**2.6 Conservation of Energy: The Energy Equation**

**2.6.1 Formulation**

The principle of conservation of energy is** **applied to an element *dxdydz*

The variables *u, v, w, p, T, *and are used to express each term in (2.14).

**Assumptions****: **(1) continuum, (2) Newtonian fluid, and (3) negligible nuclear,

electromagnetic and radiation energy transfer.

Detailed formulation of the terms A, B, C and D is given in Appendix A

The following is the resulting equation

(2.15) is referred to as the energy equation

is the coefficient of thermal expansion, defined as

The dissipation function is associated with energy dissipation due to friction. It is

important in high speed flow and for very viscous fluids. In Cartesian coordinates

is given by

**2.6.2 Simplified Form of the Energy Equation**

**Cartesian Coordinates**

**(i) ****Incompressible fluid. **Equation (2.15) becomes

**(ii) Incompressible constant conductivity fluid. **Equation (2.18) is simplified further if

the conductivity *k *is assumed constant

**Cylindrical Coordinates. **The corresponding energy equation in cylindrical

Coordinate is given in (2.24)

**Spherical Coordinates. **The corresponding energy equation in cylindrical

Coordinate is given in (2.26)