Optimum stage weight ratio?

[mbartelsm]

Is there a simple way to calculate this? considering KSP uses much simpler mechanics than real life (such as only thrust + Isp for engines instead of thrust + Isp + fuel ejection speed + fuel density). The equation's Ive found browsing the net include this data, but KSP doesn't, so I don't know how to do it.

PS: If anyone is wondering why, I'm thinking on picking up my failed attempt at modding the game.

Edited December 18, 2014 by mbartelsm

[Taki117]

There really isn't an optimum stage weight. It all depends on how much dV the stage has and what the TWR is. Other than that weight doesn't matter.

[numerobis]

I've heard that as a rule of thumb, stages with equal Isp should have equal burn times. But I've never quite understood the reasoning.

[mbartelsm]

I'm thinking of a standardized 4500 dV lift vehicle with a 1.5 TWR for the first and second stages and a 1-1.5 TWR for the third stage for X weight (which I've got to figure out). The numbers are not the problem, if I don't have them I'll figure them out, what I need are the formulas to optimize stage size.

[mbartelsm]

I've heard that as a rule of thumb, stages with equal Isp should have equal burn times. But I've never quite understood the reasoning.

If I understood correctly what I found it's because on a spacecraft with equal Isp trough all the stages the dV should be the same for all stages, that is, on a lift vehicle with 3000dV and 3 stages each stage should have 1000dV for an optimum design, since the dV and the isp are the same, so is the burn time for each stage.

Edited December 17, 2014 by mbartelsm

[numerobis]

If I understood correctly what I found it's because on a spacecraft with equal Isp trough all the stages the dV should be the same for all stages, that is, on a lift vehicle with 3000dV and 3 stages each stage should have 1000dV for an optimum design, since the dV and the isp are the same, so is the burn time for each stage.

Yes, but why is this optimal in any sense?

[mbartelsm]

Yes, but why is this optimal in any sense?

There's a whole lot of math behind it, this page explains it nicely with lots of words and few numbers

http://www.projectrho.com/public_html/rocket/multistage.php

[numerobis]

Oh, such a loud web page. But it has a page number in SMAD to look up -- I'll do that tonight!

[magnemoe]

I find the parts limit me here, you also have the recovery issue, SRB+ core who can land at spaceport is an cheap launcher.

If you launch ships with LV-N on the other hand you can save a lot by doing the last 500 m/s with the LV-N and just add extra fuel for it and you only need 4km/s for the launcher.

Same is true for Mun or Minmus missions, if you can use the landers engine and a drop tank for the last part you save a lot.

You want to use the larges SRB as they are give lots of bangs for the buck.

[mbartelsm]

I find the parts limit me here, you also have the recovery issue, SRB+ core who can land at spaceport is an cheap launcher.

If you launch ships with LV-N on the other hand you can save a lot by doing the last 500 m/s with the LV-N and just add extra fuel for it and you only need 4km/s for the launcher.

Same is true for Mun or Minmus missions, if you can use the landers engine and a drop tank for the last part you save a lot.

You want to use the larges SRB as they are give lots of bangs for the buck.

Check the OP , I'm not trying to build a ship, I'm trying to make a mod

[LethalDose]

There's a whole lot of math behind it, this page explains it nicely with lots of words and few numbers

http://www.projectrho.com/public_html/rocket/multistage.php

That page doesn't state all the state that stages should have equal dV contributions... In fact where it's discussing dV fractions at the end, it states ~ 46% dV for a two stage rocket LH2/LOx should be in the first stage.

[mbartelsm]

That page doesn't state all the state that stages should have equal dV contributions... In fact where it's discussing dV fractions at the end, it states ~ 46% dV for a two stage rocket LH2/LOx should be in the first stage.

Allocate a fraction of ÃŽâ€v to each stage

The goal is to divide up the ÃŽâ€v contributions in such a way as to minimize the vehicle mass. But the only way to discover this is by trying different combinations (see step 6 below).

In the special case where each stage has the same inert-mass fractions and the same specific impulse, they will also have the same ÃŽâ€v fraction. The fraction will be 1 / nstage. For example, if there are four stages, each stage will contribute 1/4 = 0.25 = 25% of the total ÃŽâ€v.

The section you are quoting also says a MINIMUM of .46, which means that it can be 0.5, it's also worth noting that all scenarios here are mostly idealized and as such shouldn't be taken literally.

[mbartelsm]

I found a nice table which shows weight ratios for rockets with different amounts of stages (though with constant Isp) here:

http://www.princeton.edu/~stengel/MAE342Lecture2.pdf the chart is on page 8

And, because I know the internet is volatile, here's the table

Required Mass Ratios

[table=width: 800]

[tr]

[td]Ideal dV (km/s)[/td]

[td]Single stage 240s[/td]

[td]Single stage 400s[/td]

[td]Two stages 240s[/td]

[td]Two stages 400s[/td]

[td]Three stages 240s[/td]

[td]Three stages 400s[/td]

[/tr]

[tr]

[td]7[/td]

[td]19.6[/td]

[td]6.0[/td]

[td]4.4[/td]

[td]2.4[/td]

[td]2.7[/td]

[td]1.8[/td]

[/tr]

[tr]

[td]8[/td]

[td]29.9[/td]

[td]7.7[/td]

[td]5.5[/td]

[td]2.8[/td]

[td]3.1[/td]

[td]2.0[/td]

[/tr]

[tr]

[td]9[/td]

[td]45.7[/td]

[td]9.9[/td]

[td]6.8[/td]

[td]3.1[/td]

[td]3.6[/td]

[td]2.1[/td]

[/tr]

[tr]

[td]10[/td]

[td]69.9[/td]

[td]12.8[/td]

[td]8.4[/td]

[td]3.6[/td]

[td]4.1[/td]

[td]2.3[/td]

[/tr]

[tr]

[td]11[/td]

[td]106.9[/td]

[td]16.5[/td]

[td]10.3[/td]

[td]4.1[/td]

[td]4.7[/td]

[td]2.5[/td]

[/tr]

[tr]

[td]12[/td]

[td]163.5[/td]

[td]21.3[/td]

[td]12.8[/td]

[td]4.6[/td]

[td]5.5[/td]

[td]2.8[/td]

[/tr]

[/table]

Edited December 17, 2014 by mbartelsm

[LethalDose]

I found a nice table which shows weight ratios for rockets with different amounts of stages (though with constant Isp) here:

http://www.princeton.edu/~stengel/MAE342Lecture2.pdf the chart is on page 8

And, because I know the internet is volatile, here's the table

Required Mass Ratios

Table removed for brevity

Okay, I think we had a misunderstanding somewhere, this helps clear it up.

The table you're providing appears be giving minimum mass ratios required, per stage, to get X stages into orbit with various ISPs.

I say "appears" because I wasn't sitting in the guy's lecture. That's what I hate about presentation files on the internet, its not even half the story. Regardless, let's assume that what the table is stating.

Anyway, those are minimum mass ratios, not optimum mass ratios. Hence, it can't be use support a statement like:

If I understood correctly what I found it's because on a spacecraft with equal Isp trough all the stages the dV should be the same for all stages, that is, on a lift vehicle with 3000dV and 3 stages each stage should have 1000dV for an optimum design, since the dV and the isp are the same, so is the burn time for each stage.

(emphasis mine)

I think it's interesting thinking about it: If all the stages did have the minimum mass ratio, they would all have the same dV. But that's not the point. Onto your other statement:

The section you are quoting also says a MINIMUM of .46, which means that it can be 0.5, it's also worth noting that all scenarios here are mostly idealized and as such shouldn't be taken literally.

I believe you're misquoting or misunderstand the site. This is the image from that site the information is taken from

They're looking for a fractional dV value for the first stage (f1) to minimize initial launch mass, which is not equivalent to a "minimum optimal dV fraction". The same chart shows if you increase the first stage's fraction of dV, the initial craft mass increases, meaning fractional dV at 0.5 is not optimal, as your interpretation states. In fact, if your interpretation was correct, you could set f1=1.0, 1 and still be optimal.

That same site also states that finding optimal dV fractions for 3 or more stages is, and I'm quoting here:

"a major pain in the gluteus maximus"

If the optimum design was for all 3 stages to have the same dV (f1 = f2 = f3 = 1/3 dV), this wouldn't be difficult at all.

And, okay, they're are a lot of assumptions made in these, but that doesn't mean that you can interpret them to say something else.

I really just think the "every stage should have equivalent dV" is just a myth that keeps getting perpetuated. IMO, the evidence you've cited doesn't support that myth, and, in fact, actually provides some contrary evidence.

Edited December 17, 2014 by LethalDose

[Jouni]

Let's assume that we have rocket stages with fuel mass fraction 80%, a constant Isp, and thrust that scales with stage mass. To launch a payload of mass 1, we could use an upper stage of mass 2 and a lower stage of mass 6. In this rocket, the mass ratio would be 15/7 for both stages.

Now try to transfer amount x of mass from the lower stage to the upper stage. After this transfer, the total delta-v becomes

g0 * Isp * (ln (9/(4.2+4x/5)) + ln ((3+x)/(1.4+x/5))),

which has its global maximum at x = 0. The original rocket was already optimal.

Both stages have the same Isp, the same mass ratio, and the same fuel mass fraction, so they produce the same amount of delta-v. Because engine thrust also scales with stage mass, both stages also have the same burn time.

This generalizes to any fuel mass fraction and any number of stages, as long as the stages are similar to each other. If different stages have different mass fractions or specific impulses, the result will probably be different.

Edited December 17, 2014 by Jouni
Missing parentheses

[OhioBob]

I've heard that as a rule of thumb, stages with equal Isp should have equal burn times. But I've never quite understood the reasoning.

The rules of thumb are,

1. Stages with higher I_sp should be above stages with lower I_sp.

2. More ÃŽâ€V should be provided by the stages with the higher I_sp.

3. Each succeeding stage should be smaller than its predecessor.

4. Similar stages should provide the same ÃŽâ€V.

[mbartelsm]

Okay, I think we had a misunderstanding somewhere, this helps clear it up.

The table you're providing appears be giving minimum mass ratios required, per stage, to get X stages into orbit with various ISPs.

I say "appears" because I wasn't sitting in the guy's lecture. That's what I hate about presentation files on the internet, its not even half the story. Regardless, let's assume that what the table is stating.

Anyway, those are minimum mass ratios, not optimum mass ratios. Hence, it can't be use support a statement like:

(emphasis mine)

I think it's interesting thinking about it: If all the stages did have the minimum mass ratio, they would all have the same dV. But that's not the point. Onto your other statement:

I believe you're misquoting or misunderstand the site. This is the image from that site the information is taken from

http://www.projectrho.com/public_html/rocket/images/multistage/stageGraph.jpg

They're looking for a fractional dV value for the first stage (f1) to minimize initial launch mass, which is not equivalent to a "minimum optimal dV fraction". The same chart shows if you increase the first stage's fraction of dV, the initial craft mass increases, meaning fractional dV at 0.5 is not optimal, as your interpretation states. In fact, if your interpretation was correct, you could set f1=1.0, 1 and still be optimal.

That same site also states that finding optimal dV fractions for 3 or more stages is, and I'm quoting here:

"a major pain in the gluteus maximus"

If the optimum design was for all 3 stages to have the same dV (f1 = f2 = f3 = 1/3 dV), this wouldn't be difficult at all.

And, okay, they're are a lot of assumptions made in these, but that doesn't mean that you can interpret them to say something else.

I really just think the "every stage should have equivalent dV" is just a myth that keeps getting perpetuated. IMO, the evidence you've cited doesn't support that myth, and, in fact, actually provides some contrary evidence.

Thanks for clearing that up, that is why I chose to post here and not just take what I found for granted

Also, with the help of reddit I found this document http://www.dtic.mil/dtic/tr/fulltext/u2/607412.pdf which helps clear any misunderstandings regarding "major pain in the ass"

Let's assume that we have rocket stages with fuel mass fraction 80%, a constant Isp, and thrust that scales with stage mass. To launch a payload of mass 1, we could use an upper stage of mass 2 and a lower stage of mass 6. In this rocket, the mass ratio would be 15/7 for both stages.

Now try to transfer amount x of mass from the lower stage to the upper stage. After this transfer, the total delta-v becomes

g0 * Isp * ln (9/(4.2+4x/5)) + ln ((3+x)/(1.4+x/5)),

which has its global maximum at x = 0. The original rocket was already optimal.

Both stages have the same Isp, the same mass ratio, and the same fuel mass fraction, so they produce the same amount of delta-v. Because engine thrust also scales with stage mass, both stages also have the same burn time.

This generalizes to any fuel mass fraction and any number of stages, as long as the stages are similar to each other. If different stages have different mass fractions or specific impulses, the result will probably be different.

So what you are saying is that an optimal ideal rocket would have the same mass ratios and Isp across stages, what I didn't quite understand with stage mass is, when you say stage mass do you refer to the mass of the entire remaining rocket on x stage? or the mass of the staged part alone?

Edited December 17, 2014 by mbartelsm

[Jouni]

So what you are saying is that an optimal ideal rocket would have the same mass ratios and Isp across stages, what I didn't quite understand with stage mass is, when you say stage mass do you refer to the mass of the entire remaining rocket on x stage? or the mass of the staged part alone?

If all rocket stages were identical (except for their size), then the optimal vertically staged rocket would get the same amount of delta-v from each stage. If upper stages are better than lower stages (due to having a better combination of Isp and fuel mass fraction), then the optimal rocket gets more delta-v from the upper stages than from the lower stages.

Stage mass means only the mass of the stage itself, ignoring all further stages and the payload. If engine thrust depended on the mass of the remaining rocket instead of the mass of the stage itself, the stages would not be identical, and the optimal rocket would be different.

[thereaverofdarkness]

I try to make each stage on average 3x the mass of everything above it, going no higher than 5x the mass, and no lower than twice the mass.