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My heroine has been stabbed - run through the abdomen by a foot-long dagger - and is bleeding to death. She is currently crawling toward her salvation through a very cold place. She is magically immune to the harmful effects of low temperatures, but her blood, once it has left her body, is not.

I wrote a passage stating that as she crawled on hands and knees, her blood was dripping from the point of the dagger that has impaled her, and the environment was so cold that the blood didn't splatter on the ground, but instead froze solid and bounced.

My heroine is for purposes of this question a perfectly proportioned human female 170cm tall, weighing 70kg. Despite the extremely cold environment, her body temperature is a normal 37°C. The point of the dagger is protruding from her abdomen so that it's point is level with the lowest point of her belly. She is crawling on hands and knees, with straight arms. While she has been stabbed, and major organs are involved, the continued presence of the dagger is acting as a plug, preventing a rapidly fatal loss of blood. Most of her loss of blood is going into her abdomen, but enough is leaking out around the point of the dagger that it is dripping rather than gushing. Gravity is a normal 9.8m/s^2, and atmospheric pressure is around 60kPa, about average for an altitude of a bit over 4000m ASL. Please assume that the temperature of the blood droplets start at 37°C, and cooling does not begin until the droplet is falling freely, and the droplets must be frozen completely solid before hitting the ground, which for purposes of this question is effectively a flat surface. The blood droplets have not begun to clot significantly.

So, my question is this: Just how cold would it have to be for human blood to freeze solid before hitting the ground, so that the frozen droplets would bounce rather than liquid blood splattering? Is the required temperature realistic, in that Nitrogen and Oxygen can remain gaseous, and/or the temperature above 0K, or must it be "Magically cold"?

I am looking for answers that include all necessary proof and calculations, hence the hard-science tag. I am not looking for educated guesses without a basis on fact. While I mention magic, magic is only involved perforce in setting the starting conditions, and the answer will be: Possible, Possible but x atmospheric gases would not remain gaseous at the required temperature, or Not Possible due to the required temperature being below 0K.

Edit:

The droplets of blood will be falling a distance of approximately 30cm, though this may vary from 25-35 cm according to the exact position of the victim as she attempts to reach salvation. The blood droplets should therefore freeze solid after a downward journey of 25 cm at minimum.

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This question asks for hard science. All answers to this question should be backed up by equations, empirical evidence, scientific papers, other citations, etc. Answers that do not satisfy this requirement might be removed. See the tag description for more information.

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    $\begingroup$ Are you sure you want the frozen droplets to bounce? Usually frozen things are incredibly brittle and if your temperature was that low, shattering would be much more plausible than bouncing. $\endgroup$ – Shadowzee Feb 14 at 5:25
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    $\begingroup$ So in summary you have three specific questions then : 1) how far above the ground is the abdomen of a normal human female of these proportions when on her hands & knees. 2) how long will it take for a drop of liquid to fall that distance. 3) What temperature is required to freeze a drop of blood solid inside that time? $\endgroup$ – Pelinore Feb 14 at 5:30
  • $\begingroup$ @Shadowzee, if you wanted to calculate if the frozen blood droplets would bounce or shatter on impact, go right ahead. My requirement is really only that the droplets freeze solid. However, given that they need only be near 0°C at impact, means that the droplets are unlikely to display behavior attributed to supercooled substances. In my experience, small hailstones typically bounce, and do not shatter. $\endgroup$ – Monty Wild Feb 14 at 5:35
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    $\begingroup$ @MontyWild Can you tell us how high it will be dropping from? No need for us to create an equation where both height/time and temperature are variables. Locking in the height will make it much easier, especially since the character is crawling on her hands and knees. $\endgroup$ – Shadowzee Feb 14 at 5:40
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    $\begingroup$ It's not about temperature, it is more about en.m.wikipedia.org/wiki/Heat_transfer_coefficient $\endgroup$ – Mołot Feb 14 at 7:06
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It's not possible

I don't believe it's possible to freeze a droplet in the time it takes to fall 25 cm. Here is why:

Droplet properties

Thanks to Molot's link (reproduced here), a good estimate for blood droplets from weapons varies from 4 to 6mm. We can take the average and say 5mm diameter droplets. Using the density of water (1g/mL), we get 0.06 grams per droplet.

Time of flight:

To fall 25cm from rest requires 0.225 seconds ($t = \sqrt{\frac{2\Delta y}{g}}$). Air resistance can be ignored in this case: compare the final speed before impact of 2.2 m/s ($v_f = gt$) to the terminal velocity of 11.5 m/s ($v_t = \sqrt{\frac{2mg}{\rho A C_d}}$). Or put another way, the drag force is only about 2% of the weight of the droplet at the point of impact.

heat of fusion of water:

Blood (aka water) has a heat of fusion of 334 J/g and a heat capacity of 4.148 J/gK. Therefore to reduce the temperature a droplet of water (with mass 0.06g) and freeze the entire thing would require the removal of 32 Joules of energy ($Q = m C \Delta T - m L_f$ ).

Rate of heat loss:

Removing 32 Joules in 0.225 seconds requires an average power of 141 Watts ($P = Q/t$). Blood (aka water) has an average thermal conductivity of 0.5915 W/mK between 0 and 37 degrees.

To pull 32 Joules of energy out of a droplet of blood 5mm in diameter in 0.225 seconds would require a temperature gradient of -7623K.

($P = \frac{kA\Delta T}{L}$). This assumes the entire surface area ($A = \pi D^2$) transfers heat and the distance is across the radius ($L = D/2$). It also assumes a constant temperature gradient.

Conclusions

It is not possible to pull 32 Joules of energy out of a droplet of only 5mm in diameter in 0.225 seconds. The coldest temperature gradient possible would be between absolute zero and 37C, or around -300 Kelvin. My estimate is something 25 times larger than that. Therefore, unless I have made an assumption that is over a magnitude off, I just don't see how you can freeze a droplet in such a short amount of time. Even re-running the numbers and estimating that only the first 1mm layer of blood need freeze (i.e. a shell of ice) produces an estimate of -6000K.

Addendum

I don't want to leave you with a killjoy answer. Running the numbers in reverse, a droplet 0.5mm in diameter would freeze through with a temperature gradient of -76K. That puts the ambient temperature at 37C-76 = -39C -- very believable. While your heroine wouldn't drip droplets that small, she might cough up blood in a mist that would immediately freeze, the tiny beads making tinkling noises across the icy floor like throwing sand on glass.

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  • $\begingroup$ What is the largest droplet size that could freeze solid in these conditions? $\endgroup$ – Monty Wild Feb 19 at 23:43
  • $\begingroup$ Unfortunately, my heroine was stabbed in the kidney from behind, so she won't be coughing up blood. $\endgroup$ – Monty Wild Feb 19 at 23:45
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The primary blood drop size ranged from 4.15 ± 0.11 mm up to 6.15 ± 0.15 mm (depending on the object), with the smaller values from sharper objects. Source

Wound is blunt, and dagger handle is blunt, so $6mm$ diameter, $113mm^3$ volume and, lucky coincidence, $113mm^2$ surface area, is good bet within scientific range. Of course, each drop will be different - thus, selecting higher end of sizes makes sense, if big drops will freeze, smaller ones will, too. For completeness, drops from the point of the knife will be at $4.15mm$, $37.42mm^3$ volume and $54.11mm^2$ surface

Freezing blood is a complicated matter, as we can read here, blood is frozen at $-3°C$, so we need to cool it by about $40°C$.

The specific heat capacity of water is 4.18 and of the human body (blood and tissues) $3.49 kJ/kg°C$, respectively. source

So, it is a bit easier to freeze blood than it is to freeze water.

Blood density varies from source to source see here - it is up to $1066 kg/m^3$. This gives $120.45800mg$ of blood to freeze from wound / handle, and $39.88972mg$ from the tip

This gives us $16.8159368J$ of heat to remove from drop of blood to put it at the edge of freezing foe big drops, and $5.56860491J$ for smallest ones we can expect.

As per this comment - I'm posting this as a partial answer with intent to finish when I'll have time to do further research.


Heat transfer coefficient of air is up to 1kW per square meter per 1K temperature difference. Source. This source provides rather wide ranges, so potentially can be only used to prove "impossible"

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  • $\begingroup$ The blood would be dripping from the point of a fairly sharp dagger. Doesn't that article suggest that smaller droplets would be formed? $\endgroup$ – Monty Wild Feb 14 at 11:37
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    $\begingroup$ @MontyWild I must've misunderstood, I thought it was dripping from the wound or blunt part of the dagger, and blade is inside person. I'll add calculations for both ends of drop size spectrum when ill have time. $\endgroup$ – Mołot Feb 14 at 11:57
  • $\begingroup$ Where's the latent heat of fusion? It should be around 330 kJ/kg, thus about two times as large as the heat difference between 37 °C and −3 °C. (Just cooling an amount of water won't freeze it; to freeze it needs to dissipate the latent heat of fusion.) (P.S. Units of measurement must be shown in Roman type, with a space before them; $6mm$ is incorrect, $6\,\mathrm{mm}$ is correct; although I don't understand at all what good does LaTeX formatting bring.) $\endgroup$ – AlexP Feb 15 at 17:13
  • $\begingroup$ @AlexP as you can see, i did it step by step by step and ended just before latent heat of phase change. As you can see in the answer ans comments, this answer is partial, with permission of OP and diamond mod. I'll expand it when I can. $\endgroup$ – Mołot Feb 15 at 18:50
  • $\begingroup$ Using the largest figure of 1000 J/s the cooling seems doable. But the heat transfer for a falling drop is hc = 10.45 - v + 10 SQRT(v), and in our speed range it stays between 10 and 32 W/(m^2K). Also, it seems likely that the drop would freeze on the outside, stopping internal convection and insulating the inside, with the result that the drop would splatch at the end of the fall rather than shatter or bounce as it would do if frozen solid. We need to increase the heat transfer - maybe imagine a very dense, wet fog (sulphur hexafluoride?) and low gravity. $\endgroup$ – LSerni Feb 19 at 23:07

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