# Does chamber pressure have the same impact on ISP for different types of engines?

From this answer I understand that higher chamber pressure means higher ISP - but is this always the case for similar engines? For example:

• I noticed that a lot of upper-stage engines have pretty low chamber pressure (e.g. around 40 bar). Maybe chamber pressure is not that important in the vacuum?
• It seems like chamber pressure is important for bell-nozzle engines. Is it as important for other nozzle designs - e.g. aerospike?

By similar engines I mean:

1. Engines that use the same propellants
2. Engines with the same mass flow rate
• A quick glance at Wikipedia's rocket engine table en.wikipedia.org/wiki/Comparison_of_orbital_rocket_engines will tell you that that answer is umm, oversimplified, to put it politely. Raptor @ 300 bar and Isp of 350 vs SSME @ 200 bar and Isp of 450 is a pretty striking counterexample. – Organic Marble Feb 9 at 22:31
• See also researchgate.net/figure/… "While variations in the chamber pressure itself have only a minor influence, i.e. an increase by a factor of six yields only a marginal (about 0.1 %) performance increase, an only minor increase of the combustion efficiency (by 1% from 0.96 to 0.97) yields already an increase of the performance of about 1%. " – Organic Marble Feb 9 at 22:43
• @OrganicMarble - my understanding was that Raptor and SSME cannot be compared directly as they use different fuels. But would SSME @ 300 bar be any better than @ 200 bar? And similarly, would Raptor @ 200 bar be much worse? – irakliy Feb 9 at 22:51
• Yeah, I think that's the key - it's probably true for similar engines. Not "absolutely true." But when you are varying the chamber pressure, are you keeping the mass flow rate constant?? How about the chamber temperature?? There's this: space.stackexchange.com/q/12133/6944 but the mass flow is not constant. – Organic Marble Feb 9 at 22:56
• @OrganicMarble - makes sense. I modifies the question slightly to make it more clear. – irakliy Feb 9 at 23:11

There's a probably relevant term in this equation I shamelessly stole from the wikipedia page on de Laval nozzles:

$$v_e = \sqrt{\frac{TR}{M} \cdot \frac{2\gamma}{\gamma - 1} \cdot \left[1 - \left(\frac{p_e}{p}\right)^{\frac{\gamma - 1}{\gamma}}\right]}$$

and that term is $$p_e/p$$, the ratio of exhaust pressure as it leaves the nozzle to the chamber pressure (more or less). The other terms aren't particularly interesting in this context, so I'll quietly ignore them, and hope you will, too.

All else being equal, raising $$p$$ will indeed increase your exhaust velocity and hence your Isp. There's an obvious hard limit because the bit of the equation in square brackets isn't going to rise above 1, but even before that the enclosing square root means that big increases in pressure will have more modest effects on exhaust velocity and so only serve to upset your engineers. You therefore pick the highest practical chamber pressure that suits the ambient pressure in the expected operating environment.

Obviously, if $$p_e$$ is very low, because you've got a nice vacuum-suitable rocket nozzle operating in a vacuum, $$p$$ doesn't need to be very high, and presumably it will make all the rest of the engineering much simpler if it isn't.

I've no idea about aerospikes.

• Great answer! The only bit missing is about aerospikes. Since the are not de Laval nozzles, I'm guessing that this equation doesn't apply (at least not in the same way) - but would be great to get some intuition behind it. – irakliy Feb 10 at 16:59
• @irakliy I'm not getting much useful information on aerospikes, but I'm guessing their behaviour isn't wildly different... they seem to have fairly similar operating parameters, just with a much wider range of acceptable ambient pressures. Its the same fuel developing the same energy and giving it to the same exhaust species and ultimately developing the same peak Isp. I think it is more or less handwaveable as being "close enough" in this case. – Starfish Prime Feb 10 at 21:40
• Some charts from here and here lead me to think that performance of aerospike engines is not as impacted by chamber pressure as for bell-nozzle engines. E.g. you might be able to have an efficient sea-level aerospike engine @ 40 bar, while this is probably not the case for bell-nozzle engines. But, I can't say that I fully understand all the data from the links, and might be wrong on this. – irakliy Feb 10 at 22:27
• @irakliy the thrust of the engine still rather depends on chamber pressure. You can get away with a lower chamber pressure for an aerospike because the thrust coefficient of the nozzle is much better than for a conventional bell nozzle. I'm trying to track down the working parameters of the J-2T as they'd be really handily comparable with other member sof the J-2 family, but they're proving elusive... – Starfish Prime Feb 10 at 22:36
• @irakliy Note that for Aerospikes the main difference is that p_e = p_ambient rather than a constant. They get additional losses if p_ambient < p_design but the main advantage is that they don't suffer offer expansion related thrust loss. Comparing them would have to take varying p_e into account and therefore a whole fight simulation. – Christoph Feb 11 at 12:35

There might be another day were I post an answer not using cpropep but this will certainly not be the day.

We can few this issue from multiple points of view (this only accounts for bell nozzles):

1. Increased Ae/At at same nozzle size

The density of an ideal gas is m/V = p M / (R T). This means higher pressure increases the density and therefore the mass flow per area through the throat. A smaller A_throat results in a bigger expansion ratio for the same nozzle size. Higher chamber pressure shifts the equilibrium (see 3.) and increases the temperature and therefore v_sonic = sqrt(γ R T / M) which increases throat flow but it also reduces the density so that overall the increased temperature is a negative influence in this regard and the variation is still almost linear as T is almost constant.

The following plot shows the vaccum Isp for various chamber pressures at the exact same nozzle length and exit diameter as a nozzle with p = 40 bar, Ae_At (design) = 7.7 resp. 70 and length = 0.5 * length of the full length nozzle (which would completely straighten the flow). 0.5 results in a length that nearly matches the "classical" 85-90% length of the 15 degree cone. This represents the Isp variation for a typical lower / upper stage engine at constant nozzle mass.

Note that Ae_At in the plot is the true Ae/At after cutting 50% of the full length nozzle. Find the data at plot.ly.

The nozzles and resulting Isp (including cosine loss by not fully straightening) were calculated by a tool I wrote over the last days which implements the algorithm from Supersonic Axisymmetric Minimum Length Nozzle Conception at High Temperature with Application for Air - Zebbiche, Toufik. The only difference to the paper is that I extended it to also account for shifting equilbriums using cpropep.

The fuel used is lox and propane (both at 85K) with a mass ratio of 2.8 O2:C3H8 simply because that's what I'm currently interested in. It should be pretty representive of most hydrocarbons.

2. Increased Ae/At at same exit pressure

For first stage engines expansion ratio is limited by the pressure at the nozzle exit due to flow separation. For this plot we use the same fuel as above and as reference pressure we use the pressure at the exit of a nozzle with Ae_At(true) = 16 and p_chamber = 97 MPa (matches Merlin 1D).

One can clearly see that this makes a huge difference for booster engines. Moreover notice that the Ae/At rises much less linear when compared to constant nozzle size plots.

3. Increased chamber temperature will increase efficiency at constant Ae/At

I will add a plot for this too but coding + calcualting it will probably another few hours and I'm not sure that I will find the time for it today.

• If you have ideas for more interesting plots or want plots of the nozzles or access to the source code please let me know. Source code is still quite messy and subject to change but I'm willing to share it under MIT licence. – Christoph Feb 11 at 13:49