Mathematical models, design purpose

Fluid phase equilibria and mixing properties are of primary interest for theoretical purposes (mathematical model design, parameter estimatioii, etc.), and for the development of a general proper liquid theory. In chemical irrdrrstiial processes irrvolving liquid mixtures, the optimization and adequate design of separation... 

On the basis of different assumptions about the nature of the fluid and solid flow within each phase and between phases as well as about the extent of mixing within each phase, it is possible to develop many different mathematical models of the two phase type. Pyle (119), Rowe (120), and Grace (121) have critically reviewed models of these types. Treatment of these models is clearly beyond the scope of this text. In many cases insufficient data exist to provide critical tests of model validity. This situation is especially true of large scale reactors that are the systems of greatest interest from industry s point of view. The student should understand, however, that there is an ongoing effort to develop mathematical models of fluidized bed reactors that will be useful for design purposes. Our current... 

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. 

The kinetics of a catalytic reaction is usually measured in a reactor under conditions relevant to the industrial process. The measured overall rates can then be fitted to a mathematical model, the macroscopic kinetics. This is extremely convenient for process design purposes. 

There is, however, one obvious difference between a mathematical model and a physical model (or the real system itself). The response of the former to the same set of conditions is always identical. In physical experiments, where results are measured rather than calculated, there are inevitably random errors which may be appreciable. As already pointed out, mathematical models are usually to some extent imperfect in other words, they do contain systematic errors. The important point is that these imperfections are always reproduced in the same way, even though their ultimate source may have been random errors in data on which the model was based. This point has been stressed because it is important to recognize that only partial use of methods from statistical treatments of design of experiments is involved in what follows. The use of these methods here is only for the purpose of studying the geometry of response with respect to the controllable variables. No consideration of probability or of error enters into the discussion. 

For control purposes, somewhat simplified mathematical models usually are adequate. In distillation, for instance, the Underwood-Fenske-Gilliland model with constant relative volatilities and a simplified enthalpy balance may be preferred to a full-fledged tray-by-tray calculation every time there is a perturbation. In control situations, the demand for speed of response may not be realizable with an overly elaborate mathematical system. Moreover, in practice not all disturbances are measurable, and the process characteristics are not known exactly. Accordingly feedforward control is supplemented in most instances with feedback. In a well-designed system (Shinskey, 1984, p. 186) typically 90%... 

A better understanding of the behavior of FCC units can be obtained through mathematical models coupled with industrial verification and cross verification of these models. The mathematical model equations need to be solved for both design and simulation purposes. Most of the models are nonlinear and therefore they require numerical techniques like the ones described in the previous chapters. 

The purpose of any fouling model is to assist the designer or indeed the operator of heat exchangers, to make an assessment of the impact of fouling on heat exchanger performance given certain operating conditions. Ideally a mathematical interpretation of Equation 4.6 would provide the basis for such an assessment but the inclusion of an extensive set of conditions into one mathematical model would be at best, difficult and even impossible. 

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