and consequent productivity gains in the extruder laboratory. 

It may turn out that the mathematical model is too rough, meaning numerical results of computations are not consistent with physical experiments, or the model is extremely cumbersome for everyday use and its solution can be obtained with a prescribed accuracy on the basis of simpler models. Then the same work should be started all over again and the remaining stages should be repeated once again. 

The nature of the mathematical model that describes a physical system may dictate a range of acceptable values for the unknown parameters. Furthermore, repeated computations of the response variables for various values of the parameters and subsequent plotting of the results provides valuable experience to the analyst about the behavior of the model and its dependency on the parameters. As a result of this exercise, we often come up with fairly good initial guesses for the parameters. The only disadvantage of this approach is that it could be time consuming. This counterbalanced by the fact that one learns a lot about the structure and behavior of the model at hand. 

To illustrate the actual importance of dynamic properties for the functioning of metabolic networks, we briefly describe and summarize a recent computational study on a model of human erythrocytes [296]. Erythrocytes play a fundamental role in the oxygen supply of cells and have been subject to extensive experimental and theoretical research for decades. In particular, a variety of explicit mathematical models have been developed since the late 1970s [108, 111, 114, 123, 338 341], allowing us to test the reliability of the results in a straightforward way. 

In order to obtain an in vitro-in vivo relationship two sets of data are needed. The first set is the in vivo data, usually entire blood/plasma concentration profiles or a pharmacokinetic metric derived from plasma concentration profile (e.g., cmax, tmax, AUC, % absorbed). The second data set is the in vitro data (e.g., drug release using an appropriate dissolution test). A mathematical model describing the relationship between these data sets is then developed. Fairly obvious, the in vivo data are fixed. However, the in vitro drug-release profile is often adjusted by changing the dissolution testing conditions to determine which match the computed in vivo-release profiles the best, i.e., results in the highest correlation coefficient. 

While computers are a substantial aid in statistical analysis, it is also true that statistical methods have helped in certain computer applications. In Section V the subject of mathematical models will be discussed. These are in many cases based on empirical correlations. When these have been obtained by regression methods, not only is the significance of the results better understood, but also the correlation is expressed directly in a mathematical form suitable for programming. 

A field experiment was conducted at the Canadian Air Forces Base Borden, Ontario, to study the behavior of organic pollutants in a sand aquifer under natural conditions (Mackay et al., 1986). Figure 25.9 shows the results of two experiments, the first one for tetrachloroethene, the second one for chloride. Both substances were added as short pulses to the aquifer. The curves marked as ideal were computed according to Eqs. 25-20 or 25-23. The measured data clearly deviate from the ideal curve. The nonideal curves were constructed by Brusseau (1994) with a mathematical model that includes various factors causing nonideal behavior. 

The last stage of mathematical modeling consists of interpretation of the results. Here lies its greatest weakness, at least prior to the introduction of computer sys-... 

In this chapter we have provided an overview of mathematical modeling from inception of design through specification of solution method, production of solution, and analysis of results. Additionally, we have provided a framework for including computers, particularly current and emerging application software, as vital agents in the modeling process. 

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