GT2010-23525 V2

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1 Copyright © 20xx by ASME

Proceedings of the ASME Turbo Expo 2010GT2010

June 14-18, 2010, in Glasgow, Scotland

DRAFT GT2010-23525

NUMERICAL COMPUTATION OF FORCES ACTING ON BLADES AS A FUNCTIONOF AXIAL CLEARANCE AND MINIMUM THROAT AREA

J. C. GarcíaCIATEQ

Querétaro, Querétaro, Méx.

F. AboitesCIATEQ

Querétaro, Querétaro, Méx

F. Sierra

UAEMCuernavaca, Morelos,Méx

H. Morales

CIATEQQuerétaro, Querétaro, Méx

M. Gonzalez

CIATEQQuerétaro, Querétaro, Méx

ABSTRACTIt’s well known that in any stage of steam turbine the working

fluid comes trough the nozzles, which direct the flow towards

the blades causing loads on the blade surfaces to move the rotor

and produce useful work. These loads are oscillating in time in

a harmonic pattern and could be computed by knowing the

pressure around the blades in every moment. The variation in

the loads are due because the interaction between the nozzle

wakes with the rotating blades. In this paper, a 2D numericalcomputation of forces acting on blades as a function of the

axial clearance and minimum throat area is presented. The

pressure field in a Curtis stage of a 300 MW steam turbine was

numerically computed. The Navier Stokes equations were

resolved in 2D using a commercial program based on the finite

volume method. The sliding mesh technique was used to take

into account the interaction between the nozzle wakes and the

blade motion. The forces acting on the blades were computed

for several axial clearances and throat area variations. It is

showed how these forces are affected by the variability of these

distances. Dependence of the forces from the pressure field

variation in time in the axial clearance is investigated. These

forces, which cause forced vibrations on blades, are expressedas Fourier series in order to investigate the changes in these

forces.

INTRODUCTIONDuring the operation of a steam turbine, there are dynamic

interactions between the fluid work and the internal parts like

nozzles and blades. Some of those interactions cause

unsteadiness downstream which may induce vibrations on

blades, affecting the operation and performance of turbines.

A lot of blade vibrations are caused by pressure variations in

the axial clearance. One of the most important sources of

pressure variations for a blade row is the upstream nozzle

wake [1, 2]. The nozzle wake could be affected by the length

of the axial clearance and by the nozzle throat area. If the

blade vibrations have large amplitude could cause high

alternating stresses on blades, leading to failures by fatigue.In this paper, a 2D Curtis time dependant numerical

simulation was used to compute the forces acting on blades

and caused by the passing nozzle wake. The total force on

blades as a function of axial clearance and nozzle throat area

are showed. The total force computed for different cases of

axial clearance and nozzle throat area was expressed as

Fourier series.

NOMENCLATUREρ density

Ω angular velocity

μ viscosity

ur relative velocityu absolute velocity

r vector of position

x spatial coordinate

t time

g gravity

F force

f i frequency Hz

Ai constants in a Fourier series

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2 Copyright © 20xx by ASM

φ phase angle

GOVERNING EQUATIONS AND MATHEMATICAL

MODEL

The numerical computations of the pressure field in the first

Curtis stage of a 300 MW steam turbine was performed

applying Fluent CFD code, using the RNG-κ -ε turbulence

model. To predict non-stationary phenomena, in the blade-

nozzle flow interaction, the sliding mesh technique was used

[3,4,5]. A relative reference frame was used to evaluate easilythe force acting on the blades. The flow field was numerically

resolved under steady-state and time-dependent simulations

using 2-D geometric models. The force acting on blades was

calculated using a user defined function, which integrates the

static pressure around the blade walls at every time step. These

force time-dependent data were analyzed using a Fast Fourier

Transform (FFT).

The computation domain is defined by: a) the stator zone and

b) the blade zone.

The Navier-Stokes equations in rotating domains, as the turbine

stage, include an additional term to take into account the fluid

acceleration inside the moving zones [6].

The absolute and relative velocities in the rotating domain arerelated by:

)( r uur ×Ω−= (1)

The continuity equation may be used either with the absolute or

the relative velocity:

0)( =∂

∂+

∂

∂i

i

u xt

ρ ρ

(2)

In the stationary domain the momentum equation is written as:

( ) ( )

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂−=

∂

∂+

∂

∂

j

j

i j

i

ji

i ji

j

i x

u

x x

u

x x

pguu

xu

t µ µ ρ ρ ρ

3

1

Instead in the rotating domain the momentum equation mustinclude the relative velocity

r u and the angular velocity Ω as

follows:

( ) ( )

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂−=×Ω+

∂

∂+

∂

∂

j

j

i j

i

ji

i jri

j

i x

u

x x

u

x x

puuu

xu

t µ µ ρ ρ ρ

3

1)(

GEOMETRY AND BOUNDARY CONDITIONS

The Curtis stage of a 300 MW steam turbine under study has a

mean radius of 0.94 m. The stator has 56 nozzles and a mean

height of 0.0228 m. The rotor has 84 blades and a mean height

of 0.027 m.

Only a segment of the last stage was simulated and periodic

boundary conditions were applied at every side of the stage

segment. The stage segment was defined by 2 nozzle and 3

blades (as shown in Fig. 1).

Figure 1 Geometrical aspects of the 2-D turbine Curtis

stage.

The boundary conditions at the stage were inlet pressure 15.7

MPa and outlet pressure was 11.1 MPa. The inlet and outlet

temperature of the steam were 805.55 K and 760.49 K,respectively.

Figure 2 A 2D mesh of the Curtis Stage

The geometrical model was meshed with quadrilateral cells,

using a structured mesh. For meshing wall’s vicinity a

boundary layer was used. A zoom of the 2D mesh are shown

in Fig. 2. Meshes of different sizes were used to assure mesh

independence. A profile of static pressure, located at the

clearance nozzle-rotor, was used as a convergence criterion. Amesh with 138 558 cells was selected after a convergence test

where another mesh of more than 750 thousands cells

converged to the same result.

A time step size of 1x10-5

s was used during time-dependent

simulation, whereas that the rotor speed was 173.03 m/s.

(3)

4

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RESULTSThe Figure 1 also shows as lines “a” through “f”, the spatial

location where the static pressure profiles were taken at the

axial nozzle-rotor clearance. These profiles were used to show

a 3D view of the pressure field in the axial clearance at given

instant. In order to show the pressure field as a time function,

monitors a-f were used. These profiles and monitors were located in the axial clearance

as show in Table 1. For a giver profile or monitor the

dimensionless distance was calculated using the total axialclearance and the distance between that profile or monitor and

the nozzle trailing edge.

Table 1 Dimensionless distance to locate the profiles and

monitors

Profiles or

monitors

Dimensionless distance in the axial clearance

a 0.009

b 0.17

c 0.35

d 0.5

e 0.7

f 0.9

In order to compute the total force on blades as function of

axial clearance, six different meshes were used. In each mesh a

different axial clearance was utilized as shown in Table 2. The

design axial clearance corresponds to the mesh labeled as AIII

D: 0.0163 m.

Table 2 Axial clearances used during the computations

Case Axial clearance [m]

AI 0.0113

AII 0.0123

AIII D 0.0163

AIV 0.0213AV 0.0263

AVI 0.0313

In the case of the computations of blade forces as a function of

the nozzle minimum throat area, three different meshes were

used, as shown in Table 3. The design nozzle minimum throat

area corresponds to the mesh TH D: 0.000353 m2 /nozzle. The

minimum throat area was calculated using the minimum length

“T” shown in Figure 1 and the nozzle height. The THI mesh

correspond to a increased throat area, caused by an hypothetical

nozzle wear, while the THII mesh correspond to a reduced

throat area, caused by an hypothetical deposit.

Table 3 Minimum nozzle throat area

Case nozzle minimum throat area

[m2]

TH D 0.000353

TH I 0.000418

TH II 0.000257

The static pressure profiles at a given time and for the design

axial clearance (AIII D) of the Curtis stage is shown in Fig 3.

The spatial position of the profiles inside the axial clearance

was shown in Fig. 1. The profiles show alternating zones of

low and high static pressure in the axial clearance (nozzle to

blade). The profiles of static pressure near (Fig. 3) the nozzle

trailing edge have two peaks which fall in the same position of

the nozzle. While the profiles near the blade leading edge

have three peaks which are coincident with the blade leading

edge position (Fig. 3). The profiles located at the middle ofthe axial clearance (c or d) shows a transition in the number of

peaks. It is clear that the static pressure field in the axial

clearance is described by series of harmonic profiles that

change their peaks number.

0. 0 5

0. 0 6

0. 0 7

0. 0 8

0. 0 9

0. 1 0

0. 1 1

0. 1 2

0. 1 3

0. 1 4 0. 1

5 0. 1

6

0.2

0.4

0.6

0.8

1.0

fe

dc

ba

Blade leading edge

Nozzle trailing edge

p r e s s

u r e s t a t i c

p r o f i l e s

length into the simulation group [m]

s t a t i c p r e s s u r e

[ d i m e n s i o n l e s s ]

Figure 3 Static pressure profiles in the axial clearance

design (Case AIII D)

Static pressure time variations across the axial clearance

monitored in fixed points, a-f, are shown in the Fig. 4. The

spatial location of fixed monitors was shown in Fig. 1. Fig. 4

shows how the static pressure field in front of blade is

changing as a function of time and since a relative referencepoint of view. Figure 4 also shows that the peak magnitudes of

the static pressure are higher near the nozzle trailing edge than

near the blade leading edge.

0.4570 0.4575 0.4580 0.4585 0.4590 0.4595 0.4600

0.74

0.78

0.81

0.85

0.89

0.93

0.96

1.00

s t a t i c p

r e s s u r e [ d i m e n s i o n l e s s ]

simulation time [s]

acef

time variation of static pressureacross the axial clearance

Figure 4 Static pressure profiles in the axial clearance design

as time function (Case AIII D)

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As expected, the static pressure field in the axial clearance

was affected by the variation of the length of axial clearance

and by the variation of the nozzle throat area. The Figs. 5 and

6, shows the cases with major variation in the pressure field.

Fig. 5 corresponds to the case AI which has the smallest axial

clearance used during the simulations. Fig. 6 corresponds to the

case THII which has the smallest nozzle throat area used during

the simulations. The maximum amplitude variations for the

static pressure profiles were found in these two cases: AI and

THII. These results also can be observed in Fig. 7, which is acomparison of the “f” static pressure profiles. Fig. 7 shows that

the “f” profile has the maximum amplitude for the cases AI and

THII. At this point one can infer that these two cases could

cause vibrations with major amplitude than the other cases.

0. 0 5

0. 0 6

0. 0 7

0. 0 8

0. 0 9

0. 1 0

0. 1 1

0. 1 2

0. 1 3

0. 1 4 0. 1

5 0. 1

6

0.2

0.4

0.6

0.8

1.0

fe

dc

ba

Blade leading edge


p r e s s

u r e s t a t i c

p r o f i l e

s




Figure 5 Static pressure profiles in a reduced axial

clearance (Case AI)

0. 0 5

0. 0 6

0. 0 7

0. 0 8

0. 0 9

0. 1 0

0. 1 1

0. 1 2

0. 1 3

0. 1 4 0. 1

5 0. 1

6

0.2

0.4

0.6

0.8

1.0

fe

dc

ba

Blade leading edge


p r e s s

u r e s t a t i c

p r o f i l e s




Figure 6 Static pressure profiles in the a reduced throat

area (Case THII)

0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

Nozzle trailing edgeBlade leading edge

s t a t i c p

r e s s u r e [ d i m e n s i o n l e s s ]

length into the simulation group (m)

AIII D (Desing clearance)AI (Reduced clearance)AIV (Increased clearance)THI (Increased throat area)THII (Reduced throat area)

Figure 7 Comparison of the “f” static pressure profiles

in the axial clearance.

The forces acting on the blades were calculated by integration

of the static pressure on the blade walls using a defined user

function. The calculated forces for case A III D -design case-

are shown in the Fig. 8. The curves have a harmonic patternand can be expressed as a Fourier series.

0. 5 0 1 0 0

0. 5 0 1 2 5

0. 5 0 1 5 0

0. 5 0 1 7 5

0. 5 0 2 0 0

0. 5 0 2 2 5

0. 5 0 2 5 0

0. 5 0 2 7 5

0. 5 0 3 0 0

0. 5 0 3 2 5

0. 5 0 3 5 0

0. 5 0 3 7 5

0. 5 0 4 0 0

0.73

0.77

0.82

0.87

0.91

0.96

1.00

f o

r c e [ d i m e n s i o n l e s s ]

simulation time [s]

total force

total force on blades in curtis stage as a time function

Figure 8 Total force on blades in a Curtis stage as a

time function (Case A III D –design case-)

Fig. 9 shows the total force as function of axial clearance. The

maximum total force was reached for the case AI, however

this case has the larger pressure amplitude across the axialclearance (Fig 7). In contrast Fig. 10 shows the tangential

force as function of axial clearance. The tangential force

shows a minimum for the case AI, and reaches a maximum for

the case AVI (this has the maximum axial clearance)

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0. 0 1 0 0

0. 0 1 2 5

0. 0 1 5 0

0. 0 1 7 5

0. 0 2 0 0

0. 0 2 2 5

0. 0 2 5 0

0. 0 2 7 5

0. 0 3 0 0

0. 0 3 2 5

0.975

0.980

0.985

0.990

0.995

1.000

total force

t o t a l f o r c e [ d i m e n s i o n l e s s ]

axial clearance [m]

total force on blades in a Curtis stageas functon of axial clearance

Figure 9 Total force on blades in a Curtis stage as a

function of axial clearance

0. 0 1 0 0

0. 0 1 2 5

0. 0 1 5 0

0. 0 1 7 5

0. 0 2 0 0

0. 0 2 2 5

0. 0 2 5 0

0. 0 2 7 5

0. 0 3 0 0

0. 0 3 2 5

0.950

0.955

0.960

0.965

0.970

0.975

0.980

0.985

0.990

0.995

1.000

tangential force on blades in a Curtis stageas functon of axial clearance

t a n g e n t i a l f o r c e [ d i m e n s i o n l e s s ]

axial clearance [m]

tangential force

Figure 10 Tangential force on blades in a Curtis stage

as a function of axial clearance

Fig. 11 shows the total and tangential forces on blades as

function of the nozzle throat area. The total force reach a

maximum for the increased nozzle throat area, however the

maximum tangential force is reached at the design condition.

The forces data for the case AI, AIIID, AVI, THI and THII were

analyzed using Fast Fourier Transform (FFT) and the results

are showed in the Table 4, where the results of frequency, phaseangle and amplitude are tabulated. The frequency calculated

with FFT shows good agreement with the nozzle passing

frequency (56 nozzles X 60 Hz=3360 Hz).

The frequency and constants given in the table are the first

terms of the Fourier series as indicated by the next equation:

( )[ ]∑=

−+=n

i

ii t f Cos A AF 1

0 2 φ π (5)

0.00024 0.00027 0.00030 0.00033 0.00036 0.000390.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

f o r c e [ d i m e n s i o n l e s s ]

minimum throat area

total forcetangential force

Figure 11 Total and tangential forces on blades in a

Curtis stage as a function of minimum throat area

The term Ai in the Table 4 states the force amplitude around

the term A0. Table 4 shows that the term Ai with more

amplitude variation corresponds to the case AVI (reducedaxial clearance) and to the case THII (reduced nozzle throat

area).

Table 4 Terms of the Fourier series calculated trough FFT

of the forces calculated using a 2D numerical simulation

Case A0

[dimensi

onless]

Ai

[dimensi

onless]

f i [Hz] φ

A III D 1.000 0.119 3357.860 156.524

A VI 1.004 0.025 3365.301 -146.185

A I 1.019 0.254 3357.864 -118.353

Th I 1.072 0.130 3365.301 104.443Th II 0.904 0.273 3365.301 33.231

Using the results tabulated in the Table 4 with equation (5),

one may reproduce the forces calculated using fluid dynamics

simulation in additional studies, like fatigue life estimation.

CONCLUSIONSThe two dimensional unsteady flow across the axial clearance

in a Curtis stage of 300 MW steam turbine was numerically

investigated. The computations show that the pressure profilesin the axial clearance have an oscillatory pattern. For a given

instant, a picture of pressure field shows static pressure

profiles with different number of peaks or valleys across the

clearance. Near the nozzle trailing edge the profiles have a

number of peaks equal to the nozzles and at the vicinity of the

blade leading edge, the profiles pressure have a number of

peaks equal to the number of blades at that segment.

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The pressure profiles as time function in the axial clearance

show that pressure fields in front of the blades have harmonic

variations with a frequency equal to the nozzle passing

frequency.

In the case of the axial clearance variations, the maximum total

force was reached for the minimum axial clearance, however

this case has the minimum tangential force and shows large

pressure variations across the axial clearance, which could

cause forces acting on blades with large amplitude.

Talking about of the nozzle throat variation, the maximum totalforce was reached for the case with increased throat area (THI),

while the maximum pressure variations across the axial

clearance were find for the case with reduced throat area.

However this two cases show a tangential force lesser than the

case with the design nozzle throat area.

REFERENCES

[1] Rangwalla, A.A. and Rai, M.M., A numerical analysis of

tonal acoustics in rotor stator interactions, Journal of Fluids and

Structures, 1993.

[2] Chaluvadi, V.S.P., Kalfas, A.I. and Hodson H.P., Vortex

transport and blade interactions in high pressure turbines,ASME, Journal of Turbomachinery, Vol 126, 2004.

[3] Kosowski, K. and Stepien, R., Theoretical investigations

into flows in rotor blade shroud clearance, Transactions of The

Institute of Fluid-Flow Machinery, No. 113, Gdansk, Poland,

2003.

[4] Lampart, P.et al., Unsteady forces acting on rotor blades of

a large power steam turbine control stage at different levels of

partial admission, Transactions of The Institute of Fluid-Flow

Machinery, No. 114, Gdansk, Poland , 2003.

[5] Fluent user’s guide, version, 6.1, 2003

[6] Pantakar, S.V., Numerical heat transfer and fluid flow,

McGrawHill, NY, 1980.

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GT2010-23525 V2