Mathematical modeling computer engineering aspects

Constraints in optimization arise because a process must describe the physical bounds on the variables, empirical relations, and physical laws that apply to a specific problem, as mentioned in Section 1.4. How to develop models that take into account these constraints is the main focus of this chapter. Mathematical models are employed in all areas of science, engineering, and business to solve problems, design equipment, interpret data, and communicate information. Eykhoff (1974) defined a mathematical model as a representation of the essential aspects of an existing system (or a system to be constructed) which presents knowledge of that system in a usable form. For the purpose of optimization, we shall be concerned with developing quantitative expressions that will enable us to use mathematics and computer calculations to extract useful information. To optimize a process models may need to be developed for the objective function/, equality constraints g, and inequality constraints h. 

Dente and Ranzi (in Albright et al., eds., Pyrolysis Theory and Industrial Practice, Academic Press, 1983, pp. 133-175) Mathematical modeling of hydrocarbon pyrolysis reactions Shah and Sharma (in Carberry and Varma, eds., Chemical Reaction and Reaction Engineering Handbook, Dekker, 1987, pp. 713-721) Hydroxylamine phosphate manufacture in a slurry reactor Some aspects of a kinetic model of methanol synthesis are described in the first example, which is followed by a second example that describes coping with the multiplicity of reactants and reactions of some petroleum conversion processes. Then two somewhat simplified industrial examples are worked out in detail mild thermal cracking and production of styrene. Even these calculations are impractical without a computer. The basic data and mathematics and some of the results are presented. 

Box, G. E. P., and T. L. Henson, Some aspects of mathematical modeling in chemical engineering, Proc. Inaugural Conf. of the Scientific Computation Centre and the Institute of Statistical Studies and Research, Cairo Univ. Press, Cairo (1970), 548. 

In this chapter, the current status and several major aspects of PEFC component modeling are described and discussed. Mathematical and computational modeling (both analytical and numerical) play important roles in the technology development and optimal operation of PEFCs from a fundamental understanding of underlying phenomena to engineering design and optimization that can lead to cost reductions and durability improvements. At present, the macroscopic descriptions of phenomena in the individual fuel-cell components have been formulated and... 

It is important to note that in using computer-aided models for batch distillation, the various assumptions of the model can have a significant impact on the accuracy of the results e.g., see the discussion of the effects of holdup above. Uncertainties in the physical and chemical parameters in the models can be addressed most effectively by a combination of sensitivity calculations using simulation tools, along with comparison to data. The mathematical treatment of stiffness in the model equations can also be very important, and there is often a substantial advantage in using simulation tools that take special account of this stiffness. (See the 7th edition of Perry s Chemical Engineers Handbook for a more detailed discussion of this aspect). 

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