Chapman-Enskog theory

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

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 ... 

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. 

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. 

As illustrated in the low-density limit of Fig. 3.3, the viscosity of gases increases with increasing temperature. Moreover, for pressures well below the critical pressure, there is very little pressure dependence. The kinetic theory of dilute gases provides the theoretical basis for the temperature dependence. The Chapman-Enskog theory provides an expression for dilute pure-species viscosities as... 

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]... 

At moderate pressures the diffusion coefficient of a binary gas mixture of molecules i and j is well described by the Chapman-Enskog theory, discussed in Section 12.4 ... 

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 these equations, T is the temperature, p is the pressure, X is the mole fraction of species k, m is the molecular mass, R is the universal gas constant, and / is the pure species viscosity. The T>jk are first order (in the Chapman-Enskog theory) binary diffusion coefficients, given by Eq. 12.113. It is actually inappropriate [103] to use a second-order or higher approximation [265] to the binary diffusion coefficients here. For this reason the Dixon-Lewis paper used the notation to emphasize that the first-order approxima-... 

Fig. 4.23 also indicates a slight decrease of the signal plateau which, at a first glance, was unexpected. In the following, a reactive dispersion model given in ref. [37] is applied to deduce rate constants for different reaction temperatures. A trapezoidal response function will be used. The temperature-dependent diffusion coefficient was calculated according to a prescription by Hirschfelder (e.g., [80], p. 68 or [79], p. 104] derived from the Chapman-Enskog theory. For the dimensionless formulation, the equation is divided by M/A (with M the injected mass and A the cross-section area). This analytical function is compared in Fig. 4.24 with the experimental values for three different temperatures. The qualitative behavior of the measured pulses is well met especially the observed decrease of the plateau is reproduced. The overall fit is less accurate than for the non-reactive case but is sufficient to now evaluate the rate constant. 

The binary diffusivity, D, of the reactants in which the Leonard-Jones parameters (o, Q) are available, is calculated through the Chapman-Enskog theory by,... 

We now see that the rigid sphere molecular model gives a value for a that depends only on the mixture composition and the molecular diameter and weight of all the mixture species. However, a more rigorous treatment based on Chapman-Enskog theory would yield a slight pressure and temperature dependence for the ratio D, JD,. The value of a in that case would also show a weak dependence on the process conditions for a fixed mixture composition. 

Unlike the elementary kinetic theory, the three collision integrals ( 2D AB, Qu and T2a) are introduced in the Chapman-Enskog theory. Moreover, the collision diameter (er ) is used instead of the molecular diameter (d ). 

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 ... 

The mean free path once more is usually calculated not by (9.11) because of the difficulty of directly measuring cjab> but from the binary diffusivity of A in B, DAB. This diffusivity can be either measured directly or calculated theoretically from the Chapman-Enskog theory for binary diffusivity (Chapman and Cowling 1970) by... 

The difference in the thermal conductivity of the mixture (Ha-He) and pure demonstrating the qualitative agreement between predictions of the Chapman-Enskog theory and chromatograph observations. 

Before we proceed further, we will discuss in this section briefly about correlation used to calculate binary diffusivity. The equation commonly used to calculate this diffusivity is derived from the Chapman-Enskog theory (Bird et al., 1960) ... 

Dll. We have steady-state diffusion of ammonia in air across a film that is 0.033 mm thick On one side of the film, the ammonia concentration is 0.000180 and on the other side it is 0.000257 kmol/m. We desire an ammonia flux rate of 9.60 x 10 kmol/(m s). The apparatus is at 0.90 atm. Use the Chapman-Enskog theory to estimate the diffusivity. Find the required operating temperature. 

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