This is motivated by a practical problem, but it's the pure physics that really puzzles me.
My pool is heated via a heat exchanger. Water from the boiler enters the heat exchanger through (let's call it) Pipe A and leaves through Pipe B. Water from the pool enters the heat exchanger, comes into contact with these pipes, and returns to the pool. Pipe B is consistently cooler to the touch than Pipe A; it seems clear that this temperature difference measures the rate at which heat is being transferred to the pool water.
I can turn a valve that controls the rate at which pool water enters (and leaves) the heat exchanger. My instinct is to leave this valve maximally open. My pool guy insists that this is a mistake; instead there is some optimum rate, less than the available maximum, at which water should be sent through the heat exchanger. His argument is that if the water passes through the exchanger too fast, it doesn't have time to pick up much heat. My counter-argument is that yes, if you increase the flow rate, then any given volume of water will pick up less heat per minute --- but you're heating a greater volume per minute.
Moreover, my intuition tells me that these effects should exactly cancel --- the rate of heat transfer between the PipeA/PipeB circuit and the pool water should depend only on the current temperature difference between Pipe A and the pool water, and therefore (at least above a certain minimum) the rate of pool water flow should be irrelevant. My pool guy's experience tells him otherwise.
Is he right, and if so, exactly what determines the optimal rate of pool water flow?
Edited to add: To clarify what I'm optimizing: I want to minimize the time it takes to get the pool water from some initial temperature to some (higher) desired temperature.
Edited to add: The water coming from the boiler is always kept at a fixed 180 degrees, and returns at a lower temperature. Therefore the boiler works harder when more heat is being transferred to the pool water (more heat transfer implies colder return to the boiler implies more work for the boiler to reheat that water). So answers that assume a fixed amount of work by the boiler seem to me to be at best incomplete.
(And just to clarify even further: There are two thermostats. One turns off the boiler when the water in Pipe A hits 180 degrees; the other turns off the boiler when the pool water reaches the desired temperature.)