Calculating Specific Impulse in Atmospheric and Vacuum

[QuantumInc]

ISP is a term that gets thrown around fairly regularly in these parts. Ostensibly it is a measure of a thruster's fuel efficiency, the ratio of the thrust provided vs the amount of fuel consumed. This ratio is pretty easy to calculate when you are looking at the statistics of various parts in the VAB. However the numbers didn't necessarily line up with how things were talked about on these here forums, the most notable being Nuclear engines which are known for having the best fuel efficiency, but when calculated from VAB numbers actually have the worst. Right clicking the engine of a craft it flight can show a direct measure of ISP that doesn't match up with the thrust vs fuel consumption numbers one sees in the VAB or SPH. Forum discussions also occasionally mention a difference of atmospheric ISP and Vacuum ISP and the Wikipedia shows that the two are different for almost everything, with the vacuum ISP being noticeably but not dramatically higher. Of course I assume the difference is in fact quite dramatic for Ion and Nuclear.

The question: What is the relationship between Atmospheric ISP, Vacuum ISP, and the ratio of Thrust divided by fuel consumption as shown by the parts list stats in the VAB/SPH?

Edited March 10, 2015 by QuantumInc

[Pecan]

In the VAB/SPH the Isp stats are shown for atmosphere and vacuum - right-click the part's thumbnail to expand its description.

The LV-N has a vacuum Isp of 800, being much better than any other 'normal' engine, which have a maximum of 390. The Ion's is 4,000 but its thrust is so low it's not usually considered for the same sort of vehicles where you'd use a heavy engine.

But, yes, at full atmospheric pressure the LV-N's Isp is only 220; one of the worst ... it increases from that to 800 as you climb and the atmosphere gets thinner (indicated by the barometer beneath the altimeter). On Kerbin the LV-N's Isp already exceeds the other engines by an altitude of 2km or so. Unless you're planning extended flights in atmosphere you can generally ignore atmospheric Isp except for first-stage engines that will perform their whole burn, or most of it, immediately after launch.

NB: How Isp works is one of the fundamental things that will be changed and in general there will be a re-balancing of parts it 1.0, so things may be very different in the next version.

[Red Iron Crown]

Currently, as Isp changes thrust remains constant and fuel flow changes, so a nuclear rocket motor will have 60kN in any condition but will consume about 4x the fuel to make that thrust deep in the atmosphere as opposed to vacuum.

Coming in the next update, thrust will change with Isp rather than fuel flow changing with Isp. So the fuel flow rate of that nuclear rocket will be constant, but it will only deliver about 1/4 the thrust deep in atmosphere. This more accurately simulates the way real rockets work.

Oddly, ions have the same Isp in atmo as in vacuum in KSP, so they are unaffected by it.

As for the formula, it is:

Thrust = Isp * G₀ * FuelFlow

In KSP, the units are kN for thrust, seconds for Isp, 9.82m/s² for G₀, and tons/second for fuel flow.

[OhioBob]

The question: What is the relationship between Atmospheric ISP, Vacuum ISP, and the ratio of Thrust divided by fuel consumption as shown by the parts list stats in the VAB/SPH?

In KSP the atmospheric and vacuum values of I_sp are somewhat arbitrary. It might be helpful to understand why, in real life, this difference exists. To understand we must look at the basic thrust equation:

F = q * V_e + (P_e - P_a) * A_e

where q is the rate of the ejected mass flow, V_e is the exhaust gas ejection speed, P_e is the pressure of the exhaust gases at the nozzle exit, P_a is the pressure of the ambient atmosphere, and A_e is the area of the nozzle exit.

Thrust has two components, the first, q*V_e, is called the momentum thrust, and the second, (P_e-P_a)*A_e, is called the pressure thrust. Most of a rocket's thrust comes from momentum. Pressure thrust is the result of unbalanced pressure forces at the nozzle exit.

For a given rocket engine operating under steady state conditions, the values of q, V_e, P_e, and A_e are constant. The only variable on the right hand side of the equation is the atmospheric pressure P_a, which varies from sea level pressure to zero in a vacuum. Thrust is maximum in a vacuum, when P_a is equal to zero. Thrust is minimum at sea level, when the maximum thrust is reduced by the amount P_a*A_e.

Specific impulse is given by the equation,

I_sp = F / (q * g_o)

where g_o is standard gravity, 9.80665 m/s² in real life and 9.82 m/s² in KSP.

Putting it all together, and letting P_a equal sea level pressure, we can see that

I_sp (vacuum) - I_sp (sea level) = (P_a * A_e) / (q * g_o)

As I've described above, in real life q is typically constant and it is F that changes as a rocket ascends through the atmosphere. As others have explained, in stock KSP it's the other way around, with F constant and q variable. From what we've been told, this is about to change with the release of version 1.0.

Edited March 6, 2015 by OhioBob

[AlexinTokyo]

I_sp varies with ambient air pressure.

IRL: A rocket engine's thrust decreases as air pressure increases (alternatively, as altitude decreases), while the fuel flow rate remains the same.

Since I_sp = F_thrust / á¹ x g₀ (where á¹ is the mass flow rate of the propellants, and g₀ is a conversion factor of 9.81[9.82 in KSP] m.s^-2), you can see that I_sp will vary in direct proportion to thrust. I_sp will be a minimum at sea level (max pressure) and a maximum in a vacuum.

Dimensionally, F_thrust is in kg.m.s^-2, á¹ in kg.s^-1, and g₀ in m.s^-2, giving I_sp in seconds.

Currently KSP simulates the expected change in I_sp by varying á¹ and holding F_thrust constant. This is wrong, but does give the expected change in I_sp and hence in ÃŽâ€v. In addition, the I_sp does not decrease below its Kerbin-sea-level value even if ambient pressure exceeds Kerbin-sea-level pressure, as it would on Eve or deep in Jool's atmosphere

In the next version (1.0) the I_sp calculation is being changed so that á¹ will be constant and F_thrust will change with air pressure. I cannot recall seeing any comment as to whether the curve will be extended to pressures above Kerbin-sea-level pressure.

I think this answers all your questions, please let me know if anything is not clear.

[QuantumInc]

Thank you for your answers.

In retrospect the fact that air/ambient pressure is subtracted from the exhaust's pressure seems utterly obvious.

F = q * V_e + (P_e - P_a) * A_e still doesn't seem right, I would think that F = q * V_e = P_e * A_e in vacuum, and in atmosphere V_e would be less meaningful since the exhaust immediately impacts the air. I guess I'll have to take your word for it OhioBob.

I guess this is another element of the game where they simply did it in a unrealistic way for 0.1.

[Starhawk]

One of the interesting things about this discussion to me is how ISP and thus thrust change with pressures above one atmosphere. If I understand correctly, ISP varies linearly with ambient pressure.

If this relationship were modeled accurately, the 48-7S would have an ISP of 100 at Eve sea-level. This would give it a thrust of only 8.5 rather than 30. The LV-N would produce zero thrust and other engines would be nerfed similarly to the 48-7S, more or less.

Except for the magical aerospike! Because it's ISP changes by almost nothing with one atmosphere of pressure, it would be practically unaffected by the pressure even at sea level on Eve. As was explained to me, in real life an aerospike engine is good, but in KSP the ISP stats are essentially 'magical'.

I'm quite interested to see what the aero changes and part rebalance will mean in this area.

Happy landings!

[Mister Dilsby]

If this relationship were modeled accurately, the 48-7S would have an ISP of 100 at Eve sea-level. This would give it a thrust of only 8.5 rather than 30. The LV-N would produce zero thrust and other engines would be nerfed similarly to the 48-7S, more or less.!

This is exactly why I wanted to go to Eve prior to 1.0. My lander uses mostly aerospikes, but I do carry a Kerbo 25x4 in the core. Post-1.0 and the new aero model, I'll probably explore wing assists for Eve.

[NathanKell]

Given what's already been said about changes to Isp and the Isp curve for 1.0, stay tuned.