The maths of staging

[Yasmy]

Let's do some math. This may not exactly be GoSlash27's problem, but it's well defined and tractable.

Assumptions:

1) one engine type with a constant fixed Isp

2) one fuel tank type of fixed full to dry ratio, but of arbitrary size

3) the final stage consists of payload, a fuel tank and an engine

4) a non-final state consists of a fuel tank and engine and a decoupler

5) we start with a known single stage rocket, and wish to know it can be replaced

with a two stage rocket with the same delta-v, but total lower mass.

Variables for the single stage rocket:

p = payload mass

t = dry tank mass

f = fuel mass

e = engine mass

v = engine exhaust velocity = Isp * g0

dv = v ln((p+e+t+f)/(p+e+t))

Now for the two stage rocket, define a and b as variables which scale the single stage rocket fuel tank:

a*f = final stage fuel

b*f = initial stage fuel

d = decoupler mass

dv = v ln((p+e+a*(t+f))/(p+e+a*t)) + v ln((p+2*e+d+(a+*(t+f))/(p+2*e+d+(a+*t+a*f))

Require the two rockets to have the same dv. Fun with logs:

ln(x) = ln(y) + ln(z) = ln(y*z)

x = y*z

Now do a bit of algebra to solve for b, the scale of the initial stage fuel tank, given scale a:

(p+e+t+f)/(p+e+t) = (p+e+a*(t+f))/(p+e+a*t)) * ((p+2*e+d+(a+*(t+f))/(p+2*e+d+(a+*t+a*f))

Solve for b (using a computer algebra system):

b = (1-a)*(p+e)*(p+2*e+d+a*(t+f)) / ((p+e)*(p+e+a*f) + a*(2*p+2*e+f)*t + a*t^2)

What we care about is the ratio of the initial mass of two stage rocket to the single stage rocket:

r = (p+2*e+d+(a+*(t+f)) / (p+e+t+f)

Substitute b, and do a pile of algebra:

r = (p+e+a*t)*(p+2e+d+a*(t+f)) / ((p+e)*(p+e+a*f)+a*(2p+2e+f)+a*t^2)

Plots and thoughts will come later, but here's what you do with this equation:

Pick values of p, e, t, f and d.

Plot r(a) from a = 0 to 1. If r is ever less than 1, you have a better two stage rocket.

Edit: Of course, if we have r(a), we can minimize r(a) to find the mass-optimal final stage fuel scale:

a_opt = (-(e+p)^2*t*(f+t)+((e+p)*t*(f+t)*(2*e+f+e*f+2*p+f*p-(e+p)*t+t^2)*(e^2*(4+f-t)+d*(f+e*(2+f)+2*p+f*p+t^2)+p*(f+2*p-p*t+t^2)+e*(6*p+f*(2+p)+2*t*(-p+t))))^(1/2))/(t*(f+t)*(f+e*(2+f)+2*p+f*p+t^2))

Ugly monstrosity, but it should tell you the ratio of fuel tanks in the upper stage of a two-stage rocket to fuel tanks in a single stage rocket. If the ratio is ever less than zero or greater than 1, single stage is best.

Edited June 5, 2015 by Yasmy
typos

[FancyMouse]

ok, second round of calculation shows some more interesting stuff even without TWR restriction.

First, there are cases where even staging 2 identical 1 engine per stage is still helpful due to savings of empty tank mass. HECS+Oscar-B*2+Ant [TR-2V staging] FL-T100+Ant gives 1167kg and 5464m/s, while HECS+FL-T200+Ant gives 1245kg but 5022m/s.

Next, given the amount of fuel, obviously you can calculate dV immediately, but you can also compute the optimal stage point. However, that's a pretty nasty rational function. With actual values substituted in, it should just boil down to a quadratic inequality, which isn't hard. But symbolically I think it's just a headache to deduce anything human understandable. In addition, given the same amount of fuel is the easiest assumption that I can find that is reasonable. Things like given dV or given mass will just spam the calculation with exp/ln and I won't be surprised if we need to solve some exponential equation eventually.

So yeah - original problem solable via some formula, but you probably prefer a program to compute it - it's O(1) computation, though.

EDIT: ninja'ed... sounds like the exact same calculation happened above me.

Edited June 5, 2015 by FancyMouse

[cybersol]

Now do the math for asparagus staging, where you get to have your extra engines and use them too.

*Drops mic, walks out of room*

[Yasmy]

Pretty trivial change, cybersol. Just replace 2e with e in the initial stage delta v equation.

Edited June 5, 2015 by Yasmy
Edited for tone. No need to be snarky.

[cybersol]

In a theoretical world where an asymetrical rocket would be easy to fly, I don't think asparagus staging would actually change your deltaV equations at all. It should only matter for TWR.

More on topic, I think it will turn out that it's a trade off between empty tank weight and the weight of the engine plus decoupler -- modulo the fact that the tradeoff occurs in log transformed space. Plus I was really just killing time until I could see your plots!

[Yasmy]

Javascript is disabled. View full album

payload: Mk1 command pod drained of monoprop, tank: 5 FL-T200, engine: LV-909, decoupler: TR-18A

p = 0.8, e = 0.5, f = 5 * 1.125, t = f/9, d = 0.05

The plot shows several quantities vs the final stage to single stage fuel ratio, a. Apologies for not making this more color vision deficiency friendly.

The middle solid curve (yellow) is the ratio of the masses of a two stage rocket to a single stage rocket. When r < 1, the two stage rocket is less massive. Two stages is favorable for approximately a > 0.14 (by eyeball).

I eyeballed the plot and picked a = 0.4 and a = 0.8 as values which had nice round numbers of fuel tanks for both a and b (the lower solid (blue) line). So a = 0.4, b = 0.5 is one rocket, and a slightly worse rocket is a = 0.8, b = 0.2, roughly.

b = 0.5 means 2.5 FL-T200 tanks, since the single stage rocket had 5.

I've posted a screenshot of the single stage rocket, and the two two-stage rocket. Frankly, I didn't expect to get a solution so easily for the LV-909. First attempt worked.

N.B.: My eyeballing was only so-so. b(0.4) = 0.497 (spot on). b(0.8) = 0.15, a bit less than my guess of 0.2.

Edited June 5, 2015 by Yasmy

[Green Baron]

Hello,

i found this when searchng through the web. It may contain the solution to your problem, but it's hard stuff to chew on ...

www.dtic.mil/dtic/tr/fulltext/u2/a069296.pdf

k

[cybersol]

So I was thinking about how to estimate an answer to Slashy's question of how many stages is optimal for a given dV budget. I also noticed that Yasmy's graph for the 2-stage case shows a fairly broad minimum and that an equal fuel budget between the 2 stages is at least "near" optimal. So I built a quick spreadsheet to apply Meithan's equation below:

According to Meithan, for a given desired ÃŽâ€v and payload mass (m_payload), the optimal (i.e. minimum) total mass of the ship for a given engine is:

Under the assumptions that you can divide your total dV budget into equal stages, I calculated the total mass of the ship for 1-10 stages in an ugly spreadsheet. I assumed the engine had its vacuum ISP and added the weight (.05) of a decoupler to the engine weight. The weight of the rocket was the number of stages times the weight of each stage, where the weight of the stage was calculated by Meithan's formula after dividing the dV budget up equally between the stages. I used a payload size of 10, but the results varied very little with payload. I used an alpha of 8.

Empirically I found that a given engine had a almost constant dV per stage that was optimal. I also found that you can get a back of the envelope estimate of this dV by:

"Near" optimal dV per stage = Engine_Isp * 7.25

The actual factors I observed for 5 different engines ranged from about 7.2 to 7.3 dV per ISP, including the Nuke and Dawn engines (if you also assume alpha = 8 for them, which is not true).

Edit: As you lower alpha the proportionality factor between stage dV and Isp also lowers, but it is remarkably consistent for a constant alpha. For the dawn with alpha = 1.25 I observed an optimal dV per stage of about 14,500 dV, which is a proportionality factor of 3.46 instead of 7.25.

Edit2: So, I'm not properly accounting for weight in the upper stages, but it should be possible to fix the spreadsheet. However, the assumption of equal dV should likely be equal mass instead as Yasmy points out below. Will revisit this when I can.

Edited June 5, 2015 by cybersol
Upper stage weight not properly accounted for

[Yasmy]

Nice cybersol. And thanks for sending me looking for Meithan's 1.0.2 optimal engine charts. Cool.

I was assuming n_engines = 1, basically ignoring Slashy's requirement for a minimum TWR. Though of course, assuming you start with single stage rocket of sufficient but not outrageous TWR, if you replace it with a multi-stage rocket of lower total mass, then the original number of engines per stage must automatically meet the same minimum TWR requirements.

I would be careful about inferring too much from my one plot. Other setups may produce something very different. Additionally, for the case shown, the mass optimal fuel split is more like 56%/44%. The final stage mass optimal split is a = 50.6% of the fuel of the single stage rocket, but the initial stage has only about b = 39%. a + b must be < 1 to make up for the added engine and decoupler. The plot also shows the per-stage dvs are quite different. They are the dotted lines. At a = 51%, dv1 = 1.02 * v_e, dv2 = 0.35 * v_e.

Edited June 5, 2015 by Yasmy

[GoSlash27]

Let's do some math. This may not exactly be GoSlash27's problem, but it's well defined and tractable.

Assumptions:

1) one engine type with a constant fixed Isp

2) one fuel tank type of fixed full to dry ratio, but of arbitrary size

3) the final stage consists of payload, a fuel tank and an engine

4) a non-final state consists of a fuel tank and engine and a decoupler

5) we start with a known single stage rocket, and wish to know it can be replaced

with a two stage rocket with the same delta-v, but total lower mass.

Variables for the single stage rocket:

p = payload mass

t = dry tank mass

f = fuel mass

e = engine mass

v = engine exhaust velocity = Isp * g0

dv = v ln((p+e+t+f)/(p+e+t))

Now for the two stage rocket, define a and b as variables which scale the single stage rocket fuel tank:

a*f = final stage fuel

b*f = initial stage fuel

d = decoupler mass

dv = v ln((p+e+a*(t+f))/(p+e+a*t)) + v ln((p+2*e+d+(a+*(t+f))/(p+2*e+d+(a+*t+a*f))

Require the two rockets to have the same dv. Fun with logs:

ln(x) = ln(y) + ln(z) = ln(y*z)

x = y*z

Now do a bit of algebra to solve for b, the scale of the initial stage fuel tank, given scale a:

(p+e+t+f)/(p+e+t) = (p+e+a*(t+f))/(p+e+a*t)) * ((p+2*e+d+(a+*(t+f))/(p+2*e+d+(a+*t+a*f))

Solve for b (using a computer algebra system):

b = (1-a)*(p+e)*(p+2*e+d+a*(t+f)) / ((p+e)*(p+e+a*f) + a*(2*p+2*e+f)*t + a*t^2)

What we care about is the ratio of the initial mass of two stage rocket to the single stage rocket:

r = (p+2*e+d+(a+*(t+f)) / (p+e+t+f)

Substitute b, and do a pile of algebra:

r = (p+e+a*t)*(p+2e+d+a*(t+f)) / ((p+e)*(p+e+a*f)+a*(2p+2e+f)+a*t^2)

Plots and thoughts will come later, but here's what you do with this equation:

Pick values of p, e, t, f and d.

Plot r(a) from a = 0 to 1. If r is ever less than 1, you have a better two stage rocket.

Yasmy,

I believe this is *exactly* what I'm looking for!

I'll have to review it more thoroughly.

Also, I think I know why I kept getting my odd result: I was starting from a baseline where the single stage was already optimal.

I was choosing the payload so that one engine was able to meet the requirement in a single stage and then trying to find a lighter staged solution using the same engine. Of course, this means that my upper stage has more engine than it needs, so r will never get under 1. *facepalm*

Thanks about a million!

-Slashy

Edited June 5, 2015 by GoSlash27

[GoSlash27]

All,

I ran a new scenario using 10 engines in the single stage and vacuum Isp. Here's the crossover points by engine type and minimum acceleration:

Type__|0.5G_____|1.0G_____|1.4G

______|_________|_________|_____________

LV-909|4.0km/Sec|3.5Km/Sec|3.4Km/Sec

LV-T30|3.7Km/Sec|3.4Km/Sec|3.2Km/Sec

Above these DV budgets, it's lighter to stage. Below, it's lighter to go single stage.

Of course, this does not take into account changing acceleration requirements, changes in Isp and thrust with density, and engine advantages/ disadvantages in different flight regimes.

Best,

-Slashy