## 2.17Internal energy

The conservation of energy Eq. (2.51 ) introduces the internal energy using the speciﬁc quantity , measured in in SI units. It represents the total molecular energy consisting of: kinetic energy associated with temperature; potential energy due to particle forces, both within particles as chemical bonds, and between particles, e.g. van der Waals forces.

The energies at diﬀerent scales can be summarised as:

• bulk — kinetic due to bulk motion, potential due to forces and ;
• molecular — kinetic characterised by , potential due to bonds.

To understand how energy is transferred between these scales, we can derive an equation for internal energy from Eq. (2.51 ), by cancelling terms in mechanical energy formed by taking the inner product of Eq. (2.19 ) with .

 (2.56)

The identity is the key element of the analysis. The term in Eq. (2.56 ) must represent the contribution of mechanical power to the internal energy, i.e. passing from bulk to molecular scale. Conversely, the term must contribute to mechanical energy.

Substituting from Eq. (2.41 ), Eq. (2.56) becomes

 (2.57)
The sign of depends on , i.e. whether the ﬂuid is expanding or contracting. Since the sign can change, it therefore represents a recoverable contribution to internal energy.

If we substitute the Newtonian model from Eq. (2.41 )

 (2.58)
The term is always positive since all components of are squared. Its contribution to is therefore non-recoverable and represents mechanical power that is dissipated as heat. In the majority of CFD analyses, the contribution is small and can be ignored.

### Double inner product of two tensors

The double inner product of two tensors, denoted by “”, is introduced in Eq. (2.56). It produces a scalar which is evaluated as the sum of the 9 products of the tensor components, eg:

 (2.59)
For a scalar , .
18This identity relies on the substitution of by which is equivalent due to symmetry of . can then be replaced by since .

Notes on CFD: General Principles - 2.17 Internal energy