Thermal conductivity kinetic theory expression

The ordinary multicomponent diffusion coefficients D j and the viscosity and thermal conductivity are computed from appropriate kinetic theory expressions. First, pure species properties are computed from the standard kinetic theory expressions. For example, the binary diffusion coefficients are given in terms of pressure and temperature as... 

For gases in the low-density limit, a kinetic-theory expression similar to that for viscosity can be used to evaluate single-component thermal conductivity. For a monatomic gas, meaning a gas with no rotational of vibrational degrees of freedom, the thermal conductivity is expressed as... 

It turns out to be a surprisingly difficult task to determine accurately the thermal conductivity of polyatomic gases from the viscosity. Many of the approaches are motivated by the ideas of Eucken. The so-called Eucken factor is a nondimensional group determined by dividing the kinetic-theory expression for a monatomic gas by that for viscosity, yielding... 

Using appropriate kinetic theory expressions, evaluate and plot the pure-species viscosities and thermal conductivities in the range 1000 K < T < 2000 K. 

The pure species thermal conductivities are given by the standard kinetic theory expression [178]... 

The kinetic theory expressions for the thermal conductivity of mixtures containing polyatomic molecules are very complicated and contain many essentially unknown quantities involving inelastic collisions and relaxation times (Monchick etal. 1965 see Chapter 4). Direct use of these formulas is essentially hopeless. However, by a careful application of some simplifying approximations and analysis of the major sources of errors, it was possible to obtain a fairly simple formula for predicting the composition dependence of A- ix with an uncertainty of the order of 2% (Uribe et al. 1991). 

The remaining quantity, 6 (1001), is not experimentally accessible but could be deduced using kinetic theory expressions also given in Chapter 4, and the subcorrelations for 6 (0001) and 6 (2000) as well as data for the thermal conductivity. A consistency test using Dm/D as a criterion then could be applied to exclude experimental data from the primary data set which are burdened with systematic errors (Millat Wakeham 1989a,b Millat et al. 1989b). 

Thermal Properties of Metallic Solids. In the preceding sections, we saw that thermal conductivities of gases, and to some extent liquids, could be related to viscosity and heat capacity. For a solid material such as an elemental metal, the link between thermal conductivity and viscosity loses its validity, since we do not normally think in terms of solid viscosities. The connection with heat capacity is still there, however. In fact, a theoretical description of thermal conductivity in solids is derived directly from the kinetic gas theory used to develop expressions in Section 4.2.1.2. 

The Molecular Origins of Mass Diffusivity. In a manner directly analogous to the derivations of Eq. (4.6) for viscosity and Eq. (4.34) for thermal conductivity, the diffusion coefficient, or mass diffusivity, D, in units of m /s, can be derived from the kinetic theory of gases for rigid-sphere molecules. By means of summary, we present all three expressions for transport coefficients here to further illustrate their similarities. 

Based on kinetic-theory principles and the Eucken correction, develop a general expression for the thermal conductivity of diatomic gases. Collect and combine all the constants, such that the expression depends on the molecular weight (g/mol), temperature (K), collision diameter (A), and reduced temperature T (nondimensional). 

The equation of state for a perfect gas is presented and expressions arising from this for pure gases and gas mixtures are given. The kinetic theory of gases, which is a useful model of perfect gases, is introduced and two particularly useful results are emphasised. These are the mean free path (/) and the mean or thermal velocity (c). Of particular importance is Z74, which is numerically equal to the volume rate of flow per unit area and which can be used to determine quantities such as area-related pumping speeds, conductances, etc. 

The kinetic theory of gases was briefly discussed. It enables the mean or thermal velocity (c) of gas molecules at a given temperature to be obtained and gas flux to be calculated. From the latter, effusion rates, area-related condensation rates and conductances under molecular flow can be determined (see Examples 1.5 and 1.7-1.10). Calculation of collision frequency (obtained from c, n and the collision cross-section of molecules), enables the mean free path (f) of particles to be determined. The easily obtained expression for Ip is a convenient way of stating the variation of / withp (Examples 1.11-1.15). 

At low pressure, the viscosity and thermal conductivity are independent of the pressure. This is observed experimentally, in confirmation of the kinetic theory. Therefore, these quantities can be expressed as a function of the temperature alone. Most of these correlations will also be based on the square root of the temperature, although the exact expressions tend to be more complicated. 

For binary and multicomponent mixtures, the thermal conductivity depends on the concentrations as well as on temperature, and the formulas of the accurate kinetic theory are quite complicated [5]. Empirical expressions for X are therefore more useful for both binary [9] and ternary [6], [26] mixtures, although few data exist for ternary mixtures. Tabulations of available experimental and theoretical results for thermal conductivities may be found in [5], [6], [13], and [18]-[21], for example. The thermal diffusivity, defined as 2p/Cp, often arises in combustion problems its pressure and temperature dependences in gases are XjpCp T7p ( < a < 2), and its typical values in combustion lie between 10 cm /s and 1 cm s at atmospheric pressure. 

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]. 

This is the usual expression encountered in polymer kinetic theory discussions however, for systems with temperature gradients, such as occur in the study of thermal conduction, Eq. (12.16) must be used instead. 

The kinetic theory of gases attempts to explain the macroscopic nonequilibrium properties of gases in terms of the microscopic properties of the individual gas molecules and the forces between them. A central aim of this theory is to provide a microscopic explanation for the fact that a wide variety of gas flows can be described by the Navier-Stokes hydrodynamic equations and to provide expressions for the transport coefficients appearing in these equations, such as the coefficients of shear viscosity and thermal conductivity, in terms of the microscopic prop>erties of the molecules. We devote most of our attention in this article to this problem. 

The kinetic theory of gases yields theoretical expressions for the thermal conductivity and other transport properties of gases. For ideal gases around atmospheric pressure, where the mean free path is much less than the smallest dimension of the container, the ratio of the thermal conductivities of the isotopic molecules is inversely proportional to the ratio of the square-roots of their molecular masses. At lower pressures, where the mean free path becomes comparable to, or larger than the dimensions of the container, the thermal conductivity will be strongly pressure dependent. The isotope analysis of isotopic gas mixtures by using a catharometer is based on the fact that, to a first approximation, the relationship between the thermal conductivity and isotopic composition of the mixture is linear (Muller et al. 1969). The isotope ratio of the viscosities of gases is equal, to a first approximation, to the square-root of the molecular mass ratio. 

Other models use the kinetic theory of phonons through which the thermal conductivity is expressed as a function of the volumetric heat capacity, the phonon group velocity, and the phonon mean-free path (De Boor et al. 2011). 

As the development of the kinetic theory outlined above has shown, the transport coefficients are obtained in different orders of approximations according to the number of terms included in the basis vectors of equation (4.7). Fortunately, the lowest-order approximations, at least for viscosity and thermal conductivity, are remarkably accurate and adequate for many purposes. However, for the most accurate work it is necessary to take account of higher-order kinetic theory corrections. These corrections may be expressed in the form... 

The other approach uses kinetic theory to calculate the transport coefficients in a stationary non-equilibrium situation such as shear flow. The first application of this approach to SRD was presented in [21], where the collisional contribution to the shear viscosity for large M, where particle number fluctuations can be ignored, was calculated. This scheme was later extended by Kikuchi et al. [26] to include fluctuations in the number of particles per cell, and then used to obtain expressions for the kinetic contributions to shear viscosity and thermal conductivity [35]. This non-equilibrium approach is described in Sect. 5. 

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