Does fuel feeding make sense?
Imagein a rocket with a central fuel/engine stack and four fuel/engine stacks of the same size and type attached with radial decouplers. Here are two scenarios:
Scenario 1 - The outer four fire together as your first stage. Then they are decoupled and the center one fires as your second stage.
Scenario 2 - The outer four and center one all fire together as your first stage. Fuel ducts are used to feed fuel into the center tank so that when the outer tanks are empty, you can decouple them and continue with just the center engine having full fuel. That is the second stage.
In both cases the second stage consists of one engine and a full tank of fuel. But the first stage is what I'm interested in. Everyone seems to think scenario 2 is the way to do it. But I was thinking about the math and I came up with this: scenario 2 gives you more thrust, but for less time. So you get 5/4 as much thrust, but since the four outer tanks are fueling all five engines your first stage lasts 4/5 as long. So how far does that get you? The distance should be acceleration times time squared. So 5/4 * 4/5 * 4/5, which seems to end up being only 4/5 as far as scenario 1.
Is that right or am I missing somthing? Do you actually get farther with the scenario 1 setup? I can't get to KSP right now to test this experimentally. I'd like to understand the math behind fuel ducting.
Your question is actually about how fast you are going to spend the fuel of your first stage: Are you using four or five engines simultaneously to convert it into speed? When you use all 5 engines but throttle down to 80% thrust, it's exactly the same as if you would use 4 engines at 100% thrust. To answer this question, we first need to talk about what forces affect a rocket during lift-off.
There are two forces which prevent your rocket from getting into orbit.
- Gravity
- Atmospheric Drag
The first, gravity, is accelerating your rocket constantly downward until you have enough horizontal velocity to cancel it (achieved a stable orbit). The more time you spend affected by gravity, the more speed ("delta-v") does the gravity give you which you have to cancel by expending additional fuel. For that reason it is economically to accelerate early, so you get to your orbit faster and consume less acceleration from the gravity.
But there is also the second force: atmospheric drag. Atmospheric drag depends on atmospheric pressure and speed. The relationship with speed is quadratic. The faster you go, the more speed ("delta-V") you lose through air friction. That means it might not be so good after all to go too fast while you are still in the lower atmosphere.
So where do these two factors cancel out?
The ideal speed to balance atmospheric drag and gravity drag (assuming perfectly vertical ascent) is equal to the terminal velocity on the current atmospheric density. That speed depends on how aerodynamic your vehicle is.
When you go faster than this, you are wasting fuel on atmospheric drag. When you go slower than this, you are spending unnecessary fuel to fight gravity.
To get back to your initial question "what's better: serial staging or parallel crossfeed staging": It depends on your total thrust-to-weight ratio in the lower atmosphere. But my general experience is that a rocket gets higher with cross-feeding.
But what when you are already in orbit?
The truth is, it doesn't matter. The amount of delta-v you get per liter of fuel depends on the total weight of the ship and the average fuel-efficiency (Isp) of the engines you use. It doesn't matter how fast (through how many engines) you spend it. All that matters is to avoid having more mass than necessary (get rid of fuel tanks when they are empty). An orbital transfer stage with less engine power is more efficient, only because it tends to have a lower overall mass. This, however, is bought with longer burn-times to get the same amount of delta-v. Longer burn-times can sometimes mean less efficient burns because you can't hit your maneuver nodes that exactly, but this only matters in situations where a burn is very time-critical.