Process Dynamics and Mathematical Models

General References Seborg, Edgar, and Mellichamp, Process Dynamics and Control, Wiley, New York, 2004 Marlin, Process Control, McGraw-Hill, New York, 2000 Ogunnaike and Ray, Process Dynamics Modeling and Control, Oxford University Press, New York, 1994 Smith and Corripio, Principles and Practices of Automatic Process Control, Wiley, New York, 1997. 

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He has published over 200 papers in the fields of process control, optimization, and mathematical modeling of processes such as separations, combustion, and microelectronics processing. He is coauthor of Process Dynamics and Control, published by Wiley in 1989. Dr. Edgar was chairman of the CAST Division of AIChE in 1986, president of the CACHE Corporation from 1981 to 1984, and president of AIChE in 1997. 

Known scale-up correlations thus may allow scale-up even when laboratory or pilot plant experience is minimal. The fundamental approach to process scaling involves mathematical modeling of the manufacturing process and experimental validation of the model at different scale-up ratios. In a paper on fluid dynamics in bubble column reactors, Lubbert and coworkers (54) noted ... 

This second-level modeling of the feedback mechanisms leads to nonlinear models for processes, which, under some experimental conditions, may exhibit chaotic behavior. The previous equation is termed bilinear because of the presence of the b [y (/,)] r (I,) term and it is the general formalism for models in biology, ecology, industrial applications, and socioeconomic processes [601]. Bilinear mathematical models are useful to real-world dynamic behavior because of their variable structure. It has been shown that processes described by bilinear models are generally more controllable and offer better performance in control than linear systems. We emphasize that the unstable inherent character of chaotic systems fits exactly within the complete controllability principle discussed for bilinear mathematical models [601] additive control may be used to steer the system to new equilibrium points, and multiplicative control, either to stabilize a chaotic behavior or to enlarge the attainable space. Then, bilinear systems are of extreme importance in the design and use of optimal control for chaotic behaviors. We can now understand the butterfly effect, i.e., the extreme sensitivity of chaotic systems to tiny perturbations described in Chapter 3. 

For all of these reasons, a thorough understanding of the NH3 adsorption-desorption phenomena on the catalyst surface is a prerequisite In fact, typical SCR catalysts can store large amounts of ammonia, whose surface evolution becomes the rate-controlling factor of the reactor dynamics. Also, mathematical modeling appears to be even more useful for the analysis and development of unsteady SCR processes than in the case of steady-state operation. 

The simplicity of chemical systems and the very complicated dynamics of chemical processes cause the mathematical models of chemical reactions to be an important area of applications of non-linear methods of mathematics, including catastrophe theory. 

Mathematical formulation of dynamic models and their linearisation is treated in books dealing with process dynamics and control. Here we mention the textbooks of Stephanopoulos (1984), Ogunnaike Ray (1994), Luyben (1995), and Marlin (1995). Very useful theoretical and practical Insights in dynamics and control of distillation processes, with so many implications in dynamic simulation, can be found in the monograph edited by Luyben (1992) with contribution of specialists in different areas. A useful presentation of process dynamics from a practical viewpoint can be found in the book of Ingham et al. (1994). 

The tools that we need to help the modelling of complex reaction systems have to fill the gap between chemical and mathematical modelling. They also should allow the chemist to gain practical and effecient access to the required mathematical knowledge. Finally, those tools should be able to take into account the specific features and properties of chemical reaction equations, and, at the same time, to do this using a chemical language to describe the expected behaviour and dynamical structure of the model, for instance, in terms of chemical network, reaction processes, autocatalysis, activation or inhibition. We are far from that, which indicates that we are still lacking theoretical methods to handle those problems. Moreover, even well established mathematical theories are still not usually implemented in effecient practical procedures. 

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