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Effects of Heat Capacity Lag in Gas Dynamics

By Arthur Kantrowitz

SUMMARY

The existence of energy dissipations in gas dynamics,
which must be attributed to a lag in the vibrational heat
capacity of the gas, has been established both theoreti-
cally and experimentally.

The flow about a very small impact tube is discussed.
It is shown that total-head defects due to heat-capacity
lag during and after the compression of the gas at the
nose of an impact tube are to be anticipated. Experi~
ments quantitatively verifying these anticipations in
carbon dioxide are discussed. A general theory of the
dissipations in a more general flow problem is developed
and applied to some special cases. It is pointed out
that energy dissipations due to this effect are to be
anticipated in turbines. Dissipations of this kind might
also introduce errors in cases in which the flow of one
gas is used in an attempt to simulate the flow of another
gas. Unfortunately, the relaxation times of most of the
gases of engineering importance have not been studied.

A new method of measuring the relaxation time of
gases is introduced in which the total—head defects ob-
served with a specially shaped impact tube are compared.
with theoretical considerations. A parameter is thus
evaluated in which the only unknown quantity is the re-
laxation time of the gas. This method has been applied
to carbon dioxide and has given consistent results for
two impact tubes at a variety of gas velocities.

INTRODUCTION

The heat content of gases is primarily three forms
of molecular mechanical energy. First, there is the

translational kinetic-energy which is ERT, where R
is the gas constant and T is the absolute temperature.
Secondly, there is the rotational kinetic energy. For

all gases near or above room temperature, the n rota-
tional degrees of freedom involving moments of inentia

due to the separation of atomic neuclei have energy states
close enough together that the rotational internal energy

is close to the classical value ERT. The third prin-

cipal form of internal energy is the vibrational energy
of the molecules. If the frequencies of the normal
modes of vibration of the molecule are known (say, from
spectra), the vibrational heat capacity can be'computed
by the methods of statistical mechanics. (See, for
example, reference 1.)

The possibility of dispersion and absorption of
sound due to parts of the heat capacity lagging behind
the rapid temperature changes accompanying the propaga-
tion of a sound wave in a gas was first discussed theo-
retically by Jeans and Einstein. Dispersion and ab-
sorption in carbon dioxide observed by Pierce were shown
by Herzfeld and Rice to be attributable to lagging of
the vibrational heat capacity of the gas. Kneser was
able to account quantitatively for dispersion and
absorption in 602 and oxygen on the assumption that the

vibrational heat capacity lagged.

The dispersion and absorption of sound in several
gases have been investigated and a fairly complete
bibliography is available in reference 2. It is found,
in general, that dispersion and absorption many times
larger than those attributable to vicosity and heat
conduction are to be expected in gases with vibrational
heat capacity. These effects can be described by rela-
tions such as those given by Kneser and can be attributed
to the vibrational heat capacity of the gas.

All the measurements of dispersion and absorption
have demonstrated that most impurities markedly reduce
the relaxation time of a gas; for example, Kneser and
Knudsen (references 5 and 4) concluded that the adjust-
ment of the vibrational heat capacity of oxygen was
dependent entirely on the action of impurities.