A New Thermal Model for Predicted Discharge Craters in Micro/Nano-EDM Considering the Non-Fourier Effect

Micro/nano-electrical discharge machining is an alternative preparation method for surface micro/nano-structures, but it is difficult to precisely control the size of the micro/nano-structures due to its unclear material removal mechanism. Thus, it is useful to study its machining mechanism to achieve high-efficiency and controlled processing. At present, most of the established EDM thermal models for predicting the discharge crater size are based on the classical Fourier heat conduction law, assuming that the conduction velocity of heat energy is infinite. However, the single-pulse discharge time of micro/nano-EDM is transitory (<1 μs), and thus, the steady state heat balance condition cannot be achieved in a single-pulse discharge time. In order to predict the size of the micro/nano electrical discharge craters more accurately, the non-Fourier effect was considered to study the temperature field distribution of micro/nano-EDM of single-pulse discharge machining. Firstly, the classical Fourier heat conduction law was modified by introducing a relaxation time. Secondly, several key factors were considered to establish the thermal model of micro/nano-EDM in single-pulse discharge machining. Subsequently, numerical simulation software was used to solve the thermal model for obtaining the temperature field distribution of the workpiece material and predicting the size of the discharge craters. Finally, the predicting accuracy of the new thermal model was evaluated by comparing the relative error between the simulated values and experimental values. The comparison results show that considering the non-Fourier effect can reduce the average error of the thermal model from 33% to 10%. The non-Fourier effect is more obvious under the shorter discharge time of a single pulse.


Introduction
Due to its advantages of ignoring material hardness, the ability to machine a complex curved surface, and no cutting stress, EDM is widely used in aerospace, instruments, the molding industry, and other precision manufacturing fields [1,2]. The discharge energy of a short-period single pulse is lower in micro/nano-EDM. It can produce micro-notches, micro-grooves, and micro-cavities. Micro/nano-EDM plays an important role in the manufacturing of micro-parts and micro-electromechanical systems. However, compared to traditional EDM, micro/nano-EDM is more difficult to control and depends heavily on operational skills and experience. Therefore, it is important to study the material removal mechanism and predict the size of the discharge craters to improve the controllability of micro/nano-EDM.
EDM relies on the ultra-high electric field strength between the electrode and the workpiece, which causes electrons to escape from the electrode and accelerate hitting the workpiece surface, producing a large amount of heat and making the workpiece material melt or vaporization. Hence, the material removal process of EDM is generally considered a thermal removal process. As pointed out in Refs. [3][4][5][6], due to the difference in the scale effect, the material removal mechanism of micro/nano-EDM was similar but not identical to the traditional EDM. In the traditional EDM, the process of single-pulse discharge included the channel breakdown stage, discharge stage and deionization stage. In the channel breakdown stage: when the power supply was applied to the electrode and workpiece, the electric field intensity between the electrode and workpiece increased with the decrease in the gap between the electrode and workpiece. The dielectric could be broken down and form a discharge channel when the electric field intensity exceeds the critical value. In the discharge stage: due to the accelerating effect of the electric field, a great deal of electrons bombarded the workpiece surface at a high speed. The kinetic energy of electrons transformed into thermal energy because of the collision between electrons and the positive ion. Then, a good deal of thermal energy was produced, which could rapidly melt and vaporize the workpiece material. A part of the melted and vaporized material was ejected into the dielectric due to the thermal explosion. Then, the discharge crater was formed. Moreover, the dielectric around the electrode was continuously vaporized from liquid to gas. In the deionization stage, the gap voltage between the electrode and workpiece became 0 V. A part of melted material was solidified from liquid to solid and adhered to the workpiece surface again. It was the so-called recast layer. Compared with traditional EDM, the skin effect and area effect decreased due to the smaller scale of electrode diameter in micro/nano-EDM. Then, the material removal rate and electrode wear rate decreased with the decrease in the electrode diameter. Due to the smaller scale of discharge pulse duration, the discharge energy in the single pulse decreased with the decrease in pulse-on time. Then, the thinner recast layer and heat affect zone could be obtained in micro/nano-EDM, which was beneficial for increasing the reliability of the part machined by EDM. Due to the smaller scale of the discharge crater in micro/nano-EDM, the grain size had an important influence on the material removal rate. The material near the grain boundary was easier to remove than that in grain due to the impurity atom from equilibrium segregation. Then, the material removal rate decreased with the increase in the grain size.
At present, establishing a thermal model is one of the most common methods for studying the EDM process. This method can effectively predict the discharge craters and reveal the material removal mechanism of EDM [7]. On the basis of the classical Fourier heat conduction theory, Liu Zhidong [8] assumed that the energy in the discharge channel obeys the Gauss distribution. The proportion of discharge energy absorbed by the workpiece was set to 40%. The thermal model of single-pulse discharge in EDM was established to obtain a temperature field distribution and predict the size of the discharge crater. The predicted results were in good agreement with the experimental values after correction. Singh H. [9] pointed out that the discharge energy distribution coefficient should not be constant in the study of temperature field distribution. By combining the thermal model prediction with an experimental measurement, the quantitative relationship between the discharge energy ratio absorbed by the workpiece and the machining parameters was fitted. Hargrove [10] used the method of thermal modeling to obtain the temperature field distribution. By combining the critical temperature of the material phase transition, the thermal damage layer of the workpiece was predicted, which was essentially consistent with the experimental results. In addition, Somashekhar K.P. et al. [11] built the electrothermal model of micro-EDM, deduced the mechanism of micro-EDM, and predicted the width and depth of craters in micro-EDM. Furthermore, a new numerical scheme, which was different from the commonly used discrete finite element solution technology, was proposed to a solve finite model meshed by a uniform grid with implicit flux discretization. In this scheme, the computational domain was divided into a square cell for spatial discretization. It was shown that the temperature dropped faster during the discharge; the smaller the ratio of the pulse time to rest time was, the faster the temperature dropped. Mujumdar S.S. et al. [12] established the plasma model of µ-EDM to predict the size of a molten pool in a single discharge spark. The plasma model consisted of three modules, including plasma chemistry, a power balance, and a bubble dynamics model. These three modules aimed to solve the reaction kinetic in terms of ionization and plasma temperature and evolve the plasma shape. The level set method was applied to calculate the fractions of liquid and solid of the workpiece in the molten region. The experimental results showed that this plasma model was consistent with the measurements when the discharge gap was 2 µm. Since the actual discharge crater was smaller than that of the theoretical, Jithin S. et al. [13] built a finite element model of EDM considering the plasma flushing efficiency and temperature dependency of the material physical properties. Subsequently, the simulation of multi-spark EDM was carried out. In multi-spark, arc discharge might occur due to the increasing removed material in the discharge gap. Therefore, the randomness of three variables (position, energy level, and time) of the sparks was taken into consideration in the model. It was stated that the relative error between the prediction results and the experimental results was about 11.5%. Burlayenko V.N. [14] et al. carried out a coupled thermal-mechanical simulation of functionally graded materials (FGM) to deduce the FGM equation of coupled thermal-mechanical coupling. The finite element expression of FGM under plane strain conditions was proposed. The gradient of FGM properties was defined in the elements by integrating the matrix of the finite elements. In comparison to other references, the thermal-mechanical model and equation were proven to be of high reliability in the distributions of the thermal field and thermally induced stresses. Tang L. et al. [15] focused on studying the processing mechanism of particle reinforced composites in powder mixed EDM (PMEDM) through a thermo-electrical coupling simulation model during the single-pulse discharge process. A random distribution model was adopted to build a thermo-electrical coupling model during the multi-pulse discharge process. Based on the thermo-electrical coupling model, the material removal rate (MRR) of 5%-wt particle material could be predicted. The prediction error of the MRR ranged from 4.13% to 5.18%. Moreover, the powder mixing method could improve the MRR by 16.34% and reduce the surface roughness by 29.42%. Razeghiyadaki [16] improved the prediction model for the MRR and Ra of ceramic materials by adding some assumptions, such as the residual heat effects of existing craters, on subsequent processing and instantaneous evaporation. The axisymmetric rectangular region was used to predict the MRR, while the rectangular region was used to predict the Ra. Each spark point was generated at the end of the previous crater. Better prediction accuracy was presented by this model. Tlili A. et al. [17] developed a finite element difference model that involved the changes in the plasma channels and the instantaneous removal of materials during the discharge process. The results suggested that the MRR and crater morphology were consistent with the actual situation. However, this model was not ideal for high currents. Almainha J.A. et al. [18] simulated a single-pulse discharge process in EDM by an electrothermal model based on the Joule effect. It is worth noting that the dielectric strength plays a bold role in the discharge gap and work voltage. In addition, the MRR and tool wear rate (TWR) could not be predicted well during the single-pulse discharge process. However, the prediction of the average maximum surface roughness was in good agreement with the experimental data. The prediction values of the MRR and TWR were close to the actual data during a multi-pulse discharge process. The above-mentioned studies about the thermal model of conventional EDM were based on the Fourier heat conduction law. Since the period of the single-pulse discharge process is about 10-100 µs, it is sufficient to achieve thermal equilibrium inside the workpiece material during this period. However, the period of the single-pulse discharge process of micro/nano-EDM is very short (<1 µs) [19]; thus, it is difficult to achieve thermal equilibrium inside the workpiece material during this period. Shahri et al. [20] also pointed out in their review of EDM temperature field research that the temperature gradient of micro/nano-EDM was extremely large. The Fourier heat conduction law was difficult to use to accurately describe the temperature field distribution in micro/nano-EDM because it assumes that the heat conduction speed is extremely fast. Therefore, in order to accurately reveal the discharge mechanism of micro/nano-EDM, it is necessary to consider the non-Fourier effect while building the thermal model. The goal of this study was to develop a new thermal model for predicting discharge craters in micro/nano-EDM more precisely, considering the non-Fourier effect. The relaxation time was brought into deriving the non-Fourier heat conduction law. The temperaturerelated material parameters, the latent heat of the phase change, and the parameter-related energy partition coefficient were considered to establish the thermal model during singlepulse discharge machining in micro/nano-EDM. The developed thermal model was solved by numerical simulation software. The temperature field distribution of the workpiece material was calculated to predict the size of the discharge craters. The comparison between experimental data and the simulated data of the discharge craters was completed to assess the predicting accuracy of the new thermal model.

Derivation of the Non-Fourier Heat Conduction Law for Micro/Nano-EDM
The classical Fourier heat conduction law is an empirical equation developed by Fourier while studying a large number of heat conduction phenomena. It is the basic law for the analysis of conventional heat conduction problems, as shown in Equation (1) [21]. This law has three main assumptions: (1) the speed of heat wave transmission is infinite; (2) the medium is uniform and continuous; (3) the medium belongs to isotropic material.
q(x, y, z, t) = −kgrad(T) (1) where q is the heat flux of the material, and k is the thermal conductivity of the material. In practice, many physical phenomena (such as thermal, acoustic, optical, and electrical) do not occur instantaneously. Due to the influence of inertial damping of the system, there is a time lag in response to physical phenomena. Similarly, there is also a time lag in the phenomenon of heat conduction. The heat transfer in solids is actually the collision transfer between molecules, which is limited by the velocity and the mean free path of the molecule. In some cases (such as ultrafast laser processing and rapid metal condensation), the speed of heat transfer cannot be assumed to be infinitely fast in the process of rapid temperature change.
In this study, the classical Fourier heat conduction law was modified by introducing the relaxation time (τ 0 ) of heat transfer. Then, the generalized heat conduction law was derived, as shown in Equation (2) [22].
For Equation (2), the Taylor series was used to carry out the τ 0 expansion. Since the relaxation time of the material was small, the quadratic term and the high-order term about τ 0 were ignored, and the result is as shown in Equation (3): The non-Fourier effect heat conduction equation can be obtained by simultaneously taking the derivative of both ends of Equation (3) and the law of conservation of energy, as shown in Equation (4) [22].
where ρ is the density of the material, and C is the specific heat of the material. According to the knowledge of calculus, Equation (5) is equal to Equation (6).
ρCτ 0 Then, based on Equation (6), the two-dimensional axisymmetric non-Fourier Equation can be obtained, as shown in Equation (9).

Configuration and Hypothesis of the Thermal Model
The thermal model of micro/nano-EDM in this study is based on the non-Fourier effect heat conduction equation (Equation (4)). According to the processing principle of micro/nano-EDM, the three-dimensional thermal model can be simplified to a twodimensional axisymmetric model [23], as shown in Figure 1. The R P is the radius of the discharge channel. The boundary conditions and initial values of micro/nano-EDM are shown in Equation (10) [23]. B1 edge: when the r coordinate is smaller than or equal to the radius of the discharge channel (RP), the discharge energy q(r) will be absorbed by the workpiece. When the r coordinate is larger than the radius of the discharge channel (RP), the thermal energy is diffused outward by convection. hc is the convective heat transfer coefficient, ΔQ is the heat flux, and the Tref is the ambient temperature. B2 edge is an axisymmetric side. The heat flux on the B2 edge is 0. In the simulation model, the lengths of the B1 edge and B2 edge are 1000 times the length of the radius of the discharge channel (RP). Then, the B3 edge can be considered an open boundary. The heat flux on the B3 edge is 0. The initial temperature of the workpiece is the same as the ambient temperature.
Due to the multiple physics involved, the discharge process of the micro/nano-EDM is very complicated. To simplify the calculation of the model, several assumptions are made to establish the thermal model, as follows: The boundary conditions and initial values of micro/nano-EDM are shown in Equation (10) [23]. B 1 edge: when the r coordinate is smaller than or equal to the radius of the discharge channel (R P ), the discharge energy q(r) will be absorbed by the workpiece. When the r coordinate is larger than the radius of the discharge channel (R P ), the thermal energy is diffused outward by convection. h c is the convective heat transfer coefficient, ∆Q is the heat flux, and the T ref is the ambient temperature. B2 edge is an axisymmetric side. The heat flux on the B2 edge is 0. In the simulation model, the lengths of the B1 edge and B2 edge are 1000 times the length of the radius of the discharge channel (R P ). Then, the B3 edge can be considered an open boundary. The heat flux on the B3 edge is 0. The initial temperature of the workpiece is the same as the ambient temperature. Due to the multiple physics involved, the discharge process of the micro/nano-EDM is very complicated. To simplify the calculation of the model, several assumptions are made to establish the thermal model, as follows: (1) The physical properties of the workpiece material (such as thermal conductivity and specific heat) are only related to the temperature of the workpiece material [9]; (2) The workpiece material is uniform and isotropic [23]; (3) The energy distribution of the discharge channel obeys the Gaussian distribution [24]; (4) The radius of the discharge channel is a function of the pulse current and the pulse-on time [9]; (5) The ratio of the absorbed energy of the workpiece material is a function of the pulse current and the pulse-on time [9].

Establishment of the Thermal Model
Several key factors are considered to establish the thermal model of micro/nano-EDM in single-pulse discharge machining, which includes the Gauss heat source model, the radius of the discharge channel, the proportion of the discharge energy absorbed by the workpiece, and latent heat of phase change and the temperature-related physical parameters. The radius of the discharge channel and the proportion of the discharge energy absorbed by the workpiece vary with the discharge parameters.

Gauss Heat Source Model
The heat source model of the EDM temperature field mainly includes the point heat source, the uniform surface heat source, the uniform body heat source, and the Gauss heat source. Among them, the Gauss heat source is more in line with the discharge theory of EDM and has become the most popular heat source model for the temperature field of EDM. In previous studies, the Gauss heat source of the EDM temperature field was divided into two types, as shown in Equation (11) [25] and Equation (12) where U is the discharge voltage, I is the discharge current, and the f c is the proportion of energy absorbed by the workpiece. Equation (8) is a Gauss distribution function with respect to time. However, according to previous research [25,26] and experimental experience, neither the waveforms of the discharge current nor the discharge voltage showed a Gauss distribution over time. Therefore, Equation (7) was chosen as the Gauss heat source model in this study.

Radius of the Discharge Channel
In the process of EDM, the diameter of the discharge channel is difficult to measure directly because of the short discharge time and the strong light effect. In reference [2], the empirical equation of the radius of the discharge channel was deduced by observing actual discharge craters with SEM, as shown in Equation (13). The radius of the discharge channel (R p : µm) is a function of the pulse current (I: A) and the pulse-on time (T on : µs).

Proportion of the Discharge Energy Absorbed by the Workpiece
When the electric field strength between the workpiece and the electrode reaches the critical value, the dielectric between the workpiece and the electrode will be broken down. At the same time, the free electrons inside the electrode will escape from the surface of the Crystals 2022, 12, 794 7 of 14 electrode and bombard the surface of the workpiece at a high speed after being accelerated by the electric field. Since the mass of the atom/molecule of the workpiece is much larger than that of the electrons, the kinetic energy of most electrons will be converted into thermal energy, which will melt or vaporize the workpiece materials. Discharge energy mainly flows into three parts, which consist of the workpiece, electrode and dielectric. There is no unified conclusion about the distribution coefficient of the discharge energy. However, the discharge energy distribution coefficient that varies with the discharge parameters has gradually replaced the fixed partition coefficient. In reference [2], by combining temperature field simulation with the experimental measurements, the ratio of the discharge energy absorbed by the workpiece was obtained, which was a function of the pulse current and pulse-on time, as shown in Equation (14). The proportion of energy absorbed by the workpiece in this study is consistent with that in reference [2].

Latent Heat of Phase Change
In the process of phase change, crystalline materials will absorb/release a certain amount of energy, which is called the latent heat of phase change. Two phase-change processes are involved during single-pulse electrical discharge machining in micro/nano-EDM, solid to liquid and liquid to gas. The effect of the latent heat of phase change on the temperature field of micro/nano-EDM can be converted into equivalent specific heat, as shown in Equations (15) and (16) [1,23]: where C p is the solid specific heat of the material, C m is the liquid specific heat of the material, C ev is the gas specific heat of the material, L m is the latent heat of the melting material, L ev is the latent heat of the vaporizing material, T m is the melting point of the material, and T ev is the boiling point of the material.

Physical Parameters of the Workpiece Material
Austenitic stainless steel 304 was chosen as the workpiece material in this study. Because the temperature of micro/nano-EDM varies greatly, it was necessary to consider the thermophysical parameters that vary with temperature, including thermal conductivity and specific heat. The physical parameters are shown in Tables 1 and 2.

Calculation of the Thermal Model in Micro/Nano-EDM Considering the Non-Fourier Effect
According to the above model assumptions and theoretical analysis, the finite element model of the temperature field in micro/nano-EDM can be established. On the basis of COMSOL multi-physics numerical calculation software, the temperature field distribution under the specific processing parameters can be obtained. The non-Fourier effect can be compared and analyzed.
On the basis of COMSOL multi-physics software, the partial differential equations of the coefficient form are adopted, as shown in Equation (17).
To make Equation (17) equal to Equation (5), the coefficients of Equation (17) should be set as Equation (18). Then, the relaxation time can be precisely introduced into the thermal model.
Based on the results and comparison, the influence of the non-Fourier effect on the temperature field distribution can be characterized and analyzed. Figures 2 and 3 are the temperature field distribution results without considering and considering the non-Fourier effect in EDM, respectively. The processing parameters are as follows: pulse current of 1.15 A, pulse-on time of 0.49 µs, and peak voltage of 100 V. The calculated results show that the energy absorption ratio of the workpiece was 8.3%, and the radius of the discharge channel was 5.9 µm. From Figures 2 and 3, the following can be observed: (1) The temperature distribution of the micro/nano-EDM was sequentially decreased from the center of the discharge point along the axial and radial directions; (2) the maximum temperature can reach thousands of degrees Celsius; according to the melting point/boiling point temperature lines in the figure, the volume of vaporized material of the workpiece reached about 40% of the volume of melted material, and therefore, it was necessary to consider the melting and vaporization latent heat simultaneously; (3) without considering the non-Fourier effect, the maximum temperature of the workpiece material was 5.07 × 10 3 K, the radius of the molten pool was 6.0 µm, and the depth of the molten pool was 2.03 µm; (4) considering the non-Fourier effect, the maximum temperature of the workpiece material was 8.3 × 10 3 K, the radius of the molten pool was 5.63 µm, and the depth of the molten pool was 1.87 µm; (5) from the aspect of the temperature gradient, the temperature gradient considering the non-Fourier effect was obviously higher than that without considering non-Fourier effect near the center of the discharge point. The main reason for this phenomenon is that the discharge process of micro/nano-EDM is transient. However, there was a time delay in the heat wave transfer. Thus, the discharge energy could not be transferred instantaneously to the workpiece material, which caused the discharge energy to gather in the center of the discharge point, promoting a larger temperature gradient near the discharge center.

Experimental Configuration
In this study, the predicted precision of the thermal model for micro/nano-EDM considering the non-Fourier effect was evaluated by comparing the sizes of experimental discharge craters. Figure 4 shows the schematic diagram of the single-pulse discharge machining system [2]. The power supply of a single-pulse discharge machining system was a DC power supply. The working voltage ranged from 0 V to 120 V. The current-limiting resistance was 50 Ω, which was used to adjust the discharge current. A high-speed digital signal processor (DSP2407) was applied to control the open-on and cut-off of the high-frequency triode, which could accurately regulate the duration of the single discharge pulse. The electrode and workpiece were set on the three-axis precise motion platform. The resolution of displacement was 0.2 μm. The cylindrical tungsten

Experimental Configuration
In this study, the predicted precision of the thermal model for micro/nano-EDM considering the non-Fourier effect was evaluated by comparing the sizes of experimental discharge craters. Figure 4 shows the schematic diagram of the single-pulse discharge machining system [2]. The power supply of a single-pulse discharge machining system was a DC power supply. The working voltage ranged from 0 V to 120 V. The current-limiting resistance was 50 Ω, which was used to adjust the discharge current. A high-speed digital signal processor (DSP2407) was applied to control the open-on and cut-off of the high-frequency triode, which could accurately regulate the duration of the single discharge pulse. The electrode and workpiece were set on the three-axis precise motion platform. The resolution of displacement was 0.2 μm. The cylindrical tungsten

Experimental Configuration
In this study, the predicted precision of the thermal model for micro/nano-EDM considering the non-Fourier effect was evaluated by comparing the sizes of experimental discharge craters. Figure 4 shows the schematic diagram of the single-pulse discharge machining system [2]. The power supply of a single-pulse discharge machining system was a DC power supply. The working voltage ranged from 0 V to 120 V. The currentlimiting resistance was 50 Ω, which was used to adjust the discharge current. A high-speed digital signal processor (DSP2407) was applied to control the open-on and cut-off of the high-frequency triode, which could accurately regulate the duration of the single discharge pulse. The electrode and workpiece were set on the three-axis precise motion platform. The resolution of displacement was 0.2 µm. The cylindrical tungsten stick of 200 µm diameter was used as the electrode. The stainless steel 304 was used as the workpiece. The peak voltage was 100 V, and the dielectric was kerosene. stick of 200 μm diameter was used as the electrode. The stainless steel 304 was used as the workpiece. The peak voltage was 100 V, and the dielectric was kerosene. Before discharge machining, every workpiece was polished to accurately measure the discharge craters. The procedures of the single-pulse discharge process were as follows: (a) The gap voltage between the electrode and the workpiece was constantly monitored with an oscilloscope. (b) The working voltage of the DC power supply was set to 15 V. The electrode slowly approached the workpiece step by step until the monitoring voltage of the oscilloscope became 0 V. (c) The distance between the electrode and the workpiece was adjusted to 10 μm. (d) One pulse was applied to the electrode and the workpiece, which was controlled by the DC power supply, DPS and PC. The working voltage of the DC power supply was set as 100 V. The electrode approached the workpiece step by step until the discharge voltage waveform was displayed on the oscilloscope. The displacement of the electrode in each step was 0.2 μm. Then, one discharge crater was generated on the workpiece surface. The sizes of discharge craters were measured using a high depth-of-field microscope (KEYENCE VHX-600E). The magnification was 3000×. The measurement accuracy was 0.01 μm. In addition, the shape of the molten pool was assumed to be a semi-ellipsoid [23,24], and the equation for calculating the volume of the molten pool (Vc) is shown in Equations (19) and (20), where Rc is the radius of the molten pool, and Dc is the depth of the molten pool.   Table 3 shows the experimental design and results [2]. It could be found that the experimental values of the sizes of discharge craters were always less than the simulation values. This phenomenon is consistent with the relative error between the experimental value and simulation values in Ref [23,24]. This is because the melted and vaporized workpiece material could not be completely removed due to its own gravity. Thus, the crater after the discharge spark was smaller than that at the end moment of the discharge spark. In addition, it could be found that the crater radius and crater depth increased with the increase in the single-pulse discharge energy, which is mainly because higher discharge energy can generate higher thermal energy for melting or vaporizing more workpiece material. The function of the single-pulse discharge energy with the pulse current and the pulse-on time is shown in Equation (21)   Before discharge machining, every workpiece was polished to accurately measure the discharge craters. The procedures of the single-pulse discharge process were as follows: (a) The gap voltage between the electrode and the workpiece was constantly monitored with an oscilloscope. (b) The working voltage of the DC power supply was set to 15 V. The electrode slowly approached the workpiece step by step until the monitoring voltage of the oscilloscope became 0 V. (c) The distance between the electrode and the workpiece was adjusted to 10 µm. (d) One pulse was applied to the electrode and the workpiece, which was controlled by the DC power supply, DPS and PC. The working voltage of the DC power supply was set as 100 V. The electrode approached the workpiece step by step until the discharge voltage waveform was displayed on the oscilloscope. The displacement of the electrode in each step was 0.2 µm. Then, one discharge crater was generated on the workpiece surface. The sizes of discharge craters were measured using a high depth-offield microscope (KEYENCE VHX-600E). The magnification was 3000×. The measurement accuracy was 0.01 µm. In addition, the shape of the molten pool was assumed to be a semi-ellipsoid [23,24], and the equation for calculating the volume of the molten pool (V c ) is shown in Equations (19) and (20), where R c is the radius of the molten pool, and D c is the depth of the molten pool. Table 3 shows the experimental design and results [2]. It could be found that the experimental values of the sizes of discharge craters were always less than the simulation values. This phenomenon is consistent with the relative error between the experimental value and simulation values in Ref [23,24]. This is because the melted and vaporized workpiece material could not be completely removed due to its own gravity. Thus, the crater after the discharge spark was smaller than that at the end moment of the discharge spark. In addition, it could be found that the crater radius and crater depth increased with the increase in the single-pulse discharge energy, which is mainly because higher discharge energy can generate higher thermal energy for melting or vaporizing more workpiece material. The function of the single-pulse discharge energy with the pulse current and the pulse-on time is shown in Equation (21) [1].   Figure 5 shows the relationship between the relative error of the thermal model and the pulse-on time. It can be found that: (1) In the thermal model without considering the non-Fourier effect, the simulation data of the crater radius and crater depth are larger than the experimental value. The relative error between the simulation data and experiment data of the volume of the discharge crater ranges from 16% to 55%. The average relative error is 33%. In the thermal model considering the non-Fourier effect, the simulation data of the crater radius and crater depth are slightly larger than the experimental value. The relative error between the simulation data and experiment data of the volume of the discharge crater ranges from 5% to 16%. The average relative error is 10%. It means that, in micro/nano-EDM, the thermal model considering the non-Fourier effect has higher prediction accuracy than that without considering the non-Fourier effect. (2) The relative error between the simulation data of the thermal model without and with considering the non-Fourier effect decreased with the increase in pulse-on time. A long pulse-on time ensures enough time for heat transferring within the workpiece material during the pulseon time. The temperature field can better achieve a thermal equilibrium state. The thermal hysteresis has little effect on heat conduction. The non-Fourier effect is not significant under long pulse-on time (>1 µs). Meanwhile, the thermal relaxation time of metal is of the order of magnitude of 10 −9 -10 −8 s. The duration time of discharge pulse is of the order of magnitude of 10 −7 -10 −6 s. Then, the effect of thermal hysteresis on the thermal transmission during the material removing process is more significant under the shorter pulse-on time, especially when the duration time of discharge pulse is as low as 0.1 µs. The distribution of thermal energy is more concentrated under the shorter pulse-on time. Then, the proportion of the workpiece material is removed because vaporizing increases with the decrease in pulse-on time under the same discharge energy in the single-pulse discharge process. It means the non-Fourier effect is more obvious under the shorter discharge time of a single pulse. more significant under the shorter pulse-on time, especially when the duration time of discharge pulse is as low as 0.1 μs. The distribution of thermal energy is more concentrated under the shorter pulse-on time. Then, the proportion of the workpiece material is removed because vaporizing increases with the decrease in pulse-on time under the same discharge energy in the single-pulse discharge process. It means the non-Fourier effect is more obvious under the shorter discharge time of a single pulse.

Conclusions
(1) The simulation data of the thermal model show that the maximum temperature of the temperature field distribution considering the non-Fourier effect was higher than that without considering the non-Fourier effect. On the contrary, the volume of the molten pool simulated by the thermal model considering the non-Fourier effect was smaller than that simulated by the thermal model without considering the non-Fourier effect. This is because there was a time lag in the phenomenon of heat conduction when considering the non-Fourier effect.
(2) In the thermal model without considering the non-Fourier effect, the simulation data of the crater radius and crater depth were higher than the experimental value. The relative error between the simulation data and experimental data of the volume of the discharge crater ranges from 16% to 55%. The average relative error is 33%. The effect of the non-Fourier effect is more obvious under the shorter pulse-on time.
(3) In the thermal model considering the non-Fourier effect, the simulation data of the crater radius and crater depth were slightly higher than the experimental value. The relative error between the simulation data and experiment data of the volume of the discharge crater ranges from 5% to 16%. The average relative error is 10%.
Hence, it is necessary to consider the non-Fourier effect for establishing a high-precision thermal model when the pulse-on time is small enough (for example, less than 1 μs). The results of this study can provide a reference for the theoretical research and the precise control of micro/nano-EDM in the future.

Conclusions
(1) The simulation data of the thermal model show that the maximum temperature of the temperature field distribution considering the non-Fourier effect was higher than that without considering the non-Fourier effect. On the contrary, the volume of the molten pool simulated by the thermal model considering the non-Fourier effect was smaller than that simulated by the thermal model without considering the non-Fourier effect. This is because there was a time lag in the phenomenon of heat conduction when considering the non-Fourier effect.
(2) In the thermal model without considering the non-Fourier effect, the simulation data of the crater radius and crater depth were higher than the experimental value. The relative error between the simulation data and experimental data of the volume of the discharge crater ranges from 16% to 55%. The average relative error is 33%. The effect of the non-Fourier effect is more obvious under the shorter pulse-on time.
(3) In the thermal model considering the non-Fourier effect, the simulation data of the crater radius and crater depth were slightly higher than the experimental value. The relative error between the simulation data and experiment data of the volume of the discharge crater ranges from 5% to 16%. The average relative error is 10%.
Hence, it is necessary to consider the non-Fourier effect for establishing a highprecision thermal model when the pulse-on time is small enough (for example, less than 1 µs). The results of this study can provide a reference for the theoretical research and the precise control of micro/nano-EDM in the future.