# Is temperature equilibrium actually reached in a mixer tap, or is it merely a jumbling of hot/cold droplets?

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.

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?