Fun nerd snipe! I gave it a quick go and was mostly able to deconfuse myself, though I’m still unsure of the specifics. I would still love to hear an expert take.
First, what exactly is the confusion?
For an LOX/LH2 rocket, the most energy efficient fuel ratio is stoichiometric, at 8:1 by mass. However, real rockets apparently use ratios with an excess of hydrogen to boost Isp[1] -- somewhere around 4:1[2] seems to provide the best overall performance. This is confusing, as my intuition is telling me: for the same mass of propellant, a non-stoichiometric fuel ratio is less energetic. Less energy being put into a gas with more mols should mean lower-enough temperatures that the exhaust velocity should be also be lower, thus lower thrust and Isp.
So, where is my intuition wrong?
The total fuel energy per unit mass is indeed lower, nothing tricky going on there. There’s less loss than I expected though. Moving from an 8:1 → 4:1 ratio only results in a theoretical 10% energy loss[3], but an 80% increase in products (by mol).
However, assuming lower energy implies lower temperatures was at least partially wrong. Given a large enough difference in specific heat, less energy could result in a higher temperature. In our case though, the excess hydrogen actually increases the heat capacity of the product, meaning a stoichiometric ratio will always produce the higher temperature[4].
But as it turns out, a stoichiometric ratio of LOX/LH2 burns so hot that a portion of the H2O dissociates into various species, significantly reducing efficiency. A naive calculation of the stoichiometric flame temperature is around 5,800K, vs ~3,700K when taking into account these details[5]. Additionally, this inefficiency drops off quickly as temperatures lower, meaning a 4:1 ratio is much more efficient and can generate temperatures over 3,000K.
This seems to be the primary mechanism behind the improved Isp: a 4:1 fuel ratio is able to generate combustion temperatures close enough to the stoichiometric ratio in a gas with a higher enough heat capacity to generate a higher exhaust velocity. And indeed, plugging in rough numbers to the exhaust velocity equation[6], this bears out.
The differences in molecular weight and heat capacities also contribute to how efficiently a real rocket nozzle can convert the heat energy into kinetic energy, which is what the other terms from the exhaust velocity help correct for. But as far as I can tell, this is not the dominant effect and actually reduces exhaust velocity for the 4:1 mixture (though I’m very uncertain about this).
The internet is full of 1) poor, boldly incorrect and angry explainers on this topic, and 2) and incredibly high-quality rocket science resources and tools (this was some of the most disconsonant non-CW discourse I’ve had to wade through). With all the great resources that do exist though, I was surprised I couldn’t find any existing intuitive explanations! I seemed to find either muddled thinking around this specific example, or clear thinking about the math in abstract.
… or who knows, maybe my reading comprehension is just poor!
For the fuel rich mixture, if we were somehow able to only heat the water product, the temperature would equal the stoichiometric flame temp. Then when considering the excess H2 temperatures would be necessarily lower. Charts showing showing the flame temp of various fuel ratios support this: http://www.braeunig.us/space/comb-OH.htm
See the SSME example here: https://www.nrc.gov/docs/ML1310/ML13109A563.pdf Ironically, they incorrectly use a stoichiometric ratio in their calculations. But as they show, the reaction inefficiencies explain the vast majority of the temperature discrepancy.
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.”
I wish I had a more short-form reference here, but for anyone who wants to learn more about this, Rocket Propulsion Elements is the gold standard intro textbook. We used in my university rocketry group, and it’s a common reference to see in industry. Fairly well written, and you should only need to know high school physics and calculus.
This is an impressive piece of deconfusion, probably better than I could have done. I’d be excited about seeing a post with a log of your thought process and what sources you consulted in what order.
Fun nerd snipe! I gave it a quick go and was mostly able to deconfuse myself, though I’m still unsure of the specifics. I would still love to hear an expert take.
First, what exactly is the confusion?
For an LOX/LH2 rocket, the most energy efficient fuel ratio is stoichiometric, at 8:1 by mass. However, real rockets apparently use ratios with an excess of hydrogen to boost Isp[1] -- somewhere around 4:1[2] seems to provide the best overall performance. This is confusing, as my intuition is telling me: for the same mass of propellant, a non-stoichiometric fuel ratio is less energetic. Less energy being put into a gas with more mols should mean lower-enough temperatures that the exhaust velocity should be also be lower, thus lower thrust and Isp.
So, where is my intuition wrong?
The total fuel energy per unit mass is indeed lower, nothing tricky going on there. There’s less loss than I expected though. Moving from an 8:1 → 4:1 ratio only results in a theoretical 10% energy loss[3], but an 80% increase in products (by mol).
However, assuming lower energy implies lower temperatures was at least partially wrong. Given a large enough difference in specific heat, less energy could result in a higher temperature. In our case though, the excess hydrogen actually increases the heat capacity of the product, meaning a stoichiometric ratio will always produce the higher temperature[4].
But as it turns out, a stoichiometric ratio of LOX/LH2 burns so hot that a portion of the H2O dissociates into various species, significantly reducing efficiency. A naive calculation of the stoichiometric flame temperature is around 5,800K, vs ~3,700K when taking into account these details[5]. Additionally, this inefficiency drops off quickly as temperatures lower, meaning a 4:1 ratio is much more efficient and can generate temperatures over 3,000K.
This seems to be the primary mechanism behind the improved Isp: a 4:1 fuel ratio is able to generate combustion temperatures close enough to the stoichiometric ratio in a gas with a higher enough heat capacity to generate a higher exhaust velocity. And indeed, plugging in rough numbers to the exhaust velocity equation[6], this bears out.
The differences in molecular weight and heat capacities also contribute to how efficiently a real rocket nozzle can convert the heat energy into kinetic energy, which is what the other terms from the exhaust velocity help correct for. But as far as I can tell, this is not the dominant effect and actually reduces exhaust velocity for the 4:1 mixture (though I’m very uncertain about this).
The internet is full of 1) poor, boldly incorrect and angry explainers on this topic, and 2) and incredibly high-quality rocket science resources and tools (this was some of the most disconsonant non-CW discourse I’ve had to wade through). With all the great resources that do exist though, I was surprised I couldn’t find any existing intuitive explanations! I seemed to find either muddled thinking around this specific example, or clear thinking about the math in abstract.
… or who knows, maybe my reading comprehension is just poor!
Intense heat and the dangers of un-reacted, highly oxidizing O2 in the exhaust also motivates excess hydrogen ratios.
The Space Shuttle Main Engine used a 6.03:1 ratio, in part because a 4:1 ratio would require a much, much larger LH2 tank.
20H2 + 10O2 → 20H2O: ΔH ~= −5716kJ (vs) 36H2 + 9O2 → 18H2O + 18H2: ΔH ~= −5144.4kJ
For the fuel rich mixture, if we were somehow able to only heat the water product, the temperature would equal the stoichiometric flame temp. Then when considering the excess H2 temperatures would be necessarily lower. Charts showing showing the flame temp of various fuel ratios support this: http://www.braeunig.us/space/comb-OH.htm
See the SSME example here: https://www.nrc.gov/docs/ML1310/ML13109A563.pdf Ironically, they incorrectly use a stoichiometric ratio in their calculations. But as they show, the reaction inefficiencies explain the vast majority of the temperature discrepancy.
Equation 12: http://www.nakka-rocketry.net/th_nozz.html The rough numbers I got where 4,600m/s for 4:1 and 3,800m/s for 8:1
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.”
I wish I had a more short-form reference here, but for anyone who wants to learn more about this, Rocket Propulsion Elements is the gold standard intro textbook. We used in my university rocketry group, and it’s a common reference to see in industry. Fairly well written, and you should only need to know high school physics and calculus.
This is an impressive piece of deconfusion, probably better than I could have done. I’d be excited about seeing a post with a log of your thought process and what sources you consulted in what order.