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.
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.