Pressure Characteristics of Underwater High-Voltage Pulsed Discharge Shock Waves Using Needle-Mesh Electrode

Abstract

High-Voltage Fragmentation is a novel comminution technology that utilizes shock waves generated in water by nanosecond pulsed voltages with fast rise times (<500 ns) to fracture materials, offering significant advantages in energy efficiency and environmental friendliness. This study established an underwater pulsed discharge experimental platform to meet the fast-rise-time pulse parameter requirements. It analyzed the influence patterns of the needle-mesh electrode gap distance, the needle electrode tip radius of curvature, and water conductivity on shock wave pressure intensity and time-domain characteristics. The research found that the energy conversion efficiency of underwater pulsed discharge is significantly affected by the pre-breakdown process. The peak pressure, impulse, velocity, and rise slope of the shock wave exhibit a trend of initially increasing and then decreasing with increasing needle-mesh electrode gap distance and needle electrode tip radius of curvature. The maximum pressure intensity, maximum equivalent wave velocity, maximum rise slope, and shortest wavefront time occurred at a 20 mm gap distance and a needle electrode tip curvature radius of 0.45 mm. Both pressure intensity and propagation velocity initially increased and then decreased with increasing water conductivity, reaching their maxima at a water conductivity of 340 μS/cm. Water conductivity showed no significant effect on rise slope and wavefront time.

1. Introduction

High-Voltage Fragmentation (HVF) is a novel comminution technology. In principle, HVF utilizes shock waves generated in water by nanosecond pulses with fast rise times (<500 ns) to achieve selective separation of materials. Currently, HVF has demonstrated promising application results in coalbed methane development [1,2] and ore fragmentation [3,4,5]. Researchers have also applied HVF to the recycling of waste printed circuit boards [6] and solar panels [7]. HVF offers advantages such as high peak pressure and strong mechanical damaging effects, enabling effective fragmentation and liberation of material components. This results in high recovery yields and quality for the separated components, demonstrating significant technical advantages over conventional fragmentation methods.

Researchers have conducted studies on the characteristics of shock waves generated by underwater pulsed discharges, focusing on aspects such as optical observation of the discharge, plasma channel development characteristics, and numerical simulation of shock wave pressure. Kocik [8] employed schlieren photography to capture images of the pulsed discharge streamers and the spatiotemporal distribution of shock waves between needle-plate electrodes. These images reflect the phase transition process at discharge initiation, streamer propagation, shock wave generation, and bubble formation, indicating that shock wave formation is related to the breakdown characteristics of the discharge channel. Xinpei Lu [9,10] captured optical images of the shock wave front, revealing a propagation velocity of approximately 1500 m/s. Through observation of high-speed shadowgraphs of the plasma channel, it was noted that the channel possesses a boundary layer whose thickness is influenced by the power density within the channel, and the boundary layer thickness decreases as the expansion velocity increases [11]. Yi Liu [12] observed the phenomenon of microsecond pulsed discharge in water between needle-needle electrodes. They analyzed the influence of deposited discharge energy, underwater arc characteristics, and its expansion velocity [13] on shock wave pressure characteristics. Their findings indicate that shock wave pressure intensity is positively correlated with deposited discharge energy, and improving energy conversion efficiency can significantly enhance shock wave intensity [14]. The research team led by Aici Qiu has conducted extensive research on shock waves generated by underwater wire explosion. They performed breakdown experiments on water dielectric switches under hundred-nanosecond short pulses, discovering that the breakdown voltage and time of the water increase with rising water pressure [15]. Based on the classical piston model, they established a one-dimensional computational model for shock waves from underwater wire explosion. Experiment showed that shock wave intensity attenuates with increasing radial propagation distance [16]. Furthermore, they investigated the effect of the current rise slope during microsecond-scale wire explosion on underwater shock waves, finding that the shock wave pressure amplitude is positively correlated with the current rise slope. Under wire explosion conditions, shock wave intensity is also determined by the deposited energy [17].

Existing research indicates that the pressure and development characteristics of shock waves generated by underwater pulsed discharges are intrinsically linked to the energy conversion efficiency during their formation. This efficiency, however, is influenced by factors such as power supply parameters, discharge channel length, and conductivity [18]. Nevertheless, most current studies employ microsecond or submicrosecond voltage pulses, whereas HVF requires pulses with a fast rise time (<500 ns). Systematic investigations into the pressure intensity, time-domain characteristics, and influencing factors of shock waves generated under such fast-rise-time nanosecond pulses remain scarce. Under the action of fast-rising nanosecond pulses, solid materials are more susceptible to breakdown discharge than the surrounding water medium [19]. The pulsed discharge energy propagates along the plasma channel, subjecting the solid material to extreme temperature (up to ~10⁴ K) and pressure (up to ~10¹⁰ Pa) [20]. The rapidly expanding penetrating plasma channel generates shock waves that propagate along the material interface. Consequently, the material undergoes tensile stress and fragments. The already fragmented particles are further disintegrated by the underwater shock waves, leading to complete particle breakdown. To enhance the practical efficacy of HVF, this study establishes an underwater high-voltage pulsed discharge experimental platform. Experiments on underwater pulsed discharge shock waves were conducted to investigate the influence patterns of electrode tip radius of curvature, electrode gap distance, and water conductivity on both shock wave pressure intensity (peak pressure and pressure impulse) and time-domain characteristics (propagation time, wavefront time, and rise slope).

2. Materials and Methods

The experimental platform employed for studying the pressure characteristics of shock waves generated by underwater pulsed discharge is illustrated in Figure 1. It comprises three core subsystems: a high-voltage pulsed power source, an underwater pulsed fragmentation chamber, and an integrated measurement system for electrical characteristics and pressure signals. The high-voltage pulsed power source applies pulsed voltage across electrodes within the fragmentation chamber. This significantly enhances the inter-electrode’s electric field intensity, leading to the formation of an underwater pulsed discharge channel between the electrodes. The rapid expansion of this plasma channel generates shock waves. Simultaneously, the measurement system performs real-time acquisition of voltage, current, and pressure waveform data.

The high-voltage pulsed power source incorporates a 5-stage Marx generator circuit, with each stage employing a 0.1 μF capacitor. This system delivers negative-polarity pulsed voltages ranging from 100 kV to 240 kV. Output parameters are controlled via an optoelectronic control module. The output voltage pulses feature a fast-rise-time characteristic typical of nanosecond pulses. As illustrated in Figure 2, the rise time t_vr measures 90–110 ns, while the pulse width at half-maximum t_vw is approximately 750 ns. This study defines t_vr as the time required for the voltage to increase from 10% to 90% of peak value, and t_vw as the time for the voltage to decay to 50% of peak value.

The structure of the underwater pulsed discharge chamber is shown in Figure 3. Both the end cover and cylinder body are constructed from nylon material selected for its excellent insulation properties and mechanical strength. The internal volume is approximately 1 L. A removable pressure sensor is mounted on the cylinder sidewall at a radial distance of 50 mm from the central conductor. The needle-mesh electrode is made by tungsten steel. The needle electrodes used in this study feature tip radii of curvature of 0.25 mm, 0.45 mm, and 0.95 mm, with an adjustable inter-electrode gap spanning 10 mm to 30 mm. The bottom mesh electrode has a diameter of 70 mm with 2 mm apertures, connected to the grounded terminal of the pulsed power source via shielded cabling and securely grounded. The chamber base incorporates a sample collection compartment joined to the main body through a threaded connection with a sealing ring, ensuring robust pressure containment. During fragmentation, when the particle size of the sample becomes smaller than the aperture size of the mesh electrode, the particles fall into the collection chamber at the bottom of the cavity, thereby preventing over-crushing of the sample.

The electrical measurement system consists of a voltage divider, a Rogowski coil, and an oscilloscope. The pulsed voltage across the electrodes was measured using a VD-150 high-voltage probe (North Star Inc., Bainbridge Island, WA, USA) with a measurement range of 0–240 kV, a fixed ratio of 10,000:1, an input impedance of 2000 MΩ, and an upper cutoff frequency of 20 MHz. The current was monitored with a Pearson model 4997 current transducer (Pearson Electronics, Palo Alto, CA, USA), which has a minimum rise time of 25 ns, an upper cutoff frequency of 15 MHz, and a maximum peak-current capability of 20 kA, fully meeting the accuracy and response-time requirements of this study. Voltage, current, and pressure signals were recorded using a LeCroy HDO4104A high-definition oscilloscope (LeCroy Corporation, Chestnut Ridge, NY, USA), offering 12-bit vertical resolution, a bandwidth of 1 GHz, and a sampling rate of 10 GS/s, thereby satisfying all experimental measurement needs.

The pressure measurement setup includes a pressure sensor and a matched signal conditioner. A high-frequency piezoelectric sensor (model 113B22, PCB Piezotronics Inc., Depew, NY, USA) was employed, with a range of 0–5000 PSI (1 PSI ≈ 6.895 kPa) and a resonant frequency ≥ 500 kHz. The sensor was coupled to a signal conditioner that linearly converts the pressure signal into an electrical output, providing a sensitivity of 146.5 mV/MPa.

Figure 4b shows the typical shock wave waveform captured by the pressure sensor. Because the shock wave propagation direction forms an angle with the sensor’s horizontal axis due to its fixed position on the chamber wall, the shock wave exhibits a distinct double-peak phenomenon shown in Figure 4c. Relying solely on peak pressure as a metric for shock wave intensity introduces experimental inaccuracies, as it fails to account for pressure duration. Therefore, this study introduces pressure impulse as a complementary parameter. Shock wave loading intensity is now characterized by two metrics: peak pressure P_m and pressure impulse I_p.

In this study, the pressure impulse I_p is quantified exclusively for the first pressure pulse detected by the sensor following voltage initiation. The calculation of corresponding integral area S_p (MPa·s) and I_p (N·s/m²) duration shock wave duration t_w (µs) is defined by Equations (1) and (2) as follows:

$$S_p={\displaystyle \underset{0}{\overset{t_w}{\int }}P\cdot \mathrm{dt}}$$

(1)

$$I_p=S_p\cdot {10}^6$$

(2)

The temporal characteristics of shock waves reflect their development and propagation dynamics, directly governing the duration and velocity of material fragmentation. Based on measured waveforms, this study investigates variation patterns of three key time-domain parameters under different experimental conditions: wavefront time t_pr, rise slope k_pr, and propagation time t_s. As defined in Figure 4, t_p represents the duration for pressure to rise from 0.1 P_m to 0.9 P_m, and t_pr could be calculated via Equation (3). t_s denotes the interval from voltage breakdown initiation to shock wave detection by the sensor, while k_pr is quantified using Equation (4).

$$t_{pr}=1.25\cdot t_p$$

(3)

$$k_{pr}=0.8P_m/t_p$$

(4)

In this study, the discharge frequency was fixed at 1 Hz. Given the stochastic nature of underwater pulsed discharges, five datasets were captured per voltage level across varying electrode gap distances, tip radii of curvature, and water conductivity conditions to minimize the impact of discharge randomness on experimental outcomes.

3. Results

3.1. Effect of Electrode Distance

This study used a needle electrode with a curvature radius of 0.45 mm and purified water with a conductivity of 35.2 μS/cm to analyze the peak pressure and pressure impulse of shock waves at electrode distance of 10 mm, 15 mm, 20 mm, 25 mm, and 30 mm. The experimental results are shown in Figure 5. Across different electrode distance, the peak voltage maintained an approximately linear positive correlation with both the peak pressure and pressure impulse. The peak pressure ranked from high to low as follows: P_m (20 mm) > P_m (25 mm) > P_m (15 mm) > P_m (30 mm) > P_m (10 mm), while the pressure impulse ranked as I_p (20 mm) > I_p (25 mm) > I_p (30 mm) > I_p (15 mm) > I_p (10 mm). Notably, the pressure impulses at 15 mm and 30 mm were relatively similar. Overall, within the electrode distance adjustment range of 10 mm to 30 mm, the pressure intensity initially increased and then decreased as the distance widened. At the 20 mm, the pressure intensity was significantly higher than that at other spacings. Under a voltage of 223 kV, the pressure intensity reached its maximum, with a peak pressure of 5.56 MPa and a pressure impulse of 9.51 N·s/m². Compared to the results at 10 mm under a similar voltage, where the peak pressure was 4.03 MPa and the pressure impulse was 7.44 N·s/m², these values increased by 38.0% and 27.8%. This demonstrates that optimizing the electrode distance significantly enhances the pressure intensity of shock waves.

Electrode distance effect on pressure intensity can be interpreted through the mechanism of underwater shock wave generation. During the pre-breakdown phase, streamer channels extend from the needle tip to the ground electrode. The resulting conduction current dissipates part of the capacitor’s stored energy, reducing the net energy deposition during breakdown. Assuming negligible auxiliary losses, this process obeys the energy conservation principle in Equation (5).

$$Q_C=Q_1+Q_B$$

(5)

where Q_C represents the capacitor’s stored energy, Q₁ denotes the energy loss during the pre-breakdown phase, and Q_B stands for the deposited energy during breakdown. Q_B is ultimately delivered into the discharge channel, determining the final mechanical energy conversion efficiency. The energy conversion efficiency during the breakdown process η_B can be expressed by Equation (6).

$${\eta }_B={\displaystyle \frac{Q_B}{Q_C}}\times 100\%$$

(6)

The relationship between Q_C, Q_B, and voltage can be characterized as:

$$Q_C={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{{CU}_{\mathrm{m}}}^2$$

(7)

$$Q_B={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{{CU}_B}^2$$

(8)

where U_m represents the peak voltage and U_B denotes the breakdown voltage. The applied pulse voltage waveform in this study approximately follows a double-exponential form. After reaching peak voltage, the amplitude decays exponentially as expressed in Equation (9).

$$U(t)=U_{\mathrm{m}}{e}^{-{\displaystyle \frac{t}{R_{0}C}}}$$

(9)

where R₀ represents the circuit resistance. The breakdown time delay t_B is defined as the time interval between voltage peak occurrence and breakdown initiation. By substituting t = t_B into Equation (9) and through subsequent derivation, the relationship between η_B and U_m is obtained as shown in Equation (10). Both the increase in breakdown time delay and the decrease in circuit resistance will reduce the breakdown efficiency, consequently decreasing the energy delivered into the plasma channel during the breakdown process.

$${\eta }_B=e^{-{\displaystyle \frac{2t_B}{R_{0}C}}}\times 100\%$$

(10)

The power supply used in this study generated output voltages with rise times ranging from 90 to 110 ns. This characteristic leads to an important phenomenon: longer breakdown time delays directly correlate with extended pre-breakdown durations, resulting in greater energy loss during the pre-breakdown phase, reduced breakdown efficiency, decreased deposited breakdown energy, and ultimately weaker shockwave intensity. As shown in Figure 6, the breakdown delays are 2–4 μsat different electrode distance. These microsecond delays clearly indicate that the discharges follow a dynamic breakdown mechanism where water vaporization heating and field-induced ionization processes are temporally and spatially coupled [21]. Notably, while the peak conduction currents remained relatively consistent across different distance as shown in Figure 6a, the breakdown delays increased significantly with longer streamer propagation paths in Figure 6b. This extension of breakdown delay directly leads to increased energy loss during the pre-breakdown phase and consequently reduces the breakdown efficiency as the electrode distance increases.

However, the shock wave pressure intensity demonstrates a trend of initial increase followed by decrease with growing electrode spacing. While enhanced breakdown efficiency η_B leads to increased deposited breakdown energy, in reality during the process from channel expansion and shock wave formation to pressure sensor detection, the deposited energy undergoes incomplete conversion to mechanical energy. Significant portions are transformed into thermal energy, potential energy, and radiant energy. This process is characterized by mechanical conversion efficiency η_E, with the overall shockwave conversion efficiency η being expressed as Equation (11).

$$\eta ={\eta }_B\cdot {\eta }_E\times 100\%$$

(11)

During water gap breakdown, the equivalent resistance is significantly smaller than the resistance of the external circuit, generating strong current on the order of hundreds of amperes in the circuit. Consequently, the losses caused by the external circuit cannot be neglected. Variations in electrode distance will lead to changes in the water gap resistance. Research has shown that the plasma channel initially expands in the form of a cylindrical wave during its formation and early expansion phase, with the pressure and temperature inside the channel approximately uniform [21]. In this study, the pressure sensor was positioned approximately 5 cm from the discharge point, meaning the plasma channel can be approximated as cylindrical in shape [22]. The actual resistance of the plasma channel varies dynamically and can be simplified for influencing factor analysis using the equivalent resistance R_gap as expressed in Equation (12), where L is the length of the plasma channel, r is channel radius and σ is water conductivity.

$$R_{gap}={\displaystyle \frac{L}{\sigma \cdot \pi r^2}}$$

(12)

From the morphology of the plasma channel, it does not propagate in a straight line between the electrodes [11], but the length of the plasma channel depends on the gap spacing, allowing us to equivalent L to the gap spacing d_gap in this analysis. Under this condition, the conductivity is fixed at 35.2 μS/cm, while the effective discharge area of the electrode remains unchanged, and the radius of the plasma channel can be regarded as a fixed value. Consequently, R_gap increases with the increase in d_gap. Under identical voltage conditions, the peak current during the first oscillation half-cycle of breakdown decreases as electrode spacing expands, as clearly demonstrated in Figure 7.

When the electrode distance is small, t_B is low and η_B is high. However, R_gap is also relatively small, which increases the loop current and reduces η_E due to the large external circuit loss. At this time, the overall shockwave conversion efficiency η is low, which is not conducive to the increase in the shock wave strength. With the increase in d_gap, R_gap increases, the loop current decreases, the external circuit loss decreases, and η_E increases accordingly. However, when the electrode distance is large, t_B is high, the loss in the pre-breakdown process increases, and η_B is reduced. When the electrode distance is too large, the gap is not easy to be broken down, so it is necessary to apply higher voltage to meet the electric field strength requirements of water gap breakdown, which puts forward higher requirements for power capacity. From the point of view of shock wave strength, there exists the electrode distance that makes η the highest. In this paper, the optimal electrode distance is 20 mm.

Figure 8a reveals that underwater shockwave propagation times across different electrode distances cluster within a narrow range of 40.28–42.92 μs, and the shortest time occurs at 20 mm. The changing trend of wavefront time and rise slope is shown in Figure 8b,c. Under the conditions of this article, the energy conversion efficiency is highest at the electrode distance of 20 mm. The higher discharge efficiency leads to an increase in temperature and pressure gradients inside and outside the plasma channel. When the same voltage is applied to the electrodes, the energy per unit length of the plasma channel is maximized, and the shock wave propagates at a higher speed. The wavefront thickness is smaller, the wavefront time is correspondingly shortest, and the rise slope is the largest.

3.2. Effect of Electrode Curvature Radius

With the electrode spacing fixed at 20 mm and using purified water with a conductivity of 35.2 μS/cm, shockwave pressure characteristic tests were conducted under different curvature radii. Figure 9 shows the variation in pressure intensity with voltage for different curvature radii. As the electrode curvature radius increases, higher amplitude pulse voltages are required to achieve water gap breakdown. For electrodes with a 0.95 mm curvature radius, a pulse voltage of 183 kV or higher was necessary to initiate breakdown. In contrast, when using electrodes with curvature radii of 0.45 mm and 0.25 mm, breakdown occurred at 143 kV and 140 kV, respectively. Within the voltage range applied in this study, both the peak pressure and impulse initially increased and then decreased with increasing electrode curvature radius. The maximum pressure intensity was observed at the 0.45 mm curvature radius.

Figure 10 illustrates the variation in pre-breakdown characteristics with voltage under different electrode curvature radii. The peak value of the first half-cycle of the conduction current initially decreases and then increases with increasing curvature radius, with the smallest conduction current during the pre-breakdown process observed for the 0.45 mm electrode. The breakdown time delay t_B increases with larger curvature radii, as shown in Figure 10b. Based on the dynamic breakdown mechanism and energy conversion characteristics, the influence of different curvature radii on shock wave pressure intensity was analyzed. A larger curvature radius reduces the electric field at the electrode tip, thereby decreasing the propagation speed of the streamer channel and increasing the breakdown time delay, which is unfavorable for improving practical energy conversion efficiency. However, under a larger electrode curvature radius, the effective discharge area increases, facilitating the formation of a thicker discharge channel in the water gap [23], and intensifying the vaporization heating process at the streamer head. Furthermore, a larger curvature radius leads to a reduced gradient in the inter-electrode electric field, resulting in a more uniform field distribution. This diminished electric field gradient weakens its inhibitory effect on the vaporization process at the streamer head, thereby increasing the energy input power [24] and subsequently enhancing shock wave energy conversion. Consequently, the actual shock wave energy conversion efficiency first increases and then decreases with increasing curvature radius. Among the electrodes used in this study, the 0.45 mm curvature radius electrode exhibited the highest actual conversion efficiency.

It should be noted that the influence of electrode curvature radius on shock wave conversion efficiency is highly dependent on the power supply capacity. Under the power supply conditions of this study, the electrode with a 0.95 mm curvature radius at fixed 20 mm gap only broke down at voltages of 183 kV, whereas breakdown could not be achieved at the same gap spacing with a 4.5 mm curvature radius electrode. If the power supply capacity were sufficient to meet the breakdown requirements for electrodes with larger curvature radii, the more uniform electric field distribution under larger curvature radii would increase the effective discharge area. In such cases, the attenuating effect of the electric field gradient on the channel head would become negligible compared to the intense vaporization heating at the streamer head. Consequently, increasing the curvature radius would significantly enhance shock wave intensity.

Figure 11 illustrates the time-domain characteristics of shock waves under different curvature radii. The propagation time decreases with increasing voltage. Within the curvature radius range of 0.25–0.95 mm, the wavefront time initially decreases and then increases with larger curvature radii, while the rise slope first increases and then decreases, showing a trend similar to that of the peak pressure. Under the electrode with a 0.45 mm curvature radius, the discharge efficiency is the highest, resulting in the greatest amount of energy actually converted into the shock wave. At the same applied voltage, this configuration exhibits the shortest propagation time, the briefest wavefront time, and accordingly, the steepest rise slope.

3.3. Effect of Water Conductivity

In this experiment, the conductivity of water was modulated through the addition of sodium chloride (NaCl) and quantified using a calibrated conductivity meter. Using an electrode with a curvature radius of 0.25 mm and a fixed electrode spacing of 20 mm, the variations in peak pressure and impulse with voltage of different conductivities are shown in Figure 12. Under the same applied voltage, both the peak pressure and pressure impulse in water with a conductivity of 340 μS/cm are significantly higher than those in water with conductivities of 35.2 μS/cm and 550 μS/cm. Furthermore, an increase in conductivity results in more stable shock waves generated by the pulsed discharge, as evidenced by smaller deviations in the measured peak pressure and pressure impulse data at the same voltage.

It is noteworthy that in highly conductive water, dielectric breakdown becomes difficult to achieve. Specifically, in water with a conductivity of 3.83 mS/cm, breakdown could not be initiated within the adjustable voltage range of the power supply used in this study. This phenomenon can be explained by the pre-breakdown characteristics under different conductivity conditions. As evident from Figure 13, the conduction current in the gap increases significantly with higher water conductivity. This leads to greater energy loss per unit time during streamer propagation, weakening the electric field in the water gap and reducing the streamer development speed. If the water conductivity is excessively high, the energy stored in the capacitor may be depleted before the streamer can bridge the gap to the ground electrode, preventing breakdown altogether. On the other hand, when the conductivity increases within a moderate range, the elevated ionic current enhances the vaporization and heating process of the water medium, promoting the extension of the streamer channel toward the ground electrode. As a result, the breakdown time delay t_B first decreases and then increases with increasing water conductivity. In this study, the shortest t_B and highest breakdown efficiency η_B were achieved in water with a conductivity of 340 μS/cm.

Based on the previous analysis, the intensity of the pressure wave is influenced not only by the breakdown efficiency η_B, but also by the mechanical energy conversion efficiency η_E. As observed from the breakdown currents under different conductivity conditions shown in Figure 14, although the equivalent resistance of the channel decreases with increasing conductivity, the breakdown current remains relatively consistent and shows little variation across different conductivity levels. It can thus be inferred that, under the experimental conditions of this study, the mechanical energy conversion efficiency η_E is not significantly affected by water conductivity. Instead, the energy actually converted into the shock wave is predominantly influenced by energy losses during the pre-breakdown process.

In terms of temporal characteristics, the shock wave propagation time under different conductivity conditions is shown in Figure 15a. As the water conductivity increases, the propagation time first decreases and then increases, corresponding to an equivalent wave velocity that first rises and then declines. The shortest propagation time and highest equivalent wave velocity occur in water with a conductivity of 340 μS/cm. This is because the breakdown efficiency is highest at this conductivity level, resulting in the maximum amount of energy actually converted into the shock wave. As a result, the temperature and pressure differentials inside and outside the plasma channel increase, driving faster expansion of the channel. This trend is consistent with the variation observed in peak pressure.

As for the wavefront time and rise slope, Figure 15b,c indicate that conductivity exhibits no significant influence on these parameters.

4. Discussion

• As the electrode spacing increases, the extended propagation path of the streamer channel leads to a longer breakdown time delay and reduced breakdown efficiency. However, the equivalent resistance of the water gap also increases with larger spacing, which reduces the breakdown current and thereby decreases energy loss in the external circuit, improving the mechanical energy conversion efficiency. As a result, there exists an optimal electrode spacing that maximizes the energy converted into the shock wave. Under the conditions of this study, the optimal spacing is 20 mm, which corresponds to the highest equivalent wave velocity, the shortest wavefront time, and the steepest rise slope.
• A larger curvature radius reduces the electric field intensity at the electrode tip, slowing down the development of the streamer channel and increasing the breakdown time delay t_B. On the other hand, the larger curvature radius enlarges the discharge channel radius and reduces the equivalent gap resistance, causing the conduction current to first decrease and then increase. The reduced electric field gradient also diminishes its attenuating effect on vaporization heating at the streamer head, increasing the energy input power. Therefore, the actual conversion efficiency first increases and then decreases with larger curvature radii. Under the experimental conditions herein, the 0.45 mm curvature radius yielded optimal performance, producing the highest shock wave intensity, equivalent wave velocity, and rise slope, along with the shortest wavefront time.
• When water conductivity increases within a moderate range, the elevated ionic current accelerates the vaporization and heating process of the water medium, promoting streamer propagation toward the ground electrode and reducing the breakdown time delay. In contrast, at higher conductivity levels, the conduction current in the gap increases significantly, leading to greater energy loss per unit time during streamer development. This weakens the electric field in the water gap, slows down streamer propagation, and increases the breakdown time delay. Hence, an optimal water conductivity exists that minimizes the breakdown time delay and maximizes the breakdown efficiency. In this study, a conductivity of 340 μS/cm resulted in the highest peak pressure, impulse, and equivalent wave velocity. Additionally, water conductivity had no significant influence on the rise slope or wavefront time.

5. Conclusions

This study systematically investigates the characteristics of shock waves generated by high-voltage pulsed discharge in water under fast-rising nanosecond pulses, focusing on the effects of electrode spacing, electrode curvature radius, and water conductivity on pressure intensity and temporal characteristics. The results show that the peak pressure, impulse, and propagation velocity of the shock wave initially increase and then decrease with increasing electrode spacing and curvature radius, reaching optimal values at a spacing of 20 mm and a curvature radius of 0.45 mm. Shock wave intensity peaks at a water conductivity of 340 μS/cm, beyond which it declines. This study elucidates the relationship between energy conversion efficiency and the “pre-breakdown” process, channel resistance, and electric field distribution, providing important insights into the engineering application of high-voltage pulsed fragmentation technology in recycling decommissioned solar panels.