[Math]Calculating The Max Payload of your Lifter?

[Whirligig Girl]

Any way to calculate the maximum payload of your lifter? Take the Delta-V of your rocket without the payload and use mathy goodness to figure out the maximum weight?

[Captain Sierra]

Supposedly, one way to find nominal capacity is to launch without payload. Though this only works if you get down to your last stage, something I am discovering with my Phoenix lifter project. It is reaching orbit without even touching the upper stage, which means I have no nominal capacity gauge.

[Whirligig Girl]

That is one problem I make a nice large lifter, but the upper stage isn't used, even though I thought my payload would be heavy enough. I guess one thing to do would be to build a new rocket each time, and adjust the Dv with the payload mass added to m1 and m0.

[DaveofDefeat]

Or put on kerbal enginner with the payload you want to lift and make sure it has enough thrust and delta-v

[Whirligig Girl]

I want to be very precise, but make my launcher as I please but have a definitive number for the maximum payload.

[arq]

A decent rule-of-thumb is 'payload mass' is 14-16% of the launchpad mass.

[aeronaut]

Take the equation for delta V and work backwards.

Unless I made an algebra error somewhere that should give you the payload mass. The delta V to get into LKO is around 4500 m/s according to the ksp wiki.

Edited October 10, 2013 by aeronaut
Added delta V for LKO

[NASI Director]

That is a very complicated equation! I've used a much simpler one:

deltaV = ln (M-full / M-empty) * 9.81 * Isp

Where In: Natural Logarithm

M-Full: Mass Full

M-Empty : Mass Empty

So, if you had a Command Pod Mk.1 (.8 tonnes), a FL-T100 Fuel Tank (0.5625 tonnes-f; 0.0625 tonnes-e), and a Rockomax 48-7S (0.1 tonnes; 300Isp-atm), then the equation would be:

deltaV = In(1.4625 * 0.9625) * 9.81 * 300

So your deltaV would equal roughly 1006m/s. No need that rocket science mumbo-jumbo that Aeronaut posted!

[tavert]

No need that rocket science mumbo-jumbo that Aeronaut posted!

Aeronaut's equation is the same as yours, he just took it several steps further by breaking down the individual components of the mass and solving for payload. However if your rocket is multiple stages, you have to apply the rocket equation to each stage in turn. Simplest to use MechJeb or Kerbal Engineer, add mass to your payload until the sum of all stage delta-V's prior to the payload is ~4500 m/s.

[Buzzed Aldrin]

Generally delta V isn't usually the limiting factor for payload mass. That's usually determined by your ship's initial T/W ratio:

Sum of thrust of engines / total given mass x acceleration due to gravity. (Eg. 9.81ms^2 for Kerbin at sea level)

If it's less than 1 you are definitely not going to space today. If it's damned near 1 you might get to space but it's going to take a lot more delta V.

Fix

[enki]

I use mods to track the stats of my lifters. At one point I built a line of paste-on rockets to lift payloads of various mass to LKO, but I've held off on recreating them until we get the subassembly bit from .22. I shoot for numbers like a TWR near 2.0 but not too much higher and ~4700 dV. That gives me wiggle room and some extra fuel to deorbit the lift stage.

[numerobis]

Aeronaut's equation is a good start for a ballpark number but has two minor flaws. The more important is that Isp isn't constant - it changes as you climb, and as you stage. The lesser issue is that dV depends on payload to some extent: bigger payload means lower TWR means a marginally less efficient ascent.

But it's probably good enough for kerbal work.

[capi3101]

Longer than intended post ahead:

Aeronaut's equation is spot on - it's what I use to figure out how much fuel to plan for in my stages. (Key thing about about the fuel tanks in KSP - for all but the Round 8 and Oscar-B fuel tanks, the full mass of a fuel tank is 9 times the mass of the same tank when it's dry, i.e. M = 9M_d). So M_m + M_pl in aeronaut's equation is a constant value and can be treated as such. I call it x, where x is the "deadweight" - which is everything but the fuel tanks; both the mass of all stages above the current stage, the mass of the current stage's engines and any other equipment it's hauling around (like decouplers, winglets, RCS, the works - EVERYTHING BUT THE FUEL). Actually, there are several values that become constants - you just solve them as you go.

So the equation simplifies - it's still Tsiolkovsky:

ln(M + x/M_d + x) * g_o * I_sp = dV

ln(M + x/M_d + x) = (dV/I_sp)/g_o

M + x/M_d + x = e^{(dV/I_sp)/g_o}

set a constant y = e^{(dV/I_sp)/g_o}, therefore M+x/M_d+x = y

M+x = y(M_d+x)

substitute M = 9M_d!! 9M_d+x = y(M_d+x)

9M_d+x = yM_d+yx

9M_d - yM_d = yx-x

(9-y)M_d = (y-1)x

M_d = (y-1)x/(9-y)

M = 9M_d = 9(y-1)x/(9-y)

Bear in mind that you have to do this per stage. Gets really fun when you're trying to do the math on a five-stage asparagus lifter...

I have a system for that too...Basically, you play solitaire. Let's say you have a three-stage to orbit lifter.

You give each of the stages a point, then go to the next stage up, give that stage and the one above it a point. Then go up to the next stage, give it a point.

So for three stages, the lowest stage has one point, the second stage has two and the third stage is three. Add those points together, you get six (of course).

You then divide the points for each stage by the total number of points - stage one has 1/6, stage two has 1/3 (reduced), stage three has 1/2.

Multiply that by the total amount of delta-V you want your booster to have in aggregate (4550 for a Kerbin launch)

So stage one would be 758 m/s (rounded), stage two would be 1,517 m/s and stage three would be 2,275 m/s. ]Those are your delta-V targets for each stage.

Plug those values in when you're working Tsiolkovsky backwards.

For each stage, you take the calculated fuel mass and add it to your deadweight to get the overall mass of the stage. Take your thrust, divide by that overall mass, and then divide that by the local gravity and pray you've got a value greater than one (if its a booster stage anyway). 1.2 at launch is usually the lowest I find acceptable for boosters, with 1.6-1.7 optimal.

And yeah, the procedure outlined above it isn't exactly correct because I_sp is not a constant as you go up. It's a lot easier to treat it as one, though (unless you want to figure out just where in the atmosphere you'll be after the first 758 m/s of delta-V and adjust the value to the appropriate amount for that level). As a rule, use the vacuum value for any stage you know will be in space, use the atmosphere value for any stage you're not sure about. And if you've got a mix of engines, you've got to calculate that too.

Well, hopefully I haven't confused anybody here...

Edited October 10, 2013 by capi3101

[blizzy78]

I wonder what the point of calculating the maximum payload capability of a lifter is. Wouldn't you just build your payload as usual, then design a lifter capable enough and put it underneath?

[aeronaut]

Aeronaut's equation is a good start for a ballpark number but has two minor flaws. The more important is that Isp isn't constant - it changes as you climb, and as you stage. The lesser issue is that dV depends on payload to some extent: bigger payload means lower TWR means a marginally less efficient ascent.

But it's probably good enough for kerbal work.

I don't think you will be able to come up with an expression to solve directly for the actual delta V. You can take into account that thrust is a function of the ambient pressure but then Isp becomes time dependent based on the amount of time it takes to traverse the atmosphere. Also I'm unsure how KSP actually defines thrust, is it set it as a constant and mass flow rate vaies as a function of Isp or is thrust (and Isp) calculated as a function of ambient pressure (and therefore altitude)?

My math skills are a little rusty so I'm not sure I could derive an equation that captures this right now. An alternative would be to simulate the launch profile with a small time step to minimize error, that is more work than I want to put into playing a game though!

[Buzzed Aldrin]

I have simulated with an excel spreadsheet, and it seems that the game treats thrust as a constant and ISP changes as a function of atmospheric pressure.

[numerobis]

You will also notice by right-clicking on the engine that the thrust is constant, only Isp changes. This causes the occasional YOUR GAME SUXX0RZ thread.

[aeronaut]

I didn't think of this until just now, doing the calculation using your maximum and minimum Isp will give you upper and lower bounds on payload mass. Use the minimum Isp for a conservative payload mass estimate.

Edited October 10, 2013 by aeronaut

[arq]

4500 vacuum dV is usually enough for me to get to orbit if my TWR is in order, even using mainsails. You need to remember that the atmosphere drops off very quickly, so that at 3500m you are already halfway to the vacuum ISP (by 12km it's over 90%), and even for mainsails the atmo ISP is 85% of the vacuum value. So really the atmo ISP isn't a big deal for launchers, at least not for Kerbin.

Sidenote, by 3500km NR-V's are the most efficient rockets in the game (though the terrible TWR still makes them bad for launchers).

[capi3101]

I wonder what the point of calculating the maximum payload capability of a lifter is. Wouldn't you just build your payload as usual, then design a lifter capable enough and put it underneath?

If you've got subassembly (which I think will be default in 0.22 if I remember the rumors correctly), you can save a booster and apply it to multiple payloads in the same general weight class. The idea is to save yourself the headache of having to design a booster every time you want to put something up. Temstar's got a whole line of subassembly boosters - the Zenith series - that's extraordinarily popular, probably because it has saved a lot of folks a lot of time; just build your payload, pick your booster and go.

Yes, you SHOULD customize a booster for every unique payload, but sometimes folks are lazy, and sometimes you just want to hit space as quickly as possible. I'll admit I'm guilty of this exact thing from time to time. My Mun space station? All launched off of Temstar's booster subassemblies.

[blizzy78]

If you've got subassembly (which I think will be default in 0.22 if I remember the rumors correctly), you can save a booster and apply it to multiple payloads in the same general weight class. The idea is to save yourself the headache of having to design a booster every time you want to put something up.

That's what I've been doing for quite some time now. Still I don't really get why one would need to calculate lifting capability afterwards.

[numerobis]

Building a booster and then publishing its lifting capability is how rockets are built IRL these days.

Plus, math is fun.

[blizzy78]

Plus, math is fun.

I have no doubts, I just think that it works from top to bottom, not the other way around. You don't build a lifter only to find out that it can only lift five tons, when all your customers need lifters for ten tons and up. Rather, you'd go and ask what they need, then build the lifter for it. Or build a lifter fleet with predefined maximum capabilities.

[M5000]

Do it like the Russians, trial and error baby!

[nhnifong]

I like to just eyeball it. It's not rocket science, it's rocket tomfoolery!