Energy Balance of Control Volumes

In the previous section work for the energy of a closed system was discussed. These however are not the only types of systems, there are also open systems where a control volume is setup. That control volume can have a mix of imaginary boundary layers and real boundary layers. Due to the fact these control volumes can have imaginary boundary layers mass could flow through the system. If this is occurring, conservation of mass would need to be taken in consideration to determine how the mass flow rate of the mass flowing through system. Once the mass flow rate is known, flow work can be determine by using equation 1.

(Eq 1)  $W_{flow}=Pv$

$P$ = Pressure

$v$ = velocity

For a closed system, the general energy equation would be represented by equation 2. However, for an open system the flow work needs to be added to the equation. Refer to equation 3.

(Eq 2)  $e=u+ke+pe$

$u$ = internal energy per unit mass

$ke$ = kinetic energy per unit mass

$pe$ = potential energy per unit mass

(Eq 3)  $Φ=Pv+e=Pv+u+ke+pe$

Now recall that the equation for enthalpy is flow work plus the internal energy (equation 4). This means that equation 3 can be modified to equation 5 and enthalpy can be used to represent the flow work of the mass flowing through the system.

(Eq 4)   $h=Pv+u$

(Eq 5)  $Φ=h+ke+pe$

Finally, using conservation of mass the mass flow of the fluid flowing through the system can be determined using a basic equation like equation 6. Once the mass flow value is known it can be multiplying by the energy equation to determine power required to move the mass through the control volume, refer to equation 7.

(Eq 6)  $\dot{m}=ρvA$

$ρ$ = density

(Eq 7)  $\dot{E}_{mass}=mΦ$

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