Mathematical modeling example

Theoretical Models of the Response Surface Mathematical models for response surfaces are divided into two categories those based on theory and those that are empirical. Theoretical models are derived from known chemical and physical relationships between the response and the factors. In spectrophotometry, for example, Beer s law is a theoretical model relating a substance s absorbance. A, to its concentration, Ca... 

Models Part of the foundation of statistics consists of the mathematical models which characterize an experiment. The models themselves are mathematical ways of describing the probabihty, or relative likelihood, of observing specified values of random variables. For example, in tossing a coin once, a random variable x could be defined by assigning to x the value I for a head and 0 for a tail. Given a fair coin, the probabihty of obsei ving a head on a toss would be a. 5, and similarly for a tail. Therefore, the mathematical model governing this experiment can be written as... 

Measurement Selection The identification of which measurements to make is an often overlooked aspect of plant-performance analysis. The end use of the data interpretation must be understood (i.e., the purpose for which the data, the parameters, or the resultant model will be used). For example, building a mathematical model of the process to explore other regions of operation is an end use. Another is to use the data to troubleshoot an operating problem. The level of data accuracy, the amount of data, and the sophistication of the interpretation depends upon the accuracy with which the result of the analysis needs to oe known. Daily measurements to a great extent and special plant measurements to a lesser extent are rarelv planned with the end use in mind. The result is typically too little data of too low accuracy or an inordinate amount with the resultant misuse in resources. 

Mathematical models that ignored kinetic forms may fit the experimental results very well but fail to predict critical performance attributes. For example, neglecting the well known exponential form of the Arrhenius fianction made one, entirely mathematical, model fail in predicting the thermal runaway. 

To facilitate the use of methanol synthesis in examples, the UCKRON and VEKRON test problems (Berty et al 1989, Arva and Szeifert 1989) will be applied. In the development of the test problem, methanol synthesis served as an example. The physical properties, thermodynamic conditions, technology and average rate of reaction were taken from the literature of methanol synthesis. For the kinetics, however, an artificial mechanism was created that had a known and rigorous mathematical solution. It was fundamentally important to create a fixed basis of comparison with various approximate mathematical models for kinetics. These were derived by simulated experiments from the test problems with added random error. See Appendix A and B, Berty et al, 1989. 

This brief discussion of some of the many effects and interrelations involved in changing only one of the operating variables points up quite clearly the reasons why no exact analysis of the dispersion of gases in a liquid phase has been possible. However, some of the interrelationships can be estimated by using mathematical models for example, the effects of bubble-size distribution, gas holdup, and contact times on the instantaneous and average mass-transfer fluxes have recently been reported elsewhere (G5, G9). 

Obviously, construction of a mathematical model of this process, with our present limited knowledge about some of the critical details of the process, requires good insight and many qualitative judgments to pose a solvable mathematical problem with some claim to realism. For example what dictates the point of phase separation does equilibrium or rate of diffusion govern the monomer partitioning between phase if it is the former what are the partition coefficients for each monomer which polymeric species go to each phase and so on. 

The first level of complexity corresponds to simple, low uncertainty systems, where the issue to be solved has limited scope. Single perspective and simple models would be sufficient to warrant with satisfactory descriptions of the system. Regarding water scarcity, this level corresponds, for example, to the description of precipitation using a time-series analysis or a numerical mathematical model to analyze water consumption evolution. In these cases, the information arising from the analysis may be used for more wide-reaching purposes beyond the scope of the particular researcher. 

Statistical tests incorporate mathematical models against which reality, perhaps unintentionally or unwillingly, is compared, for example ... 

Mathematical models require computation to secure concrete predictions. Successes in relatively simple cases spurs interest in more complex situations. Somewhat specialized computer hardware and software have emerged in response to these demands. Examples are the high-end processors with vector architecture, such as the Cray series, the CDC Cyber 205, and the recently announced IBM 3090 with vector attachment. When a computation can effectively utilize vector architecture, such machines will out-perform even the most powerful conventional scalar machine by a substantial margin. Such performance has given rise to the term supercomputer. ... 

It is important that chemical engineers master an understanding of metabolic engineering, which uses genetically modified or selected organisms to manipulate the biochemical pathways in a cell to produce a new product, to eliminate unwanted reactions, or to increase the yield of a desired product. Mathematical models have the potential to enable major advances in metabolic control. An excellent example of industrial application of metabolic engineering is the DuPont process for the conversion of com sugar into 1,3-propanediol,... 

Mathematical Models. As noted previously, a mathematical model must be fitted to the predicted results shown In each factorial table generated by each scientist. Ideally, each scientist selects and fits an appropriate model based upon theoretical constraints and physical principles. In some cases, however, appropriate models are unknown to the scientists. This Is likely to occur for experiments Involving multifactor, multidisciplinary systems. When this occurs, various standard models have been used to describe the predicted results shown In the factorial tables. For example, for effects associated with lognormal distributions a multiplicative model has been found useful. As a default model, the team statistician can fit a polynomial model using standard least square techniques. Although of limited use for Interpolation or extrapolation, a polynomial model can serve to Identify certain problems Involving the relationships among the factors as Implied by the values shown In the factorial tables. 

They cannot be part of a mathematical model whose purpose would be to turn the classification into a continuous quantitative variable. In particular, the example of physical factors illustrates this. Whereas for the highest degree criteria are the same as those of the NFPA code, the simple fact of wanting to add in physical factors to these calculation models forced the originators of this technique to forget about the NFPA code. 

Optimisation may be used, for example, to minimise the cost of reactor operation or to maximise conversion. Having set up a mathematical model of a reactor system, it is only necessary to define a cost or profit functionOptimisation and then to minimise or maximise this by variation of the operational parameters, such as temperature, feed flow rate or coolant flow rate. The extremum can then be found either manually by trial and error or by the use of a numerical optimisation algorithms. The first method is easily applied with ISIM, or with any other simulation software, if only one operational parameter is allowed to vary at any one time. If two or more parameters are to be optimised this method however becomes extremely cumbersome. 

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... 

This is rarely the case in engineering. Most of the time we do have some form of a mathematical model (simple or complex) that has several unknown parameters that we wish to estimate. In these cases the above designs are very straightforward to implement however, the information may be inadequate if the mathematical model is nonlinear and comprised of several unknown parameters. In such cases, multilevel factorial designs (for example, 3k or 4k designs) may be more appropriate. 

Let us reconsider the hydrogenation of 3-hydroxypropanal (HPA) to 1,3-propanediol (PD) over Ni/Si02/Al203 catalyst powder that used as an example earlier. For the same mathematical model of the system you are asked to regress simultaneously the data provided in Table 16.23 as well as the additional data given here in Table 16.28 for experiments performed at 60°C (333 K) and 80°C (353 K). Obviously an Arrhenius type relationship must be used in this case. Zhu et al. (1997) reported parameters for the above conditions and they are shown in Table 16.28. 

A number of examples from biochemical engineering are presented in this chapter. The mathematical models are either algebraic or differential and they cover a wide area of topics. These models are often employed in biochemical engineering for the development of bioreactor models for the production of bio-pharmaceuticals or in the environmental engineering field. In this chapter we have also included an example dealing with the determination of the average specific production rate from batch and continuous runs. 

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