Theoretical optimum of cost efficiency to orbit (1.2)

[rcp27]

2 minutes ago, Abastro said:

So how much does the fuel cost?

Fuel for the plane (not counting the orbiter's tank) is 270.  While getting confused about some odd cost numbers, I have now noticed that it appears the fairing itself, not just the "base" has a cost associated with it.  For example a Hammer costs 400, a fairing base 300 and a tiny decoupler 300, so you would expect the cost of the rocket in my earlier post to go down by 1000 if I remove those (those being the "lifer costs".  In fact it goes down by 1038 because that's the cost of the fairing itself.  So call the cost-of-launch (assuming I can figure out how to recover the plane) about 310 (its fairing is a bit longer).

Right, now I have to go and clear out all of these "2Cheap" "SuperCheap" and "BargainBasement" satellites from my LKO before they go all Kessler on me.

 

[Reusables]

On 2017. 3. 4. at 11:16 PM, rcp27 said:

Fuel for the plane (not counting the orbiter's tank) is 270. 

So nothing can beat reusable spaceplane in any region. 

By the way, I added analysis for SRB on the spreadsheet, it clearly shows that for big enough payloads Kickback is the way to go for anow atmispheric stage. It's not for thrust augmentation for liftoff, though.

Also switching into cost per thrust with engines from cost per ton.

[Reusables]

I found that combination of 2~3 LFOx engines could work better than sticking to one kind of engine, in theory. For instance, There is poodle&Spark for circularization stage which gets reasonable increase in TWR while the ISP and the cost don't get much effect. I'm quite sure that this is applicable to the real KSP.

I'm going to update the spreadsheet with these engine combinations.

By the way, just found that O-puff is cheap engine with great TWR. (Only cost 73.575 per 1t weight of thrust, with TWR  22.65)
 It might be worth it for low dv operations.

Edited March 9, 2017 by Reusables
Combination of 3 can be better

[MR L A]

On 03/03/2017 at 2:34 AM, Abastro said:

So it is disposable spaceplane? I thought you are launching it vertically. In the case, single divertless intake will work decently. (As speed gained with airbreathing stage will be near 1km/s)

Sort of... I really need to upload pictures, but the spaceplane itself isn't disposable, it just jettisons the Whiplash + liquid tank + intake assembly once the air is too thin (an aerospike kicks in before this). The idea came from the point in my KSP engineering skills where I was to crappy to build a proper SSTO... I've kept the design around purely because it looks cool and is fun to fly

[Reusables]

3 hours ago, MR L A said:

Sort of... I really need to upload pictures, but the spaceplane itself isn't disposable, it just jettisons the Whiplash + liquid tank + intake assembly once the air is too thin (an aerospike kicks in before this). The idea came from the point in my KSP engineering skills where I was to crappy to build a proper SSTO... I've kept the design around purely because it looks cool and is fun to fly

So it's spaceplane with drop tanks&engines. I think you can add some fuels and streamline the plane so that it would be fully reusable SSTO. Won't take much more time than disposable planes.

[Leafbaron]

I feel like the only reason to have drop tanks on a spaceplane is if your closed cycle engine can't produce the TWR to push you into orbit when the air breathers kick off.

[MR L A]

On 09/03/2017 at 1:30 AM, Abastro said:

So it's spaceplane with drop tanks&engines. I think you can add some fuels and streamline the plane so that it would be fully reusable SSTO. Won't take much more time than disposable planes.

 

17 hours ago, Leafbaron said:

I feel like the only reason to have drop tanks on a spaceplane is if your closed cycle engine can't produce the TWR to push you into orbit when the air breathers kick off.

Yeah, I realise all this. The point of the craft is nothing beyond fun and a sort of nostalgia from my early design days. If I wanted it to be a full SSTO... well I'd just use one of my SSTO designs haha

[Duxwing]

We can and should define this problem more abstractly.  The cost per ton of a launch system equals the amount of money spent, per launch, to lift one ton of payload along a desired launch profile to a desired orbit.  These latter two constraints are important because they prevent payloads from being launched by very cheap but very weak lifters, whose cost would approach that of fuel with increasing payload regardless of whether the system were recovered or air-breathing or not.  

Therefore, we must state the theoretical lower bound of efficiency in two lines rather than one.  The first line must describe the aforementioned constraints, while the second line defines the efficiency as the payload mass divided by a cost function ultimately derived from the rocket equation.  Working from the handy variables given by Abastro, we may write that this cost function must total the costs of the fuel, tankage, engines, and utilities.

Cost (System) = Cost (Fuel, Tankage, Engines, Utilities) = F + T + E + U

If, like Abastro, we assume that the cost imposed by each variable is a function of its associated mass, then we may derive these masses and thereby their associated costs from the rocket equation of our launch system.  We will write the masses of our components as the lower-case versions of their costs.  Therefore, where p is the mass of the payload, we may compactly write that,

dV = I_sp ln(m_w/m_d) = I_sp ln[(f+p+t+e+u)/(p+t+e+u)]

exp(dV / I_sp) = (f+m_d)/(m_d) 

f/m_d = exp(dV / I_sp) - 1

The masses of the payload, engine, and utilities being given alongside the necessary dV and I_sp by the first line of the statement of efficiency, we may consider e and u fixed and declare the lefthand side the mission constraint function k.

f/(p+t+e+u) = k 

f = (p+t+e+u)k

f - tk = (p+e+u)k

Since we may assume that the mass of the tank equals the mass of the fuel multiplied by a propellant-mass-fraction proportion w, we may write,

f - wfk = (p+e+u)k

f(1-wk) = (p+e+u)k

f = (p+e+u)k/(1-wk)

Note that this equation shows, just as we would expect, that the mass of the fuel depends on the masses of the engines and utilities alongside the propellant-mass-fraction and the constraint function.  This check lets us confidently declare that

Cost (System) = F(f) + T(t) + E(e) + U(u) = F((p+e+u)k/(1-wk)) + T(wk) + E(e) + U(u)

Of course, this definition of the cost equation requires the system to have only one stage, but summing it over each stage of a multi-stage system reveals the system's total cost.  In addition, one may handle different types of fuels, tanks, and engines by considering each function and variable above a function with the numbered function and mass F₁, f₁, F₂, f₂ ... of each fuel or tank or engine type as its arguments.  Lastly, to determine the efficiency, divide the payload by the cost.

Notes for recoverable launch systems:

• Subtract from the cost of the tank, engine, and utilities the recovery function, which depends on internal KSP math and the mass of the recovered components.
• To include the startup cost of building the system, multiply the recovered mass amount by 1 - 1/launches, where launches is the expected lifetime of the system.  Ignore this startup cost if you click the "Recover" button.

Having determined the efficiency of our system and organized the mission constraints and engine and utility masses in the first line of our efficiency statement, we see that the efficiency of our launch system is already determined by the first line of our efficiency statement, which defines the mission constraints and thereby the dV, Isp, and masses of engine and fuel, which must be adjusted to satisfy them.  For example, reaching Kerbin's orbit from its surface requires a powerful and therefore heavy first stage engine, decreasing efficiency.

In conclusion, we discover that launch systems have only one efficiency, and that decreasing our cost per ton will therefore require different mission constraints rather than better mathematics.  Launch profiles must be relaxed to accommodate more-efficient ascents.  Payloads must be sent to destinations requiring smaller changes in velocity.  And here we recognize that efficiency is merely one aspect of our space program, which must not only save money to do more things but also spend money to do anything at all.  In essence: no bucks, no Buck Rogers.

-Duxwing

Edited March 13, 2017 by Duxwing
Added How to Maximize Efficiency

[Reusables]

1 hour ago, Duxwing said:

great article

You got great points there! Cost per launch of specific mission, and Reliability of the launch are important indeed.

Here I neglected several factors in the calculation, and ignored important elements in gameplay. It was because the calculation gets really hard when I was aiming to do the whole calculation. It needs several simplifications to get the calculation to be actually doable. I think the same counts for calculations on real life, like scientific engineering.

Here I'd justify mine with the reason I took these simplications:

1. It's hard to calculate with every parts involved, as working with those several variables is nearly impossible. (As those has effect on each other; selection of a part often prohibits/allows/mandates another part)

2. For disposable launch, launch profiles doesn't matter much as long as you are trying to perform efficient gravity turn (there is enough margin). I discovered this while trying with rockets which lacks dv&TWR just a bit. It consistently fails to get to orbit with any kind of profile.

3. Dealing with reliability is one of the hardest thing to analyze. It is heavily dependent on the player, as good pilot can deal with some unstable rockets. Also occasional mistake/error can ruin a plan, which is even harder to take account in. (Welp; Kraken is involved in here)

Also, this is all about 'theoretical optimum' (I mean good enough bound here), so it is to determine how good one's rocket is & how good a specific part is.

 

I'll list some problems of my analysis to solve/improve.

1. Adapters(Decouplers). This one is important element which consists considerable amount of launch cost. In the calculation, Reliant is definitely better than Thud(yes that thud), but it needs mk1 adapter which will cost a lot for a Reliant or get so much drag.

2. Mission profiles like asparagus staging and SRB-augmented liquid fuel launches. This one is significant weakpoint of my calculation, as (one of?) those are where practical optimum is located.

3. Combination of several rocket engines can work better than specific engine.

 

Besides, I couldn't get this line:

1 hour ago, Duxwing said:

decreasing our cost per ton will therefore require different mission constraints rather than better mathematics.

What do you mean by this? AFAIK, mathematics only give constraints of the lower bound of launch cost. Improving practical cost per ton is about mission profiles and design decisions.

(I couldn't find the actual lower bound for any launch profiles, though. There's so much complex relations involved)

[Duxwing]

25 minutes ago, Abastro said:

You got great points there!

Thanks!

26 minutes ago, Abastro said:

For disposable launch, launch profiles doesn't matter much as long as you are trying to perform efficient gravity turn (there is enough margin). I discovered this while trying with rockets which lacks dv&TWR just a bit. It consistently fails to get to orbit with any kind of profile.

Launch profile matters for all systems going beyond LKO, but I see your point.

27 minutes ago, Abastro said:

What do you mean by this? AFAIK, mathematics only give constraints of the lower bound of launch cost. Improving practical cost per ton is about mission profiles and design decisions.

I meant, and my thesis turned out to be, that improving design can only improve cost per ton to the limit imposed by the payload and mission profile.  This limit can be called the theoretical upper bound on the efficiency of any rocket system, whether for lifting or transit.

Now that I think about it, we could state generally that the maximum efficiency of any engineering project with well-defined theoretical principles is given by applying them to the constraints of the project.  In Kerbal Space Program, for example, the maximum cost-efficiency of using an LV-N to accelerate a payload would be given by applying the Rocket Equation to the relevant propellant fractions of the tanks, properties of the LV-N, etc.

[Reusables]

On 2017. 3. 13. at 4:15 PM, Duxwing said:

Launch profile matters for all systems going beyond LKO, but I see your point.

Yeah, I missed that in the simplifications.

On 2017. 3. 13. at 4:15 PM, Duxwing said:

I meant, and my thesis turned out to be, that improving design can only improve cost per ton to the limit imposed by the payload and mission profile.  This limit can be called the theoretical upper bound on the efficiency of any rocket system, whether for lifting or transit.

Now that I think about it, we could state generally that the maximum efficiency of any engineering project with well-defined theoretical principles is given by applying them to the constraints of the project.  In Kerbal Space Program, for example, the maximum cost-efficiency of using an LV-N to accelerate a payload would be given by applying the Rocket Equation to the relevant propellant fractions of the tanks, properties of the LV-N, etc.

Right. I'm trying to find the the maximum efficiency.

 

By the way, I'm still working on the combination of the engine. Showed that combination of 3 or more engine types can't be better than a single engine or two engines. It was hard math with bunch of equations wandering around, so I can't post it here.

[Archgeek]

I did a few tests with SRBs and hyper-dense payloads (small, full or mostly full ore cans) in 1.25m fairings.

'Found that the lame ol' Thumper is actually very convenient about its mass/drag profile during an ascent from an initial tilt on a launch clamp, and can just about make it out of the atmosphere in a normal gravity turn.

The Kickback is of course more capable, but a lot less stable, needing a few fins in back, which required greatly reducing the pad tilt to halt lawndarting.  It was also a lot angrier, blowing up said fins almost invariably when trying to pick up horizonatal velocity.  Best result was 6.6 tonnes to a very low orbit for 1291/tonne using a Hammer for a Payload Assist Module, 'found that said 7.2t payload was about as much as one can get away with on a single kickback, as the turn gets too aggressive, nearly blowing up the fairing.  A similar design using a Spark instead of a PAM did more poorly, failing to make orbit due to losing too much speed to the atmosphere after MECO, or not get enough horizontal kick out of the booster and running out of gas in the 990m/s upper stage and just not getting the periapsis up.

[Reusables]

29 minutes ago, Archgeek said:

I did a few tests with SRBs and hyper-dense payloads (small, full or mostly full ore cans) in 1.25m fairings.

'Found that the lame ol' Thumper is actually very convenient about its mass/drag profile during an ascent from an initial tilt on a launch clamp, and can just about make it out of the atmosphere in a normal gravity turn.

The Kickback is of course more capable, but a lot less stable, needing a few fins in back, which required greatly reducing the pad tilt to halt lawndarting.  It was also a lot angrier, blowing up said fins almost invariably when trying to pick up horizonatal velocity.  Best result was 6.6 tonnes to a very low orbit for 1291/tonne using a Hammer for a Payload Assist Module, 'found that said 7.2t payload was about as much as one can get away with on a single kickback, as the turn gets too aggressive, nearly blowing up the fairing.  A similar design using a Spark instead of a PAM did more poorly, failing to make orbit due to losing too much speed to the atmosphere after MECO, or not get enough horizontal kick out of the booster and running out of gas in the 990m/s upper stage and just not getting the periapsis up.

Experiment! Always better than theory.

Though, I think this one is inconsistent with mine and this (some more here). It's for 1.1.3 and drag profile changed a bit on 1.2, but it shouldn't be changed these things a lot. Kickback provides good amount of dv, which is really helpful for first stage and augmentation.

 

Besides, what do you mean by Payload Assist Module? Does that mean side boosters augmenting the thrust of the center?

[Reusables]

My claim on combination of engines was wrong. Individual engine is always better than in (lifter cost) / (payload mass).

The proof involves in gradient and local minimum. (And a bit of eyeballing). It's just wall of text, so I'd rather not post it here. If you want it, ask me here or PM me.

[Archgeek]

4 hours ago, Abastro said:

Experiment! Always better than theory.

Though, I think this one is inconsistent with mine and this (some more here). It's for 1.1.3 and drag profile changed a bit on 1.2, but it shouldn't be changed these things a lot. Kickback provides good amount of dv, which is really helpful for first stage and augmentation.

Besides, what do you mean by Payload Assist Module? Does that mean side boosters augmenting the thrust of the center?

Nah, theory might be better in this case.  My experiments (that at all succeeded) have been fairly limited, one or two full small ore tanks or less payload, that 1291 number was a single kickback under a hammer under a fairing with mini decoupler, two small ore tanks, a hex and a z-100 on top.  'Biggest problem was trying to fly the crazy things -- such a rocket no longer turns for love or money once the booster's anywhere near burnout -- most likely my ascent profile is insanely sub-optimal.

A PAM's an SRB upper stage, intended for vacuum use, usually boosting a payload up to an intended orbit or out of the SOI, like the STAR-48 motor used for New Horizons.  For some reason they keep packing a bit under 1km/s delta-v with arbitrary dense payloads I've selected, and are of course very cheap (seperatron PAM's arguably the cheapest way to circularize small probes if the resulting orbital parameters don't matter much).

[Reusables]

1 hour ago, Archgeek said:

Nah, theory might be better in this case.  My experiments (that at all succeeded) have been fairly limited, one or two full small ore tanks or less payload, that 1291 number was a single kickback under a hammer under a fairing with mini decoupler, two small ore tanks, a hex and a z-100 on top.  'Biggest problem was trying to fly the crazy things -- such a rocket no longer turns for love or money once the booster's anywhere near burnout -- most likely my ascent profile is insanely sub-optimal.

A PAM's an SRB upper stage, intended for vacuum use, usually boosting a payload up to an intended orbit or out of the SOI, like the STAR-48 motor used for New Horizons.  For some reason they keep packing a bit under 1km/s delta-v with arbitrary dense payloads I've selected, and are of course very cheap (seperatron PAM's arguably the cheapest way to circularize small probes if the resulting orbital parameters don't matter much).

So SRB only rocket? That's great concept.

Although, AFAIK SRB is useless as second stage. Terrier/Reliant(yes reliant)/Poodle is good enough, for the low TWR requirement.

Also it seems that you're going for a bit high TWR. Try adding more mass(payload) over the Kickback / Thumper. I had no problem tilting them on TWR 1.6~1.7

[rcp27]

A major challenge for designing low cost staged launchers is the cost of staging itself.  You might think that a Reliant and FL-T800 fuel tank makes an efficient lower stage for a small launcher, and in mass/fuel efficiency terms, it is.  If you add in the cost of decoupling it, though, the balance goes in a funny direction.  A Reliant and a dry FL-T800 tank combined weigh 1.75 T.  A TR-18A stack decoupler, while only weighing 0.05 T, seems a convenient way to stage it off to allow a lighter, more efficient upper stage.  That coupler, though, costs 400.  If you look at the kind of cost-to-orbit that is achievable in the game, it's likely to be cheaper to not stage those off and pay the extra fuel to bring the whole thing to orbit.  That's one of the reasons why the ultra-cheap launchers I described earlier in the thread were specifically built around the idea of keeping the decoupling to an absolute minimum.  I did have a go to see if it was possible to make one fly where I ditched the decoupler entirely, put the ant on the top, pushing backwards, but I didn't achieve orbit, though it was quite funny pushing an empty Hammer backwards.

[Reusables]

@rcp27,

Decoupler is required for every staged rocket, and SSTO is almost always less efficient than staged rockets. The dry mass is nearly twice for those, so you'll need much more fuel. Also fuel is expensive in ksp, so single FL-T400 already costs more than the 1.25m decoupler. I think the only reasonable SSTO is one from Twin-Boar.

Though, the cost (including adapters) matters when comparing different sized rockets. It's definitely a weak point of my analysis to improve.

(Minor note: I recall that Reliant needs more than FL-T800 tank as first stage)