Need for mathematical modeling

Possibly the chemical industry does not have as much need for mathematical models in process simulation as does the petroleum refining industry. The operating conditions for most chemical plants do not seem subject to as broad a choice, nor do they seem to require frequent reappraisals. However, this is a matter which must be settled on the basis of individual circumstances. Sometimes the initial selection of operating conditions for a new plant is sufficiently complex to justify development of a mathematical model. Gee, Linton, Maire, and Raines describe a situation of this sort in which a mathematical model was developed for an industrial reactor (Gl). Beutler describes the subsequent programming of this model on the large-scale MIT Whirlwind computer (B6). These two papers seem to be the most complete technical account of model development available. However, the model should not necessarily be thought typical since it relies more on theory, and less on empiricisms, than do many other process models. 

By changing Fc the system s steady states are changed and consequently the gasoline yield changes, too. This shows the complexity of the interaction between the variables of the system and a need for mathematical models to optimize the operation of FCC units. The most useful models must be able to account for all varying parameters simultaneously. 

McNamara, J.P., J.E. Pettigrew, R.L. Baldwin, B. Walker, W.H. Close and J.W. Oltjen, 1991. Information needed for mathematical modelling of energy use by animals. In Eneigy Metabolism in Farm Animals, EAAP Pub. No. 58, Gruppe Emahrung, Zurich Switzerland 468-472,. 

An example of the use of this method with the constant capacitance model on the data for TiC>2 in 0.1 M KNO is illustrated in Figure 6. It appears from the figure that the problem is perfectly well determined, and that unique values of Ka and Ka2 can be determined. However, as is shown below, the values of Ka and Ka2 determined by this method are biased to fulfill the approximations made in processing the data (i) on the acidic branch, nx+, nx nx-, which yields a small value for Ka2, and (ii) on the basic branch, nx-, nx nx+, which yields a large value of Ka. Thus the approximation used to find values for Qa and Qa2 leads to values of Ka and Ka2 consistent with the approximation of a large domain of predominance of the XOH group. This constraint arose out of the need for mathematical simplicity, not out of any physical considerations. 

In terms of computer-based and chemometric approach, additional improvements were also needed in mathematical models for chromatography and in method development, in order to help identifying the correct type of model and the adequate experimental parameters then, application to high volume of generated data is possible. 

More commonly, we are faced with the need for mathematical resolution of components, using their different patterns (or spectra) in the various dimensions. That is, literally, mathematical analysis must supplement the chemical or physical analysis. In this case, we very often initially lack sufficient model information for a rigorous analysis, and a number of methods have evolved to "explore the data", such as principal components and "self-modeling analysis (21), cross correlation (22). Fourier and discrete (Hadamard,. . . ) transforms (23) digital filtering (24), rank annihilation (25), factor analysis (26), and data matrix ratioing (27). 

From a pragmatic viewpoint, there is no need for a model of the photon. One may be content with a description of the particle based entirely on the equations that it obeys. This is a very respectable scientific stance. There is another equally respectable scientific position—try to understand the mathematical equations in relation to a physical model. In previous paragraph we mentioned the attempts of several investigators [30-33]. More recent trials are those of Warburton [34], Fox [35], Scully and Sargent [36], Hunter and Wadlinger [37,38], Evans and Vigier [39], Barbosa and Gonzalez [40], and Lehnert [41]. For additional contemporary models see Hunter et al. [42]. 

Besides the mathematical improvements, the atmospheric model has been adapted to a semi-Lagrangian formulation. By following selected air masses, we avoid commitments of large quantities of memory and the incursions of artificial diffusion errors. Most important, we do not end up with stacks of computer printout that relate to regions where there are no measurements. Also, predictive calculations will become more useful, but our present levels of resources and sophistication demand that effort be concentrated on verification. Only in this way can the confidence be built that is needed for applying modeling techniques to implementation planning. 

To assess the feasibility of the BSR as a competitor of the monolithic reactor, the parallel-passage reactor, and the lateral-flow reactor, it is necessary to do case studies in which the performance and price of these reactors are compared, for certain applications. To allow such case studies, two tools are needed (1) mathematical models of the reactors that predict the reactor performance, and (2) an optimization routine that, given a mathematical reactor model and a set of process specifications, finds the optimum reactor configuration. Furthermore, data are needed on costs, safety, availability, etc. In this section, five mathematical models of different complexity for the bead-string reactor (BSR) are presented that can be numerically solved on a personal computer within a few hours down to a few minutes. The implementation of the reactor models in an optimization routine, as well as detailed cost analyses of the reactor, are beyond the scope of this text. 

The importance of molecular weight distribution in studies of polymerization, polymer processing and the physical and mechanical properties of polymers creates a need for mathematical description of the distribution. Several models are commonly used (Flory [1], Schulz-Zimm... 

Compartmental or media box models offer an alternative practical approach. They are derived from applying integrated forms of the CE. These involve volume and area integrals over the boxes. The volume integrals sum the mass accumulation and reaction terms while the area integrals direct the flux terms to account for the movement of chemicals between the boxes. Typically, the result is a set of linear ordinary differential equations capable of mathematically mimicking many of the key dynamic and other features of the chemodynamics in natural systems. This handbook provides the mass transport parameters needed for both model types. 

In this section we consider the boundary value problem for model equations of a thermoelastic plate with a vertical crack (see Khludnev, 1996d). The unknown functions in the mathematical model under consideration are such quantities as the temperature 9 and the horizontal and vertical displacements W = (w, w ), w of the mid-surface points of the plate. We use the so-called coupled model of thermoelasticity, which implies in particular that we need to solve simultaneously the equations that describe heat conduction and the deformation of the plate. The presence of the crack leads to the fact that the domain of a solution has a nonsmooth boundary. As before, the main feature of the problem as a whole is the existence of a constraint in the form of an inequality imposed on the crack faces. This constraint provides a mutual nonpenetration of the crack faces ... 

Both the need to reduce experimental costs and increasing reHabiHty of mathematical modeling have led to growing acceptance of computer-aided process analysis and simulation, although modeling should not be considered a substitute for either practical experience or reHable experimental data. 

Even when a total system analysis is unnecessai y, the methodology of mathematical modeling is useful, because by considering each component of a system as a block of a flow sheet, the interrelationships become much clearer. Additional alternatives often become apparent, as does the need for more equipment-performance data. 

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. 

We need a mathematical representation of our prior knowledge and a likelihood function to establish a model for any system to be analyzed. The calculation of the posterior distribution can be perfonned analytically in some cases or by simulation, which I... 

Over the years there have been many attempts to simulate the behaviour of viscoelastic materials. This has been aimed at (i) facilitating analysis of the behaviour of plastic products, (ii) assisting with extrapolation and interpolation of experimental data and (iii) reducing the need for extensive, time-consuming creep tests. The most successful of the mathematical models have been based on spring and dashpot elements to represent, respectively, the elastic and viscous responses of plastic materials. Although there are no discrete molecular structures which behave like the individual elements of the models, nevertheless... 

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