# Analytical approximations for the shapes of these plots of ISP vs nozzle diameter? (Raptor engine)

A comment on the currently unanswered question Is there an authoritative catalog for rocket launch identifiers? led me to this Github page which links to r/spacexlounge on which currently appears a post with a plot, titled Raptor efficiency for various nozzle sizes, also shown below.

1. For the two limiting cases of the well defined conditions of ambient pressure: sea level and vacuum, are there analytical approximations to these two curves? The curves are reminiscent of parabolas, but with the ordinate being something similar to $$1 - (1/d^2-a)^2$$ where $$d$$ might be diameter and $$a$$ might be related to ambient pressure, but that's just putting with my eyes closed.

2. How was (or at least might) the intermediate curve for effective ISP be defined?

note: Learning the original source for the plot will be interesting, but my question is primarily about rocket nozzle math and physics in general.

NOTE: As pointed out in comments below, the plot below seems to be an individual contribution rather than officially released data. Don't quote from it or use it as a premise of follow-up questions without taking that into consideration.

• I assume "effective ISP" is intended to represent the time average of the ISP over the course of ascent from sea level to vacuum; it could be either an ad hoc weighting of the two ISP figures, or determined through a discrete step simulation of ascent. – Russell Borogove Oct 16 '18 at 23:13
• In the comments on the reddit post, it looks like OP made some substantial naive errors in their (unrelated) turbopump efficiency spreadsheet, so you may want a grain of salt to go with your plot. – Russell Borogove Oct 16 '18 at 23:16
• @RussellBorogove thanks for that. I'm primarily interested in simpler analytical equations that show this kind of behavior, rather than addressing the veracity of these particular curves, but it is important to know these are individual-generated not official, so I'll note that in the question. – uhoh Oct 16 '18 at 23:54
• I'm thinking this just shows the effect of changing expansion ratio, but without knowing anything like the throat area, flow rates, etc, I will have to look and see if everything drops out or can be backed out. A task for tomorrow. – Organic Marble Oct 17 '18 at 2:50
• @OrganicMarble I think if there are equations that have the same behavior (atm upside-down curve peaks at finite diameter, vac curve peaks at infinity) that would be fine, and vac curve always greater than atm curve would be even better It wouldn't have to match exactly at all, there may be too many engine-specific details for that. – uhoh Oct 17 '18 at 8:30

The only equation needed to produce these curves is the thrust equation in a couple of different guises.

1. $$F = \dot{m}v_e + (p_e-p_{atm})A_e$$
2. $$I_{sp}= F /(g_0 \dot{m})$$

You also need isentropic flow charts or tables to calculate the nozzle parameters that change with changing area ratio. An example isentropic flow chart is shown here, from 1953 book "The Dynamics and Thermodynamics of Compressible Fluid Flow" by Ascher Shapiro, page 87.

Since the X axis of the graph in the question is nozzle exit diameter, which sets nozzle exit plane area, and therefore the area ratio, I rearranged the Shapiro chart to show pressure ratio and velocity ratio plotted against area ratio. Both axes are dimensionless, area ratio is on the X axis. I started with an isentropic flow table spreadsheet from here. The spreadsheet didn't have velocity ratio in it, so I calculated it from pressure ratio using equation 3-26 in Sutton (4th edition).

Knowing the area ratio gives us these two ratios which give us all the information we need to calculate the exit plane parameters needed for the thrust equation - if we know the throat properties. We will need to make some assumptions about the engine for that.

Assumption 0 - The Raptor's throat area is 0.6 ft2. This comes from old information giving the exit diameter as 1.7 meters and the expansion ratio as 40. I have found no reference to the expansion ratio on the latest iteration of the engine, so I used this.

Assumption 1 - The exit plane pressure where the question graph of sea level $$I_{sp}$$ reaches a maximum is 15 psi. A sensible assumption because when the exit plane pressure matches ambient, the losses are minimized.

Assumption 2 - The Raptor's sea level thrust is 380,000 lbf (Wikipedia)

With these assumptions we can calculate the mass flowrate using equation 2. I read the maximum sea level $$I_{sp}$$ from the chart as 334, this gives a mass flowrate of 35.3 slugs/sec.

Using that mass flowrate we can use equation 1 to calculate the exhaust velocity. The delta pressure term vanishes when the exit plane pressure is 15, so the exhaust velocity is 10,750 ft/s.

With these assumptions we know the area ratio, exhaust velocity, and exit pressure at the peak of the sea level curve. This lets us calculate the throat properties using the isentropic tables and gives

$$V_t$$ = 4898 ft/s

$$P_t$$ = 640,900 lbf/ft2.

We can now calculate specific impulse for any nozzle exit diameter by the following process:

1. Calculate exit plane area based on diameter; divide this by the throat area to get the area ratio.
2. Look up the pressure and velocity ratios in the isentropic tables based on area ratio.
3. Calculate the exit plane pressure using the pressure ratio and the throat pressure.
4. Calculate the exhaust velocity using the velocity ratio and the throat velocity.
5. With the mass flowrate, exhaust velocity, exit plane pressure, and exit plane area, use Equation 1 to calculate the thrust.
6. Use Equation 2 to calculate the specific impulse.

You may certainly change the assumptions and get different values. However, this demonstrates that by holding those parameters constant and calculating the isentropic flow properties solely based on area ratio, you can generate curves that are shaped like the ones in the question.

This doesn't match the graph in the question exactly; I presume the maker of that graph had better/different info about the engine.

Re: the "effective ISP"

It's supposed to be the average ISP for a booster as it goes from sea-level to main engine cutoff. I don't know enough about the flight profile to really calculate the effective ISP so it's just the sea level and vacuum ISP averaged together with the vacuum weighted as twice as important.

Source- reddit thread in the question

• On my launch sim todo list is integrating (in multiple senses) the ISP versus time curve from one of the Apollo launch vehicle reports. It's a pretty nice sigmoid, and symmetrical enough that I think the weighting function should be 1:1 rather than skewed toward vacuum. en.wikipedia.org/wiki/Sigmoid_function – Russell Borogove Oct 17 '18 at 6:17
• @uhoh I completely rewrote this to better match the graph in the question using some assumed Raptor properties; I also tried to explicitly go through the calculation process. This is probably my final attempt at this question; I really enjoyed digging into it, thanks! – Organic Marble Oct 20 '18 at 20:02
• This is perfect! Coffee is brewing now... (I liked it better when the plot went out to 2.5 for purely aesthetic reasons) – uhoh Oct 21 '18 at 2:30
• Yeah, The compressible flow table I downloaded only went out to Mach 5. So when I put in the more realistic assumptions I ran off the end of the table. – Organic Marble Oct 21 '18 at 2:34
• @uhoh I answered it, but briefly, the incoming air stream term is only important for jet engines, which accelerate that incoming air stream. The pressure thrust term should not be dropped, it just evaluates to zero if the pressures match. – Organic Marble Oct 21 '18 at 14:16