Notes on Computational Fluid Dynamics: General Principles

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2.19 Energy and temperature

Speciﬁc internal energy $\text{[math]}$ and temperature $\text{[math]}$ were described in Sec. 2.17 and Sec. 2.16 , respectively. They are related through the speciﬁc heat capacity $\text{[math]}$ , deﬁned by Eq. (2.61 ) in Sec. 2.18 .

Analyses involving heat usually incorporate both $\text{[math]}$ and $\text{[math]}$ since:

$\text{[math]}$ is the measurable quantity speciﬁed as initial and boundary data and whose data is required as part of the “results”;
$\text{[math]}$ 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 $\text{[math]}$ and $\text{[math]}$ is therefore needed, and vice versa. Incorporating Eq. (2.61 ) into a deﬁnite integral for $\text{[math]}$ , $\text{[math]}$ , gives

Z e = T c dT + e : Tref V ref \relax \special {t4ht=

(2.62)

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

$PICT\relax \special {t4ht=$

For applications that cover a reasonably narrow temperature range, $\text{[math]}$ can be assumed constant. From Eq. (2.62 ), the $\text{[math]}$ relation becomes

e = cV(T Tref) + eref: \relax \special {t4ht=

(2.63)

Alternatively, $\text{[math]}$ can be integrated analytically by representing $\text{[math]}$ by a polynomial of order $\text{[math]}$ with coeﬃcients $\text{[math]}$ ﬁtted to measured $\text{[math]}$ data

n c = X a T i: V i=0 i \relax \special {t4ht=

(2.64)

The values $\text{[math]}$ and $\text{[math]}$ ultimately add a constant component to $\text{[math]}$ . Since Eq. (2.51 ) is concerned with changes in $\text{[math]}$ and the absolute values $\text{[math]}$ are usually of no interest, the values of $\text{[math]}$ and $\text{[math]}$ are often immaterial.

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

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

If an analysis involves changes to ﬂuid composition, it can then adopt $\text{[math]}$ and $\text{[math]}$ for individual ﬂuid species, to account for the change in $\text{[math]}$ 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