2.18 Heat capacity

When the heat conduction model of Eq. (2.54 ) is substituted into Eq. (2.51 ), we produce an equation with two variables eqn and eqn. For example, starting with Eq. (2.56 ) for internal energy, and ignoring mechanical power and heat source terms, gives

 De- r rT = 0: Dt \relax \special {t4ht=
To be able to solve this equation, a relation between eqn and eqn is needed. Heat capacity provides this, describing an amount of heat required to produce a unit change in eqn. We use specific heat capacity, relating to a constant volume process, whose SI units are eqn, defined as
 cV = -@e : @T V \relax \special {t4ht=
Statistical mechanics provides a theorem known as equipartition which provides quantitative predictions of eqn.19 It calculates that any degree of freedom (DoF), such as a component eqn of particle translational velocity eqn, which appears quadratically in energy, i.e. as eqn, contributes eqn to eqn.

For monatomic gases, e.g. argon Ar, particle motion is translational only (until atoms dissociate into subatomic particles). With 3 translational DoFs, eqn is constant.

A gas which obeys the ideal gas law with constant eqn is known as calorically perfect.

PICT\relax \special {t4ht=

For diatomic gases, e.g. nitrogen eqn, molecular motion is translational (left) and rotational (centre) at lower temperature (in the gas state). At higher temperature, vibrational motion is excited along its bond (right). Heat capacity eqn for diatomic gases are then a function of temperature as shown below for eqn.

PICT\relax \special {t4ht=

Rotational motion provides 2 DoFs (rotational energy about the axis is negligible), giving eqn. Beyond 500K, vibrational motion provides 2 additional DoFs (one kinetic, one potential), causing a gradual transition to eqn. At high temperature, molecules dissociate and eqn increases further.

A thermally perfect gas is an ideal gas with eqn. Often eqn can be treated as constant over a range of eqn, e.g.eqn 600K for eqn. Otherwise, accurate calculations require a suitable model of eqn. An imperfect gas exhibits eqn.

19originating from heat capacity studies by Alexis Petit and Pierre Louis Dulong, Recherches sur quelques points importants de la théorie de la chaleur, 1819.

Notes on CFD: General Principles - 2.18 Heat capacity