Page 292 - Bird R.B. Transport phenomena

276 Chapter 9 Thermal Conductivity and the Mechanisms of Energy Transport

Values of п = fl are given for the Lennard-Jones intermolecular potential in Table E.2
к M
as a function of the dimensionless temperature кТ/е. Equation 9.3-13, together with
Table E.2, has been found to be remarkably accurate for predicting thermal conductivi-
ties of monatomic gases when the parameters a and s deduced from viscosity measure-
ments are used (that is, the values given in Table E.I).
Equation 9.3-13 is very similar to the corresponding viscosity formula, Eq. 1.4-14.
From these two equations we can then get

Ь ^ = I ^ fJL (mo^atomic gas) (9.3-14)
The simplified rigid-sphere theory (see Eqs. 1.4-8 and 9.3-11) gives k = C /JL and is thus in
V
error by a factor 2.5. This is not surprising in view of the many approximations that were
made in the simple treatment.
So far we have discussed only monatomic gases. We know from the discussion in §0.3
that, in binary collisions between diatomic molecules, there may be interchanges be-
tween kinetic and internal (i.e., vibrational and rotational) energy. Such interchanges are
not taken into account in the Chapman-Enskog theory for monatomic gases. It can there-
fore be anticipated that the Chapman-Enskog theory will not be adequate for describing
the thermal conductivity of polyatomic molecules.
A simple semiempirical method of accounting for the energy exchange in poly-
atomic gases was developed by Eucken. 3 His equation for thermal conductivity of a
polyatomic gas at low density is

+ ~ Ц у (polyatomic gas) (9.3-15)

This Eucken formula includes the monatomic formula (Eq. 9.3-14) as a special case, be-
cause C = §CR/M) for monatomic gases. Hirschfelder 4 obtained a formula similar to that
p
of Eucken by using multicomponent-mixture theory (see Example 19.4-4). Other theo-
5 6
ries, correlations, and empirical formulas are also available. '
Equation 9.3-15 provides a simple method for estimating the Prandtl number, de-
fined in Eq. 9.1-8:
C p fJL C p
Pr = -j— = -—— (polyatomic gas) (9.3-16)

This equation is fairly satisfactory for nonpolar polyatomic gases at low density, as can
be seen in Table 9.3-1; it is less accurate for polar molecules.
The thermal conductivities for gas mixtures at low density may be estimated by a
7
method analogous to that previously given for viscosity (see Eqs. 1.4-15 and 16):
N у Ь
fcmix = 1 ~ t ~ (9-3-17)
The x a are the mole fractions, and the k are the thermal conductivities of the pure chem-
a
ical species. The coefficients Ф р are identical to those appearing in the viscosity equation
а

3
A. Eucken, Physik. Z., 14, 324-333 (1913); "Eukcer" is pronounced "Oy-ken."
4
J. O. Hirschfelder, /. Chem. Phys., 26, 274-281, 282-285 (1957).
J. H. Ferziger and H. G. Kaper, Mathematical Theory of Transport Processes in Gases, North-Holland,
5
Amsterdam (1972).
6
R. C. Reid, J. M. Prausnitz, and В. Е. Poling, The Properties of Gases and Liquids, McGraw-Hill, New
York, 4th edition (1987).
7
E. A. Mason and S. C. Saxena, Physics of Fluids, 1, 361-369 (1958). Their method is an
approximation to a more accurate method given by J. O. Hirschfelder, /. Chem. Phys., 26, 274-281,
282-285 (1957). With Professor Mason's approval we have omitted here an empirical factor 1.065 in his
Ф (/ expression for i Ф j to establish self-consistency for mixtures of identical species.