Page 618 - Bird R.B. Transport phenomena

598 Chapter 19 Equations of Change for Multicomponent Systems

From this equation and a similar one for x we can get x at z = 8. Then from Eq. 19.4-35 we get
2/ 3

which gives the rate of production of carbon dioxide at the catalytic surface. This result can
then be substituted into Eqs. 19.4-35 and 36 and the three mole fractions can be calculated as
functions of z.

EXAMPLE 19.4-4 In §9.3 we pointed out that the thermal conductivities of polyatomic gases deviate from the
formula for monatomic gases, because of the effects of the internal degrees of freedom in the
Thermal Conductivity complex molecules. When the Eucken formula for polyatomic gases (Eq. 9.3-15) is divided by
of a Polyatomic Gas the formula for monatomic gases (Eq. 9.3-14) and use is made of the ideal gas law, one can
write the ratio of the polyatomic gas thermal conductivity to that of a monatomic gas as

(19.4-38)
к топ 5 15
Derive a result of this form by modeling the polyatomic gas as an interacting gas mixture, in
which the various "species" are the polyatomic gas molecules in the various rotational and vi-
brational states.

SOLUTION The heat flux for a gas mixture is given in Eq. 19.3-3. All "species" will have the same thermal
conductivity because they differ only in their internal quantum states. Therefore we expect
each k to be A: . Similarly, the mass flux for each "species" should be given by Fick's law for
a
mon
a pure gas ) = -рЯЬ У(о , with all the ЯЬ having a common value Q) . Thus we get
а
a
ас
mon
аа
^
=
q P oi y
- cQ) m 2 H Vx a (19.4-39)
a
since the molecular weights of all the "species" are the same.
If now it is postulated that the distribution over the various quantum states is in equilib-
rium with the local temperature, then Vx = (dx /dT)VT. Then we can define the effective ther-
a
a
mal conductivity of the mixture by
= H (dxJdT)VT=-k VT (19.4-40)
q P oi y a po[y
and write
AIJ
a

m
= 1 - Г
= l+|A[(C™ /C )-l] (19.4-41)
ly mon
Here the temperature-dependent quantity

Л — 4( (19.4-42)
can be calculated from the kinetic theory of gases at low density. It varies only very slowly
with temperature, and a suitable mean value is 1.106. The quantity C P/poly = dH/dT is the heat
capacity for a gas in which the equilibrium among the various quantum states is maintained