Maximum possible dV for a single stage

[WafflesToo]

I was looking through the KW rocketry parts the other day and comparing them to the vanilla parts and I noticed that the KW rocketry fuel parts had a mass ratio of 10-1 while the vanilla parts all have a ratio of 9-1.

Contemplating the consequence of this, it kind of hit me; there is an upper maximum limit to how much dV is possible with a single-stage rocket no matter what you do. Because the Tsiolkovsky rocket equation works off of this ratio it is impossible to build a rocket with a mass ratio greater than it.

So for an LV-30 or LV-45 with stock tanks this is

9.81*370*LN(9/1) or 7,975 m/s

An LV-909 gives us 8,406 m/s

And the LV-N is capable of 17,244 m/s

If you're into RCS ports you're looking at

9.81*290*LN(17/2) or 6,088 m/s

Ion engines have an upper limit of

9.81*4200*LN(12/5) which gives us 36,071 m/s

Of course we're looking at this from the impossible assumption of 100% of the mass of the vehicle being fuel tanks, but these are the limit values you should start to approach as fuel tank mass starts to dominate the mass of the stage.

I know someone plotted these out more fully than this, but I don't think they pointed out that there was an unbreachable ceiling to the equation.

Anyway, food for thought I suppose.

[Specialist290]

An astute observation, and one that's indeed not obvious to those not familiar with the math. Nicely done!

[WafflesToo]

An astute observation, and one that's indeed not obvious to those not familiar with the math. Nicely done!

Thanks. It feels like one of those things that should've been obvious, but I guess I hadn't thought about it like that before then. I'm sure that there's a point of diminishing returns that you run into well before the limit, but I haven't sat down and run the numbers to see where it lies at.

[numerobis]

This is why SSTO isn't going to happen on Eve without some strange tricks. But, when you stage off those empty tanks, you break the maximum.

[Specialist290]

Thanks. It feels like one of those things that should've been obvious, but I guess I hadn't thought about it like that before then. I'm sure that there's a point of diminishing returns that you run into well before the limit, but I haven't sat down and run the numbers to see where it lies at.

I'd imagine that there's some practical limit that's relatively easy* to calculate for an SSTO vehicle using a given engine. In an atmosphere, an ideal ascent profile's earliest stage is more or less going to follow terminal velocity, at least until it reaches the point where the atmosphere thins out faster than the engines can reasonably accelerate the craft as a whole, and a single engine's thrust, mass, and rate of fuel consumption are all known, fixed** values.

* "Relatively easy," of course, should not be taken in this case to mean "absolutely easy," as the math is probably something that requires calculus, which honestly is a bit beyond my own level of knowledge. Still, it's probably easier to calculate the ideal proportion for this one case than for every possible case, which is what I mean by way of comparison.

** In the sense that, in the particular case of fuel consumption, it's linked to Isp, which for rocket engines rises in a linear inverse proportion as atmospheric pressure falls. If we assume a fixed amount of thrust with no throttle variation, this again should become relatively easy to calculate -- though, again, see the caveat mentioned above on what "relatively easy" means in this context.

[tavert]

Max theoretical dV is just the limit of infinite fuel tanks at the best mass ratio your component choices allow. Max practical dV is a function of TWR. Flip through my charts and you'll see that for high TWR, there's an upper limit of dV on the right side of the chart, that just isn't possible at a given TWR with the engines and fuel tanks we have. Play around with the spreadsheet version and you can see that as dV gets close to the max possible for a given TWR, the craft mass increases very quickly relative to the payload.

[AuoroP]

Very clever insight!

I did notice a couple of caveats. The OP is correct that FL and Rockomax tanks have a ratio of 9:1 but there are a couple tiny LF tanks with their own ratios: ROUND-8's is 5.44 (full 0.136t and emtpy 0.025t), Oscar-B's is 5.25 (full 0.078675t empty 0.015t).

The OP also missed jet engines and jet fuel. A basic jet engine has an Isp of 2000 at sea level. The Mk1 fuelselage has a ratio of 3.14 and all the others have a ratio of 5 (exactly). Putting the best possible case together we have

31,577 m/s = 2000 s * 9.81 m/s^2 * ln(5/1)

I also wanted to more verbosely explain the OP. The ratio the OP was talking about is the mass ratio used in the rocket equation: Mfull / Mempty. The Rockomax X200-16 (handily) has a full mass of 9 t and an empty mass of 1 t.

The OP had the insight that all the FL and Rockomax fuel tanks have a ratio of 9 t / 1 t and further that as you keep adding more and more tanks the weight of the engine becomes more and more neglible unil we just throw away all other mass besides fuel tanks and we get an upper limit on an idealized stage delta-v that is, for the LV-909:

8,400 m/s = 390 s * 9.81 m/s^2 * ln(9/1)

[tavert]

Jets don't use the rocket equation in quite the same way, since they don't have to carry their own oxidizer. I think you can effectively multiply the Isp by the ratio of intake air to fuel mass (or is it intake air plus fuel mass, divided by fuel mass? I forget which). numerobis knows for sure and hopefully correct me here.

[numerobis]

Multiply Isp by 15, since you burn 15 air per unit fuel. I think -- unless it's multiply by 16, because fuel is 1/16th of the mixture. However, thrust on a KSP jet depends on your speed, so the deltaV you achieve is not what the Tsiolkovski equation predicts (it's lower). And you're operating near a planet, which means gravity and drag losses matter a lot. So I don't see much utility in this metric for jets.

You could measure max dV assuming you maintain 1 km/s and you're burning stored intake air. Intakes are very inefficient tanks (0.15 full to 0.1 empty for radials, 0.11 full to 0.1 empty for rams), so I count the LV-N as outperforming a jet on this metric.

[AuoroP]

Multiply Isp by 15, since you burn 15 air per unit fuel. I think -- unless it's multiply by 16, because fuel is 1/16th of the mixture. However, thrust on a KSP jet depends on your speed, so the deltaV you achieve is not what the Tsiolkovski equation predicts (it's lower). And you're operating near a planet, which means gravity and drag losses matter a lot. So I don't see much utility in this metric for jets.

The Isp I used was already 2.5 times higher than the magical Ion drive so I just want to check that the game hasn't already accounted for what your adjustment. Otherwise, I would dismiss gravity and drag losses not being accounted for with any other theoretical limits and arrive at 473,657 m/s.

Intakes are very inefficient tanks (0.15 full to 0.1 empty for radials, 0.11 full to 0.1 empty for rams), so I count the LV-N as outperforming a jet on this metric.

I'm not seeing why this is true. Higher ratios means higher limits so ratios 15:1 or 11:1 beat the otherwise winners FL and Rockomax ratios of 9:1 but I don't see how that relates to efficacy. A single intake is a small fraction of the mass of a LF tank but provides unbounded oxidizer but if you insist on bringing your own oxidizer with you then 55%, 110/(90+110), of your LF tank is oxidizer a jet engine doesn't need.

[numerobis]

The 15:1 ratio is the ratio of IntakeAir:LiquidFuel that the engine burns. If you multiply Isp by 16 (or 15, whichever is right) you're assuming IntakeAir is for free. Which is a halfway reasonable assumption.

My only problem is that, while the maximum deltaV for a given single-stage spacecraft is something that gives you a good idea of performance limits in interplanetary space, the max deltaV for a jet tells you very little. For the basic jet, you're measuring how much dV you can achieve as long as the jet stays at sea level and never actually moves (since thrust goes down with speed). For the turbojet, you're measuring how much dV you can achieve as long as the jet stays at precisely 6020m altitude (max Isp ~2525) and is going precisely 1 km/s (to achieve the stated thrust). Neither of these is a useful model of a flight.

On the flip side, it's definitely useful to assume you'll be getting about 10ks Isp when measuring out how much fuel you should pack for your flight.

The second ratio I mentioned, 0.15:0.1, is the ratio of full:dry mass of a radial intake, i.e. the ratio that goes into the log in your equation. Intakes are tanks of air: close them before liftoff, and they'll be full. Bring them to orbit and you can burn their stored IntakeAir in a 15:1 ratio with LiquidFuel, in the vacuum. For example.

[AuoroP]

I checked and 1km/s does not exceed terminal velocity as long as you're above 23.395 km. This is not true for 6.020 km whose terminal velocity is is 181.0 m/s. I don't have hard data on this but my intuition tells me that terminal velocity is a cruise limit for jet engines and the flight you're suggesting exceeds this limit by 552%. So I'll admit that drag loss would make this cruise only theoretical.

I got a laugh out of imagining a Bob Kermin putting two jet engines (front and back) on a fuel tank and speeding ahead until they're exceed 1m/s and yelling OMG TOO FAST!!! STOP!!! I never liked Bob because what always seemed to freak him out was being in space. I've long suspected Kerbal nepotism: a Kerbal senator's wife's useless brother needing a job? Finally a test arises which is ideally suited to him! I can also imagine how much happier Jeb would be if he were cruising along at 1km/s in a jet and then decided to stop because Bob's screaming was ruining the experience. We would know that stopping would require a delta-v of 1000 m/s.

Still if there is a maximum possible delta-v for a single stage, I still think there should be a value for jets; even with a star and a guffawing laugh beside it. What number would you pick? I'll be enjoying my first flight with Bob Kermin the jet sailor!

[WafflesToo]

Max theoretical dV is just the limit of infinite fuel tanks at the best mass ratio your component choices allow. Max practical dV is a function of TWR. Flip through my charts and you'll see that for high TWR, there's an upper limit of dV on the right side of the chart, that just isn't possible at a given TWR with the engines and fuel tanks we have. Play around with the spreadsheet version and you can see that as dV gets close to the max possible for a given TWR, the craft mass increases very quickly relative to the payload.

I think your charts were the ones I was thinking of (god the 48-7S is OP , wonder how they'd fair now?)

I'm not usually concerned with TWR for my cruise-stages (landers are a different story though ) so the limit is of some interest to me. I haven't sat down to analyze the curve yet to find where the point of diminishing returns (and thus the effective stage limit) is at.

...jets... dV calculations... etc .

Jet engines are such an enigma to me I don't even have any idea how the calculations for those work.

Edited December 7, 2013 by Specialist290
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