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

Question

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

Answer 1

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 I_sp. 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.

Answer 2

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

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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 $\frac{m}{V} = \frac{p \cdot M}{R \cdot T}$. 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 $v_{sonic} = \sqrt{\frac{γ \cdot R \cdot T}{M}}$ 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 vacuum Isp for various chamber pressures at the exact same nozzle length and exit diameter as a nozzle with $p = 40\ bar$, $Ae\text{_}At\ \text{(design)} = 7.7\ resp.\ 70$ and $\text{length} = 0.5 \cdot \text{length of the full length nozzle}$ (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 $Ae\text{_}At$ 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 equilibria using cpropep.

The fuel used is lox and propane (both at 85 K) with a mass ratio of $2.8 \text{O}_2:\text{C}_3\text{H}_8$ simply because that's what I'm currently interested in. It should be pretty representative 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 $Ae\text{_}At(true) = 16$ 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 + calculating it will probably another few hours and I'm not sure that I will find the time for it today.