[AR] Re: airbreathing engines (was Re: Re: to the stars, soon)

  • From: Henry Spencer <hspencer@xxxxxxxxxxxxx>
  • To: Arocket List <arocket@xxxxxxxxxxxxx>
  • Date: Tue, 5 Apr 2016 13:13:40 -0400 (EDT)

On Tue, 5 Apr 2016, Peter Fairbrother wrote:

...Adding momentum to air that is already flowing past you at high speed is very costly in energy. ... A quick bit of algebra suggests that airbreathing Isp should be roughly inversely proportional to flight speed,

inversely proportional to the *square* of flight speed?
jet must accelerate x lb of air by y ft/s to produce thrust z. assuming flight speed >> y, energy required to do so depends ~ on square of relative speed of air.
or did I miss something?

Alas, you did. There are two components to the energy requirement. One is proportional to y^2; the other to v*y, where v is flight speed. (And if v >> y, the second component dominates.) You're not accelerating the air from 0 to y, but from v to v+y:

E = 0.5*x*(v+y)^2 - 0.5*x*v^2
  = 0.5*x*(v^2 + 2*v*y + y^2 - v^2)
  = 0.5*x*(2*v*y + y^2)

Actually, for systems that accelerate the air by burning onboard fuel in it, there is a v^2 component as well, because the mass of the fuel has to be accelerated from 0 to v+y. However, the mass flow involved is much smaller -- especially if the fuel is hydrogen -- and for feasible airbreather speeds, the v*y component tends to dominate.

Henry

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