Chapman-Enskog formulas

The Chapman-Enskog formula for monatomic gases at low density and temperature T produces better predictions, and is given by... 

For accurate calculations the Chapman-Enskog formula has been found suitable for evaluating the bulk diffusivity at moderate temperatures and pressures. The equation is... 

When ions move under equilibrium conditions in a gas and an external electric field, the energy gained from the electric field E between collisions is lost to the gas upon collision so that the ions move with a constant drift speed v = KE. The mobility K of ions of charge e in a gas of density N is given in terms of the collision integral by the Chapman-Enskog formula [2]... 

Binary gaseous mixture of species A and B at low density The Chapman-Enskog formula (Bird et al, 2002) is given by... 

For an accurate calculation of the molecular diffusivity An, 12, the Chapman-Enskog formula (Bird et al. 1960) can be used ... 

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). 

Pressure Dependencies Equation 3.95 predicts the binary diffusion coefficient to scale as p l, which is generally true except as the pressure approaches or exceeds the critical pressure. The Takahashi formula [392], which can be used to describe the high-pressure behavior, is discussed below. The Chapman-Enskog theory also predicts that Vji, increases with temperature as T3/2. However, it is often observed experimentally the temperature exponent is somewhat larger, say closer to 1.75 [332], An empirical expression for estimating T>jk is due to Wilke and Lee [433]. The Wilke-Lee formula is [332]... 

The theoretical basis, naturally, is the kinetic theory as summarized in Chapter 4. The explicit formulas of the Chapman-Enskog theory given there are all that is needed to formulate a principle of corresponding states, and were the basis for the two-parameter correlation for the noble gases given in 1972 (Kestin et al. 1972a,b), and for the extended correlation for the noble gases (Najafi et al. 1983). 

Viscosity is a measure of fluid resisfance to mofion, and if relates fhe strain rate to applied shear stress. A functional dependence of gas viscosify on temperature at low density is given by Chapman-Enskog based on kinetic theory (Bird et al., 1960) using Lennard-Jones potentials. The theory has been also extended to multicomponent gas mixtures. For most common applications, however, a simplified semiempirical formula of Wilke (1950) is used ... 

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... 

Estimate the diffusion coefficient of oxygen in nitrogen at 80°C and 2 atm using Chapman-Enskog s theoretical formula and by Fuller s empirical correlations. 

Estimate the binary diffusion coefficient of methane in hydrogen at 200°C and 2 atm pressure using (a) Chapman-Enskog s theoretical formula, (b) Fuller s empirical correlations, and (c) Bird s correlation. 

Pore sixe distribution data obtained from gas desorption (Barret et al. 1951) and mercury porisimetry experiments together with a knowledge of adsorbate molecular size thus enables the mode of diffusive transport to be ascertained. It should be noted that both molecular and Knudsen diffusion may occur in the same porous medium when the porous medium contains both macropores and micropores (revealed from an analysis of a bimodal pore size distribution curve). Unconstrained molecular diffusion. Dm, and Knudsen diffusion, Dk, coefficients are subsequently calculated from formulae derived from transport properties of fluids (gaseous and liquid) and the kinetic theory of gases. The molecular diffusivity for a binary gas mixture of A and B is evaluated from the Chapman-Enskog theory (Chapman and Cowling 1951) equation... 

The connection between the classical and quantum formulations of the transport coefficients has been studied by applying the WKB method to the quantum formulation of the kinetic theory (B16, B17). In this way it was shown that at high temperatures the quantum formulas for the transport coefficients may be written as a power series in Planck s constant h. When the classical limit is taken (h approaches zero), then the classical formulas of Chapman and Enskog are obtained. 

As in the case of the diffusion coefficient, the thermal conductivity in fluids can be predicted with satisfactory accuracy using theoretical expressions, such as the formulas of Chapman and Enskog for monoatomic gases, of Eucken for polyatomic ones, or of Bridgman for pure liquids. The thermal conductivity of solids, however, has not yet been predicted using basic thermophysical or molecular properties, just like the analogous diffusion coefficient. Usually, the... 

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