# Steady flow energy equation

For a steady flow through a control volume with a single inlet and single outlet, the energy equation can be written as:

$\dot{Q} - \dot{W}_s -\dot{W}_v = \dot{m}\left[(h_2 + \frac{1}{2}v_2^2 + gz_2) - (h_1 + \frac{1}{2}v_1^2 + gz_1)\right]$

where $\dot{Q}$, $\dot{W}_s$, $\dot{W}_v$ are the heat transfer rate, shaft work, and viscous work performed on the flow respectively, $h_1$ and $h_2$ are the inlet and outlet enthalpies respectively, $v_1$ an $v_2$ are the inlet and outlet fluid velocities respectively, $z_1$ and $z_2$ is the change in height of the fluid at the inlet and outlet respectively and $g$ is the gravitational constant.

What the steady flow energy equation says is that the stagnation enthalpy $h_0 = h + \frac{1}{2}v^2$ only defers between the inlet and outlet if heat is transferred to the fluid or if work is done on the fluid (assuming $z_1 = z_2$).

[1] Fluid Mechanics, 5th edition, White, F. M., McGraw-Hill, 2003.