Mathematical models comparison with experiment

Throughout the chapter, the importance of network formation theories in understanding and predicting structural development is stressed. Therefore, a short expose on network formation theories is given in this chapter. Although the use of theoretical modeling of network build-up and comparison with experiments play a central role in this chapter, most mathematical relations and their derivation are avoided and only basic postulates of the theories are stated. The reader can always find references to literature sources where such mathematical relations are derived. 

D. Bradley, P.H. Gaskell and X.J. Gu, Application of a Reynolds Stress, Stretched Flamelet, Mathematical Model to Computations of Turbulent Burning Velocities and Comparison with Experiments, Comb, and Flame 96 (1994) 221. 

COMPUTATIONAL CHEMISTRY is the scientific discipline of applying computers to gain chemical information. It is the link between theoretical and experimental chemistry. Theoretical chemistry is mainly concerned with the development of mathematical models which allow one to derive chemical properties from calculations and to interpret experimental observations. The mathematical models developed in theoretical chemistry are usually validated by comparison with experiment. Theoretical chemistry existed before the arrival of electronic computers. Computational chemistry, however, relies heavily on powerful microelectronics to cope with huge computational tasks. It focuses on the application of theoretical methods which require calculational treatments which are by far too large to be done without fast computers. 

It is necessary to note the essential physical difference between the system (9) and its asymptotic approximation at <5 0 that is the equation (14). The system (9) at finite values S describes the physical waves and is suitable to comparison with experiments. The asymptotic equation (14) simulates the mathematical waves of unbounded length and infinitesimal amplitude and is not included parameters connecting with experimental conditions. This circumstance defines the preference of (9) before (14). It is important to note that mathematical model for nonlinear waves is reduced to single equation in the limiting case (5 0 only. That model includes a system of two equations for finite S values. 

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. 

Detailed quantitative analyses of the data allowed the production of a mathematical model, which was able to reproduce all of the characteristics seen in the experiments carried out. Comparing model profiles with the data enabled the diffusion coefficients of the various components and reaction rates to be estimated. It was concluded that oxygen inhibition and latex turbidity present real obstacles to the formation of uniformly cross-linked waterborne coatings in this type of system. This study showed that GARField profiles are sufficiently quantitative to allow comparison with simple models of physical processes. This type of comparison between model and experiment occurs frequently in the analysis of GARField data. 

Test of the Mathematical Model. In the mathematical model, behavior of the polymer monolayer was related to two parameters the degree of compression, (1-A/A ), and the ratio of the relaxation rate to the compression (expansion) rate, q. The results from three typical hysteresis experiments with three different polymers were chosen for comparison to the theory. 

If the model perfectly describes the experiments, the sample of residual errors does not contain systematic errors thus, it must be compatible with the statistical distribution of the random experimental errors. All the systematic discrepancies eventually observed are attributed to the mathematical model, thus allowing a comparison between alternative models, since systematic errors can be decreased if a better model becomes available. 

The situation here does have a fairly large shadow on it because of the use of the expression (3.120) in ic. It will be seen (Section 3.14) that, at concentrations as high as 1 N, there are some fundamental difficulties for the ionic-cloud model on which this ic expression of Eq. (3.120) was based (the ionic atmosphere can no longer be considered a continuum of smoothed-out charge). It is clear that when the necessary mathematics can be done, there will be an improvement on the VF expression, and one will hope to get it more correct than it now is. Because of this shadow, a comparison of Eq. (3.130) with experiment to test the validity of the model for removing solvent molecules to the ions sheathes should be done a little with tongue in cheek. 

The experimental technique adopted for this work was the measurement of weight gain and loss as a function of time when small amounts of molecular sieve are exposed to gas streams of constant composition in a flow system. Comparison of experimental data with curves based on a suitable mathematical model permitted the determination of effective diffusivities for ethane. Equilibrium data also were obtained from these experiments. The present technique was chosen in preference to fixed-bed studies because of the increased sensitivity offered by the single particle approach. 

Therefore, the solubility in a medium containing 0.1 M bile salt (X[ = 0) and an equimolar ratio of lecithin (X2 = 0) was determined in duplicate (experiments 5 and 5 as described in chapter 4) at the same time as the measurements at the factorial points. (Note that experiments at the test points should be done, if possible, at the same time as the other experiments, all of them in a random order.) The experimentally measured solubilities were 11.70 and 11.04 mg mL , a mean value of 11.37. This is a difference of 1.21 mg mL" with respect to the model calculation of 10.16, which appears large in comparison with the differences observed previously. We therefore believe that the response surface may not be an inclined plane, but a curved surface. We have detected this curvature in the centre of the domain, and we therefore require a more complex mathematical model. This conclusion for the moment is entirely subjective as we have not yet considered any statistical tests. We will demonstrate later on (section II.B) how it is possible to test if this difference is significative and we will show that in such a case it may be attributed to the existance of squared terms in the model. For the moment we will limit ourselves to the conclusion that a more complex mathematical model is necessary. 

The coefficients eja governing the mathematical transformation from normal coordinates to atomic Cartesian coordinates rj provide a transparent description of mode character. The vector eja parallels the motion of atom j in normal mode a, while the squared magnitude describes its relative mean squared amplitude. Normalization, according to J] = 1, then ensures that the mode composition factor is equal to the fraction of mode energy associated with motion of atom j. The resulting KED, together with the directional information, facilitates model-independent comparison of experiments with each other and with computational predictions. 

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