Rocket Equation Hypothesis

[Wcmille]

I have a hypothesis about the rocket equation, hoping someone can help me either disprove with counter example, or give an informal proof that it is always correct:

For every two stage rocket with total mass M, and a payload weight of P, if the bottom stage engine (called E1) with ISP of  I1 and the top stage (called E2) has an ISP of I2, and also I1 <  I2, then:

There is never a two-stage rocket design of payload P and total mass M where E1 is placed on the top and E2 is placed on the bottom that has more dV than the best two-stage design which has E1 on the bottom, and E2 on the top.

in other words, if I have a 12 ton rocket with a 1 ton payload, a swivel on the bottom stage and a terrier on the top stage, and the fuel is split optimally. There's no design for a 12 ton rocket with the terrier on the bottom and the swivel on top that will be better in terms of dV.

[Kerbart]

Even if there would be, assuming you picked the best engine for the job (with thrust being a large factor that is irrelevant to the rocket equation), why would you pick that design?

[Snark]

3 minutes ago, Kerbart said:

Even if there would be, assuming you picked the best engine for the job (with thrust being a large factor that is irrelevant to the rocket equation), why would you pick that design?

The fact that the Terrier has a lower TWR than the Swivel, and practical considerations generally favor doing higher TWR first, tends to muddy the waters a bit. 

What @Wcmille is driving at (though he states it more formally than this) is:  It's better to do your low Isp burns before you do your high Isp burns.

And the answer to that is "yes, all other things being equal."

23 minutes ago, Wcmille said:

I have a hypothesis about the rocket equation, hoping someone can help me either disprove with counter example, or give an informal proof that it is always correct:

For every two stage rocket with total mass M, and a payload weight of P, if the bottom stage engine (called E1) with ISP of  I1 and the top stage (called E2) has an ISP of I2, and also I1 <  I2, then:

There is never a two-stage rocket design of payload P and total mass M where E1 is placed on the top and E2 is placed on the bottom that has more dV than the best two-stage design which has E1 on the bottom, and E2 on the top.

in other words, if I have a 12 ton rocket with a 1 ton payload, a swivel on the bottom stage and a terrier on the top stage, and the fuel is split optimally. There's no design for a 12 ton rocket with the terrier on the bottom and the swivel on top that will be better in terms of dV.

What you're driving at is "is it always better to do your low-Isp burn before your high-Isp burn."  The answer is "yes, generally," but it's theoretically possible to go the other way.

For example, if I have two stages, one of which is Isp 350 and one of which is Isp 800, then in general I want to do the low-Isp one first, all other things being equal.  But suppose that the Isp 350 engine weighs 1 ton, and the Isp 800 engine weighs 1000 tons?  (And we're only talking about, say, 10 tons of fuel.)  Then clearly I want to ditch the heavy engine as soon as possible, it's a mass hog and accelerating it is a total waste of fuel, and in that case it would be better to do the high-Isp burn first.  So there's a counterexample to your formally-stated hypothesis. 

However, unless there's some gross disparity in the dry masses of the stages involved, it's better to do the low-Isp burn first.  That's because the high-Isp burn gets max dV out of a smaller mass, and the exponential nature of the rocket equation means that you pay a heavier penalty for mass that's closer to the payload in the burn sequence.  So being mass-efficient at the upper end is a good idea.

[Zhetaan]

I assume you're operating in space, where TWR and aerodynamic concerns don't factor so much.  From looking at the rocket equation, it's clear that Isp is a dominant factor, but the mass ratio can overcome it in extreme cases.  In the event of an extremely high engine mass or an extremely low fuel mass, the mass ratio (all other aspects of the payload being the same) negates the advantage of higher Isp.

 

EDIT:  Ninja'd!

Edited January 8, 2016 by Zhetaan

[Wcmille]

Ah yes, you're right. An extension of your counter-example is that you want NERVs to push Ants for tiny payloads, and not the other way around.

I'm writing a desktop app for myself that I hope will solve some of the issues with Gary Court's very excellent Optimal Rocket Calculator, such as including NERVs and drop tanks in the design choices. One of the problem with these calculators is that the permutations explode pretty quickly, so I want to make smart filtering choices on engine configurations and not even consider ridiculous ones (like a terrier that ignites before a swivel).

 

[numerobis]

The hypothesis has counterexamples (as you note) if I get to choose the engines mass and Isp adversarially.

I'm pretty sure the hypothesis is true if the engines are massless, or if they're constrained to have equal mass, but the proof doesn't fit in the margin here.

In practice it's a decent rule of thumb for mid-sized rockets. It breaks down with very small rockets (where engine mass becomes a large fraction of the total). It also breaks down if you consider jet engines, which only work in the atmosphere so they must be used first before the rocket.

[Wcmille]

It sounds like there must be an equation that describes the critical point where dV will be the same, regardless which engine goes first. (all masses being equal)

[Wcmille]

On ‎1‎/‎8‎/‎2016 at 4:34 PM, Snark said:

The fact that the Terrier has a lower TWR than the Swivel, and practical considerations generally favor doing higher TWR first, tends to muddy the waters a bit. 

What @Wcmille is driving at (though he states it more formally than this) is:  It's better to do your low Isp burns before you do your high Isp burns.

And the answer to that is "yes, all other things being equal."

What you're driving at is "is it always better to do your low-Isp burn before your high-Isp burn."  The answer is "yes, generally," but it's theoretically possible to go the other way.

For example, if I have two stages, one of which is Isp 350 and one of which is Isp 800, then in general I want to do the low-Isp one first, all other things being equal.  But suppose that the Isp 350 engine weighs 1 ton, and the Isp 800 engine weighs 1000 tons?  (And we're only talking about, say, 10 tons of fuel.)  Then clearly I want to ditch the heavy engine as soon as possible, it's a mass hog and accelerating it is a total waste of fuel, and in that case it would be better to do the high-Isp burn first.  So there's a counterexample to your formally-stated hypothesis. 

However, unless there's some gross disparity in the dry masses of the stages involved, it's better to do the low-Isp burn first.  That's because the high-Isp burn gets max dV out of a smaller mass, and the exponential nature of the rocket equation means that you pay a heavier penalty for mass that's closer to the payload in the burn sequence.  So being mass-efficient at the upper end is a good idea.

Could this be restated as:

A design cannot be optimal (for mass-optimal dV) if a later stage has an engine which is both a lower Isp and a higher engine mass?

[Snark]

14 minutes ago, Wcmille said:

Could this be restated as:

A design cannot be optimal (for mass-optimal dV) if a later stage has an engine which is both a lower Isp and a higher engine mass?

Alas, it's not that simple.  It's more complex than that.

Minor correction first.  Engine mass isn't what matters here.  Dry mass is what matters.  Engine mass is part of that, yes-- but so is the mass of everything else, such as empty fuel tanks and so forth.

But even replacing "engine mass" in your statement with "dry mass", it's still not always true.

Consider this example:  Let's say you have a two stage rocket.  Each stage has:  1. a certain dry mass, 2. a certain fuel mass, and 3. a certain Isp.

• If they have the same dry mass and fuel mass, but the Isp is different... you'd always want to do the low-Isp stage first.
• If they have the same fuel mass and Isp, but different dry mass... you'd always want to do the high-dry-mass stage first.
• If they have the same dry mass and Isp, but different fuel mass... you'd always want to do the low-fuel-mass stage first.

So, basically, the factors that would tend to encourage you to put a given stage earlier in the sequence would be:  low Isp; high dry mass; low fuel mass.

The problem is that in practice, it's never going to be just one of those three factors that's different between the two stages.  It's going to be some combination of all three.  So working out which one goes first will depend on the balance among the three, which is going to be a case-by-case basis.  If one stage has a slightly better Isp, and a slightly lower dry mass, but a much lower fuel mass, it still might not make sense to go first.

Yes, you can do the math.  You can even come up with an equation for a two stage rocket, with variables in place for the Isp and masses involved, to show mathematically which stage is better to go before the other.  But you can't reduce it to a simple general statement of the form "Always do the burn for <condition> first."

Given the nature of the stock parts and typical rocket design in KSP, it usually works out that Isp dominates the other factors, mainly because dV scales linearly with Isp and logarithmically with the mass ratio, so Isp usually ends up winning.  But not always.  The LV-N, in particular, makes for an interesting case study, because it has a really high Isp but also a really high mass.  If you've got a really lightweight stage powered by, say, a single Spark engine, then that stage can actually make sense to come after the LV-N stage, because even though it has a crappier dV, it kills on the dry mass.

[Wcmille]

7 minutes ago, Snark said:

Alas, it's not that simple.  It's more complex than that.

Minor correction first.  Engine mass isn't what matters here.  Dry mass is what matters.  Engine mass is part of that, yes-- but so is the mass of everything else, such as empty fuel tanks and so forth.

But even replacing "engine mass" in your statement with "dry mass", it's still not always true.

Consider this example:  Let's say you have a two stage rocket.  Each stage has:  1. a certain dry mass, 2. a certain fuel mass, and 3. a certain Isp.

• If they have the same dry mass and fuel mass, but the Isp is different... you'd always want to do the low-Isp stage first.
• If they have the same fuel mass and Isp, but different dry mass... you'd always want to do the high-dry-mass stage first.
• If they have the same dry mass and Isp, but different fuel mass... you'd always want to do the low-fuel-mass stage first.

So, basically, the factors that would tend to encourage you to put a given stage earlier in the sequence would be:  low Isp; high dry mass; low fuel mass.

The problem is that in practice, it's never going to be just one of those three factors that's different between the two stages.  It's going to be some combination of all three.  So working out which one goes first will depend on the balance among the three, which is going to be a case-by-case basis.

Yes, you can do the math.  You can even come up with an equation for a two stage rocket, with variables in place for the Isp and masses involved, to show mathematically which stage is better to go before the other.  But you can't reduce it to a simple general statement of the form "Always do the burn for <condition> first."

Given the nature of the stock parts and typical rocket design in KSP, it usually works out that Isp dominates the other factors, mainly because dV scales linearly with Isp and logarithmically with the mass ratio, so Isp usually ends up winning.  But not always.  The LV-N, in particular, makes for an interesting case study, because it has a really high Isp but also a really high mass.  If you've got a really lightweight stage powered by, say, a single Spark engine, then that stage can actually make sense to come after the LV-N stage, because even though it has a crappier dV, it kills on the dry mass.

I'm currently building a program that will build optimal rocket designs for a certain stagecount and payload mass. At present, I try all combinations of rocket engines for every stage. Then I start to add fuel to the engine that will yield the highest differential dV to the vehicle until the goal is reached or all TWR limits are reached (which is a failure). As you might imagine, trying all combinations explodes quickly, making it very important to not try designs that can never succeed.

I think I'm leveraging that nearly all the fuel tanks have the same wet/dry ratio (which, OK, they don't) and assuming everything is just 9:1 for fuel.

Spark would be OK in my example, because it does not have BOTH a lower Isp AND a higher mass than a LV-N. It's only lower Isp.

[Rocket In My Pocket]

9 minutes ago, Wcmille said:

I'm currently building a program that will build optimal rocket designs for a certain stagecount and payload mass.

Great! Now the MechPlebs can have a computer build and fly their rockets for them.

By the time they get home, everything will be done for them and they can just turn KSP off and un-install it.

[IncongruousGoat]

2 hours ago, Wcmille said:

I'm currently building a program that will build optimal rocket designs for a certain stagecount and payload mass. At present, I try all combinations of rocket engines for every stage. Then I start to add fuel to the engine that will yield the highest differential dV to the vehicle until the goal is reached or all TWR limits are reached (which is a failure). As you might imagine, trying all combinations explodes quickly, making it very important to not try designs that can never succeed.

I think I'm leveraging that nearly all the fuel tanks have the same wet/dry ratio (which, OK, they don't) and assuming everything is just 9:1 for fuel.

Spark would be OK in my example, because it does not have BOTH a lower Isp AND a higher mass than a LV-N. It's only lower Isp.

You may want to look at this thread then:

In it several people (including myself) work through the mathematics behind that exact problem.

And, yes, most of the liquid fuel tanks have the same wet/dry mass ratio. The exceptions are the Mk2 and Mk3 airplane parts, the monopropellant tanks, and the xenon tanks.

[Vanamonde]

Let's not do the pro/anti-MechJeb thing again. No one's making you use it, and no one's forcing you to complain about other people using it, either. 

[GoSlash27]

I see no reason for this to be true. e^(DV/Ve) is simply Mw/Md, or how much of the rocket is fuel. All else being equal, I don't see any difference whatsoever if the first stage is more efficient or the second. You either wind up with a heavy payload pushed by a lighter stage with more fuel to compensate, or a light payload pushed by a bloated stage. It all works out the same.

What *really* matters is that each individual stage is optimized for it's mission. There are a lot more considerations than just raw Isp, and that often isn't even the big consideration. To be sure, you have to mathematically model each stage with every engine and then pick the lightest resultant stage and then rinse and repeat. This is pretty easy to do with a spreadsheet, and how I design all my missions.

Best,
-Slashy

 

[foamyesque]

4 hours ago, Vanamonde said:

Let's not do the pro/anti-MechJeb thing again. No one's making you use it, and no one's forcing you to complain about other people using it, either. 

 

I can't believe that this is still a thing. It's been years, people!

 

Anyway, to add to the corner cases, there are circumstances -- landers being the most obvious one -- where you would desire a lower-Isp engine over a higher-Isp one purely on grounds of the TWR; so you could have a mainsail pushing nukes that push reliants or terriers or whatever.

 

Likewise, ion engines, despite their ridiculous Isp, should generally be the last thing in your stack :v

[Snark]

22 hours ago, foamyesque said:

despite their ridiculous Isp, should generally be the last thing in your stack

Actually, it's because of their ridiculous Isp that they should generally be the last things in the stack.

[foamyesque]

1 minute ago, Snark said:

Actually, it's because of their ridiculous Isp that they should generally be the last things in the stack.

 

Er, whoops. :v

[Zhetaan]

On ‎1‎/‎1‎/‎2018 at 12:42 AM, foamyesque said:

Likewise, ion engines, despite their ridiculous Isp, should generally be the last thing in your stack :v

 

19 hours ago, foamyesque said:

Er, whoops. :v

Ion engines should be last because of the ridiculous I_sp (great) and despite the ridiculous mass ratio of the tanks (bad).  Xenon tanks are terrible (awful).

Incidentally, the Mk. 0 LF tank has a wet/dry mass ratio of 10, not 9, so if you're going to min-max your Nerva-powered interplanetary transfer stage, that's the way to do it.

 

Anyway, @Wcmille, I think the wall you're going to hit with this pursuit is less to do with the existence of an 'optimal' rocket and more to do with the presence of three variables in two equations.  You have wet mass, dry mass, and I_sp (actually v_exh in the original rocket equation but KSP reports I_sp instead), but you're solving for the more optimal of two stages.  The fact that KSP is, in some ways, LegoLand In Space! serves to constrain the solution space a bit (there are only so many discrete choices for I_sp because there are only so many engines, each of which has a fixed I_sp) but you still have equal-value solutions for anything within that space.  I see that you chose to constrain the solution space further by tying the result to TWR, but TWR is not universally relevant.  It's also difficult to constrain; a Skipper cannot lift something that weighs 700 kN, but two Skippers can, for exactly the same I_sp and nearly the same dV, which means that both thrust and weight are variable--this means that ultimately, there is a number of engines that will work to lift anything provided that the engine can lift itself against its own weight in the gravity field.  It's the same problem of too many variables and not enough equations.

[Continued...]

Spoiler

 

This similarly expands the solution space as I just described with reference to TWR:  For any given engine of a particular I_sp, there is a solution in terms of wet mass that gives the same dV as any other engine of any other choice of I_sp.  I'll give the general solution now:  if you don't add any fuel (wet/dry mass ratio = 1), then the dV is zero for every choice of engine.

One thing you ought to consider is that if you have a fixed payload mass, then it may be fruitful to examine the problem in terms of related rates.  The rate of change of dV depends on the product of the ratio of I_sp to wet mass and the rate of change of wet mass (ddV = (I_sp * g₀ / m_w) * dm_w)--I could run you through the derivation, but it really ought to be intuitive:  consumption of fuel results in decreasing available dV, the amount of decrease dependent on the efficiency of the engine modified by the mass it has to propel and the amount of fuel actually consumed.  However, it works in reverse, too, when you consider loading fuel.

Of course, because different engines have different masses as well as different specific impulses, I've only taken you to a place where you have two variables and one equation; there will still need to be some kind of choosing algorithm.  On the other hand, the discovery of differential dV is made a lot easier:  rate of change of wet mass is fuel consumption, always (unless you're staging which requires a wholly different mathematical approach), so if you pick an arbitrary value for that change (say, add one orange tank's worth of fuel), then the differential dV will depend only on the relationship between the engine I_sp and the mass it (plus the orange tank) adds to the wet stack, which are both known.  Thus, the largest delta-V increase--which is not the total delta-V of the actual rocket--corresponds to the best engine for that fuel load.

Hopefully, that is more computationally straightforward, but it is worth noting that the wet mass described in this equation is the wet mass of the final rocket, not the wet mass of the payload plus your test fuel load.  In other words, the change in delta-V depends not only on the fuel added, but also the fuel already present because the fuel already present has mass that the new fuel must push from place to place.  You should know this if you intuit the rocket equation at all, but it bears repeating because there are additional features of mass allocation in rocket design that emerge from this property of the equation:  for example, this is what accounts for the Nerv being a great engine for interplanetary journeys but not so great for orbital manoeuvres around Kerbin.

For a less drastic example, the Spark and the Terrier are quite competitive with one another for small (<5 tonnes) craft; such craft can benefit more from the lower mass of the Spark than the higher efficiency of the Terrier, but there is a point where it makes more sense to use the larger engine.  In this specific case, if we consider adding one tonne of fuel to an empty rocket, that point is when the craft masses approximately four tonnes dry.  If we consider adding two tonnes of fuel, then the Terrier is the better choice at three tonnes dry.  Note the convergence; at five tonnes of fuel, the Terrier is better at zero tonnes dry--which is to say that it is always better.  In reality, this means that the Terrier gets more delta-V than the Spark if it only has to push its own mass plus the five tonnes of fuel--this is the value at which its greater efficiency overcomes its greater mass.  At five tonnes dry, the Spark loses to the Terrier even if you add zero fuel, which further confirms that the Spark is never better when the wet mass is greater than five tonnes.  However, the question of whether the vessel will have the desired wet mass depends entirely on its mission.  A lander that operates at Minmus as a shuttle from the surface to near-Minmus orbit will not need much fuel.  A Mun-based shuttle of similar capability will need more.  Woe betide the person whose lander is over five tonnes going one way and under five tonnes coming back.  (Take both engines and use them alternately.  Yes ... that's the trick.)

 

All of this is summed simply by saying that if you wish to choose the best engine for the job, then at some point, you need to know what the job is.

Edited January 2, 2018 by Zhetaan

[GoSlash27]

2 hours ago, Zhetaan said:

All of this is summed simply by saying that if you wish to choose the best engine for the job, then at some point, you need to know what the job is.

^ Exactly. This is why the mission is planned backwards before designing the hardware. In order to design the ideal stage for a job, you need to know exactly what the job is. Atmospheric density, minimum acceptable acceleration, payload, and DV. Once you have those defined, you can mathematically model stages using every engine to find the lightest or cheapest one.

Best,
-Slashy

[Wcmille]

Here's my app. The part circled in red tells an important story. There are many possible rocket combinations, but most are not going to yield useful designs. As the number of stages increases, we need ways to avoid brute force search.

@GoSlash27 I think the problem that I run into with a single-stage lookup is that there are situations where two or more stages will out-perform a single stage design. There are also situations where composite engines (e.g. Swivel + 2x Thud) will outperform a single engine (e.g. Skipper).

[GoSlash27]

wcmille,

 When you run into the question of single vs. multiple stages, it really begins to illustrate just how futile raw Isp is as a benchmark... at least in stock KSP applications. When the DV budget runs to the edges of performance for a single stage, I usually run a few iterations of single vs. 2 or even 3 stage combos just as a sanity check. After a while, you kind of get a feel for what's a good single stage solution vs. what's not.

 For hybrid combos, that's absolutely a problem. There's an insanely high number of possible combinations of random engines to check. Maybe one or several combinations are better than the single engine solutions.
 Since I'm such a lazy man, I've found that the best approach is to not look at hybrid solutions at all, just to simplify the math. There may be some superior hybrid solutions out there, but usually they're not superior enough to justify the effort of looking for them. You could perhaps do better with a hybrid stage, but not much in the overall scheme of things.

 In almost all applications (especially once you get to orbit), I've found that the most expedient solution is one that involves a single engine rather than multiples. Easier to build, more structurally sound, etc.

What this approach *really* lacks is evaluating the merits of serial staging vs. parallel and other more exotic concepts like drop tanks, asparagus, twisted candle, etc. It's just straight serial. Too hard to model that stuff in a simple fashion, but there are often substantial gains to be found there.

 I've figured out some rules of thumb for lifters. When to trade complex for cheap, etc. But once in orbit, I try not to stage at all if I can help it, 'cuz I prefer to reuse my hardware rather than throw it away.

Best,
-Slashy

 
 

[Wcmille]

If you look at this thrust versus Isp curve, I think it gives clues where composite engine combinations could be useful. I have found a single poodle in concert with many NERVs to be a valuable combo. Also, anywhere where you can't get a fraction of an engine, multiples are useful (e.g. 3x vector, since you can't have 0.75 mammoths).