Kinetic theory, Chapman-Enskog

Equation (2.71) can be compared with Eq. (2.46) for the thermal conductivity of gases, and with Eq. (2.19) for the viscosity. For binary gas mixtures at low pressure, is inversely proportional to the pressure, increases with increasing temperature, and is almost independent of the composition for a given gas pair. For an ideal gas law P = cRT, and the Chapman-Enskog kinetic theory yields the binary diffusivity for systems at low density... 

The exact form of the expressions for the diffusional fluxes jj depends on the degree of sophistication used in representing the transport phenomena. A precise approach, including also the calculation of the thermal conductivity of gas mixtures, and based on the Chapman-Enskog kinetic theory, has been described by Dixon-Lewis [122]. However, simpler approaches involving the form j = —pDiAwijAy may also give satisfactory representation in many cases [119—121,123]. 

Gas transport properties are required to apply the theory given in Sections 3.3 and 3.4. Viscosities of pure nonpolar gases at low pressures are predicted from the Chapman-Enskog kinetic theory with a Lennard-Jones 12-6 potential. The collision integrals for viscosity and thermal conductivity with this potential are computed from the accurate curve-fits given by Neufeld et al. (1972). 

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 ... 

In the DGM model, porous media are considered as arrays of heavy molecules i.e., dust) that are motionless and uniformly distributed in space. By treating the dust particles as giant molecules it is possible to use the Chapman-Enskog kinetic theory. The dust molecules are treated as an (ji - - l)th pseudo-species added to the n-component gas mixture. The dust particles are kept fixed in space (i.e., motionless) and are considered like another gas component in the Maxwell-Stefan equations. 

Alternative estimates of the transport coefficients can be obtained from the rigorous Chapman-Enskog expansion method of mono-atomic gases at low densities (e.g., [24] [25] [12] [61] (p 202) [28]). The transport coefficients deduced from the Chapman-Enskog kinetic theory with the rigid elastic spheres interaction model yield (e.g., [39] sect 8.2 [61], p 202) ... 

The Chapman-Enskog kinetic theory actually gives general expressions for the transport properties in terms of the intermolecular potential energy which is... 

Maxwell suggested that the porous material itself be described as a supplementary dust species, consisting of very large molecules that are kept motionless by some unspecified external force. The Chapman-Enskog kinetic theory is then applied to the new pseudo-gas mixture, in which the interaction between the dust and gas molecules simulates the interaction between the solid matrix and the gas species. In addition, one is no longer faced with the problem of flux and composition variations across a pore and problems related to catalyst geometry. 

The prediction of the diffusion coefficients of gases from basic thermophysical and molecular properties is possible with great accuracy using the Chapman-Enskog kinetic theory. Diffusivities in liquids, on the other hand, in spite of the absence of a rigorous theory, can be estimated within an order of magnitude from the well-known equations of Stokes and Einstein (for large spherical molecules) and Wilke (for dilute solutions). 

T is the absolute temperature in kelvin, MW is an average molecular weight, is the total absolute pressure in atmospheres, and a is average diameter of the spherical molecules in A. The more detailed and accurate Chapman-Enskog kinetic theory is valid for nonpolar molecules to about 70 atm. This equation with in mAs tCussler. 2009 Geankoplis, 2003 Wankat and Knaebel. 20081 is... 

The Chapman-Enskog kinetic theory cf gases (Hirschfelder et al., 1964) is used to describe the multicomponent diffusion flux of species i in a mixture of n gas species and expressed as the Stefan-Maxwell equation (Bird et al, 2002). The diffusion flux of species i is given as... 

A corranonly used theorehcal formula for the binary diffusion coefficient is derived based on the Chapman-Enskog kinetic theory (Bird et al., 2002 Hirschfelder et al., 1964 Sherwood et al, 1975) for low-pressure gas mixture and is given as... 

The binary diffusion coefficient is based on the Chapman-Enskog kinetic theory and is given by Equation 6.108 ... 

The Chapman-Enskog kinetic theory actually gives general expressions for the transport properties in terms of the intermolecular potential energy which is related to the intermolecular force as expressed by (2.54) and (2.55). The molecular interaction is most frequently described by the empirical Lennard-Jones 12-6 potential. 

The transport coefficients like viscosity, thermal conductivity and self-diffusivity for a pure mono-atomic gas and the diffusivity for binary mixtures obtained from the rigorous Chapman-Enskog kinetic theory with the Lennard-Jones interaction model yield (e.g., [9], Sects. 1-4, 9-3 and 17-3 [55], Sect. 8.2) ... 

Predictions from the Chapman-Enskog kinetic theory tend to be limited in two ways. First, the theory requires estimates of nonpolar gases, and this excludes compounds like water and ammonia. These interactions depend on replacing the Lennard-Jones potential used to characterize the collision with more exact potentials. Such replacement is often complex. 

The characteristics of diffusion coefficients described in this chapter are summarized in Table 5.7-1. In general, diffusion coefficients in gases and in liquids can often be accurately estimated, but coefficients in solids and in polymers cannot. In gases, estimates based on the Chapman-Enskog kinetic theory are accurate to around ten percent. In liquids, estimates are based on the Stokes-Einstein equation or its empirical parallels. These estimates, accurate to around twenty percent, can be supplemented by a good supply of experimental data. In solids and polymers, theories allow coefficients to be correlated but rarely predicted. 

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