What (if anything) limits the efficiency of a rocket engine?

Humans have developed lots of rockets. I observe that in most cases to increase the payload capacity we just increase the amount of fuel.

Is it not possible to exponentially increase the efficiency of our rocket engines by changing design of rocket, or is there a limit to the efficiency?

• to increase payload capacity we also change the fuel – user20636 Jun 9 '20 at 8:49

First of all: Thermodynamics!

A rocket is transforming potential chemical energy into kinetic energy -> You need more kinetic energy (= more mass and/or moving faster)? So you need more chemical energy at first. So yes, you are right, there is this factor "efficiency" which means: The more efficient you are, the more is the part of the chemical energy you transform into kinetic energy. BUT (at some point): It does NOT mean you can get more and more kinetic energy out of less and less fuel. So there is indeed a point where "efficiency is limited".

Actually, the efficiency in numbers is always between 0 and 1. Not only in rockets, but in everything … like really everything … clicking the keys on the keyboard, your muscles do work, they transform chemical energy (food and oxygen) into movement, they have an efficiency of (not sure exactly) 0.3. So about two thirds of the energy needed is heating your finger up. A car is burning (transforming) fuel (chemical energy) into movement (kinetic energy), efficiency about 0.1. An electric heater transforming electric energy into heat, efficiency nearly 1. Tapping on your desk to heat the bathtub, efficiency nearly 0. A classical light bulb (transforming electrical energy into light), efficiency about 0.05.

So back to topic: "Why not just increase efficiency". The efficiency of rockets is about 0.678 (source in german). Sounds bad at first, but thermodynamics has something called "carnot efficiency" which is the theoretically maximum technical efficiency for machines making work by heat. Mostly it results in something between 0.6 and 0.7 … making our rocket quite efficent.

Aaaaaaand: actually a whole bunch of people all over the the world are trying to make them more efficient. They can get some percent here and there, but they cannot reinvent the wheal.

Second: the Rocket Equation

How do rockets work? Preservation of impulse … Imagine sitting in a canoe on a lake, having hammers on board, but no rudders. To get to shore you need to throw the hammers as fast you can to the back, so that the canoe moves forward. At some point, you cannot throw the hammers faster, so to get to shore, you need more hammers.

Which actually means: At some point, you cannot simply make the exhaust gases of your rocket faster (because: thermodynamics), so you need to take more fuel with you in the first place.

You want to make a larger tour with your car: Visit a fuel station first! For groceries around the next corner, the rest in your tank will be sufficient.

And in the end, allow me a question: Am I understanding your question wrong, or did you really think nobody has thought of "simply improving" rocket designs? (I am not the downvoter, but I think that is what she/he might have thought)

• Yah you understood it right . Thank you very much. – Akshat Jun 9 '20 at 6:38
• I don't understand "Tipping on your desk to heat the bathtube" – user20636 Jun 9 '20 at 8:34
• @JCRM it is not a "code", it is just the first example i had for something VERY inefficient. So when sitting in front of a desk and you tip the desk, air will start to vibrate. This vibration will travel all the way to the bathroom, there the water will start to vibrate. Now damping will cause the water to heat.... I am quite sure: its so ineffective, you cannot even measure the transfer. So just an example for something very ineffective. – CallMeTom Jun 9 '20 at 8:39
• with the explanation I understand that "tipping your desk to heat the bathtub" is inefficient, but I'm not sure it makes sense on its own. – user20636 Jun 9 '20 at 8:47
• You might want to say "stirring your tea to heat it" what-if.xkcd.com/71 – ikrase Jun 9 '20 at 9:17

The existing answer is good, but I'd like to touch on part of the thermodynamics: the nozzle.

A rocket creates thrust essentially by converting pressure (coming from combustion) into velocity of a stream of gas. Most any rocket worth its salt does this with a converging-diverging nozzle that constricts the high pressure flow until it becomes supersonic, then expands it until the pressure drops enough that it isn't worth expanding it any more (because eventually you need very large nozzles to keep catching the expanding gas--see for example the difference between the sea-level Merlin engine and the Merlin Vacuum engine.)

Chemistry and materials conspire to limit reasonable nozzle performance. Any hot gas can essentially only push so hard, limiting the exhaust velocity of the rocket engine (thus ultimately its overall efficiency).

This is a significant factor in why an ion engine can be so much more efficient than combustion engines: exhaust velocities for an ion engine can be much higher than achievable with any nozzle engine. Unfortunately existing ion engines can't currently generate sufficient thrust to replace combustion engines for lofting a rocket from Earth's surface, and the required power ranges for doing so are many orders of magnitude beyond existing designs.

• This ignores the potential for fusion rockets with magnetic nozzles. – ikrase Jun 9 '20 at 8:43
• I admit I've never heard of one. I assume that a magnetic nozzle would fall into the points I was making in the last paragraph, which could maybe be made more general. – Erin Anne Jun 9 '20 at 8:46
• @ikrase that's pretty easy to ignore since controlled fusion is years or decades away. – Organic Marble Jun 9 '20 at 12:48

If you are willing to be a bit flexible about what you define as a 'rocket' and embark on some fairly (OK, very) mad science, then there is indeed a well-defined limit to efficiency. The most efficient possible 'rocket' you can make is something which emits light, and which does it by consuming mass with complete efficiency, so all the energy implicit in the mass is turned into light. The way to do that is to have your rocket be fuelled by equal amounts of matter and antimatter (but even then you probably can't get very close).

If you can do the annihilation completely efficiently, so that all you get out is photons, and you can arrange life that all these photons contribute to your momentum (see notes below), then you get this equation:

$$\frac{m_0}{m_f} = \frac{1 + v/c}{\sqrt{1 - v^2/c^2}} - 1$$

Where $$m_0$$ is the launch mass, $$m_f$$ is the final mass and $$v \lt c$$ is the final velocity.

That's the best you can do: a rocket which consumes mass completely efficiently and spits out light. Here's a plot of the mass ratio ($$m_0/m_f$$, the plot calls this 'mass fraction' which is wrong, sorry) of this for $$v \in [0, 0.9]$$ with $$v$$ in units where $$c = 1$$ (or alternatively I forgot to label the axis as $$v/c$$).

Why this isn't practical. Well, there are a very large number of reasons why a system like this is not practical: this is very much a mad science idea.

Production and storage of macroscopic amounts of antimatter is challenging, to put it rather mildly. I don't know how much antimatter has ever been produced but it's a small amount. Storing it in large quantities is going to be really hard (I really don't have any idea how you'd do it at all), and the cost of making a mistake is appalling: you really do not want the power to fail in your antimatter storage system if you have any macroscopic amount of the stuff.

The 100% efficiency thing is a problem. If you collide a proton with an antiproton what you actually get is some shower of unstable particles, which I think eventually must all decay to light, but only if you can arrange for them to be near each other for long enough. And some of them, for instance, are neutrinos, which are not easy to contain, to put it rather mildly.

If you collide electrons and positrons, you do just get light. But you now have the problem that a lot of the light that is produced is in the form of very high energy photons (gamma rays) and these are very hard to reflect efficiently with any kind of mirror you can plausibly make. And also you need to store a lot of electrons and positrons, either by attaching them to protons / antiprotons as hydrogen / antihydrogen, in which case what do you do with the protons and antiprotons (which are almost all the mass!) left over, or by some approach mad even by the standards of mad science.

But, well, this is the theoretical limit: you can't do better than this, even in principle, with anything that might count as a rocket, short of radically new physics.

• I think this is a neat way to think about it, but I'm having trouble understanding what the practical takeaway is here since I'm not practiced in the rocket equation. Say the exhaust is at or near the speed of light due to perfect annihilation...so what? Is the resulting mass fraction maximized? What does that mean in practical terms? I think my understanding is missing something fundamental here. – aranedain Jun 9 '20 at 19:55
• @aranedain: yes, the thing which should be the mass fraction, which is the ratio of payload mass to launch mass is maximised for a given $\Delta v$. So this is the most efficient 'rocket' you can make. – tfb Jun 10 '20 at 12:43