Chapman-Enskog

The Chapman-Enskog solution of the Boltzmaim equation [112] leads to the following expressions for the transport coefficients. The viscosity of a pure, monatomic gas can be written as... 

A3.1.3.2 THE CHAPMAN-ENSKOG NORMAL SOLUTIONS OF THE BOLTZMANN EQUATION... 

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 tenns of the collision integral by the Chapman-Enskog fomuila [2]... 

Chapman-Enskog (Bird et al.) and Wilke-Lee The inherent assumptions of these equations are quite restrictive (i.e., low density. 

Binary Mixtures—Low Pressure—Polar Components The Brokaw correlation was based on the Chapman-Enskog equation, but 0 g and were evaluated with a modified Stockmayer potential for polar molecules. Hence, slightly different symbols are used. That potential model reduces to the Lennard-Jones 6-12 potential for interactions between nonpolar molecules. As a result, the method should yield accurate predictions for polar as well as nonpolar gas mixtures. Brokaw presented data for 9 relatively polar pairs along with the prediction. The agreement was good an average absolute error of 6.4 percent, considering the complexity of some of... 

Riazi-Whitson They presented a generahzed correlation in terms of viscosity and molar density that was apphcable to both gases and liqmds. The average absolute deviation for gases was only about 8 percent, while for liquids it was 15 percent. Their expression relies on the Chapman-Enskog correlation [Eq. (5-194)] for the low-pressure diffusivity and the Stiel-Thodos correlation for low-pressure viscosity ... 

Chapman-Enskog Expansion As we have seen above, the momentum flux density tensor depends on the one-particle distribution function /g, which is itself a solution of the discrete Boltzman s equation (9.80). As in the continuous case, finding the full solution is in general an intractable problem. Nonetheless, we can still obtain a useful approximation through a perturbative Chapman-Enskog expansion. 

Bather than using the Chapman-Enskog procedure directly, we shall employ the technique of Burnett,12 which involves an expansion of the distribution function in a set of orthogonal polynomials in particle-velocity space. 

The Burnett Expansion.—The Chapman-Enskog solution of the Boltzmann equation can be most easily developed through an expansion procedure due to Burnett.15 For the distribution function of a system that is close to equilibrium, we may use as a zeroth approximation a local equilibrium distribution function given by the maxwellian form ... 

Chapman-Enskog Solution.—The solution of the Boltzmann equation obtained by Chapman and Enskog involves the assumption... 

Block relaxation, 61 Bogoliubov, N., 322,361 Boltzmann distribution, 471 Boltzmann equation Burnett method of solution, 25 Chapman-Enskog method of solution, 24... 

Chapman-Enskog solution, 35 coefficicent equations, 28 derivation from Liouville s equation, 41... 

The Chapman-Enskog method has been used to solve for steady state tracer diffusion (. ). According to the method the singlet distribution function for the diffusing species 1, present In a trace amount n nj, 1 1) In an otherwise equilibrium fluid. Is approximated by... 

The Chapman-Enskog theory of flow In a one-component fluid yields the following approximation to the momentum balance equation (Jil). 

In principle, one should solve the Boltzmann equation Eq. (65) in order to arrive at explicit expressions for the pressure tensor p and heat flux q, which proves not possible, not even for the simple BGK equation Eq. (11). However, one can arrive at an approximate expression via the Chapman Enskog expansion, in which the distribution function is expanded about the equilibrium distribution function fseq, where the expansion parameter is a measure of the variation of the hydrodynamic fields in time and space. To second order, one arrives at the familiar expression for p and q... 

The binary diffusion coefficient, D k, can be either experimentally measured or calculated using the Chapman—Enskog equation. The dependence of the diffusion coefficient on temperature and pressure is generally given by ... 

Equation (105) is the basis for the determination of gas-phase diffusion coefficients and ultra low vapor pressures using the methods proposed by Davis and Ray (1977), Ravindran et al. (1979), and Ray et al. (1979). Additional information can be gained by writing the Chapman-Enskog first approximation for the gas-phase diffusivity (Chapman and Cowling, 1970),... 

The Chapman-Enskog equation (see Chapman and Cowling, 1970) is semi-empirical because it uses equation (3.11) and adjusts it for errors in the observations of diffusivity in gases. It also includes a parameter, S2, to account for the elasticity of molecular collisions ... 

Wilke and Lee (1955) found that the Chapman-Enskog equation could not estimate the diffusivity of lower molecular weight compounds, as well as those with a higher molecular weight. They therefore adjusted the constant, p, as follows ... 

The Chapman-Enskog equation missed the measurement by 19% and the Wilke-Lee adjustment by 12%. Both of these are greater than the 8% mentioned previously, but water is a highly polar molecule. 

Estimate the diffusion coefficient for 10 compounds through air at 1 atmosphere pressure from the Wilke-Lee adjustment to the Chapman-Enskog theory and compare your results with measurements. What is the percent error of the estimation (assuming that the measurements are correct) What is the primary cause of the differences between the estimated diffusivities ... 

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