Enskog theory, transport

The Chapman-Enskog theory was developed for dilute, monatomic gases for pure substances and for binary mixtures. The extension to multicomponent gas mixtures was performed by Curtiss and Hirschfelder (C12, Hll), who in addition have shown that the Chapman-Enskog results may also be obtained by means of an alternate variational method. Recently Kihara (K3) has shown how expressions for the higher approximations to the transport coefficients may be obtained, which are considerably simpler than those previously proposed by Chapman and Cowling these simpler formulas are particularly advantageous for calculating the coefficients of diffusion and thermal diffusion (M3, M4). 

For the light molecules He and H2 at low temperatures (below about 50°C.) the classical theory of transport phenomena cannot be applied because of the importance of quantum effects. The Chapman-Enskog theory has been extended to take into account quantum effects independently by Uehling and Uhlenbeck (Ul, U2) and by Massey and Mohr (M7). The theory for mixtures was developed by Hellund and Uehling (H3). It is possible to distinguish between two kinds of quantum effects— diffraction effects and statistics effects the latter are not important until one reaches temperatures below about 1°K. Recently Cohen, Offerhaus, and de Boer (C4) made calculations of the self-diffusion, binary-diffusion, and thermal-diffusion coefficients of the isotopes of helium. As yet no experimental measurements of these properties are available. 

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. 

Expressions for the transport coefficients suitable for use in computational simulations of chemically reacting flows are usually based on the Chapman-Enskog theory. The theory has been extended to address in detail transport properties in multicomponent systems [103,178]. 

Chapman-Enskog theory provides the basis for the multicomponent transport properties laid out by Hirschfelder, Curtiss, and Bird [178] and by Dixon-Lewis [103]. The multi-component diffusion coefficients, thermal conductivities, and thermal diffusion coefficients are computed from the solution of a system of equations defined by the L matrix [103], seen below. It is convenient to refer to the L matrix in terms of its nine block submatrices, and in this form the system is given by... 

In the liquid phase also, the Enskog theory has also proved quite effective for the representation and prediction of the properties of pure and mixed substances. In this case, the theory has been modified using the results of computer simulations of hard spheres, which have indicated the limitations of the assumptions of entirely random motions at elevated densities. In combination, the Enskog theory corrected in this way provides a good description of the properties of some pure liquids. An even more general result of the Enskog theory is that the transport properties of a fluid or... 

Note that the viscosity parameter p has been introduced as a prefactor in front of the tensor functions by substitution of the kinetic theory transport coefficient expression after comparing the kinetic theory result with the definition of the viscous stress tensor o, (2.69). In other words, this model inter-comparison defines the viscosity parameter in accordance with the Enskog theory. 

Garz6, V., Dufty, J. W. Hrenya, C. M. 2007 Enskog theory for polydisperse granular mixtures. I. Navier-Stokes order transport. Physical Review E 76, 031303. 

The model considers that the porous media is composed of giant molecules fixed and uniformly distributed in space known as dust and hence these dust particles are treated as one component of the gas mixture. The Chapman-Enskog theory (Ferziger, 1972) is then applied to this pseudo-gas mixture. The dusty gas model separates the problem of transport into three independent parts ... 

Here we will not go through the detailed calculations that lead to the Enskog theory values for the transport coeflicients of shear viscosity, bulk viscosity, and thermal conductivity appearing in the Navier-Stokes hydro-dynamic equations. Instead we shall merely cite the results obtained and refer the reader to the literature for more details. One finds that the coefficient of shear viscosity 17 is given by ... 

Despite the fact that real molecules are not hard spheres, the Enskog theory has been used to describe transport properties of real fluids over a wide range of densities and temperatures with a considerable degree of success. To apply the Enskog theory to real systems one must assume that (a) the mechanisms for the transport of energy and momentum in a real system do not differ in any essential way from the mechanisms of transport in a hard-sphere fluid, and (b) the expressions for the transport coefficients of a real fluid at a given temperature and density are identical to those of a hard-sphere fluid at the same density, provided one replaces a and x(ti) in the hard-sphere expressions by quantities d and x(T) where d is an effective hard-sphere diameter of the molecules at temperature T, and x(T) is an effective radial distribution function that takes into account the temperature dependence of the collision frequency in the real fluid. ... 

The Enskog theory can be used to describe the transport properties of the fluid over a much wider range of temperatures and densities if one determines d and X in consistent way from equilibrium properties of the In... 

The modified Enskog equation can easily be generalized to apply to dense mixtures of hard-sphere gases/ The transport coefficients that result satisfy the Onsager reciprocal relations. Thus the principal difficulty in generalizing the Enskog theory to mixtures has now been removed. 

If gaseous systems have high densities, both the kinetic theory of gases and the Chapman-Enskog theory fail to properly describe the transport coefficients behavior. Furthermore, the previously derived expression for viscosity and... 

We have suggested expressions for X l and X. Based on the behavior of the transport coefficients according to the Modified Enskog Theory ( 1), one can derive for the pure fluid, (and similarly for the mixture)... 

The expressions for the transport coefficients given by Enskog theory (equations (5.2)-(5.4)) lead to the following results for the second transport virial coefficients... 

Hanley, H. J. M., McCarty, R. D. Cohen, E. G. D. (1972). Analysis of the transport coefficients for simple dense fluids Application of the modified Enskog theory. 

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. 

Fig. 5.8. Computed corrections to Enskog theory for transport property X.

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