Page 290 - Bird R.B. Transport phenomena

274 Chapter 9 Thermal Conductivity and the Mechanisms of Energy Transport

§93 THEORY OF THERMAL CONDUCTIVITY
OF GASES AT LOW DENSITY

The thermal conductivities of dilute monatomic gases are well understood and can be de-
scribed by the kinetic theory of gases at low density. Although detailed theories for poly-
atomic gases have been developed, 1 is it customary to use some simple approximate
theories. Here, as in §1.5, we give a simplified mean free path derivation for monatomic
gases, and then summarize the result of the Chapman-Enskog kinetic theory of gases.
We use the model of rigid, nonattracting spheres of mass m and diameter d. The gas
as a whole is at rest (v = 0), but the molecular motions must be accounted for.
As in §1.5, we use the following results for a rigid-sphere gas:

и = J^j = mean molecular speed (9.3-1)
Z = \пп = wall collision frequency per unit area (9.3-2)

A = —— = mean free path (9.3-3)
2
V2ird n
The molecules reaching any plane in the gas have had, on an average, their last collision
at a distance a from the plane, where
a = | Л (9.3-4)
In these equations к is the Boltzmann constant, n is the number of molecules per unit
volume, and m is the mass of a molecule.
The only form of energy that can be exchanged in a collision between two smooth
rigid spheres is translational energy. The mean translational energy per molecule under
equilibrium conditions is
> i ? = §KT (9.3-5)
as shown in Prob. 1C.1. For such a gas, the molar heat capacity at constant volume is

= Ш (9.3-6)

in which R is the gas constant. Equation 9.3-6 is satisfactory for monatomic gases up to
temperatures of several thousand degrees.
To determine the thermal conductivity, we examine the behavior of the gas under a
temperature gradient dT/dy (see Fig. 9.3-1). We assume that Eqs. 9.3-1 to 6 remain valid
in this nonequilibrium situation, except that \mu 2 in Eq. 9.3-5 is taken as the average ki-
netic energy for molecules that had their last collision in a region of temperature T. The
heat flux q y across any plane of constant у is found by summing the kinetic energies of
the molecules that cross the plane per unit time in the positive у direction and subtract-
ing the kinetic energies of the equal number that cross in the negative у direction:
q y = Z{\rnu \ . y a -\mu \ )
2
2
y+a
= |KZ(T| _ -TL ) (9.3-7)
y
fl
+fl
1
C. S. Wang Chang, G. E. Uhlenbeck, and J. de Boer, Studies in Statistical Mechanics, Wiley-
Interscience, New York, Vol. II (1964), pp. 241-265; E. A. Mason and L. Monchick, /. Chem. Phys., 35,
1676-1697 (1961) and 36,1622-1639, 2746-2757 (1962); L. Monchick, A. N. G. Pereira, and E. A. Mason,
/. Chem. Phys., 42, 3241-3256 (1965). For an introduction to the kinetic theory of the transport properties,
see R. S. Berry, S. A. Rice, and J. Ross, Physical Chemistry, 2nd edition (2000), Chapter 28.