# What is the relation between chamber pressure and expansion ratio?

I've always assumed that the expansion ratio of an engine equaled the pressure inside the engine over the atmospheric pressure, but I'm not sure this is right, could someone check me on this?

• @OrganicMarble true but you could theoretically make an infinitely long nozzle designed for an absolute vacuum getting diminishing results. if this is the case, however, do you know how to calculate the actual expansion ratio? – Reuben Farley-Hall Nov 25 '20 at 3:29
• Yes, you pick the exit plane pressure you want and back out the area ratio from the pressure ratio. I answered a question on here about that, but I'm not able to search for it now. – Organic Marble Nov 25 '20 at 4:05
• I found it, this shows how to get area ratio given the pressure ratio space.stackexchange.com/a/46091/6944 you have to know gamma as well. – Organic Marble Nov 25 '20 at 4:09
• Note it has pressure ratio of 100, area ratio of 11. – Organic Marble Nov 25 '20 at 4:13

## 1 Answer

If you're referring to nozzle expansion ratio, it's defined as the ratio of the nozzle exit area and throat area (the widest to the narrowest) and is often designated by greek letter epsilon.

$$\epsilon = A_2 / A_t$$

Where $$A_2$$ is the exit plane area and $$A_t$$ is the throat area. Usually nozzle expansion ratios are written in form 10:1 or just 10 (exit plane area being ten times the throat area). It is only a parameter of the nozzle design, not related to the chamber pressure.

You could also mean pressure ratio. When we talk about pressure ratio, it usually means the ratio between chamber pressure and exit plane pressure.

$$p_1/p_2$$

Where $$p_1$$ is the chamber pressure and $$p_2$$ exit plane pressure. Note, however, that the pressure on the exit plane is not necessarily equal to the pressure of the environment. The nozzle is designed to be at optimum expansion at certain altitude (so that athmospheric pressure equals the exit plane pressure), otherwise the nozzle is over- or underexpanded so the exit plane pressure would differ from that of the environment.

Rocket Propulsion Elements by G.P. Sutton explains these concepts very well.