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In zero gravity, the liquid in a fuel tank tends to form a blob in a random location, so you'd have to take precautions to make sure the fuel pump can draw fuel. In a recent question about spinning spacecraft, @Erik suggested that spinning can be used to get the fuel in place for an engine start.
But if you spin a spacecraft along its thrust axis, the fuel ends up towards the outside of the craft, or at 90 degrees relative to the thrust line. So when you start the engine, the fuel will move violently towards the engine, potentially (if the fuel tank is less than 50% filled) moving away from the fuel intake.
How is this done in practice?

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Starting liquid-fueled rocket engines in zero-g conditions is a non-trivial problem that should be explicitly addressed during design. There are a number of common approaches:

  1. Use an auxiliary propulsion system to perform what is often called a settling burn. The purpose of this maneuver is to provide just enough thrust to force the propellant towards the sump, or feed-line from the tank(s) to the engine, and the ullage to the opposite end. The trick is ensuring the system provides enough positive thrust for enough time to ensure good wetting of the sump. After ignition, the main engine thrust / acceleration should keep the propellant in the right place. (see below)

    • If there is a reaction or attitude control system on the stage that produces axial thrust this is a good option.
    • The Saturn V used solid ullage motors for a similar purpose.
  2. Maintain positive acceleration during coast. For engines that shut down and then restart (e.g. after an in-space coast phase), a very small amount of thrust can be used, not to settle propellants per se, but to keep them settled. In this case, the thrust can be micro-g levels. The downside is that positive thrust should be maintained for the entire coast, so even though the thrust is small it has to be active for a long time.

  3. Propellant management devices. Non-rocket systems, for example liquid propellant systems on satellites, typically use other methods to keep enough propellant near the sump to get the system started. Often these can be screens or other internal tank features that use capillary or surface tension forces to keep at least some of the liquid from floating away.

  4. Lastly would be if the propulsion system was robust to the having liquid, gas, or two-phase flow enter the engine. This is pretty rare, but if it was the case then the design could theoretically ignore this issue.

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The final position of the liquid fuel and/or oxidizer in a spinning fuel tank depends on the shape of the walls of the tank. if the tank tapers towards the top, the fuel will spin out to the bottom. Google "Erlenmeyer flask", chemistry glassware designed to allow swirling of liquids without the risk of spilling.

I can recall from the early days of the US space program, reference to what was called an "ullage burn". This involves firing a very small rocket (solid or gas fuel). The small thrust encourages the main liquid fuel to settle to the "bottom" of the tank, where the fuel intakes are located.

I remember this particularly since the explanation of the term included getting the last drops out of a wine or whiskey barrel!

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    $\begingroup$ Also there is a wicking approach, where there is a mesh or somesuch that uses surface tension to move enough fuel to the engine to get it started, and once under thrust, the fuel settles towards the intakes. $\endgroup$ – geoffc Oct 17 '13 at 16:29
  • $\begingroup$ Going by the scale model of the ATV I'm currently building, spheres are a popular choice for small fuel tanks. Logical, as the sphere is the strongest and therefore the lightest choice. $\endgroup$ – Hobbes Oct 17 '13 at 16:59
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    $\begingroup$ Nowadays an "ullage burn" its called a "settling burn", and is usually done with smaller hydrazine thrusters to settle the propellant of the main motor to the end it will go to on the main burn which is where the intakes are. Though liquid, the hydrazine system does not itself need a settling burn (Catch-22!), since it has propellant management that uses e.g. surface tension to effectively wick the propellant to the intakes in zero-G. $\endgroup$ – Mark Adler Oct 17 '13 at 21:21
  • $\begingroup$ @MarkAdler are these smaller hydrazine thrusters specifically for this purpose? Or are they usually part of the attitude control system? $\endgroup$ – user29 Oct 18 '13 at 1:34
  • $\begingroup$ In my experience, they also serve the purpose of attitude control thrusters. $\endgroup$ – Mark Adler Oct 18 '13 at 5:21
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If the fuel and oxidizer tanks use a positive-displacement system, such as a piston or bellows, there is no need for a settling or ullage burn. This is the case with many (most?) attitude control thrusters, and even some main propulsion systems.

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