Q. Best efficiency point and operating point for pumps?

In a pumping system, a system curve can be derived based on static head needed to lift up the fluid and variable head because of possible head losses. Pump curves that relate the performance of pumping to head against discharge can be attained from pump suppliers. When system curve is superimposed on the pump curve, intersection point is defined as operating point (or duty point). Operating point may not be essentially the same as best efficiency point. Best efficiency point is a function of the pump itself and it's the point of lowest internal friction inside pump at the time of pumping. These losses are persuaded by adverse pressure, shock losses and friction. 

Losses owing to adverse pressure gradient take place in pumps as the pressure of flow increases from inlet to outlet of pumps and flow travels from a region of low to high pressure. It causes the formation of shear layers and flow separation. Flow oscillation can also take place that accounts for noise and vibration of pumps. Effect of adverse pressure gradient is more substantial in low flow condition. 

For shock losses, they are persuaded when inflow into pumps isn't radial and comprises swirl. In an ideal situation, flow within the pump must be parallel to impellers such that flow angle is very close to impeller angle. Deviation of the above situation from design causes energy losses and vibration.

Fig: A diagram showing point of best efficiency ≠ operating point