2.17 Internal energy

The conservation of energy Eq. (2.51 ) introduces the internal energy using the specific quantity eqn, measured in eqn 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 eqn due to bulk motion, potential due to forces eqn and eqn;
  • molecular — kinetic characterised by eqn, 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 eqn.

PICT\relax \special {t4ht=

|--------------------------| |De- | Dt| = r q + r + ☐☐☐ ru | ---------------------------- \relax \special {t4ht=

The identity eqn is the key element of the analysis. The eqn 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 eqn term must contribute to mechanical energy.

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

 De- = r q + r + ☐☐☐ ru p(r u): Dt \relax \special {t4ht=
The sign of eqn depends on eqn, 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 )

☐☐☐ ru 2 (devD) ru 2 (dev D) (dev D):18 \relax \special {t4ht=
18The eqn term is always positive since all components of eqn are squared. Its contribution to eqn is therefore non-recoverable and represents mechanical power that is dissipated as heat. In the majority of CFD analyses, the eqn contribution is small and can be ignored.

Double inner product of two tensors

The double inner product of two tensors, denoted by “eqn”, 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:

T S = TxxSxx + TxySxy + TxzSxz + TyxSyx + TyySyy + TyzSyz + T S + T S + T S zx zx zy zy zz zz \relax \special {t4ht=
For a scalar eqn, eqn.
18This identity relies on the substitution of eqn by eqn which is equivalent due to symmetry of eqn. eqn can then be replaced by eqn since eqn.

Notes on CFD: General Principles - 2.17 Internal energy