On the Mechanism of High-Voltage Pulsed Fragmentation from Electrical Breakdown Process

Abstract

With the deepening of the development of natural resources, the difficulty of rock breaking is increasing. Novel efficient rock-breaking technology and matching rock-breaking tools are urgently needed. The high-voltage pulsed fragmentation (HVPF) method has high efficiency and great development potential. So, it has attracted wide attention. However, the unclear understanding of the relationship between the mechanism of HVPF, the design and parameter optimization of HVPF tools, and rock-breaking energy consumption also impede the commercial progress of this technology. In this paper, the electrical crushing is subdivided into “partial electrical breakdown” (PEB) and “complete electrical breakdown” (CEB) from electrical breakdown process (EBP), and the mechanism of PEB is proposed. Finally, a method, which is named as voltage partitioned method (VPM), for designing and optimizing parameters of the HVPF tools is provided. The results of numerical simulation and electrical breakdown experiment can well support the mechanism of PEB. The mechanism of PEB provides a basis for the application of multi-pulse electrical breakdown technology. The VPM establishes the relationship between voltage loading parameters, electrode structure parameters and rock electrical parameters, which can provide reliable help for the design and parameter optimization of HVPF tools.

Appendix

1.1 Appendix A

1.1.1 Calibration process of material parameters in simulation

The calibration process of material parameters in simulation will be introduced in this section. And the relative permittivity of the rock ε_R to measure the “electrical damage” in the simulation.When the amplitude of electric field strength inside the rock |E_p| is less than the partial TBS, the rock can’t be broken down, and the relative permittivity inside the rock is ε_min. There is no damage inside the rock at this time. When the amplitude of electric field strength inside the rock |E_p| is larger than the partial CBS, the local rock area is completely electrically broken down, forming some “plasma channel branches” (it can be regarded as “electric fracture” or “electric damage”). At this time, to make the simulation converge, it is assumed that the relative permittivity of the breakdown domain of rock is a constant value ε_max which is much larger than ε_min. In the process of electrical breakdown, the relative permittivity inside the rock ε_R is between ε_min and ε_max, namely:

$$\varepsilon _{R} = \left\{ {\begin{array}{*{20}l} {\varepsilon _{{\max }} } \hfill & {\left| {E_{{\text{p}}} } \right| \ge E_{{{\text{ps}}}} } \hfill \\ {\left[ {\frac{{\left( {\left| {E_{{\text{p}}} } \right| - E_{{{\text{pc}}}} } \right)}}{{E_{{{\text{ps}}}} - E_{{{\text{pc}}}} }}} \right] \times \left( {\varepsilon _{{\max }} - \varepsilon _{{\min }} } \right)} \hfill & {E_{{{\text{pc}}}} < \left| {E_{{\text{p}}} } \right| < E_{{{\text{ps}}}} } \hfill \\ {\varepsilon _{{\min }} } \hfill & {E_{{{\text{pc}}}} < \left| {E_{{\text{p}}} } \right| < E_{{{\text{ps}}}} } \hfill \\ \end{array} } \right.,$$

(5)

where ε_min is the lower limit of relative permittivity, dimensionless; ε_max is the upper limit of the relative permittivity, dimensionless; ε_R is the relative permittivity of the rock; |E_p| is the amplitude of electric field strength inside the rock; E_pc is the partial TBS (V/m); E_ps is the partial CBS (V/m).

In the process of electrical breakdown, the applied pulse voltage (the curve in Fig. 9) and the governing equation Eq. (4) drive the electric field (such as charge) inside the rock to change; the electric field distortion caused by the rock material equation and the “electrical damage” governing equation Eq. (5) reacts to the governing equation Eq. (4). In this way, the self-coupling between the electric field and the “electrical damage” is realized during the electrical breakdown, which reflects the non-uniform response of the rock to the electric field.

From the above analysis, the parameters that need to be determined in the electrical breakdown simulation of rock are the lower limit of relative permittivity ε_min, the upper limit of relative permittivity ε_max, the partial TBS and the partial CBS. The parameter calibration process is shown in Fig. 

Fig. 24

Schematic diagram of rock parameter calibration process

24. The specific calibration process is:

• (1) Setting the voltage as 50 kV in the electrostatic field, and the lower limit of the relative permittivity ε_min is fixed at 6.5 (the relative permittivity of mica is 6–8), then the maximum field strength in the rock domain is measured at this time as 1.94 × 10⁷ V/m, that is, the partial TBS (E_pc = 1.94 × 10⁷ V/m) is obtained.

• (2) Setting the voltage as 100 kV in the electrostatic field, and giving a testing value of the upper limit of relative permittivity ε_max*, then the maximum field strength, which is called the testing value of the partial CBS E_ps*, in the rock domain is extracted from the result.

• (3) Giving a pulse voltage with a peak value of 100 kV (i.e., U_p = 100 kV), and substituting the parameters, which include the lower limit of the relative permittivity ε_min, the partial TBS, the testing value of the upper limit of relative permittivity ε_max* and the partial CBS E_ps*, obtained in steps (1)-(2) into the rock material equation Eq. (5) to perform transient electrical breakdown simulation. Observing the “electrical damage” inside the rock after the electrical breakdown simulation (i.e., plasma channel). If the plasma channel does not penetrate the rock, slowly increase the testing value of the upper limit of relative permittivity ε_max*; if the plasma channel penetrates the rock, slowly decrease the testing value of the upper limit of relative permittivity ε_max*.

• (4) Repeating steps (2)-(3) until the plasma channel just penetrates the rock when U_p = 100 kV, then the calibrated value of the upper limit of the relative permittivity ε_max and the partial CBS can be obtained (i.e., ε_max = ε_max*, E_ps = E_ps*).

After multiple “trails and errors” calibration experiments, the calibration values of the upper limit of the relative permittivity and the partial CBS are ε_max = 206.5 and E_ps = 3.85 × 10⁷ V/m, respectively.

1.2 Appendix B

1.2.1 Porosity tests of two kind of sandstones

The porosity tests of two kind of sandstones will be introduced in this section. And the measured porosity of the two kinds of sandstones is obtained using tap water immersion method. First, measure the masses M₁ of the two remaining sandstones after coring. Then, the two rocks are soaked in tap water for 48 h and then taken out, and the mass M₂ at this time is measured. Finally, the porosities of the two kinds of rocks are obtained according to the mass difference after and before the rock samples are immersed in water. The masses of the two rocks measured in the test are shown in Tables

Table 3 Data in porosity test of red sandstone

3 and

Table 4 Data in porosity test of lime sandstone

4, respectively. The calculation formula of measured porosity is

$$p = \frac{{\Delta M}}{{M_{1} }} \times \frac{{\rho _{R} }}{{\rho _{w} }} \times 100\% ,$$

(6)

where p is measured porosity, %; $\Delta M$ is mass difference after and before the rock sample is immersed in water, kg; ρ_w is the density of tap water, and taking ρ_w = 1000 kg/m³ here; ρ_R is the density of the rock, and the density of red sandstone is 2086.7 kg/m³, while the density of lime sandstone is 2239.9 kg/m³.

Substituting the data in the “Total” row in Tables 3 and 4 into Eq. (6), the measured porosities of red sandstone and lime sandstone are 8.39% and 12.92%, respectively.