How cold does it need to be for spit to freeze before hitting the ground?

Question

What is the dominant form of heat transfer between warm water and cold air?

If a $100 mg$ drop of water falls through $-40 C$ air, how quickly could it freeze?

Is it credible that in very cold weather spit freezes in the half a second it takes to reach the ground?

Answer 1

Consider a spherical drop of water, initial temp 40C, radius 3mm, mass 0.1g

To get it down to 0C, you need to remove 4.18 (J/gK) * 0.1 g * 40 K = 17 J

then, to freeze it solid, you need to remove latent heat of fusion 333 (J/g) * 0.1 g = 33 J

for a total of 50 J.

The heat conductivity equation is

$H=\frac{\Delta Q}{\Delta t} = k A\frac{\Delta T}{x}$

where $k$ is the thermal conductivity of water ($0.6 W/m\cdot K$), $A$ is the surface area, and $x$ is the thickness. Take $A=4\pi R^2$ and $x=R=3$mm, and you find that it would take 27 sec to freeze the drop of water in -40C.

Now in practice, the drop will be elongated, increasing $A/x$, and really only the surface layer needs to freeze, possibly eliminating 50-90% of the required latent heat of fusion, so in practice, I think it should be possible to freeze in about a second.

For a real answer, I think we need to go to Mythbusters!!