Cathode structure optimization and process experiment in electrochemical machining of multi-stage internal cone hole

In order to solve the problem of uneven gap distribution and flow pattern in the complex parts with multi-stage internal cone hole in electrochemical machining, a method of computer simulation–assisted cathode design is proposed. The electric field and flow field models of machining gap are established respectively, and the simulations of different cathode profiles are carried out. When the cathode cone angle is 2°, the electric field distribution between the cathode and the workpiece is reasonable, and the electrolyte distribution in the machining gap is uniform. With the conditions of processing voltage 10 V, electrolyte inlet pressure 1.5 MPa, electrolyte temperature 28 ℃, and cathode feed speed 5 mm/min, the electrochemical machining (ECM) processing of multi-stage internal cone hole is carried out by using the optimized cathode. The results show that the surface of the workpiece has no flow pattern, the dimensional forming accuracy is better than 0.1 mm, and the surface roughness reaches Ra 0.697 μm. Research shows that the optimization of cathode structure with computer simulation can shorten the cathode development cycle and reduce the cost of cathode design effectively in ECM, which provides an efficient and feasible method for the optimization of complex cathode structure.


Introduction
With the rapid development of modern industry, in many fields such as aerospace, weapon equipment, biomedicine, and mechanical engineering, the demand for complex and integral parts with hole structures is increasing [1][2][3][4]. As an important machining process that accounts for a large proportion of machining, about one-third of the machining volume is used for hole structure processing, which takes about a quarter of the total machining time [5]. Hole making technology has become an important part of modern manufacturing technology, especially the processing of large aspect ratio, complex structure, and taper hole structure. The integral components with multi-stage internal cone hole are formed by connecting internal holes of different diameters through the taper angle, which have the advantages of light and compact. However, the processing is very difficult. Not only the processing surface is required to be complete and defect-free, but also the radial and axial direction have very high dimensional accuracy requirements. The existing machining technology has some problems, such as low production efficiency, poor surface quality, and difficulty in precise control of taper, which cannot meet the actual needs. There have been some researches on the processing methods and processes of tapered hole parts with variable cross-sections, but new processing methods for tapered hole structures with variable cross-sections of special difficult-tomachine materials are rarely involved [6,7]. Electrochemical machining (ECM) has the advantages of one-time forming, high processing efficiency, good surface quality, and no loss of cathode in theory, which determines that ECM is one of the effective methods to realize the machining of conical hole structure parts [8][9][10][11][12].
A large number of scholars have carried out a lot of researches on the problem of low forming precision and poor surface quality for hole parts. For processing technology, Ma et al. fabricated the non-circular hole structure on the surface of a turbine rotor blade by using the combined electrical machining technology [13]. Sathish used the injector nozzle to drill micro-holes in pure stainless steel. Considering the process parameters such as feed speed, voltage, and duty cycle, the process parameters of stainless steel micro-drilling were optimized, and the taper of micro-holes was analyzed by VMS series optical image measuring instrument [14]. Zhao designed a flow field model with a variety of flow channel characteristics for the electrochemical machining of diamond-shaped holes. The method of combining short arc flow channel and vibrating feed was adopted, which significantly improved the accuracy of machining positioning and the stability of the machining process [15]. Zhao et al. proposed a packet pulse-matched composite feed mode based on linear feed and composite feed modes, which optimizes the process stability and localization of diamond holes markedly [16]. Zhu et al. proposed a new method of machining small holes called the ultrasonic-assisted electrochemical drill-grinding, and studied the simulation of electrochemical drill-grinding process to illustrate the effect of ball-end electrode on reducing the hole taper. Finally, small holes with taper of less than 0.6 degree were machined [17]. Sarkar et al. presented a novel combination of micro-ECDM drilling and micro-ECM finishing for producing micro-holes in SS-304 stainless steel, and carried out the experiment which indicates a reduction in hole taper angle, and improved circularity and surface quality [18]. Xu et al. presented a hybrid process called tube electrode high-speed electrochemical discharge drilling that combines electrical discharge high-speed drilling and electrochemical machining, which can fabricate a small hole with low tool wear and almost no recast layer by the hybrid process [19]. In addition, some scholars have proposed various algorithms for optimizing process parameters to improve the accuracy of predicting processing results, which helps to shorten the cycle of optimizing process parameters and save manufacturing costs [20][21][22].
For cathode designing and structure, Uchiyama and Kunieda carried out the numerical analysis of the electrostatic field and large deflection in the ECM of the curved hole structure, designed and optimized the cathode, reduced the radius of curvature, and improved the processing efficiency [23]. Burger et al. studied the electrochemical machining characteristics of LEK94 nickel-based single crystal material, optimized the processing parameters, and designed a special electrode for tapered holes, which realized the finishing of the tapered micro-holes of the LEK94 material [24]. By simulating and analyzing the distribution of potential and current density on the cathode surface and electrolyte, Mi and Wataru designed a copy-shaped tool cathode with a gradually decreasing conductive area, and successfully processed a reverse tapered hole [25]. Kong et al. presented a novel method for preparing an insulation layer by asymmetric-timed bipolar electrophoretic coating in an aqueous epoxy acrylic solution to solve the problem that the prepared insulation layer is easily damaged under the influence of high-pressure liquid flushing and bubble tearing, the taper of machined holes is reduced by more than 60%, and the surface quality is improved by about 30%, compared to the holes machined with the traditional electrodes [26]. Li et al. designed the tubular electrode and used the pulse electrochemical machining method to process the hole with variable cross-section, whose parameters meet the design requirements [27]. Fan and Hourng designed a cathode with a diameter of 100 μm and used a rotating feed method to conduct electrochemical machining of micropores [28]. Selvarajan et al. used copper electrodes to process Al7075 material by pulse electrochemical machining, and studied the influence of electrolyte concentration, pulse duty cycle, and machining voltage on the actual machining effect [29]. Ramezanali and Mohammadreza. carried out the cathode design of the special tool for electrochemical machining of the spiral in a small-caliber barrel [30]. In addition, the application of computer-aided design and finite element method in cathode design speeds up the design work, which can effectively reduce the cost of electrochemical processing and improve processing efficiency [31][32][33].
The structure of the multi-stage internal cone hole studied in this paper is shown in Fig. 1. The movable cathode Fig. 1 Structure of multi-stage internal cone hole is used to realize the processing of this structure. For the electrochemical machining of the multi-stage internal cone hole, due to the large span of the taper hole size, it is easy to cause uneven distribution of the gap between the machining gap, which affects the forming accuracy and surface quality. As shown in Fig. 2, the surface of the sample processed with an equal gap cathode produces serious flow lines, and the second cone tail end has an extremely poor molding size. In order to improve the forming accuracy and surface quality of the parts, this paper uses computer simulation methods to simulate the electric field and flow field of the cathodes with different processing profiles, optimized the cathode profiles, and carried out process experiments. A multi-stage internal cone hole part that meets the design requirements was processed finally.

Establishment of cathode model
The size of multi-stage internal cone hole is narrowed by width, and the gap flow field is easy to change and uneven in the process of machining. When designing the cathode, the pre-machined cone section is designed according to the initial blank size of the workpiece, and then, the first and second cone processing sections are designed according to the design size of the parts. The processing of the multi-stage internal cone hole is a two-dimensional derivative electrolytic processing, where the workpiece does not move and the cathode needs to feed. At the end of the processing, the cathode forming section is required to be in the cone hole of the workpiece, and it must ensure that the cathode stops feeding and the size meets the requirements of designing. Compared with through hole processing, in addition to the continuous change of radial processing gap, there are also side wall cone angles in the axis. The difficulty is to ensure the shape size of the pre-processed section, realize the orderly connection of the first and second cones, and finally ensure that each section meets the requirements. Based on the principle that the removal depth of one side is proportional to the length of the cathode during full-cone cathode processing, under equal gap processing conditions, the cathode working section is divided into four sections, namely the pre-processed cone, the insulating section, the first cone processing section, and the second cone processing section, as shown in Fig. 3.
During the machining process, the electrolyte enters the machining gap from the outlet hole, and the sealing groove is matched with the sealing ring to ensure the sealing performance. The pre-processing cone cooperates with the first and second machining cones to complete the machining of the multi-stage internal cone hole. The front and rear guide ensure that the cathode is coaxial with the workpiece preventing short-circuit between the two poles. Finally, the electrolyte flows back to the circulation system through the return hole.

Simulation of gap electric field in ECM of multi-step internal cone hole
In the multi-stage internal cone hole machining, the forming of the second cone surface of workpiece is the most difficult to control. After the reaming is completed, the angle of the second cone processing section determines the real-time machining gap and the distribution of electric field in ECM, which affect the dissolution rate of workpiece directly, so the formation of the second cone surface is mainly guaranteed by the angle of the second cone processing section. By analyzing the current density distribution of different cathode profiles during processing, the cathode profile structure which suits for this research object is selected.

Establishment of simulation model of the gap electric field
1. Establishment of mathematical model The model studied in this paper does not include the power. The gap electric field is treated as a passive electric field, and the electric field potential distribution of the gap satisfies the Laplace equation.  The electric field distribution of the gap in ECM of the multi-stage internal cone hole is shown in Fig. 4.
In the formula, x, y, and z are respectively the coordinates of each point in the machining gap (unit: mm).
The current density of the machining gap is In the formula, � ⃗ J is the gradient of current density distribution in machining gap, unit: A/cm 2 ; � ⃗ E is the gradient of the potential distribution of the machining gap, unit: V/m.
In the formula, , electric potential, unit: V; n, normal component. When reaching the equilibrium processing state, the removal rate of the workpiece on the processed surface is: When the electrolytic surface of the workpiece is perpendicular to the direction of the cathode feeding, θ = 0°, that is, Considering only the relationship between the values of the conductivity κ and � ⃗ E , the potential gradient around the surface of the artifact anode is:

Establishment of finite element model
Due to the large taper difference between the two ends of the multi-stage internal cone holes, the cathode profile was gradually modified, which requires continuous optimization and iteration. According to the angle between the extension line of the second cone surface and the axis, the cathode model with equal clearance normal cone (−1°) and the angle of the processing section of the second cone were designed with 0°, 1°, and 2° respectively. The four cathode models are shown in Fig. 5.
In the process of electrochemical machining, the positive and negative poles of the DC power supply are respectively connected with the workpiece and cathode, and the two equipotential surfaces form electric fields in the machining gap. Therefore, the electric field boundary conditions are as follows: the cathode potential is 0, and the anode potential is the processing voltage V 0 .

Analysis of current density simulation results
Under the condition of voltage of 10 V, the electric field simulation was carried out on the machining gap after  ) -1°b) 0°c ) 1°d) 2°3 the cathode with different machining surfaces completely entered the workpiece. The current density distribution of the machining gap of the second cone at different angles is shown in Fig. 6. As can be seen from Fig. 6a, when the cathode is a normal cone with equal gap, the current density of machining gap is evenly distributed, and the current density at the tail end of the second cone is 1.3 × 10 5 A/m 2 . In mobile electrochemical machining, the second cone tail end of the workpiece is processed for the longest time. With the continuous feeding of the cathode, the current density of the second cone tail end does not decrease, so as the second cone will continue to be processed after forming, resulting in too much erosion of the second cone tail end, and the size of the workpiece is too large after processing.
When the second cone angle is 0°, namely the second cone cathode body for the standard cylinder, the current density distribution of the machining gap is shown in Fig. 6b; the second cone tail current density is 8.1 × 10 4 A/m 2 . It can be seen that with the gradual increase of the angle of the second cone, the machining gap gradually expands, and the current density distribution from the first cone to the second cone presents a decreasing trend, the removal velocity of the tail end of the second cone decreases, which can improve the problem of the end size of the workpiece after machining. When the angle of the second cone of the cathode is 1°, the simulation result of the electric field of the machining gap is shown in Fig. 6c. The current density of the tail end of the second cone of the workpiece is 3.7 × 10 4 A/m 2 , and the removal rate of the second cone end of the workpiece continues to decrease during the processing. When the angle of the second cone of the cathode is 2°, the electric field simulation result of the machining gap is shown in Fig. 6d, and the current density of the tail end of the second cone of the workpiece is 2.4 × 10 4 A/m 2 . It can be seen from the current density distribution cloud that the removal rate of the second cone tail of the workpiece will decrease greatly with the process of machining.
With the increase of the angle of the second cone, the electric field distribution becomes more concentrated and the gradient of the current density distribution in the machining gap becomes more obvious. In view of this result, points on the workpiece surface within the machining gap of the second cone tail end were selected. The current density of the second cone tail end at different angles is shown in Fig. 7.

Simulation of gap flow field in ECM of multi-stage internal cone hole
The taper size span of multi-stage internal cone holes is large, and the forming rules of different cone angles are different in the process of ECM. The electrolyte flow field is easy to mutate in the machining gap and form beam, resulting in flow lines on the surface of multi-stage internal cone holes. The simulation analysis of the influence for different cathode types on the flow field of ECM provides theoretical guidance for the later cathode optimization process.

Establishment of the simulation mathematical model of the gap flow field
In mobile electrochemical machining, machining gap is constantly changing, affected by electrolyte flow, current, temperature, and other factors. In order to facilitate the study, the machining gap is assumed to be an ideal state, the electrolyte is incompressible fluid without bubbles, and the influence of solid particles in the gap is ignored. The dynamic viscosity does not change with the change of velocity. Ignoring the influence of temperature changes, assuming that the machining gap temperature is constant at 30 °C, the electrolyte is 5%NaCl + 16%NaNO 3 + 4%NaClO 3 composite electrolyte, the dynamic viscosity coefficient μ is 1.01 × 10 −3 Pa·s, and the density ρ is 1.11 × 10 3 kg/m 3 , and the machining process is steady. The electrolyte does not travel a long distance in the processing area, and its flow is restricted by the law of conservation of mass and the law of conservation of momentum, and the flow model satisfies the Navier-Stokes equation, where is the electrolyte density; u is the electrolyte flow velocity; I is the unit tensor; and F is the volume force of the fluid per unit mass. The volume force F received by the electrolyte in the machining gap is extremely small and can be ignored.
Based on the above assumptions, the standard k− model was used to calculate the flow field distribution in machining gap. The model introduces two additional dependent variables: turbulent kinetic energy k and turbulent dissipation rate . Turbulent viscosity is where C μ is the model constant.The transmission equation of k is as follows: The transmission equation of is as follows: where turbulence generates term is The model constant values in the formula are shown in Table 1.

Analysis of simulation results of gap flow field
The simulation results are shown in Fig. 8. In Fig. 8a, when the cathode second cone processing section is normal cone −1° of equal gap, the electrolyte flow velocity of the second cone in the machining gap is polarized seriously. In Fig. 8b, when the angle of the second cone processing section is 0°, the polarization degree of electrolyte flow velocity is alleviated compared with normal cone −1° of equal gap, and the electrolyte flow velocity increased. In Fig. 8c, when the second cone processing section is a 1° inverted cone profile, the electrolyte flow rate continued to increase, and the flow rate became more uniform in the second cone section. In Fig. 8d, when the second cone processing section is a 2° inverted cone profile, the electrolyte flow rate of the second cone section in machining gap is more uniform compared with the other three cathode profiles, and presents a trend of continuous increase. The simulation results show that with the change of the second cone angle from −1° to 2°, the flow velocity of electrolyte increases gradually, and the flow field presents a gradually uniform trend.
The flow field simulation results of different cathode profiles fully show that changing the second cone angle of the cathode can significantly improve the gap flow field distribution for ECM and effectively reduce the beam generation caused by the uneven distribution of the electrolyte in the machining gap.
The tail current density of the second cone at different angles

Experiments on ECM of multi-stage internal cone holes with different cathode profiles
The testing equipment for multi-stage internal coneshaped integral components is a large-scale horizontal CNC electrochemical machining system independently developed, as shown in Fig. 9a. The system mainly includes machine tool, power supply system, control system, cooling system, electrolyte circulation system, filtration system, etc. In addition, the details of the clamping part during the machining process are introduced, as shown in Fig. 9b. The normal cone cathode with equal gap before optimization and the cathode with reversed-cone after optimization were used respectively, as shown in Fig. 10, to carry out the experimental research on the process of multi-stage internal cone hole. Using 5%NaCl + 16%NaNO 3 + 4%NaClO 3 composite electrolyte, under the conditions of processing voltage of 10 V, electrolyte inlet pressure of 1.5 MPa, electrolyte temperature of 28.5nditions of processing volta mm/min, the section of the processed sample is shown in Fig. 11.
Under the same processing parameters, the surface of the sample processed by the equal gap normal cone cathode has serious flow lines, and the tail end size is out of tolerance; the surface of the sample processed by the optimized inverted cone 2° cathode has no flow lines; the surface of the sample processed by the optimized inverted cone 3° cathode has light flow lines, and the tail end erosion is insufficient; the processing size does not meet the design requirements. The surface of the sample processed by the inverted cone 2° cathode was cleaned, and the internal diameter dial indicator was used to measure the sampling points in sections. The measurement result of the sample size is shown in Fig. 12. The sample size processed by the inverted cone 2° cathode is close to the design size of the multi-stage internal cone hole, and the maximum error is better than 0.1 mm.
To facilitate measurement, a small piece of the sample machined by the inverted cone 2° cathode was cut on the linear cutting machine and placed on the ZYGO Plus test platform of a white light interferometer, as shown in Fig. 13, to measure the surface roughness. The result is shown in Fig. 14 with the value of Ra 0.697 μm.

Conclusions
In this paper, the gap electric field and flow field simulation research are carried out for multi-stage internal cone hole ECM, the cathode was optimized, and the multi-stage internal cone hole ECM process test was carried out. The following conclusions were obtained: 1. When the inverted cone angle is 2°, it can effectively reduce the secondary machining of the second cone end of the workpiece as the cathode feeding, and making the electrolyte flow field uniformly distributed in the gap at the same time, which can reduce or even eliminate the flow pattern on the surface of the multi-stage internal cone hole; 2. Under the processing conditions of composite electrolyte 5% NaCl + 16% NaNO 3 + 4% NaClO 3 , the machining voltage 10 V, electrolyte temperature 28 s of composite electrolyte 5% MPa, the cathode feed rate of 5 mm/min, the typical parts with multi-stage internal cone hole with surface roughness of Ra 0.697 μm were processed stably, and the dimensional accuracy is better than 0.1 mm; 3. The cathode optimization assisted by computer simulation can effectively shorten the development cycle and reduce the design cost, which provides an efficient and feasible method for the complex cathode structure optimization in ECM.
Author contribution Lin Tang was the main contributor and is the corresponding author of the manuscript. Professor Lin Tang designed the research process, and led Wenli Yang and Kaige Zhai to conduct the cathode structure optimization and process experiment in electrochemical machining of multi-stage internal cone hole. Chengjin Shi and Lifeng Zhang participated in the experimental study and guided the manuscript writing. Data availability All data generated or analyzed during this study are included in this published article.

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Conflict of interest
The authors declare no competing interests.