Page 291 - Bird R.B. Transport phenomena

§9.3 Theory of Thermal Conductivity of Gases at Low Density 275

Fig. 9.3-1. Molecular transport
^Temperature profile T(y) f (kinetic) energy from plane
o
at (y - a) to plane at y.
Г| I + а \

Typical molecule arriving
\у-а from plane at (y - a)
with energy-к Т\ _
у а

Equation 9.3-7 is based on the assumption that all molecules have velocities representa-
tive of the region of their last collision and that the temperature profile T(y) is linear for a
distance of several mean free paths. In view of the latter assumption we may write

Цу-а ~ 3 ^ т (9.3-8)

T Т (9.3-9)
\y+a = \у + 3 ^ ^ -
Ву combining the last three equations we get

(9.3-10)

This corresponds to Fourier's law of heat conduction (Eq. 9.1-2) with the thermal con-
ductivity given by
к = \пш\ = lpC u\ (monatomic gas) (9.3-11)
v
in which p = nm is the gas density, and C = |K/W (from Eq. 9.3-6).
v
Substitution of the expressions for п and A from Eqs. 9.3-1 and 3 then gives
(monatomic gas) (9.3-12)
ird 2
which is the thermal conductivity of a dilute gas composed of rigid spheres of diameter
d. This equation predicts that к is independent of pressure. Figure 9.2-1 indicates that this
prediction is in good agreement with experimental data up to about 10 atm for most
gases. The predicted temperature dependence is too weak, as was the case for viscosity.
For a more accurate treatment of the monatomic gas, we turn again to the rigorous
Chapman-Enskog treatment discussed in §1.5. The Chapman-Enskog formula 2 for the
thermal conductivity of a monatomic gas at low density and temperature T is
, _ 25 УтгткТ л or к = 1.9891 X 10" (monatomic gas) (9.3-13)
32 2 п ^

In the second form of this equation, к [=] cal/cm • s • К, Т [=] К, а [=] A, and the "colli-
sion integral" for thermal conductivity, Q , is identical to that for viscosity, П^ in §1.4.
k

J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, Wiley, New
2
York, 2nd corrected printing (1964), p. 534.