Expansion Modelling

The molar thermal expansion model of polymers (MTE-model)... 

Inferences from the thermal expansion model 4.4.5.I. Numerical values of the molar thermal expansivities... 

FIG. 4.2 Thermal expansion model of polymers (based on a concept of Simha and Boyer, 1962). 

At the same time we may consider the mutual agreement as a verification of the quantitative applicability of the molar thermal expansion model, given in Fig. 4.2. 

Response surface models are local Taylor expansion models which are valid only in the explored domain. It is often found that the stationary point on the response surface is remote from the explored domain and in the model may not describe any real phenomenon around the stationary point. Mathematically, a stationary point can be a maximum, a minimum, or a saddle point but it sometimes corresponds to unrealistic reponses (e.g. yield > 100%) or unattainable experimental conditions (e.g. negative concentrations of reactants). When the stationary point is outside the explored domain, the response surface is monotonous in the explored experimental domain and zx directions which correspond to small coefficients will describe rising or falling ridges. Exploring such ridges offers a means for optimizing the response even if the response surface should be oddly shaped. 

The multipole expansion model has seen use in the examination of solvation effects on both reaction coordinates and conformational equilibria, including the isomerization of push-pull ethylenes o (e,g., nitroenamines), the ketene-imine [2-f 2]-cycloaddition to form p-lactam,24i and the Diels-Alder reaction.242,243 Again, only the ENP terms are considered in general. 

The aqueous solvation free energies of the four tautomers available to the 5-(2H)-isoxazolone system have also been studied using a variety of continuum models (Table 7). Hillier and co-workers - " have provided data at the ab initio level using the Born-Kirkwood-Onsager model, the classical multipolar expansion model (up to I = 7), and an ab initio polarized continuum model. We examined the same BKO model with a different cavity radius and the AMl-SMl and AMl-SMla o- models, and Wang and Ford have performed calculations with the AMl-PCM model. 

Fig. 7.15. Changes in pressure and mass fraction burned during compression and expansion. Modelled and experimental results compared. From [132].

For recirculation flow the Taylor dispersion mechanism was introduced by Shyu and Miyauchi (S13). Equation (4-12) is a revised result for it. For this flow regime, Ohki and Inoue (02) developed an expansion model with parameters adjusted to the data available, and also introduced the Taylor dispersion mechanism for the low-gas-velocity region of uniform bubble flow. 

As long as one deals with small clusters, beam analysis is possible by combining spectroscopy with expansion modeling. It is possible to use, for example, the soft ionization methods to obtain a better idea of the relative concentration of different clusters in the beam. Soft ionization can be achieved either by direct photoionization or by applying the multiphoton ionization methods (see Cheshnovsky and Leutwyler as a recent example). This technique does not solve completely the fragmentation problem, since if the positively charged species formed is not stable, it will fall apart. However, combining it with sp>ectroscopy, satisfactory results can be obtained. 

Figure 3 shows the eomparison of the normalized bed height from the H-Oil reactor data and ANN model predicted values after two millions training events. The maximum ARD% is 13.8% with an AAD% of 1.92% for the 85 sets of input data employed. However, if only data with bed height values below the allowable upper level are considered, the max. ARD% and AAD% reduce to 10.7% and 1.43% respectively. It is clearly demonstrated the predicted results from the ANN ebullated-bed expansion model are very close to the literature values. This model by no mean limits its applications just to predict the interface level. It can be extended to eover heat generation in terms of exotherms, spread temperature and/or catalyst average temperature (CAT) from data recorded in the technieal report [15]. 

Prediction models for ionic conductivity and viscosity of ILs using quantitative structure property relationships coupled with the descriptors of group contribution type were introduced [155], The polynomial expansion model based on the type of cation, length of side chain, and type of anion was applied to the expression of IL properties. Parameters of these polynomial expansion models were determined by means of a genetic algorithm. The reverse design of ILs was also tested [155],... 

The two-stage clonal expansion model (TSCE) was originally developed by Moolgavkar and Venzon (1979) and Moolgavkar and Knudson (1981). This model assumes that malignant transformation of susceptible stem cells oeeurs as a result... 

Hazelton, W. D., Luebeck, E. G., Heidenreich, W. F., and Moolgavkar, S. H. (2001). Analysis of a historical cohort of Chinese tin miners with arsenic, radon, cigarette smoke, and pipe smoke exposures using the biologically based two-stage clonal expansion model. Radiat Res 156, 78-94. 

Zielinski, J. M., Kodell, R. L., and Krewski, D. (2001). Interaction between two carcinogens in the two-stage clonal expansion model of carcinogenesis. J Epidemiol Biostat 6, 219-228. 

The concept of free volume has been of more limited use in the prediction of solubility coefficients although, Peterlin (H) has suggested that the solubility coefficient is directly proportional to the free volume available in the polymer matrix. In many respects, the free volume expressions closely resemble the relationships developed in the activated state approach. In fact for the case of diffusivity, the two models can be shown to be mathematically equivalent by incorporating thermal expansion models such as the one proposed by Fox and Flory (12). The usefulness of the free volume model however, lies in the accessibility of the fractional free volume, through the use of group contribution methods developed by Bondi (12.) and Sugden (li), for correlation of barrier properties of polymers of different structure as demonstrated by Lee (15.). ... 

The Margules expansion model has been tested on some ionic systems over very wide ranges of composition, but over limited ranges of temperature and pressure (33,34). In this study, the model is applied over a wider range of temperature and pressure, from 25-350 C and from 1 bar or saturation pressure to 1 kb. NaCl and KCl are major solute components in natural fluids and there are abundant experimental data from which their fit parameters can be evaluated. Models based on the ion-interaction ajiproach are available for NaCl(aq) and KCl(aq) (8,9), but these are accurate only to about 6 molal. Solubilities of NaCl and KCl in water, however, reach 12 and 20 m, respectively, at 350 C, and ionic strengths of NaCl-KCl-H20 solutions reach more than 30 m at this temperature (35). The objective of this study is to describe the thermodynamic properties, particularly the osmotic and activity coefficients, of NaCl(aq) and KCl(aq) to their respective saturation concentrations in binary salt-H20 mixtures and in ternary NaCl-KCl-H20 systems, and to apply the Margules expansion model to solubility calculations to 350 C. 

Figure 4. Solubilities of halite (NaCl) in water to 350°C. The curve represents values calculated using the Margules expansion model for activity coefficients (infinite dilution reference state), and standard state Gibbs energies for NaCl(aq) derived from the equations of Pitzer et al. to 300°C, and of Tanger and Helgeson above 300 C.

The lifetime variation of Cr + with pressure has also been considered in alexandrite (BeAl204) [114,249] and several garnets [113,139-141]. Jia et al. [114] observed a non-linear increase in the R-line lifetime of Cr + in the mirror sites of alexandrite from 0.5 ms (Rj, R2) at ambient pressure to 3.5 ms (Rj) and 3.0 ms (R2) at 68 kbar. They attributed the lifetime increase to a decreasing thermal population of the T2 state resulting from an increase in the energy of the T2 state with pressure. Jovanic [2491 reported a similar increase in the R-line lifetime in alexandrite and interpreted the increase in the context of the radial expansion model described in their analysis of ruby [244,2451. 

---