Water Drop Erosion on Turbine Blades Numerical Framework and Applications

Water Drop Erosion on Turbine Blades:Numerical Framework and Applications

Qulan Zhou 1;2;*,Na Li 1;2,Xi Chen 2,Akio Yonezu 3;*,Tongmo Xu 1,Shien Hui 1and Di Zhang 1

1The State Key Laboratory of Power Engineering Multiphase Flow,Xi’an Jiaotong University,Xi’an 710049,P.R.China 2

Department of Civil Engineering and Engineering Mechanics,Columbia University,New York,NY 10027,USA 3

Department of Mechanical Engineering,Osaka University,Suita 565-0871,Japan

When small droplets are formed in the wet steam stage of a steam turbine,they may impact the blade surface at a high velocity and repetitive impacts cause water drop erosion,which emerges as one of the primary reliability concerns of the turbine.We propose an e?ective numerical framework that couples ?uid mechanics with solid mechanics.The movements of water drops in a blade channel are analyzed based on the solution of the ?ow ?eld of water steam in turbine,and impact statistics such as impact frequency,velocity,and position are obtained as the working condition and particle size are varied.A nonlinear wave model is established for high velocity liquid-solid impact,from which the characteristic impact pressure in liquid and peak impact stress in solid are obtained;the solutions are then superimposed with the pathways of water particles,and a fatigue analysis is carried out to elucidate the mechanisms of water drop erosion.The lifetime map on a blade surface with two di?erent materials (1Cr13and Ti-6Al-4V)under typical working conditions are obtained,in terms of operation hours,and the most dangerous water drop erosion regions and operating conditions of the steam turbine are deduced.[doi:10.2320/matertrans.MRA2008025]

(Received January 17,2008;Accepted April 15,2008;Published June 11,2008)Keywords:water drop erosion,liquid-solid impact,numerical simulation

1.Introduction

Near the exhaust of a steam turbine engine (also known as the penult stage or ?nal stage),the low-pressure working condition favors phase transformation,and small water drops are produced by the condensation of steam.Once formed,the water drops move with the ?ow and some of them may impact the blade surface with a velocity over 200m/s —upon consecutive impact on the turbine blades,the surface material may spall o?and this is known as the water drop erosion which severely a?ects the system reliability (Fig.1).According to experimental observations,the most favorable impact locations include the leading or trailing edges of back arc and inner arc,and the erosion rate is around 0.1micron per hour.1)Therefore,water drop erosion emerges as one of the primary reliability concerns in a turbine and its mechanism must be su?ciently understood.

The water drop erosion is also a very complex problem that couples multiphase ?uid ?ow with impact mechanics and fatigue analysis,which requires a seamless coupling between ?uid and solid mechanics.First,it is important to solve for the trajectories and velocities of water drops,as well as the distribution of mass and size of the particles.While some water drops may follow the steam ?ow and exit the turbine,others may impact the blade surface and key information needs to be collected,including the impact frequency,location,velocity,which depend on the aforementioned variables.Finally,the liquid-solid impact problem imposes a tremendous challenge and an appropriate model is needed to solve for the distribution and magnitude of transient impact stresses in the solid.When the solution of the fundamental liquid-solid impact problem is superimposed with the impact statistics,an fatigue analysis may be carried out to evaluate the lifetime,most dangerous areas and working conditions.In this paper,we establish multidisciplinary computational and

mathematical models toward such an objective.The ?exible framework may also be extended to similar problems encountered in practice,such as rain drop erosion on civil structures or airplane engines.2.

Flow Field in a Wet Steam Turbine Stage

2.1Model and computation method

For a long blade turbine stage,the Ma number varies from low at the inlet to high ($supersonic).The conventional approach 2,3)cannot be directly applied to simulate the ?ow in a turbine stage that involves both static nozzles and rotating blades.In order to reduce computational cost,in this section,the compressible pressure correction method (SIMPLE)is combined with the mixing plane model;4,5)the static and moving regions are treated separately and connected through the ‘‘mixing plane’’.A series of techniques are proposed to take into account the various boundary con-

Fig.1Schematic of the water erosion problem.The turbine stage,where the ?uid ?ows through nozzle and blade channels,connected by the mixing plane.The stator consists of nozzles,while the rotor (which rotates to export mechanical work)incorporates blades.As the stator is static and the rotor is rotating,there is a narrow gap between them (mixing plane).The water drops (from condensation of wet steam)carried by the ?ow tend to impact on the blade and induce water drop erosion.

*Corresponding

authors,E-mail:qz2129@https://www.360docs.net/doc/cd11596912.html, (Q.Zhou),

akio yonezu831@https://www.360docs.net/doc/cd11596912.html, (A.Yonezu)

Materials Transactions ,Vol.49,No.7(2008)pp.1606to 1615#2008The Japan Institute of Metals

ditions that are related with di?erent speed ranges in the nozzle and blade.

The traditional K à"turbulence model 2,6)is employed in numerical study,which takes the general form of N-S equation:@@x e u Tt@@y e v Tt@

e w T?@@x à@ @x t@@y à@ @y t@@z à@ @ tS 0tS 1e1Twhere u ,v ,w are the velocity components in x ,y ,and z directions,respectively; is the density of ?uid,àis the generalized viscosity factor,S 0is the source term caused by turbulence model;S 1is the source term caused by other factors,and is the ?eld variable,which can be a velocity component (u ,v ,w ),temperature (T ),turbulence energy (K ),and turbulence energy dissipation ("),etc.,see Table 1.The ?rst column is the ?eld variable of the standard K à"turbulence model;the second column is the generalized viscosity factor;and the third column is the source term caused by turbulence model.In Table 1,c ?0:09,c 1?1:44,c 2?1:92, k ?1:0, "?1:3, T ?1:0.Pr is the Prandtl number,p is pressure, is the molecular viscosity factor,and t is the turbulence viscosity factor:6) t ?c K 2=":e2T

eff ? t t

e3T

G ? t (2@u @x 2t@v @y 2t

@w @z 2

t@u @y t@v @x 2t@w @x t@u @z 2t@v @z t@w @x

2):e4T

Since there are 8variables in (1),e ;p ;u ;v ;w ;K ;";T T,the state equation is needed to close the equation set.For wet steam,the second Virial coe?cient equation is used:

p ?RT = e1tB = T;

e5T

where is the kinematic-viscosity coe?cient,and R is the gas constant,and B ?2:0624?10à3

à2:61204T ?10100800

.In this work,the coordinate system is ?xed on the rotating blade.Consider a ?uid block in a rotary system with angular velocity !,both the centrifugal force (f e )and the Coriolis force (f c )are acting on the block:

f e ex T?0;f e ey T?!2ey ày 0T;f e ez T?!2ez àz 0Te6Tf c ex T?0;f c ey T?2!w ;f c ez T?à2!v

e7T

where the direction of angular velocity vector is aligned

with the x-axis,i.e.~

?e!;0;0T.The coordinates of the rotary axis is ey 0;z 0T,thus,the polar radius vector is ~r

?e0;y ày 0;z àz 0T.The source terms of the momentum equations become

S u ?0;S v ?f c ey Ttf e ey T;S w ?f c ez Ttf e ez T:

e8T

The inlet conditions include the temperature and velocity of the ?uid.If the speed at the outlet approaches supersonic,the static inlet pressure must be given.The distribution of static pressure is given on the outlet,and the outlet velocity is assumed to be unidirectional;6)other outlet variables are obtained by extrapolation.The ?uid ?ow near the wall follows the wall function.6)The static and dynamic regions are solved separately based on time-averaged parameters,and interacted through a mixing plane model.7)In the physical space,the mixing plane (Fig.1)is the narrow gap between the nozzle and blade,where the wet steam exits the nozzles and ?ows into the blade channels.

2.2Flow ?eld of wet steam ?ow in a turbine stage

The numerical framework established above is employed to simulate the ?ow ?eld in a whole wet steam turbine stage,which is a penult stage of a 125MW turbine (Dongfang Steam Turbine Company,China).The geometry and dimen-sion are given in Fig.2,where the height of nozzle,l 1?450mm and the height of blade is l 2?460mm.The outlet of nozzle channel and the inlet of blade channel are aligned with a plane in the radial direction,so as to apply the mixing plane model.Two operating conditions (Table 2)are simulated,under full designed and reduced (75%)loads,respectively.For simplicity,we consider only the ?ux of the working ?uid (i.e.the inlet Ma number)is varied and all other parameters are ?xed between these two working conditions.Note that the numerical protocol can be readily extended to other turbines with di?erent dimensions and operating conditions.

The ?ow ?elds under the two operating conditions are shown in Fig.3(a)and (b),respectively,for the top section of the stage.The color bar and arrow length denote the

Table 1The meanings of terms in eq.(1)when the standard K à"turbulence model is used. àS 0

u t t à@p @x t

@@x eff @u

t@@y eff @v @x t@@z eff @w @x

v t t à@p t

@ eff @u

t@ eff @v t@ eff @w w t t à@p t@ eff @u

@ eff @v t@ eff @w T t t

K t

t K G à ""

t "

ec 1G àc 2 "T

Top

Bottom

Top

Bottom

(a)(b)

Fig.2The pro?le and dimension of the nozzle (a)and blade (b)used in the simulation.

Water Drop Erosion on Turbine Blades:Numerical Framework and Applications 1607

magnitude of velocity vectors.While the patterns of ?ow ?eld in the nozzle channel are fairly close for the two working conditions,the ?ow ?elds in the blade channel are distinct.When the load of turbine varies,the ?ow speed at the inlet/outlet of the nozzle changes yet the angular velocity of blade is invariant.When these two factors are combined,both the magnitude and direction of the ?ow ?eld in the blade channel become very sensitive to the load.Subsequently at reduced load,the inlet velocity vectors become opposite to the direction of rotation,which cause the inlet steam ?ow toward the back arc,and a back?ow region is formed on the top section (Fig.3(b)).3.

Motion of Water Drops in Blade Channel

3.1Model and computation method

The diameter distribution of particles entering the blade channel is given by Shi.8)The particle trace model in the Lagrangian coordinates 9)is employed in this study to obtain the statistics of movement of water drops in the blade channel.

Since the typical humidity of wet steam is lower than 10%1,10)and the volume fraction of water drops in steam is under 0.01%,it is reasonable to assume that the motion of water drop would not counter-a?ect the ?uid ?ow.We fur-ther neglect the crash and coalescence of water drops since the transient stress develops very quickly in the solid upon impact,and it is the most critical for erosion (see below).We use a semi-random trace model which incorporates the correction of turbulence dissipation and with moderate computational cost.In the rotary coordinate system attached to the blade,the motion of particles is described as:du p dt ?1 eu g tu 0g àu p Tf eRe Tdv p dt ?1 ev g tv 0g àv p Tf eRe Tt2!w p dw p dt

?1 ew g tw 0g

àw p Tf eRe Tà2!v p t!2r p 8>>>>>><>>>>>>:e9Twhere !is the angular velocity of the coordinates (same as that of the blade),and r p is the radial coordinate of the

water drop,t ? ? p d 2

p =e18 Tis the relaxation time of particle

f eRe T?1Re <11t1

6Re 2=31 Re 10000:01854Re Re >10008><>:

e10T

Re ? g eu g àu p Td p = ;e11Twhere p is the particle density,d p is the particle diameter,

is the dynamic-viscosity coe?cient of gas.u p ,v p ,w p are the velocity components of the particle,u g ,v g ,w g are

the velocity components of the ?uid,and u 0g ,v 0g ,w 0

g are the pulsant velocities of the turbulent ?ow.Assuming the turbulence is isotropic,

u 0g ? ??????"u 02p ?23K ;v 0g ? ??????"v

02p ?23K ;w 0g ? ???????"w

02p ?2

3K e12Twhere is a random number with normal distribution,and it remains constant during a time step which tracks the movement of the particle.The turbulence energy K in the K à"turbulence model can be solved from the steam ?ow ?eld (see Section 2).The velocities eu p ;v p ;w p Tof the particle are obtained from (9);we then integrate the velocities eu p ;v p ;w p Tto obtain the position coordinates ex p ;y p ;z p Tof the particle.

3.2Trace of water drops in blade channel

With the turbine stage and working conditions speci?ed in Section 2.2,sample traces are shown in Figs.4and 5at the bottom and top of blade channel,https://www.360docs.net/doc/cd11596912.html,binations

Table 2

The inlet parameters of the turbine stage.

Operating condition Ma number Temperature T (K)

Static pressure p (MPa)

No.1(the rated load)0.2163560.03No.2(75%load)

0.16

356

0.03

y x

(a)

(b)

Fig.3The ?ow ?eld on the top section of the stage,for both nozzle (left)and blade (right)under (a)operating condition No.1(full designed load),(b)operating condition No.2(75%reduced

load).

y x y x y x y

(a)(c)(b)(d)

Fig.4The traces of water drops at the bottom of blade channel (view in x -y

plane):(a)5m m particles with operating condition No.1;(b)100m m particles with operating condition No.1;(c)5m m particles with operating condition No.2;(d)100m m particles with operating condition No.

y x y x y

x y

(a)(b)(c)(d)

Fig.5The traces of water drops at the top of blade channel (view in x -y plane):(a)5m m particles with operating condition No.1;(b)100m m particles with operating condition No.1;(c)5m m particles with operating condition No.2;(d)100m m particles with operating condition No.2.

1608

Q.Zhou et al.

of di?erent particle size and working conditions are ex-plored.Most 5m m particles move along with the steam ?ow consistently,and only very few particles impact the inner arc of blade with small normal impact velocities.Because the inlet angles of ?ow are di?erent,more 5m m particles tend to impact the blade under the full load than that under reduced load (although the percentage of particles colliding with the blade is fairly small in both cases).By contrast,with the dominance of their inertia,the 100m m particles can be easily separated from steam ?ow and impact the blade surface at a high normal velocity;in particular,under the reduced load,the head of back arc appears su?ering to high-possibility of water drop impacts.We introduce a dimensionless impact rate s :

s ?lim

áA !0áN áA 0

áA áN 0

e13T

where áN is the number of drops collected by the element area,áA ;N 0is the total number of drops entering the channel,and A 0is the area of the entry plane of the channel.Figure 6gives the maps of s of 100m m on the inner and back arcs of the blade,and under both operating conditions.The x and z coordinates are normalized by the total length of blade in the x direction (x b )and height in the z direction (z b ),respectively.The maximum impact rate with the reduced load is 2.6,which is greater than under full load.Even though the number of drops launched on the entry plane is same,the impact frequency on the head of back arc with operating condition No.2(Fig.6(b))is almost 5times of that of No.1(Fig.6(a)).In practice,the number of drops on the entry plane under the low load operating condition should be more than that of the full load condition,because the steam under the low load operating condition has lower enthalpy and more water drops will be produced.Thus,the head of back arc of blade is the most dangerous region that will be eroded by drops under the low load operating conditions.For the inner arc,the maximum impact frequency with full load is larger than that of reduced load.Under full load,the most critical erosion region is close to the diagonal line of the blade,whereas the region of concern upon reduced load is at the bottom of the blade.

Figures 7and 8give the maps of normal impact velocity distributions on the back arc/inner arc under the operating

conditions No.2,as the particle size is varied.In all cases,the impact velocity patterns of the larger particles are essentially the same when the diameter of drops is bigger than 50m m .Since the larger particles are more critical for water drop erosion,once the diameter of drop exceeds 50m m ,the impact frequency and velocity distributions can be regarded as invariant so as to reduce the computational cost.

Therefore,for larger particles the distributions of impact velocity are independent of water drop diameters,and since they are also more critical,we mainly focus on the impact by larger particles in the following sections.At reduced load,the maximum impact velocity (Fig.7)appears at the head of blade top,consistent with the highest impact frequency (Fig.6(b)).Thus,this region is quite likely to subject to the most severe erosion;this agrees with.10)The maximum impact velocity also increases signi?cantly with larger particle size,which may be as high as 160m/s for large particles on the inner arc (Fig.8).Such critical region which spans from the front bottom to the middle height of trailing edge,is partially overlapped with that of highest frequency (Fig.6(d)),which may be regarded as the second possible erosion region.

The impact velocity and frequency maps are only partly responsible for erosion mechanism.The blade fails by transient stress during impact,and the liquid-solid impact problem must be solved so as to evaluate the blade lifetime in a more reliable manner.The liquid-solid impact stress ?eld is analyzed in the next section,followed by erosion life analysis in Section 5.

(a)(c)(d)

Head Bottom

Top

00.055000.1100

0.16500.2200

0.2750

0.33000.38500.4400

0.49500.5500

X Z

00.26000.5200

0.7800

1.0401.300

1.5601.820

2.080

2.3402.600

Top

Bottom

Head T rail

Head Bottom

T op

00.35000.70001.050

1.400

1.750

2.1002.4502.800

3.1503.500

X Z

Head Bottom

Top

00.24000.4800

0.72000.9600

1.200

1.4401.6801.920

2.1602.400

(b)Fig.6The distribution of s of 100m m particles under operating condition (a)the back arc of blade of No.1;(b)the back arc of blade of No.2;(c)the inner arc of blade of No.1;(d)the inner arc of blade of No.2(view in x -z plane).

(a)(b)(c)(d)

(e)

Bottom

Top

5.00010.0015.00

20.00

25.00

30.0035.0040.00

45.0050.00

Bottom

Top

12.00

24.0036.00

48.0060.00

72.00

84.0096.00

108.0120.0

Bottom

Top

014.00

28.0042.0056.0070.00

84.0098.00112.0126.0140.0

Bottom

Top

14.0028.0042.00

56.0070.00

84.00

98.00112.0

126.0140.0

Bottom

Top

14.0028.0042.00

56.0070.00

84.0098.00112.0126.0140.0

Fig.7The distribution of normal impact velocity (m/s)on the back arc under the operating condition No.2(view in x -z plane),with the particle diameter:(a)5m m ,(b)10m m ,(c)50m m ,(d)150m m ,(e)250m m .

(a)(b)(c)(d)(e)

Bottom

Top

8.000

16.0024.00

32.0040.00

48.00

56.0064.00

72.0080.00

Bottom

Top

16.00

32.0048.0064.0080.00

96.00

112.0128.0

144.0160.0

Bottom

T op

18.0036.0054.00

72.0090.00

108.0126.0144.0162.0180.0

Bottom

Top

16.0032.0048.00

64.0080.00

96.00

112.0128.0

144.0160.0

Bottom

Top

16.0032.0048.00

64.0080.00

96.00112.0128.0144.0160.0

Fig.8The distribution of normal impact velocity (m/s)on the inner arc under the operating condition No.2(view in x -z plane),with the particle diameter:(a)5m m ,(b)10m m ,(c)50m m ,(d)150m m ,(e)250m m .

Water Drop Erosion on Turbine Blades:Numerical Framework and Applications

1609

4.Liquid-Solid Impact

4.1Fundamental considerations of modeling

There are very limited experimental studies on high velocity liquid-solid impact.11–13)Moreover,these experi-ments focused on the pressure?eld in the?uid,however the stress?eld in the solid is the most critical for causing water drop erosion.Since the surface material spalls o?upon consecutive impact,the transient peak stress causing such damage due to impact must be below the surface.Therefore, e?ective theoretical model and numerical simulation are necessary for exploring the stress?eld and related erosion mechanisms.Due to the di?culty of coupling between liquid and solid phases,the study of liquid-solid impact is far less comparing with solid-solid impact counterparts.14,15)

Along previous analytical attempts,many researchers established simple models upon liquid-rigid surface inter-action(many of them are1D)and obtained the steady-state pressure,16–19)and then applied that pressure to obtain stress characteristics in an elastic half-space.16,17,20–23)Apparently, in these works,the pressure?eld in liquid was not simulta-neously coupled with the stress wave in solid,the solutions were not valid for the realistic3D condition,and the transient peak pressure in liquid and peak stress in solid,which are much more severe than the steady-state values,15)were not pursued.

In this section,we develop a fully-coupled,3D dynamic solid-liquid model.On the interface between liquid and solid, all transient parameters,include pressure,force,mass point position and movement,are exchanged between the two regions and solved in situ.When a spherical water drop impacts on a solid plane,the shock wave forms inside the water drop due to the compressibility of liquid.24)The water drop thus imposes a pressure distribution on the surface of solid which varies with time and space,inducing stress waves which transmit through the solid,with possible formation and propagation of microcracks.

The Navier-Stokes(N-S)equation is the basic mathemat-ical model for?uid:

@ @t tráe VT?0;

@ V

tráeTtp ItDT?0e14T

where V is the velocity vector of mass point in liquid;the?rst equation of(14)is the continuity equation and the second equation is the momentum equation.Here,T is the momentum tensor,p I is the pressure tensor,and D is the viscosity force tensor.I is the Kronecker delta.Due to the complexity of the liquid-solid impact,a few basic assump-tions are necessary for our model:

(1)We assume normal impact is much more critical than

tangential impact,and thus we focus only on the normal component of the impact velocity(since the tangential component does not produce impact stress).Thus we only focus on normal axisymmetric problem(Fig.9) which simpli?es formulation.

(2)Due to the short characteristic impact duration($ns)on

turbine blade,the deformation in the metal is elastic for high-velocity impacts due to the strain rate e?ect.15) This greatly simpli?es the current analysis and linear elasticity theory with small deformation can be adopted

for the solid,and the coupling between the Euler coordinate system in liquid and Lagrange coordinate system in solid becomes easier.

(3)Since the maximum pressure appears before the shock

wave leaves the body of water drop,25,26)which will be validated in this paper,we do not consider subsequent events after the shock wave leaves the water drop.If the sonic speed c E at the point e(Fig.9,inside the edge cell

E)is larger than the moving speed of the point(v e),that

implies the disturbing pressure wave(i.e.the shock wave front)is moving faster than the edge of liquid,and the shock wave will break o?the liquid edge,thus determining t SB,the critical moment when the shock wave breaks o?.

(4)From Section3,at a high impact velocity up to300m/s,

the contact angle (Fig.9)of the shock wave leaving the water drop is about10 .26)Moreover,the water drop retains its spherical shape before the shock wave leaves;24)later we will also show that the sonic wave is con?ned to a very small region before it leaves the water drop.Therefore,both the shape and volume variations of water sphere are negligible before the shock wave leaves the liquid drop,25,26)and thus the impact is an acoustic procedure with negligible viscos-ity e?ect.We may thus ignore the viscosity of liquid drop during high velocity impact(while retaining the compressibility of?uid),and reduce the N-S equation to the nonlinear wave equation.

4.23D Nonlinear wave model for liquid-solid impact The coordinates in this problem are?xed at the interface of the undisturbed liquid,and thus we consider the problem where the solid is‘‘impacting’’the liquid(Fig.9).The undisturbed?uid is static,V0?0.The local density, pressure,and velocity in the liquid can be expressed as the summation of the values of undisturbed liquid(with subscript 0)and the relevant perturbation,which are all functions of time and space coordinates:

? 0tá ;p?p0táp;V?V0táVe15TSubstitute eq.(15)into eq.(14),note that@ 0=@t?0, r 0?0,r p0?0.ráT%0after neglecting the higher order terms.After ignored the viscosity D,eq.(14) becomes

Fig.9The coordinates system of axisymmetric liquid-solid impact and coupling and.schematic of the moment when the shock wave breaks o?the liquid.

1610Q.Zhou et al.

@eá T=@ttráe VT?0e16aT

@e VT=@ttráeáp IT?0e16bTFor a?uid,á can be written as27)

á ?e@ =@pTáp?áp=c2e17Twhere c is the sonic speed in the?uid.Subtract the time derivative of eq.(16a)from the divergence of eq.(16b),and also subtract the time derivative of eq.(16b)from c2times the divergence of eq.(16a),and use eq.(17):

r2p?1

@2p

@t2

;r2V?

@2V

@t2

e18T

These are the wave equations.Introduce the velocity potential functionéwith V?réand p?à á@é=@t, the wave equations of p and V are merged to one wave equation

r2é?1

@2é

@t2

e19T

Since the present mathematical model of liquid-solid impact is based on the wave equation,it is also termed as‘‘the wave model of liquid-solid impact’’,or the‘‘wave model’’.Since the sonic speed c is a variable of the density(and thus pressure),the wave equation is a nonlinear second order PDE.According to the Tait state equation of water28)

p?A àBe20Twhere A?1:0147663?10à19,B?2:858987?108Pa,and ?7:15.c can be calculated from(17)as:

??????

????????????????

ptB

e21T

Combine eqs.(19)–(21),the governing equation set of the nonlinear wave model is derived,which dominates the physical procedure in the liquid region of the liquid-solid impact:

r2é?

@2é

@t2

p?à

@é

p?A àB c?

????????????????

ptB

>><

>>:e22T

There are4equations and4variables,é,c,p, .The equation set is closed and can be solved.The compressibility of liquid is embedded in the sonic wave speed c.When the wave model for liquid is coupled with elasticity,the stress ?eld in the solid can also be solved.

Speci?ed for axisymmetric impact(Fig.9),the origin O is located at the point where the liquid sphere and solid plane just contacts each other at the beginning of impact.The x axis is normal to solid plane and also along the impact direction, and the r axis is radial direction.Due to elastic deformation, the solid wall is moving with an impact velocity v0.In the solid region,the displacement solution of elastodynamics (the Lame equations)is used so as to ensure consistent movement of mass points along the interface.The governing equations and coupling conditions of axisymmetric liquid-solid impact are:

Liquid

r2é?

@2é

@t2

;éj t?0?0

@é

j x?0?v0à

@U s

j x?0

>><

>>:Solid

e t Trerá~U sTt r2~U s? s

@2~U s

@t2

j t?0?0; xr j x?0?0; x j x?0?p j x?0

>:e23T

Here,~U s is the elastic displacement vector of solid mass point,with components U s and V s in x and r directions,respectively. The velocity in hoop( )direction is zero due to axisymmetry.Denote the solid Young’s modulus as E s and Poisson’s ratio as

s, ? s E s

e1t sTe1à2 sTand ?E s

2e1t sT

are the Lame elastic coe?cients of solid. x j x?0and xr j x?0are the normal and shear stress

components at the solid surface.

The in situ coupling between liquid and solid are realized through the last equation in liquid region(via velocity coupling) and last equation in solid region(via stress-pressure coupling)inside the contact area;outside the contact area the solid normal stress is zero at the interface.In each time step,the geometry of contact area is updated using iteration to ensure consistency of the above equations.The equations are solved using the?nite di?erence method,and the boundary conditions are exchanged on the interface.

Once the transient displacement?eld inside the solid is obtained,the strain?eld is readily obtained,followed by the stress ?eld.A useful measure of the magnitude of the multiaxial stress?eld at a material point is the equivalent stress:

????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????

e xà rT2=2te rà T2=2te à xT2=2t3e 2

t 2

e24T

In what follows,the equivalent stress in solid and pressure in liquid are both normalized by the water hammer force 0c0v0 (with 0and c0the undisturbed density and sonic speed in liquid,Section3.3),and the dimensionless stress and pressure are:

?? e=e 0c0v0Tand P?p=e 0c0v0Te25TIt will be shown that the maximum pressure and maximum equivalent stress occur within the liquid and solid,respec-tively.The dimensionless pressure and stress are expressed as the dimensionless coordinates X and R,and the dimension-less time T:

X?x=r0;R?r=r0;T?c0t=r0e26Twhere r0is the initial radius of liquid drop.

4.3Simulations of water drop-1Cr13/Ti-6Al-4V impact Although the?exible model and numerical framework developed in this study can be readily extended to any liquid and elastic solid,we use two representative blade materials,

Water Drop Erosion on Turbine Blades:Numerical Framework and Applications1611

the high strength stainless steel1Cr13(ASTM#S41000)after quenching and tempering,and Ti-6A1-4V,to demonstrate the results of the liquid-solid impact.The main material properties(measured from experiments)are shown in Table3,along with that of water.

As the particle size is varied,the dimensionless parame-ters,including the dimensionless time when shock wave breaks o?the solid,maximum interface pressure and the time of its appearance,maximum stress and the time/position of its occurrence,are found to be invariant and they are only functions of impact velocity.This?nding is supported by experimental data in,24)where the water drops with di?erent diameters have the same contact angle of lateral jetting.Thus, in the following examples,we only use one water drop size (1mm)during simulation and all results are expressed in dimensionless terms.

We?rst?x the impact velocity at a representative v0?100m/s.Figure10shows a snap shot of the pressure distribution inside water drop and the stress?eld inside elastic solid,at9ns after impact.The pressure in water drop increases away from the origin,and within the contact area, the highest pressure is found at the edge.Semicircular-shaped pressure waves are formed in the disturbed liquid region because of the moving disturbing source(contact point),and the cap-like shock wave front is the envelope of the pressure wave.The stress distribution in solid exhibits an interference ?eld of two symmetrical wave sources.There are two prominent high stress regions:the?rst one is near the contact edge,where the peak impact pressure is also found which acts as the primary wave source;the second one(which is important for erosion)is near the axis and below the surface, caused by the superposition of stress waves.For these two solid materials,the patterns of pressure and stress are slightly di?erent,because their sonic speeds are relatively close.The stress in Ti-6Al-4V is a little bit lower than1Cr13because of its smaller modulus(see Table3).

After the stress?eld of every time step is solved,the maximum equivalent stress point can be found and within each time step,this point should be regarded as the most dangerous point.If we plot the trace of such critical dangerous point for every time step,then the in?uence range of the critical impact stress can be known.Figure11shows the moving traces of the maximum equivalent stress point at each time step,with an intervalát?0:05ns.The basic length against the dimensionless depth X and dimensionless

radius R is the diameter of water drop.The in?uence zone of impact in the solid mainly develops in the redial direction and near to the surface.The two materials have similar traces and in?uence zone because of their close sonic speed.

The solution of liquid-solid impact depends strongly on the impact speed v0.With c0the sonic speed in the undisturbed liquid,the impact Mach number is Ma0?v0=c0.We de?ne3 important dimensionless parameters to characterize the main features of impact:

(i)The maximum dimensionless stress in solid,?inf,is the

maximum dimensionless stress in solid(at all time steps before the shock wave breaks o?),?inf? e,max=

Table3The physical properties of1Cr13and Ti-6Al-4V.

1Cr13Ti-6Al-4V Water

Density(m3/kg)770046201000

Sonic speed(m/s)509650101430 Acoustic impedance(kg/m2s)3:924?1074:255?1071:430?106 Young’s modulus(Pa)2?10111:16?1011—Fatigue limit/threshold(MPa)468586—

0.2360

0.4720

0.7080

0.9440

1.180

1.416

1.652

1.888

2.124

2.360

(a)

0.2240

0.4480

0.6720

0.8960

1.120

1.344

1.568

1.792

2.016

2.240

(b)

0.29300.58600.8790 1.172 1.465 1.758 2.051 2.344 2.637 2.930

0.26800.53600.8040 1.072 1.340 1.608 1.876 2.144 2.412 2.680

Fig.10The dimensionless pressure in liquid and equivalent stress in solid (for r0?1mm,v0?100m/s,and t?9ns)with di?erent solid materials: (a)1Cr13;(b)Ti-6Al-4V.

Dimensionless radius, R

(a)

Dimensionless radius, R

(b)

Fig.11The moving trace of the maximum equivalent stress points (for r0?1mm,v0?100m/s):(a)1Cr13;(b)Ti-6Al-4V.

1612Q.Zhou et al.

e 0c 0v 0T,where e,max is the maximum equivalent stress in solid at all time steps.

(ii)The dimensionless in?uence radius,R inf ,is the max-imum dimensionless radius reachable by the maximum stress (before the shock wave breaks o?),R inf ?r max =r 0.

(iii)The dimensionless in?uence depth,L inf ,is the max-imum depth at which the maximum stress may achieve before the shock wave breaks o?,L inf ?l max =r 0.

Figure 12(a)shows the maximum stress in the solid,which varies almost monotonically with the impact Mach number.Ti-6Al-4V has lower stress than 1Cr13,because it has lower Young’s modulus.The data may be ?tted as a power-law relationship:

1Cr13:?inf ?9:89132Ma 0:4188

0e27TTi-6Al-4V:?inf ?9:12974Ma 0:402020

e28T

The dimensionless in?uence radius and depth are given in

Fig.12(b).The two materials have almost same curves because of their similar sonic speeds.The in?uence radius basically depends on the movement of liquid-solid contact area rather than the properties of solid material,thus di?erent solid materials should have almost the same in?uence radius.Whereas,the in?uence depth primarily depends on the sonic speed of solid,and the in?uence depths of 1Cr13and Ti-6Al-4V are somewhat di?erent.

When the impact velocity is low (Ma 0<0:07),the impact-a?ected region in solid develops in both radial and axial (depth)directions.Although the maximum stress can reach a depth up to about 0.08,the magnitude of such stress is low which may not yet be su?cient to cause damage.When the impact velocity is moderate (Ma 0?0:07$0:2),the in?uenced region mainly develops in the redial direction and near the solid surface.When the impact velocity is very high (Ma 0>0:2),the a?ected zone again expands on both directions —the deep depth achievable plus the large magnitude of resulting stress make the high velocity impact most dangerous,and repetitive impact may cause the material near the surface to spall o?(with a thickness several percent of the water drop radius);this is water drop erosion.A qualitative ?tting can be obtained by:

1Cr13and Ti-6Al-4V:R inf ?0:37957Ma 0:6904

0e29T1Cr13:L inf ?1:21536Ma 0à9:24615Ma 20

t19:82268Ma 3

e30T

Ti-6Al-4V:L inf ?1:10598Ma 0à8:414Ma 20

t19:03864Ma 30

e31T

In this Section,we established a nonlinear wave model to solve the fully coupled liquid-solid impact problem,and detailed information of the stress ?eld in solid and pressure ?eld in liquid are obtained.The main features of the solutions,eqs.(27)$(31),which characterizes the peak stresses occurred during water drop-1Cr13and Ti-6Al-V impact,are expressed in dimensionless terms.When analyz-ing the fatigue limit,such explicit information can be used to calculate the dimensional values of the impact stress and in?uence range,and then substituted into a fatigue model to evaluate the erosion process in a typical blade channel made by a typical material.If one uses a di?erent material for the blade,similar results may be generated by following the framework in this paper.5.

Water Drop Erosion Analysis in a Typical Blade Channel

5.1A fatigue model

Upon repetitive and high-frequency impact,the erosion process of a typical turbine blade are similar to fatigue 29,30)and thus water drop erosion can be explored via fatigue analysis.31)For water drop-1Cr13and water drop-Ti-6Al-V impacts,the maximum impact stress and in?uence zone size (Section 4)can be superimposed with the statistics of impact (Section 3).Since the stress ?eld is multiaxial and relatively complex,we use the equivalent stress as a caliber to analyze the fatigue of solid.Although the magnitude of maximum equivalent impact stress is found to be irrelevant of the diameter of water drop,the position of critical stress point scales with the diameter,which is also related with fatigue analysis.Since the sizes of water drop as well as its in?uence zone are much smaller than the actual blade dimension,for simplicity,we ignore the di?erence of the in?uence zone size of di?erent particle sizes,and assume each particle would only cause damage directly beneath it.Thus,the problem of water drop erosion is transformed to the problem of fatigue cracking of the solid material under the in?uence of impact stress.

A simple fatigue model is established,from which the lifetime of the specimen is 32)

N i ?C i á 21tn eqv àá eqv àá21tn

th !à2

e32THere,N i is the fatigue lifetime;C i is the crack resistance

coe?cient;eá eqv Tth is the stress range of fatigue crack threshold,which is related with the fatigue limit of material.When á eqv eá eqv Tth ,N i !1.

5.2Erosion lifetime map

The material and fatigue parameters are obtained from our experiment.For 1Cr13:eá eqv Tth ?468MPa,C i ?24:3?1014,and n ?0:085.For Ti-6Al-V,eá eqv Tth ?586MPa,C i ?24:3?1014,and n ?0:https://www.360docs.net/doc/cd11596912.html,ing 1Cr13as an exam-ple,from normal impact tests,it is found that when the speed of incident water drop is below v ds ?150m/s for 1Cr13,no apparent erosion could be observed experimentally.Accord-

D i m e n s i o n l e s s s t r e s s , Σi n f

Dimensionless Mach number, Ma 0

(a)(b)

D i m e n s i o n l e s s i n f l u e n c e r a d i u s a n d d e p t h , R i n f , L i n f

Fig.12Important parameters in the solid with respect to the impact Mach number Ma 0:(a)the maximum dimensionless stress;(b)in?uence radius and depth.

Water Drop Erosion on Turbine Blades:Numerical Framework and Applications

1613

ing to eq.(32),if the water erosion fatigue limit is set as N i?107,the threshold value of the equivalent stress range can be calculated as:

eá eqvTth?468MPae33TAnd thus the threshold of the nominal stress range are obtained from the fatigue model:

eá nTds?726:85MPae34TWe rewrite eq.(27)in dimensional form:

inf?0:674738v1:4188

eMPaTe35TFinally,the threshold impact speed v ds of1Cr13can be calculated:

v ds?137m/se36TThe value predicted from our model is9%lower than the experimental value,which should be regarded as a good agreement given uncertainties(e.g.surface roughness,water ?lm on blade surface,stress concentration,etc.)encountered during the experiment,and thus validated that our multi-disciplinary models and numerical framework could e?ec-tively capture the most dominant aspects of water drop erosion.After same procedure,the threshold impact speed v ds of Ti-6Al-4V can be obtained as159m/s.It is higher than 1Cr13because of its higher fatigue limit value.

By employing the impact velocity/distribution of water drops on the blade surface(Section3),the erosion lifetime map turbine blade in the whole wet steam turbine stage is obtained as a function of impacted times.The result can be further translated to that represented by operating hours.With respect to eq.(13),if the entrance?ux of water drop(N0=A0) is given,the impact times per hour on an area of interest(áA) can be calculated by:

áN? s N0áA=A0(1/hour)e37TSince the number?ux(N0)and diameter of water drops are di?cult to be measured accurately at the entrance,for convenience,an empirical coe?cient C n is introduced and the lifetime counted by operating hours is(N f is the impact times to failure)

át?N f=áN?C n N f= s(hour)e38TThe value of C n can be obtained from engineering practice or experimental data.We estimate the value of C n of the target turbine blade is about10?105hour(under operating condition No.1).Due to the value of enthalpy of steam of reduced operating condition No.2,the number of water drops N0is estimated to be2times of that of No.1.

Figure13shows lifetime distributions on back/inner arcs upon di?erent operating conditions of1Cr13.On both arcs with full load(Fig.13(a),(b)),because the impact speed is less than the threshold speed,the minimum lifetime of whole surface is more than40?105hours.However under the operating condition No.2(Fig.13(c),(d)),due to high impact speed and high impact frequency,the minimum lifetime of the surface is about1:6?105hours.For Ti-6Al-4V(Fig.14),the corresponding value is2:2?105hours which is37%longer than1Cr13.For both materials,the reduced operating condition is always more dangerous,and the inner arc su?ers to more severe damage than the back arc. Finally,the most dangerous water drop erosion region and operating condition can be deduced:Under reduced load condition,the head of back arc of the blade and the band that spans from the bottom front to the middle of trailing edge are likely to su?er the most severe water drop erosion, with the second region more critical.

6.Conclusion

In this study,we developed multidisciplinary models and numerical tools to study the water drop erosion in turbine https://www.360docs.net/doc/cd11596912.html,putational?uid dynamics and particle model are employed to simulate the trajectories of water drops in a?ow of wet steam in the blade channel,and the most important statistics of impact are obtained.A nonlinear wave model and relevant analytical and numerical algorithms are established to explore the fundamental aspects of liquid drop-solid impact,which leads to the erosion analysis based on a fatigue model.Results are speci?ed for water drop normal impact on two representative blade alloys,1Cr13and Ti-6Al-4V,and the lifetime distribution on blade surface under typical working conditions are obtained.Important conclu-sions include:

(1)Once the initial diameter of water drops gets larger

than50m m,the distribution maps of impact frequency

Bottom

Top

4.000

8.000

12.00

16.00

20.00

24.00

28.00

32.00

36.00

40.00

Bottom

Top

4.000

8.000

12.00

16.00

20.00

24.00

28.00

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36.00

40.00

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8.000

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36.00

40.00

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4.000

8.000

12.00

16.00

20.00

24.00

28.00

32.00

36.00

40.00

(a)(b)(c)(d)

(Unit: 105 operating hours Water drop diameter50m)

Fig.13Lifetime map of metal surface in terms of operating hours,for water-1Cr13impact:(a)back arc upon operating condition No.1;(b)inner arc upon operating condition No.1;(c)back arc upon operating condition No.2;(d)inner arc upon operating condition No.2.

Bottom

Top

4.000

8.000

12.00

16.00

20.00

24.00

28.00

32.00

36.00

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4.000

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(a)(b)(c)(d)

(Unit: 105 operating hours Water drop diameter50μm)

Fig.14Lifetime map of metal surface in terms of operating hours,for water-Ti-6Al-4V impact:(a)back arc upon operating condition No.1;(b) inner arc upon operating condition No.1;(c)back arc upon operating condition No.2;(d)inner arc upon operating condition No.2.

1614Q.Zhou et al.

and velocity on the blade surface are essentially the same.In other words,the statistics of impact fre-quency and velocity of large water particles may be replaced by that of a?xed particle size(e.g.50m m), which may save computational time considerably.

Regions where high impact velocity and frequency coexist may represent potentially dangerous areas for water drop erosion.

(2)The nonlinear wave model could capture the most

essential characteristics of high-velocity liquid-solid impact,by ignoring the viscosity of?uid.The3D model fully couples liquid pressure/compressibility and solid stress/deformation,and the solution focuses on the transient e?ects which dominate the high-velocity impact process.The transmission speed of shock wave in the radial direction is faster than that in the axial direction in the water drop.Two high transient stress regions are identi?ed in the solid: one near the edge of the contact area and the other one near the normal axis(and several m m below the surface).

(3)The dimensionless parameters de?ned in this paper can

well describe the impact procedure for particles of all sizes.The characteristic dimensionless scale(e.g.

in?uence zone radius and depth),dimensionless time

(e.g.time when shock wave breaks o?the liquid body),

dimensionless pressure(e.g.the maximum pressure at contact surface),and dimensionless stress(e.g.the maximum transient equivalent stress inside solid)are nonlinear functions of the impact https://www.360docs.net/doc/cd11596912.html,par-isons between impacts on1Cr13and Ti-6Al-4V are carried out,whose di?erence mainly arises from their di?erent moduli.The stress patterns are close in these two materials,because their sonic speeds are close.

The maximum dimensionless stress in the solid varies nonlinearly with the impact Mach number,in a form

close to power-law,?inf?9:89132Ma0:4188

0for water-

1Cr13impact and?inf?9:12974Ma0:40202

0for water-

Ti-6Al-4V impact.For other materials similar functions may be?tted by following the same procedures using the numerical framework.

(4)With the development of a fatigue model,the most

dangerous water drop erosion regions are deduced for a representative steam turbine upon typical operating conditions.The maps of erosion lifetime in terms of operations hours are obtained.Under a reduced load, the head of blade top(back arc)and the band that spans from the bottom front to the middle part of trailing edge are most critical regions for water drop erosion,which agrees with previous experimental observations.Thus, in order to alleviate water erosion,the blade should avoid to be operated under reduced load.The erosion resistance is material-dependent:although having sim-ilar sonic speed,the life time of Ti-6Al-4V is37% longer than1Cr13.

The analytical and numerical framework established in this paper is quite?exible and it may be extended to other liquid-solid impact/erosion problems,such as rain drop erosion.Acknowledgement

The work is supported in part by Hitachi Ltd.,in part by the National Natural Science Foundation of China (50276051),the National Basic Research Program of China(2005CB221206),U.S.National Science Foundation CMS-0407743and CAREER-CMMI-0643726,and in part by the Department of Civil Engineering and Engineering Mechanics,Columbia University.

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