Stodola's cone law

File:Zoelly turbine, section (Heat Engines, 1913).jpg

File:Stodola's cone not choked.svg

File:Stodola's cone choked.svg

Stodola's cone law, consider a multistage turbine, like in the picture. The design calculation is done for the design flow rate ($\backslash scriptstyle\; \backslash dot\; m\_0\; \backslash ,$, the flow expected for the most uptime). The other parameters for design are the temperature and pressure at the stage group intake, $\backslash scriptstyle\; T\_0\; \backslash ,$ and $\backslash scriptstyle\; p\_0\; \backslash ,$, respectively the extraction pressure at the stage group outlet $\backslash scriptstyle\; p\_2\; \backslash ,$ (the symbol $\backslash scriptstyle\; p\_1\; \backslash ,$ 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 $\backslash scriptstyle\; \backslash dot\; m\_\{01\}\; \backslash ,$, respectively, the temperature and pressure at the stage group intake are $\backslash scriptstyle\; T\_\{01\}\; \backslash ,$ and $\backslash scriptstyle\; p\_\{01\}\; \backslash ,$ and the outlet pressure is $\backslash scriptstyle\; p\_\{21\}\; \backslash ,$.

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.Creța, p. 300Leyzerovich, p. 175 For a constant initial pressure $\backslash scriptstyle\; p\_\{01\}\; \backslash ,$ the flow rate depends on the outlet pressure $\backslash scriptstyle\; p\_\{21\}\; \backslash ,$ as an arc of an ellipse in a plane parallel to $\backslash scriptstyle\; \backslash dot\; m\_\{01\}\; \backslash ,0\backslash ,\; p\_\{21\}\; \backslash ,$

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

Usually, Stodola's cone does not represent absolute flow rates and pressures, but rather maximum flow rates and pressures, with the maximum values of the diagram having in this case the value of 1. The maximum flow rate has the symbol $\backslash scriptstyle\; \backslash dot\; m\_\{0m\}\; \backslash ,$ and the maximum pressures at the inlet and outlet have the symbols $\backslash scriptstyle\; p\_\{0m\}\; \backslash ,$ and $\backslash scriptstyle\; p\_\{2m\}\; \backslash ,$. The pressure ratios for the design flow rate at the intake and outlet are $\backslash scriptstyle\; \backslash epsilon\_0\; =\; p\_0\; /\; p\_\{0m\}\; \backslash ,$ and $\backslash scriptstyle\; \backslash epsilon\_2\; =\; p\_2\; /\; p\_\{2m\}\; \backslash ,$, and the off-design ratios are $\backslash scriptstyle\; \backslash epsilon\_\{01\}\; =\; p\_\{01\}\; /\; p\_\{0m\}\; \backslash ,$ and $\backslash scriptstyle\; \backslash epsilon\_\{21\}\; =\; p\_\{21\}\; /\; p\_\{2m\}\; \backslash ,$.

If the speed of sound is reached in a stage, the group of stages can be analyzed until that stage, which is the last in the group, with 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 $\backslash scriptstyle\; 0\backslash ,\; p\_\{02\}\; \backslash ,$ axis direction, appearing as a triangular surface, depending on the critical pressure ratio $\backslash scriptstyle\; \backslash epsilon\_c\; =\; p\_c\; /\; p\_\{01\}\; \backslash ,$, where $\backslash scriptstyle\; p\_c\; \backslash ,$ is the outlet critical pressure of the stage group.Creța, p. 301Leyzerovich, p. 176

The analytical expression of the flow ratio is:Creța, p. 303

:$\backslash frac\{\backslash dot\; m\_0\}\{\backslash dot\; m\_\{01\}\}\; =\; \backslash sqrt\{\backslash frac\{T\_\{01\}\}\{T\_0\}\}\; \backslash sqrt\{\backslash frac\; \{\backslash epsilon\_0^2\; (1\; -\; \backslash epsilon\_c)^2\; -\; (\backslash epsilon\_2\; -\; \backslash epsilon\_c\; \backslash epsilon\_0)^2\}\{\backslash epsilon\_\{01\}^2\; (1\; -\; \backslash epsilon\_c)^2\; -\; (\backslash epsilon\_\{21\}\; -\; \backslash epsilon\_c\; \backslash epsilon\_\{01\})^2\}\}$

For condensing turbine the ratio $\backslash scriptstyle\; \backslash epsilon\_c\; \backslash ,$ is very low, previous relation reduces to:

:$\backslash frac\{\backslash dot\; m\_0\}\{\backslash dot\; m\_\{01\}\}\; =\; \backslash sqrt\{\backslash frac\{T\_\{01\}\}\{T\_0\}\}\; \backslash sqrt\{\backslash frac\; \{\backslash epsilon\_0^2\; -\; \backslash epsilon\_2^2\}\{\backslash epsilon\_\{01\}^2\; -\; \backslash epsilon\_\{21\}^2\}\}$

simplified relationship obtained theoretically by Gustav Flügel (1885–1967).Leyzerovich, p. 174

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

:$\backslash frac\{\backslash dot\; m\_0\}\{\backslash dot\; m\_\{01\}\}\; =\; \backslash sqrt\{\backslash frac\{\backslash epsilon\_0^2\; -\; \backslash epsilon\_2^2\}\{\backslash epsilon\_\{01\}^2\; -\; \backslash epsilon\_\{21\}^2\}\}$

For condensing turbines $\backslash scriptstyle\; \backslash epsilon\_2\; \backslash ,\; \backslash approx\; \backslash ,\; \backslash epsilon\_\{21\}\; \backslash ,\; \backslash approx\; \backslash ,\; 0$, so in this case:

:$\backslash frac\{\backslash dot\; m\_0\}\{\backslash dot\; m\_\{01\}\}\; =\; \backslash frac\{\backslash epsilon\_0\}\{\backslash epsilon\_\{01\}\}\; =\; \backslash frac\{p\_0\}\{p\_\{01\}\}$

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

• {{in lang|ro}} 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 lang|ro}} Alexander Leyzerovich, Large Steam Power Turbines, Tulsa, Oklahoma: PennWell Publishing Co., 1997, Romanian version, București: Editura AGIR, 2003, {{ISBN|973-8466-39-3}}

• {{in lang|de}} Aurel Stodola, Die Dampfturbinen, Berlin: Springer Verlag, 1903–1924 (six editions)
• Aurel Stodola, Steam and Gas Turbines, New York: McGraw-Hill, 1927
• {{in lang|de}} 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

{{Reflist}}

• [http://famos.scientech.us/Papers/1983/1983section11.pdf Modeling of Off-Design Multistage Turbine Pressures by Stodola's Ellipse] {{Webarchive|url=https://web.archive.org/web/20100705074219/http://famos.scientech.us/Papers/1983/1983section11.pdf |date=2010-07-05 }}

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