The Value of Models

Just as the scientific method is fundamental to the woik of scientific research, mathematical modeling is fnndamental to engineering research. A mathematical model may be simple or complex it may consist of no more than one mathematical equation or may involve hundreds of equations. Its subject may be a comprehensive overview of a total system or it may be the tiniest piece of a microminiature subsystem. 

The archival function of models implies that they should also serve a valuable teaching function, as indeed they do in the physical sciences. Dynamic respiratory models, especially in their computerized interactive format, should be very valuable in teaching physiologists, medical students, and physicians the essence of normal and pathological pulmonary physiology. 

Finally, models provide a mechanism for rigorously exploring the observable implications of physiological hypotheses and thus can help to design experiments to test them. Investigators must know what a particular hypothesis commits them to in terms of experimental observations before they can test it. 

In a complex system with many interacting variables which cannot be experimentally isolated, rigorous modeling may be the only way to obtain them. Such predictions may sometimes turn out to be unexpected and counterintuitive. If they survive an exhausting recheck of model formulation and computation, this surprising behavior of models is one of their most valuable attributes in hypothesis testing. 

Starfield et al. (1990) state that mathematical models are like caricatures they overly emphasize some aspects at the expense of others to make conspicuous those results due to the emphasized aspects. Thus, models are not always general descriptions of a phenomenon. Indeed, a thorough mathematical description of some scientific phenomenon would be as complicated as the original phenomenon itself, and serve very little purpose. It is often difficult for a scientist to truly believe what value is contained in a model that does not predict all scientific observations related to a particular phenomenon. 

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Table 2 shows transition moments calculated by the different EOM-CCSD models. As has been discussed above, the right-hand transition moment 9 is size intensive but the left-hand transition moment 9 in model I and model II is not size intensive. Model II is much improved as far as size intensivity is concerned because of the elimination of the apparent unlinked terms. The apparent unlinked terms are a product of the size-intensive quantity ro and size-extensive quantities and therefore are size extensive. The difference between the values of model I and model II, as summarized in the fifth column, reveals strict size extensivity. Complete elimination of unlinked diagrams by using A amplitudes brings strict size intensivity for the transition moment and therefore the transition probabilities calculated by model III are strictly size intensive. 

ECETOC, The Value of Model Ecosystem Studies in Ecotoxicology. Technical Report, Brussel, 1997. 

For the values of model parameters given in the capture to Fig. 6 we shall estimate now the following quantities ... 

Blum, U., 1995. The value of model plant-microbe-soil systems for understanding processes associated with allelopathic interactions One example. In Inderjit, Dakshini, K. M. M., Einhellig, F. A. (Eds.), Allelopathy organisms, processes, and applications. ACS Symposium Series No. 582. American Chemical Society, Washington D.C, 127-131... 

When mathematical models are used to draw inferences, the values of the model parameters may be a source of uncertainty. As the values of model parameters are not measured by direct observation (they are estimated as part of the model fitting process), the uncertainty of a parameter cannot be characterized by simply recording variability in a series of measurements. However, once the best model criterion has been established, the variability associated with a parameter can be linked to the variability in the data. If standard statistical assumptions are employed, the variability of and correlations among the parameters may be calculated directly. 

Through the use of a model for a batch reactor for a particularly complex reaction, we have demonstrated the value of modeling in optimization of process conditions and in evaluation of possible hazards. For a very complex system like the present one, it is most probably easier and more cost effective to do the modeling than to run the experiments needed for proper analysis. To save laboratory data acquisition time, it is always better to plan an experimental strategy based on the anticipated need in advance. This model has been successfully used in two scale-ups. Data from these scale-ups have been used to refine the model. These refinements included a better understanding of the chemistry of the process. Plots similar to the ones presented in Figures 6-10 were used in the Reactive Chemicals Review of the present process. 

The value of model potentials is not the availability of analytic formulas for vibrational overlap integrals rather, it is the ability to pull together fragmentary information of diverse types into a coherent picture in order to test the observable consequences (location and strength of other perturbations) of alternative assignment schemes (Section 5.1.5). RKR and numerical integration comprise the most convenient way to obtain vi vj) factors and other quantities derivable from V(R) [e.g., centrifugal distortion constants (Albritton, et al, 1973)]. 

Figure 3 shows an adjustment of the Bertholon model made from a sample of 100 simulated values. The curves show the distribution functions of the theoretical model, the experimental model (simulated data) and the fitted model. We get approximately the values of model parameters used for simulation. 

Again, we find approximately the values of model parameters used for simulation as shown in Fig. 4. 

This appendix contains a series of computer simulations that are thought to represent the most significant basic kinetic models in bioprocessing. The models are summarized in Table 11.1. The simulations in the figures also contain the values of model parameters chosen for demonstration. Mainly two different kinds of plots are presented, the first showing concentration/time curves and the second the corresponding time curves of specific rates of bioprocesses. The models are as follows ... 

Using this Equation 3.308 and the values of mean and variance calculafed from the E-curve data, the value of model parameter N is estimated for the given reaction vessel. It may be seen from Equation 3.308 that the variance (o = 0) is minimum for an ideal PER (N = CXD ) and is maximum (o = 9 ) for an ideal CSTR N = 1). Although N is defined as a whole number, that is, integer, it can also take a fractional value. 

To emphasize the value of model integration to obtain comprehensive and robust configuration decisions. 

The values of model parameters at 300°C are listed in Table I. The negative values of the apparent adsorption constant K. indicate that the product butanes and hydrogen sulphide are more strongly adsorbed than thiophene. 

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