Initial density dependence

Presently, we are initiating density-dependent experiments on this system, elucidating the origin of the distribution, and determining if the observed results are probe dependent. 

Strehlow, T., and Vogel, E., Temperature Dependence and Initial Density Dependence of the Viscosity of Sulfur Hexafluoride, Physica A 161,101,1989,... 

This present volume, which is complementary to the previous publication, discusses the present state of theory with regard to the dilute-gas state, the initial density dependence, the critical region and the very dense gas and liquid states for pure components and mixtures. In all cases, the intention is to present the theory in usable form and examples are given of its application to nonelectrolyte systems. This will be of particular use to chemical and mechanical engineers. The subtitle of this volume Their correlation, prediction and estimation reflects the preferred order of rqrplication to obtain accurate values of transport properties. Careful correlation of accurate experimental data gives reliable values at interpolated temperatures and pressures (densities), and at different compositions when the measurements are for mixtures. Unfortunately, there are only a limited number of systems where data of such accuracy are available. In other cases, sound theoretical methods are necessary to predict the required values. Where information is lacking - for intermolecular forces, for example - estimation methcxls have to be used. These are of lower accuracy, but usually have more general tq)plicability. 

However, important information on the initial density dependence can be obtained by calculation of the Bx coefficients (Friend Rainwater 1984 Rainwater Friend 1987). This is described in Section 5.2. For higher gas densities, use is made of the observation that the transport properties show a regular dependence on density (see Figure 5.2). When the excess property AX, equal to X p, T) — X T) — AXc p, T), is plotted versus density the points fall very close to a single curve. The assumption that the excess transport properties can be described as a function of density alone has been a useful basis for correlating data within stated density and temperature intervals. It... 

Recent advances in the theoretical description of the initial density dependence of the transport properties justify a separate treatment. If moderately dense gases are considered, only the linearized equations (5.1) are needed that is, the virial form of the density expansion can be truncated after the term linear in density. This means that the deviation from the dilute-gas behavior can be represented by the second transport virial coefficients Bx or alternatively by the initial-density coefficients which are... 

In general, there are two methods that can be applied in order to describe Bx(T) (x = T], X) The most up-to-date theory, proposed by Friend Rainwater (1984 Rainwater Friend 1987), models the moderately dense gas as a mixture of monomers and dimers which interact according to the Lennard-Jones (12-6) potential. Besides the fact that this potential is only a rough approximation of the real physical situation, this model has the disadvantage that it has not yet been extended to describe the internal contribution to the initial density dependence of thermal conductivity. 

The initial density dependence of the thermal conductivity of a polyatomic gas is given by expression (5.7). Neither the Rainwater-Friend model nor the modified Enskog theory accounts for the contribution of internal degrees of freedom, but it is assumed that this can be modeled as a purely diffusive process following an idea originally introduced by Mason Monchick (1962) for dilute gases... 

Calculation of the initial density dependence of the thermal conductivity for polyatomic gases requires knowledge of the translational and internal mode contributions to the dilute-gas thermal conductivity. Thus,... 

Bich, E. Vogel, E. (1991). The initial density dependence of transport properties Noble gases. Int. J. Thermophys., 12,27-42. 

Hendl, S. Vogel, E. (1992). The viscosity of gaseous ethane and its initial density dependence. Fluid Phase Equil, 76,259-272. 

Ross, M., Szczepanski, R., Trengove, R. D. Wakeham, W. A. (1986). The initial density dependence of the transport properties of gases. AIChE Annual Winter Meeting, paper 8, held in Miami, Florida, USA. 

The absence of a rigorous theory for the transport properties of fluids in the intermediate-density range means that it has been necessary to employ methods of evaluation based upon an approximate theory, the principle of corresponding states (Chapter 12) or empiricism (see Section 5.3.3). The only approximate theory to have been used to any extent is the Enskog theory, outlined in Section 5.1 and discussed in a modified form in Section 5.2 in the context of the initial density dependence of the transport properties. 

The initial-density dependence r for the viscosity, defined in equation (14.8) - and represented by in equation (5.6) - is shown versus temperature in Figure 14.6b. The... 

The dilute-gas contribution to the transport properties is for most practical purposes an experimentally accessible quantity. Some techniques were developed in order to measure it directly, but in view of the possible effect of the initial density dependence (see Chapter 5) it is advisable, especially at subcritical temperatures, to analyze transport coefficients as a function of density along isotherms. Appropriate extrapolation schemes (see 14.1) can be used to deduce the zero-density value of the transport property, which can be identified with the dilute-gas value (see Chapter 4). It is thus always possible to analyze X T) independently of the remaining terms in equation (14.23). 

The excess contribution AA (p, T) can - at least in principle - be separated into the initial-density dependence and higher-density contributions AhJC(p, T) (see Chapter 5). 

Because of the existence of internal degrees of freedom and the anisotropy of the intermolecular potential the prediction of the initial-density dependence of thermal conductivity is still very uncertain (see Section 5.3). For this reason, an equation analogous to equation (14.35) was not applied for thermal conductivity. 

Table 14.10. Coefficients for the representation of the viscosity and thermal conductivity of ethane (dilute gas and initial-density dependence).

At present for polyatomic gases, this is possible only for viscosity, since the results for the thermal conductivity are not yet at the stage where they can be used for correlation or prediction purposes. In principle, the best approach to produce the correlation of viscosity at low densities is to analyze the available experimental data in conjunction with theory. Unfortunately, for ethane the available experimental data on the viscosity in the vapor phase at low density are very scarce (Hendl et al 1994), and it has not been possible to take advantage of these data in the development of the initial-density contribution. Thus the theory has been used in a predictive mode to generate the initial-density dependence of the viscosity. This was deemed necessary for ethane, since the vapor phase covers an industrially important and easily accessible region where the need for accurate transport properties is significant. 

In order to separate the initial-density dependence of the viscosity from higher-density terms, the excess viscosity is expressed as... 

Once the excess viscosity data have been generated, the initial density dependence can be evaluated as V and subtracted from the excess to obtain the higher-density contribution, Ah , by use of equation (14.55). ThCTe is no theoretical guidance to the functional form of Ah , but it is customary to express it in terms of power series in the density and in the reciprocal reduced temperature. Thus,... 

The initial—density dependence can be desaibed as follows (Section 5.2) ... 

X is introduced to make the terms in the equation dimensionless. It has the value 0.4945 W m K . The reduced temperature is equal to T/T, where T equals 647.27 K. The coefficients in the equation, which were originally determined by Aleksandrov Matveev (1978), are given in Table 14.19. The initial-density dependence of the thermal conductivity in the gaseous phase at low pressures can be represented by... 

The Rainwater-Friend theory that has proved so successful in the representation of the initial-density dependence of the viscosity of pure gases has not been extended to mixtures. It is therefore necessary to make use of the Thome-Enskog equations... 

Secondly, the local pressure must be considered. As shown in Figure 4, the released neutrals and ions give rise to a gradually expanding microcloud. Its initial density depends on the volatility of the analyte, i.e. the number of species released at a given power density. The importance of ion- molecule interactions... 

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