Effect of TWR, TVR, and DVR on Orbital Launch Efficiency (deltaV)

[arkie87]

[tl:dr]To obtain efficiency greater than or equal to 90%, a thrust-to-weight ratio (TWR) of at least 1.4 at launch on an atmosphereless, Kerbin-sized body (like Tylo) is required with standard ISP engines; but for planets with smaller orbital velocity or more efficient engines (nuclear or Ion), TWR at launch to obtain efficiency greater than 90% increases. [/tl:dr]

I have made a computer model in Matlab (EDIT: Excel version now available! See Below!) to simulate optimal TWR (thrust-to-weight ratio), TVR (thrust-velocity-ratio), and DVR (delta-V-ratio). I provide non-dimensional contour plots to show results which are globally applicable for the same non-dimensional parameters.

Link to the Paper

Results:

This graph shows the independence of efficiency on DVR. The slope of the curve is essentially zero, until efficiency drops to zero instantly when delta-V-ratio no longer provides enough deltaV to get into orbit due to efficiency.

Since efficiency is not a function of DVR (i.e. how much extra fuel you carry into orbit), we only need to vary TWR and TVR. The above plot contains all the information a player needs when designing craft.

First, increasing TWR increases efficiency while increasing TVR decreases efficiency. Increasing TWR increases efficiency by allowing the craft to aim more horizontal, thereby, using more of its fuel to accelerate into orbit instead of fighting gravity. Increasing TVR decreases efficiency since, for a given DVR, less fuel is burned. This, in turn, results in a more constant TWR during the flight, and therefore, a longer flight as well as a steeper angle above vertical, causing more fuel to be wasted fighting gravity.

Second, for low TVR, lower TWR are needed for a given efficiency; similarly, for a higher TVR, a higher TWR is needed to obtain the same efficiency. This is the direct result of the trends described above, and is the most important result of the simulation: thus, for two crafts with the same ISP engine, the craft on Minmus will require a higher TWR for the same efficiency as a craft on Mun, and so on.

Finally, this model should be easy to test. Since efficiency is not a function of DVR (i.e. size or scale), two craft with the same TWR can be compared on the same planet with different ISP engines or on different planets with the same engine, and the craft with the larger ISP engine or equivalently on the planet with a smaller orbital velocity should have reduced efficiency.

And in case you dont believe that efficiency is independent of DVR for all TWR and TVR:

Excel Versions of the Simulator:

Version 1

Version 2

Version 3 with Macros

Edited December 29, 2014 by arkie87

[Anquietas314]

Uh, did you not already post this in what is really the more appropriate forum? This appears to be almost a copy-paste.

[arkie87]

Uh, did you not already post this in what is really the more appropriate forum? This appears to be almost a copy-paste.

In the science forum, i developed, edited, tested in-game, and refined the paper.

Now it is fully-refined, and serves merely as information for game play. Maybe tutorials is the more proper forum.

[Superfluous J]

This has GOT to be the most time anybody has ever put into justifying MOAR BOOSTERS.

[arkie87]

This has GOT to be the most time anybody has ever put into justifying MOAR BOOSTERS.

+1

But the results show that MOAR BOOSTERS doesnt pay very much... (at least once TWR > 1.6 and for airless bodies)

But it is MOAR FUN!

[Jouni]

Two things:

1. Define TVR and provide an intuitive description for it. Despite having played KSP for over a year, I've never heard of the term before.
   
2. Define efficiency and justify why your definition is useful. Are you using payload fraction, vehicle cost per tonne of payload or something else?
   

[GoSlash27]

Jouni,

I've been volunteering gofer work on this project and can answer these questions.

1. Define TVR and provide an intuitive description for it. Despite having played KSP for over a year, I've never heard of the term before.
   

Yeah, that had me scratching my head as well at first.

It is the ratio of exhaust velocity to orbital velocity, or 9.82Isp/Vo.

An intuitive description... As your Isp goes up, TVR goes up proportionally. As your orbital velocity goes up, TVR goes down inversely.

Here's a handy table I knocked together:

Name / DVmin / orbit height/ local G/ orbit velocity/

Mun / 580 / 10 km / 1.63 / 547

Minmus/ 180 / 6 km / 0.491 / 164

Gilly / 30 / 8 km / 0.049 / 19.9

Moho / 870 / 30 km / 2.70 / 776

Ike / 390 / 10 km / 1.10 / 364

Dres / 430 / 30 km / 1.13 / 358

Pol / 130 / 5 km / 0.373 / 121

Bop / 220 / 25 km / 0.589 / 166

Tylo / 2,270 / 60 km / 7.85 / 2,070

Vall / 860 / 25 km / 2.31 / 799

Eeloo / 620 / 20 km / 1.69 / 569

So as an example, using 48-7S engines (Isp= 350) to launch from the Mun (Vo= 547) would give you a TVR of 350*9.82/547= 6.3

2. Define efficiency and justify why your definition is useful. Are you using payload fraction, vehicle cost per tonne of payload or something else?

It's the ratio of "perfect" Dv expenditure to how much Dv you'd actually expend to achieve orbit. It's useful because it allows us to define where adding "Moar boosters" is no longer beneficial due to cost and weight penalty.

Using the example above, we can see that using the contour plot at DVR= 6.3, we achieve about 58% efficiency at t/w=1 and about 90% efficiency at t/w=2. If the fuel and tankage costs and weighs less than the engines, it's not worth it to double the t/w.

Best,

-Slashy

Edited December 26, 2014 by GoSlash27

[Jouni]

It's the ratio of "perfect" Dv expenditure to how much Dv you'd actually expend to achieve orbit. It's useful because it allows us to define where adding "Moar boosters" is no longer beneficial due to cost and weight penalty.

So efficiency is essentially (orbital velocity at altitude 0) / (optimal delta-v required to reach that orbit), which depends on the initial TWR and the ratio of Isp to orbital velocity. This is useful for estimating how much delta-v is required for the ascent with a given TWR, but less useful for determining the optimal TWR. After all, a higher TWR usually means more engine mass and more fuel required to get the same amount of delta-v.

[GoSlash27]

arkie,

The above graph is lacking range in the y axis for smaller bodies and higher Isp engines (unless my math is off).

It doesn't extend far enough for LV-N launches from any body smaller than Vall and doesn't accomodate PB-ION engines anywhere.

It also can't accomodate the use of any engines (no matter what the Isp) on tinier planets like Pol.

-Slashy

Edited December 26, 2014 by GoSlash27

[Sean Mirrsen]

Wouldn't this need to at least be in terms that actually exist within KSP in order to be useful as a KSP gameplay aid? I.e. I_sp rather than TVR, at least. And DVR is... payload? As you defined it, it's literally "ratio of how much fuel you carry to how much fuel you need to actually make orbit". With TWR and I_sp collectively defining the rocket as a whole, of course DVR has no impact - excess fuel reduces TWR instead, but if that is assumed constant then changing DVR mostly just changes the size of the rocket.

Basically I'm questioning the practical usefulness of the graphs as designed. It's neat as a theoretical exercise, but I'm seeing no practical applications to gameplay. It's hardly any help to building a good rocket. More dV being "wasted" as I_sp increases is an interesting effect in principle, however, again, inapplicable - a craft losing less mass in spent fuel with a more efficient engine means that while it "wastes" dV in principle, it also means that in typical conditions it also carries more dV in the first place. There's also the theoretical maximum TWR of existing KSP engines, and tankage. Without accounting for all variables, I see little use for this graph in the gameplay tutorials section.

[GoSlash27]

Jouni,

So efficiency is essentially (orbital velocity at altitude 0) / (optimal delta-v required to reach that orbit), which depends on the initial TWR and the ratio of Isp to orbital velocity.

Not quite. It would be the "perfect DV to achieve your desired orbit"/ "how much DV you actually expend".

As an example, referring to the Mun in my table above, it would theoretically take 580 m/sec DV to achieve a stable 10kM orbit. At 50% efficiency, you would actually expend 1,160 m/sec.

This is useful for estimating how much delta-v is required for the ascent with a given TWR, but less useful for determining the optimal TWR. After all, a higher TWR usually means more engine mass and more fuel required to get the same amount of delta-v.

I disagree, although I said essentially the same thing.

It requires more engine mass, but only marginally more fuel (the only added mass is the engine itself). Were engines massless, you could double the t/w and not require any additional fuel to make the same DV.

Best,

-Slashy

[arkie87]

Wouldn't this need to at least be in terms that actually exist within KSP in order to be useful as a KSP gameplay aid? I.e. I_sp rather than TVR, at least.

The reason I cannot put it in terms of ISP is because ISP, alone, is not what is important. What is important for estimating efficiency is ISP*g0/v0. So having a certain ISP engine on one planet will perform differently than on others. Thus, i would need to provide a graph of efficiency vs. ISP and TWR for every body in KSP. Defining TVR, on the other hand, allows me to show just one graph, which can be used for all planets and moons as long as the player can do some basic math to find out what their TVR is. This is the advantage of non-dimensionalization.

And DVR is... payload? As you defined it, it's literally "ratio of how much fuel you carry to how much fuel you need to actually make orbit". With TWR and I_sp collectively defining the rocket as a whole, of course DVR has no impact - excess fuel reduces TWR instead, but if that is assumed constant then changing DVR mostly just changes the size of the rocket.

I dont think that result is necessarily obvious, given the complicated, non-linear nature of the equations.

Basically I'm questioning the practical usefulness of the graphs as designed. It's neat as a theoretical exercise, but I'm seeing no practical applications to gameplay. It's hardly any help to building a good rocket. More dV being "wasted" as I_sp increases is an interesting effect in principle, however, again, inapplicable - a craft losing less mass in spent fuel with a more efficient engine means that while it "wastes" dV in principle, it also means that in typical conditions it also carries more dV in the first place.

It is important to know that if you increase ISP, you efficiency drops, and so while you might gain more deltaV, you will also need more deltaV to get into orbit. If a player didnt know this, they might improve their engine ISP, and reduce FMR (amount of fuel) to compensate to maintain the same total deltaV they had before, without realizing they will need more fuel to get into orbit..

There's also the theoretical maximum TWR of existing KSP engines, and tankage. Without accounting for all variables, I see little use for this graph in the gameplay tutorials section.

I'm not sure what this has to do with anything. My graphs dont suggest you go to TWR = infinity. They suggest quite the opposite, actually, that TWR > 2.5 are quite pointless (though very fun).

Edited December 26, 2014 by arkie87

[Jouni]

Basically I'm questioning the practical usefulness of the graphs as designed. It's neat as a theoretical exercise, but I'm seeing no practical applications to gameplay.

I find it quite useful for those playing with RSS. For stock planets, there are many good estimates of the delta-v requirements for landing and taking off. For the different rescalings of the Kerbol System and the real solar system, the estimates are less accurate. This method gives a good way of estimating the delta-v requirements, once you know the velocity at a low orbit.

It requires more engine mass, but only marginally more fuel (the only added mass is the engine itself). Were engines massless, you could double the t/w and not require any additional fuel to make the same DV.

The differences in efficiency are also marginal, as people rarely build landers with TWR less than 1.5. Still, knowing which initial TWR maximizes the payload fraction for a given engine type and a given planet would be interesting.

[GoSlash27]

The differences in efficiency are also marginal, as people rarely build landers with TWR less than 1.5. Still, knowing which initial TWR maximizes the payload fraction for a given engine type and a given planet would be interesting.

Actually, I do and I always have (except where I can achieve higher t/w for free) for this very reason. I came to this conclusion but never had hard numbers to go from before.

Hopefully I can work with all this data and boil it down to a much simpler set of recommendations; like "for this engine, it's not worth it to add engines beyond such-and-such a t/w".

Gonna take me a while to crunch the numbers, tho'.

Best,

-Slashy

[Torquemadus]

The paper mentions taking off via a Hohmann transfer. This implies taking off horizontally and giving myself a periapsis on the opposite side of the planet. I would then coast to periapsis and circularise.

In practice, to take off from the equivalent of sea level from an airless body, I need to clear any mountains or other terrain that might be in the way. Therefore, I would perform a short burn to go sub-orbital, with a periapsis at a safe altitude. I would then coast to periapsis and circularise there.

[Anquietas314]

The paper mentions taking off via a Hohmann transfer. This implies taking off horizontally and giving myself a periapsis on the opposite side of the planet. I would then coast to periapsis and circularise.

I'm pretty sure you meant apoapsis - periapsis is the low point in your orbit.

[Torquemadus]

I'm pretty sure you meant apoapsis - periapsis is the low point in your orbit.

Yes, I meant to say apoapsis!

[arkie87]

The paper mentions taking off via a Hohmann transfer. This implies taking off horizontally and giving myself a periapsis on the opposite side of the planet. I would then coast to periapsis and circularise.

In practice, to take off from the equivalent of sea level from an airless body, I need to clear any mountains or other terrain that might be in the way. Therefore, I would perform a short burn to go sub-orbital, with a periapsis at a safe altitude. I would then coast to periapsis and circularise there.

First, i think by periapsis, you mean apoapsis? EDIT

This paper mentions it doesnt include the effects of terrain. However, unless you are right next to a vertical ridge, there is almost always likely a relatively horizontal ascent approach.

I'm not sure what you mean by a "short burn to go sub-orbital". If you mean burn vertically to go a few km up, this is a really bad approach (unless you are literally next to a vertical wall) since all deltaV used to jump up will be wasted when you start to burn horizontally.

You can do that math assuming infinite TWR (impulse burns) and jumping up even a short distance is much worse than aiming as horizontal as possible.

I had another thread http://forum.kerbalspaceprogram.com/threads/102947-Vertical-Ascent-vs-To-LXO-First that tried to see if there was a single instance in the Kerbol solar system of this being practical, and the only place i found was Minmus (since there are regions which are surrounded by tall hills).

Edited December 26, 2014 by arkie87

[Torquemadus]

First, i think by periapsis, you mean apoapsis? EDIT

This paper mentions it doesnt include the effects of terrain. However, unless you are right next to a vertical ridge, there is almost always likely a relatively horizontal ascent approach.

I'm not sure what you mean by a "short burn to go sub-orbital". If you mean burn vertically to go a few km up, this is a really bad approach (unless you are literally next to a vertical wall) since all deltaV used to jump up will be wasted when you start to burn horizontally. You can do that math assuming infinite TWR (impulse burns) and jumping up even a short distance is much worse than aiming as horizontal as possible.

I try to make my ascent path as horizontal as possible, but I rarely take off from locations that are completely flat.

Normally, after take off, I would aim for a near horizontal sub-orbital trajectory that just barely clears any nearby terrain, with an apoapsis at the lowest safe altitude for safe orbital flight. I would then coast to apoapsis and then circularise.

[arkie87]

I try to make my ascent path as horizontal as possible, but I rarely take off from locations that are completely flat.

Normally, after take off, I would aim for a near horizontal sub-orbital trajectory that just barely clears any nearby terrain, with an apoapsis at the lowest safe altitude for safe orbital flight. I would then coast to apoapsis and then circularise.

I dont think coasting to apoapsis is fuel efficient, particularly for low TWR. You have already spent fuel to get apoapsis up, and bought yourself time to burn horizontally, I would think you should do that...

[Sean Mirrsen]

The reason I cannot put it in terms of ISP is because ISP, alone, is not what is important. What is important for estimating efficiency is ISP*g0/v0. So having a certain ISP engine on one planet will perform differently than on others. Thus, i would need to provide a graph of efficiency vs. ISP and TWR for every body in KSP. Defining TVR, on the other hand, allows me to show just one graph, which can be used for all planets and moons as long as the player can do some basic math to find out what their TVR is. This is the advantage of non-dimensionalization.

Okay, but then you have to provide the TVR value for every planet/engine combo you encounter. For which you need to know the mass of fuel (which isn't necessarily in any real units), the I_sp of the block of engines on your craft, and the orbital velocity at the orbit you need around the planet you're at. Having an I_sp/TWR graph per each airless body in KSP, especially since it's pretty much a matter of plugging numbers into a formula, seems like the more useful method.

I dont think that result is necessarily obvious, given the complicated, non-linear nature of the equations.

There is, on the other hand, the plain, simple nature of the definition you use for DVR. Delta-V Ratio, as in "the ratio of delta-V you have versus the delta-V you need". Given that definition, it's quite easy to see why its effect on the efficiency is so binary. Either you make it or you don't.

It is important to know that if you increase ISP, you efficiency drops, and so while you might gain more deltaV, you will also need more deltaV to get into orbit. If a player didnt know this, they might improve their engine ISP, and reduce FMR (amount of fuel) to compensate to maintain the same total deltaV they had before, without realizing they will need more fuel to get into orbit..

Er, no. They'll need less fuel to get into orbit. More time than ideal perhaps, yes, and thus more dV lost to gravity, but less fuel - that's what I_sp does. I see how it might be useful in extremely dV-tight situations though, I guess. The concept of dV needed increasing because TWR does not increase as much with less initial fuel mass fraction is rather often overlooked.

I'm not sure what this has to do with anything. My graphs dont suggest you go to TWR = infinity. They suggest quite the opposite, actually, that TWR > 2.5 are quite pointless (though very fun).

Not starting TWR. Final TWR. A craft with total mass of 3 tons, a dry mass of 1 ton, and a starting TWR of 2, will have a final TWR of 6 just as its fuel runs out - since its mass decreases threefold. Let's say .2 tons of that is tankage, .8 tons is the engine. The engine thus has a theoretical maximum TWR of 7.5, in an ideal craft without tankage. An LV-N, however, has high I_sp, but awful TWR - around 3.5, IIRC. Whatever the starting TWR and FMR, it can never go beyond that 3.5 TWR because that's how much the engines themselves weigh, even disregarding tankage.

If your formula assumes an ideal craft with no tankage and massless engines, with TWR approaching infinity as fuel runs out, it may very well skew your results.

[arkie87]

Okay, but then you have to provide the TVR value for every planet/engine combo you encounter. For which you need to know the mass of fuel (which isn't necessarily in any real units), the I_sp of the block of engines on your craft, and the orbital velocity at the orbit you need around the planet you're at. Having an I_sp/TWR graph per each airless body in KSP, especially since it's pretty much a matter of plugging numbers into a formula, seems like the more useful method.

Why on earth would i have to provide this? That would defeat the whole point of non-dimensionalization! The player can calculate what their TVR is for the engines they are using and the body they are departing from.

It might be more useful to use ISP and then have separate plots for each body in KSP, but it is definitely more convenient to have one chart, which can be used to generate all those graphs for all those bodies in KSP, and in RSS, and for any new planets that are made etc...

Besides, the definition of TVR teaches us something about the physics: that a craft with ISP=400 on a planet with V_orb = 1000 m/s will have the same efficiency as craft with ISP=800 on a planet with V_orb = 500 m/s.

There is, on the other hand, the plain, simple nature of the definition you use for DVR. Delta-V Ratio, as in "the ratio of delta-V you have versus the delta-V you need". Given that definition, it's quite easy to see why its effect on the efficiency is so binary. Either you make it or you don't.

That's not the definition of DVR. DVR is the ratio of deltaV your craft carries if it were floating in space in the absence of any gravitational field to the amount of deltaV required to get into orbit using an impulse burn i.e. minimum possible. It could easily have been the case that carrying more deltaV would effect efficiency and require higher TWR. Regardless, it is good to know and show it doesnt.

Er, no. They'll need less fuel to get into orbit. More time than ideal perhaps, yes, and thus more dV lost to gravity, but less fuel - that's what I_sp does. I see how it might be useful in extremely dV-tight situations though, I guess. The concept of dV needed increasing because TWR does not increase as much with less initial fuel mass fraction is rather often overlooked.

We are talking past each other. I agree, they will need less mass of fuel-- that is what increasing ISP does. However, knowing this, players are likely to decrease the mass of their fuel accordingly (!) such that they will have the same total deltaV, not realizing that they need more deltaV to get into orbit due to reduced efficiency (assuming they also adjusted their engines to maintain the same TWR)....

Not starting TWR. Final TWR. A craft with total mass of 3 tons, a dry mass of 1 ton, and a starting TWR of 2, will have a final TWR of 6 just as its fuel runs out - since its mass decreases threefold. Let's say .2 tons of that is tankage, .8 tons is the engine. The engine thus has a theoretical maximum TWR of 7.5, in an ideal craft without tankage. An LV-N, however, has high I_sp, but awful TWR - around 3.5, IIRC. Whatever the starting TWR and FMR, it can never go beyond that 3.5 TWR because that's how much the engines themselves weigh, even disregarding tankage.

That is true but not relevant at all. My model requires inputs of TWR, TVR, and DVR. The players can get these numbers from their craft and use it to assess whether their design has enough deltaV.

It might not be possible to get any arbitrary combination of TWR, TVR, and DVR from the parts available in KSP, as you pointed out. The player either (a) supplies these numbers using current design to assess current design, ( uses the results to generally guide TWR selection (i.e. keep it 1.4 < TWR < 1.6) or © uses the results to try to improve design, but has to cleverly tradeoff TWR, TVR, and DVR to improve efficiency since adding removing one part might improve TWR, but reduce TVR and/or DVR...

If your formula assumes an ideal craft with no tankage and massless engines, with TWR approaching infinity as fuel runs out, it may very well skew your results.

My formula assumes nothing about the craft, except its TWR, TVR, and DVR. If you can get those values, then the model will predict results accurately. For the case you described above, where the only mass of the ship is fuel, then DVR = infinity since m_wet/m_dry = infinity since m_dry = 0. Thus, in order for my model to "make" those assumptions, I will have to input DVR = infinity. Otherwise, it assumes realistic values for DVR.

Edited December 26, 2014 by arkie87

[Torquemadus]

I dont think coasting to apoapsis is fuel efficient, particularly for low TWR. You have already spent fuel to get apoapsis up, and bought yourself time to burn horizontally, I would think you should do that...

So, I'm trying to get into orbit from an airless body using minimum fuel. At take off, I burn as horizontal as I possibly can, taking care not to collide with any nearby mountains or other intervening terrain. This puts me on a sub-orbital trajectory with an apoapsis high enough to avoid crashing, but no higher.

As I approach apoapsis, I begin a burn to circularise into orbit. I try to time the burn so that it occurs as close to apoapsis as I can within the limits of the thrust my spacecraft can produce. This puts me in a minimum altitude orbit that only just barely clears the terrain. In practice, flying just barely above the terrain is pretty scary! I usually give myself a small margin of error for the sake of safety.

[arkie87]

So, I'm trying to get into orbit from an airless body using minimum fuel. At take off, I burn as horizontal as I possibly can, taking care not to collide with any nearby mountains or other intervening terrain. This puts me on a sub-orbital trajectory with an apoapsis high enough to avoid crashing, but no higher.

As I approach apoapsis, I begin a burn to circularise into orbit. I try to time the burn so that it occurs as close to apoapsis as I can within the limits of the thrust my spacecraft can produce. This puts me in a minimum altitude orbit that only just barely clears the terrain. In practice, flying just barely above the terrain is pretty scary! I usually give myself a small margin of error for the sake of safety.

Yes, but if you have low TWR (which in general, should be cheaper in terms of funds, since it requires fewer engines), you cannot wait until you reach apoapsis to burn horizontal, since you cannot accelerate fast enough. Thus, you probably have to start burning horizontally before, and once you reach apoapsis, then pitch up to the angle necessary to keep "time to apoapsis" as close to zero seconds as possible

Also, yeah it is scary. Sometimes you pass over a hill by only a few meters Gets the blood pumping...

[numerobis]

Nice work.

I'm curious about a further question: say you're on the lip of a crater, infinitely deep, and you won't reach orbital velocity by the time you get to the other side. Should you follow this grazing approach, or should you fall into the crater a bit (thrusting due sideways) and recover later on?