I was running the washing up water this morning, and started to think about why the cold tap isn't hot, and why the water doesn't get hotter the faster it is flowing (if anything, the cold tap gets colder the faster it flows).
From my understanding
$K.E = \frac{mv^2}{2}$
and temperature is directly proportional to kinetic energy.
I know that the $v$ in the above equation is really the mean speed of the particles and therefore some are moving backwards and some moving forwards, it is the speed that is used. But surely the particles of water in the tap are all moving faster, therefore they should all be hotter. Perhaps the particles in the stream are moving at a much higher mean speed than the water is flowing, so the temperature increase is negligible… Am I correct in thinking this? or otherwise, why doesn't the cold tap get hot the more you turn it on?
Best Answer
The water gets colder the longer you run it (in the UK at least) because the water mains pipes buried in the ground are colder than the ones in your house, so sadly this isn't evidence for any fundamental physical effect.
In principle any fluid flowing in a pipe gets hotter because energy is dissipated in viscous flow. You could in principle calculate the energy dissipated using the pressure drop per length of pipe, which is described by the Darcy-Weisbach equation, but this would be a somewhat involved calculation for real pipes/taps and in any case it isn't relevant to the core of your question.
When you relate velocity to temperature you're presumably thinking of the Maxwell-Boltzmann distribution for the temperature dependance of the velocity profile in gases. The trouble is this distribution is arrived at by considering redistribution of energy between gas molecules due to collisions between them. If you simply add a constant velocity to every gas molecule you aren't making any difference to the way the gas molecules collide with each other, because it's only their relative velocities that matter.
Although water is a liquid not a gas the same argument applies. It's the velocities of the water molecule relative to each other that determine the temperature. So just adding a constant velocity to every water molecule makes no difference.