Enskog theory coefficients

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. 

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 hard sphere diameters were then used to calculate the theoretical Enskog coefficients at each density and temperature. The results are shown in Figure 3 as plots of the ratio of the experimental to calculated coefficients vs. the packing fraction, along with the molecular dynamics results (24) for comparison. The agreement between the calculated ratios and the molecular dynamics results is excellent at the intermediate densities, especially for those ratios calculated with diameters determined from PVT data. Discrepancies at the intermediate densities can be easily accounted for by errors in measured diffusion coefficients and calculated diameters. The corrected Enskog theory of hard spheres gives an accurate description of the self-diffusion in dense supercritical ethylene. 

The most important result of the model is the weak dependence of D/D on particle mass for Z > 1. First this suggests that the Enskog theory will provide a good estimate for heavy atoms in a molecular fluid at all densities. Second the Enskog diffusion coefficient itself is only weakly dependent on mass, namely, as [(1 -I- Z)/Z] and if this factor is used to estimate the mass... 

Note that the viscosity parameter p has been introduced as a prefactor in front of the tensor functions by substitution of the kinetic theory transport coefficient expression after comparing the kinetic theory result with the definition of the viscous stress tensor o, (2.69). In other words, this model inter-comparison defines the viscosity parameter in accordance with the Enskog theory. 

This pressure tensor closure was derived by Gidaspow [22] in accordance with the Enskog theory presented by Chapman and Cowling [11], chap 16. That is, with the restitution coefficient e equal to one, the y factor substituted by go, and bp = Aad this relation corresponds to equation (16.34 — 2) in Chapman and Cowling [11]. 

The direct calculation of the collective contribution DJDs to the self-diffusion coefficient is complicated by the inadequate temperature dependence of the shear viscosity in ref. [ ]. Indeed, it is easy to verify that the ratio r / r g for the model argon increases with temperature on isochors. From the physical viewpoint, this result is inadequate. It is worth noting that for ( ) < 0.4 the values of r from ref. f ] and those determined on the basis of the Enskog theory for hard spheres diameter of which coincides with the effective diameter... 

Figure 6. Self-diffusion coefficient of argon as a function of density. Solid lines correspond to experiments, slim line - to modified Enskog theory. Squares-results of calculations according to (2).

The estimation of low pressure diffusivity is based on the corresponding states theory. The dense gas diffusion coefficient estimation is based on the Enskog theory. The binary diffusion coefficient D jj at high pressures as modeled by the Dawson-Khoury-Kobayashi correlation, is next given as a representative model. For a binary system, the equations are ... 

Here we will not go through the detailed calculations that lead to the Enskog theory values for the transport coeflicients of shear viscosity, bulk viscosity, and thermal conductivity appearing in the Navier-Stokes hydro-dynamic equations. Instead we shall merely cite the results obtained and refer the reader to the literature for more details. One finds that the coefficient of shear viscosity 17 is given by ... 

Despite the fact that real molecules are not hard spheres, the Enskog theory has been used to describe transport properties of real fluids over a wide range of densities and temperatures with a considerable degree of success. To apply the Enskog theory to real systems one must assume that (a) the mechanisms for the transport of energy and momentum in a real system do not differ in any essential way from the mechanisms of transport in a hard-sphere fluid, and (b) the expressions for the transport coefficients of a real fluid at a given temperature and density are identical to those of a hard-sphere fluid at the same density, provided one replaces a and x(ti) in the hard-sphere expressions by quantities d and x(T) where d is an effective hard-sphere diameter of the molecules at temperature T, and x(T) is an effective radial distribution function that takes into account the temperature dependence of the collision frequency in the real fluid. ... 

The modified Enskog equation can easily be generalized to apply to dense mixtures of hard-sphere gases/ The transport coefficients that result satisfy the Onsager reciprocal relations. Thus the principal difficulty in generalizing the Enskog theory to mixtures has now been removed. 

Moreover, one can also show that the time integral of (248) for PD,o(t) leads to exactly the same values for the coefficient of self-diffusion D as given by the Boltzmann equation. Similarly the time integral of po e(0 Eq- (250), leads to an expression for D that is identical to the Enskog theory result. ... 

The uncertainties Ay/yhave been calculated taking into account the errors on Io/Iat and L (thickness of the recombination boundary layer) but also on the flow parameters the diffusion coefficient Do,air determined using the Chapman-Enskog theory, the mean square atomic velocity V determined using the gas kinetic theory (rarefied gas). The accuracy on these two last values is due essentially to that of the gas temperature, measured by emission spectroscopy (N2 rotational temperature), this leads to a total accuracy of 35%. 

If gaseous systems have high densities, both the kinetic theory of gases and the Chapman-Enskog theory fail to properly describe the transport coefficients behavior. Furthermore, the previously derived expression for viscosity and... 

We have suggested expressions for X l and X. Based on the behavior of the transport coefficients according to the Modified Enskog Theory ( 1), one can derive for the pure fluid, (and similarly for the mixture)... 

The molecular diffusion coefficient of sulfur dioxide can be predicted from Chapman-Enskog theory and its value is 1.13 cmVs at 840 °C and 1 atm and for the temperature range of interest, it can be approximated from... 

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