2.16 Temperature

In the conservation of energy Eq. (2.51 ), the mechanical kinetic energy, power flux and sources can be calculated from eqn, eqn and eqn from the momentum Eq. (2.19 ). Heat sources can contribute to eqn, e.g. from thermal radiation, chemical reactions etc.

That leaves the heat flux term eqn which represents conduction of heat. It is commonly modelled by Fourier’s law14 which states eqn is proportional to the negative gradient of temperature eqn, i.e.

|----------| q =-------rT-- \relax \special {t4ht=
where the constant of proportionality is thermal conductivity eqn.

Temperature scale

The heat flux Eq. (2.54) requires temperature to be defined and measurable. Measurement requires a scale. Empirical scales correlate temperature with a measured physical property of a working substance, e.g. EMF at a junction of two metal alloys. Empirical scales have the drawbacks of: being dependent on the working substance; and, not actually defining temperature.

PICT\relax \special {t4ht=


Instead, the thermodynamic scale defines temperature as a measure of the average kinetic energy of random motions of particle constituents of matter. It provides an absolute measure of temperature that is independent of the choice of working substance and includes a zero point16. It must be measured in units with a zero point, such as the SI unit Kelvin, eqn.

Substitution of our model Eq. (2.54) into Eq. (2.51 ) yields the term eqn. It is logical that this is a Laplacian term since it represents diffusion which is associated with random motions of submicroscopic particles, as we we established in Sec. 2.14 .

Ideal gas

The behaviour of many gases under typical working conditions is captured by the ideal gas equation of state

p = RT; \relax \special {t4ht=
where the specific gas constant is eqn. It is calculated from the Universal Gas Constant eqn in SI units, and the molar mass eqn of the gas, with units of eqn.

The ideal gas equation originates from classical thermodynamics as a combination of empirical laws17. Later, it was derived from first principles from both statistical thermodynamics and kinetic theory, with temperature representing average kinetic energy.

The derivations assume that molecules have no volume, undergo purely elastic collisions and there are no inter-molecular forces.

A scale of temperature defined by the ideal gas equation of state is exactly equivalent to the thermodynamic temperature scale.

14Joseph Fourier, Théorie analytique de la chaleur, 1822.
15in accordance with the zeroth law of thermodynamics.
16in accordance with the third law of thermodynamics.
17Benoît Clapeyron, Mémoire sur la puissance motrice de la chaleur, 1834.

Notes on CFD: General Principles - 2.16 Temperature