Control volume with one inlet and one outlet
A control volume allows mass exchange through inlets and outlets. The mass flux at each inlet and outlet is equivalent to the product of the density, cross sectional area, and velocity. To avoid confusion with volume or specific volume, we will use $\bold v$ for velocity.
$$ \dot{m}=\rho A\bold{v} $$
For a control volume in steady-state, all properties throughout the control volume are unchanging with respect to time. This gives rise to the mass rate balance and energy rate balance. Note that unlike a closed system, which exchanges energy with its surroundings through work and heat transfer, the control volume also exchanges energy with its surroundings thorough the energy exchange at the inlets and outlets. This energy exchange by mass-flux includes internal energy, kinetic energy, and gravitational potential energy.
$$ \Sigma \dot{m}=0=\dot m_i-\dot m_e $$
$$ \dot{U}=0=\dot{Q}{{CV}}-\dot{W}+\Sigma\dot{m}(u+\frac{\bold{v}^2}{2}+gz) $$
The total power of the system is comprised of the control volume work (such as power generated by a turbine) plus work done at inlets and outlets. The pressure maintained at the inlets and outlets does work on the fluid incoming and outgoing, so this must be considered.
$$ \dot{W}=\dot{W}{{CV}}+\Sigma \dot{m}pv $$
Substituting enthalpy for internal energy gives a more direct form of the energy rate balance, because the $pv$ term cancels with the inlet and outlet work. Note that going forward, this form will be used, and so the $_{CV}$ subscripts will be left off.
$$ \dot{U}=0=\dot{Q}{{CV}}-\dot{W}{{CV}}+\Sigma\dot{m}(h+\frac{\bold{v}^2}{2}+gz) $$