Rocket engines use different cycles to produce thrust. On this site the most common cycles are explained.
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Gas Generator Cycle
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The Gas Generator cycle is the most simple cycle. In this cycle portions from the Main Fuel and oxidizer steram are fed into the preburner. In the preburner fuel and oxidizer are burned, and the hot gases are directed in to the Turbine. The Turbine then drives the fuel and oxidizer pumps. The exhaust of the turbine is dumped overboard, through a small nozlle which also creates a small amount of thrust. The pressurized fuel and oxidizer main streams are led into the combustion chamber and burned. If the engine is regen cooled the fuel flows, beofre entering the combustion chamber, around the nozzle walls. For example the merlin engine of falcon 9 uses this cycle.
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Staged Combustion Cycle
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A more complex cycle but similar to the gas generator cycle is the staged combustion cycle. Here the complete fuel stream flows into the preburner and is burned fuel rich with a protion of the oxidizer. The gases enter then the turbine, which in turn drives the pumps. The  partially burned fuel out of the turbine is not dumped over board but instead directed into the main combustion chamber where it is fully burned with the complete oxidizer flow.
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There are some variations of this cylce, for example instead of the fuel rich preburner also an oxidizer rich preburner can be used. Another subcycle is the full flow staged combustion cycle. Here both oxidizer and fuel a directed into fuel rich and oxidizer rich preburner. This is the most complex cycle with the best performance and efficiency. One advantage is that the propellants are better mixed because they are entering the combustion chamber in gaseous form. Only two companies have succesfully hotfired such an engine, namely spacex with its raptor enigne and stoke space with its nova launcher rocket engine.
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Tap Off Cycle
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The Next cycle is the tapp off cylce. In principal this cylce is reallly simple. Here both fuel and oxidizer enter directly the combustion chamber and are burned. Then a portion from the hot gas inside the combustion chamber is directed in to the turbine, which drives the turbopumps. The exahaust of the turbine is dumped overboard afterwards. This type of engine does not need that many parts but the materials of the components have to withstand really high temperatures. The reaver enginer from firefly aerospace and the BE-3 engine from blue origin are using this cycle.
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Expander Cycle
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Another cycle is the expander cycle. This type of engine is regen cooled, the complete fuel stream flows along the combustion chamber and nozzle walls, heats up and expands. This expanding of the fuel is used to drive turbines, which in turn drives the turbopumps. The expanded fuel is afterwards directed into the combustion chamber and burned with th complete oxidizer stream. In this engine design, the combustion chamber is often extended, so that the fuel gains more heat while running through the cooling pipes. This scheme presents the closed expander cycle, another subcycle is the open expander cycle where only a portion of the expanend fuel is used to drive the turbines and afterwards dumped overboard. An example for this type of engine with an extended combustion chamber is the vinci engine.
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Electric pump fed cycle
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The last engine cycle is the Electric pump fed cycle. In this simple cycle the oxidizer and fuel pumps are driven by an electric motor. Oxidizer and fuel are then burned in the combustion chamber. Rocket labs electron rocket uses this cycle.