You’ve got the nail on the head here. Aside from the practical limits of high temperature combustion (running at a lower chamber temperature allows for lighter combustion chambers, or just practical ones at all) the various advantages of a lighter exhaust most than make up for the slightly lower combustion energy. the practical limits are often important: if your max chamber temperature is limited, it makes a ton of sense to run fuel rich to bring it to an acceptable range.
One other thing to mention is that the speed of sound of the exhaust matters quite a lot. Given the same area ratio nozzle and same gamma in the gas, the exhaust mach number is constant; a higher speed of sound thus yields a higher exhaust velocity.
The effects of dissociation vary depending on application. It’s less of an issue with vacuum nozzles, where their large area ratio and low exhaust temperature allow some recombination. For atmospheric engines, nozzles are quite short; there’s little time for gases to recombine.
I’d recommend playing around with CEA (https://cearun.grc.nasa.gov/), which allows you to really play with a lot of combinations quickly.
I’d also like to mention that some coefficients in nozzle design might make things easier to reason about. Thrust coefficient and characteristic velocity are the big ones; see an explanation here
Note that exhaust velocity is proportional to the square root of (T_0/MW), where T_0 is chamber temperature.
Thrust coefficient, which describes the effectiveness of a nozzle, purely depends on area ratio, back pressure, and the specific heat ratio for the gas.
You’re right about intuitive explanations of this being few and far between. I couldn’t even get one out of my professor when I covered this in class.
To summarize:
Only gamma, molecular weight, chamber temp T0, and nozzle pressures affect ideal exhaust velocity.
Given a chamber pressure, gamma, and back pressure, (chamber pressure is engineering limited), a perfect nozzle will expand your exhaust to a fixed mach number, regardless of original temperature.
Lower molecular weight gases have more exhaust velocity at the same mach number.
Dissociation effects make it more efficient to avoid maximizing temperature in favor of lowering molecular weight.
This effect is incredibly strong for nuclear engines: since they run at a fixed, relatively low engineering limited temperature, they have enormous specific impulse gains by using as light a propellant as possible.
One other thing to mention is that the speed of sound of the exhaust matters quite a lot. Given the same area ratio nozzle and same gamma in the gas, the exhaust mach number is constant; a higher speed of sound thus yields a higher exhaust velocity.
My understanding is this effect is a re-framing of what I described: for a similar temperature and gamma, a lower molecular weight (or specific heat) will result in a higher speed of sound (or exit velocity).
However, I feel like this framing fails to provide a good intuition for the underlying mechanism. At the limit, Te→0K,Ma→∞anyways, so it’s harder (for me at least) to understand how gas properties relate to sonic properties. Yes, holding other things constant, a lower molecular weight increases the speed of sound. But crucially, it means there’s more kinetic energy to be extracted to start with.
Aside from dissociation/bond energy, nearly all of the energy in the combustion chamber is kinetic. Hill’s Mechanics and Thermodynamics of Propulsion gives us this very useful figure for the energy balance:
A good deal of the energy in the exhaust is still locked up in various high-energy states; these states are primarily related to the degrees of freedom of the gas (and thus gamma) and are more strongly occupied at higher temperatures. I think that the lighter molecular weight gasses have equivalently less energy here, but I’m not entirely sure. This might be something to look into.
Posting this graph has got me confused as well, though. I was going to write about how there’s more energy tied up in the enthalpy of the gas in the exhaust, but that wouldn’t make sense—lower MW propellants have a higher specific heat per unit mass, and thus would retain more energy at the same temperature.
The one thing to note: the ideal occurs where the gas has the highest speed of sound. I really can’t think of any intuitive way to write this other than “nozzles are marginally more efficient at converting the energy of lighter molecular weight gases from thermal-kinetic to macroscopic kinetic.”
You’ve got the nail on the head here. Aside from the practical limits of high temperature combustion (running at a lower chamber temperature allows for lighter combustion chambers, or just practical ones at all) the various advantages of a lighter exhaust most than make up for the slightly lower combustion energy. the practical limits are often important: if your max chamber temperature is limited, it makes a ton of sense to run fuel rich to bring it to an acceptable range.
One other thing to mention is that the speed of sound of the exhaust matters quite a lot. Given the same area ratio nozzle and same gamma in the gas, the exhaust mach number is constant; a higher speed of sound thus yields a higher exhaust velocity.
The effects of dissociation vary depending on application. It’s less of an issue with vacuum nozzles, where their large area ratio and low exhaust temperature allow some recombination. For atmospheric engines, nozzles are quite short; there’s little time for gases to recombine.
I’d recommend playing around with CEA (https://cearun.grc.nasa.gov/), which allows you to really play with a lot of combinations quickly.
I’d also like to mention that some coefficients in nozzle design might make things easier to reason about. Thrust coefficient and characteristic velocity are the big ones; see an explanation here
Note that exhaust velocity is proportional to the square root of (T_0/MW), where T_0 is chamber temperature.
Thrust coefficient, which describes the effectiveness of a nozzle, purely depends on area ratio, back pressure, and the specific heat ratio for the gas.
You’re right about intuitive explanations of this being few and far between. I couldn’t even get one out of my professor when I covered this in class.
To summarize:
Only gamma, molecular weight, chamber temp T0, and nozzle pressures affect ideal exhaust velocity.
Given a chamber pressure, gamma, and back pressure, (chamber pressure is engineering limited), a perfect nozzle will expand your exhaust to a fixed mach number, regardless of original temperature.
Lower molecular weight gases have more exhaust velocity at the same mach number.
Dissociation effects make it more efficient to avoid maximizing temperature in favor of lowering molecular weight.
This effect is incredibly strong for nuclear engines: since they run at a fixed, relatively low engineering limited temperature, they have enormous specific impulse gains by using as light a propellant as possible.
My understanding is this effect is a re-framing of what I described: for a similar temperature and gamma, a lower molecular weight (or specific heat) will result in a higher speed of sound (or exit velocity).
However, I feel like this framing fails to provide a good intuition for the underlying mechanism. At the limit, Te→0K,Ma→∞ anyways, so it’s harder (for me at least) to understand how gas properties relate to sonic properties. Yes, holding other things constant, a lower molecular weight increases the speed of sound. But crucially, it means there’s more kinetic energy to be extracted to start with.
Is that not right?
Aside from dissociation/bond energy, nearly all of the energy in the combustion chamber is kinetic. Hill’s Mechanics and Thermodynamics of Propulsion gives us this very useful figure for the energy balance:
A good deal of the energy in the exhaust is still locked up in various high-energy states; these states are primarily related to the degrees of freedom of the gas (and thus gamma) and are more strongly occupied at higher temperatures. I think that the lighter molecular weight gasses have equivalently less energy here, but I’m not entirely sure. This might be something to look into.
Posting this graph has got me confused as well, though. I was going to write about how there’s more energy tied up in the enthalpy of the gas in the exhaust, but that wouldn’t make sense—lower MW propellants have a higher specific heat per unit mass, and thus would retain more energy at the same temperature.
I ran the numbers in Desmos for perfect combustion, an infinite nozzle, and no dissociation, and the result was still there, but quite small:
https://www.desmos.com/calculator/lyhovkxepr
The one thing to note: the ideal occurs where the gas has the highest speed of sound. I really can’t think of any intuitive way to write this other than “nozzles are marginally more efficient at converting the energy of lighter molecular weight gases from thermal-kinetic to macroscopic kinetic.”