Dumb rocket question

[VQ]

If supersonic flows are not affected by resistances downstream, how come the effective velocity of a rocket is reduced in the atmosphere even though the flow is supersonic downstream of the throat?

[Kiwi]

I also have a dumb rocket question. Much dumber than VQ's. 

Exactly how efficient is a rocket engine in terms of total energy generated and actual pushing power to lift the rocket?

The way I understand it is probably best described in simplistic terms as follows:

We'll view a spherical combustion chamber as a vertical slice through the middle of a real one, a clock face. Put very simply, the following happens:

1. The fuel and oxidiser are burnt at the centre of the clock.

2. The burnt gases expand rapidly in four directions, to 3, 6, 9, and 12 o'clock, where they press against the sides and top of the chamber.

3. The pressures of the gases at 3 and 9 o'clock cancel each other out, so are, in fact, wasted.

4. There is little or no pressure at 6 o'clock because the gases continue out the chamber's exhaust nozzle, so are also wasted.

5. Finally, the gas pressure against the top of the chamber is the propulsive force which lifts the rocket. And that's why a rocket works in a vacuum. The gases don't push against the air when they come out of the nozzle, as some people think.

I realise that simplifications can be dangerous and am deliberately ignoring that the burnt gases expand in all directions mainly for the sake of other laypersons like myself.

So does this mean that, in this case, about 75% of the energy generated is wasted at 3, 6, and 9 o'clock?

Are the gases at 3, 9 and 12 o'clock next used in some way, or do they just bounce around and finally hurtle out the exhaust nozzle?

Does the shape of the chamber have any effect on the energy generated and how much is used and how much is wasted?

[gwiz]

If supersonic flows are not affected by resistances downstream, how come the effective velocity of a rocket is reduced in the atmosphere even though the flow is supersonic downstream of the throat?

If correctly designed, the conditions inside the nozzle and the velocity at the nozzle exit are the same in atmosphere or vacuum.  The loss of performance in an atmosphere is due to atmospheric pressure acting on the outside of the nozzle to produce a drag force.

However, if the nozzle is designed for a lower atmospheric pressure than the one it is operating in, then a fully supersonic flow cannot be established in the nozzle and there is a shockwave and a change to subsonic flow part way down the nozzle.  This gives a performance loss due to the lower velocity downstream of the shock.  If this situation is on a rocket that is climbing, then the shockwave moves downstream as the external pressure reduces.

[gwiz]

I also have a dumb rocket question. Much dumber than VQ's. 

Exactly how efficient is a rocket engine in terms of total energy generated and actual pushing power to lift the rocket?

The pressure inside the combustion chamber isn't all, there is also pressure on the inside of the nozzle.  Ignoring the problem of external atmospheric pressure, see post above, the more the nozzle expands, the higher the velocity of the flow at the exit and the more efficient the rocket.  The nozzle is a system to convert the high internal energy of the hot gas inside the combustion chamber into high kinetic energy.  For maximum efficiency, the nozzle is large enough to drop the gas exit pressure to zero, but this comes with a nozzle mass penalty, so there is an optimum nozzle size.

[ka9q]

The difference in thrust between a rocket operating in vacuum and the same rocket operating in an atmosphere is exactly equal to the nozzle exit area times the ambient pressure, with the thrust always lower in an atmosphere.

It's easy to see intuitively why this happens. The plume inside the expanding section of a rocket nozzle is supersonic, so it is completely unaffected by anything outside the rocket. The expanding gases exert pressure on the inside of the nozzle, and because the nozzle flares toward the exit that gas pressure exerts a net forward force on the nozzle that becomes part of the thrust. (The rest of the thrust is up against the top of the combustion chamber).

The plume exerts the only force on the nozzle when it operates in a vacuum. But in an atmosphere you also have ambient pressure pushing on the outside of the nozzle, and again because of the nozzle's shape that generates a net rearward force. The magnitude of that force is just the ambient pressure acting on the projection of the nozzle along its axis.

Think of a nozzle squashed flat along its axis. You'd have a disk with a diameter equal to that of the nozzle exit, which is the largest part of the nozzle. So that's the effective area on which the atmosphere applies its pressure in a rearward direction.

This assumes the ambient pressure isn't so high as to slow the flow down to subsonic, and it also assumes the nozzle isn't so overexpanded in the atmosphere that you get plume flow separation. I.e., that the rocket operates properly in both environments.

[ka9q]

A rocket engine is fundamentally a heat engine, specifically an internal combustion engine. But instead of driving an external load, the rocket's job is to convert the heat of combustion into the kinetic energy of its exhaust products. In the process, rearward momentum is transferred to the exhaust and, to obey Newton's Third Law, an equal amount of forward momentum is applied to the rocket -- and that's the effect we're actually looking for.

The efficiency of a heat engine is limited by the Second Law of Thermodynamics, specifically the Carnot Limit:

efficiency <= 1 - Tc/Th

where Th is the temperature of the heat source (e.g., the hot gas in the combustion chamber) and Tc is the temperature of the cold sink, which in this case is the exhaust plume at the nozzle exit. This sets a theoretical limit on efficiency; real engines always perform worse.

Rockets actually do quite well in this conversion because rocket combustion produces some of the hottest flames in any artificial device. (Remember that the fuels are typically burned with pure oxygen, rather than air that's only 21% oxygen.) So the conversion of heat energy to kinetic energy is actually quite good; I think I estimated ~70% for the F-1 engine though I might be remembering that wrong.

But our real goal is to impart momentum to the rocket, and accelerating the rocket exhaust is only a means to that end. Here we run into the most depressing law of rocketry, which is that the amount of power it takes to produce a given amount of thrust increases with the exhaust velocity, while the mass flow rate needed to produce that same thrust decreases with exhaust velocity. If you want an energy efficient rocket, you have to carry lots of propellant mass, which then takes more thrust and more energy and more propellant to lift, and so on. And if you want a light rocket, then you need to generate a lot more power (with less fuel!) to generate the same thrust with a lower propellant flow rate.

So we typically just use the most energetic fuels we can find and not worry about the dollar cost of each joule of energy or how efficiently that energy is converted into mechanical energy of the payload. Within limits, a more energetic fuel results in a lower overall launch cost and that's what matters.

[VQ]

If correctly designed, the conditions inside the nozzle and the velocity at the nozzle exit are the same in atmosphere or vacuum.  The loss of performance in an atmosphere is due to atmospheric pressure acting on the outside of the nozzle to produce a drag force.

However, if the nozzle is designed for a lower atmospheric pressure than the one it is operating in, then a fully supersonic flow cannot be established in the nozzle and there is a shockwave and a change to subsonic flow part way down the nozzle.  This gives a performance loss due to the lower velocity downstream of the shock.  If this situation is on a rocket that is climbing, then the shockwave moves downstream as the external pressure reduces.

You sure? I think in a nozzle optimized for vacuum operating at, say 1 atm there will be a separation boundary where the supersonic stream detaches from the nozzle wall, but not a shock where the flow slows back down to below mach 1. Even at sea level the exhaust stream velocity from the bell should be much, much faster than that!

[gwiz]

You sure? I think in a nozzle optimized for vacuum operating at, say 1 atm there will be a separation boundary where the supersonic stream detaches from the nozzle wall, but not a shock where the flow slows back down to below mach 1. Even at sea level the exhaust stream velocity from the bell should be much, much faster than that!

A flow will separate from a solid boundary when it is decelerating and the boundary layer runs out of energy.  The supersonic flow in a nozzle is accelerating, so can only separate if it meets a shock.

[bknight]

Thread resurrection with a different question concerning the F-1 engines.  I know from previous threads and some reading that fuel was used to cool the engine housings and then burned after exiting.  What percentage of fuel was used in this procedure?

[Geordie]

Thread resurrection with a different question concerning the F-1 engines.  I know from previous threads and some reading that fuel was used to cool the engine housings and then burned after exiting.  What percentage of fuel was used in this procedure?

  I came across this article, in which it says, in reference to the J-2X engine, "As compared to the large quantities of propellants being pumped through the whole engine, the amount going to the gas generator is small (less than 3% for J-2X)."

  I also found this PDF, which says that "Hot gas from the [F-1] gas generator enters the turbine at a flowrate of 170 pounds per second through the inlet manifold [....]"

  And from here "Hot gases for the turbopump turbine originated in the gas generator and entered the turbine at 77 kilograms per second."

  SLS researchers have been firing up original F-1 gas generators recently, but everbody seems to be coy on the actual propellant consumption.

  I can't convert kg/sec of gas output to propellant volume input, but at least the two sources agree.

  Another number bandied about is 55,000 HP for the turbine. Maybe someone can work backwards from that.

  Consolation prize: there's lots of new footage of F-1 gas generator testing on YouTube.

[Geordie]

Thread resurrection with a different question concerning the F-1 engines.  I know from previous threads and some reading that fuel was used to cool the engine housings and then burned after exiting.  What percentage of fuel was used in this procedure?

  I came across this article, in which it says, in reference to the J-2X engine, "As compared to the large quantities of propellants being pumped through the whole engine, the amount going to the gas generator is small (less than 3% for J-2X)."

  I also found this PDF, which says that "Hot gas from the [F-1] gas generator enters the turbine at a flowrate of 170 pounds per second through the inlet manifold [....]"

  And from here "Hot gases for the turbopump turbine originated in the gas generator and entered the turbine at 77 kilograms per second."

  SLS researchers have been firing up original F-1 gas generators recently, but everbody seems to be coy on the actual propellant consumption.

  I can't convert kg/sec of gas output to propellant volume input, but at least the two sources agree.

  Another number bandied about is 55,000 HP for the turbine. Maybe someone can work backwards from that.

  Consolation prize: there's lots of new footage of F-1 gas generator testing on YouTube.

  Here's another number, from http://www.wired.co.uk/article/f-1-moon-rocket:

  "[...] The [F-1] gas generator [...] churns out about 138 kilonewtons of thrust[....]"