Apologies in advance if this has been addressed before... it's something if a daily puzzle during my morning ablutions and I haven't gotten a handle on it yet.
Consider the hot water pipe from my basement to second floor sink. Between uses this cools to ambient temperature; call it 15 degrees C.
When I turn on the hot water tap in this state, I am both flushing that cooled water out of the pipe and losing heat until the pipe comes up to (approximately, ignoring gradient and ongoing losses to the surroundings) the temperature of the incoming hot water (let's say 60Cish for simplicity?), at which point I get a steady stream of as-hot-as-it's-going-to-get water until I turn off the tap and the system starts cooling again.
The question is, do I waste less water and/or energy before that steady state is reached by running the tap at a lower flow rate, a higher flow rate, something in the middle, undeterminable without a complete model of the system, or does it actually not make much difference?
Currently I'm leaning towards "too many unknown variables, just try to measure it." But I'd be delighted to be wrong!