Transport properties critical enhancement

The transport properties of a near-critical system contain an enhancement or a reduction due to critical fluctuations in addition to the contributions of molecular transport processes which are strictly a function of the thermodynamic state. Therefore, the transport coefficients in the critical region are usually... 

Saim and Subramaniam [38] and Ginosar and Subramaniam [39] also found that the in situ extraction of the coke compounds by near-critical or supercritical reaction mixtures prevents pore plugging that otherwise occurs at subcritical (gas-like) conditions. Although the coke laydown decreased at supercritical (liquid-like) conditions, the isomerization rates were lower and deactivation rates were higher due to pore diffusion limitations in the liquid-like reaction mixtures. It was therefore concluded that near-critical reaction mixtures provide an optimum combination of solvent and transport properties that is better than either subcritical (gas-like) or dense supercritical (liquid-like) mixtures for maximizing the isomerization rates and for minimizing catalyst deactivation rates. These findings indicate that catalytic reactions which require liquid-like reaction media for coke extraction and heat removal, yet gas-like diffusivities for enhanced reaction rates, can benefit from the use of near-critical reaction media. 

A supercritical fluid is defined as a material above its critical temperature and critical pressure (see Table 2.1). These fluids are characterized by gas-like transport properties and liquid-like densities. They also offer a greatly enhanced solvating capability in comparison with gases. Recently, the use of supercritical fluids has been applied to the generation of ceramic powders. The rapid expansion of a supercritical fluid solution results in the formation of a powder. As the pressure is reduced, the solubility of the solute decreases and supersaturation occurs. Stable... 

Although the contribution is written with an emphasis on polyatomic gases, it also includes results for pure monatomic gases and mixtures containing monatomic species. Chapters 5 and 6 consider fluids at moderate and high densities as well as the critical enhancements to the transport properties and, therefore, complete this brief summary of the statistical theory of fluids. 

Thermodynamic states of systems near a critical point are characterized by the presence of large fluctuations of the order parameter associated with the critical phase transition. For one-component fluids near the vapor-liquid critical point the order parameter can be identified with the fluctuating density p. That is, near the vi r-liquid critical point the local density becomes a function of the position r. The density fluctuations have a pronounced effect on the behavior of both the thermodynamic and the transport properties of fluids in the critical region. Specifically, they lead to a strong enhancement of the thermal conductivity and to a weak enhancement of the viscosity of fluids in the critical region, as will be elucidated in this chapter. 

If the terms of equation (7.13) are truly independent, then it may be appropriate to fit the dilute-gas transport properties to primary data at low density the dilute-gas contribution calculated from the resultant correlation can then be subtracted from the experimental data at higher densities to study the remaining excess and critical enhancement terms. Since the dilute-gas term is a function of temperature alone (or a reduced temperature such as T = T/Tc or T = k T/i) and both the excess and critical enhancement terms are zero in the limit of zero density, the dilute-gas term is expected to be mathematically independent of the excess and critical enhancement terms. However, if the dilute-gas term has some cross correlation with respect to density (because of its determination from experimental data at nonzero density), then it may not be proper to fit the dilute gas data independently. Any cross correlation which is present in the data is not necessarily due to the behavior of the transport property itself, but may be due to systematic effects. This uncertainty may be a function of the other independent variables such as density or composition. 

Once good fits for the background properties are found, they can be evaluated at each state point and subtracted from the data to obtain data for the critical enhancement. The asymptotic form of the critical enhancement for transport properties has been understood for some time (Sengers Keyes 1971 Sengers 1972 see Chapter 6). Correlations which make use of this theory, as well as empirical approximations, such... 

The excess functions of equation (7.13) are not unique but depend strongly on both the functional form and the optimized parameters of the enhancement terms. The optimized parameters of both terms depend on the selected equation of state used to determine the density (especially in the critical region) as well as the compressibility and heat capacity, which are required in the theoretically based critical enhancement models. It is necessary to have data over an extremely wide range of temperature and pressure to resolve any temperature dependence of the transport property excess functions. Detailed examples of such pure fluid correlations are presented in Chapter 14. 

In general for mixtures, a more predictive rather than a validated algorithm must be used for obtaining the required densities in a transport-property correlation, and a more empirical transport-property surface may be appropriate. It is convenient to use corresponding-states methods to obtain the equilibrium properties in order to establish the correlation in terms of density, temperature and composition. An approach which gives accurate equilibrium properties for the pure fluid limits is desirable in establishing a general transport-property mixture surface. To use the critical enhancement expressions above, the mixture critical point needs to be calculated at any composition. These... 

Several empirical methods to extend the critical enhancement term to mixtures have been explored (see also Chapters 6 and 15). The parameters in the mode-coupling approach mentioned above have been made composition-dependent and the results are reasonable. Luettmer-Strathmann (1994) has recently reported a new mode-coupling solution which describes the critical enhancements to the transport properties of fluid mixtures. Corresponding states algorithms, based on the mode-coupling solutions, have also been used to describe the thermal conductivity of mixtures (Huber et al 1992). 

For pure fluids, it is most common to represent the saturated vapor and saturated liquid transport properties as simple polynomial functions in temperature, although polynomials in density or pressure could also be used. Exponential expansions may be preferable in the case of viscosity (Bmsh 1962 Schwen Puhl 1988). For mixtures, the analogous correlation of transport properties along dew curves or bubble curves can be similarly regressed. In the case of thermal conductivity, it is necessary to add a divergent term to account for the steep curvature due to critical enhancement as the critical point is approached. Thus, a reasonable form for a transport property. 

The purpose of this chapter, in a book about transport properties, is to give advice to the reader on the best methods for converting the data, which are usually measured as a function of P and T, to a function of p and T, which is the form required for the correlating equations and, in addition, to provide sources for values of the ideal-gas isobaric heat capacities, which are also required for the transport-property calculations. Both of these purposes can be fulfilled by calculations from a single equation of state which has been fitted to the whole thermodynamic surface. Heat capacities of the real fluid are required only for the calculation of the critical enhancement of the thermal conductivity and viscosity, as described in Chapter 6 discussion of these properties in this chapter will be restricted to Section 8.4.4. 

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. 

Hence, in order to evaluate the critical enhancement of the thermal conductivity by means of equation (14.58) apart from the background transport properties, a knowledge of three thermodynamic properties is needed, namely, that of the isochoric and isobaric heat capacities and of the correlation length, all of which can be evaluated from the relevant equation of state. Furthermore, one needs to determine four adjustable parameters (see Chapter 6) which enter the expressions for Q and Qq- Two of these, namely, the system-dependent amplitudes fo and r, have been obtained by the application of the... 

The form of equation (14.46) indicates that it is relatively straightforward to calculate the excess transport property fi om the experimental data by subtracting the dilute-gas contribution and the critical enhancement. As mentioned before, in practice an iterative procedure is often necessary. Here, for the purpose of discussion it will be assumed that... 

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