order of magnitudephase-transitionthermodynamicswater
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?
The equation you give:
$$ Q = m C_p \Delta T $$
just tells us the total amount of heat transferred, and does not tell us anything about the rate at which the heat transfer occurs. To calculate heat flow we have to solve the heat equation. If you do a physics degree this apparently innocent equation will cause you many hours of frustrated head scratching, so in practice we tend to use simple approximations. In everyday life heat flow tends to be well described by Newton's equation:
$$ \frac{dQ}{dt} \propto \Delta T $$
so, as you suggest, the greater the temperature difference the faster the heat flow.
The experiment you describe isn't really a good way of showing this, because if you mix hot and cold liquid in practice the rate of temperature change will be controlled by how fast you do the mixing. Newton's equation would be more useful if you place the two liquids in contact but don't allow them to mix e.g. have a metal (or some other high thermal conductivity) divider between them. In that case you're quite correct that the initial heat flow will be faster at 100°C than at 70°C. However the 100°C system will take longer to cool because the amoutn of heat, $Q \propto \Delta T$, that needs to be transferred is greater with a 100°C difference. The temperature difference as a function of time will look like:
$$ \Delta T(t) = \Delta T_0 e^{-\alpha t} $$
where $\Delta T_0$ is the initial temperature and $\alpha$ is a constant related to the thermal conductivity (large $\alpha$ means high thermal conductivity). If I use this equation to graph the cooling for 70°C and 100°C initial temperatures (choosing a random value of $\alpha$) I get:
So even though the 100°C difference initially cools faster the 70°C difference always reaches any specified temperature difference before the 100°C does.
As we don't know anything about the intial temparature of the water, another fun but unlikely explanation would be to consider what we call supercooled water.
Basically, at atmospheric pressure, water below 0°C doesn't have to be solid. Sure enough, it would be its most natural state. But actually if you cool down some very pure, very steady water, you can pretty easily get... a very cold liquid below 0°C. This is all because in order for the water molecules to re-arrange into Crystal of ice, they need some support to rely on, a little seed to grow around (we call that a nucleus). So if there is no impurity in that water, then you can cool it down without freezing it ! You have to be careful to keep it steady too, because the energy of a shock would give the kick it needs to start the crystallization. You can find plenty of YouTube videos demonstrating this phenomenon. So if we assume that your water is already below zero, and in a supercooled state, then the swimmer in it would kickstart the crystallization, which would happen very quickly. For all I know, it is plausible that she'd then get trapped Under ice.
Your stroy makes me think of that because there is a horrible but pretty cool legend about a lake in Russia called the Ladoga lake. Alledgedly, during the Siege of Leningrad in 1942, in a cold winter's night, a fire started in the forest, scaring off a thousand of horses. Those who could run away from the flames plunged into the cold waters of the lake. And suddenly, with a sound of broken glass, the lake froze as the horses were jumping in it, trapping hundreds of them them in ice. I think it's Hubert Reeves who first suggested that the lake might have been in supercooled state, which could explain this sudden freezing, with the carateristic sound of shattered glass. However most people suggest it's just bullshit, as the water would have to be insanely pure and steady (which is not very natural). Furthermore, I have never seen an experiment about supercooled wtaer on such big scales. I don't know if this would hold.
So in short, supercooled water might be an (unlikely) way to justify it. But not so bad regarding series standards :)
Best Answer
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!!