Shock-wave propagation in supercritical CO₂ induced by nanosecond-pulsed arc plasma

Figure 2 shows the voltage and current waveforms in the SC phase (T = 305 K, P = 7.53 MPa). The pulsed voltage increased with the rise in the voltage rate of 0.5 kV ns⁻¹. We previously reported that the branched streamer discharge initiates at the needle tip and develops toward the plane electrode as the applied voltage increases [19–22]. The large pulsed current suddenly began to flow when the voltage collapsed at the moment the streamer reached the plane electrode. A local thermodynamic equilibrium was established ranging from 1 ps to 1 ns [18]. Owing to the high conductivity of the arc channel, the inductance–capacitance–resistance series damped oscillation of the waveforms continued during approximately 700 ns.

Zoom In Zoom Out Reset image size

The shadowgraph images were taken after the ignition of the pulsed arc discharge. The pulsed arc ignition time t = 0 was determined by the start of the pulse-arc current, as shown in figure 2. Figure 3 shows the shadowgraph images of the post-breakdown phenomena in the SC phase (T = 305 K, P = 7.50 MPa) during the period between t = 64 ns and t = 7 μs. Figures 3(b)–(f) show images from different arc discharges. The cylindrical shock wave began to propagate before t = 64 ns, as shown in figures 3(b) and (g). The delay from the ignition of the pulsed arc plasma to the separation of the shock wave was almost independent of the CO₂ density. Furthermore, in figure 3(b), the residual branched streamer channels were observed with the pulsed arc plasma. The shock wave expanded together with the dark-region caused by the discharge plasma, as shown in figure 3(c). The dark-region was larger than the light emission region of the plasma, as shown in figure 3(d). The schematic diagram of figure 3(d) is shown in figure 3(h). The black portion of the shadowgraph image can be seen because the backlight did not reach the detector due to refraction. The reason for the appearance of the dark-region could be due to the extreme difference refraction index between dark-region and background (SC-CO₂). The variation of the refraction index in dark-region was caused by the pulsed arc plasma; however, further investigation of the plasma parameter should be done to elucidate the detailed conditions of the dark-region in the future. The dark-region collapsed after the end of its expansion (2 μs ∼), as shown in figures 3(e) and (f).

Zoom In Zoom Out Reset image size

Figure 4 shows the characteristics of the shock-wave propagation and the expansion of the dark-region in the high-pressure gas phase (figure 4(a); T = 305 K, P = 1.59 MPa) and the SC phase (figure 4(b); T = 305 K, P = 7.53 MPa). In general, the propagation characteristics of the cylindrical shock wave generated by lightning or a spark discharge can be described by the weak shock-wave theory (1.16 < Mach number < 3) [23–27]. The overpressure (where the pressure ratio through the shock front is less than 10) and the trajectory of the shock wave can be formulated by application of the weak shock-wave theory. However, the application of this theory to pressurized CO₂ is problematic because the theory assumes perfect gases at constant specific heat levels. In this study, the shock-wave propagation with respect to the time t was approximated by the quadratic polynomial approximation (equation (1)) to estimate the Mach number in the pressurized CO₂.

$$\begin{equation}R = - a{t^2} + bt + c,\end{equation} \tag{ 1 }$$

Zoom In Zoom Out Reset image size

where a, b, and c are constant, and R is the propagation length of the shock wave. Although the physical characteristics of the gas and SC phases are completely different, it was found that the quadratic polynomial approximation fits the shock-wave propagation of both gas and supercritical phases well. This agreement suggests that shock-wave propagation in both gas and SC phases is governed by the same physical mechanism. The velocity of the shock wave v was expressed by differentiating equation (1) as follows:

$$\begin{equation}v = \frac{{dR}}{{dt}} = - 2at + b.\end{equation} \tag{ 2 }$$

The Mach number for t was calculated by dividing equation (2) by the sound speed and is shown in figure 4. The t_s, which is the separation time between the shock wave and the dark-region, was then assigned to t in equation (2) to determine the initial Mach number.

Figure 5 shows the initial Mach number as a function of CO₂ density ranging from 1.59 to 8.0 MPa. The error ranges indicate the maximum and minimum values of the shot-to-shot variations. The sound speed of CO₂ was calculated based on the literature [28]. The Mach number reached a local maximum (T = 305 K, P = 7.53 MPa) at close to the critical anomaly condition in the SC phase. The anomaly of the Mach number is explained by the specific heat capacity ratio γ, which is one of the physical parameters related to shock-wave propagation [23]. The γ is a dimensionless quantity. Figure 5 shows the γ reached a local maximum at critical density, which can affect shock-wave propagation. The γ of CO₂ was calculated by the equation of state expressed in form of the Helmholtz energy [28].

Zoom In Zoom Out Reset image size