dV Payload question

[Stargate525]

This thought occurred to me while I was constructing my all-in-one Duna mission.

Suppose we have a payload of, say, 40 tons. All else being equal (dV needed, positions, ignoring (un)wieldiness of the design), is it cheaper in terms of fuel expenditure to ship one payload of 40 tons, or two payloads of 20? Does this answer depend on the payload size or the dV? Basically, is there a point at which a mission, base, etc, becomes so large it's more efficient to ship it out in parts than it is to do one large burn?

[UmbralRaptor]

In terms of ÃŽâ€V (and therefore fuel), it's equally efficient, whether done as 1 40 tonne mission, or 40 1 tonne missions. The real reason to split up would be the practical problems (unwieldiness, parts breaking, etc) of large launchers being greater than docking.

[Decent Weasel]

Okay, I went futzing around on Wolfram Alpha and formatted

dv = isp*g*ln((m(total)) / (m(full) - m(propellant)))

which is what i have written for the rocket equation

into

dv = isp*g*ln((payload+fuel)/(payload))

assuming 2500 m/s of dv and 800s isp...

20 tons gives me 7.5027 propellant per mission.

40 tons gives me 15.005t propellant.

Yeah, Raptor's right on.

Edited November 9, 2013 by Decent Weasel
delimiter mistake that gave me weird numbers that were terrible

[Stargate525]

That doesn't seem to make sense...

Although, the way I worded it... Right. If you're moving just the payload, and just the fuel. The engines used will matter, correct, as well as the containers? Because in one trip you're moving x engines and y fuel, whereas in the other you're moving 2x engines, which will bring the total 'payload' (of drive engines and miscellanea rather than useful materials) up. Is that right?

The numbers you used, Weasel, make me think that you're using the nukes. If you have one nuke, the fuel to get the 2500dv, and the 40t payload, that will be more efficient than two vessels. In the other scenario you, by necessity, are hauling out 2.25 tons of extra engine. Less efficient, but a better thrust to weight ratio.

[700NitroXpress]

It depends on the size of base you want to get there and the transfer stage used. This is my Duna Command Center base. I lifted it into orbit as one whole part. Then I sent up 5 nuclear engines and fitted them to the base and sent it to Duna. So with something like this, it's easier to piece the engines to it as separate missions, but it would be more efficient to launch it all up as one with the nuclear engines already attached. But my orbital vehicle was simple enough and once you get into higher tonnages, the lifters get more complex to rig. So it really depends on what you want to do. I'm a pro at docking anything in orbit so I found this way to be easier than building a complex lifter because I already had lifters that could lift the things that I had rather than build a new one just for this mission.

The Duna Command Center is about 80 tons or so with full fuel in the landing engines.

Here is the lifter. This configuration got the base into orbit with fuel to spare.

Here is the base after it landed on Duna and deployed.

Here's a video of the mission and the landing:

The video also shows the engines I used and the docking process if it helps your inspiration.

You can download this base here: https://drive.google.com/file/d/0B-vTRL2n8wvzalY4b0VZc0Fsdk0/edit?usp=sharing

[UmbralRaptor]

Although, the way I worded it... Right. If you're moving just the payload, and just the fuel. The engines used will matter, correct, as well as the containers? Because in one trip you're moving x engines and y fuel, whereas in the other you're moving 2x engines, which will bring the total 'payload' (of drive engines and miscellanea rather than useful materials) up. Is that right?

Looks right. Everything in the craft scales up by a factor of 2, so you get the same mass ratio, TWR, etc.

Ignoring how design details can make ascents and maneuvers more/less efficient, ÃŽâ€V requirements for a given mission are always the same no matter what the craft or payload mass is. Design details (available TWR, staging options, control authority, lag, etc) are why you want to choose between one large payload and several small ones.

[arq]

As the others are suggesting, the total dV (and even fuel) cost will be the same, assuming you use the same TWR on each stage. If you put the two together but only add engines for one, it will actually be cheaper because of the lower engine mass (though equivalent to if you had only used half the engines on each payload). dV is agnostic to scale, it depends only on fuel fraction and ISP.

[Seanner]

The discrete nature of engine and fuel selection prevents a perfect launch, but a single law of number theory shows that it's likely better to launch whole:

a | b and a | c -> a | (b+c)

This means that splitting up your launch into sub-payloads that can be lifted efficiently by e.g. a mainsail + orange tank, or whatever that block of engines and fuel is (from here on called a "mainsail") is the same as lifting the whole thing efficiently, since the inefficiency is the amount not divided evenly by mainsails. For example if you can lift 10 to orbit with a mainsail, and your payload is 30, you can just use 3 mainsails in 1 launch or 1 for 10 payload and 2 for 20 payload and it all works out. The contrapositive means that if you *cannot* efficiently lift a payload, there is no choice of sub-payloads that will increase your efficiency! For example, a 24 payload is either:

whole at 24 - 20 for 2 mainsails = 4 over

or

2 launches at 10 + 14 = 1 efficient mission + 4 over on the other

or

2 at 12 each = 2 over on each = 4 total

There's no way to remove inefficiency, however you can make it worse by choosing poorly how you divide up the payload:

1 + 23 = 9 over + 3 under = 12 total inefficiency, presuming the penalty for lifting too little mass is the same as too much

Finally, because it is not a bi-implication, you might be able to lift a certain payload efficiently without necessarily being able to lift its components that way:

whole at 60 for 6 mainsails = efficient

or

2 launches at 24+36 = 4 over and 4 under with 2 and 4 engines respectively for 8 total inefficiency even though the whole thing can be lifted efficiently!

Splitting up your launch may not hurt, but definitely cannot help, and it *may* hurt. If you can handle the part counts, don't split it up.