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Numerical investigation of the solid particle erosion rate in a steam turbine nozzle
Applied Thermal Engineering 27 (2007) 2394–2403 www.elsevier.com/locate/apthermeng
Numerical investigation of the solid particle erosion rate in a steam turbine nozzle Alfonso Campos-Amezcua a, Armando Gallegos-Mun˜oz b, C. Alejandro Romero b, Zdzislaw Mazur-Czerwiec a,*, Rafael Campos-Amezcua a a
Instituto de Investigaciones Ele´ctricas, Gerencia de Turbomaquinaria, Av. Reforma No. 113, Col Palmira, Cuernavaca, Mor. C.P. 62490, Mexico b Department of Mechanical Engineering, University of Guanajuato, Av. Tampico 912, Col. Bellavista, Salamanca, Gto. C.P. 36730, Mexico Received 6 December 2005; accepted 1 March 2007 Available online 19 March 2007
Abstract A numerical investigation of solid particle erosion in the nozzle of 300 MW steam turbine is presented. The analysis consists in the application of the discrete phase model, for modelling the solid particles flow, and the Eulerian conservation equations to the continuous phase. The numerical study employs a computational fluid dynamics (CFD) software, based on a finite volume method. The investigation permits us to know the influence of the parameters such as: particle diameter, impact angle, particle velocity and particle distribution on the erosion rate in the surface of the nozzle. These parameters are analyzed to different operational conditions in the turbine. The results show the eroded zone which increases the throat area through the nozzle provoking changes in the operation conditions. When the throat area increases the turbine demands an increase of steam flow to maintain the power supply. On the other hand, it is shown that the solid particles flow cause the most severe the erosion rate whereas the steam mass flow rate is the most sensitive parameter. Finally it is obtained that the erosion rate decreases as the diameter of the particle increases in a nearly linear form. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Solid particle erosion; Abrasive wear; Steam turbine; Numerical simulation
1. Introduction The erosion of steam turbine blades has been one of the important problems in power generation systems. It was proved that the collision of solid particles have brought about severe erosion problems in steam turbine blades, causing a high cost of maintenance and repair as well as a safety problem and low efficiency of power generation [1]. Many of the factors which control the rate of erosion, such as particle velocity or particle mass flow rate, particle diameter, impact angle and particle distribution can be studied at different flow conditions of the system. A lot of practical examples may be found when a change in flow conditions has greatly increased or decreased erosion. In *
Corresponding author. Tel.: +52 7773623811; fax: +52 7773623834. E-mail address: [email protected] (Z. Mazur-Czerwiec).
1359-4311/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2007.03.010
general where the flow direction changes rapidly, the erosion is usually considerably more severe than in straight lines, though it has also been reported that local turbulence due to a roughened surface or misalignment can increase the rate of erosion damage [2]. There are several numerical works of solid particle erosion in blades, but simplifying the event or under other conditions and geometry [3–5], which changes the obtained results. The experimental study on the dynamic behaviour of solid particles entrained within a steam flow in complex geometries requires special equipment and methodology to pursue this goal. Also, the erosion process is a complex problem to obtain a mathematical formula to account for some of the factors which control the rate of erosion of the blades. This paper presents a numerical study of the erosion process, applying Computational Fluid Dynamics (CFD) and considering different parameters such as
A. Campos-Amezcua et al. / Applied Thermal Engineering 27 (2007) 2394–2403
2395
Nomenclature Aface CD C1e C2e DH dp F FD Fother f(a) Gk k L m_ v m_ p Re Rerosion Sm
area (m2) drag coefficient (–) coefficient (–) coefficient (–) hydraulic diameter (m) particle diameter (m) force (N) drag force (N) additional forces (N) impact angle function (–) turbulent kinetic energy generated due to mean velocity gradients (kJ/kg) turbulent kinetic energy (kJ/kg) turbulent length scale (m) steam mass flow rate (kg/s) particle mass flow rate (kg/s) Reynolds number (–) erosion rate (kg/ms2) mass of the dispersed second phase (kg/s)
particle mass flow rate, particle diameter and steam flow conditions, in the first steam turbine stationery stage (nozzle) of 300 MW to determine the erosion rate. 2. Description of analysis The numerical study of the erosion process applying CFD, considers a mathematical model with Eulerian conservation equations in the continuous phase and a Lagrangian frame to simulate a discrete second phase. The dispersion of particles in the fluid phase can be predicted using a stochastic tracking model. This model includes the effect of instantaneous turbulent velocity fluctuations on the particle trajectories. 2.1. Governing equations The computational domain considers the mass conservation and momentum equations for incompressible flow in a 3D geometry in a steady state. The mass conservation is r ðqvÞ ¼ S m
ð1Þ
where Sm is the mass added to the continuous phase from the dispersed second phase. The momentum equation is r ðqvvÞ ¼ rp þ r s þ F;
ð2Þ
where s is the stress tensor and F are the forces that arise from interaction with the dispersed phase. For turbulence, the numerical study includes the standard k–e model [6], where the turbulent kinetic energy equation, k, is
TL t u up v x
integral time scale (–) time (s) velocity (m/s) particle velocity (m/s) velocity vector (m/s) distance coordinate (m)
Greek symbols a impact angle e turbulent kinetic energy dissipation rate (kJ/kg-s) l molecular dynamic viscosity (kg/m s) lt turbulent viscosity (kg/m s) q density (kg/m3) r Prandtl number (–) rk turbulent Prandtl number for k (–) re turbulent Prandtl number for e (–) s stress tensor (N/m2) $ operator nabla (–)
oðqkui Þ o ¼ oxi oxj
lþ
lt ok þ Gk qe rk oxj
and the dissipation rate equation, e, is oðqeui Þ o l oe e e2 ¼ lþ t þ C 1e Gk C 2e q oxi oxj k re oxj k
ð3Þ
ð4Þ
where Gk represents the generation of turbulent kinetic energy due the mean velocity gradients, C1e and C2e are constants (C1e = 1.44, C2e = 1.92) and rk, re are the turbulent Prandtl numbers for k and e (rk = 1.0, re = 1.3). 2.2. Discrete phase model (DPM) This model permits us to simulate a discrete second phase in a Lagrangian frame of reference, where the second phase consists of spherical particles dispersed in the continuous phase. The coupling between the phases and its impact on both the discrete phase trajectories and the continuous phase flow is included. The turbulent dispersion of particles is modelled using a stochastic discrete-particle approach. This approach predicts the turbulent dispersion by integrating the trajectory equations for individual particles, using the instantaneous fluid velocity. The prediction of particle dispersion makes use of the concept of the integral time scale, TL, which describes the time spent in turbulent motion along the particle path. This time scale can be approximated in the standard k–e model as T L 0:15
k e
ð5Þ
2396
A. Campos-Amezcua et al. / Applied Thermal Engineering 27 (2007) 2394–2403
The trajectory of a discrete phase particle can be predicted by integrating the force balance on the particle, which is written in a Lagrangian reference frame. This force balance equates the particle inertia with the forces acting on the particle, dup ¼ F D ðu up Þ þ F other dt
ð6Þ
where the drag force per unit particle mass is FD(u up), and FD is defined as FD ¼
18l C D Re qd d 2p 24
ð7Þ
The additional forces, Fother, in this case is compounded by the force required to accelerate the fluid surrounding the particle, defined as F other ¼
1 q d ðu up Þ 2 qp dt
ð8Þ
which is important when q > qp, and an additional force due to the pressure gradient in the fluid F other ¼
q ou up qp ox
qd p ðup uÞ l
Finally, to evaluate the erosion rate at the wall of the blades, applying the discrete phase model, it is important to define parameters such as: the mass flow rate of the particle stream, m_ p , impact angle of the particle path with the wall face a, function of the impact angle f(a), and the area of the wall face where the particle strikes the boundary Aface. The erosion rate is defined as N particle
Rerosion ¼
X m_ p Cðd p Þf ðaÞvbðvÞ Aface p¼1
ð12Þ
ð9Þ Table 1 Mesh refinement on the blade surface
The relative Reynolds number, Re, is defined as Re ¼
where the drag coefficient, CD, is applied for smooth spherical particles. To incorporate the effect of the discrete phase trajectories on the continuum it is important to compute the interphase exchange of momentum from the particle to the continuous phase. This exchange is computed by examining the change in momentum of a particle as it passes through each control volume in the computational domain. This momentum change is computed as ! X 18l C D Re ðup uÞ þ F other m_ p Dt F ¼ ð11Þ qd d 2p 24
Mesh density
Cells
Erosion rate (kg/m2 s)
ð10Þ
Fig. 1. 3D computational domain of the turbine control stage.
9462
2.03
12,312
1.86
24,624
3.00
49,248
3.02
A. Campos-Amezcua et al. / Applied Thermal Engineering 27 (2007) 2394–2403
where C(dp) is a function of particle diameter, a is the impact angle of the particle path with the wall face, f(a) is a function of impact angle, v is the relative particle velocity, b(v) is a function of relative particle velocity, and Aface is
the area of the cell face at the wall. In Eq. (12) the impact angle function was defined by a piece-linear profile [7] and the diameter function and velocity exponent function are 1.8E09 and 2.6, respectively [8].
Table 2 Mesh refinement in the fluid (near blade surface) Mesh density
2397
Cells
Erosion rate (kg/m2 s)
24,624
3.00
29,712
2.91
31,452
2.54
35,892
2.28
40,980
2.17
Mesh detail
2398
A. Campos-Amezcua et al. / Applied Thermal Engineering 27 (2007) 2394–2403
2.3. Computational domain and grid Fig. 1 shows the 3D computational domain of the steam turbine nozzle (stationary control stage). It shows the grid on the pressure surface in the front. Geometric construction and meshing were performed with GAMBIT applying 40,980 hexahedron cells. 2.4. Boundary conditions The mass flow rate and total temperature was imposed as well as the components of flow direction. The rotational periodic type boundary condition is applied for all the variables. The turbulent kinetic energy, k, and the dissipation
ratio, e, were also specified from the turbulence intensity (I = 5%) and the turbulence length scale, L = DH, based on the hydraulic diameter of the steam path area. For the base running, 50lm diameter particles are released from the inlet, and the solid mass flow rate is 1 kg/s. 3. Results and discussion The numerical analysis was conducted using a commercial computer CFD code Fluent V6.2.16 on the basis of the finite volume method. Tables 1 and 2 summarize the analyses of grid sensibility. Table 1 shows the results on grid refinement on the blade surface whereas Table 2 shows the grid refinement in the fluid (near blade surface).
Fig. 2. The contours of velocity in the passage at the mid-plane (m/s).
Fig. 3. The contours of pressure in the passage at the mid-plane (Pa).
A. Campos-Amezcua et al. / Applied Thermal Engineering 27 (2007) 2394–2403
The independence of the results on the mesh was obtained with the last model. A converged solution was obtained for a base running ðm_ v ¼ 4:5 kg=sÞ in 818 iterations. For this case, the volume fraction of particles with respect to the steam is 0.12%, whereas the mass fraction is 22.22%. The convergence of residuals for continuity, j and e equations were resolved to levels of 1E3 while the energy equation was set to a level of 1E06. The interactions between continuous and discrete phase models were set at 15. Figs. 2–6 show numerical results obtained by the simulation for a base running.
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Fig. 2 shows the contours of velocity at the mid-span (50% vane height) passage. The maximum steam flow velocity (continuous phase) reaches 404 m/s in the throat passage. The minimum value is in the leading edge, which corresponds to the stagnation point. The pressure distribution at the mid-span is shown in Fig. 3. The maximum pressure on the leading edge has a value of 13.8 MPa, while the minimum is on the suction side in the throat region. Fig. 4 presents the particle tracks released from inlet domain. The particle tracks and the path lines for base running are shown in Figs. 4 and 5. Five particles that impact on
Fig. 4. Five particle tracks for base running.
Fig. 5. Path lines for base running.
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Fig. 6. Contours of DPM erosion rate for base running.
the leading edge where selected to follow their trajectory and locate the impact zone. Fig. 6 presents the contours of erosion rate for the base running. It is observed that maximum value is located at the trailing edge, on the pressure side. This value reaches
2.27 kg/m2s and corresponds to the throat zone. In the leading edge the erosion rate is minimum compared to the trailing edge, near to 0.05 kg/m2 s. Variation in steam mass flow rate, solid mass flow rate and diameter of the particles were done to predict the
Table 3 Number of particles and their fractions for each case m_ v (kg/s)
Particles/m3 steam
Variation of steam mass flow rate 1.5 5.44E+10 2 4.08E+10 2.5 3.26E+10 3 2.72E+10 3.5 2.33E+10 4 2.04E+10 4.5 1.81E+10 m_ p (kg/s)
Particles/m3 steam
Variation of solid mass flow rate 0.6 1.09E+10 0.7 1.27E+10 0.8 1.45E+10 0.9 1.63E+10 1 1.81E+10 1.1 1.99E+10 1.2 2.17E+10 1.3 2.36E+10 1.4 2.54E+10 Diameter (m)
Particles/m3 steam
Variation of particle diameter 5.00E05 1.81E+10 6.00E05 1.05E+10 7.00E05 6.60E+09 8.00E05 4.42E+09 9.00E05 3.11E+09 1.00E04 2.27E+09
Volume fraction (%)
Mass fraction (%)
0.36 0.27 0.21 0.18 0.15 0.13 0.12
66.67 50.00 40.00 33.33 28.57 25.00 22.22
Volume fraction (%)
Mass fraction (%)
0.07 0.08 0.09 0.11 0.12 0.13 0.14 0.15 0.17
13.33 15.56 17.78 20.00 22.22 24.44 26.67 28.89 31.11
Volume fraction (%)
Mass fraction (%)
0.12 0.12 0.12 0.12 0.12 0.12
22.22 22.22 22.22 22.22 22.22 22.22
A. Campos-Amezcua et al. / Applied Thermal Engineering 27 (2007) 2394–2403 2.5
2401
4
Maximum erosion rate [kg/m2s]
maximum erosion rate [kg/m2s]
3.5 2
1.5
1
0.5
3 2.5 2 1.5 1 0.5
0
0 1.5
2
2.5
3
3.5
4
4.5
steam mass flow rate [kg/s]
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
solid particle mass flow rate [kg/s]
Fig. 7. Maximum erosion rate vs. steam mass flow rate.
Fig. 9. Maximum erosion rate vs. particle mass flow rate.
erosion rate. Table 3 shows, for each case, the number of particles per unit volume of steam and their percentage in volume and mass with respect to the steam. In the first case, seven boundary conditions were set: from m_ v ¼ 1:5 kg/s to m_ v ¼ 4:50 kg=s with increases of 0.5 in addition their appropriate output pressure is updated for each steam mass flow rate. Fig. 7 illustrates the variation of maximum erosion rate with respect to steam mass flow rate. As the vapor mass flow diminishes, the maximum erosion rate also diminishes, due to the particle velocity decrease with the velocity of the continuous phase. Fig. 8 presents the contours of erosion rate for the maximum value of erosion rate in the range analyzed, which correspond to the part-load m_ v ¼ 3:5 kg=s. In the second case, solid particles mass flow rate injected changes from 0.6 kg/s to 1.4 kg/s with increments of 0.2, maintaining the rest of the variables of the base running.
Fig. 9 illustrates the effect of the discrete phase flow rate over the maximum erosion rate. As the solid particles mass flow rate increases, the maximum erosion rate also increases reaching a maximum value of 3.34 kg/m2 s, diminishing slightly later with the increment of the solid particle mass flow rate. Fig. 10 presents the contours of erosion rate for a solid particles mass flow rate of 1.2 kg/s, which shows the maximum erosion just in the upper border of the trailing edge. Finally, a third case is analysed, the effect of diameter of the particles over the maximum erosion rate, Fig. 11. The diameter changes from 50 lm up 100 lm at increments of 10. It is observed that when the particle diameter increases the erosion rate diminishes in a nearly linear form, changing from 2.27 to 1.38 kg/m2 s, this performance is caused by the fact that as the diameter increases, the amount of particles diminishes for a fixed solid particles mass flow
Fig. 8. Contours of DPM erosion rate for a part-load ðm_ v ¼ 3:5 kg=sÞ.
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Fig. 10. Contours of DPM erosion rate for a solid mass flow rate of 1.2 kg/s.
rate. Fig. 12 shows the erosion rate predicted for a diameter of particle of 50 lm, where the area eroded is larger than the other cases, and is not only at the trailing edge.
4
Maximum erosion rate [kg/m2s]
3.5 3 2.5
4. Conclusions
2 1.5 1 0.5 0 50
60
70
80
90
diameter of particles [microns]
Fig. 11. Maximum erosion rate vs. diameter of the particles.
100
In the present work continuous–discrete phase models are used to predict the erosion rate on surface blades. The effect of vapor mass flow rate, diameter of the particle and solid mass flow rate is studied by comparing the results, obtained by solving CFD problem. The comparison shows that the effect of solid mass flow rate, in the range presented, is more important than steam mass flow rate on erosion rate. On the other hand, the erosion rate obtained varying diameter of particles shows that
Fig. 12. Contours of DPM erosion rate for a diameter of particle of 10 lm.
A. Campos-Amezcua et al. / Applied Thermal Engineering 27 (2007) 2394–2403
the maximum values of erosion rate decrease, while the particles diameter increases. However, a definite conclusion in this regard may await more experimental data on erosion rate in blades, and comparison with other erosion models. References [1] W. Tabakoff, A. Hamed, R. Wenglarz, Particulate Flows, Turbomachinery Erosion and Performance Deterioration, Von Karman Lecture Series 1988–89, Brussels, Belgium, 1988. [2] G.I. Parslow, D.J. Stephenson, J.E. Strutt, S. Tellow, Investigation of solid particle erosion in components of complex geometry, Wear 23 (1999) 737–745.
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[3] S. Ivanov, A numerical simulation of particle laden flow through airfoil cascade, in: International Conference on Computer Systems and Technologies – CompSysTech’2005. [4] R.W. Lyczkowski, J.X. Bouillard, State-of-the-art review of erosion modelling in fluid/solid systems, Progress in Energy and Combustion Science 28 (2002) 543–602. [5] A. Hamed, W. Tabakoff, R.B. Rivir, K. Das, P. Arora, Turbine blade surface deterioration by rosion, Journal of Turbomachinery 127 (2005) 445–452. [6] B.E. Launder, D.B. Spalding, The numerical computation of turbulent flows, Computer Methods in Applied Mechanics and Engineering 3 (1974) 269–289. [7] FLUENT V6.2.16 User’s Guide, Fluent Inc., Canterra Resource Park, 10 Covendish Court, Lebanon NH03766, EUA, 2003. [8] K.C. Cotton, Evaluating and Improving Steam Turbine Performance, Cotton Fact Inc., New York, 1993.
Numerical investigation of the solid particle erosion rate in a steam turbine nozzle Alfonso Campos-Amezcua a, Armando Gallegos-Mun˜oz b, C. Alejandro Romero b, Zdzislaw Mazur-Czerwiec a,*, Rafael Campos-Amezcua a a
Instituto de Investigaciones Ele´ctricas, Gerencia de Turbomaquinaria, Av. Reforma No. 113, Col Palmira, Cuernavaca, Mor. C.P. 62490, Mexico b Department of Mechanical Engineering, University of Guanajuato, Av. Tampico 912, Col. Bellavista, Salamanca, Gto. C.P. 36730, Mexico Received 6 December 2005; accepted 1 March 2007 Available online 19 March 2007
Abstract A numerical investigation of solid particle erosion in the nozzle of 300 MW steam turbine is presented. The analysis consists in the application of the discrete phase model, for modelling the solid particles flow, and the Eulerian conservation equations to the continuous phase. The numerical study employs a computational fluid dynamics (CFD) software, based on a finite volume method. The investigation permits us to know the influence of the parameters such as: particle diameter, impact angle, particle velocity and particle distribution on the erosion rate in the surface of the nozzle. These parameters are analyzed to different operational conditions in the turbine. The results show the eroded zone which increases the throat area through the nozzle provoking changes in the operation conditions. When the throat area increases the turbine demands an increase of steam flow to maintain the power supply. On the other hand, it is shown that the solid particles flow cause the most severe the erosion rate whereas the steam mass flow rate is the most sensitive parameter. Finally it is obtained that the erosion rate decreases as the diameter of the particle increases in a nearly linear form. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Solid particle erosion; Abrasive wear; Steam turbine; Numerical simulation
1. Introduction The erosion of steam turbine blades has been one of the important problems in power generation systems. It was proved that the collision of solid particles have brought about severe erosion problems in steam turbine blades, causing a high cost of maintenance and repair as well as a safety problem and low efficiency of power generation [1]. Many of the factors which control the rate of erosion, such as particle velocity or particle mass flow rate, particle diameter, impact angle and particle distribution can be studied at different flow conditions of the system. A lot of practical examples may be found when a change in flow conditions has greatly increased or decreased erosion. In *
Corresponding author. Tel.: +52 7773623811; fax: +52 7773623834. E-mail address: [email protected] (Z. Mazur-Czerwiec).
1359-4311/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2007.03.010
general where the flow direction changes rapidly, the erosion is usually considerably more severe than in straight lines, though it has also been reported that local turbulence due to a roughened surface or misalignment can increase the rate of erosion damage [2]. There are several numerical works of solid particle erosion in blades, but simplifying the event or under other conditions and geometry [3–5], which changes the obtained results. The experimental study on the dynamic behaviour of solid particles entrained within a steam flow in complex geometries requires special equipment and methodology to pursue this goal. Also, the erosion process is a complex problem to obtain a mathematical formula to account for some of the factors which control the rate of erosion of the blades. This paper presents a numerical study of the erosion process, applying Computational Fluid Dynamics (CFD) and considering different parameters such as
A. Campos-Amezcua et al. / Applied Thermal Engineering 27 (2007) 2394–2403
2395
Nomenclature Aface CD C1e C2e DH dp F FD Fother f(a) Gk k L m_ v m_ p Re Rerosion Sm
area (m2) drag coefficient (–) coefficient (–) coefficient (–) hydraulic diameter (m) particle diameter (m) force (N) drag force (N) additional forces (N) impact angle function (–) turbulent kinetic energy generated due to mean velocity gradients (kJ/kg) turbulent kinetic energy (kJ/kg) turbulent length scale (m) steam mass flow rate (kg/s) particle mass flow rate (kg/s) Reynolds number (–) erosion rate (kg/ms2) mass of the dispersed second phase (kg/s)
particle mass flow rate, particle diameter and steam flow conditions, in the first steam turbine stationery stage (nozzle) of 300 MW to determine the erosion rate. 2. Description of analysis The numerical study of the erosion process applying CFD, considers a mathematical model with Eulerian conservation equations in the continuous phase and a Lagrangian frame to simulate a discrete second phase. The dispersion of particles in the fluid phase can be predicted using a stochastic tracking model. This model includes the effect of instantaneous turbulent velocity fluctuations on the particle trajectories. 2.1. Governing equations The computational domain considers the mass conservation and momentum equations for incompressible flow in a 3D geometry in a steady state. The mass conservation is r ðqvÞ ¼ S m
ð1Þ
where Sm is the mass added to the continuous phase from the dispersed second phase. The momentum equation is r ðqvvÞ ¼ rp þ r s þ F;
ð2Þ
where s is the stress tensor and F are the forces that arise from interaction with the dispersed phase. For turbulence, the numerical study includes the standard k–e model [6], where the turbulent kinetic energy equation, k, is
TL t u up v x
integral time scale (–) time (s) velocity (m/s) particle velocity (m/s) velocity vector (m/s) distance coordinate (m)
Greek symbols a impact angle e turbulent kinetic energy dissipation rate (kJ/kg-s) l molecular dynamic viscosity (kg/m s) lt turbulent viscosity (kg/m s) q density (kg/m3) r Prandtl number (–) rk turbulent Prandtl number for k (–) re turbulent Prandtl number for e (–) s stress tensor (N/m2) $ operator nabla (–)
oðqkui Þ o ¼ oxi oxj
lþ
lt ok þ Gk qe rk oxj
and the dissipation rate equation, e, is oðqeui Þ o l oe e e2 ¼ lþ t þ C 1e Gk C 2e q oxi oxj k re oxj k
ð3Þ
ð4Þ
where Gk represents the generation of turbulent kinetic energy due the mean velocity gradients, C1e and C2e are constants (C1e = 1.44, C2e = 1.92) and rk, re are the turbulent Prandtl numbers for k and e (rk = 1.0, re = 1.3). 2.2. Discrete phase model (DPM) This model permits us to simulate a discrete second phase in a Lagrangian frame of reference, where the second phase consists of spherical particles dispersed in the continuous phase. The coupling between the phases and its impact on both the discrete phase trajectories and the continuous phase flow is included. The turbulent dispersion of particles is modelled using a stochastic discrete-particle approach. This approach predicts the turbulent dispersion by integrating the trajectory equations for individual particles, using the instantaneous fluid velocity. The prediction of particle dispersion makes use of the concept of the integral time scale, TL, which describes the time spent in turbulent motion along the particle path. This time scale can be approximated in the standard k–e model as T L 0:15
k e
ð5Þ
2396
A. Campos-Amezcua et al. / Applied Thermal Engineering 27 (2007) 2394–2403
The trajectory of a discrete phase particle can be predicted by integrating the force balance on the particle, which is written in a Lagrangian reference frame. This force balance equates the particle inertia with the forces acting on the particle, dup ¼ F D ðu up Þ þ F other dt
ð6Þ
where the drag force per unit particle mass is FD(u up), and FD is defined as FD ¼
18l C D Re qd d 2p 24
ð7Þ
The additional forces, Fother, in this case is compounded by the force required to accelerate the fluid surrounding the particle, defined as F other ¼
1 q d ðu up Þ 2 qp dt
ð8Þ
which is important when q > qp, and an additional force due to the pressure gradient in the fluid F other ¼
q ou up qp ox
qd p ðup uÞ l
Finally, to evaluate the erosion rate at the wall of the blades, applying the discrete phase model, it is important to define parameters such as: the mass flow rate of the particle stream, m_ p , impact angle of the particle path with the wall face a, function of the impact angle f(a), and the area of the wall face where the particle strikes the boundary Aface. The erosion rate is defined as N particle
Rerosion ¼
X m_ p Cðd p Þf ðaÞvbðvÞ Aface p¼1
ð12Þ
ð9Þ Table 1 Mesh refinement on the blade surface
The relative Reynolds number, Re, is defined as Re ¼
where the drag coefficient, CD, is applied for smooth spherical particles. To incorporate the effect of the discrete phase trajectories on the continuum it is important to compute the interphase exchange of momentum from the particle to the continuous phase. This exchange is computed by examining the change in momentum of a particle as it passes through each control volume in the computational domain. This momentum change is computed as ! X 18l C D Re ðup uÞ þ F other m_ p Dt F ¼ ð11Þ qd d 2p 24
Mesh density
Cells
Erosion rate (kg/m2 s)
ð10Þ
Fig. 1. 3D computational domain of the turbine control stage.
9462
2.03
12,312
1.86
24,624
3.00
49,248
3.02
A. Campos-Amezcua et al. / Applied Thermal Engineering 27 (2007) 2394–2403
where C(dp) is a function of particle diameter, a is the impact angle of the particle path with the wall face, f(a) is a function of impact angle, v is the relative particle velocity, b(v) is a function of relative particle velocity, and Aface is
the area of the cell face at the wall. In Eq. (12) the impact angle function was defined by a piece-linear profile [7] and the diameter function and velocity exponent function are 1.8E09 and 2.6, respectively [8].
Table 2 Mesh refinement in the fluid (near blade surface) Mesh density
2397
Cells
Erosion rate (kg/m2 s)
24,624
3.00
29,712
2.91
31,452
2.54
35,892
2.28
40,980
2.17
Mesh detail
2398
A. Campos-Amezcua et al. / Applied Thermal Engineering 27 (2007) 2394–2403
2.3. Computational domain and grid Fig. 1 shows the 3D computational domain of the steam turbine nozzle (stationary control stage). It shows the grid on the pressure surface in the front. Geometric construction and meshing were performed with GAMBIT applying 40,980 hexahedron cells. 2.4. Boundary conditions The mass flow rate and total temperature was imposed as well as the components of flow direction. The rotational periodic type boundary condition is applied for all the variables. The turbulent kinetic energy, k, and the dissipation
ratio, e, were also specified from the turbulence intensity (I = 5%) and the turbulence length scale, L = DH, based on the hydraulic diameter of the steam path area. For the base running, 50lm diameter particles are released from the inlet, and the solid mass flow rate is 1 kg/s. 3. Results and discussion The numerical analysis was conducted using a commercial computer CFD code Fluent V6.2.16 on the basis of the finite volume method. Tables 1 and 2 summarize the analyses of grid sensibility. Table 1 shows the results on grid refinement on the blade surface whereas Table 2 shows the grid refinement in the fluid (near blade surface).
Fig. 2. The contours of velocity in the passage at the mid-plane (m/s).
Fig. 3. The contours of pressure in the passage at the mid-plane (Pa).
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The independence of the results on the mesh was obtained with the last model. A converged solution was obtained for a base running ðm_ v ¼ 4:5 kg=sÞ in 818 iterations. For this case, the volume fraction of particles with respect to the steam is 0.12%, whereas the mass fraction is 22.22%. The convergence of residuals for continuity, j and e equations were resolved to levels of 1E3 while the energy equation was set to a level of 1E06. The interactions between continuous and discrete phase models were set at 15. Figs. 2–6 show numerical results obtained by the simulation for a base running.
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Fig. 2 shows the contours of velocity at the mid-span (50% vane height) passage. The maximum steam flow velocity (continuous phase) reaches 404 m/s in the throat passage. The minimum value is in the leading edge, which corresponds to the stagnation point. The pressure distribution at the mid-span is shown in Fig. 3. The maximum pressure on the leading edge has a value of 13.8 MPa, while the minimum is on the suction side in the throat region. Fig. 4 presents the particle tracks released from inlet domain. The particle tracks and the path lines for base running are shown in Figs. 4 and 5. Five particles that impact on
Fig. 4. Five particle tracks for base running.
Fig. 5. Path lines for base running.
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Fig. 6. Contours of DPM erosion rate for base running.
the leading edge where selected to follow their trajectory and locate the impact zone. Fig. 6 presents the contours of erosion rate for the base running. It is observed that maximum value is located at the trailing edge, on the pressure side. This value reaches
2.27 kg/m2s and corresponds to the throat zone. In the leading edge the erosion rate is minimum compared to the trailing edge, near to 0.05 kg/m2 s. Variation in steam mass flow rate, solid mass flow rate and diameter of the particles were done to predict the
Table 3 Number of particles and their fractions for each case m_ v (kg/s)
Particles/m3 steam
Variation of steam mass flow rate 1.5 5.44E+10 2 4.08E+10 2.5 3.26E+10 3 2.72E+10 3.5 2.33E+10 4 2.04E+10 4.5 1.81E+10 m_ p (kg/s)
Particles/m3 steam
Variation of solid mass flow rate 0.6 1.09E+10 0.7 1.27E+10 0.8 1.45E+10 0.9 1.63E+10 1 1.81E+10 1.1 1.99E+10 1.2 2.17E+10 1.3 2.36E+10 1.4 2.54E+10 Diameter (m)
Particles/m3 steam
Variation of particle diameter 5.00E05 1.81E+10 6.00E05 1.05E+10 7.00E05 6.60E+09 8.00E05 4.42E+09 9.00E05 3.11E+09 1.00E04 2.27E+09
Volume fraction (%)
Mass fraction (%)
0.36 0.27 0.21 0.18 0.15 0.13 0.12
66.67 50.00 40.00 33.33 28.57 25.00 22.22
Volume fraction (%)
Mass fraction (%)
0.07 0.08 0.09 0.11 0.12 0.13 0.14 0.15 0.17
13.33 15.56 17.78 20.00 22.22 24.44 26.67 28.89 31.11
Volume fraction (%)
Mass fraction (%)
0.12 0.12 0.12 0.12 0.12 0.12
22.22 22.22 22.22 22.22 22.22 22.22
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4
Maximum erosion rate [kg/m2s]
maximum erosion rate [kg/m2s]
3.5 2
1.5
1
0.5
3 2.5 2 1.5 1 0.5
0
0 1.5
2
2.5
3
3.5
4
4.5
steam mass flow rate [kg/s]
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
solid particle mass flow rate [kg/s]
Fig. 7. Maximum erosion rate vs. steam mass flow rate.
Fig. 9. Maximum erosion rate vs. particle mass flow rate.
erosion rate. Table 3 shows, for each case, the number of particles per unit volume of steam and their percentage in volume and mass with respect to the steam. In the first case, seven boundary conditions were set: from m_ v ¼ 1:5 kg/s to m_ v ¼ 4:50 kg=s with increases of 0.5 in addition their appropriate output pressure is updated for each steam mass flow rate. Fig. 7 illustrates the variation of maximum erosion rate with respect to steam mass flow rate. As the vapor mass flow diminishes, the maximum erosion rate also diminishes, due to the particle velocity decrease with the velocity of the continuous phase. Fig. 8 presents the contours of erosion rate for the maximum value of erosion rate in the range analyzed, which correspond to the part-load m_ v ¼ 3:5 kg=s. In the second case, solid particles mass flow rate injected changes from 0.6 kg/s to 1.4 kg/s with increments of 0.2, maintaining the rest of the variables of the base running.
Fig. 9 illustrates the effect of the discrete phase flow rate over the maximum erosion rate. As the solid particles mass flow rate increases, the maximum erosion rate also increases reaching a maximum value of 3.34 kg/m2 s, diminishing slightly later with the increment of the solid particle mass flow rate. Fig. 10 presents the contours of erosion rate for a solid particles mass flow rate of 1.2 kg/s, which shows the maximum erosion just in the upper border of the trailing edge. Finally, a third case is analysed, the effect of diameter of the particles over the maximum erosion rate, Fig. 11. The diameter changes from 50 lm up 100 lm at increments of 10. It is observed that when the particle diameter increases the erosion rate diminishes in a nearly linear form, changing from 2.27 to 1.38 kg/m2 s, this performance is caused by the fact that as the diameter increases, the amount of particles diminishes for a fixed solid particles mass flow
Fig. 8. Contours of DPM erosion rate for a part-load ðm_ v ¼ 3:5 kg=sÞ.
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Fig. 10. Contours of DPM erosion rate for a solid mass flow rate of 1.2 kg/s.
rate. Fig. 12 shows the erosion rate predicted for a diameter of particle of 50 lm, where the area eroded is larger than the other cases, and is not only at the trailing edge.
4
Maximum erosion rate [kg/m2s]
3.5 3 2.5
4. Conclusions
2 1.5 1 0.5 0 50
60
70
80
90
diameter of particles [microns]
Fig. 11. Maximum erosion rate vs. diameter of the particles.
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In the present work continuous–discrete phase models are used to predict the erosion rate on surface blades. The effect of vapor mass flow rate, diameter of the particle and solid mass flow rate is studied by comparing the results, obtained by solving CFD problem. The comparison shows that the effect of solid mass flow rate, in the range presented, is more important than steam mass flow rate on erosion rate. On the other hand, the erosion rate obtained varying diameter of particles shows that
Fig. 12. Contours of DPM erosion rate for a diameter of particle of 10 lm.
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the maximum values of erosion rate decrease, while the particles diameter increases. However, a definite conclusion in this regard may await more experimental data on erosion rate in blades, and comparison with other erosion models. References [1] W. Tabakoff, A. Hamed, R. Wenglarz, Particulate Flows, Turbomachinery Erosion and Performance Deterioration, Von Karman Lecture Series 1988–89, Brussels, Belgium, 1988. [2] G.I. Parslow, D.J. Stephenson, J.E. Strutt, S. Tellow, Investigation of solid particle erosion in components of complex geometry, Wear 23 (1999) 737–745.
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[3] S. Ivanov, A numerical simulation of particle laden flow through airfoil cascade, in: International Conference on Computer Systems and Technologies – CompSysTech’2005. [4] R.W. Lyczkowski, J.X. Bouillard, State-of-the-art review of erosion modelling in fluid/solid systems, Progress in Energy and Combustion Science 28 (2002) 543–602. [5] A. Hamed, W. Tabakoff, R.B. Rivir, K. Das, P. Arora, Turbine blade surface deterioration by rosion, Journal of Turbomachinery 127 (2005) 445–452. [6] B.E. Launder, D.B. Spalding, The numerical computation of turbulent flows, Computer Methods in Applied Mechanics and Engineering 3 (1974) 269–289. [7] FLUENT V6.2.16 User’s Guide, Fluent Inc., Canterra Resource Park, 10 Covendish Court, Lebanon NH03766, EUA, 2003. [8] K.C. Cotton, Evaluating and Improving Steam Turbine Performance, Cotton Fact Inc., New York, 1993.