Design tools from mathematical modeling

Let s summarize our analysis of the desalinator so far. We started with the conservation of mass and derived operating equations for the process. We then used the operating equations as design tools to predict the performance of the desalinator. Creating design tools is the essence of chemical engineering design. 

---

The rest of the knowledge, to the extent possible, must be generated from actual experiments (at both lab and pilot plant scale) and from mathematical modeling. Ideally the modeling should be done in intimate conjunction with the experimental testing. Models are useful tools to help design experiments since a reasonable model captures not only process understanding but also process uncertainty. 

The present chapter is not meant to be exhaustive. Rather, an attempt has been made to introduce the reader to the major concepts and tools used by catalytic reaction engineers. Section 2 gives a review of the most important reactor types. This is deliberately not done in a narrative way, i.e. by describing the physical appearance of chemical reactors. Emphasis is placed on the way mathematical model equations are constructed for each category of reactor. Basically, this boils down to the application of the conservation laws of mass, energy and possibly momentum. Section 7.3 presents an analysis of the effect of the finite rate at which reaction components and/or heat are supplied to or removed from the locus of reaction, i.e. the catalytic site. Finally, the material developed in Sections 7.2 and 7.3 is applied to the design of laboratory reactors and to the analysis of rate data in Section 7.4. 

The detailed course of a polymerization is determined by the nature of the particular reaction as well as by the characteristics of the reactor which is used. The design and control of the operation are greatly aided by mathematical modeling of the process. Such models may be based on empirical relations between the independent and dependent operating variables. This is not as satisfactory, however, as a model that is derived from accurate knowledge of the polymerization process and reactor operation, because only the latter tool permits extrapolation to reaction conditions that have not yet been tried. 

To this aim, the current development situation is analyzed via the product state of several source databases or tools. The scenario shown here is a simplification of a part of the design scenario described in Subsect. 1.2.2. Here, the selection and adaption of a mathematical model is treated, to simulate the separation of the final product Polyamide-6 from the residue monomer Caprolactam. 

The rest of the chapter is organized as follows. Section 9.2 describes the FCC process in detail and the corresponding multi-ohjective mathematical model. The multi-ohjective evolutionary toolbox used as the optimization tool in this work is presented in Section 9.3, and the results obtained from the evolutionary design are discussed in Section 9.4. Section 9.5 involves the decision-making and economic evaluation of the solutions for practical implementation. Finally, conclusions are drawn in Section 9.6. 

Mathematical models from stochastic geometry are useful tools to achieve this goal since they provide methods allowing for a quantitative description of the correlation between microstructure and functionality. Moreover, systematic modifications of model parameters, in combination with numerical transportation models, offer the opportunity to identify morphologies with improved physical properties by model-based computer simulations, that is, to perform a virtual material design. 

---