# Droptank Staging Theory

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So working on the XenonStorm Mk3's fuel pod design has once again lead me to an old question: "Given a bunch of drop tanks of equal mass and a certain overhead per stage in the form of decouplers (whose presence in upper stages negatively affect all stages below), how many stages before decoupler mass starts really getting in the way, and how many tanks should be in each stage?"

Absent decoupler mass, the clear answer is purge dry mass at every opportunity, staging early and often, however, decouplers do have mass, and in the case of my fuel pod, over 1/10 the mass of an empty tank.  Also, if later stages are too large, the dry mass can damage the mass ratio more than it would on an earlier stage.  To explore the problem further, I built a silly spreadsheet, here dealing with 100 tanks across 10 stages:

I put a few important quantities here and there, notably the full and empty numbers for my fuel tanks, decoupler mass (stack decoupler), thrust, Isp, g, and the mass of the ship itself as the ultimate payload.  "excess ratio" comes about because when the same engines are used all the way, it's the ln(wet/dry) part of things that makes the delta-v happen, so I took the mass ratio of each stage and subtracted 1 to make them more easy to compare, and allow them to be summed in a way that wouldn't be affected by the number of stages.  The fuel pod's actually 7 stacks of tanks stuck on the back of a 1.25-2.5m adapter plate, but for simplicity I've chosen to study the case of 1 long string of xenon tanks with decouplers in experimental locations.

A bit of experimentation bore some odd results -- running with the 100 tank concept, I first tried to keep all stages about equal in mass ratio and thus delta-v, winding up with 3/4/5/6/9/11/14/17/23 yielding a sum exccess ratio of 1.1623 and a delta-v of 45.3km/s, as seen above.

Reversing this pattern (on the stage early and often concept) gave 1.1612 and 42km/s, supporting the too much dry mass in a late stage argument.

Trying something odd, I tested 10/10/10/10/10/10/10/10/10/10, getting 1.1726 and 44.78km/s.  I was expecting that to be worse, but very confusingly, the mass ratio sum is better than the even dv split distribution while the delta-v is worse.  I'm no longer sure that the sum of the mass ratios really means that much...that or I messed up a formula.

I then tried messing around with less stages, trying 12/15/19/23/31 for 1.1784 and 43km/s and 7/11/17/26/39 for 1.1788 and 43.6km/s

All of this is of course just half-blind experimentation, though.  I get the sense that there's something I picked up in algebra 2 or possibly differential equations that'd lead to a far better solution, as well as answer related questions like whether the fundamental results change noticably with different fuel tanks for decoupler masses.

@GoSlash27's stuff on the reverse rocket equation is excellent, but doesn't answer weirder questions like "how much delta-v can I get out of a set number of identical fuel tanks by varying the staging?"

Edit: wow, that table looked like a toilet happened.  Have a screenshot instead, or a copy of the spreadsheet itself: https://dl.dropboxusercontent.com/u/59091477/Monstrosities/Staging Theory.xlsx

Edited by Archgeek
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Alright brains. Crawl out of your brain-holes so this thread can become life-encompassingly interesting. I aint good enough at math for me to have any input! >:[

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Archgeek,

The reverse rocket equation is a hammer- simple tool for designing an optimal single expendable stage. It only answers the question "If I want to impart a certain DV to a known payload mass using a given engine, how much fuel tankage is required?"

It can be expanded to comparison test different engines and different t/w requirements simultaneously, thus providing a shortcut to simple cheap launchers... but that's about as far as it goes.

When you get into complicated staging schemes, all that goes out the window. When looking at drop tanks, the goal is to balance the penalty of dry tankage with the penalty of decouplers. The penalty will be in the form of mass, drag, and cost. Plus in the early career, total part count. Algebra can answer all of these questions simultaneously, but the optimal points would be different for each criteria.

I haven't dug into the nuts 'n' bolts of this problem, so I'm keen to see what your exploration turns up.

Best,

-Slashy

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13 minutes ago, GoSlash27 said:

Archgeek,

The reverse rocket equation is a hammer- simple tool for designing an optimal single expendable stage.

[...]

When you get into complicated staging schemes, all that goes out the window. When looking at drop tanks, the goal is to balance the penalty of dry tankage with the penalty of decouplers. The penalty will be in the form of mass, drag, and cost. Plus in the early career, total part count. Algebra can answer all of these questions simultaneously, but the optimal points would be different for each criteria.

I haven't dug into the nuts 'n' bolts of this problem, so I'm keen to see what your exploration turns up.

Yep, excellent tool, but doesn't work for weird problems like mine.

That's just it, I can smell a curve, possibly three intersecting curves forming a partial 3-space (tank mass ratio, ratio of empty tank mass to decoupler mass, and possibly the ratio of the final payload mass to the dry mass of each stage or something), but it's just out of range.  I'm not even sure if I need my half-forgotten algebra II, my multivariable calculus, or my painfully patchy diff eq. knowledge ('not even sure if this can be expressed as a rate problem or not) for this.

Before deciding it was spreadsheet time, I tried a simplistic equation featuring the sum of all the mass ratios, but it's actually a bit monstrous and does not want to be simplified in any sane manner: (9.375a+.05)/(.4175a+.05) + (9.375b+.1)/(.4175b+.1) + (9.375c+.15)/(.4175c+.15) + (9.375d+.2)/(.4175d+.2) + (9.375e+.25)/(.4175e+.25) + (9.375f+.3)/(.4175f+.3)

What's odd is that it almost looks familiar, like one side of a Bernoulli equation or something, especially when rendered shorter and more simplistic:

(aX+3)/(bX+3)+(aY+2)/(bY+2)+(aZ+1)/(bZ+1)

However, that's still not quite in my grasp, and wolfram alpha produces some serious horror when asked to maximize that for X+Y+Z = 100.

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What you need to do is to let your brain rest then think of how to simplify all the gibberish you said so far.

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2 hours ago, Alpha 360 said:

What you need to do is to let your brain rest then think of how to simplify all the gibberish you said so far.

Perhaps you should try to understand the "gibberish" or let the people who do talk about it without this sort of interjection.

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2 hours ago, Alpha 360 said:

What you need to do is to let your brain rest then think of how to simplify all the gibberish you said so far.

I don't understand it either, so I get your point, but for those who do it can be as clear as the atmosphere on Eve without a 'clouds' mod installed.

Yes simplifying it may help those of that don't understand to at least follow it a bit, but quite often technical discussions need to use the appropriate technical language.

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Yes. The brains are starting to slither over to this thread now. They can smell the math.

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Did you use identical drop tanks for a reason, or just didn't want to deal with the idea of 10 stages made up of different drop tanks (Squad keeps tanks at remarkably similar efficiency)?

The reason you should ignore mass ratio (except as an issue of total mass/cost you put on the pad) is that they need to be averaged as logarithms, just like Jeb and Tsiolkovsky intended.  Equal delta-v isn't guaranteed to be optimal (nobody has shown a general solution, and I suspect there isn't one short of a truly hairy beast that takes aero into consideration), but it is typically one of the best places to start.

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A while ago, I derived the result that for optimal staging (i.e. highest overall payload ratio),

v(E exp(dV/ve) - 1)

is the same for each stage (where ve is the stage's effective exhaust velocity, dV is its delta-V and E is its structural coefficient). I can't be 100% sure that it's correct, but it might be worth calculating it for each stage on your spreadsheet to see whether it matches up with what you expect.

Edited by ferrer
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On 6/27/2016 at 5:10 PM, Archgeek said:

@GoSlash27So working on the XenonStorm Mk3's fuel pod design has once again lead me to an old question: "Given a bunch of drop tanks of equal mass and a certain overhead per stage in the form of decouplers (whose presence in upper stages negatively affect all stages below), how many stages before decoupler mass starts really getting in the way

I think you can simplify that question by rephrasing it in terms of dry weight.

Imagine a stack of ROUND-8's connected by TR-18A's.  The dry weight of the fuel tanks is effectively doubled.

So the question just becomes, how much extra weight can your bottom stage tolerate.

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This sounds like a fun differential equation assuming you have access to tanks of any size and an unlimited budget (or unlimited supply of free decouplers).  I may try to carve some good old procrastination time in at work for this....

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On 6/29/2016 at 11:34 AM, wumpus said:

Did you use identical drop tanks for a reason, or just didn't want to deal with the idea of 10 stages made up of different drop tanks (Squad keeps tanks at remarkably similar efficiency)?

The reason you should ignore mass ratio (except as an issue of total mass/cost you put on the pad) is that they need to be averaged as logarithms, just like Jeb and Tsiolkovsky intended.  Equal delta-v isn't guaranteed to be optimal (nobody has shown a general solution, and I suspect there isn't one short of a truly hairy beast that takes aero into consideration), but it is typically one of the best places to start.

I was just simplifying the problem by going with the simple case of identical tanks (making this a discrete problem) purely in vaccuum, as drag considerations would complicate things pointlessly.  Ah, so it's the average mass ratio I should be considering?

22 hours ago, ferrer said:

A while ago, I derived the result that for optimal staging (i.e. highest overall payload ratio),

v(E exp(dV/ve))

is the same for each stage (where ve is the stage's effective exhaust velocity, dV is its delta-V and E is its structural coefficient). I can't be 100% sure that it's correct, but it might be worth calculating it for each stage on your spreadsheet to see whether it matches up with what you expect.

Interesting.  V_e's a constant, so I can treat that as 1 for comparison purposes, bringing use to E e^(dv).  Now I just need to look up structural coeffcient, then see how that acts.  It's no general solution, but it may reveal something.

1 hour ago, Corona688 said:

I think you can simplify that question by rephrasing it in terms of dry weight.

Imagine a stack of ROUND-8's connected by TR-18A's.  The dry weight of the fuel tanks is effectively doubled.

So the question just becomes, how much extra weight can your bottom stage tolerate.

That'd be nice, but I'm not worried about the bottom stage or TWR (Though I am keeping an eye on that, to catch impractical solutions before taking them too far.)  That does give me an idea, though.  Perhaps it'd be better to think moar philosophically.  That is, at what point is the current dry tankage plus its decoupler greater than some threshold value based on the dry mass of the next stage and its decoupler, or something like that.  Notably for the first two stages it's a problem of splitting the tankage such that the dry mass that you'll not be able to ditch in the future is no more than the dry mass you ditch when you stage, which comes to a not quite even split and is moot when the decoupler is half or more of the dry mass of the combined tankage.  Beyond that point, I've been spinning my wheels pretty badly between tasks at work, with several genius-sounding ideas that later make no sense; which is too bad, as I thought I'd spotted a hidden simplicity that wouldn't require a differential equation and that could be pretty easilly brought to bear on the question of how many stages, as it seemed to give the point where it was worth "splitting" the new stage.

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16 hours ago, Archgeek said:

Interesting.  V_e's a constant, so I can treat that as 1 for comparison purposes, bringing use to E e^(dv).  Now I just need to look up structural coeffcient, then see how that acts.  It's no general solution, but it may reveal something.

Just had a look at this again and realised I'd forgotten something - should be

v(E exp(dV/ve) - 1)

though like you said, that doesn't matter if your ve is the same. (Here are definitions for the various ratios, including structural coefficient, by the way.) Now if you're using drop tanks with equal structural coefficients, it results that using equal delta-V for each stage is optimal (as wumpus suggested earlier on).

I ended up solving the problem numerically, the final result being the way a certain amount of delta-V should be distributed between a given number of stages so as to minimise the total mass of the spacecraft (interestingly, this turns out to be independent of payload mass, seeing as it's not part of the expression above). Not sure how useful all this is to you, but it's better written here than on some scrap paper at the bottom of my drawer

Edited by ferrer
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14 hours ago, ferrer said:

[...]like you said, that doesn't matter if your ve is the same. (Here are definitions for the various ratios, including structural coefficient, by the way). Now if you're using drop tanks with equal structural coefficients, it results that using equal delta-V for each stage is optimal (as wumpus suggested earlier on).

I ended up solving the problem numerically, the final result being the way a certain amount of delta-V should be distributed between a given number of stages so as to minimise the total mass of the spacecraft (interestingly, this turns out to be independent of payload mass, seeing as it's not part of the expression above).

Ah, so the structural coefficient is just the mass ratio, in this case, tankage and decouplers for a stage.  Considering the big tanks have a worse mass ratio than the smaller ones, I'll have to see if can't eeke out some more performance by starting with the .04 and .07 tonne tanks, with their 2.2738:1 and 2.2727:1 ratios being a bit better than 2.245:1.   Curious that my instinctual draw to the equal dv arrangement was not off.  Hooray for math!

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16 hours ago, Archgeek said:

That'd be nice, but I'm not worried about the bottom stage or TWR

That's when the vast majority of your drop tanks leave, and the one time TWR really dominates the interaction, but if you're thinking of droptanks for extraorbital things, here's a really simple way to gauge the maximum benefit:

Put all that fuel in one stage and calculate delta-V.  Then subtract the weight of the tanks but not the fuel and calculate delta-V again.  This is your "upper bound", the theoretical maximum benefit;  you'll never manage better than this number no matter how you drop your tanks.  So if the number's not that impressive, it's not worth bothering with.

Edited by Corona688
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9 hours ago, Corona688 said:

That's when the vast majority of your drop tanks leave, and the one time TWR really dominates the interaction, but if you're thinking of droptanks for extraorbital things, here's a really simple way to gauge the maximum benefit:

Put all that fuel in one stage and calculate delta-V.  Then subtract the weight of the tanks but not the fuel and calculate delta-V again.  This is your "upper bound", the theoretical maximum benefit;  you'll never manage better than this number no matter how you drop your tanks.  So if the number's not that impressive, it's not worth bothering with.

Oooh, neat technique, there.  HEH, going with the 12.125 tonne mk3 + project Helios Pioneer capsule/return system, that 100 tanks comes up 68.6km/s.  I think I might be packing on a bit too much fuel, considering a nice, tight, 610km circular solar orbit's only 13.7km/s with a direct injection from Kerbin, less if one goes bi-elliptic with Jool.

Now I just need to figure out the best stage dv for mass efficiency.  I'm willing to guess that for as light as the decouplers are, I can get away with whatever the top stage tank gives me, and still be over the optimal fuel per stage, mass-wise, but I'm curious where the line might lie.

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18 minutes ago, Archgeek said:

Oooh, neat technique, there.  HEH, going with the 12.125 tonne mk3 + project Helios Pioneer capsule/return system, that 100 tanks comes up 68.6km/s.

Yikes, that's a lot, did you accidentally subtract the weight of the fuel?  you're supposed to leave it in, i.e. calculate the weight of the craft and the fuel but not the fuel tanks.

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18 minutes ago, Corona688 said:

Yikes, that's a lot, did you accidentally subtract the weight of the fuel?  you're supposed to leave it in, i.e. calculate the weight of the craft and the fuel but not the fuel tanks.

Did that.  Each of the big tanks weighs in at .9375 tonnes, .52 of which is xenon.  Move the decimal point over twice and that's 52 tonnes of gas.  ln(64.125/12.125)*4200g = 68.6246km/s.  Without the tankage, the mass ratio is out of hand at over 5.  It gets more out of hand if I use the smaller tanks, but so does the part count.

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Oh, you're using ion engines!  Wow, yeah, I can see that.  Xenon tanks are heavy suckers even when empty.

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Let's imagine another spherical cow:  A ship made of nothing but xenon tanks.

One full tank weighs as much as 2 empties, so, by the time this stack is one-third empty, that's 20% dead weight and (I think) 125% benefit to dropping the empty tanks.  The more often you drop tanks, the smaller the benefit, and the fuller your craft is, the less the benefit.  Decouplers every tank would add 10-25% dead weight so it'd be easy to overdo it.

In other words -- this is just the rocket pyramid again, looked at upside down.  That works exponentially.  Drop an exponentionally smaller number of tanks the emptier your craft gets.  I'm arbitrarily picking one-third here since that's a nice clean number with xenon tank weights, starting with 81 tanks eject 27 tanks, then 18 tanks, then 6 tanks, then 4 tanks, etc.  Don't bother ejecting either of the last two since the dead weight of the decoupler will ruin the benefit.

Hypothetically speaking, anyway.  A real craft will have engines too.

Edited by Corona688
My tank counts were garbage.
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• 4 weeks later...

Sorry for resurrecting this thread, but I've encountered another wrinkle to the problem that I was actually unaware of.  Thanks to the discussion here, I've actually ginned up a pretty serviceable solution:

Spoiler

However, in practice, I have to make geometric concessions to prevent or reduce oscillation -- and so've taken to using the rockomax adapter plate 2 and some cubic octagonal struts to make one stack into seven, six staged pairwise, 1 central core, to reduce decouplers per stage, as seven would get annoying.
Unfortunately, [STAGE_PRIORITY_FLOW] does not work the way I thought it did.  Rather than drain the tanks in the highest-numbered stage, it drains every tank in the group with the highest number of decouplers between them and the root part:

Spoiler

So, if I want to stage pairwise, I could either do something dumb and stack extra decouplers to force the order I want, or I could do something weird, and split a stage's fuel such that some of it is in the next stage's tanks, effectively sticking the upper stage with some of the lower stage's dry mass...except I can't do that either, as it drains every individual tank with the same decoupler number at the same rate.  This means that stages with the same type of tank will empty at the same time, preventing their treatment as separate stages.

The one bit of control I can get is that a stage with smaller tanks will drain before a stage with bigger tanks, such that for 700 unit xenon tanks around a core of 5200 unit tanks, every big tank will still have 4500 units in it when the outer tanks are staged, effectively turning the first .07 tonnes in each into a part of the outer stage, and dumping some of the outer stage's dry mass on the inner stage.

Spoiler

The question is, what does this shenanigan do the optimum mass ratio?  I'm guessing since this increases the inner stages' structural coefficient, that the answer is no longer "match them as closely as possible".

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Okay, that might not make a whole lot of sense, so here's a trivial example with a 5 ton payload/panels/engines:

wet mass payload + 5200(wet)-700(fuel)+stack decoupler) 700(wet)*6 + 700(fuel) + mini decoupler*6 + S0
mass ratio wet mass/(wet mass - .52+.07) wet mass/(wet mass - .07*7)
5.9375-.07 + .05/(wet mass - .52+.07) = 5.9175/5.4675 = 1.0823 .88 + 5.9375/(wet mass - .07*7) = 6.8175/6.3275 = 1.0774
dv ln(1.0823)*41202 = 3259m/s ln(1.0774)*41202 = 3073m/s

I just noticed that the gain in structural coefficient suffered by S0 from the theft of some of its fuel comes with a loss of it enjoyed by S1...I don't suppose they balance out entirely, considering the later stage would feel an increase more accutely than a lower one.

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