Principles of mathematical modelling

Kapur (1988) has listed thirty-six characteristics or principles of mathematical modelling. These are very much a matter of common sense, but it is very important to have them restated, as it is often very easy to lose sight of the principles during the active involvement of modelling. They can be summarised as follows ... 

Principles of mathematical modelling 2 Probability density function 112 Process control examples 505-524 Product inhibition 643, 649 Production rate in mass balance 27 Profit function 108 Proportional... 

A system approach will be adopted which treats the fixed bed reactor as a system consisting of subsystems with their properties and interactions giving the overall system (the reactor) its characteristics. Before describing details, it is important to give a brief discussion of system theory and the principles of mathematical models with emphasis on fixed bed catalytic reactors. 

Basic Principles of Mathematical Modelling for Industrial Fixed Bed Catalytic Reactors... 

BASIC PRINCIPLES OF MATHEMATICAL MODELLING FOR INDUSTRIAL FIXED BED CATALYTIC REACTORS... 

The approach chosen here is to use the H-principle of mathematical modelling (Hoskuldsson (1996)). The basic idea is to carry out the modeling in steps and at each step compute a rank one approximation to the solution. This rank one solution is based on optimizing the balance between the improvement in fit and the associated precision that can be obtained by such an improvement in the solution. Thus, each of the tank one part is a result of optimization task involving fit and precision, such that all parts are in certain sense optimal at the respective step of the analysis. 

The present approach is concerned with finding stable solution in the case the data show low rank like in the case of NIR data. The algorithm proposed is independent of V. Thus, V can be zero or any other prior choice. The solution is based on the H-principle of mathematical modeling that we shall consider closer. 

Sidnyaev, N. I., Hrapov, P. V., Melnikova, Y. S., Principles of mathematical modeling of temperature fields in multiphase media. Collection of reports of IV all-Russian... 

There are two basic classes of mathematical models (see Fig. 5.3-18) (1) purely empirical models, and (2) models based on physicochemical principles. 

A standard continuous-time job-shop scheduling formulation [3] can be used to model the basic aspects of the production decisions, such as sequencing and assignment of jobs. Here, the key of the mathematical solution is to capture the durations of each processing step and to relate it to the amounts of material. Therefore, only a top-down approach will be presented to illustrate some main principles of the model. 

At the time of this writing, it must be conceded that there have been no fundamental principles-based mathematical model for Nafion that has predicted significantly new phenomena or caused property improvements in a significant way. Models that capture the essence of percolation behavior ignore chemical identity. The more ab initio methods that do embrace chemical structure are limited by the number of molecular fragments that the computer can accommodate. Other models are semiempirical in nature, which limits their predictive flexibility. Nonetheless, the diversity of these interesting approaches offers structural perspectives that can serve as guides toward further experimental inquiry. 

As noted in the Molecular Simulation of Structure and Properties section, there have been no fundamental principle-based mathematical models for Nafion that have predicted new phenomena or caused property improvements in a significant way. This is due to a number of limitations inherent in one or the other of the various schemes. These shortcomings include an inability to sufficiently account for chemical identity, an inability to simulate and predict the long-range structure as would be probed by SAXS or TEM, and the failure to simulate structure over different hierarchy levels. Certainly, advances in this important research front will emerge and be combined with advances in experimentally derived information to yield a much deeper state of understanding of Nafion. 

The principles and methods of scale-up can be applied to chemical reactors. In the absence of significant thermal effects, i.e., when the ratio <2r/ Vr may be considered negligible, ideal batch reactors do not show any problem of scale-up, because the volume Vr does not appear in the mathematical model (2.17), so that their performance is only determined by chemical kinetics (see Sect. 2.3). On the contrary, a very complex behavior is expected for real reactors in fact, this behavior cannot be analyzed in terms of mathematical models, and the design procedures must be largely based on semi-empirical rules of scale-up. 

The term chemometrics was hrst coined in 1971 to describe the growing use of mathematical models, statistical principles, and other logic-based methods in the held of chemistry and, in particular, the held of analytical chemistry. Chemometrics is an interdisciplinary held that involves multivariate statistics, mathematical modeling, computer science, and analytical chemistry. Some major application areas of chemometrics include (1) calibration, validation, and signihcance testing (2) optimization of chemical measurements and experimental procedures and (3) the extraction of the maximum of chemical information from analytical data. 

Mention has already been made of mathematical models which simulate partitioning in the environment. This has been facilitated by the introduction of fugacity principles to environmental modelling, which simplifies the linking of complex partition and rate constants in many of the current multimedia environmental models. A detailed explanation of the ideas involved, and their application, has recently been published by Mackay.39... 

In theory, by feeding the MWD and experimental rate data into a mathematical model containing a variety of polymerization mechanisms, it should be possible to find the mechanism which explains all the experimental phenomena and to evaluate any unknown rate constants. As pointed out by Zeman (58), as long as there are more independent experimental observations than rate parameters, the solution should, in principle, be unique. This approach involves critical problems in choice of experiments and in experimental as well as computational techniques. We are not aware of its having yet been successfully employed. The converse— namely, predicting MWD from different reactor types on the basis of mathematical models and kinetic data—has been successfully demonstrated, however, as discussed above. The recent series of interesting papers by Hamielec et al. is a case in point. 

Apart from its usefulness in the construction of mathematical models, the shortsightedness principle packs notable predictive power. As an example, an olefinic double bond can exist in only six configurations ... 

Maloszewski P. and Zuber A. (1993) Principles and practice of calibration and validation of mathematical models for the interpretation of environmental tracer data in aquifers. Adv. Water Resour. 16, 173—190. 

This short review of zeolite and zeotype synthesis is written for those who are relatively new to the field. It aims to present an overall introduction to some fundamental aspects of the subject and to indicate where further information can be found. An account of experimental practice is followed by a summary of mathematical modelling procedures. Observations from crystallisation studies then introduce basic principles of the synthesis process. 

Here the basic concepts of system theory and principles for mathematical modelling are presented for the development of diffusion reaction models for fixed bed catalytic reactors. 

Chapter 1 introduces the reader to system theory and the classification of systems. It also covers the main principles for the development of mathematical models in general and for fixed bed catalytic reactors in particular. 

Part II (Chapters 4 and 5) introduces the reader to the modeling requirements for process control. It demonstrates how we can construct useful models, starting from basic principles, and determines the scope and difficulties of mathematical modeling for process control purposes. 

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