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.)
Best Answer
I will assume your heat exchanger uses the common counter flow principle - that is, the direction of flow of the "cold" water opposes that of the "hot" water, so the hottest water in the heating loop (entering the heat exchanger) is in touch with the hottest water of the pool loop (just before exiting the heat exchanger).
The heat flow across the exchanger is proportional to the temperature difference. Since the "input temperature" is fixed at 180 F, the only variable is the temperature of the "sink" - the pool water. The colder the pool water, the greater the heat flow.
At the input of the heat exchanger, the temperature is the temperature of the pool; at the exit, it will be somewhat warmer. The slower the water flows, the more heat it will pick up, and the hotter the water that re-enters the pool. However, the hotter the pool water in the exchanger, the smaller the thermal gradient, and therefore the smaller the heat flux into the pool water.
The water will heat most rapidly if the pool water runs quickly - this keeps the temperature difference greatest.
There is just one caveat: the power of the pump moving the water. If the pump is working harder to move water through a constricted valve, it would generate a little bit more power; if the water flow is set up so heat from the pump is dumped to the water, you will get a small amount of additional heating; but I don't believe that would ever offset the benefit of the faster gradient.
One other consideration: what happens to the surface of your pool. This relates to the way the output of your heat exchanges returns to the pool. If you have a jet that dumps deep inside the pool, there will be little disturbance at the surface; if it's aimed at the surface, you will cause some "stirring". As you may know, the greatest heat loss from a pool happens through evaporation - so if there is anything in your setup that increases evaporation as a function of flow rate through the heat exchanger, that will affect the total heating time.
If it were my pool, I would probably rig up a thermocouple and a data logger, and look at the evolution of temperature. Turn the flow rate up and down every two hours or so, and see if you can observe a change in heating rate on the temperature trace.
I am sorry - according to the laws of physics, your pool guy is wrong. Open that valve, and let the water flow!