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LS3: Combustion & Propellants
Original Author: Matt Morningstar '21, matt_m@mit.edu
Lecture Zoom Recording
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Specific Impulse and the Rocket Equation, Revisited
Remember when we told you that Eq. 3 (see below) was the equation for the delta-v of a rocket?
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This simplified form is what informed the original rocket equation we showed you. In this case, Isp is directly proportional to exhaust velocity. So, we didn’t really lie. Exhaust velocity still comes out as a very important quantity and the primary driver of Isp.
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Exhaust Velocity
As we discovered in LSET 1, exhaust/exit velocity is an extremely important quantity for rocket engine performance. How can we predict what the exit velocity is, and how do we design our engine to achieve a high Ve?
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R *
/ (
This is simply a function of what your exhaust products are, which is a function of your propellant choice (and mixture ratio). We can express R as Runiversal / M, where M is the molecular weight of the exhaust products. Don’t confuse this with Mach number (I’ve italicized molecular weight to distinguish the two). (Runiversal is the constant value that you may have seen in high school w/ PV = nRT, Ru = 8314.4 J/(kmol.K)). This tells us that a propellant choice that yields exhaust products with a lower molecular weight is more efficient.
Tc
This is the temperature of combustion, which is a function of propellant choice (and mixture ratio). A propellant choice that yields a higher combustion temperature is more efficient.
[ 1 - (Pe/Pc)^((
This term is a little more complex. If our Pe/Pc is zero, then this term becomes 1 (maximum theoretical efficiency). However, in general, this term tells us that the larger our Pe/Pc ratio becomes, the less efficient we get. (As an example, if our = 1.2, Pe = 14.7 psi, and Pc = 1000 psi, this term becomes roughly ½. A 50% drop from maximum theoretical efficiency!) Rockets are most efficient when their nozzles are ‘matched’: the exit pressure equals the ambient pressure (you can find this result yourself by doing a lot of algebra with equations you already know). Thus, our Pe/Pc ratio is really a Pa/Pc ratio, where Pa is the ambient pressure. The only ‘knob’ we can really turn here is Pc. This equation tells us that a higher chamber pressure yields a more efficient engine, especially if our ambient pressure is higher. If our ambient pressure is very low (e.g. vacuum), then our Pe/Pc ratio will already be very low, and thus increasing Pc will not have a very sizable effect.
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Propellants
The first two terms that we discussed in the last section tell us that propellant choice has a huge effect on exhaust velocity, and thus engine efficiency. In fact, if we neglect the last term and talk only about theoretical maximum efficiency, the propellant choice is the only thing that affects exhaust velocity.
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These two characteristics are the primary drivers for modern propellant choices. As you may have heard before, hydrogen and oxygen is the propellant choice with the highest theoretical efficiency. We can see why! Hydrogen and oxygen produce a very high combustion temperature, and hydrogen has a very low molecular weight, leading to a low molecular weight of the exhaust products.
Other factors that impact propellant choice:
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Liquid Hydrogen (LH2) - Cryogenic
The most efficient fuel. Very low density. Used frequently in second stages of launch vehicles to achieve high exhaust velocities.
Examples: Saturn V’s J-2, Blue Origin’s BE-3, Aerojet Rocketdyne’s RS-68
Hydrocarbon Fuels
Methane (usually called LNG - ‘Liquid Natural Gas’) - Cryogenic
Historically not a common fuel, but has garnered tons of recent attention and development. No orbital rocket using methane has been flown to date. Can be created in situ on Mars.
Examples: SpaceX’s Raptor, Blue Origin’s BE-4, Relativity Space’s Aeon
RP-1 (Kerosene) - Storable
Very common propellant historically.
Examples: Saturn V’s F1, SpaceX’s Merlin, Rocket Lab’s Rutherford
Ethanol - Storable
Common propellant in the early days of rocketry.
Examples: Redstone engine, RS-88, ~Helios~
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Mixture Ratio
The mixture ratio is the mass flow ratio of the oxidizer to the fuel:
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As we discussed in the first section, a high combustion temperature is most efficient. So you might expect that rocket engines use a stoichiometric mixture ratio. Unfortunately, the combustion temperatures achieved at a stoichiometric mixture ratio are so hot that they exceed the limits of our material/cooling capabilities. As a result, nearly all rocket engines use a fuel-rich mixture ratio (ox-rich combustion presents a number of issues, which is why an ox-rich mixture ratio is not commonly used in the main combustion chamber). It is also true that a stoichiometric mixture ratio (even if we could withstand its temperature) does not always yield maximum Isp. In the case of hydrogen/oxygen, it is more optimal to operate fuel-rich, as it leaves more light H2 molecules in the exhaust.
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