The Law of the Ellipse, or Stodola's cone law, is a method for calculating highly nonlinear dependence of extraction pressures with a flow for multistage turbine with high backpressure, when the turbine nozzles are not choked. It is important in turbine off-design calculations.
Description
Stodola's cone law, consider a multistage turbine, like in the picture. The design calculation is done for the design flow rate ( m ˙ 0 {\displaystyle \scriptstyle {\dot {m}}_{0}\,} , the flow expected for the most uptime). The other parameters for design are the temperature and pressure at the stage group intake, T 0 {\displaystyle \scriptstyle T_{0}\,} and p 0 {\displaystyle \scriptstyle p_{0}\,} , respectively the extraction pressure at the stage group outlet p 2 {\displaystyle \scriptstyle p_{2}\,} (the symbol p 1 {\displaystyle \scriptstyle p_{1}\,} is used for the pressure after a stage nozzle; pressure does not interfere in relations here). For off-design calculations, the Stodola's cone law off-design flow rate is m ˙ 01 {\displaystyle \scriptstyle {\dot {m}}_{01}\,} , respectively, the temperature and pressure at the stage group intake are T 01 {\displaystyle \scriptstyle T_{01}\,} and p 01 {\displaystyle \scriptstyle p_{01}\,} and the outlet pressure is p 21 {\displaystyle \scriptstyle p_{21}\,} . Stodola established experimentally that the relationship between these three parameters as represented in the Cartesian coordinate system has the shape of a degenerate quadric surface, the cone directrix being an ellipse. For a constant initial pressure p 01 {\displaystyle \scriptstyle p_{01}\,} the flow rate depends on the outlet pressure p 21 {\displaystyle \scriptstyle p_{21}\,} as an arc of an ellipse in a plane parallel to m ˙ 01 0 p 21 {\displaystyle \scriptstyle {\dot {m}}_{01}\,0\,p_{21}\,}
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