Simulation and experiment research on liquid channel of diffuser blade by electrochemical machining

Aiming to solve the problems of the low electrolyte flow rate at leading edge and trailing edge and poor uniformity of the end clearance flow field during the electrochemical machining (ECM) of diffuser blades, a gap flow field simulation model was established by designing three liquid-increasing channels at the leading edge and the trailing edge of the cathode. The simulation results indicate that the liquid-increasing hole channel (LIHC) with an outlet area S of 1.5 mm2 and a distance L from channel center to edge point of 3.2 mm achieves optimal performance. In addition, the experiment results show that the optimized cathode with liquid-increasing hole channel (LIHC) significantly improves the machining efficiency, accuracy, and surface quality. Specifically, the feed speed increased from 0.25 mm/min to 0.43 mm/min, the taper decreased from 4.02° to 2.45°, the surface roughness value of the blade back reduced from 1.146 to 0.802 µm. Moreover, the roughness of the blade basin decreased from 0.961 to 0.708 µm, and the roughness of the hub reduced from 0.179 to 0.119 µm. The results prove the effectiveness of the proposed method and can be used for ECM of other complex structures with poor flow field uniformity.


Introduction
The diffuser is one of the core components used in the aero engine [1,2], and its main function is to convert the kinetic energy of high-speed air flow at the impeller outlet into pressure energy. In order to meet the high performance requirements and adapt to harsh working environments, the diffuser is usually made of titanium alloy or nickel-based superalloy which is resistant to high temperature and corrosion, but has high hardness, low thermal conductivity, and is difficult to process. Traditional manufacturing methods will lead to high cutting temperature, low machining surface quality, serious tool wear, and high residual stress [3][4][5][6]. Electrochemical machining (ECM), however, is a non-traditional machining method based on the anodic electrochemical dissolution mechanism, which is not only suitable for machining difficult-to-cut materials and complex structures, but also has the advantages of no tool wear, no residual stress, high machining efficiency, theoretically no cathode loss and feasible mass manufacturing [7][8][9][10]. Therefore, ECM has been widely used in automotive, aerospace, mold industry, medical, and other fields [11][12][13].
In ECM, the distribution of the flow field directly affects the machining stability, machining accuracy, and surface quality [14,15]. In recent years, many researchers have focused their efforts on the flow field of ECM, including ultrasonic-assisted ECM to improve the electrolyte flow state in the machining gap, and used pulsating electrolyte to improve the surface quality and material removal rate, and employed progressive pressure electrolyte to increase the flow rate in the electrode gap [16]. However, the above references have higher requirements for auxiliary external factors. In addition, in the study of flow field mode, in 2013, Xu et al. used the Π shape flow mode to ECM the integrated blade cascade channel, and a more uniform flow field improved the processing quality, stability, and efficiency [17]. Two years later, the reverse flow field was used to ECM for the closed integral impeller, showing that the whole process was stable, and the machining quality was high [18]. In 2020, Wang et al. designed a new tangential flow field, which effectively eliminated the defect of sudden change of flow channel in ECM for large size blade [19]. In the same year, Liu and Qu controlled the flow direction of the electrolyte in the machining area by changing the type of flow channel inside the tool electrode, which can significantly reduce stray corrosion and improve the surface processing quality of TB6 titanium alloy [20]. In 2021, Lei et al. proposed an edge electrolyte supply mode and optimized the structure of the insulating sleeve, which effectively improved the machining accuracy and machining quality of the overall blade cascade (Ti6Al4V) processed by electrochemical trepanning [21]. Although changing the flow field mode has many advantages, it requires complex tooling. To solve the problems of low electrolyte flow rate at the leading edge and the trailing edge and poor uniformity of the end clearance flow field during ECM of diffuser blades, this work designs three different liquid-increasing channels at the leading edge and the trailing edge of cathode to improve the flow field without external auxiliary conditions and complex tooling. Through flow field simulation, the three liquid-increasing channels were tested and analyzed. By comparison, the liquid-increasing hole channel (LIHC) has been proven to promote the end clearance flow field to obtain a better flow state. In order to further improve the uniformity, it is optimized by modifying the structural parameters. Finally, comparative experiments were carried out to verify the effectiveness of the proposed method.

Design of flow field mode with liquid-increasing channel
Generally speaking, the flow mode of electrolyte is divided into three categories: side flow, positive flow, and reverse flow [22]. In this work, the electrolyte flow is positive flow.
In the process of positive flow ECM, the assembled cathode consists of cathode body, insulating layer, and cathode piece. The electrolyte is pumped from the top inlet of the cathode, flows through the inner cavity of the cathode body coated with insulating layer, enters the cathode sheet forming groove, and finally flows out from the outer side gap and the end clearance of the leading edge and the trailing edge, as shown in Fig. 1a.
With the continuous feeding of the cathode, the blade of the diffuser is finally processed into a desired shape. However, for

Inlet
Insulating layer  , a low speed area was observed at the leading edge and the trailing edge in this processing method. This can be understood that the leading edge and the trailing edge witness a small diameter but large radian and area, and hence the accessibility of the flow field is low, as shown in Fig. 1b. Thus, the low speed leads to non-uniform flow field, which makes the machining unable to continue due to spark short-circuit, thereby causing an uneven surface of the flow channel and poor processing quality.
In order to solve this problem, three kinds of liquidincreasing channels were designed at the leading edge and the trailing edge of the cathode: liquid-increasing curved seam channel (LICSC), liquid-increasing straight seam channel (LISSC), and liquid-increasing hole channel (LIHC). The inlet of the three kinds of liquid-increasing channels is also at the top of the cathode, passing through the cathode body and cathode piece, and their inlet area is larger than the outlet area, as shown in Fig. 1c, d, and e. Therefore, the electrolyte can flow into the end clearance from both the original inlet and the inlet of the liquid-increasing channel at the same time. Thanks to this design, the liquid-increasing channel is on the cathode, which eliminates the complex fixture and tooling and directly supplies the liquid at the edge. What is more, it shortens the flow distance of the electrolyte, and hence increases the electrolyte flow rate at the edge, improving the accessibility and uniformity of the flow field.

Establishment of mathematical model
In ECM, the flow field in processing gap is very complex. To simplify the simulation of the flow field, the following assumptions were made: (1) the fluid is an incompressible and constant Newtonian fluid, meaning that no matter how the velocity gradient changes, the dynamic viscosity remains unchanged. (2) The influence of bubbles and solid particles in the fluid is not considered, and the bubbles and Joule heat generated during the processing can be ignored. (3) There is no movement and slip at the solid boundary.
The electrolyte is required to be in a turbulent flow state in ECM so as to allow the heat and products generated in the ECM area to be taken away to the greatest extent. The state of laminar flow and turbulent flow is usually distinguished by the Reynolds number Re. Therefore, the condition needs to be satisfied to ensure a turbulent flow state: where u is the flow rate of the electrolyte, D h is the hydraulic diameter, is the kinematic viscosity of the electrolyte, A is the area of the flow channel section, x is the wet circumference of the flow channel section. According to the above assumptions, the motion control equation of the electrolyte in the flow field simulation can adopt the Navier-Stokes equation [23]: where u represents the component of the time-averaged velocity in the i direction, x i is the coordinate in the i-axis direction of the coordinate system, x j is the same, p represents the time-averaged pressure, and ij is the component of the stress tensor in the ij plane.
In the internal flow channel of ECM of diffuser blades, the channel shape is complex, and the electrolyte streamline is curved. Therefore, the RNG k-ε turbulence flow model suitable for the flow on the curved wall was selected to solve the simulation of the threedimensional ECM gap flow field [24,25]. The turbulent kinetic energy k and the dissipation rate equations are as follows: where G k is the turbulent kinetic energy generation term caused by the average velocity gradient, is the viscosity coefficient, is the density of the electrolyte, t is the turbulent viscosity, t is the time, x i and x j are coordinate positions, u i is the velocity in the x i direction, and C 1 , C 2 , C , , k is the model constant,

Three-dimensional flow field model and boundary conditions
According to the actual processing conditions, the original model (Fig. 2a), the LICSC model (Fig. 2c), the LISSC model (Fig. 2d), and the LIHC model (Fig. 2e) were established in the SolidWorks software. Figure 2b is the cross-sectional view of plane B. The above four models were selected when the machining depth h was 5 mm, the end clearance Δb 1 and the outer side gap Δb 2 were both set to 0.5 mm, the inlet area and outlet area of the three liquid-increasing channels were the same. In the simulation process, the original inlet and the inlet of the LICSC, the inlet of the LISSC, and the inlet of the LIHC had the same pressure of 0.73 MPa, while the outlet pressure was set as 0 MPa.

Flow field simulation and analysis
FLUENT software was used to simulate the flow field of the original model, the LICSC model, the LISSC model, and the LIHC model. The section A in the middle of the end clearance was selected as the reference section of the processing area, and the effects of the three liquid-increasing channels on the velocity distribution of electrolyte in the ECM of diffuser blades were compared. Figure 3 shows the velocity distribution on section A of four different models. It can be seen from the velocity cloud diagram that in the original model flow field (Fig. 3a), the electrolyte flow rate was low at the leading edge and the trailing edge, the anode dissolved product could not be quickly taken away during processing, and hence the uniformity of the flow field was poor, resulting in an unstable machining and even short circuits. By contrast, in the flow field of three liquid-increasing channels, the electrolyte velocity at the leading edge and the trailing edge was improved, and the overall flow field distribution was more uniform (Fig. 3b-d).
In order to understand the flow velocity at the edge and the uniformity of the flow field of the entire section more intuitively, 12,569 points were uniformly extracted at each of the two edges, and 182,473 points were evenly extracted across the entire section. Their velocities were obtained in the post-processing step. The average flow velocities at the leading edge and the trailing edge are set to u a1 and u a2 , and the variance of the flow velocity across section A was set to σ u , and the results are shown in Table 1. According to Fig. 3 and Table 1, compared with the flow field of the LICSC and the LISSC, the flow velocity at the edge of the LIHC flow field was faster, the overall variance was smaller, and the velocity distribution was more uniform. This may be because the shape of the LIHC is more sufficient for the low speed area and can be better integrated with the original inlet electrolyte.

Optimization of the LIHC
Under the condition of the flow field with the LIHC, the uniformity of the flow field can be effectively controlled by adjusting the outlet area and the position of the LIHC. Next, therefore, the above two influencing factors were optimized.

Optimization of outlet area of the LIHC
Due to the limitation of the cathode structure, the inlet area of the LIHC was fixed at 4.2 mm 2 , and then the outlet area S was set to 0.9 mm 2 , 1.1 mm 2 , 1.3 mm 2 , 1.5 mm 2 , and 1.7 mm 2 , respectively. The velocity cloud diagrams under each outlet area are obtained, as shown in Fig. 4.
The same number of points as above were extracted at the leading edge (region 1), at the trailing edge (region 2), and the whole section A through the velocity distribution cloud diagram, and the u a1 , u a2 , σ u under different outlet areas S are shown in Fig. 5. It is seen that the electrolyte supply of the LIHC with small outlet area to the large low speed area was insufficient, and the electrolyte flow rate of the LIHC with the large outlet area failed to meet the requirements. When the outlet area S was 1.5 mm 2 , the electrolyte flow rate in region 1 and region 2 reached the highest, and the overall variance was the lowest (18.64 m/s, 18.62 m/s, and 32.52 m 2 /s 2 , respectively). Therefore, 1.5 mm 2 was considered to be the optimal outlet area of the LIHC.

Optimization of the position of the LIHC
The center point of the LIHC is located on the line connecting the edge point with the intersection of the blade basin and the leading edge and the trailing edge. The distance L between the edge point and the center point of the LIHC (Fig. 6a) directly affects the electrolyte flow in the edge area, so six flow models were established based on different distances: 2.3 mm, 2.6 mm, 2.9 mm, 3.2 mm, 3.5 mm, and 3.8 mm. Figure 6b shows the simulation results under different L values. It can be seen from the figure that the increasing distance L caused the electrolyte flow rates u a1 and u a2 in the two edge areas to first increase and then decrease. When L was 3.2 mm, u a1 and u a2 reached their maximum values of 19.15 m/s and 19.29 m/s, respectively, and σ u was 31.47 m 2 /s 2 , the minimum. As a matter of fact, if L was small, the electrolyte supply of the LIHC to the low speed area was uneven. On the other hand, if L was large, the LIHC would have a great influence on the original inlet flow field. Therefore, the distance L was selected to be 3.2 mm.
By optimizing the outlet area and position of the LIHC, the optimal parameters were obtained: the outlet area S was 1.5 mm 2 , and the distance L between the edge point and the center point of the LIHC was 3.2 mm. This optimized LIHC can effectively increase the electrolyte flow rate at the edge and improve the uniformity of the flow field in the processing area.

Experiment and discussion
For the purpose of verifying the effectiveness of flow field simulation, the original cathode without the LIHC and an optimized cathode with the LIHC having the outlet area S of 1.5 mm 2 and the distance L of 3.2 mm (Fig. 7b) were fabricated. The material of the cathode body was stainless steel, and the material of the cathode piece was red copper. The comparative experiment was conducted on the ECM equipment (PECM, 800S, Germany). The experimental setup is shown in Fig. 7a. For analyzing the effect of the LIHC, the analysis was carried out from three aspects: machining efficiency, taper angle, and surface roughness. The experimental conditions are shown in Table 2. The voltage was set to 20 V because it is more suitable for processing nickel-based superalloy materials. The inlet pressure was the same as the simulation value, and the machining depth was equal to the axial height of the diffuser blade.

Machining efficiency analysis
The feed rate was introduced in the comparative experiment to study the influence of the LIHC on the machining efficiency. Under the condition of using the original cathode and the feed rate of 0.25 mm/min (Fig. 7c), the whole machining process was relatively stable without short circuit, but the current fluctuation was large, and there were slight ablative marks at the edge. The surface morphology of the hub at the leading edge and the trailing edge was observed by a scanning electron microscope  EVO 20,Germany). Due to the poor uniformity of the flow field in the end clearance, the electrolyte flowed divergently, and obvious flow marks were observed on the hub at the leading edge and the trailing edge. When the feed speed of the original cathode was increased to 0.3 mm/min (Fig. 7d), the actual feed depth of the cathode was 1.2 mm, there were severe ablation marks at the edge, resulting in short circuit during the process. In contrast, when the optimized cathode was used and the feed speed was 0.43 mm/min (Fig. 7e), the current fluctuation was small, no short circuit occurred in the whole machining process, the hub surface was very smooth, and there were no obvious flow marks at the leading edge and the trailing edge. The results show that the optimized cathode can effectively improve the machining efficiency.

Taper analysis
Blade 1 was machined with the original cathode at a feed rate of 0.25 mm/min, while, blade 2 was processed using the optimized cathode at a feed rate of 0.43 mm/min. To measure the machining accuracy of the two groups of blades, three sections were selected at the leading edge, the trailing edge, and the middle position of the two groups of machined blades. A three-dimensional optical microscope (Zeiss-Smart Zoom 5, Germany) was used to measure the profile of each section. The leading edge sections of blade 1 and blade 2 are shown in Fig. 9a, b. The taper angle (Fig. 8) was calculated according to the formula, which was defined as: where L 1 is the transverse distance of the blade tip, L 2 is the transverse distance of the blade root, ΔL is the transverse distance error between the blade root and the blade tip on one side of the blade, H is the height of the machined blade. The taper angle measurement results are shown in Fig. 9. The comparative results of the two groups of blades illustrated that the taper angle of the leading edge decreased from 5.16° to 2.42°, the taper angle of the middle position decreased from 3.61° to 2.59°, the taper angle of the trailing edge decreased from 3.29° to 2.34°, and the average taper angle of the three sections decreased from 4.02° to 2.45°, which indicated that the taper angle was significantly reduced. This can be explained by the fast feed speed. Specifically, the fast feed rate causes the blade to leave the cathode edge and enter the cathode body with an insulating layer in a fast manner, and hence the time of stray corrosion caused by the residual current on the cathode edge was short. In addition, the machining gap becomes smaller due to the fast rate, which is conducive to improving the machining accuracy. Besides, it can be seen from the taper angles of the three sections that the uniform flow field in the process of machining blade 2 could make the machining allowance more consistent.

Surface roughness analysis
The surface roughness of the machined blades was obtained by using a surface profilometer (Mahr, Marsur-fLD 120, Germany). To ensure the measurement accuracy, five measurement lines were evenly selected on the blade basin surface, blade back surface, and hub surface of blade 1 and blade 2, respectively. The average results of the five measurement lines are listed in Fig. 10a. It can be seen that the surface roughness of the blade back was reduced from 1.146 to 0.802 µm, the surface roughness of the blade basin was reduced from 0.961 to 0.708 µm, and the surface roughness of the hub was reduced from 0.179 to 0.119 µm. The surface quality of blade 2 was better than that of blade 1. This is because fabricating LIHC helps accelerate the flow rate of electrolyte at the edge, and hence the uniformity of the flow field was improved, which causes the electrolytic products and bubbles affecting the surface roughness to be quickly taken away. On the other hand, due to the faster feed speed and smaller (8) = arctan(ΔL∕H) processing area, the current density becomes larger and the workpiece dissolves evenly, resulting in a significantly reduced surface roughness.

Conclusions
In this paper, a method of liquid-increasing channel was proposed to improve the uniformity of flow field in ECM, and the conclusions can be drawn as follows: 1. Three kinds of liquid-increasing channels for direct liquid supply were designed at the leading edge and the trailing edge of the cathode, which improves the flow field distribution of the end clearance and eliminates the complex fixture and tooling. 2. According to different structures of the liquid-increasing channel, the corresponding flow field model was established, and the flow field simulation was carried out. The results show that the LIHC achieves the best improvement effect on the flow field of the end clearance. The electrolyte velocity at the leading edge and the trailing edge was increased, and the flow field uniformity of the end clearance was improved. 3. The optimal design of the LIHC was carried out, and the optimal parameters were obtained. The optimal values of the outlet area S and the distance between the center of the LIHC and the edge point L were 1.5 mm 2 and 3.2 mm, respectively. 4. The comparative experiments show that the optimized cathode with the LIHC increased the feed rate from 0.25 to 0.43 mm/min, and the taper decreased from 4.02˚ to 2.45˚. Besides, the surface roughness value of the blade back was reduced from 1.146 to 0.802 µm, the value of the blade basin decreased from 0.961 to 0.708 µm, and the value of the hub reduced from 0.179 to 0.119 µm. This method can be applied to the ECM of other complex structures with low speed flow field, creating social and economic value in engineering applications.