Enskog values dependence

All of the transport properties from the Chapman-Enskog theory depend on 2 collision integrals that describe the interactions between molecules. The values of the collision integrals themselves, discussed next, vary depending on the specified intermolecular potential (e.g., a hard-sphere potential or Lennard-Jones potential). However, the forms of the transport coefficients written in terms of the collision integrals, as in Eqs. 12.87 and 12.89, do not depend on the particular interaction potential function. 

Here ji(qa) is the spherical Bessel function of order l,g(a) is the radial distribution function at contact, and f = /fSmn/Anpo2g a) is the Enskog mean free time between collisions. The transport coefficients in the above expressions are given only by their Enskog values that is, only collisional contributions are retained. Since it is only in dense fluids that the Enskog values represents the important contributions to transport coefficient, the above expressions are reasonable only for dense hard-sphere fluids. Earlier Alley, Alder, and Yip [32] have done molecular dynamics simulations to determine the wavenumber-dependent transport coefficients that should be used in hard-sphere generalized hydrodynamic equations. They have shown that for intermediate values of q, the wavenumber-dependent transport coefficients are well-approximated by their collisional contributions. This implies that Eqs. (20)-(23) are even more realistic as q and z are increased. 

Fig. 4.23 also indicates a slight decrease of the signal plateau which, at a first glance, was unexpected. In the following, a reactive dispersion model given in ref. [37] is applied to deduce rate constants for different reaction temperatures. A trapezoidal response function will be used. The temperature-dependent diffusion coefficient was calculated according to a prescription by Hirschfelder (e.g., [80], p. 68 or [79], p. 104] derived from the Chapman-Enskog theory. For the dimensionless formulation, the equation is divided by M/A (with M the injected mass and A the cross-section area). This analytical function is compared in Fig. 4.24 with the experimental values for three different temperatures. The qualitative behavior of the measured pulses is well met especially the observed decrease of the plateau is reproduced. The overall fit is less accurate than for the non-reactive case but is sufficient to now evaluate the rate constant. 

We now see that the rigid sphere molecular model gives a value for a that depends only on the mixture composition and the molecular diameter and weight of all the mixture species. However, a more rigorous treatment based on Chapman-Enskog theory would yield a slight pressure and temperature dependence for the ratio D, JD,. The value of a in that case would also show a weak dependence on the process conditions for a fixed mixture composition. 

The direct calculation of the collective contribution DJDs to the self-diffusion coefficient is complicated by the inadequate temperature dependence of the shear viscosity in ref. [ ]. Indeed, it is easy to verify that the ratio r / r g for the model argon increases with temperature on isochors. From the physical viewpoint, this result is inadequate. It is worth noting that for ( ) < 0.4 the values of r from ref. f ] and those determined on the basis of the Enskog theory for hard spheres diameter of which coincides with the effective diameter... 

In Figure 5.3, values for B calculated in this way (MET-I) for the Lennard-Jones (12-6) potential are shown to have the same general temperature dependence as the experimental data (there are other modifications of Enskog s theory which are not considered here.). In Figure 5.4, an analogous comparison is shown for thermal conductivity. The calculated values (MET-I) underestimate the experimental results when the association correction (see Table 5.1) is omitted from the MET approach but are... 

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