Untitled - Aerobib - Universidad Politécnica de Madrid

56 CHAPTER 2. TRANSPORT PHENOMENA IN GAS MIXTURES
right hand side of (2.56) is small compared to the other two and can be neglected.
Then ¯q reduces to
¯q = −λ∇T + ρ ∑ Y i h i¯v di . (2.57)
i
In a pure gas ¯q reduces to
¯q = −λ∇T. (2.58)
If, moreover, the gas is monatomic, thermal conductivity coefficient can be
computed through successive approximations as done for diffusion and viscosity. The
first approximation [λ] 1
is
√
T/M
10 7 [λ] = 1989.1
σ 2 Ω (2,2)∗ (T ∗ ) . (2.59)
Here [λ] is measured in cal cm −1 s −1 K −1 and the other magnitudes in the same units
as in Eq. (2.17).
When comparing (2.59) and (2.47) it is seen that for monatomic gases thermal
conductivity and viscosity coefficients are proportional. Similarly to what happens for
other transport coefficients the first approximation is, generally, sufficient for practical
applications.
Expression (2.59) does not take into account for polyatomic molecules energy
transfer between internal and external degrees of freedom during collisions. Such
transfer is not important in the study of viscosity and diffusion but is fundamental in
thermal conductivity. 28 In polyatomic molecules [λ] must be substituted by the following
approximate expressions, due to Eucken, which takes into account the influence
of such energy transfer
[λ] = 15
4
(
R 4
M [µ] 1
15
C v
R + 3 )
, (2.60)
5
where C v is molar heat of the gas at constant volume. Table 2.5 compares experimental
values with those given by Eucken’s formula (2.60) for several gases. 29
H 2 O 2 CO 2 CH 4 N 2
Calculated 4 140 615 386 741 619
Experimental 4 227 635 398 819 619
Table 2.5: Values of the thermal conductivity coefficient (10 7 [λ] cal cm −1 s −1 K −1 ) for
several gases at T=300 K .
It is seen that agreement is fair even if not as good as for other transport coefficients.
28 See Ref. [2], p. 489.
29 See Ref. [2], p. 574.