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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?

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Do you really need to combine questions in mechanics (free fall in homogeneous gravity) and the completely orthogonal question in heat transfer here? I understand why this is interesting, but you should instead ask just for the nontrivial part (speed of freezing of water containing some compounds) and provide the current question just as a motivation. –  Marek Dec 29 '10 at 23:16
    
Okay. Good suggestion. –  Mark Eichenlaub Dec 29 '10 at 23:19
    
The n-th hand story always mention temperatures in the neighborhood of -30 or -40 $^\circ F$ (-35 to -40 $^\circ C$). Be they for spit or the dregs of a cup of coffee. Never had to deal with anything below about -10 $^\circ F$ myself, so I've no first hand knowledge. –  dmckee Dec 29 '10 at 23:26

2 Answers 2

up vote 3 down vote accepted

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!!

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Is this really correct? I always thought the heat transfer by conduction is only relevant for fluids and solids that are in contact. Gases are not dense enough to cause enough conductive cooling (think about the most extreme case: vacuum). The drop of water should mainly cool by evaporation, I think. There will also be some thermal radiation. I am not sure which effect is the most important, but your analysis definitely feels incomplete. –  Marek Dec 30 '10 at 17:35

Yes, I just did it. Arced a small spheriod of spit (no phlem) and it hit the ground and rolled instead of splattered. -15 degrees F, light wind in parking garage (not sure if that matters) attempted with numerous quantaties of spit and trajectories but small quantity with upward trajectory is what got the tiny frozen droplet.

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You might want to write your experiment up a bit more thoroughly: what was the temperature, at least! –  WetSavannaAnimal aka Rod Vance Jan 6 at 14:36
    
You can do that now with all the frozen weather we are having in the US. Go to anywhere around Chicago its at least -15F –  JPM Jan 6 at 21:27

protected by Qmechanic Jan 6 at 16:53

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