Parallel Computation of Turbine Blade Clocking

Paul G. A. Cizmas

Department of Aerospace Engineering

Texas A&M University

College Station, Texas 77843-3141

Daniel J. Dorney

Department of Mechanical Engineering

Virginia Commonwealth University

Richmond, Virginia 23284-3015

ABSTRACT

This paper presents a numerical study of airfoil clocking of a six-row test turbine configuration withequal pitches. Since the rotor-stator interaction flow is highly unsteady, the numerical simulation ofairfoil clocking requires the use of time marching methods, which can be computationally expensive.The large turnaround time and the associated cost for such simulations makes it unacceptable forthe turbomachinery design process. To reduce the turnaround time and cost/MFLOP, a parallelcode based on Message-Passing Interface libraries was developed. The relative circumferentialpositions of the three stator and three rotor rows in an industrial steam turbine were varied toincrease turbine efficiency. A grid density study was performed to verify the grid independence ofthe computed solutions. The clocking of the second-stage airfoils gave approximately a 50% greaterefficiency variation than the clocking of the third-stage airfoils. This was true for clocking bothrotor and stator airfoils. Rotor clocking produces an efficiency variation which is approximatelytwice the efficiency variation produced by stator clocking. For both stator and rotor clocking, themaximum efficiency is obtained when the wake impinges on the leading edge of the clocked airfoil.

NOMENCLATURE

p PressureT Temperatureη Efficiencyγ Ratio of specific heats of a gas

SUBSCRIPTS

ca Circumferential-averagedt − t Total-to-totalta Time-averaged

1

SUPERSCRIPTS

∗ Total (or stagnation)

INTRODUCTION

The requirement to further increase performance and improve reliability in turbomachinery hasmotivated designers to better understand unsteady effects. An important part of the unsteady ef-fects in turbomachinery results from the rotor-stator interaction. The main sources of unsteadinesspresent in the rotor-stator interaction are potential flow interaction and wake interaction. Ad-ditional sources of unsteadiness include vortex shedding, hot streak interaction, shock/boundarylayer interaction, and flutter.

Potential flow interaction is a purely inviscid interaction due to the pressure variation caused bythe relative movement of the blades and vanes. Potential flow interaction mainly affects adjacentairfoil rows. Wake interaction is the unsteadiness generated by the vortical and entropic wakesshed by one or more upstream rows. These wakes interact with the downstream airfoils and otherwakes. Wake interaction is the primary contributor to unsteady forces on the blade for a largerotor-stator gap.

The process of varying the circumferential relative position of consecutive stator airfoils isreferred to as airfoil “indexing” or “clocking”. Consecutive rotor airfoils can be clocked as well. Theeffects of airfoil clocking on compressor performance have been investigated both experimentally andnumerically. Capece [1] was among the first to show the potential performance benefits of clocking.Saren et al. [2, 3, 4] have used theoretical, experimental and computational techniques to show thatairfoil clocking can reduce unsteady forces on airfoil and increase compressor performance. Hsuand Wo [5], by clocking the downstream rotor, have experimentally shown a reduction of statorunsteady loading. Barankiewicz and Hathaway’s experimental investigation of stator row clockingon a four-stage axial compressor showed a change in overall performance of about 0.2% [6]. Theexperimental investigation of Walker et al. studied the effects of inlet guide vanes on the boundarylayer quantities and losses of a downstream stator row [7]. However, no firm conclusion could bedrawn about the stator losses since the observed variation in losses was comparable in magnitudeto the uncertainty in the data. The numerical results reported by Gundy-Burlet and Dorney [8, 9]for a 2-1

2stage compressor predict efficiency variations between 0.5% and 0.8%, as a function of

stator clocking position.For turbines, the effects of airfoil clocking have been experimentally investigated by Huber et

al. [10]. The experimental results showed a 0.8% efficiency variation due to clocking. For thesame turbine, a two-dimensional numerical analysis for the midspan geometry by Griffin et al. [11]correctly predicted the maximum efficiency clocking positions. However, the predicted efficiencyvariation was only 0.5%. Clocking effects in a 1-1

2stage turbine have also been numerically simulated

by Eulitz et al. [12] and Dorney and Sharma [13]. In all these analyses, the highest efficienciesoccurred when the first-stage stator wake impinged on the leading edge of the second-stage stator,while the lowest efficiencies were observed when the first-stage stator wake was convected throughthe middle of the second-stage stator passage.

The focus of the current investigation has been to study the effects of “fully clocking” a three-stage industrial steam turbine. This paper presents for the first time the effects of clocking rotorrows, including the effects of clocking three rotor rows. This is also the first time that the effectsof clocking three stator rows are presented. Finally, this paper shows the cumulative benefits ofsimultaneously clocking rotor and stator rows.

NUMERICAL MODEL

2

The computer code used to simulate the flow in the turbine is presented in detail in [14]. Thenumerical approach used in the code is based on the work done by Rai [15]. The code was developedas a parallel version of the STAGE-2 analysis, which was originally developed at NASA AmesResearch Center. The numerical approach is briefly described here.

The quasi three-dimensional, unsteady, compressible flow through a multistage axial turboma-chine with arbitrary blade counts is modeled by using the Navier-Stokes and Euler equations. Thecomputational domain associated with each airfoil is divided into an inner region, near the airfoil,and an outer region, away from the airfoil. The thin-layer Navier-Stokes equations are solved inthe regions near the airfoil, where viscous effects are strong. Euler equations are solved in the outerregion, where the viscous effects are weak. The flow is assumed to be fully turbulent. The eddyviscosity is computed using the Baldwin-Lomax model and the kinematic viscosity is computedusing Sutherland’s law.

The Navier-Stokes and Euler equations are written in the strong conservation form. The fullyimplicit, finite-difference approximation is solved iteratively at each time level, using an approximatefactorization method. Two Newton-Raphson sub-iterations are used to reduce the linearizationand factorization errors at each time step. The convective terms are evaluated using a third-orderaccurate upwind-biased Roe scheme. The viscous terms are evaluated using second-order accuratecentral differences and the scheme is second-order accurate in time.

Grid Generation

Two types of grids are used to discretize the flow field surrounding the rotating and stationarygrids. An O-grid is used to resolve the Navier-Stokes equations near the airfoil, where the viscouseffects are important. An H-grid is used to discretize the Euler equations away from the airfoil.The O-grid is generated using an elliptical method. The H-grid is algebraically generated. The O-and H-grids are overlaid. The flow variables are communicated between the O- and H-grids throughbilinear interpolation. The H-grids corresponding to consecutive rotors and stators are allowed toslip past each other to simulate the relative motion.

Boundary Conditions

Since multiple grids are used to discretize the Navier-Stokes and Euler equations, two classesof boundary conditions must be enforced on the grid boundaries: natural boundary conditions andzonal boundary conditions. The natural boundaries include inlet, outlet, periodic and the airfoilsurfaces. The zonal boundaries include the patched and overlaid boundaries.

The inlet boundary conditions include the specification of flow angle, average total pressureand downstream propagating Riemann invariant. The upstream propagating Riemann invariant isextrapolated from the interior of the domain. At the outlet, the average static pressure is specified,while the downstream propagating Riemann invariant, circumferential velocity, and entropy areextrapolated from the interior of the domain. Periodicity is enforced by matching flow conditionsbetween the lower surface of the lowest H-grid of a row and the upper surface of the top mostH-grid of the same row. At the airfoil surface, the following boundary conditions are enforced: the“no slip” condition, the adiabatic wall condition, and the zero normal pressure gradient condition.

For the zonal boundary conditions of the overlaid boundaries, data is transferred from the H-grid to the O-grid along the O-grid’s outermost grid line. Data is then transferred back to theH-grid along its inner boundary. At the end of each iteration, an explicit, corrective, interpolationprocedure is performed. The patch boundaries are treated similarly, using linear interpolation toupdate data between adjoining grids [16].

3

Parallel Computation

The parallel code uses Message-Passing Interface (MPI) libraries and runs on symmetric multi-processors (e.g., Silicon Graphics Challenge) and massively parallel processors (e.g., Cray T3E).The quasi three-dimensional parallel code was developed such that it could be easily extended to athree-dimensional parallel version. As a result, one processor was allocated for each airfoil in thetwo-dimensional simulation. Consequently, the number of processors necessary for a typical three-dimensional turbomachinery configuration does not exceed the number of processors available ontoday’s computers.

The processor allocation is presented in Figure 1. One processor is allocated for each inlet andoutlet H-grid. One processor is allocated for the O- and H-grids corresponding to each airfoil. Inter-processor communication is used to match boundary conditions between grids. Periodic boundaryconditions are imposed by cyclic communication patterns within rows. Inter-blade-row boundaryconditions are imposed by gather-send receive-broadcast communication routines between adjacentrows. Load imbalance issues need to be considered at grid generation time to reduce synchronizationoverhead.

NUMERICAL RESULTS

The results reported in this section are for a three-stage test turbine. This test turbine has 58first-stage stators, 46 first-stage rotors, 52 second-stage stators, 40 second-stage rotors, 56 third-stage stators and 44 third-stage rotors. A dimensionally accurate simulation of this geometrywould require the modeling of 29 first-stage stators, 23 first-stage rotors, 26 second-stage stators, 20second-stage rotors, 28 third-stage stators and 22 third-stage rotors. To reduce the computationaleffort, it was assumed that there were an equal number of blades (58) in each turbine row. As aresult, the airfoils were rescaled by the factors shown in Table 1. Note that modifying the bladecount represents a form of airfoil clocking.

The effects of clocking a turbine with an equal number of airfoils per row are larger than theeffects of clocking a turbine with a different number of airfoils per row. Consequently, the efficiencyvariation obtained for the airfoil count 1:1:1:1:1:1 will be larger than the efficiency variation forthe airfoil count 29:23:26:20:28:22. However, the focus of this paper is to estimate the relativecontribution of clocking different stator and rotor rows.

The inlet temperature in the test turbine is 293 degrees Kelvin and the inlet Mach number is0.073. The inlet flow angle is 0 degrees and the inlet Reynolds number is 53494 per inch, based onthe axial chord of the first-stage stator. The rotational speed of the test turbine is 2400 RPM.

The results presented in this paper were computed using two Newton sub-iterations per time-step and 3200 time-steps per cycle. Here, a cycle is defined as the time required for a rotor totravel a distance equal to the pitch length at midspan. Since the airfoil count is 1:1:1:1:1:1, theflow repeats after each cycle, i.e. the flow period is equal to the duration of a cycle. To ensuretime-periodicity, each simulation was run in excess of 80 cycles. All time-averaged quantities werecalculated over 20 cycles.

Three computational grids were used to verify that the numerical solutions were grid indepen-dent. The number of grid points per row for the coarse, medium and fine grids are shown in Table2. For a given grid size, the same number of grid points per airfoil were used for all the rows.Details of the grid around the first-stage stator are shown in Fig. 2 for the three different griddensities.

Before verifying that the solution is grid independent, one has to verify that the solution isperiodic. Since the flow in the last row of airfoils is more likely to be the last to become periodic,

4

the flow periodicity will be monitored in this last row. To assess periodicity, the pressure variationon the row-six airfoil is compared for three consecutive cycles, as shown in Fig. 3. Maximum,minimum and time-averaged pressures denote maximum, minimum and averaged pressures over ablade-passing cycle. The close agreement of the pressure values indicates that the solution is not afunction of the cycle number, i.e., the solution is periodic.

To verify that the solution is grid independent, the pressure variation obtained using the threedifferent size grids is compared for the last row airfoil. The results presented in Fig. 4 indicate verygood agreement for the time-averaged pressure and good agreement for the results correspondingto the maximum and minimum pressure variation. The overall good agreement among the resultscorresponding to the three grids gives confidence that the solution is grid independent. To reducethe computational effort, the coarse grid shown in Fig. 5 is used in the remainder of the paper.

The average value of y+, the non-dimensional distance of the first grid point above the surface,is less than 1 for all the rows. Approximately 25 grid points are used to discretize the boundarylayers.

In this analysis, the effects of airfoil clocking are estimated by performing simulations with theclocked stator or rotor located at five different locations equidistantly spaced over one pitch. Figure6 shows the clocking locations for the second-stage stator.

The computations were performed on a twelve-processor SGI Challenge computer. Eight pro-cessors were used for this analysis, as suggested by Fig. 1. The computation time for this simulationwas 6.24x10−6 secs/grid point/iteration.

Instantaneous Mach contours are presented in Fig. 7 in order to visualize the velocity distribu-tion. For the given flow conditions, the maximum Mach number is approximately 0.5 and is locatedon the suction side of the third-stage rotors.

In this investigation, the total-to-total efficiency is defined as [17]:

ηt−t =

(

1 −

T ∗

exit,ca,ta

T ∗

inlet,ca,ta

)

1

1 −

(

p∗exit,ca,ta

p∗inlet,ca,ta

)γ−1

γ

where subscript “ca” denotes circumferential-averaged, “ta” denotes time-averaged, and the super-script “*” denotes total (or stagnation). The efficiency variation as a function of the second-stagestator clocking position is shown in Fig. 8. The time-averaged (over a cycle) entropy contourson the second-stage stators are shown in Figures 9-13. The red contours correspond to the highentropy value and the blue contours correspond to the low entropy value. The maximum efficiencycorresponds to the case in which the wakes impact the stator at the leading edge, slightly shifted tothe pressure side. The minimum efficiency corresponds to the case in which the wake is located inthe passage between the stator airfoils. The correlation between the wake impact and the efficiencyvalue agrees well with the results of previous experimental and numerical studies [8, 9].

A slight increase in entropy can be observed on the suction side of the airfoil, at about 80% of thechord. This slight increase in entropy corresponds to the inner part of the H-grid. The magnitudeof the entropy rise is exaggerated because the entropy levels were restricted, in order to increasethe wake contrast. This local entropy rise does not affect the conclusions of this investigation.

The efficiency variation due to clocking the third-stage stator is shown in Fig. 14. To allowfor easy comparison between the efficiencies corresponding to clocking different rows, the samescale is used for all the efficiency plots. The efficiency variation obtained by clocking the third-stage stator is only 48% of the efficiency variation obtained by clocking the second-stage stator.The clocking of the third-stage stator is done with the second-stage stator clocked for maximumefficiency. Consequently, the maximum efficiency in Fig. 8 is equal to the efficiency at clocking

5

position “0” in Fig. 14. As in the case of clocking the second-stage stator, the maximum efficiencyis obtained when the wake impinges on the leading edge of the stator, as shown in Fig. 15. Theminimum efficiency is obtained when the wake is located in the passage between the stators, asshown in Fig. 16.

The maximum efficiency clocking position for the second-stage stator is modified when theindexing of the third-stage stator is modified. As a result, the maximum efficiency obtained by firstclocking the second-stage stator, and then the third-stage stator, may not be the absolute globalmaximum. To obtain the absolute global maximum efficiency, the clocking of the second-stageshould be repeated with the third-stage stators clocked in clocking position “1”. This iterativeprocess is very computationally intensive and the analysis will be limited here to one iteration only.

During the clocking of the second-stage rotor, the second- and third-stage stators were indexedfor maximum efficiency. The efficiency variation due to clocking the second-stage rotor is presentedin Fig. 17. The efficiency variation in this case is 1.83 times larger than the efficiency variationobtained by clocking the second-stage stator.

Similar to clocking stators, the maximum efficiency is obtained when the wake impinges on theleading edge of the rotor airfoil, as shown in Fig. 18. The minimum efficiency corresponds to thecase when the wake is located in the passage between the rotor airfoils, as shown in Fig. 19.

The clocking of the third-stage rotor is accomplished using the second-stage rotor indexed inposition “2”. The efficiency variation due to clocking the third-stage rotor is shown in Fig. 20.In this case, the efficiency variation is 128% of the efficiency variation obtained by clocking thesecond-stage stator. However, no absolute increase in efficiency is obtained by clocking the third-stage rotor, since the third-stage rotor was already in the optimal position. As in the previousclocking cases, the minimum efficiency corresponds to the case where the wake is located in thepassage between the rotor airfoils, as shown in Fig. 21.

The summary of efficiency variation and efficiency increase is presented in Table 3. The totalincrease in efficiency obtained by clocking the rotors and the stators is 2.45 times larger thanclocking the second-stage stator only. This additional increase in efficiency represents a significantreward for clocking multiple stator and rotor rows.

CONCLUSIONS

A quasi three-dimensional unsteady Euler/Navier-Stokes analysis, based on a parallel code,has been used to investigate the effects of “fully clocking” a three-stage turbine. The effects ofsimultaneously clocking rotor rows, including the effects of clocking three rotor rows, are presentedfor the first time. This is also the first time that the effects of clocking three stator rows have beenpresented.

Previous experimental and numerical investigations have shown that in the case of stator clock-ing, maximum efficiency is obtained when the wake impinges on the leading edge of the clockedstator. The present numerical simulation reconfirmed this observation for stator clocking and ex-tended it for rotor clocking. The fact that the wake impinging on the leading edge produces thehighest efficiency holds true for clocking multiple stator or rotor rows as well.

For the turbine investigated, the clocking of the second-stage gives larger efficiency variationsthan the clocking of the third-stage. This conclusion is true for both rotor and stator clocking.The predicted results also showed that rotor clocking produces an efficiency variation which isapproximately twice the efficiency variation produced by stator clocking.

ACKNOWLEDGMENTS

6

The authors wish to thank the Westinghouse Power Generation and Westinghouse Science & Tech-nology Center for supporting this work. The authors are thankful to Pittsburgh SupercomputingCenter for making the computing resources available. The authors are especially grateful to Dr.Karen Gundy-Burlet of NASA-Ames Research Center for her assistance with code related issuesand interpretation of the physics. The authors would also like to thank Mr. Harry Martin and Dr.Shun Chen for the helpful discussions and suggestions during this project.

References

[1] Capece, V. R., “Forced Response Unsteady Aerodynamics in a Multistage Compressor,” Ph.D.Thesis, Purdue University, West Lafayette, IN, 1987.

[2] Saren, V. E., “Some Ways of Reducing Unsteady Loads Due to Blade Row HydrodynamicInteraction in Axial Flow Turbomachines,” Second International Conference EAHE, Pilsen,Czech Republic, pp. 160-165, 1994.

[3] Saren, V. E., “Relative Position of Two Rows of an Axial Turbomachine and Effects on theAerodynamics in a Row Placed Between Them,” Unsteady Aerodynamics and Aeroelasticityof Turbomachines, Elsevier, pp. 421-425, 1995.

[4] Saren, V. E., Savin, N. M., Dorney, D. J., and Zacharias, R. M., “Experimental and NumericalInvestigation of Unsteady Rotor-Stator Interaction on Axial Compressor Stage (with IGV)Performance,” 8th International Symposium on Unsteady Aerodynamics and Aeroelasticity of

Turbomachines, Stockholm, Sweden, September 1997.

[5] Hso, S. T., and Wo, A. M., “Reduction of Unsteady Blade Loading by Beneficial Use ofVortical and Potential Disturbances in an Axial Compressor with Rotor Clocking,” ASMEPaper 97-GT-86, Orlando, FL, June 1997.

[6] Barankiewicz, W. S., and Hathaway, M. D., “Effects of Stator Indexing on Performance in aLow Speed Multistage Axial Compressor,” ASME Paper 97-GT-496, Orlando, FL, June 1997.

[7] Walker, G. J., Hughes, J. D., Kohler, I., and Solomon, W. J., “The Influence of Wake-WakeInteractions on Loss Fluctuations of a Downstream Axial Compressor Blade Row,” ASMEPaper 97-GT-469, Orlando, FL, June 1997.

[8] Gundy-Burlet, K. L., and Dorney, D. J., “Physics of Airfoil Clocking in Axial Compressors,”ASME Paper 97-GT-444, Orlando, FL, June 1997.

[9] Gundy-Burlet, K. L., and Dorney, D. J., “Investigation of Airfoil Clocking and Inter-BladeRow Gaps in Axial Compressors,” AIAA Paper 97-3008, Seattle, WA, July 1997.

[10] Huber, F. W., Johnson, P. D., Sharma, O. P., Staubach, J. B., and Gaddis, S. W., “Perfor-mance Improvement Through Indexing of Turbine Airfoils: Part1 - Experimental Investiga-tion,” ASME Journal of Turbomachinery 118, 1996, pp. 630-635.

[11] Griffin, L. W., Huber, F. W., Sharma, O. P., “Performance Improvement Through Indexingof Turbine Airfoils: Part2 - Numerical Simulation,” ASME Journal of Turbomachinery 118,1996, pp. 636-642.

[12] Eulitz, F., Engel, K., and Gebbing, H., “Numerical Investigation of the Clocking Effects in aMultistage Turbine,” ASME Paper 96-GT-26, Birmingham, UK, 1996.

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[13] Dorney, D., and Sharma, O. P., “A Study of Turbine Performance Increases Through AirfoilClocking,” AIAA Paper 96-2816, Lake Buena Vista, FL, 1996.

[14] Cizmas, P., and Subramanya, R., “Parallel Computation of Rotor-Stator Interaction”, 8th

International Symposium on Unsteady Aerodynamics and Aeroelasticity of Turbomachines,Stockholm, Sweden, 1997.

[15] Rai, M. M., and Chakravarthy, S., “An Implicit Form for the Osher Upwind Scheme,” AIAA

Journal 24, pp. 735-743, 1986.

[16] Rai, M. M., “Navier-Stokes Simulation of Rotor-Stator Interaction Using Patched and OverlaidGrids,” AIAA Paper 85-1519, Cincinnati, Ohio, 1985.

[17] Lakshminarayana, B., Fluid Dynamics and Heat Transfer of Turbomachinery, Wiley, NewYork, 1996, p. 58.

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Figure 1: Processor allocation.

Airfoil Rescaling Factor

first-stage stator 1first-stage rotor 46/58second-stage stator 52/58second-stage rotor 40/58third-stage stator 56/58third-stage rotor 44/58

Table 1: Airfoil rescaling factors.

Coarse Medium FineGrid Grid Grid

H-grid inlet 75x45 100x60 100x60H-grid airfoil 67x45 90x60 108x72O-grid airfoil 112x37 150x50 180x60H-grid outlet 75x45 100x60 100x60

Total 49704 89400 123456

Table 2: Number of grid points

9

Figure 2: Detail of the coarse, medium and fine grids around the row-one airfoil

10

−9.0

−8.5

−8.0

−7.5

−7.0

−6.5

−6.0

Pre

ssur

e co

effic

ient

, ave

rage

d

Coarse grid, Cycle 19Coarse grid, Cycle 20Coarse grid, Cycle 21

−9.0

−8.5

−8.0

−7.5

−7.0

−6.5

−6.0

Pre

ssur

e co

effic

ient

, max

imum

Coarse grid, Cycle 19Coarse grid, Cycle 20Coarse grid, Cycle 21

7.0 7.2 7.4 7.6 7.8 8.0 8.2 8.4Axial distance, X/Chord

−9.0

−8.5

−8.0

−7.5

−7.0

−6.5

−6.0

Pre

ssur

e co

effic

ient

, min

imum

Coarse grid, Cycle 19Coarse grid, Cycle 20Coarse grid, Cycle 21

Figure 3: Pressure variation on the row-six airfoil during three consecutive cycles

11

−9.0

−8.5

−8.0

−7.5

−7.0

−6.5

−6.0

Pre

ssur

e co

effic

ient

, ave

rage

d

Coarse gridMedium gridFine grid

−9.0

−8.5

−8.0

−7.5

−7.0

−6.5

−6.0

Pre

ssur

e co

effic

ient

, max

imum

Coarse gridMedium gridFine grid

7.0 7.2 7.4 7.6 7.8 8.0 8.2 8.4Axial distance, X/Chord

−9.0

−8.5

−8.0

−7.5

−7.0

−6.5

−6.0

Pre

ssur

e co

effic

ient

, min

imum

Coarse gridMedium gridFine grid

Figure 4: Pressure variation on the coarse, medium and fine grids

12

Figure 5: Computational grid (every other grid point in each direction shown)

01234

Figure 6: Clocking position of second-stage stator airfoils

Figure 7: Instantaneous Mach contour distribution

13

0 1 2 3 4 5Clocking Position

1

1

1

1

1

1

1

1

1

1

1

Effi

cien

cy

Steam turbineClocking stator2

Figure 8: Efficiency variation for clocking the second-stage stator.

14

Figure 9: Entropy variation on second-stage stator, clocking 0.

Figure 10: Entropy variation on second-stage stator, clocking 1.

Figure 11: Entropy variation on second-stage stator, clocking 2.

15

Figure 12: Entropy variation on second-stage stator, clocking 3.

16

Figure 13: Entropy variation on second-stage stator, clocking 4.

0 1 2 3 4 5Clocking position

1

1

1

1

1

1

1

1

1

1

1

Effi

cien

cy

Steam turbineClocking stator3

Figure 14: Efficiency variation for clocking the third-stage stator.

Figure 15: Entropy variation on third-stage stator, clocking 1.

17

Figure 16: Entropy variation on third-stage stator, clocking 3.

0 1 2 3 4 5Clocking

1

1

1

1

1

1

1

1

1

1

1

Effi

cien

cy

Steam TurbineClocking rotor2

Figure 17: Efficiency variation for clocking of second-stage rotor.

Figure 18: Entropy variation on second-stage rotor, clocking 2.

18

Figure 19: Entropy variation on second-stage rotor, clocking 4.

0 1 2 3 4 5Clocking position

1

1

1

1

1

1

1

1

1

1

1

Effi

cien

cy

Steam turbineClocking rotor3

Figure 20: Efficiency variation for clocking the third-stage rotor.

19

Figure 21: Entropy variation on third-stage rotor, clocking 2.

Efficiency EfficiencyClocking Variation Increase

second-stage stator 100% 100%third-stage stator 48% 114%second-stage rotor 183% 245%third-stage rotor 128% 245%

Table 3: Efficiency variation and efficiency increase.

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