Research Papers: Gas Turbines: Turbomachinery

Enhancement to the Traditional Ellipse Law for More Accurate Modeling of a Turbine With a Finite Number of Stages

[+] Author and Article Information
W. F. Fuls

Department of Mechanical Engineering,
Eskom Power Plant Engineering Institute:
Specialization in Energy Efficiency,
University of Cape Town,
Private Bag X3, Rondebosch,
Cape Town 7701, South Africa
e-mail: wim.fuls@uct.ac.za

Contributed by the Turbomachinery Committee of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received January 9, 2017; final manuscript received June 11, 2017; published online July 19, 2017. Assoc. Editor: Klaus Dobbeling.

J. Eng. Gas Turbines Power 139(11), 112603 (Jul 19, 2017) (12 pages) Paper No: GTP-17-1008; doi: 10.1115/1.4037097 History: Received January 09, 2017; Revised June 11, 2017

This paper studies the origin and applicability of the traditional Stodola ellipse law and demonstrates its deficiencies when applied in certain conditions. It extends the equation by Cooke and Traupel through the definition of a semi-ellipse law. This new law produces more accurate results as compared to the ellipse law (EL), especially for turbines with a low number of stages. It does, however, require knowledge of the choking behavior of the turbine, as well as an appropriate pressure ratio exponent. Through numerical studies and careful application of nozzle flow equations, correlations were developed to predict the critical pressure ratio of a multistage turbine, taking nozzle and blade efficiency into account. Correlations are also presented to obtain an appropriate pressure ratio exponent to use in the semi-ellipse law. A methodology is proposed through which the necessary semi-ellipse law terms can be calculated using only design base conditions and estimates of efficiencies. This was successfully validated on a steam turbine. The semi-ellipse law is believed to be the most accurate way of modeling an axial-flow multistage steam or gas turbine from design base conditions, without requiring a stage-by-stage analysis.

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Fig. 1

Potential error between the ellipse law and nozzle flow if the pressure ratio is fairly large

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Fig. 2

Critical pressure ratio at given inlet conditions for steam. Line b represents the transition when supersaturation is taken into account.

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Fig. 3

Effect of nozzle efficiency on the critical pressure ratio of three common fluids1

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Fig. 4

Nomenclature for a series of nozzles used to derive the critical pressure ratio

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Fig. 5

Pseudo algorithm to iteratively calculate the inlet pressure of a series of ideal stages, assuming the last stage is at choked condition

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Fig. 6

Five-stage impulse steam turbine modeled in flownex using restrictor components and Excel calculations to extract the work from the flow after each stage

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Fig. 7

Variation in nozzle throat area for three different turbine arrangements

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Fig. 8

Normalized mass flow versus pressure ratio for turbines with different number of stages, showing the critical pressure ratio where choking occurs for each

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Fig. 9

Results for critical pressure ratio versus number of stages for a variety of turbine configurations and inlet fluids

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Fig. 10

Choking behavior of a five-stage turbine with a governing stage

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Fig. 11

Pseudo algorithm to iteratively calculate the outlet pressure of a series of ideal stages, assuming the first stage is at choked condition

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Fig. 12

Critical pressure ratios for turbines with a governing stage, showing the numeric result from flownex, as well as results calculated assuming either the first or last stage being the limiting factor

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Fig. 13

Effect of stage efficiency on choking condition as measured using total pressures

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Fig. 14

(Left) Deviation between the semi-ellipse law (31) and the exact normalized nozzle flow (11) for different values of α. (Right) Optimum values of α for a single nozzle with different critical pressure ratios.

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Fig. 15

Pressure ratio exponent values, which produce the best fit for use in the semi-ellipse law for various turbine configurations

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Fig. 16

Validation model results compared with predictions by Stodola's ellipse as well as the semi-ellipse using approximate coefficients




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