Tait model

The analytical model that was used to do the parameter studies discussed in this chapter retains the essential assumptions of the Bethe-Tait model. These are (a) that the pressure tending to disassemble the core as a function of time can be calculated from the energy density of the fuel at that point and the initial material density, and (b) that the reactivity effects... 

APPENDIX A MODIFIED BETHE-TAIT MODEL A. Introduction... 

FIGURE 14.7 PVT plots for PLA polymers. Top 7000D (NatureWorks LLC, 4.25% D-isomer) [53] bottom Lacea H-IOOE (Mitsui Chemicals, Inc., % D-isomer unknown) [54]. The continuous lines represent the fitted results based on the two-domain-modified Tait model (Equation 14.1). 

The PVT relationship can be modeled mathematically using the modified two-domain Tait model [55-57, 112] ... 

The estimated parameter values for the modified two-domain Tait model are shown in the inserts in Figure 14.7. This model is often used for numerical simulation of injection molding processes involving finite element analysis for predicting the shrinkage behavior of injection molded articles. 

The equation derived from the Tait model is of the form... 

Burmell, P. L., Tait, D. E. N., Flanagan, P. W. and Van Cleve, K. (1977). Microbial respiration and substrate weight loss, I. A general model of the influences of abiotic variables. Soil Biol. Biochem. 9, 33-40. 

Jaenicke L, Starr RC (1996) The lurlenes, a new class of plastoquinone-related mating pheromones from Chlamydomonas allensworthii (Chlorophyceae). Eur J Biochem 241 581-585 Janowitz GS, Kamykowski D (2006) Modeled Karenia brevis accumulation in the vicinity of a coastal nutrient front. Mar Ecol Prog Ser 314 49-59 Joint I, Tait K, Callow ME, Callow JA, Milton D, Williams P, Camara M (2002) Cell-to-cell communication across the prokaryote-eukaryote boundary. Science 298 1207 Kinzie RA (1974) Experimental infection of aposymbiotic gorgonian polyps with zooxanthellae. J Exp Mar Biol Ecol 15 335-342... 

Van Rossum, A.J., Brophy, P.M., Tait, A., Barrett, J. and Jefferies, J.R. (2001) Proteomic identification of glutathione S-transferases from the model nematode Caenorhabditis elegans. Proteomics 1, 1463-1468. 

A widely accepted form of modeling the density or specific volume is the Tait equation. It is often used to represent the pvT-behavior of polymers and it is represented as,... 

In a numerical solution, we can include temperature dependent density and thermal conductivity. The temperature dependent density can be modeled interpolating throughout a pvT diagram. The temperature dependence of the thermal conductivity is not always available, as is the case for many properties used in modeling. Chapter 2 presents the Tait equation, which can be used to model the pvT behavior of a polymer. 

The Tait equation predicts that the isothermal compressibility of a liquid approaches zero as the pressure becomes infinitely large. Is this reasonable in terms of the van der Waals model of real fluids ... 

Leyendekkers J. V. and Hunter R. J. (1976) The Tammann-Tait-Gibson model for aqueous electrolyte solutions application to the refractive index. J. Phys. Chem. 81, 1657-1663. 

Chang, R. Y, Chen, C. H., and Su, K. S., Modifying the Tait equation with cooling-rate effects to predict the pressure-volume-temperature behaviors of amorphous polymers modeling and experiments, Polym. Eng. Sci., 36, 1789-1795 (1996). 

Since the spectra of ferric agrobactin and ferric parabactin do show minor differences, a number of experiments were tried in which an attempt was made to mimic these differences via examination of the 1 3 ferric complexes of model salicyl and 2,3-dihy-droxyphenyloxazolines and the 1 1 1 ferric complexes of the model oxazolines with Tait s "Compound II", -bis-(2,3-dihyd oxyben-... 

The Kirkwood-Bethe-Gilmore (KBG) model is the most elegant model so far. It incorporates an equation of state for the liquid, the so-called Tait equation that describes the pressure at the bubble wall p in terms of the density of the liquid at the bubble wall compared to the equilibrium density po. 

Starting from a different viewpoint, one can reach empirically justified EoS by phenomenological arguments used to build universal behaviors (functions) and fit them to experimental PVT data. The most common such approach for polymers is an isothermal compressibility V-P model (Tait 1888) known as the Tait equation ... 

The results discussed in this section were obtained using the modified Bethe-Tait spherical model to analyze an equivalent spherical version of the cylindrical reactor core described in Section III. The equivalent... 

It would be expected that the results of a calculation for a cylindrical core with a diameter-to-height DjH) ratio of 1 would be similar to that of a spherical core having the same core volume. Calculations were performed using the spherical and cylindrical versions of the modified Bethe-Tait method, and the results show good agreement between the two models for this case. One example of such a comparison is shown in... 

A spherical version of the basic equations used in the modified Bethe-Tait calculations is summarized below. The simplicity of this model (compared to the cylindrical model) makes it especially attractive for doing parameter studies. In addition, the results obtained with the spherical model compare favorably (see Section V) with those obtained using the more complicated cylindrical model. The set of equations given below includes a heat of fusion correction for the Doppler feedback calculations and a calculation of the explosive energy release. (The latter calculation is not coupled directly to the total energy calculation.)... 

Redlich-Kwong equation of state and Soave modification Peng-Robinson equation of state Tait equation for polymer liquids Flory, Orwoll, and Vrij models Prigogine square-well cell model Sanchez-Lacombe lattice fluid theory... 

An empirical representation of PVT data of polymer systans is given by the Tait equation. It can be viewed as an isothermal compressibility model. The general form of the Tait equation can be written as follows ... 

In Figure 2.3 the isotherms are shown at 25°C and 100°C for polystyrene in the pressure volume diagram for polystyrene as predicted by the Tait equation given in Table 2.1. It can be seen that the model does not predict the change at the glass transition temperature of the polymer. Further, the effects of molecular weight of the polymer are also not taken into account. The specific volume as a function of pressure for polystyrene in shown in Figure 2.3. 

Au contraire to the empirical equation of Tait for EOS predictions, theoretical models can be used but generally require an understanding of forces between the molecules. These laws, strictly speaking, need be derived from quantum mechanics. However, Lenard-Jones potential and hard-sphere law can be used. The use of statistical mechanics is an intermediate solution between quantum and continuum mechanics. A canonical partition function can be formulated as a sum of Boltzmann s distribution of energies over all possible states of the system. Necessary assumptions are made during the development of the partition function. The thermodynamic quantities can be obtained by use of differential calculus. For instance, the thermodynamic pressure can be obtained from the partition function Q as follows ... 

EOS theories for polymer liquids were grouped into three categories (1) lattice-fluid theory, (2) the hole model, and (3) the cell model. The Tait equation was first developed 121 years ago in order to fit compressibility data of freshwater and seawater. The Tait equation is a four-parameter representation of PVT behavior of polymers. Expressions for zero pressure isotherms and Tait parameters were provided, and values of Tait parameters for 16 commonly used polymers were tabulated. 

Leyendekkers and Himter (1977a,b) and Leyendekkers (1979) have applied the TTG (Tammann-Tait-Gibson) model to the prediction of the viscosity of aqueous electrolyte solutions at high pressures. According to this scheme, the value of the viscosity of water in solution is given by the relation... 

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