Engine Layouts and Stage Duration

[septemberWaves]

I have experimented with an enormous variety of rocket configurations in my time playing KSP. So far (at least as far as I recall), the best rocket I have ever made in terms of several aspects of its flight profile (various things contributing to the overall delta-v efficiency) involves a total of seventeen identical engines across three stages, with 11, 5, and 1 engines from bottom to top (there are also a pair of SRBs which fire for almost exactly half of the first stage duration). Each stage has a similar burn time (only including the part of the final stage burn that is required to reach orbit; it retains additional delta-v for in-orbit maneuvers). I suspect there is some significance here. I am wondering if anyone knows of particularly good engine layouts when using identical engines on each stage, and of why it might be that such engine layouts work so well.

Edited February 20, 2019 by septemberWaves

[mikegarrison]

The number of engines will depend on the required thrust for each stage, and that depends on how you want to spread your delta-V from stage to stage.

There's probably an optimum ratio, but I think practical considerations like cost and engineering complexity likely keep the real world answer only vaguely similar to the optimum answer.

It's pretty clear that getting to LEO is generally a two-stage process, so you're probably only looking at a single ratio (second stage to first stage).

Both Falcon 9 and New Glenn are being designed at about 17:1 thrust between first stage and second stage. See below. More like 8:1 for Falcon FT and 11:1 for New Glenn.

Edited February 21, 2019 by mikegarrison

[Xd the great]

Well, looking a falcon 9 and electron, I believe that the optimum engine ratio in real life is about 9 to 1.

 The difference can be contributed by the different drag, flight profile.

[mikegarrison]

2 hours ago, Xd the great said:

Well, looking a falcon 9 and electron, I believe that the optimum engine ratio in real life is about 9 to 1.

 The difference can be contributed by the different drag, flight profile.

Yeah, but while SpaceX uses a Merlin engine for the second stage and the first stage, they aren't the same Merlin engines. Each first stage Merlin 1D makes about twice the thrust of the second stage Merlin 1C. See below.

Edited February 21, 2019 by mikegarrison

[sevenperforce]

1 hour ago, mikegarrison said:

Yeah, but while SpaceX uses a Merlin engine for the second stage and the first stage, they aren't the same Merlin engines. Each first stage Merlin 1D makes about twice the thrust of the second stage Merlin 1C.

Second stage uses a Merlin 1D Vacuum at 934 kN; SL thrust on the first stage Merlin 1Ds is 845 kN.

[Cunjo Carl]

I had exactly the same question a couple years back and mathed the heck out of rockets. For the most mass optimal rocket, there's a conveniently simple equation that will calculate the optimal number of engines on each stage relative to the next:

Efficiency = ln(T+(1-T)(E+P)) / ln(P)

Where T is the fuel tank's drymass fraction (1/9 for KSP), E is the desired TWR of your craft divided by the TWR of your engine, and P is the stage's payload fraction (a number between 0 and 1). Putting it in another way, 1/P is how much bigger each stage will be relative to the next, so in your case 1/P is 5 for the smaller two stages. The idea is to plot efficiency versus P (for example by typing the equation into google, replacing 'P' with 'x') and picking the optimal efficiency. This equation works for a case that has one kind of engine, but it can be generalized to include multiple kinds of engines, drop tanks, SRBs, LFBs, and even cost efficiency. The math ramps up pretty quick though!

As for why there's an optimum, we can consider that our rocket has two ways of wasting fuel: 1. is to have too few stages such that we're pushing around empty fuel tanks for too long before dropping them, and 2. is to have too many stages such that we're always pushing around all the heavy extra engines each stage needs. Somewhere between these two effects is the optimum, and if you tell the math all this stuff, you can crunch out what that optimum will be.

Let me know if you'd like more and I'll send it along. Otherwise, happy flying!

Edited February 21, 2019 by Cunjo Carl
Emberassingly remembered the equation wrong. Fixed!

[mikegarrison]

1 hour ago, sevenperforce said:

Second stage uses a Merlin 1D Vacuum at 934 kN; SL thrust on the first stage Merlin 1Ds is 845 kN.

OK, my bad. I was mixing Falcon versions. 1.0, 1.1, FT....

So the thrust ratio for first and second stage is more like 8:1.

And when I look closer at New Glenn, the two upper stage engines are supposed to make 710 kN each, so 1420 kN. The first stage is supposed to be 7x2400 = 16800 kN, making the ratio 11:1.

Edited February 21, 2019 by mikegarrison

[kerbiloid]

If take two-staged ICBMs, they have ~4:1 thrust ratio, and ~1:1 duration ratio.
First stage T/W ~2.

As a civil rocket doesn't need to lift so fast, the 1st stage may have T/W ~1.2, so I:II thrust ratio ~6..7:1

Edited February 21, 2019 by kerbiloid

[magnemoe]

8 hours ago, Cunjo Carl said:

I had exactly the same question a couple years back and mathed the heck out of rockets. For the most mass optimal rocket, there's a conveniently simple equation that will calculate the optimal number of engines on each stage relative to the next:

Efficiency = ln(T+(1-T)(1-E-P)) / ln(P)

Where T is the fuel tank's drymass fraction (1/9 for KSP), E is the desired TWR of your craft divided by the TWR of your engine, and P is the stage's payload fraction (a number between 0 and 1). Putting it in another way, 1/P is how much bigger each stage will be relative to the next, so in your case 1/P is 5 for the smaller two stages. The idea is to plot efficiency versus P (for example by typing the equation into google, replacing 'P' with 'x') and picking the optimal efficiency. This equation works for a case that has one kind of engine, but it can be generalized to include multiple kinds of engines, drop tanks, SRBs, LFBs, and even cost efficiency. The math ramps up pretty quick though!

 As for why there's an optimum, we can consider that our rocket has two ways of wasting fuel: 1. is to have too few stages such that we're pushing around empty fuel tanks for too long before dropping them, and 2. is to have too many stages such that we're always pushing around all the heavy extra engines each stage needs. Somewhere between these two effects is the optimum, and if you tell the math all this stuff, you can crunch out what that optimum will be.

Let me know if you'd like more and I'll send it along. Otherwise, happy flying!

This is thrown off a bit by engine cost is higher than fuel tanks. Also upper stage TWR is far lower than first. This is especially true if you use SRB as you will fire upper stage higher and has more time getting up to speed. 

KSP give a bit of an wrong perspective here as the orbital speed and orbital height is both far lower than on earth so diference between suborbital and orbital speed is less and the time you stay suborbital is also low. 
One tricks in KSP is to burn a bit downward to walk AP ahead of you. This way you can get into orbit without an circulation burn at all. This is also how lots of upper stages RL is launched. 

[septemberWaves]

6 hours ago, magnemoe said:

KSP give a bit of an wrong perspective here as the orbital speed and orbital height is both far lower than on earth so diference between suborbital and orbital speed is less and the time you stay suborbital is also low.

The rocket I referred to in the original post of this thread is actually designed to work with Kerbin at 3.2x the usual scale, with atmosphere height at 1.4x the usual value (so at 98km), and with day length doubled. With around 6000m/s of vacuum delta-v required to reach a 125km parking orbit (with orbital velocity well over 3000m/s), launch vehicles I design with these scale settings perform much more similarly to a real rocket, although they do use a circularization burn since that is simpler with the autopilot (which I use because I get tired of launching everything manually).

[Cunjo Carl]

9 hours ago, magnemoe said:

This is thrown off a bit by engine cost is higher than fuel tanks. Also upper stage TWR is far lower than first. This is especially true if you use SRB as you will fire upper stage higher and has more time getting up to speed.

The equation can be extended to include costs, but grows a bit in size accordingly.

Where dV is deltaV, Ce is the cost per ton of engines, Cf is the cost per ton of fuel (including tanks) and C is the cost per ton you've spent on launching the rocket to where it is right now. We can extend it still further to include all sorts of real life complexities like launch losses (gravity drag) or changing TWRs/engines with still more bits tacked on. It all needs a solver in the end anyways, but it's nice to have the equations like this so you can really see why one option is more efficient than another.

The tricky part about cost is that it changes while you fly. On the ground, fuel is cheap per ton and engines are expensive. But by the time you're landing on the moon the main cost is just the $1M/kg it took to get there and the cost difference of fuel vs engine is marginal. Everything's equally expensive! All of these cases are rolled up in the equation above. More about it in this thread: https://forum.kerbalspaceprogram.com/index.php?/topic/178576-progress-in-theoretical-rocket-cost-optimization/ 

The intuitive explanation is: Cost efficiency _is_ mass efficiency, but with the added wrinkle that if you make your stages too small, you need more engines in total, which're expensive! So to counteract this, early stages are best made a bit larger (big dumb booster) and later stages a bit smaller as cost and mass efficiency eventually aligns.

 

I originally made the cost equation to try proving the supremacy of 2/2.5-stage kerolox to stage-and-a-half SRB/Hydrolox, but despite a really thorough search, I could never find any specifics about real life rocket costs, making the analysis a little tricky ...
Who am I kidding, I just want to see someone launch a Pyrios booster!

Pyrios, the refined and modernized big dumb booster. It's a twin F1 minus the fancy bits!
Picture from Dynetics

Edited February 21, 2019 by Cunjo Carl

[sevenperforce]

20 hours ago, mikegarrison said:

OK, my bad. I was mixing Falcon versions. 1.0, 1.1, FT....

So the thrust ratio for first and second stage is more like 8:1.

And when I look closer at New Glenn, the two upper stage engines are supposed to make 710 kN each, so 1420 kN. The first stage is supposed to be 7x2400 = 16800 kN, making the ratio 11:1.

The two upper stage engines will be BE-3Us at 530 kN each, so 15.8:1.

[mikegarrison]

27 minutes ago, sevenperforce said:

The two upper stage engines will be BE-3Us at 530 kN each, so 15.8:1.

You are starting to make me think Wikipedia isn't always reliable.

[Cunjo Carl]

@sevenperforce Do we have any idea what the mass ratios are between the first and second stages for Falcon 9 & New Glenn?

[Aegolius13]

I'm gonna go out on a limb and say the SLS's transition from the core stage (4 RS-25 / SSMEs giving over 80,000 kN thrust) to the second stage (one RL10 giving ~100 kN of thrust) is not quite the ideal ratio.  

[sevenperforce]

3 hours ago, Cunjo Carl said:

@sevenperforce Do we have any idea what the mass ratios are between the first and second stages for Falcon 9 & New Glenn?

We know the Falcon numbers to a high degree of accuracy. We'd need to estimate for NG based on published imagery and propellant bulk density. Bezos is tight-lipped.

To the OP: there is an optimization function for any proposed mission profile or configuration set, but there are a lot of variables. A vehicle intended primarily for launching BLEO will ten to be underpowered for large LEO payloads, and vice versa. Plus, if you use different propellants on the first and second stages, you get an entirely new set of variables. Recovery also complicates.

 

[FleshJeb]

On 2/20/2019 at 2:51 PM, septemberWaves said:

why it might be that such engine layouts work so well.

In addition to the other really fine answers in this thread, this should help: http://www.projectrho.com/public_html/rocket/multistage.php