Mathematical modeling general considerations

The examples that are presented above are meant to illustrate the possibilities and limitations of various assays for constituents of complex networks. Although analytical expressions in general may be rather complicated, conditions frequently can be found that considerably simplify the results. Thus, appropriate mathematical models can yield important insights into assay design. 

Without definite examples to focus our thinking, it is easy to get entangled in a quasi-philosophical discussion of just what a mathematical model might be. To avoid this, I present a dogmatic statement on modeling and proceed to consider an elementary example, returning later to the philosophical caveats and more general considerations. 

There are many types of chemical reactors which operate under various conditions, such as batch, flow, homogeneous, heterogeneous, steady state, etc. Thus, one general mathematical description which would apply to all types of reactors would be extremely complex. The general approach for reactor design, therefore, is to develop the appropriate mathematical model which will describe the specific reaction system for that particular form of reactor under consideration. For example, if the reaction system is to be evaluated for steady-state... 

When models that describe oscillating reactions are presented, one can distinguish between abstract mechanisms, representing no specific observable reaction, and models for particular reactions that contain additional steps describing surface processes unique to the experimental system under consideration. In some cases, an abstract mechanism might have been developed for a certain reaction, but can be easily generalized to other cases, because no unique, system-specific steps are involved. These abstract mathematical models have become prototypes for classes of oscillation mechanisms often referred to in publications wherein a more detailed model for a certain reaction has been developed. 

The purpose of the Section 3.1 is to analyze the implications for mathematical models, each of a practical value, because of the introduction of this force term. A number of known fluid mechanics models can be generalized in such a way. Their consideration highlights most of the properties of the obstructed flows, allows the verification of this theoretical approach, and is a useful methodological tool for teaching and learning the general problem. 

In section C we summarize the recent contributions under subsections. The general considerations on the mechanism of the B-Z reaction are in four subsections. Following these general considerations, we have sections C.2. Mathematical models and techniques, C.3. Experiments with different substrates, C.4. Experiments with different catalysts, and C.5. Horatian oscillations in bromate oxidations. The name horatian has recently been proposed for replacing the term chaotic. Here we simply state that throughout this article instead of chaotic oscillations, the term horatian oscillations will be used. 

To employ statistics, the first step is to develop a mathematical model for the unit or process. With a well-established process that has been operating for a considerable period of time, a mathematical model that is highly fundamental based on well-determined steps or reactions can often be developed. For example, Hougen and Watson [16] many years ago showed how the numerous steps in catalytic reactions can be modeled in a relatively fundamental manner. For distillation units, relatively fundamental models can generally be developed to indicate how changes of the following affect operations reflux ratio, process conditions, cost of feed streams, product and feed compositions, demand for product, etc. 

A considerable amount of work has been published during the past 20 years on a wide variety of emulsion polymerization and latex problems. A list of 11, mostly recent, general reference books is included at the end of this chapter. Areas in which significant advances have been reported include reaction mechanisms and kinetics, latex characterization and analysis, copolymerization and particle morphology control, reactor mathematical modeling, control of adsorbed and bound surface groups, particle size control reactor parameters. Readers who are interested in a more in-depth study of emulsion polymerization will find extensive literature sources. 

The selection of the most appropriate assumptions is critical in any environmental risk assessment. Often, as few as one, two or three assumptions or factors can dominate the results of the assessment. Clearly, in the health assessment of dioxin in soil, the most critical parameters are the quantity of soil ingested by children and adults, the results of the mathematical modeling of the bioassay data, consideration for dioxin s lack of genotoxicity, and dioxin s lower bioavailability in a soil matrix. Other assumptions used in the assessment are also important but generally they do not alter the results by 2-3 orders of magnitude as do those which have been discussed. 

Prediction by first and second degree models simplified approach Some statistical considerations in RSM General mathematical model for RSM... 

The apparent simplicity of this approach is, however, deceptive. For measurement of intracrystalline diffusion the method works well when diffusion is relatively slow (large crystals and/or low diffusivity), but when sorption rates are rapid the uptake rate may be controlled by extracrystalline diffusion (through the interstices of the adsorbent bed) and/or by heat transfer. The intrusion of such effects is not always obvious from the shape of the uptake curve, but it may generally be detected by changing the sample quantity and/or the sample configuration. It is in principle possible to allow for such effects in the mathematical model used to interpret the uptake curves (Fig. 2), and indeed the modeling of nonisothermal systems has been studied in considerable detail [8-12]. However, any such intrusion will obviously diminish the accuracy and confidence with which the intracrystalline diffusivities can be determined. 

---