Enskog approach

Various attempts have been made to obtain approximate solutions to the Boltzmann equation. Two of these methods were suggested independently by Chapman [10] [11] and by Enskog [24] giving identical results. In this book emphasis is placed on the Enskog method, rather than the Chapman one, as most modern work follows the Enskog approach since it is less intuitive and more systematic, although still very demanding mathematically. 

An analytical expression for the heat flux vector can be derived in a similar manner using the Enskog approach. That is, we introduce the first order approximation of the distribution function from (2.246) into the heat flux definition (2.72) and thereafter substitute the partial solution for flux vector integrand as follows [39] ... 

The pressure dependence expressed by these relations is fairly accurate for pressures up to about 10 — 20 atmospheres. At higher pressures, multi-body collisions become important and the pressure dependence is greater. Experimental evidence also show that a temperature dependence of is too weak leading to the search for more accurate relations as derived by use of the Enskog approach. 

Example 1-1 The viscosity of isobutane at 23°C and atmospheric pressure is 7.6 X 10 pascal-sec. Compare this value to that calculated by the Chap-man-Enskog approach. 

The kinetic flux is determined by the Chapman-Enskog approach [25] ... 

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. 

An alternate approach has been attempted for describing the transport phenomena in dense gas and liquid systems by means of the methods of nonequilibrium statistical mechanics, as developed by Kirkwood (K7, K8) and by Born and Green (B18, G10). Although considerable progress has been made in the development of a formal theory, the method does not at the present time provide a means for the practical calculation of the transport coefficients. Hence in this section we discuss only the applications based on Enskog s theory. 

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. 

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]... 

Solution of the Boltzmann equation gives the velocity distribution function throughout the gas as it evolves through time, for example, due to velocity, temperature, or concentration gradients. A practical solution to the Boltzmann equation was found by Enskog [114], which is discussed in the next section. This approach is used to calculate rigorous expressions for gas transport coefficients. 

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. 

Theories based on the Enskog collision time (84) or other solid-like approaches do not have a strongly temperature-dependent frequency correlation time. But they do have a temperature-dependent factor resulting from the need to create the solvent fluctuations in the first place. Thus, all fast-modulation theories predict that the dephasing rate will go to zero at 0 K. 

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]. 

The forces of attraction and repulsion between molecules must be considered for a more accurate and rigorous representation of the gas flow. Chapman and Enskog proposed a well-known theory in which they use a distribution function, the Boltzmann equation, instead of the mean free path. Using this approach, for a pair of non-polar molecules, an intermolecular potential, V (r), is given in the potential function proposed by the Lennard-Jones potential ... 

The equation resulting from this approach is sometimes referred to as the Enskog s equation of change (e.g., [83], p. 527). 

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. 

Likewise, in accordance with the Enskog [20] dense gas approach, Cidaspow [22] proposed a closure for the collisional pressure tensor. By use of the Chapman-Enskog approximate solution method [11], the collisional pressure tensor can be written as ... 

Up to the early 1970s a kinetic approach to the time-dependent properties of fluids was synonymous with a framework based on the Boltzmann equation and its extension by Enskog, in which a central role is played by those dynamical events referred to as uncorrelated binary collisions [29]. Because of this feature the Boltzmann equation is in general not applicable to dense fluids, where the collisions are so frequent that they are likely to interfere with each other. The uncorrelation ansatz is clearly equivalent to a loss of memory, or to a Markov approximation. As a result, for dense fluids the traditional kinetic approach should be critically revised to allow for the presence of non-Markovian effects. 

He et al. [12] developed a more physically consistent LBM approach based on Enskog theory of dense fluids [5], in which the force interactions are represented by van der Waals theory [35]. In the following, we will briefly summarize this approach. As before, we start with continuous Boltzmann equation, but now for the nonideal fluids, which may be written as [12]... 

The next important advance in the theory, and the one that provided the foundation for all later work in this field, was made by Boltzmann, who in 1872 derived an equation for the time rate of change of the distribution function for a dilute gas that is not in equilibrium—the Boltzmann transport equation. (See Boltzmann and also Klein. " ) Boltzmann s equation gives a microscopic description of nonequilibrium processes in the dilute gas, and of the approach of the gas to an equilibrium state. Using the Boltzmann equation. Chapman and Enskog derived the Navier-Stokes equations and obtained expressions for the transport coefficients for a dilute gas of particles that interact with pairwise, short-range forces. Even now, more than 100 years after the derivation of the Boltzmann equation, the kinetic theory of dilute gases is largely a study of special solutions of that equation for various initial and boundary conditions and various compositions of the gas.t... 

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