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7.4.4.1: Energy Equation in Accelerated Coordinate with Uniform Flow

One of the way to simplify the general equation (105) is to assume uniform flow. In that case the time derivative term vanishes and equation (105) can be written as

Energy Equation in steady state

\[
    \label{ene:eq:AccCVgeneralss1}
    \dot{Q} - \dot{W} =  
         \int_{cv} \left( h + \dfrac{U^2}{2\dfrac{}{}} + a_x\,x + a_y\, y + (a_z + g) - z\, \dfrac{\omega^2 \,r^2}{2} \right)
             U_{rn}\, \rho\,dA\\
    \nonumber
        + \int_{cv} P\,U_{bn} \,dA   \tag{106}
\]

Further simplification of equation (106) by assuming uniform flow for which
\[
    \label{ene:eq:ene:AccCVgeneralss}
    \dot{Q} - \dot{W} =  
             \left( h + \dfrac

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_{rn}\, \rho\,dA\\
        + \int_{cv} P\,{\overline{U}}_{bn} \,dA   \tag{107}
\]
Note that the acceleration also have to be averaged. The correction factors have to introduced into the equation to account for the energy averaged verse to averaged velocity (mass averaged). These factor make this equation with larger error and thus less effective tool in the engineering calculation.

Contributors

  • Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.