Thermal Conductivity

The thermal conductivity of pure monatomic gases can be determined in the frame of the Chapman-Enskog theory [2]

к = 8.3225 -1(Г2у/^М. (4.16)

azl2k

The dimension of к is [W/mK]. The dimensionless collision integral Qk is identical with that for the viscosity QM. With this identity it can be shown for monatomic gases:

The Chapman-Enskog theory gives no relation similar to eq. (4.16) for polyatomic gases. An approximate relation, which takes into account the exchanges of rotational as well as vibration energy of polyatomic gases, is the semi-empirical Eucken formula, where cp is the specific heat at constant pressure

к={Ср + ш)^ (4Л8)

The monatomic case is included, if for the specific heat cp = 2.5 R0/M is taken.

From eq. (4.18) the relation for the Prandtl number Pr can be derived:

Pr =f^ =____________ (4 19)

к cp + 1.25Д0/М 9y — 5′ V ;

This is a good approximation for both monatomic and polyatomic gases [3]. у is the ratio of the specific heats: у = cp/cv.

For temperatures up to 1,500 K to 2,000 K, an approximate relation due to C. F. Hansen—similar to Sutherland’s equation for the viscosity of air—can be used [8]

T 1.5

kHan = 1-993 • 10-3—————- . (4.20)

T+112.0 v ;

A simple power-law approximation can also be formulated for the ther­mal conductivity: к = ckTШк. For the temperature range T ^ 200 K the approximation reads—with the constant ck1 computed at T = 100 K—

к1 = ck1 TUkl = 9.572 • 10-5 T, (4.21)

and for T ^ 200 K—with the constant ck2 computed at T = 300 K—

k2 = ck2TUk2 = 34.957 • 10-5 T0-75. (4.22)

In Table 4.4 and Fig. 4.4 we compare the results of the four above relations in the temperature range up to T = 2,000 K, again with the understanding, that a more detailed consideration might be necessary due to possible disso­ciation above T « 1,500 K. The data computed with eq. (4.18) were obtained for non-dissociated air with vibration excitation from

For the determination of the specific heats at constant volume cVvibr see Section 5.2. The mass fractions wO2 = 0.26216, and uN2 = 0.73784 were taken from Sub-Section 2.2 for air in the low-temperature range.

Table 4.4. Comparison of thermal-conductivity data к [W/mK] computed with eqs. (4.18) to (4.22) for some temperatures T [K] of air (see also Fig. 4.4). Included are the specific heat at constant pressure cp (second column), and the Prandtl number Pr based on eq. (4.19) (last column). T c-p/R к TO2 eq.(4.18) к Han TO2 О i—1 fe2 – to2 Pr 50 3.500 0.476 0.435 0.478 – 0.7368 75 3.500 0.715 0.692 0.717 – 0.7368 too 3.500 0.957 0.940 0.957 1.100 0.7368 200 3.500 1.820 1.807 1.914 1.859 0.7368 300 3.508 2.521 2.514 2.872 2.519 0.7373 400 3.538 3.131 3.114 3.833 3.126 0.7389 600 3.667 4.256 4.114 – 4.238 0.7458 800 3.830 5.329 4.945 – 5.258 0.7539 1,000 3.974 6.335 5.668 – 6.216 0.7607 1,500 4.204 8.535 7.183 – 8.425 0.7708 2,000 4.242 10.453 8.440 – 10.454 0.7724

The table shows that the data from the Hansen relation compare well with the Eucken data except for the temperatures above approximately 600 K, where they are noticeably smaller. The power-law relation for T ^ 200 K fails for T ^ 200 K. The second power-law relation gives good data for T ^ 200 K.

It should be noted, that non-negligible vibration excitation sets in already around T = 400 K. This is reflected in the behavior of the Prandtl number, Fig. 4.5, where also cp/R is given. To obtain, as it is often done, the thermal conductivity simply from eq. (4.19) with a constant Prandtl number would introduce errors above T « 400 K. The Prandtl number of air generally is Pr < 1 in a large temperature and pressure range [8].[30]