Zero-velocity exhaust

[pauldear]

This is probably crazy, but anyway. There was a discusson on the Halfbakery (http://www.halfbakery.com/idea/swoop_20 ... 1306701675) which led to some interesting discussions on things like the Oberth effect. One particular point that struck me is that, when a rocket is getting near orbital speeds, its fuel contains more kinetic energy than it does chemical energy, which is freaky but true. (1kg of TNT moving at 4km/s has a kinetic energy of 8MJ, and a chemical energy of 4.6MJ)

The bottom line is that a lot of the fuel's energy is wasted, because the exhaust has a high velocity relative to the air around it. This velocity represents wasted kinetic energy, since it is eventually absorbed by the air as the exhaust gases slow down.

If you can arrange for the reaction mass (ie, the exhaust) to have zero velocity relative to the surrounding air, then in theory everything becomes more efficient. However, this also means that your thrust is negligible, if you're exhaust is a gas.

However, you can get around this by (a) starting with the rocket at some reasonable speed (say, 500m/s - it's released from an aircraft or whatever), and then (b) ejecting mass at a velocity which is equal and opposite to that of the rocket, so that the exhaust velocity relative to the atmosphere is zero. Because you're aiming for low exhaust velocities (but increasing in proportion to the velocity of the rocket), it actually makes most sense to eject lumps of stuff (ie "ballast") as the reaction mass, using a relatively small amount of propellant.

I know this all sounds weird. But, if you do the maths, you can get things like 5% of the launch mass delivered to LEO (ie, around 7km/s), using LOX/H2 as the propellant (and not a lot of propellant either). With a solid fuel, you're not much worse off (eg, a 100kg launch-mass giving about 2kg at 6.9km/s, from a starting velocity of 500m/s). Of course the calculations are simplified, but then again so are those for the standard rocket equation, so the comparison is fair. And before you tell me I'm nuts, the maths goes back a long way, and doesn't break any paws of physics.

If you're interested, take a look at the discussion on the Halfbakery and try to figure out where I've gone wrong.

[DaveHein]

The ISP of a rocket engine is basically it's exhaust velocity expressed in units of seconds, or more commonly the exhaust velocity divided by 9.8, which has units of Newton-Seconds/Kilogram. A solid fuel rocket has an ISP aroud 220 to 280. A liquid fuel rocket is from 320 to 450. At a velocity of 7.5 km/s you would need a rocket motor with an ISP of 7500/9.8 = 765 to yield zero net velocity on the exhaust. From an energy perspective, it might be efficient to design a rocket engine that can change its ISP. It could start out at an ISP of 200, and once the rocket reaches a speed of 2 km/s it could increase it's ISP up to 450 at 4.5 km/s. Another possibility would be to use a solid fuel rocket for the first stage, and then a liquid fuel rocket for the second stage.

However, It seems that rocket designers generally use the highest ISP possible, even if it isn't the most efficient use of energy. It's probably because they can get more energy per kilogram that way, or it just simplifies the rocket engine design.

[lavalamp]

The more stages for this you have, the higher the fraction of energy in the fuel you put into the payload.

Letting the number of stages go to infinity, all of the energy from the fuel goes into the payload. This allows to calculate the absolute maximum theoretical mass fraction assuming no dry weight. The mass fraction of the rocket then is determined entirely by the chemical energy stored in the fuel (ignoring such factors as speed of sound in combusting propellants, aero drag and gravity drag).

So let's do a quick analysis for a unit mass (1 kg) payload.

Energy density of TNT ~= 4.7 MJ / kg
Energy density of H2/O2 ~= 20 MJ / kg

Required KE for Δv = 9.00 km/s
KE = 0.5 * 1 * 9000^2 = 40.5 MJ

Mass of TNT = 40.5 / 4.7 = 8.617 kg
Mass of H2/O2 = 40.5 / 20 = 2.025 kg

Mass fraction for TNT = 1 / (1+8.617) = 0.104
Mass fraction for H2/O2 = 1 / (1+2.025) = 0.331

Remember, these are the MAXIMUM theoretical mass fractions for 9 km/s. It is quite unlikely that it is possible to achieve anything close to them.

Just for fun, one more for matter-antimatter and Δv =0.9c.

Energy density = 89.9 PJ / kg

Required KE = (γ-1)mc^2 = ((1 / sqrt(1 - 0.9^2)) - 1) * 1 * 299792458^2 = 116.3 PJ

Mass of matter-antimatter = 116.3 / 89.9 = 1.294 kg

Mass fraction = 1 / (1+1.294) = 0.436

[DaveHein]

lavalamp, your calculations are interesting, but they don't address the issue of producing a zero-velocity exhaust. If you don't produce a zero-velocity exhaust then all of the energy will not go into the potential/kinetic energy of the payload. The exhaust will be traveling at a nonzero velocity, and it will contain a significant amount of kinetic energy.

[lavalamp]

Perhaps I wasn't clear, I was considering the scenario for an infinite number of stages with zero velocity exhaust. Another way of thinking about it would be as you said, an engine that can vary it's specific impulse from 0 to Δv/g. Those calculations are for the theoretical maximum performance of such a system.

As far as the practicality of such a system, well it isn't really practical.

[DaveHein]

lavalamp, thanks for the clarification. It's interesting that the most energy-efficient method would require a low ISP when the velocity is low. This seems counterintuative, but I can see how that would work in free space.

[rick m]

[pauldear]

I worked up a spreadsheet to do some simulations, for a system that ejects inert masses using a fairly modest propellant, with the propellant charges sized to leave zero velocity in each ejected mass. What struck me was that:
(a) You wind up using a relatively small mass of propellant, and a large total of ejected "passive" mass.
(b) You can use fairly mundane propellants. (Caveat: I don't know much about detonation rates etc. But even if you have a "slow" propellant, you can presumably use a burst-disc of some sort to deliver impulse rapidly to an ejected mass. Maybe.)

The payload fractions aren't immense, but they're respectable*, and the system is fairly simple (hah! yes, I know everything is simple when you don't understand the details). I imagined a rocket in which each "stage" is a solid cylindrical mass, with a "spigot" on the top and a hole in the bottom, so that they stack, with explosive charges on the top of each spigot. Each mass is then the "bullet" fired by the next stage up.

*a penalty is that you have a series of high-G accelerations, which limits the type of payload you can carry.

[lavalamp]

You cannot expel a mass out of a barrel with a velocity higher than the speed of sound in the expanding gasses. What this means is that for a higher speed, you need a larger explosive charge (higher temp = higher speed of sound). For big artillery, and even some guns, the mass of the gun powder is actually larger than the round. They top out around 1 km/s (Mach 3), maybe 1.6 km/s (Mach 5) for some particularly outrageous guns.

Clearly 1.6 km/s is not enough to maintain zero-velocity exhaust all the way to orbit, you need around 7 km/s for that.

What this means is that for ever higher and higher exhaust velocities, the mass of the round becomes negligible, and you may as well simply rely on the hot exhaust gases. This is what a traditional rocket does, and even they top out at around 4.5 km/s for the exhaust velocity.

However, shooting the whole rocket stack out of a cannon is potentially a good first stage (providing you can make a sturdy enough rocket). It would be like a mini space gun.

What you could say then is use really crap rockets at the start, and progressively improve performance with each stage. Sure, that might be more efficient in terms of energy use, but that's not really the goal when building a rocket. The goal is to carry the biggest payload you can with as small a rocket as you can get away with. To do that you, generally speaking, put the highest specific impulse fuel you have available (or sugar, whatever ) into as small, low mass, tanks as you can.

For any given speed/altitude, a higher ISP will allow you to use less propellant for the same thrust, or the same amount of propellant for more thrust.

[rick m]