Mathematical modeling procedure

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. 

Not all our knowledge of microporous materials synthesis has come from laboratory experimentation. Valuable understanding has also been achieved from the development and use of a variety of mathematical modelling procedures. These have increased greatly in... 

The relationship between output variables, called the response, and the input variables is called the response function and is associated with a response surface. When the precise mathematical model of the response surface is not known, it is still possible to use sequential procedures to optimize the system. One of the most popular algorithms for this purpose is the simplex method and its many variations (63,64). 

No single method or algorithm of optimization exists that can be apphed efficiently to all problems. The method chosen for any particular case will depend primarily on (I) the character of the objective function, (2) the nature of the constraints, and (3) the number of independent and dependent variables. Table 8-6 summarizes the six general steps for the analysis and solution of optimization problems (Edgar and Himmelblau, Optimization of Chemical Processes, McGraw-HiU, New York, 1988). You do not have to follow the cited order exac tly, but vou should cover all of the steps eventually. Shortcuts in the procedure are allowable, and the easy steps can be performed first. Steps I, 2, and 3 deal with the mathematical definition of the problem ideutificatiou of variables and specification of the objective function and statement of the constraints. If the process to be optimized is very complex, it may be necessaiy to reformulate the problem so that it can be solved with reasonable effort. Later in this section, we discuss the development of mathematical models for the process and the objec tive function (the economic model). 

Parameter estimation is a procedure for taking the unit measurements and reducing them to a set of parameters for a physical (or, in some cases, relational) mathematical model of the unit. Statistical interpretation tempered with engineering judgment is required to arrive at realistic parameter estimates. Parameter estimation can be an integral part of fault detection and model discrimination. 

More appropriate mathematical models must be specifically incorporated into a test, or the data must be transformed so as to make it testable by standard procedures. 

Ultrasound can thus be used to enhance kinetics, flow, and mass and heat transfer. The overall results are that organic synthetic reactions show increased rate (sometimes even from hours to minutes, up to 25 times faster), and/or increased yield (tens of percentages, sometimes even starting from 0% yield in nonsonicated conditions). In multiphase systems, gas-liquid and solid-liquid mass transfer has been observed to increase by 5- and 20-fold, respectively [35]. Membrane fluxes have been enhanced by up to a factor of 8 [56]. Despite these results, use of acoustics, and ultrasound in particular, in chemical industry is mainly limited to the fields of cleaning and decontamination [55]. One of the main barriers to industrial application of sonochemical processes is control and scale-up of ultrasound concepts into operable processes. Therefore, a better understanding is required of the relation between a cavitation coUapse and chemical reactivity, as weU as a better understanding and reproducibility of the influence of various design and operational parameters on the cavitation process. Also, rehable mathematical models and scale-up procedures need to be developed [35, 54, 55]. 

A survey of the mathematical models for typical chemical reactors and reactions shows that several hydrodynamic and transfer coefficients (model parameters) must be known to simulate reactor behaviour. These model parameters are listed in Table 5.4-6 (see also Table 5.4-1 in Section 5.4.1). Regions of interfacial surface area for various gas-liquid reactors are shown in Fig. 5.4-15. Many correlations for transfer coefficients have been published in the literature (see the list of books and review papers at the beginning of this section). The coefficients can be evaluated from those correlations within an average accuracy of about 25%. This is usually sufficient for modelling of chemical reactors. Mathematical models of reactors arc often more sensitive to kinetic parameters. Experimental methods and procedures for parameters estimation are discussed in the subsequent section. 

Computer-aided mathematical modeling is a useful tool to supplement monitoring studies and to evaluate the environmental fate of agrochemicals under various conditions. A simulation procedure with a mathematical model using parameters observed in the monitoring study could be helpful for the interpretation of the data obtained in the study. 

Denormalization of data is needed when a statistical procedure requires that the information to be analyzed must be on the same observation. Procedures in SAS that perform data modeling are often the ones that require denormalized data, as they require that the dependent variable be present on the same observation as the independent variables. For example, imagine that you are trying to determine a mathematical model that predicts under what conditions a therapy is successful. That model might look like this ... 

The optimisation procedure presented in this chapter entails two stages as summarized in Fig. 5.3. In the first stage, a mathematical model for minimisation of freshwater requirement is solved based on maximum potential reusable water storage, gf. For clarity, this model will be referred to as model Ml in this chapter. In the second stage, the minimum freshwater requirement obtained from model Ml is used as an input parameter in another mathematical model for which the objective function is the minimisation of reusable water storage. This model will be referred to as model M2 in this chapter. Since different amounts of reusable water will be stored at various intervals within the time horizon of interest, the minimum reusable water storage capacity will correspond to the maximum amount of reusable water stored at any point within the time horizon of interest as obtained from model M2 (Constraints (5.40)). 

Note, however, there are two critical limitations to these "predicting" procedures. First, the mathematical models must adequately fit the data. Correlation coefficients (R ), adjusted for degrees of freedom, of 0.8 or better are considered necessary for reliable prediction when using factorial designs. Second, no predictions outside the design space can be made confidently, because no data are available to warn of unexpectedly abrupt changes in direction of the response surface. The areas covered by Figures 8 and 9 officially violate this latter limitation, but because more detailed... 

The design q>roblem can be approached at various levels of sophistication using different mathematical models of the packed bed. In cases of industrial interest, it is not possible to obtain closed form analytical solutions for any but the simplest of models under isothermal operating conditions. However, numerical procedures can be employed to predict effluent compositions on the basis of the various models. In the subsections that follow, we shall consider first the fundamental equations that must be obeyed by all packed bed reactors under various energy transfer constraints, and then discuss some of the simplest models of reactor behavior. These discussions are limited to pseudo steady-state operating conditions (i.e., the catalyst activity is presumed to be essentially constant for times that are long compared to the fluid residence time in the reactor). 

Chapter 2 summarizes the characteristics of process models and explains how to build one. Special attention is focused on developing mathematical models, particularly empirical ones, by fitting empirical data using least squares, which itself is an optimization procedure. 

However, if you extend this notion to an extreme and make 100,000 production runs of one unit each (actually one unit every 315 seconds), the decision obviously is impractical, since the cost of producing 100,000 units, one unit at a time, will be exorbitant. It therefore appears that the desired operating procedure lies somewhere in between the two extremes. To arrive at some quantitative answer to this problem, first define the three operating variables that appear to be important number of units of each run (D), the number of runs per year (n), and the total number of units produced per year (Q). Then you must obtain details about the costs of operations. In so doing, a cost (objective) function and a mathematical model will be developed, as discussed later on. After obtaining a cost model, any constraints on the variables are identified, which allows selection of independent and dependent variables. 

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