1984 · 49 citations · 0 references
EngineeringMechanical EngineeringAerodynamicsComputational Heat TransferFluid DynamicsEngine Performance CharacteristicsNozzle FlowBackpressureBackpressure AnalysisBackpressure EffectsCogenerationNozzlesTurbine DesignTurbine PerformanceTurbines
Extraction pressures in high‑backpressure multistage turbines vary nonlinearly with flow, making the Constant Flow Coefficient method unsuitable for off‑design loads, and the Law of the Ellipse—developed experimentally by Stodola in 1927—provides a more general framework. The study derives a generalized nozzle analogy, the Law of the Ellipse, to predict extraction pressures in multistage turbines under off‑design conditions. The authors provide a graphic interpretation of the Law of the Ellipse, derive codable relations for controlled and uncontrolled expansions—including sonic choking and reduced stages—and propose solution methods integrated with heat‑balance iterations, validated by practical experience. The generalized Law of the Ellipse subsumes the Constant Flow Coefficient method and yields pressure predictions that agree well with manufacturer data for both high‑backpressure cogeneration turbines and low‑pressure utility ends.
The variation of extraction pressures with flow to the following stage for high backpressure, multistage turbine designs is highly nonlinear in typical cogeneration applications where the turbine nozzles are not choked. Consequently, the linear method based on Constant Flow Coefficient, which is applicable for uncontrolled expansion with high vacuum exhaust, as is common in utility power cycles, cannot be used to predict extraction pressures at off-design loads. The paper presents schematic examples and brief descriptions of cogeneration designs, with background and theoretical derivation of a more generalized “nozzle analogy” which is applicable in these cases. This method is known as the Law of the Ellipse. It was originally developed experimentally by Stodola and published in English in 1927. The paper shows that the Constant Flow Coefficient method is really a special case of the more generalized Law of the Ellipse. Graphic interpretation of the Law of the Ellipse for controlled and uncontrolled expansions, and variations for sonic choking and reduced number of stages (including single stage) are presented. The derived relations are given in computer codable form, and methods of solution integral with overall heat balance iteration schemes are suggested, with successful practical experience. The pressures predicted by the relations compare favorably with manufacturers’ data on four high-backpressure, cogeneration cycle turbines and three large utility low-pressure ends.