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7.4.5: Energy Losses in Incompressible Flow

In the previous sections discussion, it was assumed that there are no energy loss. However, these losses are very important for many real world application. And these losses have practical importance and have to be considered in engineering system. Hence writing equation (15) when the energy and the internal energy as a separate identity as
\[
    \label{ene:eq:governingELossIni}
     \dot{W}_{shaft} = \dfrac{d}{dt}  
            \displaystyle \int_V  \left( \dfrac{U^2}{2\dfrac{}{}} + g\,z\right) \,\rho\,dV + \\
          \displaystyle \int_A \left( \dfrac{P}{\rho} + \dfrac{U^2} {2\dfrac{}{}} + g\,z \right) U_{rn}\, \rho \,dA +
        \displaystyle \int_A P U_{bn} dA + \\  
             \overbrace{\dfrac{d}{dt} \displaystyle \int_V  E_u  \,\rho\,dV +   
                    \displaystyle \int_A  E_u \, U_{rn}\, \rho \,dA
                    -\dot{Q} - \dot{W}_{shear} }^{\text{ energy loss}} \tag{108}
\]
Equation (108) sometimes written as
\[
    \label{ene:eq:governingELoss}
     \dot{W}_{shaft} = \dfrac{d}{dt}  
            \displaystyle \int_V  \left( \dfrac{U^2}{2\dfrac{}{}} + g\,z\right) \,\rho\,dV + \\
          \displaystyle \int_A \left( \dfrac{P}{\rho} + \dfrac{U^2} {2\dfrac{}{}} + g\,z \right) U_{rn}\, \rho \,dA +
        \displaystyle \int_A P U_{bn} dA +   {\text{energy loss}} \qquad    \tag{109}
\]
Equation can be further simplified under assumption of uniform flow and steady state as
\[
\label{ene:eq:govSTSFUfixMassLoss}
     \dot{w}_{shaft}  =   
           \left.\left( \dfrac{P}{\rho} + \dfrac{U^2} {2\dfrac{}{}} + g\,z \right)\right|_{out}  -  
           \left.\left( \dfrac{P}{\rho} + \dfrac{U^2} {2\dfrac{}{}} + g\,z \right)\right|_{in}  
        + \mbox{energy loss} \tag{110}
\]
Equation (110) suggests that term \(h + \dfrac{U^2} {2} + g\,z\) has a special meaning (because it remained constant under certain conditions). This term, as will be shown, has to be constant for frictionless flow without any addition and loss of energy. This term represents the ``potential energy.'' The loss is the combination of the internal energy/enthalpy with heat transfer. For example, fluid flow in a pipe has resistance and energy dissipation. The dissipation is lost energy that is transferred to the surroundings. The loss is normally is a strong function of the velocity square, \(U^2/2\). There are several categories of the loss which referred as minor loss (which are not minor), and duct losses. These losses will be tabulated later on. If the energy loss is negligible and the shaft work vanished or does not exist equation (??) reduces to simple Bernoulli's equation.

Simple Bernoulli

\[
\label{ene:eq:SimpleBernolli}
     0 =  \left.\left( \dfrac{P}{\rho} + \dfrac{U^2} {2\dfrac{}{}} + g\,z \right)\right|_{out}  -  
           \left.\left( \dfrac{P}{\rho} + \dfrac{U^2} {2\dfrac{}{}} + g\,z \right)\right|_{in}   \tag{111}
\]

Equation (111) is only a simple form of Bernoulli's equation which was developed by Bernoulli's adviser, Euler. There also unsteady state and other form of this equation that will be discussed in differential equations Chapter.

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.