A method and at least two devices demonstrate improvements to energy extraction from a compressible working fluid in a liquid ring heat engine, which has a rotor mounted in a case. A space in the case is occupied by a liquid that establishes a liquid ring piston for the rotor. The rotor defines at least a first and a second operating zone. In the first zone, the working fluid is expanded against the liquid and, in the second zone, the working fluid is re-compressed. Between the two zones, the working fluid is cooled. In one device, the cooling step occurs on the rotor in a third zone. In another device, the cooling occurs outside of the case.