Enskog values

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. 

The equations of motion in the extended hydrodynamic theory (Section IV) are obtained from the relaxation equation, where the correlation function is normalized. As mentioned before, in the extended hydrodynamic theory, the memory kernel matrix is considered to be independent of frequency thus the transport coefficients are replaced by their corresponding Enskog values. 

In the critical phenomena, the contribution from the different hydro-dynamic modes to the transport coefficients are calculated. On the other hand, in the extended hydrodynamic theory, only the Enskog values of the transport coefficients are used. Thus, while the critical phenomena considers only the long-time part of the memory function, the extended hydrodynamic theory uses only the short-time part of the memory function. None of the theories involve any self-consistent calculation. 

In the formulations developed from the renormalized kinetic theory approach, these self-consistencies were avoided either by using values obtained from computer simulation and experiments or by using some exactly known limiting values for the transport coefficient. For example, in the treatment of Mazenko [5-7], and of Mehaffey and Cukier s [8] the transport coefficients are replaced by their Enskog values. In the theory developed by Sjogren and Sjolander [9], the velocity autocorrelation function is required as an input that was obtained from the computer simulated values. This limits the validity of the theories only to certain regimes and for certain systems where the experimental or computer-simulated results are available. 

When the mean free path is small compared with pore diameter, the dominating experience of molecules is that of collision with other molecules in the gas phase. In that respect, the situation is much the same as that which exists in the bulk gas. The appropriate diffusion coefficient Dm may be obtained from published experimental values, or calculated from a theoretical expression. For molecular diffusion in a binary gas, the Chapman and Enskog equation may be used, as discussed by Bird, Stewart and Lightfoot(32). This takes the form ... 

Because of the excellent agreement between experimental measurements and the values calculated on the basis of the Enskog theory, empirical formulas are not needed. Sometimes, however, it is convenient to have empirical formulas available for rapid calculations or for use in analytical solutions to differential equations. Some empirical relations have been assembled by Partington (PI). 

The Chapman-Enskog theory (Chapman and Cowling, 1970) is a model which is positioned between the two approaches, the empirical relation by Fuller et al. (Eq. 18-44) and the theoretically stringent Equation 18-43. This theory improves the absolute size of the expression by taking into account the individual sizes and interactions of the diffusing molecules. However, the numerical values obtained with the model by Fuller et al. (Eq. 18-44) are still better than both the Chapman-Enskog theory and Eq. 18-43. 

Using the viscosity versus temperature table evaluated from the Chapman-Enskog expression in the previous problem, determine a best fit for the S parameter in the form of a Sutherland viscosity expression. Assume reference values of 7o = 273 K and i o = 1.716 x 10-5 N-s/m2. 

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. 

It is useful to note here that the value of a is determined by the Grote-Hynes formula, which to some extent limits the sensitivity of the rate to viscosity (rj). On the other hand, if the binary part of the friction is replaced by the Enskog friction, then larger values of a are obtained [164]. This is because the Enskog friction is directly proportional to g(increases with increasing density. The same correlations also lead to an increase in solvent... 

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. 

Estimate the viscosity and thermal conductivity of carbon dioxide using the Chapman-Enskog model at 1 atm and 250,300, and 400K and compare with the experimental values (Welty et al. (1984)) in the table below ... 

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. 

Binary diffusivities Vae at low pressures are predicted from the Chapman-Enskog kinetic theory with binary Lennard-Jones parameters predicted as follows from the pure-component values ... 

As Oxtoby showed, this last expression gives an absolute value very close to his hydrodynamic result. More recently Schweizer and Chandler derived a binary expression for dephasing by repulsive forces. The dephasing time they find is proportional to the hard-sphere Enskog collision time. 

The superlinear density dep>endence of the simulation results for listed in Table IV is well accounted for by the estimate of the Enskog enhancement factors of the Br-Ar pair. For a packing fraction of 0.01, this factor is close to 1. At this density Nordholm et al. evaluate a Br2-Ar collision frequency from analysis of the time variation of the Br2 internal energy of 2.0 x 10 which is comparable to a value of 1.5 x 10 estimated from a collision diameter fferj.Ar = Br.Ar + The collision number Nyj for V-T transfer is kg/Z and is 65 at low density. We conclude that at low densities where collisions are well resolved, the model of this work and the simulations give similar collision numbers. 

That is, instead of determining the transport properties from the rather theoretical Enskog solution of the Boltzmann equation, for practical applications we may often resort to the much simpler but still fairly accurate mean free path approach (e.g., [12], section 5.1 [87], chap. 20 [34], section 9.6). Actually, the form of the relations resulting from the mean free path concept are about the same as those obtained from the much more complex theories, and even the values of the prefactors are considered sufficiently accurate for many reactor modeling applications. 

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