Calculating Δv of Asparagus/Onion Booster Staging?

[Checkers]

Does anybody know how to calculate the Δv of a rocket of which the fuel in the boosters are drained into a core stage via the FTX-2 External Fuel Duct?

   

[Laie]

Short answer: You have to calculate the dV for every single stage, then add them all up.

More comprehensive answer:
deltaV = Vex * ln(full mass/dry mass)
where "Vex" is the exhaust velocity, that is, (Engine Isp)*9.81.

The dry mass changes every time when you throw away a spent stage, so you have to calculate the dV for every stage individually:

1. If you use different engine types, you first have find the ISP of your rocket: For every engine, add ISP*Thrust, then divide by the total thrust of the whole vessel.
2. Find the full and dry mass of your rocket stage. That is, "full mass" at the time the stage starts using it's fuel, then subtract the mass of however much fuel you're going to burn in that stage.
3. calculate dV for this stage.

An just by the way, here's my favorite "online" calculator.

 

 

Edited November 10, 2019 by Laie

[Pecan]

@Laie has completely covered it for every rocket you will ever build, not just asparagus.  Except that it's all a bit more complicated for a launch vehicle (everything is).  i) Only your first stage will use sea-level ISP and thrust so if you're trying to be completely accurate ... have fun, no one else is.  ii) Drag, especially with very wide designs (more than one tier around the core), iii) Steering losses on gravity turn

But there are slightly easier ways ...

• Just build it and believe what KSP tells you in the VAB
• Use KER/MJ and believe their estimates
• Make-up a 'dV to orbit' figure, launch your design and add the remaining dV in orbit to the number you first thought of

[Checkers]

I understand the standard equation for Δv, but I was wondering about how to calculate the Δv of a craft where the radially attached boosters and the core booster are firing their engines at the same time, but the fuel is being drained out of the radial boosters and is being pumped into the core booster via the FTX-2 External Fuel Duct so that the core booster is carrying as much fuel as it can at all times.

As you can see in this picture of the Kerbal X, when all the engines are firing, only the outermost booster will have the fuel drained from it.

Is it possible to calculate the Δv of this craft while keeping this factor in mind? (I play on the Enhanced Edition, which is a bit behind in terms of the recent updates to KSP, still calculating Δv by hand.)

Edited November 12, 2019 by Checkers

[AHHans]

On 11/12/2019 at 2:52 AM, Checkers said:

I understand the standard equation for Δv, but I was wondering about how to calculate the Δv of a craft where the radially attached boosters and the core booster are firing their engines at the same time, but the fuel is being drained out of the radial boosters and is being pumped into the core booster via the FTX-2 External Fuel Duct so that the core booster is carrying as much fuel as it can at all times.

The math is the same as if you use a more traditional staging. The rocket equation doesn't care how many engines you use to accelerate your craft, or where each engine gets its fuel from. It only cares about "total mass at start of the stage", "total mass at end of the stage", and "effective velocity with which the lost mass was ejected" (the latter usually being calculated as "average Isp times 9.81 m/s^2"). So calculating the dV for each stage is straight forward (but tedious):

1. record full mass and average Isp at the beginning of stage
2. empty all tanks of stage in editor
3. record empty  mass at the end of stage
4. remove all parts that will be staged away in editor
5. rinse and repeat until you did all stages

In reality there are two complications: one(*) is that the dV calculations of KSP get easily confused when you set up complicated fuel transfer schemes. (It usually does well enough when you use the external fuel duct, but often screws up when you manually change the fuel transfer priority.) The other is that you probably don't manage to (manually) stage away the empty parts at the exact moment when the tanks are empty, so you probably still have some more empty mass with you than the idealized calculations were assuming.

(*) O.K. I wrote that before I remembered that you don't have that feature (yet, hopefully!) in the Enhanced(*cough*) Edition, but I didn't want to delete it.

P.S. If you use engines with different Isp at the same time, then you need to calculate the average Isp, by doing a weighted average over all (simultaneously firing) engines with the rate of fuel flow as the weight.

Edited November 15, 2019 by AHHans
Added P.S.

[Snark]

22 hours ago, Checkers said:

I understand the standard equation for Δv, but I was wondering about how to calculate the Δv of a craft where the radially attached boosters and the core booster are firing their engines at the same time, but the fuel is being drained out of the radial boosters and is being pumped into the core booster via the FTX-2 External Fuel Duct so that the core booster is carrying as much fuel as it can at all times.

The earlier comment from @Laie perfectly covers this, exactly as already stated:

On 11/10/2019 at 8:33 AM, Laie said:

deltaV = Vex * ln(full mass/dry mass)
where "Vex" is the exhaust velocity, that is, (Engine Isp)*9.81.

The dry mass changes every time when you throw away a spent stage, so you have to calculate the dV for every stage individually:

1. If you use different engine types, you first have find the ISP of your rocket: For every engine, add ISP*Thrust, then divide by the total thrust of the whole vessel.
2. Find the full and dry mass of your rocket stage. That is, "full mass" at the time the stage starts using it's fuel, then subtract the mass of however much fuel you're going to burn in that stage.
3. calculate dV for this stage.

Exactly this.

 

However, there is one additional complication.  If you're firing multiple engines at the same time, and those engines have different Isp values, then coming up with the combined Isp of the engines (in order to work out the dV) requires some additional calculations.

If you have some set of engines E₁, E₂, E₃, etc., such that their thrust values are T₁, T₂, T₃, etc. and Isp values are Isp₁, Isp₂, Isp₃, etc., then the combined Isp from firing them all together is calculated thus:

Isp_total = sum(T_n) / sum(T_n/Isp_n)

You can simplify the math by treating a set of identical engines as one big engine.  For example, consider your craft that you pictured that above.  If I read that correctly, you're lifting on:

• one Mainsail
• six Reliants

Let's say you want to work out what your Isp is for that rocket, sitting on the launchpad in atmospheric pressure.

Well, each Reliant has ASL thrust of 205.16 kN, and an Isp of 265 s.  So, you can treat them all together the same as if it were a single engine that's 1230.96 kN, Isp 265 s.

The Mainsail?  At sea level, its thrust is 1379 kN and Isp is 285 s.

So, to find their combined Isp, we just plug into the above equation:

Isp_total = (1230.96 + 1379) / ( (1230.96/265) + (1379/285)) = 275.2.

So for the combination of engines here, the launchpad Isp would be 275.2 s.

[Checkers]

Thank you all! I was way overthinking and probably should have figured a lot earlier what you all were trying to get across to me.

 

So, having all engines firing with fuel being pumped into the central booster is the equivalent (in terms of Δv) of having only the outermost boosters firing and then firing the core booster after the side boosters are decoupled. So the only difference between the two is the thrust of the ship.

Edited November 15, 2019 by Checkers
grammatical mistakes

[Snark]

17 hours ago, Checkers said:

So, having all engines firing with fuel being pumped into the central booster is the equivalent (in terms of Δv) of having only the outermost boosters firing and then firing the core booster after the side boosters are decoupled.

No, because Isp.

The rocket equation (which determines dV) does not care at all about thrust.  It doesn't care how big your engines are, or how many engines you have.  It only cares about the Isp of whatever-it-is that you're firing.

If your central-core engine is firing at the same time as your radial-booster engines, and if it has a different Isp from them, then yes, the fact that it's firing does matter because the Isp of the whole group taken together won't be the same as the radial boosters' Isp, or the same as the central-core Isp.  It'll be somewhere in between, as described above.