Mathematical models, design

Although several physiologically based mathematical models designed to predict absorption properties have been available for several years (see Sect. 21.3), there is still only a limited number of studies describing their evaluation with respect to their ability to predict human Tabs and the relevance of how the input data (e.g., solubility and permeability) is generated in the drug discovery process. 

Despite the great importance of intrinsic kinetic models and the extensive research effort spent on developing sophisticated kinetic models for different important catalytic reactions, it remains the most difficult problem facing the development of rigorous mathematical models (design equations) for industrial fixed bed catalytic reactors (Agnew, 1985). 

The process of classification and building of mathematical models (design equations) has been simply discussed earlier. Here, we give more details about this process which add to the buildup of knowledge for the reader in this direction. 

It is natural that, at this stage, students be introduced to the formulation of some simple mathematical models (design equations) for certain systems. When the students are familiar with the laws governing the rates of different processes in terms of the state variables, they will be ready for an intensive applied course on mathematical modeling of chemical engineering... 

This chapter concentrates on the transformation of material and energy balance equations of homogeneous lumped systems into mathematical models (design equations). 

Fluid phase equilibria and mixing properties are of primary interest for theoretical purposes (mathematical model design, parameter estimatioii, etc.), and for the development of a general proper liquid theory. In chemical irrdrrstiial processes irrvolving liquid mixtures, the optimization and adequate design of separation... 

Generally speaking, mathematical models can describe only relatively simple physical phenomena. Mathematical models designed to describe complex phenomena tend to be so intractable as to be of little value. 

Once the flowsheet structure has been defined, a simulation of the process can be carried out. A simulation is a mathematical model of the process which attempts to predict how the process would behave if it was constructed (see Fig. 1.1b). Having created a model of the process, we assume the flow rates, compositions, temperatures, and pressures of the feeds. The simulation model then predicts the flow rates, compositions, temperatures, and pressures of the products. It also allows the individual items of equipment in the process to be sized and predicts how much raw material is being used, how much energy is being consumed, etc. The performance of the design can then be evaluated. 

K. K. Boon, "A Flexible Mathematical Model for Analy2ing Industrial P. F. Furnaces," M.S. thesis. University of Newcasde, AustraUa, Sept. 1978. R. H. Essenhigh, "A New AppHcation of Perfectly Stirred Reactor (P.S.R.) Theory to Design of Combustion Chambers," TechnicalEeport FS67-1 (u), Peimsylvania State University, Dept, of Euel Science, University Park, Pa., Mar. 1967. 

Transport Models. Many mechanistic and mathematical models have been proposed to describe reverse osmosis membranes. Some of these descriptions rely on relatively simple concepts others are far more complex and require sophisticated solution techniques. Models that adequately describe the performance of RO membranes are important to the design of RO processes. Models that predict separation characteristics also minimize the number of experiments that must be performed to describe a particular system. Excellent reviews of membrane transport models and mechanisms are available (9,14,25-29). 

Considerable work has been done on mathematic models of the extmsion process, with particular emphasis on screw design. Good results are claimed for extmsion of styrene-based resins using these mathematical methods (229,232). With the advent of low cost computers, closed-loop control of... 

Classification Process simulation refers to the activity in which mathematical models of chemical processes and refineries are modeled with equations, usually on the computer. The usual distinction must be made between steady-state models and transient models, following the ideas presented in the introduction to this sec tion. In a chemical process, of course, the process is nearly always in a transient mode, at some level of precision, but when the time-dependent fluctuations are below some value, a steady-state model can be formulated. This subsection presents briefly the ideas behind steady-state process simulation (also called flowsheeting), which are embodied in commercial codes. The transient simulations are important for designing startup of plants and are especially useful for the operating of chemical plants. 

Cycles Design methods for cycles rely on mathematical modeling (or empiricism) and often extensive pilot plant experiments. Many cycles can be easily analyzed using the methods described above apphed to the collection of steps. In some cycles, however, especially those operated with short cycle times or in shallow beds, transitions may not be very fully developed, even at a periodic state, and the complexity may be compounded by multiple sorbates. 

Some investigators have proposed, mostly on the basis of mathematical modeling, to optimize the design of scrubbers to obtain a given efficiency with a minimum power consumption (e.g., Goel and Hollands, op. cit.). In fact, no optimum in performance appears to exist apart from some avoidable regions of unfavorable operation, increased contacting power yields increased efficiency. 

On the base of the developed mathematical models was developed regression model of the atomizer efficiency via main design pai ameters such as linear dimensions and operation temperatures. 

In control engineering, the way in which the system outputs respond in changes to the system inputs (i.e. the system response) is very important. The control system design engineer will attempt to evaluate the system response by determining a mathematical model for the system. Knowledge of the system inputs, together with the mathematical model, will allow the system outputs to be calculated. 

Catalytic crackings operations have been simulated by mathematical models, with the aid of computers. The computer programs are the end result of a very extensive research effort in pilot and bench scale units. Many sets of calculations are carried out to optimize design of new units, operation of existing plants, choice of feedstocks, and other variables subject to control. A background knowledge of the correlations used in the "black box" helps to make such studies more effective. 

Design by experiment - a technique where product characteristics are established by conducting experiments on samples or by mathematical modeling to simulate the effects of certain characteristics and hence determine suitable parameters and limits. 

Mathews and Rawlings (1998) successfully applied model-based control using solids hold-up and liquid density measurements to control the filtrability of a photochemical product. Togkalidou etal. (2001) report results of a factorial design approach to investigate relative effects of operating conditions on the filtration resistance of slurry produced in a semi-continuous batch crystallizer using various empirical chemometric methods. This method is proposed as an alternative approach to the development of first principle mathematical models of crystallization for application to non-ideal crystals shapes such as needles found in many pharmaceutical crystals. 

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