Bernoulli equation with pump and losses

What happens to the Bernoulli equation when there is a pump and when there are friction losses along the path?

Fluid-dynamics
Answer

The general equation, when applied between point 1 and 2, will look like this:

$$P_1 + \frac{\rho}{2} \cdot V_1^2 + \rho \cdot g \cdot h_1 = P_2 + \frac{\rho}{2} \cdot V_2^2 + \rho \cdot g \cdot h_2$$

The effects of the pump and friction losses can be included like follows:

$$P_1 + \frac{\rho}{2} \cdot V_1^2 + \rho \cdot g \cdot h_1 + H_{\text{Pump}}= P_2 + \frac{\rho}{2} \cdot V_2^2 + \rho \cdot g \cdot h_2 + h_{\text{friction}}$$

The pump contributes to increase the pressure available in the origin \(H_{\text{Pump}}\), while the pressure losses reduce the friction available in the down-stream point \(h_{\text{friction}}\).