Advanced Methods in Mathematical Modeling

Physical (or a priori ) modeling which aims at description in the form of equations of (mass, momentum, energy) conservation leads to the most general description of the phenomenon, but in practice it may require too much interdisciplinary research and often results in a very comphcated mathematical description, which is not feasible in the time allowed (e.g. for the purposes of control and optimization). Usually, we finally obtain a mathematical description that contains a number of parametric coefficients, which are determined in the identification of the model. 

Empirical (or a posteriori ) modeling derives descriptive dependencies on the basis of an analysis of historical input and output data sets. In this respect it is nothing but an attempt to imitate nature by adopting a purely experimental approach, i.e. the use of large quantities of historical data to create process models which then form the basis for forecasting, control and optimization. 

Milewski et al., Advanced Methods of Solid Oxide Fuel Cell Modeling, Green Energy and Technology, DOI 10.1007/978-0-85729-262-9 3, 

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This chapter is concerned with the design and improvement of chemically-active ship bottom paints known as antifouling paints. The aims have been to illustrate the challenges involved in working with such multi-component, functional products and to show which scientific and engineering tools are available. The research in this field includes both purely empirical formulation and test methods and advanced tools including mathematical modelling of paint behaviour. 

Solubihties of 1,3-butadiene and many other organic compounds in water have been extensively studied to gauge the impact of discharge of these materials into aquatic systems. Estimates have been advanced by using the UNIFAC derived method (19,20). Similarly, a mathematical model has been developed to calculate the vapor—Hquid equiUbrium (VLE) for 1,3-butadiene in the presence of steam (21). 

Classical methods of mathematical physics are employed at the first stage. Numerous physical problems lead to mathematical models having no advanced methods for solving them. Quite often in practice, the user is forced to. solve such nonlinear problems of mathematical physics for which even the theorems of existence and uniqueness have not yet been proven and some relevant issues are still open. 

In the last 25 years, with continuous development of faster computers and sophisticated numerical methods, there have been many published work that have used detailed mathematical models with rigorous physical property calculations and advanced optimisation techniques to address all the issues mentioned above. These have been the motivating factors to write this book in which excellent and important contributions of many researchers around the globe and those by the author and coworkers are accommodated. 

Today contractors and licensors use sophisticated computerized mathematical models which take into account the many variables involved in the physical, chemical, geometrical and mechanical properties of the system. ICI, for example, was one of the first to develop a very versatile and effective model of the primary reformer. The program REFORM [361], [430], [439] can simulate all major types of reformers (see below) top-fired, side-fired, terraced-wall, concentric round configurations, the exchanger reformers (GHR, for example), and so on. The program is based on reaction kinetics, correlations with experimental heat transfer data, pressure drop functions, advanced furnace calculation methods, and a kinetic model of carbon formation [419],... 

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