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 ii i i i j D uu ug u Dt x τ ρρ =+ Conservative form: ( ) 1 2 1 2 ij ji i i i i j j τ ρ u ρ ρ u tx x += + ∂∂ or ij i i j j τ E uE ρ u x + where 2 1 2 mV is the kinetic energy (the conserved quantity), 1 2 E ρ is the kinetic energy per unit volume, and 1 2 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 i i j i jj u E ρ τ up x x j φ + +− where i ij j u σ x = rate of viscous dissipation of kinetic energy per unit volume (increases internal energy at expense of kinetic energy). is always positive since friction is an irreversible process. For a Newtonian fluid, with Stokes’ assumption that 2 3 0 λμ , becomes 2 3 2 j i ij ij ij u u μ ee μ x x =− .
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This note was uploaded on 07/23/2008 for the course ME 521 taught by Professor Cimbala during the Fall '07 term at Pennsylvania State University, University Park.

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