I understand that air friction cools off an object at low speeds. For example, if you blow on a spoon of hot soup, it cools off. Or if you swing a hot frying pan in the air, it cools off faster.
But at higher speeds, the situation changes in the opposite. For example, consider a meteor falling on Earth. It is so fast that it heats up to such a high temperature that it burns into ashes.
What is the critical speed in which an object starts to heat up?
For example, consider a spherical object with radius of 1 meter. Let its density be 1 g/cm3 if needed. And let the air pressure be 1 atm. Assume that the temperature of the object is 400 K and the temperature of the air is 300 K. Also assume that the specific heat constant of the material is 1 cal/(gr.K). What is the critical speed for this object after which its temperature starts to rise above 500 K?
Best Answer
John's answer is a good one, I just wanted to add some equations and addition thought. Let me start here:
The question asks specifically about a $200^{\circ} C$ increase in temperature in the atmosphere. This qualifies as "significant" heating, and the hypothesis that this would only happen at supersonic speeds is valid, which I'll show here.
When something moves through a fluid, heating happens of both the object and the air. Trivially, the total net heating is $F d$, the drag force times the distance traveled. The problem is that we don't know what the breakdown is between the object and the air is. This dichotomy is rather odd, because consider that in steady-state movement all of the heating goes to the air. The object will heat up, and if it continues to move at the same speed (falling at terminal velocity for instance), it is cooled by the air the exact same amount it is heated by the air.
When considering the exact heating mechanisms, there is heating from boundary layer friction on the surface of the object and there are forms losses from eddies that ultimately are dissipated by viscous heating. After thinking about it, I must admit I think John's suggestion is the most compelling - that the compression of the air itself is what matters most. Since a $1 m$ ball in air is specified, this should be a fairly high Reynolds number, and the skin friction shouldn't matter quite as much as the heating due to stagnation on the leading edge.
Now, the exact amount of pressure increase at the stagnation point may not be exactly $1/2 \rho v^2$, but it's close to that. Detailed calculations for drag should give an accurate number, but I don't have those, so I'll use that expression. We have air, at $1 atm$, with the prior assumption the size of the sphere doesn't matter, I'll say that air ambient is at $293 K$, and the density is $1.3 kg/m^3$. We'll have to look at this as an adiabatic compression of a diatomic gas, giving:
$$\frac{T_2}{T_1} = \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}}$$
Diatomic gases have:
$$\gamma=\frac{7}{5}$$
Employ the stagnation pressure expression to get:
$$\frac{P_2}{P_1} = \frac{P1+\frac{1}{2} \rho v^2}{P1} = 1+\frac{1}{2} \rho v^2 / P1 $$
Put these together to get:
$$\frac{T_2}{T_1} = \left( 1+\frac{1}{2} \rho v^2 / P1 \right)^{2/7}$$
Now, our requirement is that $T2/T1\approx (293+200)/293 \approx 1.7$. I get this in the above expression by plugging in a velocity of about $2000 mph$. At that point, however, there might be more complicated physics due to the supersonic flow. To elaborate, the compression process at supersonic speeds might dissipate more energy than an ideal adiabatic compression. I'm not an expert in supersonic flow, and you can say the calculations here assumed subsonic flow, and the result illustrates that this is not a reasonable assumption.
addition:
The Concorde could fly at about Mach 2. The ambient temperature is much lower than room temperature, but the heatup compared to ambient was about $182 K$ for the skin and $153 K$ for the nose. This is interesting because it points to boundary layer skin friction playing a bigger role than I suspected, but that is also wrapped up in the physics of the sonic wavefront which I haven't particularly studied.
You have to ask yourself, what pressure is the nose at and what pressure is the skin at. The flow separates (going under or above the craft) at some point, and that should be the highest pressure, but maybe it's not the highest temperature, and I can't really explain why. We've pretty much reached the limit of the back-of-the-envelope calculations.
(note: I messed up the $\gamma$ value at first and then changed it after a comment. This caused the value to go from 1000 mph to 2000 mph. This is actually much more consistent with the Concorde example since it gets <200 K heating at Mach 2.)