Question Regarding Calculating Lifting capability of a booster

[Grotoiler]

Greetings all,

I have a question that's been boggling me for a while now, and I'm wondering if any of you could help me out.

I'm constructing a fairly large multi-planet exploration craft, with multiple launches to be assembled in orbit. I've come up with a fairly reliable single-stage heavy lifter to get the sections into orbit, with a spot of fuel left to de-orbit it when it's served its purpose. Here are the stats:

Vessel Mass (Full) - 347.6t

Surface TWR - 1.61

Stage Dv - 5182m/s Atmospheric / 6432m/s Vacuum

Launching to a 75km orbit,, with zero payload (did a test-flight for starters), I'm left with a Dv surplus of 1833m/s. I'm not sure how relevant that would be to my question, but there it is.

My question is: Is there any way I can calculate the maximum practical payload that can be lifted, based on the capabilities of the lifting craft? So, for example, with the aforementioned design, how can I calculate that it would boost a maximum of (for example) 20t into a 75km orbit?

Please note that when it comes to mathematical equations, I'm a smidge above the neanderthal level of understanding rocket science. I grasp the basic tenets, but I've yet to work with Tsiolkovsky's rocket equation (or any of a thousand others for that matter) on any of my designs . A touch of basics would be appreciated!

Thanks in advance for your help all!

[purpletarget]

Tsiolkovsky's rocket equation is your best friend. Go introduce yourself to it, get to know it, learn it, love it.

If you're starting with a basic launcher, run your current design though the rocket equation (stage by stage) and find out what it's capable of with zero payload.

You already know what excess dV you have, so you can recalculate the Rocket Equation the same way with larger and larger payloads until the dV answer you get is reduced by that surplus you're getting.

With atmospheric ascents, the calculations are very rough estimated, because a lot of gravity and atmospheric drag losses are highly variable depending on design and flight profile.

My preferred method for lifters however, is to plan on a specific payload from the beginning. That is, if I want a lifter that can put 20 tons into orbit, I add 20 tons of dry mass to the equation before I even start building the first stage! And then build up the lifter from there to have the required dV with the payload already accounted for...saves lots of tears and heartache.

Anything and everything pretty much that you could want to know about the Rocket Equation and other topics, check out the Drawing Board in the tutorials section.

[Th3F3aR]

You could do this with the following equation (?Rocket equation?) :

dV = Ve * LN((Mf+P)/(Md+P))

dV = you know that already

Ve = basically Isp but with the g0 in it (Isp*g0)

LN = Logarithm n

Mf = Mass full (without payload)

Md = Mass dry (without payload)

P = Payload

There you go, do this equation for every stage and sum it up to get your output dv and you see if you got enough power to lift it up

EDIT: Important, also watch your TWR on adding a payload, it could happen you get a good dv and a TWR less than 1, so check that always

Edited April 2, 2015 by Th3F3aR

[capi3101]

If you're not interested in calculating a potential payload mass, you could install KER or Mechjeb for a delta-V and TWR readout in the VAB, and then you need to look for a mod called NRAP. NRAP would let you put a "proofing payload" on your design that can a) be adjusted for size and more importantly can be adjusted for mass. You simply stick that on the top of your booster and then increase the mass in whatever increments you want until either the TWR goes below 1.2 (generally considered the lowest "safe" TWR for a successful launch) or the delta-V goes below 4500; whatever the mass of the NRAP payload is at that time is what your booster can handle.

Alternatively, you could just load some fuel tanks on top of the booster and go with that measure. That's a more "stock" way of figuring it out.

EDIT: To answer your question mathematically, I'd need to know more about the characteristics of the booster itself - big one there is how many stages it has and the TWR of each stage, but the types of fuel tanks happen to be important in this case (the really big tanks have a wet-to-dry ratio of 8.2, while the FL-T100 through the Jumbo 64 have a wet-to-dry ratio of 9).

I suppose I could guess at your booster configuration by what information you've given us. Lessee...347.6 tonnes with 1.6 TWR, and 5182 atmospheric. You've got 5450 kN of thrust (simply based on the general T = F_t/mg TWR equation, where T is the TWR and FF_t is the thrust force of the engines, thus F_t = Tmg = 1.6 * 9.8 * 347.6 = 5450.368). Assuming a single stage at that thrust level and solving the same equation for mass (m = F_t/Tg) , you hit a 1.2 TWR at 463.47 tonnes. So your payload limit based on TWR would be 115.87 tonnes (463.47 - 347.6 = 115.87). I very much doubt that TWR is your limiting factor.

Now, there's no single engine that produces that much thrust on its own. I'ma guessing you've got a cluster of engines - and I'd need to know which specific ones so I could tell what your Isp is; that's a bit of data I'd need to tell you the delta-V limit. So I guess I'd need to know both the fuel tank and engine configuration. But what I would do is work Tsiolkovsky backwards, setting the delta-V to 4500 (of course if it's a multi-stage rocket, it'll be a might hairier, but not as bad as you might think).

Edited April 2, 2015 by capi3101

[Grotoiler]

Thanks for the replies everyone!

I'm at work right now, but i've started plugging away at figuring out Tsiolkovsky's rocket equation.... It's hurting my noggin!

@purpletarget, that's how I usually build my rockets (with help from MechJeb to check my TWR and Dv stats) - Payload first, and then build the lifter underneath as needed until I get the desired Dv and TWR. I'm more specifically trying to figure out what the practical lifting capacity of my lifter would be, so that I can start creating sub-assemblies and just picking the right lifter for the given payload instead of rebuilding a lifter for every launch...

@Th3F3aR, thanks for your slight simplification of the rocket equation. It's really helped me to start understanding it a bit more... I think I'm now on the right track to getting there. Just have to work out the Isp thing. When it comes to g0, is that relative to which celestial body you're taking off from? so, for example, on Kerbin it would be 9.81 (m/s), whereas on Minmus it would be 0.491m/s?

Also, are the Isp values quoted in-game specific impulse in seconds, or specific impulse in meters per second?

@capi3101, my configuration is as follows:

2x Kerbodyne S3-14400 tanks stacked

1x Kerbodyne KR-2L Advanced engine (2500kN thrust)

Outboard on 2x radial decouplers:

4x Rockomax Jumbo-64 Tanks (2 per side)

2x Rockomax X200-16 Tanks (1 per side)

2x Mainsails (1500kN Thrust Each, 3000kN together)

Total Maximum thrust is 5500kN

It can get to a 75km orbit without having to stage the outer engines (there's a spot of fuel left in them once it gets to 75km orbit).

I hope that helps you!

Once again, thanks for all the help all!

[Th3F3aR]

You are right in terms of the first stage capi3101, but what is with stage 2 if there is one?

I doubt that he created a one staged lifter with 5000+ dv? It's possible yeah, but i doubt that.

If you assume that stage 2 has 150t less (with a basic TWR of 1,6) of the mass left and therefore approximately a thrust of ~3200kN. Then it might be different with the TWR, that results in approximately 1,03 which is quite low. So i won't say TWR is no limitation for that lifter

Edit:

My 50T lifter (well it might not be the best configuration)

Mfull : 430250Kg

dV: 5200m/s (with Payload set to 50Tonnes)

s1 : 430t/302t | 8000kN | 1,9 TWR -> with payload 1,7

s2 : 262t/118t | 3200kN | 1,24 TWR -> with payload !1,04!

s3 : 88,5t/16,5t | 2500kN | 2,88 TWR -> with payload 1,84

My limiting factor is stage 2, that definately needs a rework on thrust. But the 5200m/s are okay, and i can squeze out bit more by reducing thrust while in stage 1 and 3

Edit 2:

okay Grotoiler, Isp is given in seconds just like on the ksp wiki parts page, i use the Ve with is given in m/s (Isp is Ve basically but not in terms of units ), you can read that by hovering over the isp values on the wiki

Yes g0 is the gravity of the body you are taking off from

Edit 3:

If i didn't mess up and you use a 112,4t Payload you'll get a total dv of lousy 2493,8m/s So yeah TWR is not limiting in this case but you just don#t have enough fuel

Equation parameters

Ve : 3142m/s | Mfull : 460t | Mdry : 208t (fuel used 252t)

With a payload of 50t you'll get at least 3123,9 m/s dV

With a payload of 25t you'll get 3659m/s dV

You see, without staging you lose way too much dV cause you transport a some 100t of b**** with you all the way up

Edited April 2, 2015 by Th3F3aR

[Brainlord Mesomorph]

I always though thought it was Robert Goddard's Rocket Equation.

or is that a different equation?

[capi3101]

You are right in terms of the first stage capi3101, but what is with stage 2 if there is one?

I doubt that he created a one staged lifter with 5000+ dv? It's possible yeah, but i doubt that.

I also doubted it was a single-stage lifter given the earlier parameters - but it's not impossible to build something like that. Just unlikely. In any event, with the new data I know it's a multi-stage rocket and we can work with that; I'll do the calculations here in a little bit.

I always though thought it was Robert Goddard's Rocket Equation.

or is that a different equation?

No, Tsiolkovsky came up with the equation; Goddard invented the first working liquid fuel rocket.

[capi3101]

Okay...so for boosters I always use the atmospheric Isp when I'm designing things - I treat it as the "worst case", because it at least accounts for some of the atmospheric effects. In reality, you'll have a smidge more delta-V than what I calculate here - just that disclaimer before I get started.

Okay - so the core stage is 2x Kerbodyne S3-14400 tanks stacked and a 1x Kerbodyne KR-2L Advanced engine. That's 82+82+6.5 = 170.5 tonnes full and 10+10+6.5 = 26.5 tonnes empty with a 280 Isp and 2500 kN thrust. That stage has 5,108 delta-V by itself (?) with a TWR of 1.496. The outboard "booster stage" adds two Mainsails, which have a different Isp - you do the math there (divide the Thrust by Isp for each engine and tally that up, then divide the total thrust by the result) and the effective stage Isp is 300.4878. Then you've got - 4x Rockomax Jumbo-64 Tanks (2 per side), 2x Rockomax X200-16 Tanks (1 per side) and 2x Mainsails. That stage has 36+36+36+36+9+9+6+6+170.5 = 344.5 tonnes full and 4+4+4+4+1+1+6+6+170.5 = 200.5 tonnes empty, for a TWR of 1.63 and 1,593.95 m/s of delta-V. Total delta-V would be 6,701.95 m/s...

From that math, I'm assuming that you're not running fuel lines from your outboard units to the core and that you're running both stages concurrently. You'd get FAR better performance from the booster if you ran the fuel lines; at this point the outboard units aren't contributing all that much to your overall delta-V.

Okay then. So things are even trickier than I thought - I'll need to calculate about how far into the flight it is when the Mainsails run out of gas and are cut loose, and then figure out what the mass of the core unit is at that point.

So let's see...Mainsails have a fuel flow rate in atmo of 0.4778 t/s. You've got two, so their fuel flow is .9576 t/s. You've got 144 tonnes of fuel in the core, so you'll be empty after 150.376 seconds (two and a half minutes into the flight, sound right?)

A KR-2L has a fuel flow rate of 0.7037 t/s in atmosphere - so after 150.376 seconds it has burned through 105.8195 tonnes of fuel. Thus at that point in the launch the core has a mass of 64.6805 tonnes.

All this assumes that there is NO payload on the craft whatsoever. I'm out of time to type at the moment - I'll try to pick this up later; maybe one of y'all can check my math in the meantime and/or pick up the discussion.

[EtherDragon]

HINT:

Take Tsiolkovsky's Rocket Equation:

Dv = Isp*G*ln((M+P)/M)

Solve for M, given a known Dv.

You know it takes 4500-ish Dv to reach orbit.

You also know how much propellant you have.

You know the Isp of your lifting engines (use the average)

G is 9.8 (m/s^2)

We get:

Dv / Isp*g = Ln((M+P)/M)

Unraveling Ln is a bit of work - but the short-cut is:

e^(Dv/Isp*g) = (M+P)/M

I leave the rest for you...

[Pecan]

For a single-stage rocket launch vehicle:

Launch payload capability = ((Launch_Ratio * Mass_Dry) - Mass_Wet) / (1 - Launch_Ratio)

Mass_Dry and Mass_Wet are without fuel and with it, respectively.

Launch_Ratio = EXP(dV_Launch / Exhaust_Velocity)

dV_Launch is the vacuum dV the vehicle needs at launch - at least 4,500m/s to get to orbit, plus probably 200m/s-400m/s to de-orbit and land - 4,900m/s

Exhaust_Velocity = Isp * 9.82 (or 9.81, I can never remember ^^).

Mainsails have an Isp of 360, giving the SSTO 40 design in Chapter 7 of 'Exploring The System' a theoretical maximum payload capability of 40.4t, as follows:

Exhaust_Velocity = 360 * 9.82 = 3,535.20

dV_Launch = 4,900

Launch_Ratio = EXP(4,900 / 3,535.2) = 4

Mass_Dry = 66.3

Mass_Wet = 386.3

Launch payload capability = ((4 * 66.3) - 386.3) / (1 - 4) = 40.4t (rounded to 1 decimal place, which I drop as 'margin for error' so 40t usable as far as I'm concerned)

ETA: What I call 'Launch_Ratio' is the same as EtherDragon ninja'd me with and that's the key to reversing the rocket equation :-) EXP() is the exponential function for LibreOffice and Excel spreadsheets.

Edited April 2, 2015 by Pecan

[capi3101]

Alright - since I was able to prove earlier that almost 98% of the delta-V of your stack was coming from the core, I'm going to focus my efforts there. We have a stage that's 170.5 tonnes full and 26.5 tonnes empty, with 2500 kN of thrust and an Isp of 280. We're attempting to figure out how much is the maximum payload this rocket can lift to orbit.

So, two possible limiting factors there - thrust and delta-V. We can solve the thrust factor using the same formula I was using earlier (T = F_t/mg) solving for mass and assuming a TWR of 1.2 (again, the generally accepted minimum figure). That gives us m = F_t/Tg = 2500 / (1.2 * 9.82) = 212.152 tonnes. Since we know the booster is 170.5 tonnes full, the maximum payload is the difference: 212.15 - 170.5 = 41.65 tonnes.

Now to see about delta-V. For this, we'll add a payload mass to both the mass factors of Tsiolkovsky (which we'll call P), and set the target delta-V to 4500. So we have the following:

dV= ln(M/Md)*9.82*Isp = ln((170.5+P)/(26.5+P)) * 9.82 * 280 = 4500

ln((170.5+P)/(26.5+P)) = 4500 / (9.82 * 280) = 1.636602

(170.5+P)/(26.5+P) = e^1.636602 = 5.13768

170.5+P = 5.13768(26.5+P)

170.5+P = 136.14852 + 5.13768P

170.5 - 136.14852 = 5.13768P - P = (5.13768-1)P

4.13768P = 34.35148

P = 8.3021, the maximum allowable payload due to delta-V limitation.

So the limiting factor of the booster's effectiveness is delta-V.

We can confirm this by adding the two indicated limiting factor masses to the mass and dry mass of the booster and calculating delta-V:

dV= ln(M/Md)*9.82*Isp

dV= ln((170.5+8.302)/((26.5+8.302))*9.82*280 = ln(178.802/34.802) * 9.82 * 280 = 4500.007

dV= ln((170.5+41.65)/((26.5+41.65))*9.82*280 = ln(212.15 / 68.15) * 9.82 * 280 = 3122.397

I'm going to make one more pass at this just to see if the booster's performance can really be expected to be this bad; I have neglected the (negligible) delta-V contribution of the outboard engines after all. But I wouldn't be too hopeful.

[Admac]

One weird trick that I assume Dermatologists hate

There is a massive advantage to adding a SRB to a low TWR vehicle, even if the overall DV hardly changes, in that you can cut a lot of time off the ascent, and save a heap(500+ DV) on gravity losses.

By getting your first 25-35 seconds of launch up to(or even above) a 2.0 TWR means you may be shaving a minute off your ascent and that saves hundred of metres of DV. I use FAR, so going fast early is only a minor aerodynamic penalty(less than 50 M/s DV), Stock will be different. By changing a single stage vehicle TWR from (1.2 -> 5) to (2->1.1->3 ) you can shave off a bunch of tons of engine, and a minute of ascent, allowing a leaner of a launch vehicle. While it's no longer a VSSTO, adding 5-20k funds of early SRBs can still be cheaper than scaling up your launch vehicle and recovering everything.

[OhioBob]

When it comes to g0, is that relative to which celestial body you're taking off from? so, for example, on Kerbin it would be 9.81 (m/s), whereas on Minmus it would be 0.491m/s?

No, g_o is always 9.81 m/s² (though it is my understanding that in KSP it actually works out to 9.82 m/s² when back calculated). It has to do with the defined relationship between Newtons and kilograms. It has nothing to do with local gravity.

- - - Updated - - -

For the record, I have a few simple guidelines that I like to follow when designing a rocket that work very well. I describe those guidelines and provide an example in the following post:

http://forum.kerbalspaceprogram.com/threads/107763-Designing-Launch-Vehicles?p=1678756&viewfull=1#post1678756

[capi3101]

Okay...so that set of calculations I made earlier? Yeah - forget it; you've got a reasonable booster just the way it is:

Hell, I even modded the damn thing a bit for FAR - extra weight for the fins nosecones, see - and it still worked well.