What limits the speed at which rocket fuel is expelled?

Rocket propulsion is based on Newton's laws. The faster propellant is expelled, the faster the rocket is accelerated.

At what speed is propellant expelled in a normal rocket? What are the factors that limit that speed? How does that compare to vehicle speeds appropriate to interstellar travel?

thanks

• – uhoh May 10 '18 at 2:22
• Actually there's a couple of useful and related questions here: the last paragraph. If they'll edit out the speed-of-light stuff it's OK, and the first and last paragraphs are independent of that. – Tom Spilker May 10 '18 at 3:19
• @TomSpilker great edit, I've withdrawn my close vote. I only found out this happened because this question is linked to in a newer question. You can help get rid of close votes after editing by leaving a message to the close voters if their username shows up here. – uhoh May 10 '18 at 11:19
• For a rocket in space, the center of mass (rocket + propellant) actually not does move at all. – Tero Lahtinen May 11 '18 at 6:29

3 Answers

The core of your question is about real, "normal" engine exhaust velocities. The fastest we get so far from practical, in-production engines is about 4,440 m/s, Space Shuttle main engines with liquid oxygen and liquid hydrogen as propellants, producing water as the exhaust product (https://en.wikipedia.org/wiki/Specific_impulse). That speed is limited by the molecular mass of the product(s) (the lighter the faster), the amount of energy available from the chemical reaction that produces them, how much of that energy comes out as thermal energy of the products (e.g., isn't lost to emitted light), and to a smaller extent, the efficiency of the engine design, specifically the combustion chamber and nozzle. Of course the combustion products must be gaseous, too! The higher the combustion energy per molecule, the faster you can expel that molecule out of the nozzle. For historical reasons, rocket engineers refer to specific impulse, which is just the exhaust velocity divided by Earth's surface gravitational acceleration. The specific impulse of the ox/hydro combination given above is ~453 s.

Electric engines can get much higher exhaust velocities, 20-50 km/s, and thus higher specific impulses, because the ions being accelerated aren't limited to the energy of a combustion reaction. They are being accelerated by electric fields. But that's another discussion.

• "The higher the combustion energy per molecule, the faster you can expel that molecule out of the nozzle." Let me add, also the lower the particle mass. That's why even very high energy reactions that involve heavy atoms aren't good. That's why shuttle runs fuel-rich - even if you get less energy from not burning all the hydrogen, you're still getting more specific impulse, as hydrogen is ejected at much higher speed than water steam. – SF. May 10 '18 at 10:00
• @SF. Well, it also runs fuel rich because oxygen is absurdly reactive. A bit of excess hydrogen isn't going to do any appreciable damage; a bit of excess oxygen, on the other hand... I'd guess this is the more important concern, though certainly hydrogen can be used as propellant even for rockets that don't use it as fuel. – Luaan May 10 '18 at 10:23
• The SSME turbine blades couldn't take anywhere near stochiometric combustion temps, the mixture ratio in the preburners was even lower than the main combustion chamber. – Organic Marble May 10 '18 at 11:33
• I'm not sure how but the answer feels like it could do with some tidying up. I'm not sure how the speed of light stuff is relevant, but after the 'molecular mass' part it looks much better. – Pureferret May 10 '18 at 11:58
• @Pureferret , the original question, before editing, referred to a scenario where 1% of the vehicle mass is expelled at the speed of light; does the remaining vehicle mass achieve 1%*c*? In Newtonian mechanics, yes, but Einstein put the kibosh on that. – Tom Spilker May 10 '18 at 16:59

The nozzle is the part of a rocket that limits the speed of the exhaust velocity. (It's also the part that converts the pressure and temperature of the expanding propellant into velocity.)

The speed of sound in the exhaust likewise regulates the expansion of the propellant gas.

For rockets using nozzles, the exhaust velocity can be expressed as

$$V=\sqrt{\frac{2 \gamma R_{{}^{\circ}} T_{{}^{\circ}}}{(\gamma -1) \mu }\left(1-\left(\frac{P_e}{P_c}\right){}^{\frac{\gamma -1}{\gamma }}\right)}$$

(in the form from Hash's self-answer elsewhere; k is used instead of $\gamma$ and M instead of $\mu$ in Basics of Spaceflight.)

Restrictions on the exhaust temperature $T$ are implied by the temperature that the nozzle can withstand. $\gamma$, $\mu$, and $R$ have to do with the choice of propellant. $\gamma$, $R$, and $T$ also directly relate the speed of sound in a gas.

As in Tom Spilker's answer, ion engines avoid that limitation because they don't rely on the gas's expansion to provide the exhaust velocity. The directed application of electromagnetic fields to an ionized exhaust stream allows higher velocities to be imparted.

• This covers a lot of the same ground that Tom Spiker's answer does, but I tried to emphasize the parts I remembered being emphasized in the Thermodynamics of Propulsion class as taught by Prof. Varghese at the University of Texas. That was a good course. – Erin Anne May 10 '18 at 7:32
• This is far more readable than other answers. – Pureferret May 10 '18 at 11:59

At what speed is propellant expelled in a normal rocket?

A very widely used metric called specific impulse addresses exactly this issue. Chemical rockets (presumably that's what you mean by a "normal rocket") have specific impulses that range from about 2000 to 4500 meters per second.

What are the factors that limit that speed?

The laws of thermodynamics.

For example, hydrogen and oxygen combine to produce water, yielding 286 kilojoules of energy for every mole of water produced. If every last little bit of the energy released by that reaction is converted into usable kinetic energy, a rocket that burns hydrogen and oxygen would have an exhaust velocity of 5630 meters/second. That's the limit placed by the first law of thermodynamics. The second law of thermodynamics says you'll get less than that if the engine or the exhaust aren't at absolute zero (they aren't) or if the exhaust isn't perfectly collimated (it isn't). The 4500 meters/second value I sited in the answer to the first question is for hydrogen and oxygen.

How does that compare to vehicle speeds appropriate to interstellar travel?

Not anywhere close. Voyager 1, which was launched by chemical rockets just over 40 years ago, is currently about 1/1880th the distance to the nearest star. Voyager 1 will eventually travel beyond that nearest star -- in about 74100 years.

Needing 74100 years to reach the nearest star is a time frame that is not appropriate to interstellar travel.

• What would be the highest speed that could be achieved with multiple gravity sling shots? – Thorbjørn Ravn Andersen May 10 '18 at 22:57
• The first paragraph here incorrectly refers to specific impulse. You're talking about exhaust velocity. – Erin Anne May 15 '18 at 21:11
• @ErinAnne - Exhaust velocity, or rather effective exhaust velocity, is how sane people (i.e., those who use the metric system) measure specific impulse. Multiply specific impulse expressed in seconds by 9.80665 m/s^2 and you have specific impulse in meters per second. That's the sane way to talk about specific impulse because it relates to the chemical potential energy of the fuel, less some inefficiencies in the rocket. Expressing specific impulse in seconds is a consequence of a choice of goofy units. – David Hammen May 16 '18 at 1:29
• My comment seems rude on reflection, so sorry about that. I got taught the American convention and didn't even stop to think there might be a different one. – Erin Anne May 16 '18 at 5:04
• @ThorbjørnRavnAndersen Once proposed solution for a fast extrasolar vehicle (faster than any of the five extrasolar vehicles built to date) is to have a vehicle do a gravity slingshot of Jupiter (possibly multiple slingshots, cycling between Earth and Jupiter, with the last Jupiter flyby removing almost all velocity with respect to the Sun, thereby sending the vehicle diving toward the Sun. The vehicle then does a powered flyby of the Sun, gaining a huge Oberth effect. This could conceivably be done only using solar sails. – David Hammen Mar 3 at 19:38