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7.3.2: Energy Equation in Frictionless Flow and Steady State

In cases where the flow can be estimated without friction or where a quick solution is needed the friction and other losses are illuminated from the calculations. This imaginary fluid reduces the amount of work in the calculations and Ideal Flow Chapter is dedicated in this book. The second low is the core of "no losses'' and can be employed when calculations of this sort information is needed. Equation (??) which can be written as   
\[
    \label{ene:eq:2law}
    dq_{rev} = T\,ds = dE_u + P\, dv \tag{76}
\]
Using the multiplication rule change equation (76)
\[
    \label{ene:eq:2lawMulti}
    dq_{rev} = dE_u + d\left(P\,v\right) - v\,dP = dE_u + d \left(\dfrac{P}{\rho}\right) - v\,dP    \tag{77}
\]
integrating equation (77) yields
\[
    \label{ene:eq:2lawMultiInt}
    \int dq_{rev} = \int dE_u + \int d \left(\dfrac{P}{\rho\dfrac{}{}}\right) - \int v\,dP    \tag{78}
\]
\[
    \label{ene:eq:2lawMultiIntA}
    q_{rev} = E_u +  \left(\dfrac{P}{\rho\dfrac{}{}}\right) - \int \dfrac{dP}{\rho}    \tag{79}
\]
Integration over the entire system results in  
\[
    \label{ene:eq:2lawSys}
    Q_{rev} = \int_V \overbrace{\left( E_u + \left(\dfrac{P}{\rho\dfrac{}{}}\right) \right)}^{h} \,\rho\,dV -
         \int_V \left( \int \dfrac{dP}{\rho\dfrac{}{}} \right) \,\rho\, dV \tag{80}
\]
Taking time derivative of the equation (80) becomes
\[
    \label{ene:eq:2lawSysRate}
    \dot{Q}_{rev} = \dfrac{D}{Dt} \int_V \overbrace{\left( E_u + \left(\dfrac{P}{\rho\dfrac{}{}}\right) \right)}^{h} \,\rho\,dV -
         \dfrac{D}{Dt} \int_V \left( \int \dfrac{dP}{\rho\dfrac{}{}} \right) \,\rho\, dV  \tag{81}
\]
Using the Reynolds Transport Theorem to transport equation to control volume results in
\[
    \label{ene:eq:2lawCVRate}
    \dot{Q}_{rev} = \dfrac{d}{dt} \int_V {h} \,\rho\,dV  
    + \int_A h\,U_{rn} \,\rho\,dA +  
         \dfrac{D}{Dt} \int_V \left( \int \dfrac{dP}{\rho\dfrac{}{}} \right) \,\rho\, dV \tag{82}
\]
As before equation (81) can be simplified for uniform flow as
\[
    \label{ene:eq:2lawU}
    \dot{Q}_{rev} = \dot{m} \left[ \left( h_{out} - h_{in} \right)  -  
        \left( \left. \int \dfrac{dP}{\rho\dfrac{}{}} \right|_{out} - \left. \int \dfrac{dP}{\rho} \right|_{in} \right)   
    \right]  \tag{83}
\]  
or  
\[
    \label{ene:eq:2lawh}
    \dot{q}_{rev} = \left( h_{out} - h_{in} \right)  -  
        \left( \left. \int \dfrac{dP}{\rho\dfrac{}{}} \right|_{out} - \left. \int \dfrac{dP}{\rho} \right|_{in} \right)   \tag{84}
\]  
Subtracting equation (84) from equation (75) results in
\[
    \label{ene:eq:frictionlessEne}
    0 = w_{shaft} +  
    \overbrace{\left( \left. \int \dfrac{dP}{\rho\dfrac{}{}} \right|_2 - \left. \int
\dfrac{dP}{\rho\dfrac{}{}} \right|_1 \right)  }
                ^{\text{change in pressure energy}} \\  
        + \overbrace{\dfrac

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^{\text{change in kinetic energy}}  
        + \overbrace{g\,(z_2 - z_1)}^{\text{change in potential energy}}   \tag{85}
\]  
Equation (85) for constant density is
\[
    \label{ene:eq:frictionlessEneRho}
    0 = w_{shaft} +  
        {\dfrac{P_2 - P_1}{\rho} } + {\dfrac
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    at (Textbook_Maps/Map:_Fluid_Mechanics_(Bar-Meir)/07:_Energy_Conservation/7.3_Approximation_of_Energy_Equation/7.3.2:_Energy_Equation_in_Frictionless_Flow_and_Steady_State), /content/body/p/span[2], line 1, column 4
+ {g\,(z_2 - z_1)} \tag{86}
\]
For no shaft work equation (86) reduced to
\[
    \label{ene:eq:frictionlessEneRhoShaft}
    0 = {\dfrac{P_2 - P_1}{\rho} } + {\dfrac
ParseError: EOF expected (click for details)
Callstack:
    at (Textbook_Maps/Map:_Fluid_Mechanics_(Bar-Meir)/07:_Energy_Conservation/7.3_Approximation_of_Energy_Equation/7.3.2:_Energy_Equation_in_Frictionless_Flow_and_Steady_State), /content/body/p/span[3], line 1, column 4
+ {g\,(z_2 - z_1)}  \tag{87}
\]

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.