In article <firstname.lastname@example.org> email@example.com writes:
To simplify our derivation we assume that the rocket is frictionless (except for engine drag)...
Fairly unrealistic for an airbreather, alas.
The June 1994 JBIS had a very interesting paper by Bob Zubrin, suggesting a methane-fueled variant of NASP. (The trick was that he used friction heat to turn the methane into an acetylene/hydrogen mix before burning it.) What I found more interesting, though, was his derivation of an extended version of the rocket equation for high-speed airbreathers.
The one key assumption required is that you have to know how the specific impulse varies with speed. Turns out, though, that a relatively good approximation -- valid from turbojets to scramjets, starting at about Mach 2 and going up a long way from there -- is to simply assume that Isp is inversely proportional to speed.
Oh yes, we also assume a standing start, V_initial = 0.
Given that, with some algebraic manipulation one ends up with a fairly simple equation. I'll state it a bit differently than he did:
V_final = Isp_half * g * ln(mass_ratio) * 1/(1 + 1/(L/D * A/g))
Isp_half is the Isp at V_final/2. This in itself is interesting -- it says that the huge Isp advantages to be had at low speeds buy you very little, because so much of your accelerating is done at high speeds where airbreathing Isp falls off badly, to 1000s or so.
The really fun part, though, is that last term. The second term in its denominator, 1/(L/D * A/g), is what you might call the Air Breather's Burden. L/D is average lift/drag, a familiar basic measure of aerodynamic performance for winged vehicles. And A/g is just average forward acceleration in Gs. If either of these goes to infinity -- you have either miraculous aerodynamics or tremendous acceleration -- the value of the ABB goes to 0, the value of the whole last term goes to 1, and you get the familiar rocket equation.
But if they don't go to infinity, what you get is trouble. Hypersonic L/D ratios typically are not good. Zubrin guesses L/D of 5 for NASP and 7 for his design (better because methane tanks are more compact than hydrogen tanks, permitting a slimmer, less draggy shape). If NASP's average acceleration is 0.2G -- nothing for rockets, but fairly impressive for airbreathers at high speeds -- the ABB equals 1.0, and NASP gets only about half the V_final you would predict from the bare rocket equation. This reduces its effective Isp to about 500s, which means it needs nearly the same mass ratio as a good oxyhydrogen rocket SSTO, despite not having to carry any oxidizer. (Worse, that means it needs *seven times* as much hydrogen, since the LOX/LH2 ratio is 6:1 for the rocket.)
Even if we assume an acceleration of 0.5G as Zubrin suggests -- which is really fierce acceleration for an airbreather, given how lousy the T:W ratio of airbreathing engines is -- the ABB only drops to 0.4 and the airbreather still needs 40% more delta-V than the rocket.
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