Designing a Planetary Gearbox to increase speed?

[Ether]

For a given required mechanical power output at the engine crankshaft, there is a continuum of engine_speed + fuel_flow_rate combinations which will produce that power. Only one of those pairs minimizes the fuel_flow_rate.

[Clem1640]

Engine efficiency is not (necessarily/generally) a function of incomplete fuel combustion. Engines do not provide 100% of the energy produced by burning fuel as useful mechanical energy. The balance of the energy is released as heat.

A modern gasoline engine has an efficiency typically in the range of 25-30%.

Engines operate most efficiently within a certain rpm range. This provides the connection between gear ratios and engine efficiency.