Energy in the air goes as the square of airspeed. Energy to accelerate
a fixed absolute amount is roughly the derivative of that curve, and
is linear with the speed.
energy added =
1/2 * x * (v + y) ^ 2 - 1/2 * x * v ^ 2 =
1/2 * x * (2 * v * y + y ^ 2)
where v is the original airspeed, and v + y is the air speed after you
accelerate it by y ft/s. If you hold y fixed, the result is roughly
linear in v.
Evan
On Tue, Apr 5, 2016 at 3:30 AM, Peter Fairbrother <zenadsl6186@xxxxxxxxx> wrote:
On 05/04/16 05:04, Henry Spencer wrote:
On Tue, 5 Apr 2016, Ian Woollard wrote:
The cost is carrying the extra mass of the LOX.
Partly, but if the issue was just carrying oxidiser you'd only lose
2/3 of the Isp. Going to a rocket the Isp drops by a whole order of
magnitude...
No, that's not a fundamental law; it depends on what kind of jet engine
you're using as your base, and what speed it's operating at. Subsonic
turbofans can have very impressive Isp, yes. Supersonic ones, less so.
Up in ramjet territory, typical airbreathers don't have much more than
that 2/3 advantage. Scramjets don't do even that well.
The advantage is energy; throwing reaction mass at a speed comparable to
flight speed is much more efficient than the fixed speed rockets give
you,
unless the flight speed is comparable to rocket exhaust's speed.
Unfortunately for the relevance of that theory, the extra advantage of
airbreathing is lost long before then -- ramjet speeds are still far
below the exhaust velocity of a decent rocket engine.
A *rocket* engine is most energy efficient with an exhaust velocity
matched to its flight speed, yes. (This is, however, of no practical
importance, since energy efficiency is not the dominant design issue for
most types of rocket.) But jets are different. Adding momentum to air
that is already flowing past you at high speed is very costly in energy.
Having *ambient* nitrogen to use as reaction mass is useful at low
speed, but becomes much less so as speed builds up. 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?
-- Peter Fairbrother
and indeed if you plot Isp vs. speed for a
wide range of real airbreathers, and draw a curve through the best-Isp
peaks, the curve is pretty much a simple 1/x shape.
Henry
