Bernoulli from Euler with head losses

Euler's equation to Bernoulli's equation

Euler's equation of motion along a streamline $s$ for a steady flow of an ideal fluid:

$$\frac{dp}{\rho} + V\,dV + g\,dz = 0$$

Integrating along the streamline for an incompressible fluid ($\rho$ constant):

$$\frac{p}{\rho} + \frac{V^2}{2} + gz = \text{constant}$$

Dividing by $g$:

$$\frac{p}{\gamma} + \frac{V^2}{2g} + z = H = \text{constant}$$

This is Bernoulli's equation: the sum of pressure head, velocity head and elevation head is constant along a streamline (steady, incompressible, frictionless flow).

With head loss, turbine and pump

Between points 1 and 2, with energy supplied by a pump $H_p$ (added to the fluid), energy extracted by a turbine $H_T$ and head loss $h_L$:

$$\frac{p_1}{\gamma} + \frac{V_1^2}{2g} + z_1 + H_p - H_T - h_L = \frac{p_2}{\gamma} + \frac{V_2^2}{2g} + z_2$$

• $h_L$ = head lost to friction and minor losses between 1 and 2 (always positive).
• $H_p$ = head added to the flow by the pump (positive).
• $H_T$ = head taken from the flow by the turbine (positive).

Power: pump $P = \gamma Q H_p/\eta_p$ (input), and turbine $P = \eta_T\gamma Q H_T$ (output).