# In the context of calculating mass flow rate from thrust and Isp, how would an additional efficiency be defined?

In this answer I show a quickie derivation for calculating a mass flow rate of a rocket from a known thrust and Isp.

A comment there says:

That assumes 100% efficiency and ignores the fuel required for the turbo pump, though.

I understand that there might a (relatively, at least) small amount of propellants used to generate power for pumps or other things, and I've certainly ignored that in my quickie calculation, but what other effects might be considered to understand this efficiency?

I'd assumed that Isp expresses the axial component of the velocity, so even if there is some transverse flow in the expanding exhaust, that wouldn't have to be accounted for beyond Isp, but is there something else?

• I was referring to 'any energy content in the fuel that doesn't end up contributing to thrust'. In an internal combustion engine, that's more than 50%, I don't know how much is lost in a rocket. – Hobbes Sep 6 '18 at 14:55
• @Hobbes Energy in the fuel that doesn't contribute to thrust probably does not contribute to Isp either? So that should be already accounted for when you try to get fuel mass flow from ISP. Indeed I have found now that Wikipedia (ok, not the greatest source) says on ISP "By definition, it is the total impulse (or change in momentum) delivered per unit of propellant consumed[1] and is dimensionally equivalent to the generated thrust divided by the propellant mass flow rate or weight flow rate.[2]" en.wikipedia.org/wiki/Specific_impulse – BlueCoder Sep 6 '18 at 15:22
• @BlueCoder et al. see this comment – uhoh Sep 6 '18 at 15:26
• @BlueCoder I agree that this effect is already baked into Isp. – Organic Marble Sep 6 '18 at 15:37

I'd assumed that Isp expresses the axial component of the velocity, so even if there is some transverse flow in the expanding exhaust, that wouldn't have to be accounted for beyond Isp, but is there something else?

To be pedantic, Isp is specific impulse, or more fully "mass specific impulse": the impulse delivered per unit of mass flow. Impulse is force times time. The connection to exhaust velocity is incidental. The proper (metric) unit for specific impulse is newton-seconds per kilogram; this happens to be dimensionally equivalent to meters per second.

For an ideal abstract rocket, emitting reaction mass perfectly linearly, specific impulse happens to be the same as exhaust velocity, but in real rockets the actual (mass weighted, average) exhaust velocity will be a tiny bit higher than the effective exhaust velocity because of the transverse component.

If you have a terrible nozzle design, spraying exhaust in all directions, your Isp will be low even though the exhaust velocity is high.

In practice, specific impulse ratings for rocket engines are determined from the flow rate and the thrust, both of which are more easily measured on the test stand than the actual exhaust velocity; all the inefficiencies involved are accounted for thereby.

Note that for jet engines, Isp/effective exhaust velocity will be much higher than actual exhaust velocity, because airborne oxygen contributes to thrust but not to expenditure of onboard mass.

• This is an excellent answer and of course you are right. I've updated the linked answer, linking back to here. Thanks for expressing this so clearly! – uhoh Sep 6 '18 at 17:00

If I understand correctly that your question is asking about inefficiencies other than those caused by propellants that bypass the main combustion chamber (like gas generator propellants) that aren't accounted for in your calculation:

Assuming you're calculating for a booster, the largest one is probably variation of specific impulse with ambient pressure / altitude.

Here is a representative graph of the variation. The liftoff specific impulse in this case is about 15% lower than the vacuum value.

• In my linked answer: "The specific impulse of a Falcon 9 Merlin 1D is about 262 seconds in the atmosphere." So it looks like I've already used the lowest value for full thrust condition. Since throttling happens, I wonder if the Isp is actually lower when an engine throttles? You can't scale the combustion chamber and throat dynamically to track throttling so easily. That might be quite a noticeable penalty. – uhoh Sep 6 '18 at 15:25
• This question addresses Isp vs throttling, there is usually a small drop: space.stackexchange.com/questions/12133/… The reason why your quick calculation is superior for back of the envelope answers is that if you try to include Isp/altitude variation in the calculation, you have to know about the trajectory as well. – Organic Marble Sep 6 '18 at 15:35
• What's the source of the graph? (Or at least are the units feet or meters?) – Russell Borogove May 28 '19 at 22:18
• I don't remember, that was a long time ago. The x axis label looks like F to me. This is an easy calculation anyway, just use the thrust equation with the exit plane pressure term included. – Organic Marble May 28 '19 at 22:26

Russell Borogove has already answered, but I wanted to point out more explicity a few things:

• ISP is a derived quantity
• You do not need to add any efficiency factor
• If the thrust and ISP figures come from the same test, by doing Thrust/ISP you are recovery the exact fuel mass flow measured at that test

I do not think you need to add any efficiency factor at all.

Specific impulse (usually abbreviated Isp) is a measure of how effectively a rocket uses propellant or jet engine uses fuel. By definition, it is the total impulse (or change in momentum) delivered per unit of propellant consumed [1] and is dimensionally equivalent to the generated thrust divided by the propellant mass flow rate or weight flow rate. [2]

So $$ISP = Thrust / mass\ flow\ rate$$ and this means $$mass\ flow\ rate = Thrust / ISP$$ So your calculation in the related answer is perfect.

One might wonder if this relationship is just theorical and if in reality some efficiency or other factors would apply when these three variables are measured.

As Russell Borogove pointed out, ISP is related to the exhaust velocity in an ideal abstract rocket, but more properly ISP is actually the effective exhaust velocity, which might be different from the velocity that you'd measure at the exhaust. For example, in jet engines, the effective velocity will be higher than the exhaust velocity because they get "free oxygen" that contributes to thrust beyond what the fuel mass they carry would do with that exhaust velocity.

So how would you measure ISP/effective velocity in pratice? From what I gathered so far, I guess that you do not measure ISP, you derive it from the other two quantities.

For sure when testing a rocket engine you can measure its thrust in a load cell and you can measure the fuel you are using running the engine ([3],[4])

So if those 7,607,000 N thrust and 162s ISP numbers come from empirical measurements done in the same session (and they are not estimates from simulations or calculated in another way), it is much likely that thrust was measured along with fuel mass flow and impulse derived simply by dividing thrust by fuel mass flow.

So when you do 7,607,000 / 162 you are really recovering the fuel mass flow measured in that test.

In this case, your fuel mass flow already accounts for any "inefficiency" or fuel that did not contribute to thrust, since it is the thrust obtained divided by the fuel really used (regardless if it was efficiently used or not).

Disclaimer: I never tested any rocket engine nor any other engine, this was just guesswork - however, this quora answer agrees: https://www.quora.com/How-do-we-measure-rocket-thrust-like-specific-impulses

• Thanks! However, I did forget that some propellants may go to another combustion area for things like electrical power or mechanically driving the turbo pumps that push the rest into the combustion chamber, so no, it was not "perfect". – uhoh Sep 7 '18 at 8:55
• No, that's already included. When you test the rocket in the static test rig, you would measure ALL the fuel going into the rocket engine, including the fuel used to spin turbine or other things not directly generating thrust. Given that due to the tiranny of the rocket equation every single Kilogram counts when launching things into space, I would be very surprised if they kept separate tanks for the fuel going to combustion chamber and the fuel "for the other purposes" and then measure only the former. – BlueCoder Sep 7 '18 at 9:58
• It would be like saying: "Oh, look, I have this very efficient rocket engine with 400 ISP, do not mind the fact that you have to carry an extra 1000 Kg of fuel to make turbopumps work". To put it again in other words, ISP is the efficiency of the engine in terms of thrust per fuel spent, so it already includes all the efficiencies and inefficiencies of the system (when empirically tested by measuring thrust output vs the fuel you have used). – BlueCoder Sep 7 '18 at 9:58
• I'm not paying attention because I'm writing two other answers at the same time. Now that I've re-read your comments, I think you are saying that Isp values for engines *always include any mass flow associated with the engine, including power to drive pumps." – uhoh Sep 7 '18 at 10:14
• For more in-depth info on fuel mass flow measurements see also this link (that I had included as a source of "mass flow is measured" but only now I've full read it). blogs.nasa.gov/J2X/2012/09/14/post_1347557419753 All fuel that goes into the engine is counted. Interestingly, there is actually a caveat - it seems we only want to measure fuel that goes overboard (but it doesn't matter from where I think). Fuel that is recycled back i.e. to keep pressurization in the tank is not counted (but that's ok to me, because you didn't really spent it, it's still in the system). – BlueCoder Sep 7 '18 at 10:18

The energy that doesn't contribute to thrust is already accounted for in ISp and thrust.

All the burned propellant from turbopump propulsion etc is injected back into the combustion chamber, contributing to reaction mass, but not to propulsion of said reaction mass. The propellant flow rate is unchanged, but its specific energy at the point of reaching combustion chamber is lower.

Calculating just flow rate from ISp and thrust is unchanged; the ISp and thrust values are a little lower than it would be had you burned pure bipropellant.

You'd need to account for these things if you were trying to derive ISp and thrust from mass flow and chemical/physical properties of the combustion process; the sort of equation you find in chapter 7 of "Ignition!" ("Performance"). Normally though, your engine will be running a little fuel-rich (or oxidizer-rich) anyway, so you just move the balance a bit towards stoichiometric, replacing unburnt ejected fuel with corresponding amount of pre-burned combustion products and you have very similar results... maybe even a little better in case of heavier fuels, as CO2 and H2O will make a better inert propellant than RP1 vapor (lighter particles = higher exhaust speed at same energy).

Still, in your calculations, you use $F = v \dot{m}$. Your $\dot{m}$ remains unchanged due to the overhead you mention - $v$ changes, and so does the published specific impulse and thrust, relative to theoretical value you'd achieve injecting pure RP1-O2 mix into the chamber.

I believe I recall there was some rocket, where the turbopump propulsion exhaust would be ejected through a separate nozzle, with considerably lower performance than the "main" one. I can't recall the details though, never mind I might have confused that with Soyuz's verniers.

• Many rockets have separate exhaust nozzles for the turbopump, including the Falcon 9. Only staged combustion engines run the turbopump exhaust through the combustion chamber. Some gas generator engines inject the GG exhaust into the nozzle for film cooling. – Hobbes Sep 7 '18 at 9:24
• @uhoh: Thing is raw unburnt fuel in a tank on the launchpad or test stand doesn't have momentum or velocity or such. It has specific energy and mass/volume, which combine into total energy and then all the rest comes from that. It's not momentum or thrust that turns the turbopump. Once you're in the realm of momentum, thrust and velocity, you're already past the point where the losses occurred. – SF. Sep 7 '18 at 11:57
• @SF. ya I see what you mean – uhoh Sep 7 '18 at 12:15