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Most kitchen and bathroom sink faucets have a mixer tap which blends the hot and cold inflows into a warm, uniform outflow stream. It's difficult to believe that complete heat transfer and temperature equilibrium is reached within the fraction of a second in which the hot/cold water flows through the mixer mechanism.

It makes me wonder if the mixer is doing nothing more than breaking the inflows into droplets and jumbling those droplets together, which yields the sensation (illusion) of a warm, uniform outflow.

Does the difference matter? In the extreme, it would. We can imagine some sort of super mixer tap that can take in an ultrahigh temperature inflow (like mafic lava) and a very low temperature inflow (like liquid helium) and blend them together. If complete heat transfer and equilibrium is actually reached within the mixer, then, with the right proportion of inflows, the warm outflow should be safe to touch. However, if the outflow is just a jumble of discrete hot/cold droplets, then it would be very dangerous to touch. I imagine the still-hot and still-cold droplets would cause both burning and freezing damage to the skin.

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When each of the fluids is broken down into small parcels in intimate contact, the surface to volume ratio of each parcel becomes large, and the conductive heat transfer between parcels becomes enormously enhanced. The time for a parcel temperature to equilibrate with that of its neighbors is on the order of $t=D^2/\alpha$, where D is the nominal parcel diameter and $\alpha$ is the thermal diffusivity of the liquid. The thermal diffusivity of water is about 0.0015 cm^2/sec. For a 0.01 cm diameter parcel, what do you calculate for the equilibration time?

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  • $\begingroup$ This is a very clear and succinct answer, thank you! I don't know what parcel sizes a mixer tap creates internally, but if they are just slightly bigger than your example, let's say 0.03 cm or 0.04 cm in diameter, then the equilibration time becomes 0.6 seconds and 1.07 seconds, respectively. This is more time than water actually spends in the mixer tap, so we better hope that our mixer taps do a good job breaking the inflows into very, very tiny parcels! $\endgroup$ – SlowMagic Nov 4 '18 at 16:04
  • $\begingroup$ The breakup is also partially comprised of thin striations of hot and cold fluids. This is the usual model of mixing that is often envisioned. The striations have to be thin, of course. Also, the times calculated from the approximate equation are those calculated to produce virtually complete equilibration. Partial equilibration will require less time, but will fortunately exhibit a reduced temperature range. So you don't have to wait to full time to obtain a greatly reduced temperature range. $\endgroup$ – Chet Miller Nov 4 '18 at 17:06

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