Transport properties correlation schemes

The topic of this chapter is the description of a quantum-classical approach to compute transport coefficients. Transport coefficients are most often expressed in terms of time correlation functions whose evaluation involves two aspects sampling initial conditions from suitable equilibrium distributions and evolution of dynamical variables or operators representing observables of the system. The schemes we describe for the computation of transport properties pertain to quantum many-body systems that can usefully be partitioned into two subsystems, a quantum subsystem S and its environment . We shall be interested in the limiting situation where the dynamics of the environmental degrees of freedom, in isolation from the quantum subsystem [Pg.521]

We will deal with the permeability in greatest detail, because it is the quantity of most direct interest in applications. In developing a more fundamental theoretical understanding of transport, however, it will be crucial to consider the diffusivity and the solubility separately. Many of the shortcomings of simple structure-property relationships for the permeability and selectivity may possibly be overcome by a more fundamental understanding, which may therefore also be useful in future refinements and practical applications of correlative schemes. 

In conclusion, Kolmogorov s theory is a realistic description of decay of turbulence. However, its application in a sophisticated form of dimensional analysis (Middleman 1965) is stretching the theory beyond its aim. Such attempts and the resulting correlations cannot lead to a unique scheme for predicting transport properties. 

When the above methods fail, estimation methods become important. Schemes based on the Corresponding-States Principle which are particularly important in this respect are described. In order to demonstrate clearly just when the methods of correlation, the theoretical expressions and estimation techniques are applicable, examples are given of transport-property data representation for systems of different complexity simple monatomic fluids, diatomic fluids, polyatomic fluids (specifically, water and refrigerant R134a), nonreacting mixtures and (dilute) alkali-metal vapors as an example of a reacting mixture. 

Rapid access to transport property data is essential for the efficient use of proposed correlation and prediction schemes. As a result, experimental data have been stored in many data banks worldwide and the final section of this volume describes a number of... 

In order to be able to develop practical correlation schemes for the transport properties, as detailed below, it is convenient to introduce the coefficient of self-diffusion that is related to an effective cross section by... 

To apply the above scheme, accurate experimental measurements for the transport properties of the monatomic fluids were collected. In Table 10.1 the experimental measurements of diffusion, viscosity and thermal conductivity used for the correlation scheme are shown. This table also includes a note of the experimental method used, the quoted accuracy, the temperature range, the maximum pressure and the number of data sets. The data cover the range of compressed gas and the liquid range but not the critical region, where there is an enhancement (Chapter 6) which cannot be accounted for in terms of this simple molecular model. 

The correlation scheme has been applied to seven simple aromatic hydrocarbons benzene, toluene, o- m-, and p-xylene, mesitylene and ethylbenzene (Assael etal. 1992d). In addition to the large number of data used in the development of the correlation, new measurements can now be included (Kaiser et al. 1991 Krall et al. 1992 Harris et al. 1993 Yamada et al. 1993). The temperature and pressure ranges of the data used in this analysis are typically from 250 to 400 K and from 0.1 to 400 MPa. Transport coefficient measurements up to high pressures are available for most of these compounds. The roughness factors Rx were found to be temperature-independent, as in the case of alkanes. Slight adjustments were made for each compound, so that the Rx factors have a constant value and Vq is property-independent and decreases smoothly with... 

Dense fluid transport property data are successfully correlated by a scheme which is based on a consideration of smooth hard-sphere transport theory. For monatomic fluids, only one adjustable parameter, the close-packed volume, is required for a simultaneous fit of isothermal self-diffusion, viscosity and thermal conductivity data. This parameter decreases in value smoothly as the temperature is raised, as expected for real fluids. Diffusion and viscosity data for methane, a typical pseudo-spherical molecular fluid, are satisfactorily reproduced with one additional temperamre-independent parameter, the translational-rotational coupling factor, for each property. On the assumption that transport properties for dense nonspherical molecular fluids are also directly proportional to smooth hard-sphere values, self-diffusion, viscosity and thermal conductivity data for unbranched alkanes, aromatic hydrocarbons, alkan-l-ols, certain refrigerants and other simple fluids are very satisfactorily fitted. From the temperature and carbon number dependency of the characteristic volume and the carbon number dependency of the proportionality (roughness) factors, transport properties can be accurately predicted for other members of these homologous series, and for other conditions of temperature and density. Furthermore, by incorporating the modified Tait equation for density into... 

Although some successful models exist for the prediction of transport properties in the liquid region (see Chapters 5 and 10), it has not proved possible, so far, to incorporate these semiempirical formulations into a scheme for correlating the excess transport property covering the complete dense fluid range. Hence, the development of the excess transport properties for ethane, like that for other polyatomic fluids, is based entirely on fitting experimental data. 

As a consequence of the size limitations of the ab initio schemes, a large number of more-approximate methods can be found in the literature. Here, we mention only the density functional-based tight binding (DFTB) method, which is a two-center approach to DFT. The method has been successfully applied to the study of proton transport in perov-skites and imidazole (see Section 3.1.1.3). The fundamental constraints of DFT are (i) treatment of excited states and (ii) the ambiguous choice of the exchange correlation function. In many cases, the latter contains several parameters fitted to observable properties, which makes such calculations, in fact, semiempirical. 

The significance, if any, of these complexes in sugar transport is not yet understood. The specificity pattern however has some suggestive correlations with those observed for transport, and the complexes may have some secondary role in determining the overall specificity (similar to that perhaps played by the hypothetical transporter or T substance of Figure 2) in the overall proposed scheme for the permease system. Considered in this sense the primary specificity of the system would be determined by the permease protein (P) in accelerating the formation of the substrate-transporter complex, but the overall specificity of the system would reflect the properties of all components. 

We found earlier that a theoretical consideration of displacement and transport exerts a unifying influence on separation science, bringing diverse methods under a common descriptive umbrella. The theory leads in a natural way to the formation of categories of separations which can be considered the beginning of a fundamental classificatory system. Here we generalize the results of transport theory to develop a fundamental basis for classification. While the resulting scheme will not be a complete polythetic classification, it will be based upon some of the most fundamental features of the separation process. These basic features, incorporated in the classification, should correlate well with other properties of separations in the same way that the number of outer-shell electrons is directly related to the diverse properties of the elements of the periodic table. This transport-oriented... 

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