Tradeoffs in Space Propulsion

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In the simple math of the rocket equation, I introduced the twin ideas of exhaust velocity and specific impulse. These are really the same concept, just expressed differently to allow for greater flexibility around unit systems. The particular figure for exhaust velocity will differ between English and SI units; specific impulse will be the same in both.

The exhaust velocity is the average speed consumed propellant leaves the rocket nozzle. From aerothermodynamics, the exhaust velocity v_{ex} of an engine will be:

\displaystyle v_{ex} = \sqrt{\frac{2k}{k-1}\frac{R'T_c}{\mathfrak{M}} \bigg[1 - \bigg(\frac{p_{ex}}{p_c}\bigg)^\frac{k-1}{k}\bigg] + v_c^2 }

where k is the specific heat ratio, R' is the universal gas constant, \mathfrak{M} is the molecular mass of the used propellant, T_c is the temperature in the combustion chamber, p_{ex} is the pressure at the nozzle exit, p_c is the pressure inside the combustion chamber, and v_c is the gas velocity inside the combustion chamber.

That’s a lot of different variables, but it’s not as bad as it looks. We can treat a few parameters as constants: the universal gas constant is, well, universal, and the chamber velocity is usually low enough—in comparison to the exhaust velocity—that most versions of this equation omit it entirely.

Additionally, the specific heat ratio doesn’t vary over a wide range, and is basically a dependent variable on other factors. Generally speaking, k falls between about 1.2 (for extremely hot gases) and 1.4 (the value of room-temperature air). I’m not fully qualified to discuss the thermochemistry behind these values; for our purposes we can just look k up in a table for a given propellant-oxidizer combination.

Thus we’re left three basic ways to modify the specific impulse of a liquid-propellant rocket engine: increasing the pressure ratio, raising the combustion temperature, and lowering the molecular mass of the exhaust gas.

The pressure ratio p_c/p_{ex} is a function of the pressure in the combustion chamber and how much the exhaust gas can expand before reaching the nozzle exit. Ideally, the gas expands to equal the external pressure. This can be achieved at any given altitude within the sensible atmosphere, but becomes impossible in the vacuum of space. Perfect expansion to a vacuum, John Clark tells us, “would require an infinitely long nozzle, which might involve some difficulties in fabrication.” Generally, we settle for pressure ratios on the order of 50–100 for vacuum engines.

Raising the combustion temperature is another attractive option, but it can be complex, because the primary way to do this is to choose a more exothermic propellant combination. Many, many possible combinations exist that produce hotter combustion, but aren’t worth the extra weight, safety risks, and difficulty to store.

Simultaneously, we would prefer combustion products to have a low molecular mass (that is, lighter molecules on average). Balancing this with combustion temperature and other factors is a key aspect of optimizing the propellant combination for a given propulsion system. The ideal propellant is monatomic hydrogen, with a molecular mass of 1. This is usually not practical; monatomic hydrogen prefers to become diatomic hydrogen when left to its own devices, and cannot combust on its own. Nuclear thermal propulsion gets around this problem by heating the exhaust gas from an external source rather than burning it—but that naturally presents its own challenges.1

All that said, specific impulse is far from the only performance parameter engineers need to optimize. Perhaps the most obvious is the thrust force developed by the rocket. Each of the strategies discussed above will affect the thrust produced by or required from the engine.

The force F produced by a rocket engine can be calculated using the equation:

F = \dot{m}v_{ex} + (p_{ex} - p_{amb})A_{ex}

where \dot{m} is the mass flow rate, p_{amb} is the ambient pressure outside the rocket vehicle, and A_{ex} is the area of the nozzle exit. (In a vacuum, the ambient pressure is zero.) The first term on the right hand side of this equation is the momentum thrust, and the second term is the pressure thrust.

What does this tell us? A couple of things.

For one, there may be a reduction in thrust when we reduce the molecular mass of the propellant, depending on how we do it. If, for instance, we modify the combustion chamber geometry—lengthening it, perhaps—to facilitate more complete reactions and thus end up with smaller combustion products, then that wouldn’t necessarily affect the mass flow rate.

However, the more amenable approach is to use a lighter propellant—replacing kerosene with liquid hydrogen, for instance. In that case, the engine design probably changes radically. Among other things, the propellant density drops considerably, and so we have to pump much more fuel through the engine to get the same amount of thrust.

Similarly, lowering the exit pressure improves the specific impulse, but reduces the pressure thrust. Extending the nozzle is the best way to lower the exit pressure, which generally means increasing the mass which the reduced-thrust rocket needs to accelerate. A longer nozzle will have a greater exit area, the exact figure depending on the shape of the nozzle in question. The same principle applies when raising combustion temperature: hotter combustion likely necessitates heavier combustion chamber materials or coolant systems.

There is no clear, direct tradeoff between the rather multivariate functions for rocket thrust and exhaust velocity. Depending on the input values, plotting specific impulse versus thrust can produce some extremely strange curves. In practice, it’s better to compare individual flight units or classes of engines.

Sutton and Biblarz provide a useful diagram showing the exhaust velocity and acceleration ranges for different types rocket propulsion systems. rocketcomp

Sutton, G. P., Biblarz, O., Rocket Propulsion Elements, 9th ed., Wiley, Hoboken NJ, 2017.

Acceleration or thrust-to-weight ratio (TWR) is a good way to normalize thrust between different-sized engines. Such comparisons are necessary, given that the same rocket engineering principles apply to everything from attitude control thrusters on small satellites to the Rocketdyne F-1. Engine and vehicle masses vary wildly and it’s entirely possible that no one has tested one that falls into the particular thrust, efficiency, and propellant-type combination in question. TWR thus lets us see the similarities and differences between disparate systems.

Here we see an approximate inverse relationship between exhaust velocity and acceleration, but that a range of options exists within each class of propulsion system. As a general rule, we can expect some sort of disadvantage to come along with any approach that increases both thrust and specific impulse—otherwise, engineers would simply default to it. And, similarly, other technical advantages to lowering one or both parameters.

For an obvious example: switching to more energetic propellants can increase the chamber temperature and thus exhaust velocity, but may exceed the material allowables of the combustion chamber materials. In that case, the engine will need either a thicker (and therefore heavier) chamber wall, or another sort of material with a higher failure point, or a coolant system which adds mass and complexity. Naturally, such options are more expensive and often more difficult to deal with—otherwise they would already be standard.

More energetic fuels may also be difficult to store. Cryogenic fuels like liquid hydrogen produce better thrust and exhaust velocities than, say, kerosene—but require significantly heavier tankage to store, and evaporate relatively quickly. Cryogenics really aren’t storable propellants, but thermal engineers have done heroic things designing spacecraft that can carry liquid hydrogen to the outer Solar System.

If a more energetic propellant isn’t cryogenic, it probably has other serious downsides. Hydrazine is already a nasty substance to work with; anything more powerful is probably going to be nastier, or else it would already be in use. If it’s not toxic, then it might unstable and prone to self-ignition. Of course, plenty of propellants are both toxic and unstable. Ignition! in many places felt like a laundry list of fuel/oxidizer combinations that provide great performance on a first-pass analysis, but turn out to be entirely unusable for one reason or another.

At a certain point, we run up against the limits of what can be done practically with exothermic chemical processes—cost, of course, often determining what is actually practical. Retooling an entire production run and introducing additional safety procedures is hardly worth the effort compared to using a bigger hydrazine tank. In the long term, perhaps such a change-over could be justified, but switching costs are usually too daunting for organizations to accept the risk.

More advanced propellants and propulsion concepts are consequently reserved for niche applications or when designing entirely new systems. Nuclear propulsion, for instance, really only competes on large-scale lunar or crewed planetary missions. Using methane as a propellant has been considered  for a long time, but methane-fueled engines are just barely reaching the development phase.2 Various forms of electric propulsion are a little older, but due to their low thrust are currently limited to small probes or satellite attitude control.

Going in the other direction, cold gas systems make up for their low thrust and efficiency with simplicity and reliability. The mechanisms weigh less and have fewer points of failure—extremely strong selling points for small spacecraft or projects with a tight budget. Nitrogen is often the main propellant: relatively nontoxic, easy to handle, and easy to store compared to many other monopropellants.

There’s no simple answer to deciding where we want to fall on any particular tradeoff curve—the answer depends on the exact application we’re designing for. If we’re buying or adapting an existing rocket, which of the dozens or hundreds should we take of the shelf? If we’re designing an entirely new system, where in the highly-multidimensional possibility space should we be aiming for?

The ideas I’ve discussed here are heuristics for making that decision. In practice, it can be quite complicated, usually dependent on the mission profile in question and the budgets for mass, power, system volume, and (of course) funding. There’s rarely a single answer, often many answers depending on how flexible the design parameters actually are. The engineer’s task is to find the most optimal solution with the resources available to them.

That applies in every sort of engineering, not just space propulsion.

References

  • Brown, C. D., Elements of Spacecraft Design, edited by J. S. Przemieniecki, AIAA Education Series, AIAA, Reston VA, 2003.
  • Clark, J. D., Ignition! An Informal History of Liquid Rocket Propellants, 1st ed., Rutgers University Press, New Brunswick NJ, 1972.
  • Dewar, J. A., To the End of the Solar System: The Story of the Nuclear Rocket, 1st ed. University Press of Kentucky, Lexington KY, 2004.
  • Farokhi, S., Aircraft Propulsion, 2nd ed., Wiley, Chichester UK, 2014.
  • Plachta, D. W., Christie, R. J., Jurns, J. M., Kittel, P., “Passive ZBO storage of liquid hydrogen and liquid oxygen applied to space science mission concepts,” Cryogenics, Vol. 46, No. 2, 2006, pp. 89–97.
  • Sutton, G. P., Biblarz, O., Rocket Propulsion Elements, 9th ed., Wiley, Hoboken NJ, 2017.

1Namely, a massive reactor system, often producing temperatures beyond the allowables of most widely-available materials, and the risk of radiation leakage. These last two, coupled together, makes nuclear thermal propulsion much riskier than using nuclear energy for electricity generation. Nuclear power plants use closed-loop thermal cycles, with no incentive to drive up temperatures and drive down mass. Ground-based projects can just eat all sorts of costs that aerospace engineers find prohibitive.

2There are a number of reasons for this, including the pace of launch vehicle development, but perhaps the most interesting is that methane’s advantages over kerosene or liquid hydrogen are most significant in the context of reusability. Hence, in addition to SpaceX and Blue Origin, a number of companies working on smaller rockets are pursuing methane-fueled engines. Oh, and methane is easy to make on Mars.

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