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#1
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Air resistance and aerodynamic heating
I'm trying to get a handle on aerodynamic heating. I'd like to be able
to answer questions like: How fast could the space shuttle go at sea level to experience the same heating as reentry (assuming same angle of attack)? If you used a electromagnetic rail gun as a first stage for a rocket launch, how fast could you have the rocket going when it left the rails at sea level....how much faster could it go if the rail gun were at 15000 ft elevation? Does anyone out there have any equations I can use to answer these questions? The perfect equation would let me input altitude and it would give me the fastest that a vehicle could travel at that altitude. A more complicated but more useful equation would let me input altitude and velocity and get out leading edge temperature. I assume that both of these equations are impossible because the geometry and construction of the vehicle would play a large part in the temperature. Does anyone have suggestions for how I can proceed? Here is what I have so far: From http://www.e31.net/luftwiderstand_e.html I got the equation: FAir = A/2 × Cd × D × v^2 FAir = force from drag (which I'm assuming is proportional to aerodynamic heating A and Cd = constants related to vehicle geometry and construction, so I'll lump them together into one constant D = density of air (which I can find from http://www.pdas.com/e2.htm) v = velocity Using this information I can't figure out anything about temperature, but I can compare the performance of the same vehicle at two different altitudes. For example if I assume my theoretical vehicle can go 1000 miles/hour at sea level without getting too hot, I can figure out how fast it can go at 15000ft elevation without getting too hot. Does anyone have any better way of calculating things that can get me some actual temperatures? Also, I'm concerned with my assumption that drag is proportional to heating. A pointy reentry vehicle would have less drag than a blunt reentry vehicle...but they discovered long ago that pointy things burn up on reentry. But I'm not really interested in reentry speeds. I'm more interested in speed around 2000mph, and elevations below regular commercial airliners. Thanks for any suggestions on how to proceed! |
#2
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Well, to start out: you cannot directly calculate the temperature -
you calculate the amount of energy coming in to the vehicle, you calculate the energy being radiated by the vehicle, and the rest of the energy goes into raising the vehicle's temperature. So, if you have a decellaration caused by air the energy put into the vehicles velocity (E=0.5*m*v^2) goes into the air. In your case, the energy put into the air is on the order of constant*Density*velocity^2 (at least that's what my quick math said, please check by calculating the derivitive of the kinetic energy of the vehicle being decellarated). Of course, the trick then is to figure out how much of the energy put into the air gets put back on the vehicle! At high altitude and hypersonic, the answer is about half. For pointy things, the answer is "lots". But for large curves a shockwave sets up in front of the vehicle, partially shielding it from the heat. So less of the heat gets to your vehicle (I usually guess 10%, but that is really only a guess - it depends on a lot of things). Typical aproach to dealing: Use large curves to lower the heat load, then use either: 1) things that can get hot enough to radiate the energy 2) things that have enough heat capacity to store the energy (ballastic trajectories have a low enough total heat load for this, orbit probably doesn't) 3) you let parts of your craft vaporize and take the energy away (ablation or boiling water) Hope that helps! -David |
#3
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I just realized that there is a far easier answer - the energy transfer
rate is approximately Force*velocity. So you can start there, and guestimate how much energy ends up in the vehicle. -David |
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