## 2.19Energy and temperature

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

Analyses involving heat usually incorporate both and since:

• is the measurable quantity speciﬁed as initial and boundary data and whose data is required as part of the “results”;
• 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 and is therefore needed, and vice versa. Incorporating Eq. (2.61 ) into a deﬁnite integral for , , gives

 (2.62)
The terms in Eq. (2.62 ) are illustrated below on a graph. Energy is represented by the area under the curve, in which represents a reference energy up to a reference temperature , and the integral from to is shown by the shaded area.

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

 (2.63)
Alternatively, can be integrated analytically by representing by a polynomial of order with coeﬃcients ﬁtted to measured data
 (2.64)
The values and ultimately add a constant component to . Since Eq. (2.51 ) is concerned with changes in and the absolute values are usually of no interest, the values of and are often immaterial.

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

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

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