Thermal conductivity of a polyatomic gas

Formula (21) does not give good results for polyatomic gases, due to the participation of the internal movements of the molecules (vibrations and rotations) to the thermal conductivity. It is therefore necessary to generalize relationship (21) in the following form  

Sc is the SCHMIDT number for a pure gas. The value of Sc l is close to 1.32. For a non-polar gas, the following value is taken  

Note that formula (22), which is meant for polyatomic gases, can also be applied to monoatomic gases, for which Cp/R = 5/2, which gives formula (21) again. 

Expressions (22) to (27) will be used with basic SI units  

Calculate the thermal conductivity of isopentane at 10 Pa and 373 K. An experimental value 2.2x10 2 j s has been published. 

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A further case of some interest is the higher-order approximation to the thermal conductivity of a polyatomic gas based on the expansion vectors and This leads to... 

In practice, the correction factor fr usually departs by at most 1% from unity. This means that equation (4.29) provides at the same time a simple and accurate description of the thermal conductivity of a polyatomic gas. 

On the other hand, the two-flux approach to the thermal conductivity of a polyatomic gas offers a rather greater opportunity for its prediction from viscosity and other experimental data because it is possible to make use of exact and approximate relationships between various effective cross sections and, in some cases, theoretically known behavior in the high temperature limit. The starting point for such an approach is... 

The initial density dependence of the thermal conductivity of a polyatomic gas is given by expression (5.7). Neither the Rainwater-Friend model nor the modified Enskog theory accounts for the contribution of internal degrees of freedom, but it is assumed that this can be modeled as a purely diffusive process following an idea originally introduced by Mason Monchick (1962) for dilute gases... 

For the thermal conductivity of a polyatomic gas one further extension of the MET is necessary to account for the internal degrees of freedom of the molecule and the transport of internal energy. This is accomplished with the aid of the assumption that the transport of internal energy is diffusive and that there is no relaxation of internal energy. This approximation is similar to that discussed in Chapter 4 for dilute gases and enables one to write... 

The first term of equation (4.127) is an approximation to the translational contribution to the thermal conductivity of the mixture. It is obtained by making use of equations (4.122)-(4.125) for the thermal conductivity of a monatomic gas mixture. For this purpose approximate translational contributions to the thermal conductivity of each pure component X, tr and an interaction thermal conductivity for each unlike interaction Xqq are evaluated by the heuristic application of equation (4.125) for monatomic species to polyatomic gases. Thus, the technique requires the availability of experimental viscosity data for pure gases and the interaction viscosity for each binary system or estimates of them. As the discussion of Section 4.2 makes clear, the use of... 

Thermal conductivity and thermal transpiration measurements provide two additional methods by which information may be obtained on inelastic molecular collisions. Mason and Monchick [146] have derived an expression for the thermal conductivity X of a polyatomic gas that includes a term having an explicit dependence upon the relaxation rates of the internal degrees of freedom. Their expression can be written [147]... 

The first Chapman-Cowling approximation to the thermal conductivity of a dilute polyatomic gas within the two-flux approach leads to a total value that is the sum of two contributions related to translational and internal degrees of freedom respectively ... 

As an example of the practical utility of MEMO simulations of the thermal conductivity, a summary of a calculation by Ravi et al. (1992) is given. The object was to explore the contribution of internal degrees of freedom to the conductivity of a polyatomic molecule. This contribution has not been quantitatively demonstrated until very recently (Murad et al. 1991). In fact, it is usually assumed that this contribution is independent of number density and is given by the dilute-gas value at the corresponding temperature. In the paper of Ravi et al. (1992), heat flow is discussed for a model benzene-like liquid with a six-centered Lennard-Jones pair potential. 

As with viscosity, the theoretical predictions become more complex as the atoms themselves become more complex and more dense. For a polyatomic gas, the thermal conductivity is given by extension of Eq. (4.35) ... 

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... 

A semiempirical Eucken equation is used to estimate the thermal conductivity of polyatomic gas with R = 1.987 cal/(molK). 

As outlined in Chapter 4, the thermal conductivity in the dilute-gas limit, is related to a number of effective cross sections, which are associated with the transport of translational and internal energy, and with their interaction. In principle, a similar analysis as given for nitrogen in Section 14.2 can be performed for polyatomic molecules. In practice, such an analysis is often hampered by a lack of experimental information and insufficient knowledge of the behavior of cross sections describing the diffusion of internal energy at high temperatures. 

Notwithstanding the further modification of the Enskog theory required to achieve this result, the procedure yields thermal conductivity predictions for a dense polyatomic gas which are little worse than for monatomic systems. 

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