The brief
The 640i was running hotter than normal, the temperature gauge fluctuating up and down rather than sitting steady. And the owner could hear a faint whirring from the front of the engine bay.
But there was no visible coolant on the underbody, and no drop in the expansion tank. That last detail is the giveaway: an overheating problem with no coolant loss is not a leak, it is a flow problem. On this engine the coolant is pumped by an electric water pump rather than a belt-driven one, and when that pump starts to fail, flow drops, the engine runs hotter than it should, and the failing pump bearings whir.
The diagnosis
We pressure-tested the cooling system: clean, no external leaks, which fit the no-coolant-loss picture. The scan tool told the rest of the story, flagging the cooling system for the coolant pump's motor speed and flow being outside tolerance, and an intermittent fault for the electric coolant pump dropping off the bus entirely.
So the pump was being asked to deliver flow it could no longer manage, and sometimes it was cutting out altogether. The whirring was the failing bearings. Pump replacement, the obvious fix once the diagnosis pointed there, no point patching an electric pump that is on its way out.
The work
Released the system pressure, drained the coolant into a clean catch, and removed the failed electric pump. Fitted a new BMW-spec replacement with a fresh O-ring.
Refilled with the correct coolant mix at the right ratio, ran the bleed cycle through the scan tool to clear the air pockets a modern cooling system traps after a refill, and verified the new pump was delivering proper flow under load before the car went out.
The outcome
Temperature gauge steady at normal, no fluctuation. The whirring gone.
The 640i went home with the cooling system reading clean. For the owner, that means a car that runs at the temperature it is designed to, with no more watching the gauge wander.
And catching a failing electric coolant pump at this stage matters: an electric pump that finally stops dead can overheat an engine in minutes with no warning, which on a 6 Series is the worst-case end of this story.