Thought Experiment: What would a graph diminishing returns of lifting stages look like?

[neamerjell]

If you could graph the diminishing returns of lifting stages, what would it look like?

I have had this idea for a graph in which I could see if there is a "sweet spot" for lifter design given payload weight, desired orbital altitude, what else would I need to consider?

Basically, at what point are you spending most of your fuel hauling your boosters into space? And don't say "at launch"! At what point do you hit the law of diminishing returns?

I've been watching DasValdez's Twitch streams and his YouTube videos and noticed that he tries to keep his TWR around 1.7 for his lifters, because atmospheric drag is an issue early on and airspeed must be limited to keep from wasting fuel fighting against it. He apparently arrived at this number via trial and error and possibly some research of his own. Is this ideal, or is it just the best that he has found so far? He also mentioned wanting to make a table or graph of altitude vs ideal airspeed. Yet another set of variables to add to the mix!

What data would I need to put into a spreadsheet in order to get a graph of fuel amount, engine ISP, TWR? Where would you even begin to create this graph for asparagus staging?

UPDATE: I seem to be either biting off more than I can chew or being too vague - probably both. Therefore, I propose this question:

I have a payload of x tons and I want a delta-V of y.

How much fuel and what kind of engine do I need in order to get that and maintain a TWR ≈ 1.7 and a Total mass to Dry mass ratio ≈ 2.5?

Edited January 20, 2015 by neamerjell
Rethinking my problem

[Dazpoet]

To me it seems TWR is king. I usually try to keep a "high" (1.2<TWR<2) TWR when below 10k metres and then it seems ~1.2 is enough up to about 35k metres and then whatever I have left is used before I switch to either a poodle or nukes to circularize.

This thread seems awesome though since I'm guessing my methods are crap and can be optimized a lot. But as I said, it seems TWR is king.

[Slam_Jones]

I read somewhere on the forum that the optimal TWR is 1.65 at launch, and 1.45 in upper atmo. Goin by that, I get into orbit with under 5k dV pretty consistently.

[neamerjell]

According to the orbital map I found a long time ago, an 80km orbit around Kerbin takes 4550m/s delta-V. TWR isn't the only factor, though.

optimal TWR is 1.65 at launch, and 1.45 in upper atmo.

- Slam_Jones

That must be where DasValdez got his numbers.

[problemecium]

^ I won't say you're outright wrong, but I have my doubts based on my experimental results using rockets with a TWR around 1.2.

[neamerjell]

^ I won't say you're outright wrong, but I have my doubts based on my experimental results using rockets with a TWR around 1.2.

Well as DasValdez explained it, you have to escape gravity fast enough so you don't waste fuel fighting gravity, but not so fast that you end up wasting fuel fighting atmospheric drag. So there is definitely a sweet spot as far as air speed is concerned.

[Dazpoet]

1.65 eh, sounds interesting I guess it's time for some testing

[neamerjell]

New information I learned from this video:

According to the rocket equation, delta v = engine ISP * 9.81 x ln(total mass / dry mass), the mass ratio (total mass / dry mass) is the part that creates the diminishing returns. The video's author, Tyler Raiz, says the mass ratio should be between 2 and 3. (EDIT: This may only be valid for lifting off from Kerbin...) If it is less than 1.5, more fuel is needed, if it is greater than 4 an additional stage is needed.

This may be the thing I was looking for! This is the missing piece!

I would need a separate graph for each engine. The Y-axis would be the fuel needed, the X-axis would be the payload size, a Z-axis could be added for TWR.

Edited January 20, 2015 by neamerjell
Spelling error and an epiphany

[Pecan]

What mass are you launching?

What ascent-profile are you using?

What staging strategy are you using?

What are the Isps of the engines you are using during each stage of your ascent?

How much dV does each stage have, and is that optimised for the the Isp of each stage?

Are you optimising for mass, cost, part-count, reliability, ... ?

There IS no single answer to the Goddard Problem.

That said; the beginner advice is to try to follow the terminal-velocity curve. I've been moving more towards simpler builds with lower launch TWR and higher thrust at burn-out.

[neamerjell]

What mass are you launching?

variable (imagine an app with a slider for this)

What ascent-profile are you using?

MechJeb standard

What staging strategy are you using?

that's one thing I'm trying to find out with this graph idea

What are the Isps of the engines you are using during each stage of your ascent?

that's another thing I'm trying to find out

How much dV does each stage have, and is that optimised for the the Isp of each stage?

variable (imagine an app with a slider for this)

Are you optimising for mass, cost, part-count, reliability, ... ?

None of the above. Not really optimizing for anything other than fuel efficiency. I just want a lifter or stage that is not overkill and trys to avoid the "MOAR BOOSTERS!" trap. I don't want a lifter spending (wasting) all its delta-V hauling itself into space...

Edited January 20, 2015 by neamerjell
More specific answers based on rethinking my problem

[Streetwind]

This is basically related to the question of your fuel mass fraction. There was already a graph on this, once. I vaguely remember it but I have no idea how to search for it because I don't recall what the topic was called.

The reason you get diminishing returns is then fuel/mass fraction of the fuel tanks. The rocket equation is basically a convoluted way of saying "multiply your fuel mass fraction with some constants", therefore this figure is critically important. Stock fuel tanks carry 8 tons of propellant for every 1 ton of dry mass. Therefore, even with a massless engine and zero payload, the maximum dV per stage can never exceed:

dV = 9.82 * 370 * ln(9/1) = 7983.396 m/s (Using 370 Isp because that is a number found on many launch engines.)

Or, in other words:

7983,396 / 370 = 21.5767 times your Isp

Simplified, even with an infinite number of fuel tanks containing the entire mass of the observable universe, you are not going to realistically get more than 21.5 times your Isp out of a single stage. Period.

And the closer you get to this ideal mass fraction, the harder it is to further improve it by adding more fuel tanks, and therefore: diminishing returns. Getting even 20 times Isp is insane - just try it ingame! Most rocket stages end up around or slightly below 10 times Isp.

Of course, at the lower end, there are also diminishing returns: having hundreds of stages is inefficient because the added weight of decouplers and engines (engines in KSP are insanely heavy for what they do!) outstrips the efficiency gains from running with very low dV per stage. Thus, there is an efficiency curve for a given tank fuel/mass fraction, a given engine Isp and a given dry mass penalty per stage, which starts low and ends low, but has a peak at a certain dV value. This is the optimal dV value that each stage should have to construct the optimal launch vehicle with these parameters.

Of course it gets more complicated than that, because in practical application, the dry mass penalty is going to be different for each stage, and you may be unable to hit the dV number exactly because you can't hit your TWR targets otherwise, and your rocket may end up with different dV numbers for each stage due to imprecision (and oversized upper stages tend to 'choke' stages below them), and so on and so forth. Welcome to rocket science!

Edited January 15, 2015 by Streetwind
Fixed major grammar derp

[OhioBob]

I read somewhere on the forum that the optimal TWR is 1.65 at launch, and 1.45 in upper atmo. Goin by that, I get into orbit with under 5k dV pretty consistently.

I agree with the 1.65, but I use about 1.3 for the upper atmosphere. I arrived at these numbers using a computer simulation. I kept changing and tweaking the numbers until I couldn't improve the rocket performance any further. I figured at that point I had pretty much hit on the optimum design. When I tried out my theoretical design in the game, it performed exactly like the simulations. Here are my basic rules of thumb:

Stage 1 TWR = 1.65

Stage 2 TWR = 1.30

Ratio of Stage 2 thrust to Stage 1 thrust = 0.35

Of course it's not always possible to hit those numbers exactly, but I just try to get as close as possible. I've been happy with the results. Using a 2-stage rocket I can routinely get a payload fraction of about 0.16, and sometimes better.

Just as important as the rocket design is the ascent trajectory. I usually start my turn at an altitude of 5000 m. I then turn gradually while keeping the nose of my rocket within a few degrees of the prograde marker on the NAVball. I accelerate up to about 2300 m/s at an altitude of about 50 km, with my rocket horizontal at that point. I cut the engine once my apoapsis reaches my intended altitude (it's actually necessary to overshoot the target altitude a bit because drag will lower the apoapsis by the time I get there). I typically require a 50-100 m/s burn at apoapsis to circularize the orbit. Using this technique I routinely get to orbit using no more than 4550 m/s delta-v.

Note that this is based on using stock aerodynamics. It's my understanding that when using NEAR/FAR, lower TWRs are ideal.

Edited January 15, 2015 by OhioBob

[Streetwind]

The TWR requirement of the upper stage depends on the dV of the lower stage. The higher up you ignite stage 2, the less TWR it needs. If you ignite it fairly early, it's going to need more. You can observe this happening in three-stage designs.

Therefore, 1.3 TWR for the second stage is not an absolute truth, but rather a vehicle-dependant, rough rule of the thumb.

[OhioBob]

The TWR requirement of the upper stage depends on the dV of the lower stage. The higher up you ignite stage 2, the less TWR it needs. If you ignite it fairly early, it's going to need more. You can observe this happening in three-stage designs.

Agreed.

Therefore, 1.3 TWR for the second stage is not an absolute truth, but rather a vehicle-dependant, rough rule of the thumb.

True. But if I follow all three* of my rules of thumb, my first stage will burn out at such time that 1.3 is the ideal TWR for my second stage. The rules only work collectively, not in isolation.

* I should say four rules of thumb because the ascent trajectory is an integral part of it. The rocket won't perform optimally if you don't fly it right.

Edited January 15, 2015 by OhioBob

[Vanamonde]

Discussions about the details of KSP flight belong in the Gameplay section. And so, moved.

[neamerjell]

Okay, thread moved, thread disappeared from view...

I've worked on some formulas lately and here is what I have come up with:

Given:

2 < R < 3 (ideally, R = 2.5, but nice, round numbers rarely happen)

TWR ≈ 1.7

If dV = Isp x 9.81 x ln( R )

Then Isp = dV / ( 9.81 x ln( R ) )

If R = Total Mass / Dry Mass, and ideally R = 2.5

Then Total Mass ≈ Payload Mass x 2.5, actually a bit more since this does not factor in the masses of fuel tanks and engines**

If TWR = Thrust / ( Total Mass x 9.81 ), ideally TWR ≈ 1.7

Then Thrust ≈ 1.7 x Total Mass x 9.81

Inputs:

Delta-V

Payload Mass

**Anyone want to guess what percentage I should multiply by to give an estimate of Total Mass including mass of engines and tanks?

Also I would invite you to check my math, my algebra is a little rusty...

Spreadsheet results using the above formulas:

Constants:

TWR = 1.7

R = 2.5

ln( R ) = 0.91629073

Inputs:

Payload Mass = 15

Delta-V = 700

Outputs:

Total Mass = 37.5

Fuel Mass = 22.5

Thrust = 625.3875

Isp = 77.8745838

These numbers look about right to you?

Edited January 20, 2015 by neamerjell
Found errors

[Zuqq]

I've really just delved into the mathematics behind KSP, but have a small input:

Are you using payload mass as the total dry mass of the rocket? Or is the payload mass seperate? (say, a sattelite being launched from a rocket once the rocket reaches the 80km orbit?)

It seems to me using a catch-all percentage to calculate total mass via payload mass would yield quite innacurate results.

[neamerjell]

I've really just delved into the mathematics behind KSP, but have a small input:

Are you using payload mass as the total dry mass of the rocket? Or is the payload mass seperate? (say, a sattelite being launched from a rocket once the rocket reaches the 80km orbit?)

It seems to me using a catch-all percentage to calculate total mass via payload mass would yield quite innacurate results.

Yes I know. That's why I asked "**Anyone want to guess what percentage I should multiply by to give an estimate of Total Mass including mass of engines and tanks?"

I'm trying to answer the question: I have a payload of x tons and I want a delta-V of y. How much fuel and what kind of engine do I need in order to get that and have a TWR ≈ 1.7 and a Total mass to Dry mass ratio ≈ 2.5?

[Zuqq]

Yes I know. That's why I asked "**Anyone want to guess what percentage I should multiply by to give an estimate of Total Mass including mass of engines and tanks?"

I'm trying to answer the question: I have a payload of x tons and I want a delta-V of y. How much fuel and what kind of engine do I need in order to get that and have a TWR ≈ 1.7 and a Total mass to Dry mass ratio ≈ 2.5?

My apologies, just trying to wrap my head around the question to try and provide some assistance... Will give it some thought

[neamerjell]

@ Zuqq: I have my head wrapped about 75% around the math involved, like I said, my algebra is a little rusty! Earlier in the thread people went of on tangents not really related to what I asked, so I decided to be more specific.

[OhioBob]

**Anyone want to guess what percentage I should multiply by to give an estimate of Total Mass including mass of engines and tanks?

Most of my rocket designs are about 80% propellant and about 20% dry mass when the payload is excluded. With the payload its more like 67% propellant and 33% dry mass.

The 80/20 breakdown is typical of my larger launchers. On smaller launch vehicles it tends to be more like 75/25.

[neamerjell]

Okay, "real world" example:

I made a skycrane for a Modular Kolonization System command unit with the solid base (not the landing frame) that has the landing legs on it. I designed it to go from orbit around Mun to landing on Mun. The results are strictly from trial and error. I tried to back calculate the numbers to see if my formulas worked, most did, Isp was way off (not sure how Kerbal Engineer calculates this for multiple engines). I also suspect Kerbal Engineer's delta-V is calculated using Kebin's gravity regardless of what planet you set it to.

Most of these numbers came from Kerbal Engineer set to Mun.

Kolony Skycrane Prototype

Max Thrust: 80kn

Isp: 300s

Total Mass: 20.844t

Dry Mass: 16.044t

LFO Mass: 4.8t

TWR: 2.36

Burn Time: 2m 56.8s

Delta-V: 771

@OhioBob: Your estimate may be accurate for a launch from Kerbin to orbit. My ultimate goal is to create something that would be useful in any scenario.

[Pecan]

Okay, "real world" example:

...I also suspect Kerbal Engineer's delta-V is calculated using Kebin's gravity regardless of what planet you set it to.

What does this example show?

deltaV does not depend on gravity (except as a constant to convert the units used for Isp).

[OhioBob]

@OhioBob: Your estimate may be accurate for a launch from Kerbin to orbit. My ultimate goal is to create something that would be useful in any scenario.

The fuel percentages are based on the parts that we have to work with in game. For example, most of the fuel tanks have an 8/1 ratio of propellant to dry mass, meaning they are 89% propellant. That's an absolute upper limit. By the time we add engines and decouplers, this lowers to about 80%. I think 80% is probably the practical upper limit for a single stage, after which it's better to add another stage rather than stacking on more fuel tanks. Of course, as you say, there are some applications where the ÃŽâ€V requirement is low enough that it's not necessary to have a very high mass ratio. To be honest, I'm not really following what it is you are trying to do. I just gave you some numbers based on my experience for you to do with as you please.

[FleshJeb]

Isp was way off (not sure how Kerbal Engineer calculates this for multiple engines).

KER's calcs flame out on me frequently. I always compare to MechJeb. If they don't agree, I throw it in a spreadsheet.

Combined ISP: http://wiki.kerbalspaceprogram.com/wiki/Cheat_sheet#Combined_specific_impulse_.28Isp.29