Most of the fuel of a rocket is its oxidizer. For the air, ramjets are an appealing alternative: just carry the reducer, squirt it into a combustion chamber, and let the hot compressed incoming air maintain the combustion!
The autoignition temperature of heptane is 223°, and it’s nearly as energy-dense as diesel or regular jet fuel: gasoline is 34 MJ/l versus diesel’s 39 or kerosene’s 35. So if the incoming air can get over 250° or so, a bit less than doubling its input temperature, we’re golden; from there it’s just a matter of adding the heptane or whatever gradually enough to avoid cooling the air below ignition temperature.
How much compression does that need? For an ideal gas, PV = nRT; in an isothermal process, where nRT is constant, PV = some constant C. So the short answer is that we need to double PV. But when we increase P, V decreases. By how much?
Well, in adiabiatic heating and cooling, PVn = C, where n is the adiabatic index, 7/5 for diatomic gases. So, I guess, if the volume is cut in half, then the pressure needs to increase by a compensating factor of 27/5 = 2.64, which means that the product PV and therefore the temperature increased by 32% (22/5 = 1.32). So to double the temperature we need to decrease the volume by 25/2 = 5.66, which will increase the pressure by 5.667/5 = 11.314, and 11.314/5.66 = 1.999.
(I had to write 24 lines of Python to figure that out.)
So we need about 11 atmospheres of pressure on the front of the ram in order to run the jet. How fast is that?
As I understand it, in isentropic compressible flow, the stagnation pressure is (1 + ½(n-1)M²)(n/(n-1)) times the static pressure of the surrounding air, at Mach M. Here n is 7/5, n-1 is 2/5, n/(n-1) is thus 7/2, so this simplifies to (1 + M²/5)7/2. So, to get 11 times higher stagnation pressure:
11 = (1 + M²/5)7/2
112/7 = 1 + M²/5
5(112/7 - 1) = M²
M = (5(112/7 - 1))½
This works out to be about Mach 2.22, about 760 m/s at sea level, if I’ve calculated everything correctly. But I suspect that it isn’t correct because Wikipedia talks about subsonic ramjets, and they surely aren’t using fuel that ignites at a much lower temperature than heptane, right? Indeed, WP says that they’ve been run as low as 45 m/s, but need to run at at least Mach 0.5 to be self-sustaining.
A crucial thing here is that the stagnation pressure and thus the stagnation temperature doesn’t depend on the scale or shape of the ramjet in any way; it’s the same for a millimeter-wide ramjet or a kilometer-wide ramjet. I’m not sure if that’s part of my error, though. The ideal-gas assumptions break down in the transonic region, as I understand it, but I don’t think that’s my problem.