The Energy Equation

2.6 Conservation of Energy: The Energy Equation

2.6.1 Formulation

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

clip_image002

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The variables u, v, w, p, T, and clip_image006 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

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(2.15) is referred to as the energy equation

clip_image010is the coefficient of thermal expansion, defined as

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The dissipation function clip_image014 is associated with energy dissipation due to friction. It is

important in high speed flow and for very viscous fluids. In Cartesian coordinates clip_image014[1]

is given by

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2.6.2 Simplified Form of the Energy Equation

Cartesian Coordinates

(i) Incompressible fluid. Equation (2.15) becomes

clip_image018

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

the conductivity k is assumed constant

clip_image020

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)

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