Calculating the nozzle exit pressure

[Guest]

I've ran into a problem. I need an equation to determine exit pressure of a rocket engine (de Laval) nozzle, given the chamber pressure and nozzle expansion ratio. It doesn't need to use any terribly detailed model, but should be reasonably accurate. It might also depend on the fuel mixture and it's OF ratio. I'm not sure if assuming an ideal gas would be quite right.

Alternatively, there's an equation on Wikipedia that includes nozzle exit pressure, but it depends on exhaust velocity, not expansion ratio. I'd need actual exhaust velocity (the one that doesn't depend on external pressure), not effective exhaust velocity to use that one, though. I'd need to know a way of the former finding out, because it's the latter is the one that is easily calculated from Isp.

The reason I need this is to have an easy way of finding altitude at which any given engine is at an ambient pressure. I'd like to be able to calculate it off data that is available for most engines.

[TheShadow1138]

I'm slightly confused by what you want to calculate, but if I understand you correctly you want to calculate the altitude at which the exit pressure is equal to the ambient pressure. According to the Wikipedia article for De Laval Nozzles, which I assume is where you found the equation you mentioned, states that to analyze the flow it is often assumed that the exhaust is an ideal gas and adiabatic. From this I would think (someone feel free to correct me on this if my assumption is incorrect) that you could use the Ideal Gas Law to perform your calculation.

The calculation would be fairly simple: P_1*V_1 = P_2*V_2, rearrange to get P_1/P_2 = V_2/V_1. V_2/V_1 should be, if I am thinking correctly, the expansion ratio. P_1 would be the chamber pressure leaving us only to solve for P_2, which should be the pressure at the nozzle exit.

I don't claim to be an expert in analyzing fluid flows, but this might be a decent starting place.

[Guest]

Yes, this was my first thought, but I'm concerned about extreme temperatures at the nozzle. The problem with ideal gas law is that it stops applying in extreme conditions, and inside of a rocket engine is one such condition. I'm not quite sure if the Wiki approach to this is good, though it's certainly easy to work with. Though maybe I'm worrying too much. Taking an inverse of the expansion ratio and multiplying by chamber pressure does seem like the simplest thing to do.

[sal_vager]

Dragon01, will this help you?

[Guest]

Not much, unfortunately. There is a single equation that could be of use (calculating throat pressure from chamber pressure), but that's it. The book just assumes exit pressure is equal to atmospheric pressure, which is not the case most of the time. Also, it assumes gaseous not liquid oxygen being used, making it rather useless. The only reason this pressure equation can be any good is that by the time they get to the throat, both propellants actually are gaseous.

That said, this equation could improve on rather poor results Ideal Gas Law gives when used with chamber pressure.

EDIT: Nope, it doesn't improve much. The pressure for SSME is still much higher than 1 atm, which is, needless to say, very wrong.

Edited November 11, 2014 by Guest

[architeuthis]

I've ran into a problem. I need an equation to determine exit pressure of a rocket engine (de Laval) nozzle, given the chamber pressure and nozzle expansion ratio. It doesn't need to use any terribly detailed model, but should be reasonably accurate. It might also depend on the fuel mixture and it's OF ratio. I'm not sure if assuming an ideal gas would be quite right.

Alternatively, there's an equation on Wikipedia that includes nozzle exit pressure, but it depends on exhaust velocity, not expansion ratio. I'd need actual exhaust velocity (the one that doesn't depend on external pressure), not effective exhaust velocity to use that one, though. I'd need to know a way of the former finding out, because it's the latter is the one that is easily calculated from Isp.

The reason I need this is to have an easy way of finding altitude at which any given engine is at an ambient pressure. I'd like to be able to calculate it off data that is available for most engines.

If you do a bit of algebra you can solve these equations for p_a, then assume the exhaust is ideally expanded such that p_a=p_e.

These are a few of the isentropic gas flow relations. You can find them in engineering fluid mechanics texts pertaining to compressible flow. Note that A_e/A_t is the expansion ratio. Then if you like you can get the altitude from p_a using the standard atmosphere model (assuming these calculations are for Earth, and not Kerbin). The equations are from braeunig.us's section on rocket propulsion.

Edited November 11, 2014 by architeuthis

[MajorThomas]

Couple of comments on rocket engine expansion which you probably already know, but might as well...

(added later) 0. The more you can expand your exhaust stream, the more efficient your engine, the more thrust force you can capture.

1. Ideally a rocket nozzle would always have optimum expansion such that the exit pressure is equal to ambient, but of course that's not possible with a fixed-size nozzle. So you'll start out over-expanded (exhaust pressure less than ambient) on your first stage engine and end up under-expanded. While over-expanded the flow may separate from the inside of the engine bell until the altitude increases/pressure drops to the point that the exhaust stream can exit the bottom of the nozzle fully expanded. Then near burnout of stage 1, the engines are under-expanded, and the pressure is higher, so the exhaust stream "balloons out" as you can see in footage of rocket flights.

2. First stage engines, operating at (let's say) half an atmosphere on average during their use, tend to be shorter, stubbier and don't expand the exhaust stream too much. But upper stage engines tend to be much bigger, to expand the exhaust much more, because they're operating in a vacuum or near-vacuum. But, there's a limit to how large you want to make your engine bells, and so they'll always be smaller than ideal in terms of expansion, but not so large as to be heavy and unwieldy. Hence upper stage engines usually operate a bit under-expanded.