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Engineering LibreTexts

13.7.1.2: Acceleration Pressure Loss

The acceleration pressure loss can be estimated by
\[
    -\left. \dfrac{dP}{dx} \right|_a = \dot{m} \, \dfrac{dU_m} {dx}  
     \label{phase:eq:accPL}  \tag{35}
\]
The acceleration pressure loss (can be positive or negative) results from change of density and the change of cross section. Equation (35) can be written as
\[
    -\left. \dfrac{dP}{dx} \right|_a =  
        \dot{m}\, \dfrac{d} {dx} \left( \cfrac{\dot{m}} {A\,\rho_m} \right)  
     \label{phase:eq:accPLa}  \tag{36}
\]
Or in an explicit way equation (36) becomes
\[
    -\left. \dfrac{dP}{dx} \right|_a =  
        {\dot{m}}^2  \left[
                \overbrace{\dfrac{1}{A} \, \dfrac{d} {dx} \left( \dfrac{1} {\rho_m} \right)}  
                    ^{\text{pressure loss due to density change}} +  
                \overbrace{\dfrac{1}{\rho_m\,A^2} \dfrac{dA} {dx}}  
                    ^{\text{pressure loss due to area change}}  
        \right]
     \label{phase:eq:accPLae}  \tag{37}
\]
There are several special cases. The first case where the cross section is constant, \(\left. dA \right/ dx = 0\). In second case is where the mass flow rates of gas  and liquid is constant in which the derivative of \(X\) is zero, \(\left. dX \right/ dx = 0\). The third special case is for constant density of one phase only, \(\left. d\rho_L \right/ dx = 0\). For the last point, the private case is where densities are constant for both phases.

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.