Ellipse Law

The Law of the Ellipse, or Stodola's cone law,[1][2] provides a method for calculating the highly nonlinear dependence of extraction pressures with a flow for multistage turbine with high backpressure, when the turbine nozzles are not choked.[3] It is important in turbine off-design calculations.

Description

Multistage turbine.

Stodola's cone for a turbine not choked.

Stodola's cone for a turbine with the last stage choked.

We consider a multistage turbine, like in the picture. The design calculation is done for design flow rate (${\displaystyle \scriptstyle {\dot {m}}_{0}\,}$, the flow expected for the most uptime). The other parameters for design are the temperature and pressure at the stage group intake, ${\displaystyle \scriptstyle T_{0}\,}$ and ${\displaystyle \scriptstyle p_{0}\,}$, respectively the extraction pressure at the stage group outlet ${\displaystyle \scriptstyle p_{2}\,}$ (the symbol ${\displaystyle \scriptstyle p_{1}\,}$ is used for the pressure after a stage nozzles, pressure does not interfere in relations here).

For off-design calculations, the off-design flow rate is ${\displaystyle \scriptstyle {\dot {m}}_{01}\,}$, respectively the temperature and pressure at the stage group intake are ${\displaystyle \scriptstyle T_{01}\,}$ and ${\displaystyle \scriptstyle p_{01}\,}$ and the outlet pressure is ${\displaystyle \scriptstyle p_{21}\,}$.

Stodola established experimentally that the relationship between these three parameters represented in Cartesian coordinate system has the shape of a degenerate quadric surface, the cone directrix being an ellipse.[4][5] For a constant initial pressure ${\displaystyle \scriptstyle p_{01}\,}$ the flow rate depends on the outlet pressure ${\displaystyle \scriptstyle p_{21}\,}$ as an arc of ellipse in a plane parallel to ${\displaystyle \scriptstyle {\dot {m}}_{01}\,0\,p_{21}\,}$

For very low outlet pressure ${\displaystyle \scriptstyle p_{21}\,}$, like for condensing turbines, flow rates do not change with the outlet pressure, but drops very quickly with the increase of the backpressure. For a given outlet pressure ${\displaystyle \scriptstyle p_{21}\,}$, flow rates changes depending on the inlet pressure ${\displaystyle \scriptstyle p_{01}\,}$ as an arc of hyperbola in a plane parallel to ${\displaystyle \scriptstyle {\dot {m}}_{01}\,0\,p_{01}\,}$.

Usually, Stodola's cone do not represent absolute flow rates and pressures, but relative to the maximum flow rate and pressures, the maximum values of the diagram having in this case the value of 1. The maximum flow rate has the symbol ${\displaystyle \scriptstyle {\dot {m}}_{0m}\,}$ and the maximum pressures at the inlet and outlet have the symbols ${\displaystyle \scriptstyle p_{0m}\,}$ and ${\displaystyle \scriptstyle p_{2m}\,}$. The pressure ratios for design flow rate at the intake and outlet are ${\displaystyle \scriptstyle \epsilon _{0}=p_{0}/p_{0m}\,}$ and ${\displaystyle \scriptstyle \epsilon _{2}=p_{2}/p_{2m}\,}$, and the off-design ratios are ${\displaystyle \scriptstyle \epsilon _{01}=p_{01}/p_{0m}\,}$ and ${\displaystyle \scriptstyle \epsilon _{21}=p_{21}/p_{2m}\,}$.

If the speed of sound is reached in a stage, the group of stages can be analyzed till that stage, which is the last in the group, the remaining stages forming another group of analysis. This division is imposed by the stage working in limited (choked) mode. The cone is shifted in the ${\displaystyle \scriptstyle 0\,p_{02}\,}$ axis direction, appearing a triangular surface, depending on the critical pressure ratio ${\displaystyle \scriptstyle \epsilon _{c}=p_{c}/p_{01}\,}$, where ${\displaystyle \scriptstyle p_{c}\,}$ is the outlet critical pressure of the stage group.[6][7]

The analytical expression of the flow ratio is:[8]

${\displaystyle {\frac {{\dot {m}}_{0}}{{\dot {m}}_{01}}}={\sqrt {\frac {T_{01}}{T_{0}}}}{\sqrt {\frac {\epsilon _{0}^{2}(1-\epsilon _{c})^{2}-(\epsilon _{2}-\epsilon _{c}\epsilon _{0})^{2}}{\epsilon _{01}^{2}(1-\epsilon _{c})^{2}-(\epsilon _{21}-\epsilon _{c}\epsilon _{01})^{2}}}}}$

For condensing turbine the ratio ${\displaystyle \scriptstyle \epsilon _{c}\,}$ is very low, previous relation reduces to:

${\displaystyle {\frac {{\dot {m}}_{0}}{{\dot {m}}_{01}}}={\sqrt {\frac {T_{01}}{T_{0}}}}{\sqrt {\frac {\epsilon _{0}^{2}-\epsilon _{2}^{2}}{\epsilon _{01}^{2}-\epsilon _{21}^{2}}}}}$

simplified relationship obtained theoretically by Gustav Flügel (1885–1967).[8][9]

In the event that the variation of inlet temperature is low, the relationship is simplified:

${\displaystyle {\frac {{\dot {m}}_{0}}{{\dot {m}}_{01}}}={\sqrt {\frac {\epsilon _{0}^{2}-\epsilon _{2}^{2}}{\epsilon _{01}^{2}-\epsilon _{21}^{2}}}}}$

For condensing turbines ${\displaystyle \scriptstyle \epsilon _{2}\,\approx \,\epsilon _{21}\,\approx \,0}$, so in this case:

${\displaystyle {\frac {{\dot {m}}_{0}}{{\dot {m}}_{01}}}={\frac {\epsilon _{0}}{\epsilon _{01}}}={\frac {p_{0}}{p_{01}}}}$

During operation, the above relations allow the assessment of the flow rate depending on the operating pressure of a stage.

References

• (in Romanian) Gavril Creța, Turbine cu abur și cu gaze [Steam and Gas Turbines], București: Ed. Didactică şi Pedagogică, 1981, 2nd ed. Ed. Tehnică, 1996, ISBN 973-31-0965-7
• (in Romanian) Alexander Leyzerovich, Large Steam Power Turbines, Tulsa, Oklahoma: PennWell Publishing Co., 1997, Romanian version, București: Editura AGIR, 2003, ISBN 973-8466-39-3

Further reading

• (in German) Aurel Stodola, Die Dampfturbinen, Berlin: Springer Verlag, 1903–1924 (six editions)
• Aurel Stodola, Steam and Gas Turbines, New York: McGraw-Hill, 1927
• (in German) Constantin Zietemann, Die Dampfturbinen, 2nd ed., Berlin-Göttingen-Heidelberg: Springer-Verlag, 1955
• Walter Traupel, New general theory of multistage axial flow turbomachines. Translated by Dr. C.W. Smith, Washington D.C. Published by Navy Dept.
• Sydney Lawrence Dixon, Fluid Mechanics and Thermodynamics of Turbomachinery, Pergamon Press Ltd., 1966, 2nd ed. 1975, 3rd ed. 1978 (reprinted 1979, 1982 [twice], 1986, 1986, 1989, 1992, 1995), 4th ed. 1998

Notes

1. Timot Veer, Andreas Ulvestad, Olav Bolland, FRAME, a Tool for Predicting Gas Turbine Condition as well as Reability, Availability Performance Archived 2011-07-18 at the Wayback Machine, ASME/IGTI TURBOEXPO Conference 2004 June 14–17, Vienna, Austria, GT-2004-53770
2. TU Delft Cycle-Tempo, Reference guide, tudelft.nl, accessed 2010-11-29
3. D. H. Cooke, On Prediction of Off-Design Multistage Turbine Pressures by Stodola's Ellipse, J. Eng. Gas Turbines Power, July 1985, Volume 107, Issue 3, 596 (11 pages), doi:10.1115/1.3239778
4. Creța, p. 300
5. Leyzerovich, p. 175
6. Creța, p. 301
7. Leyzerovich, p. 176
8. Creța, p. 303
9. Leyzerovich, p. 174

External links

• Modeling of Off-Design Multistage Turbine Pressures by Stodola's Ellipse