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Equations_conservation_energy - Equations for the...

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Equations for the Conservation of Energy Author: John M. Cimbala, Penn State University Latest revision: 25 September 2007 Mechanical Energy Equation Non-conservative form: ( ) 1 2 ij i i i i i j D u u u g u Dt x τ ρ ρ = + Conservative form: ( ) ( ) 1 2 1 2 ij i i j i i i i i j j τ ρ u u u ρ u u ρ u g u t x x + = + or ( ) ij j i i i j j τ E u E ρ u g u t x x + = + where 2 1 2 mV is the kinetic energy (the conserved quantity), 1 2 i i E u u ρ is the kinetic energy per unit volume, and 1 2 i i u u is the kinetic energy per unit mass. The terms on the right are sources (or sinks) of kinetic energy per unit volume. Alternate conservative form: ( ) ( ) j j i i ij i j j u E u E ρ u g τ u p t x x x j φ + = + + where i ij j u σ x φ = rate of viscous dissipation of kinetic energy per unit volume (increases internal energy at expense of kinetic energy). φ is always positive since friction is an irreversible process. For a Newtonian fluid,
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