Will increase in flow rate increase the rate of cooling down? Suppose a hot body is surrounded by a coil of very good thermally conducting pipe with cold water flowing inside which comes at a certain fixed low temperature in the pipe and leaves at some higher temperature after absorbing the heat. Will the rate of the hot body’s cooling down increase or decrease with increase in the water flow rate?
I think the rate of cooling will increase. But this is also in opposition to the intuition that the water will have less time to take away heat from the hot body, hence decreasing the cooling rate. I doubt that this is wrong since though the water is moving faster, water behind with lower temperature is following immediately after.
 A: The cooling rate will increase as the flow rate is increased.
The water in the pipe will heat up as it absorbs heat from the hot body, setting up a temperature distribution in the water within the pipe. Near the inlet it will have the temperature you pump it in with, and the temperature near the outlet will depend on how much heat it has absorbed from the hot body. The greater the flow rate, the less time water will have to absorb heat until it exits the pipe, and the colder it will be near the outlet.
According to Newton's law of cooling, the heat transfer rate to the water along a given piece of the pipe is proportional to the temperature difference between the hot body and the water. Since the water within the pipe is cooler on average when the flow rate is higher, this temperature difference is also higher on average, and so is the cooling rate.
The problem with your intuition is that although a given mass of water does have less time to absorb heat when water is being pumped in faster, now more water is being pumped in, by the same factor. The difference is that this mass of water does not get as hot by the time it leaves the pipe, so it is capable of absorbing more heat from the pipe per unit time as it flows.
