Because multi-stage boiler feed water pumps can generate excessive heat/temperatures at low/no-flow conditions, for example against closed discharge valves - a leakoff (recirc/bleed) is used to ensure there is a minimum amount of flow through the pump, thusly preventing damage to the pump.
......... the "balance" part of your question was lost by me ..........
did you mean a balance device that is used to determine if there is wear in the annular space by determining if there is an increase in flow through this line? If this is the case, then the balance line is used to determine the relative condition of the pump - more flow through the balance line is indicative of more internal wear/clearances.
Or another explanation:
For between bearings pumps, there are two seal chamber and a leak off or balance line system may be included so both seal chambers operate at the same pressure
Lastly a third explanation:
In the case of multistage pumps, the differential pressure across the impellers builds with each stage and some method must be employed to reduce the load on the thrust bearing. In some designs, axial thrust is balanced by means of a balance piston/leak-off arrangement; in others, an opposed impeller arrangement helps to balance axial loads.
I think you are writing about auto recirculation valve (ARV). This valve is typically mounted at pump outlet to ensure minimum fluid flow through pump in all operating conditions.
If you know anything about balance leak-off, please revert
balance leak off
Re: balance leak off
