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Simulation of fabrication for gas turbine blade turbulated cooling hole in ECM based on FEM
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 1747–1751
journal homepage: www.elsevier.com/locate/jmatprotec
Simulation of fabrication for gas turbine blade turbulated cooling hole in ECM based on FEM M.H. Wang a,∗ , D. Zhu b,1 a b
College of Mechanical Engineering, Zhejiang University of Technology, 310032 Hangzhou, China College of Mechanical Engineering, Nanjing Aeronautics and Astronautics, 210016 Nanjing, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history:
Electrochemical machining (ECM) is an important manufacture technology in machining
Received 24 November 2006
difficult-to-cut materials without tool wear and residual stress. In this study, ECM is used
Received in revised form
to machine the turbulated cooling hole on gas turbine blade for enhance efficiency of air-
25 March 2008
craft engine. However, because of the eroded size is hard to be determined in ECM, a new
Accepted 12 April 2008
approach by employing computer simulation method is applied to overcome this difficulties. Mathematical model based on the various parameters is developed. Finite element method (FEM) is selected to analyze the electric field distribution and compute the corrosion process
Keywords:
of the material by using the time-dependent simulation method. Minimum deviation of the
Electrochemical machining
simulated anode profile shape from the experimentation is performed. Furthermore, this
Computer simulation
proposed method could reduce the number of trials and save the expense greatly.
Turbulator
Crown Copyright © 2008 Published by Elsevier B.V. All rights reserved.
Time-dependent Cooling hole
1.
Introduction
With the development of high performance for gas engine, there is a tendency to augment the inlet temperature of blade to improve thermal efficiency and power output of combustion chamfer, thus gas turbine must be exposed to enormous heat. Temperature in advanced engine is higher more than 1600 K. In order to ensure the engine blade working safely, Sargison et al. (2005), Gritsch et al. (2000) and Garg (2002) studied the external cooling method, (Han and Dutta, 2001) researched the internal cooling method. Here a kind of internal cooling method by machining ribs (turbulators) on the hole wall is proposed, which is named turbulated cooling hole. These ducts form a thermal barrier between the blade and hot gases traveling through a main flow path of the engine. Accordingly, the blade experiences a cooler temperature and the engine is per-
∗
mitted to working in high flow temperature without affecting the blade. This cooling hole was certified to increase the heat transfer coefficient greatly (Jorge and Luis, 2003). Therefore, it is very promising to manufacture this kind of cooling hole in blade. Electrochemical machining (ECM) is a manufacturing process based on the principles of electrolysis (Lu and Leng, 2005). It removes electrically conductive materials regardless of their hardness and toughness (Bhattacharyya et al., 2005; Zhu et al., 2002) and is often used to machine hard alloy. For cooling hole with turbulators, ECM is proper to prepare it (Pa, 2007; Noot et al., 1998). However, because of the shape of the workpiece is difficult to determine and the experiment process is tedious and costly, computer simulation was applied in some researcher’s study, such as Kenney and Hwang (2006), Zhitnikov et al. (2004), Marius et al. (2004) and Harmen (2004). They present a simulation method to pre-
Corresponding author. Tel.: +86 25 84895912; fax: +86 25 84891077. E-mail addresses: [email protected] (M.H. Wang), [email protected] (D. Zhu). 1 Tel.: +86 25 84895912. 0924-0136/$ – see front matter. Crown Copyright © 2008 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2008.04.035
1748
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 1747–1751
Fig. 1 – Principle of turbulated cooling hole forming by ECM: (a) turbulated cooling hole machining in ECM; (b) local analysis of ECM; (c) electric potential distribution of physical model.
dict the machined profile of the workpiece during ECM by using a shaped electrode as the cathode. The shape electrode is a metal hollow tube, with the insulated materials coated partition on the outer skin of the tube (see Fig. 1). Material removal takes place under the influence of an electric field. The new profile of the cooling hole could be obtained by computing the changing coordinates of the anode surface. Mathematical model is built up to describe the development of the machined profile as a function of time and voltage. Finite element method (FEM) is selected to analyze the electric field and calculate the erode rate of the materials. The approach of the model is verified experimentally.
2.
Mathematical model
In ECM, the corrosion of the anode bases on the electrolysis. The principle for preparation of the turbulated cooling hole is illustrated in Fig. 1. A shaped electrode is used as the cathode, which is a conducting hollow tube with the epoxy resin coating on the outside discontinuous (see Fig. 1a). The cathode is lowed into the end of a smooth hole and keeps stationary while a voltage U is applied to it and electrolyte is pumped into the machining chamfer from the center of cathode. In addition, the smooth hole is processed in advance. Material near the electric part of cathode is etched and other part remains. Ultimately, some ribs form on the wall of the smooth hole (see Fig. 1b). For ECM, many parameters have effect on forming of the materials, such as voltage, concentration of electrolyte, machining speed, temperature of electrolyte and machining time, etc. Among these factors, voltage is absolutely necessarily to drive the electrochemical reactions of the workpiece and elapsed time always determines the quantity of dissolved materials. So, the mathematical model is developed as a function of voltage and machine time. For the sake of symmetry and simplicity, the twodimensional model comprising two turbulators is analyzed. According to theories of electric field and electrochemistry,
on the assumptions of ECM is in the state of equilibrium, electrical parameters not change with the time, the electric potential ϕ in the inter-electrode gap domain ˝ can be approximately described by Laplace’s equation (Hardisty et al., 1993; Domanowski and Kozak, 2001) (see Fig. 1c): 2 ϕ = 0
in ˝
(1)
Boundary conditions are as follows: ϕ|7 = V ϕ|2,4 = 0 ∂ϕ |1,3,5,6,8 = 0 ∂n
anode cathode
(2)
additional boundary
In ECM, changes of workpiece shape are determined by the material removal rate on machining surface. And material removal rate is depending on potential drop within the thin electrochemical double layers. The potential drop on the material geometry profile changes with the machining process goes on. On the basis of analyzing the electric potential ϕ in the inter-electrode gap domain ˝, the potential drop in the double layers could be obtained, the anode recession rate could be computed according to Faraday’s law. According to electric field theory, current density j is proportional to the electric field intensity E: j = E
(3)
where is conductance of electrolyte. According to Faraday’s law, the volume of the anode material dissolved V is defined by V = ωIt
(4)
where ω is volume electrochemical equivalent of material, I is the current and t is the machine time,
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j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 1747–1751
Furthermore, the volume of material removed in ECM could be described as follows: V = va St
(5)
where a is the velocity of anode dissolution and S is the small area of anode surface eroded. Current is described by I = S · j
(6)
Substitution of (3), (4) and (5) to (6), the metal removal rate could be expressed by va = ωj = ωE
(7)
As for one initial point of P0 on material, its coordination changes according to the theory of electric field principle. Pi is the point after the i × t machine time and Pi+1 is the next point of Pi . The recursion formula as follows: − → xi+1 = xi + w Ex · t,
− → yi+1 = yi + w Ey · t
(8)
Fig. 2 – Meshing model of domain ˝ by FEM.
− → − → where Ex and Ey are vector values of electric field and t is the step time. Based on FEM and iterative principle, the new profile of anode could be obtained.
mizes function G(ϕ):
3.
Numerical approach
G(ϕ)
3.1.
Discretization of gap domain
The problem discussed above is a typical value boundary problem and FEM is selected to solve it. Thus the gap domain ˝ can be divided into a series of small triangular elements as shown in Fig. 2. For each triangular element, there are three vertex points as marked i, j and k. For each vertex point, there is a linear function of x and y described as e e = aei,j,k + bei,j,k x + ci,j,k y Ni,j,k
(9)
A potential function ϕe (x, y) which varies linearly inside each triangular element is defined as ϕe (x, y) = ϕi Nie + ϕj Nje + ϕk Nke
(10)
As basis function for the FEM on each element, the potential function ϕ(x, y) of whole domain ˝ could be written as
ϕ(x, y) =
n
1 2
=
1 2
2 ∂ϕ ∂x ˝ N ˝
∂ϕ 2
+
∂y
+ ϕj Nje
+ ϕk Nke ]
(11)
Solution of equations ϕ(x, y)
Based on the FEM and variation principle, the solution of Eq. (10) is identical to finding a potential function which mini-
d˝
2 ϕi Nie
+ ϕj Nje
+ ϕk Nke
(12) d˝
e=1
Potential for each node can be calculated from minimization of G(ϕ):
∂Ge j
∂ϕj
= 0, j = 1, 2, . . . , n
(13)
where n is the number of nodal points. Assembling all elemental equations together gives: [K][ϕ] = 0
(14)
where [K] is n × n matrix with nonzero determinants assembled from ϕe (x, y). [ϕ] is column vector potential matrix. On basis of recursion principle the potential distribution could be obtained · (ki,i−1 · ϕi−1 + ki,i · ϕi ), ϕi+1 = −k−1 i,i+1
[ϕi Nie
e=1
3.2.
=
ϕ2 = −k−1 12 · (k11 · ϕ1 − b1 ),
i=1
i>1
(15) (16)
is an inverse matrix of kij. Consequently vector value where k−1 ij of electric field of domain ˝ can be expected. Substitution vector value of electric field into (8) and then new coordination of each node is obtained.
1750
4.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 1747–1751
Simulation process
In order to show the changing of anode profile, computer simulation is used. The domain ˝ is expanding step by step with the elapsed time increases. Fig. 3 is a flowchart illustrating the sequence of the simulation process of turbulated cooling hole machined in ECM based on FEM. Firstly, physical model of domain ˝ is built, which is the initial geometry profile containing several lines on the basis of the main points. Secondly, initial values of variables, such as voltage, step time and loads applied on the boundary are set. Then, FEM is used to mesh the domain ˝ into elements, and potential values of each element are computed. In FEM computing process, the anode profile are formed by some nodes, so potentials of these nodes are researched. Then coordinates of these nodes could be calculated according to Eqs. (8), (15) and (16). Subtract the coordinates of selected nodes to their initial values and get the error. Recycle process stops only if the error is smaller than tolerance. Otherwise, physical model is reshaped using changed material profile and the domain is re-meshed again. When computer simulation stops, the changing anode profiles for each step are saved. As illustrated in Fig. 4, the eroded profile expands with the increase of machining time. At the beginning, the gap between the electrode and the workpiece is narrow, electric field is large and the material near the anode is eroded much according to Faraday’s law. With the process goes on, the erosion cavity gets deeper and the gap becomes larger, the electric field intensity decreases and the machining speed slows down. After computing the sizes of ribs, the loads
Fig. 3 – Simulation process of turbulated hole by FEM.
Fig. 4 – Simulation results of eroded anode profile.
set in the simulation process for preparation of the turbulated cooling hole are adopted in our experimentation. Because of the size of the rib’s height (h) for the turbulated cooling hole is important, it is calculated at each step after the simulation (see Fig. 5). The figure denotes h is enlarged significantly at the initial stage of machining and increased slowly with increase of the elapsed time. The erosion depth produced by the shaped electrode will saturated after fifteen minutes. It is because of the cathode is not moved during the machining process. In ECM, with the increasing of the inter-electrode gap the electrolytic reaction is decreasing. When the electric field intensity is below a threshold value, no machining occurs and the shape of the workpiece not changes. However, it is detected that when the gap is smaller than 0.1 mm or when the voltage is set too high, electric discharge will occur. To avoid formation of the electric arc, the initial gap applied is larger than 0.1 mm and voltage set are from 6 V to 15 V. Simulation process also stops when the profile of anode is detected not changing more, at this time a large load should be re-applied again. For comparing for simulative experimental results, some points are measured and experimental data are shown in Table 1. Comparison between experimental results and the simulation data at the same parameters is shown in Fig. 6. It shows the eroded anode profile and the cross-sectional
Fig. 5 – Height of ribs at different machining time.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 1747–1751
Table 1 – Some measured points of rib’s height for the turbulated cooling hole Positions, x (mm) 0.195 0.580 0.920 1.275 1.520 1.800
h (mm) Positions x (mm) 0.143 0.242 0.243 0.176 0.140 0
2.080 2.325 2.680 3.020 3.450 3.700
h (mm) 0.156 0.174 0.238 0.182 0.165 0
1751
ical and experimental results. The proposed method is valid to simulate the erosion process of the turbulated cooling hole in ECM.
Acknowledgement This research was supported by the Aviation Science Foundation of China (No. 04H52055).
references
Fig. 6 – Comparison of simulation and experimental data. photograph of the turbulated cooling hole after ten minutes machining time with 10 V voltages, 200 g/L aqueous solution of NaNO3 at room temperature and 0.1 mm inter-electrode gap. The size along two ribs is measured. The comparison denotes that simulation data deviates from the experimental results in less than 10%. Accordingly the good agreement between theory and experimental data verifies the validation of the computer simulation of the turbulated cooling hole machining in ECM.
5.
Conclusions
A kind of cooling hole with ribs on its wall is designed to enhance the efficiency of aircraft engine and ECM is applied to machine the cooling hole by using a shaped electrode. Because of the eroded size is hard to be determined in ECM, a method is proposed to predict the erosion profile of the turbulated cooling hole by using FEM. Mathematical model is built up and the formation of anode profile varying with time is computed and observed by using the time-dependent simulation method. Results indicate that at the beginning of the ECM process, the electric field intensity is large and the material is removed quickly. During the process, the erosion gets deeper and the gap becomes larger, the machining process slowing down with increasing the electrode gap and decreasing the electric field intensity. On the basis of simulation results, experimentation was done and results show close agreements between theoret-
Bhattacharyya, B., Malapati, M., Munda, J., 2005. Experimental study on electrochemical micromachining. J. Mater. Process. Technol. 169 (3), 485–492. Domanowski, P., Kozak, J., 2001. Inverse problems of shaping by electrochemical generating machining. J. Mater. Process. Technol. 109, 347–353. Garg, V.K., 2002. Heat transfer research on gas turbine airfoils at NASA GRC. Int. J. Heat Fluid Flow 23, 109–136. Gritsch, M., Schulz, A., Wittig, S., 2000. Film-cooling holes with expanded exits: near-hole heat transfer coefficients. Int. J. Heat Fluid Flow 21, 146–155. Han, J.C., Dutta, S., 2001. Recent developments in turbine blade internal cooling. Ann. NY Acad. Sci. 934, 162–178. Hardisty, H., Mileham, A.R., Shirvarni, H., 1993. Finite element simulation of the electrochemical machining process. Ann. CIRP 42, 201–204. Harmen, S.J.A., 2004. EDM and ECM for mass production Philips DAP. J. Mater. Process. Technol. 149 (1–3), 18–21. Jorge, D., Luis, R., 2003. Developments to manufacture structural aeronautical parts in carbon fibre reinforced thermoplastic materials. J. Mater. Process. Technol. 143–144 (12), 342–346. Kenney, J.A., Hwang, G.S., 2006. Etch trends in electrochemical machining with ultrashort voltage pulses predictions from theory and simulation. Electrochem. Solid-State Lett. 9, 1–4. Lu, X., Leng, Y., 2005. Electrochemical micromachining of titanium surfaces for biomedical applications. J. Mater. Process. Technol. 169 (2), 173–178. Marius, P., Leslie, B., Bart, V.B., Johan, D., 2004. 3D electrochemical machining computer simulations. J. Mater. Process. Technol. 149 (1–3), 472–478. Noot, M.J., Telea, A.C., Jansen, J.K.M., 1998. Real time numerical simulation and visualization of electrochemical drilling. Comput. Vis. Sci. 1, 105–111. Pa, P.S., 2007. Electrode form design of large holes of die material in ultrasonic electrochemical finishing. J. Mater. Process. Technol. 192–193 (10), 470–477. Sargison, J.E., Oldfield, M.L., Guo, G.S.M., 2005. Flow visualization of the external flow from a converging slot-hole film-cooling geometry. Exp. Fluids 38, 304–318. Zhitnikov, V.P., Fedorova, G.I., Zinatullina, O.V., Kamashev, A.V., 2004. Simulation of non-stationary processes of electrochemical machining. J. Mater. Process. Technol. 149 (1–3), 398–403. Zhu, D., Lei, W.N., Qu, N.S., Xu, H.Y., 2002. Nanocrystal line electroforming process. Ann. CIRP 51, 173–176.
journal homepage: www.elsevier.com/locate/jmatprotec
Simulation of fabrication for gas turbine blade turbulated cooling hole in ECM based on FEM M.H. Wang a,∗ , D. Zhu b,1 a b
College of Mechanical Engineering, Zhejiang University of Technology, 310032 Hangzhou, China College of Mechanical Engineering, Nanjing Aeronautics and Astronautics, 210016 Nanjing, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history:
Electrochemical machining (ECM) is an important manufacture technology in machining
Received 24 November 2006
difficult-to-cut materials without tool wear and residual stress. In this study, ECM is used
Received in revised form
to machine the turbulated cooling hole on gas turbine blade for enhance efficiency of air-
25 March 2008
craft engine. However, because of the eroded size is hard to be determined in ECM, a new
Accepted 12 April 2008
approach by employing computer simulation method is applied to overcome this difficulties. Mathematical model based on the various parameters is developed. Finite element method (FEM) is selected to analyze the electric field distribution and compute the corrosion process
Keywords:
of the material by using the time-dependent simulation method. Minimum deviation of the
Electrochemical machining
simulated anode profile shape from the experimentation is performed. Furthermore, this
Computer simulation
proposed method could reduce the number of trials and save the expense greatly.
Turbulator
Crown Copyright © 2008 Published by Elsevier B.V. All rights reserved.
Time-dependent Cooling hole
1.
Introduction
With the development of high performance for gas engine, there is a tendency to augment the inlet temperature of blade to improve thermal efficiency and power output of combustion chamfer, thus gas turbine must be exposed to enormous heat. Temperature in advanced engine is higher more than 1600 K. In order to ensure the engine blade working safely, Sargison et al. (2005), Gritsch et al. (2000) and Garg (2002) studied the external cooling method, (Han and Dutta, 2001) researched the internal cooling method. Here a kind of internal cooling method by machining ribs (turbulators) on the hole wall is proposed, which is named turbulated cooling hole. These ducts form a thermal barrier between the blade and hot gases traveling through a main flow path of the engine. Accordingly, the blade experiences a cooler temperature and the engine is per-
∗
mitted to working in high flow temperature without affecting the blade. This cooling hole was certified to increase the heat transfer coefficient greatly (Jorge and Luis, 2003). Therefore, it is very promising to manufacture this kind of cooling hole in blade. Electrochemical machining (ECM) is a manufacturing process based on the principles of electrolysis (Lu and Leng, 2005). It removes electrically conductive materials regardless of their hardness and toughness (Bhattacharyya et al., 2005; Zhu et al., 2002) and is often used to machine hard alloy. For cooling hole with turbulators, ECM is proper to prepare it (Pa, 2007; Noot et al., 1998). However, because of the shape of the workpiece is difficult to determine and the experiment process is tedious and costly, computer simulation was applied in some researcher’s study, such as Kenney and Hwang (2006), Zhitnikov et al. (2004), Marius et al. (2004) and Harmen (2004). They present a simulation method to pre-
Corresponding author. Tel.: +86 25 84895912; fax: +86 25 84891077. E-mail addresses: [email protected] (M.H. Wang), [email protected] (D. Zhu). 1 Tel.: +86 25 84895912. 0924-0136/$ – see front matter. Crown Copyright © 2008 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2008.04.035
1748
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 1747–1751
Fig. 1 – Principle of turbulated cooling hole forming by ECM: (a) turbulated cooling hole machining in ECM; (b) local analysis of ECM; (c) electric potential distribution of physical model.
dict the machined profile of the workpiece during ECM by using a shaped electrode as the cathode. The shape electrode is a metal hollow tube, with the insulated materials coated partition on the outer skin of the tube (see Fig. 1). Material removal takes place under the influence of an electric field. The new profile of the cooling hole could be obtained by computing the changing coordinates of the anode surface. Mathematical model is built up to describe the development of the machined profile as a function of time and voltage. Finite element method (FEM) is selected to analyze the electric field and calculate the erode rate of the materials. The approach of the model is verified experimentally.
2.
Mathematical model
In ECM, the corrosion of the anode bases on the electrolysis. The principle for preparation of the turbulated cooling hole is illustrated in Fig. 1. A shaped electrode is used as the cathode, which is a conducting hollow tube with the epoxy resin coating on the outside discontinuous (see Fig. 1a). The cathode is lowed into the end of a smooth hole and keeps stationary while a voltage U is applied to it and electrolyte is pumped into the machining chamfer from the center of cathode. In addition, the smooth hole is processed in advance. Material near the electric part of cathode is etched and other part remains. Ultimately, some ribs form on the wall of the smooth hole (see Fig. 1b). For ECM, many parameters have effect on forming of the materials, such as voltage, concentration of electrolyte, machining speed, temperature of electrolyte and machining time, etc. Among these factors, voltage is absolutely necessarily to drive the electrochemical reactions of the workpiece and elapsed time always determines the quantity of dissolved materials. So, the mathematical model is developed as a function of voltage and machine time. For the sake of symmetry and simplicity, the twodimensional model comprising two turbulators is analyzed. According to theories of electric field and electrochemistry,
on the assumptions of ECM is in the state of equilibrium, electrical parameters not change with the time, the electric potential ϕ in the inter-electrode gap domain ˝ can be approximately described by Laplace’s equation (Hardisty et al., 1993; Domanowski and Kozak, 2001) (see Fig. 1c): 2 ϕ = 0
in ˝
(1)
Boundary conditions are as follows: ϕ|7 = V ϕ|2,4 = 0 ∂ϕ |1,3,5,6,8 = 0 ∂n
anode cathode
(2)
additional boundary
In ECM, changes of workpiece shape are determined by the material removal rate on machining surface. And material removal rate is depending on potential drop within the thin electrochemical double layers. The potential drop on the material geometry profile changes with the machining process goes on. On the basis of analyzing the electric potential ϕ in the inter-electrode gap domain ˝, the potential drop in the double layers could be obtained, the anode recession rate could be computed according to Faraday’s law. According to electric field theory, current density j is proportional to the electric field intensity E: j = E
(3)
where is conductance of electrolyte. According to Faraday’s law, the volume of the anode material dissolved V is defined by V = ωIt
(4)
where ω is volume electrochemical equivalent of material, I is the current and t is the machine time,
1749
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 1747–1751
Furthermore, the volume of material removed in ECM could be described as follows: V = va St
(5)
where a is the velocity of anode dissolution and S is the small area of anode surface eroded. Current is described by I = S · j
(6)
Substitution of (3), (4) and (5) to (6), the metal removal rate could be expressed by va = ωj = ωE
(7)
As for one initial point of P0 on material, its coordination changes according to the theory of electric field principle. Pi is the point after the i × t machine time and Pi+1 is the next point of Pi . The recursion formula as follows: − → xi+1 = xi + w Ex · t,
− → yi+1 = yi + w Ey · t
(8)
Fig. 2 – Meshing model of domain ˝ by FEM.
− → − → where Ex and Ey are vector values of electric field and t is the step time. Based on FEM and iterative principle, the new profile of anode could be obtained.
mizes function G(ϕ):
3.
Numerical approach
G(ϕ)
3.1.
Discretization of gap domain
The problem discussed above is a typical value boundary problem and FEM is selected to solve it. Thus the gap domain ˝ can be divided into a series of small triangular elements as shown in Fig. 2. For each triangular element, there are three vertex points as marked i, j and k. For each vertex point, there is a linear function of x and y described as e e = aei,j,k + bei,j,k x + ci,j,k y Ni,j,k
(9)
A potential function ϕe (x, y) which varies linearly inside each triangular element is defined as ϕe (x, y) = ϕi Nie + ϕj Nje + ϕk Nke
(10)
As basis function for the FEM on each element, the potential function ϕ(x, y) of whole domain ˝ could be written as
ϕ(x, y) =
n
1 2
=
1 2
2 ∂ϕ ∂x ˝ N ˝
∂ϕ 2
+
∂y
+ ϕj Nje
+ ϕk Nke ]
(11)
Solution of equations ϕ(x, y)
Based on the FEM and variation principle, the solution of Eq. (10) is identical to finding a potential function which mini-
d˝
2 ϕi Nie
+ ϕj Nje
+ ϕk Nke
(12) d˝
e=1
Potential for each node can be calculated from minimization of G(ϕ):
∂Ge j
∂ϕj
= 0, j = 1, 2, . . . , n
(13)
where n is the number of nodal points. Assembling all elemental equations together gives: [K][ϕ] = 0
(14)
where [K] is n × n matrix with nonzero determinants assembled from ϕe (x, y). [ϕ] is column vector potential matrix. On basis of recursion principle the potential distribution could be obtained · (ki,i−1 · ϕi−1 + ki,i · ϕi ), ϕi+1 = −k−1 i,i+1
[ϕi Nie
e=1
3.2.
=
ϕ2 = −k−1 12 · (k11 · ϕ1 − b1 ),
i=1
i>1
(15) (16)
is an inverse matrix of kij. Consequently vector value where k−1 ij of electric field of domain ˝ can be expected. Substitution vector value of electric field into (8) and then new coordination of each node is obtained.
1750
4.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 1747–1751
Simulation process
In order to show the changing of anode profile, computer simulation is used. The domain ˝ is expanding step by step with the elapsed time increases. Fig. 3 is a flowchart illustrating the sequence of the simulation process of turbulated cooling hole machined in ECM based on FEM. Firstly, physical model of domain ˝ is built, which is the initial geometry profile containing several lines on the basis of the main points. Secondly, initial values of variables, such as voltage, step time and loads applied on the boundary are set. Then, FEM is used to mesh the domain ˝ into elements, and potential values of each element are computed. In FEM computing process, the anode profile are formed by some nodes, so potentials of these nodes are researched. Then coordinates of these nodes could be calculated according to Eqs. (8), (15) and (16). Subtract the coordinates of selected nodes to their initial values and get the error. Recycle process stops only if the error is smaller than tolerance. Otherwise, physical model is reshaped using changed material profile and the domain is re-meshed again. When computer simulation stops, the changing anode profiles for each step are saved. As illustrated in Fig. 4, the eroded profile expands with the increase of machining time. At the beginning, the gap between the electrode and the workpiece is narrow, electric field is large and the material near the anode is eroded much according to Faraday’s law. With the process goes on, the erosion cavity gets deeper and the gap becomes larger, the electric field intensity decreases and the machining speed slows down. After computing the sizes of ribs, the loads
Fig. 3 – Simulation process of turbulated hole by FEM.
Fig. 4 – Simulation results of eroded anode profile.
set in the simulation process for preparation of the turbulated cooling hole are adopted in our experimentation. Because of the size of the rib’s height (h) for the turbulated cooling hole is important, it is calculated at each step after the simulation (see Fig. 5). The figure denotes h is enlarged significantly at the initial stage of machining and increased slowly with increase of the elapsed time. The erosion depth produced by the shaped electrode will saturated after fifteen minutes. It is because of the cathode is not moved during the machining process. In ECM, with the increasing of the inter-electrode gap the electrolytic reaction is decreasing. When the electric field intensity is below a threshold value, no machining occurs and the shape of the workpiece not changes. However, it is detected that when the gap is smaller than 0.1 mm or when the voltage is set too high, electric discharge will occur. To avoid formation of the electric arc, the initial gap applied is larger than 0.1 mm and voltage set are from 6 V to 15 V. Simulation process also stops when the profile of anode is detected not changing more, at this time a large load should be re-applied again. For comparing for simulative experimental results, some points are measured and experimental data are shown in Table 1. Comparison between experimental results and the simulation data at the same parameters is shown in Fig. 6. It shows the eroded anode profile and the cross-sectional
Fig. 5 – Height of ribs at different machining time.
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 1747–1751
Table 1 – Some measured points of rib’s height for the turbulated cooling hole Positions, x (mm) 0.195 0.580 0.920 1.275 1.520 1.800
h (mm) Positions x (mm) 0.143 0.242 0.243 0.176 0.140 0
2.080 2.325 2.680 3.020 3.450 3.700
h (mm) 0.156 0.174 0.238 0.182 0.165 0
1751
ical and experimental results. The proposed method is valid to simulate the erosion process of the turbulated cooling hole in ECM.
Acknowledgement This research was supported by the Aviation Science Foundation of China (No. 04H52055).
references
Fig. 6 – Comparison of simulation and experimental data. photograph of the turbulated cooling hole after ten minutes machining time with 10 V voltages, 200 g/L aqueous solution of NaNO3 at room temperature and 0.1 mm inter-electrode gap. The size along two ribs is measured. The comparison denotes that simulation data deviates from the experimental results in less than 10%. Accordingly the good agreement between theory and experimental data verifies the validation of the computer simulation of the turbulated cooling hole machining in ECM.
5.
Conclusions
A kind of cooling hole with ribs on its wall is designed to enhance the efficiency of aircraft engine and ECM is applied to machine the cooling hole by using a shaped electrode. Because of the eroded size is hard to be determined in ECM, a method is proposed to predict the erosion profile of the turbulated cooling hole by using FEM. Mathematical model is built up and the formation of anode profile varying with time is computed and observed by using the time-dependent simulation method. Results indicate that at the beginning of the ECM process, the electric field intensity is large and the material is removed quickly. During the process, the erosion gets deeper and the gap becomes larger, the machining process slowing down with increasing the electrode gap and decreasing the electric field intensity. On the basis of simulation results, experimentation was done and results show close agreements between theoret-
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