2.19 Energy and temperature

Specific internal energy eqn and temperature eqn were described in Sec. 2.17 and Sec. 2.16 , respectively. They are related through the specific heat capacity eqn, defined by Eq. (2.61 ) in Sec. 2.18 .

Analyses involving heat usually incorporate both eqn and eqn since:

  • eqn is the measurable quantity specified as initial and boundary data and whose data is required as part of the “results”;
  • eqn is the calculated quantity solved in energy conservation, e.g. Eq. (2.51 ), but whose data is usually of no interest.

Conversion of values between eqn and eqn is therefore needed, and vice versa. Incorporating Eq. (2.61 ) into a definite integral for eqn, eqn, gives

 Z e = T c dT + e : Tref V ref \relax \special {t4ht=
The terms in Eq. (2.62 ) are illustrated below on a eqn graph. Energy eqn is represented by the area under the curve, in which eqn represents a reference energy up to a reference temperature eqn, and the integral from eqn to eqn is shown by the shaded area.

PICT\relax \special {t4ht=

For applications that cover a reasonably narrow temperature range, eqn can be assumed constant. From Eq. (2.62 ), the eqn relation becomes

e = cV(T Tref) + eref: \relax \special {t4ht=
Alternatively, eqn can be integrated analytically by representing eqn by a polynomial of order eqn with coefficients eqn fitted to measured eqn data
 n c = X a T i: V i=0 i \relax \special {t4ht=
The values eqn and eqn ultimately add a constant component to eqn. Since Eq. (2.51 ) is concerned with changes in eqn and the absolute values eqn are usually of no interest, the values of eqn and eqn are often immaterial.

The eqn and eqn values become important when the composition of a fluid changes due to the mixing of constituent fluid species, e.g. eqn, eqn, or chemical reactions, e.g. with eqn. Each fluid specie possesses a different eqn so any change to the specie concentrations will change eqn of the overall fluid.

In those circumstances, eqn is commonly represented by the heat of formation per unit mass, eqn. The standard heat of formation eqn is the change of enthalpy during the formation of 1 mole of a substance from its constituent elements at standard temperature eqn. Measured heats of formation are available for numerous fluid species.20

If an analysis involves changes to fluid composition, it can then adopt eqn and eqn for individual fluid species, to account for the change in eqn due to changes in the concentrations of fluid species.

20Alexander Burcat and Branko Ruscic, Third Millennium ideal gas and condensed phase thermochemical database for combustion, 2005.

Notes on CFD: General Principles - 2.19 Energy and temperature