Ultrasonic cavitation did not occur in high-pressure CO₂ liquid

1 Introduction

Ultrasonic cavitation technology has important application value in chemical reaction acceleration and the development of novel reaction pathways. However, the current research mainly focuses on normal temperature and pressure conditions, and there are still many unanswered questions about the cavitation phenomenon in high-pressure liquids, especially supercritical fluids [1,2]. Therefore, exploring ultrasonic cavitation in high-pressure liquid CO₂ is not only of great significance for expanding the application of cavitation technology but also of academic value for understanding the fluid dynamics and thermodynamic processes in high-pressure environments [3].

2 Analysis

Bubbles in a liquid may contain air, impurity gases, vapors, or simply void spaces, and their inner pressure contains all of them. The external pressure on a bubble is derived from the hydrostatic pressure and the bubble’s surface tension [8,9]. The static pressure of liquid carbon dioxide significantly exceeds the surface tension [10], so the external pressure around a bubble approximately equals the static pressure. It is generally accepted that under cavitation, the bubble’s radius changes considerably [9], while its volume changes according to the third power of the radius. The increase in the bubble’s volume causes a reduction in the inner pressure. The ultrasound must provide sufficient negative pressure to sustain the growth of the bubble volume.

Kuijpers et al. [6] discussed high-pressure carbon dioxide liquid, noting that the bubbles therein might contain carbon dioxide vapor. Based on observations of corrosion phenomena and polymer formation experiments, the authors asserted that cavitation had occurred, necessitating significant internal pressure within the bubbles. According to Kuijpers et al., the pressures inside and outside a bubble are in equilibrium at a specific moment; if the bubble grows, the inner pressure will decrease rapidly. In addition, they theorized that this pressure resulted solely from the extremely rapid vaporization of the external liquid carbon dioxide into vapor, thereby maintaining a substantial vapor pressure. The same is true in the positive pressure phase of the ultrasound. The difference is that the vapor inside the bubble must liquefy quickly enough to become liquid. However, this interpretation is considered unreliable.

The acoustic process is an isentropic process. During this investigation, we consider that the swift oscillation between the negative and positive pressure phases of the sound wave limits the heat exchange between the media [10]. The CO₂ vapor inside the bubble and the liquid CO₂ outside cannot exchange mass through vaporization and liquefaction. In Kuijpers et al. [6], during the compression phase of the bubbles, it is implausible for the vapor inside the bubbles to rapidly release heat and liquefy into the external liquid. Similarly, during the expansion phase, it is unlikely for the external liquid to rapidly absorb heat and vaporize into gas within the bubble. There is no exchange of material between the inside and outside of the bubble. As the bubble volume increases, the pressure generated by the vapor decreases; as it decreases, the pressure increases. The contribution of the vapor to the internal pressure is similar to that of air or other gases inside the bubble; the pressure decreases as the bubble expands and increases as it contracts. Vapor cannot provide a consistent pressure.

An ultrasonic wave with a pressure lower than the static pressure cannot sustain a continuously growing bubble. The same holds true during the positive pressure phase of the ultrasound. Therefore, as discussed in Kuijpers et al. [6], the bubble volume would not change significantly with low-intensity ultrasound, and cavitation could not occur.

In liquid carbon dioxide, assuming that the vapor pressure in a bubble does not change, its motion under the action of ultrasound can be solved with the Kyuchi-Yasui equation. Its solution, represented by a curve showing the significant change of the bubble radius with time, gives the false impression of cavitation under ultrasound.

Therefore, the simulation result of Kuijpers et al. does not represent a bubble’s real movement. This fact raises the question of how the experiment’s metal corrosion and polymer formation occurred. We investigated the situation in supercritical CO₂.

Undoubtedly, the dynamic behavior of bubbles in supercritical CO₂ with the action of ultrasound satisfies Newton’s equation, and the following formula allows us to estimate the Blake pressure threshold [9]:

where R ₀ is the bubble’s initial radius, P ₀ is the hydrostatic pressure, P _v is the vapor pressure, and σ is the surface tension coefficient. Supercritical CO₂ is a single-phase fluid with P _v = 0, and its surface tension σ is extremely small [11]. Therefore, the Blake pressure threshold of supercritical carbon dioxide is P _B = P ₀. In other words, to induce cavitation in supercritical CO₂, the amplitude of the ultrasonic pressure must reach the static pressure P ₀.

3 Experiment

Kuijpers’s experiments demonstrate the corrosion of metallic plates and the formation of polymers under ultrasound in supercritical CO₂. The setup consists of a high-pressure chamber, a pressurization system, and an ultrasonic vibration system. Figure 1(a) shows the schematic of the experimental device, and Figure 1(b) shows the experimental device.

Figure 1

(a) Diagrammatic sketch of the experimental device. (b) Experimental device.

Legend 1 represents the carbon dioxide gas tank, legend 2 signifies the air compressor, legend 3 denotes the booster pump, legend 4 designates the ultrasonic generator, and legend 5 corresponds to the high-pressure chamber.

Figure 2(a) displays the structure of the high-pressure chamber, and Figure 2(b) displays the actual high-pressure device.

Figure 2

(a) Diagrammatic sketch of the hyperbaric chamber structure. (b) Actual high-pressure device.

Legend 1 denotes a cylindrical stainless-steel chamber, its inner radius is 4 cm, outer radius is 6 cm, and length is 25 cm. An incorporated side window facilitates observation within the chamber. Upon launching ultrasonic waves, we observed distinct turbulence in the fluid through the chamber’s PVC window.

Legend 2 marks the input port for carbon dioxide. Legend 3 designates a temperature controller that detects the chamber’s temperature and regulates heating via a heater wrapped around the exterior. Legend 4 represents the nucleation gas inlet, which allows the creation of an artificial nucleus in carbon dioxide. Interestingly, during the experiment, it was discovered that impurities in the carbon dioxide generated bubbles even without the addition of nucleation gas, suggesting that nucleation gas might not have been necessary.

Legend 5 indicates the exhaust port, and Legend 6 marks the carbon dioxide inlet valve. Legend 7 represents the nucleation gas inlet valve, Legend 8 is the exhaust valve, and Legend 9 points to the ultrasonic transducer. The radiation area of the ultrasonic transducer is π × r ² = 3.14 × 1.5² cm² = 7.065 cm², the operational frequency is 20 kHz, and the sound power is 1,000 W. The resulting acoustic power density was 141.5 W/cm², comparable to the 125 W/cm² observed in Kuijpers’ experiment. The ultrasonic transducer operated in cycles of 30 s, with a working time of 10 s followed by a 20-s pause.

Legend 10 designates the ultrasonic generator, and Legend 11 indicates a pressure gauge. The pressure and temperature inside the chamber slightly exceeded the critical point values for carbon dioxide, placing it in a supercritical state. The critical pressure and temperature for CO₂ were 7.16 MPa and 31.4°C.

The experimental procedure unfolded as follows:

1. Cut a set of four clean zinc sheets of the same size. Immersed the first one in the hyperbaric chamber and fixed it firmly to prevent it from falling off.

2. Ensured the closure of each power and valve. Connected the joint surface of the ultrasonic transducer to the high-pressure cavitation chamber. Connected the carbon dioxide gas tank, the air compressor, the booster pump, the ultrasonic generator, and the hyperbaric chamber in order.

3. Initiated the air compressor to supply driving air for the booster pump. Subsequently, activated the booster pump, opened the carbon dioxide valve and the hyperbaric chamber valve, and introduced CO₂ into the hyperbaric chamber. The booster pump increased the pressure in the hyperbaric chamber to 8 MPa, exceeding the critical pressure of 7.16 MPa.

4. Applied heat to the hyperbaric chamber and monitored the temperature, setting the temperature detector to 33℃, above the critical temperature of 31.4℃, which ensured that the carbon dioxide was supercritical.

5. Closed the carbon dioxide inlet and vent valves. Activated the ultrasonic generator and directed the ultrasonic transducer to emit ultrasonic waves into the supercritical carbon dioxide. Configured the experimental parameters, setting the ultrasonic transducer to operate for 200 cycles, with one working period of 30 s, including working for 10 s and pausing for 20 s.

6. Powered down the equipment and closed all valves, disconnected the ultrasonic transducer and the high-pressure chamber, and extracted the zinc sheet.

7. Placed the second zinc plate into the chamber, set the corresponding working cycle of the ultrasonic transducer 400 times, and then repeated experiments B–F.

8. Placed the third zinc plate into the chamber, set the corresponding working cycle of the ultrasonic transducer 600 times, and then repeated experiments B–F.

9. The fourth zinc plate was inserted into the chamber without the ultrasonic transducer being turned on, left in the supercritical carbon dioxide for 8 h, and then taken out. Used a scanning electron microscope to examine the four zinc sheets’ surface morphology.

10. Introduced the first 15 ml of methyl methacrylate (MMA) into the hyperbaric chamber. Set the working time of the ultrasonic instrument for 2 h.

11. Powered up the instruments and opened all valves. Activated the ultrasonic generator, and directed the ultrasonic transducer to emit ultrasonic waves into the supercritical carbon dioxide. Then, powered down the instruments, closed each valve, and removed the sample.

12. Introduced the second and third 15 ml of MMA into the chamber in order, setting the working time of the ultrasonic instrument for 5 and 7 h, respectively, and then repeated experiments J and K.

13. Introduced the fourth 15 ml of MMA into the chamber without activating the ultrasonic transducer, allowed it to remain in the supercritical carbon dioxide for 8 h, and then removed it.

14. Removed 5–8 ml of residual liquid from the chamber, evaluated the MMA reactants, and characterized the polymer by gel permeation chromatography (GPC).

Scanning electron microscope images revealed corrosion on metal plates exposed to ultrasound in supercritical CO₂. The extent of corrosion on the metal plates was related to the duration of ultrasonic exposure; the longer the exposure time, the more pronounced the corrosion. This contrasts with the non-corroded metal plates that were not exposed to ultrasound. The corrosion pattern on the metal plates closely resembled that observed in Kuijpers’ experiment, as illustrated in Figure 3.

Figure 3

Corrosion experiment results of metallic sheets under varying ultrasonic action cycles. (a) Without ultrasonic action. (b) Two hundred cycles of ultrasonic action. (c) Four hundred cycles of ultrasonic action. (d) Six hundred cycles of ultrasonic action.

Placed 15 ml of methacrylate into the high-pressure chamber. In comparative experiments, the ultrasonic radiation durations were 2, 5, and 7 h. Removed 5–8 ml of residual liquid from the high-pressure chamber and characterized the polymer by GPC.

Figure 4 shows the GPC spectrum; it has a peak with a normal distribution and a narrow range. Hence, there might be large molecular weight substances produced in the reaction. With the extension of ultrasonic irradiation time, the peak position of GPC was advanced, and the molecular weight increased continuously. The average of the large molecular weight substance generated is 60,000–90,000 g/mol. In the contrast experiment, we placed pure MMA in the cavitation chamber; we did not detect the macromolecular polymer formation under the same experimental pressure and temperature conditions without an ultrasonic wave.

Figure 4

Comparison of molecular weight distribution of polymers.

To keep the carbon dioxide in a supercritical state during the experiment, we kept the temperature and pressure above the critical point. There is only one phase of supercritical carbon dioxide; as Kuijpers et al. [6] suggest, there is no CO₂ vapor in the supercritical CO₂ bubbles and, hence, no constant high vapor pressure. Additionally, ultrasonic waves with sound pressure amplitudes far below the critical pressure cannot induce cavitation in supercritical CO₂. Nevertheless, similar to the metal corrosion and polymer formation experiments in Kuijpers et al. [6], we observed analogous effects, which were clearly not caused by cavitation. We conclude that cavitation is not responsible for the corrosion of metal sheets and the generation of polymers.

4 Sound field simulation

We simulated the acoustic wave equation using COMSOL Multiphysics 5.2a software. The simulated high-pressure chamber corresponded well with the actual chamber, as shown in Figure 5. We considered the walls of the hyperbaric chamber to be rigid and did not account for the acoustic absorption by the supercritical CO₂. Due to the characteristic impedance of stainless steel, ρc (the product of density and speed of sound), being approximately 80 times greater than that of supercritical carbon dioxide, the assumption of rigid boundaries for the inner surfaces of the hyperbaric chamber is reasonable. The simplification did not affect the calculation results.

Figure 5

Simulation model of the hyperbaric chamber.

Due to the good symmetry of the hyperbaric chamber structure and the sound pressures within the entire chamber, ZOX and ZOY planes were primarily analyzed. The sound pressure amplitude on the central axis of the ultrasonic transducer was also evaluated. At the end face of the 180 mm amplitude lever on the Z-axis coordinate, a pressure of 1.22 × 10⁵ Pa was applied. Effect diagrams are shown in Figure 6.

Figure 6

(a) Sound pressure levels inside the hyperbaric chamber. (b) ZOX cross-sectional sound pressure. (c) ZOY cross-sectional sound pressure. (d) Sound pressure on the cross-section of the hyperbaric chamber at Z = 185 mm.

In Figure 6(a), the sound pressure level inside the hyperbaric chamber exhibits strong symmetry, consistent with expectations. The maximum sound pressure level is 217.92 dB, corresponding to a sound pressure of 2.22 × 10⁶ Pa, but still less than the supercritical carbon dioxide cavitation threshold of 10⁷ Pa. The color at the end face of the amplitude-changing rod of the ultrasonic transducer is the darkest, indicating that the acoustic energy is primarily concentrated near the end face.

From Figure 6(b) and (c), it is observed that the amplitude of sound pressure exhibits periodic changes, due to the standing waves formed by the superposition of 20 kHz ultrasound with reflected waves. The extreme sound pressure values at the end surface and the middle of the transducer are relatively higher. The maximum sound pressure values of the ZOY and ZOX sections are 1.36 × 10⁶ and 2.22 × 10⁶ Pa, respectively, all within the range of 10⁶ Pa. The amplitude of sound pressure in the 180–230 mm range is decreasing, indicating thermal viscous absorption during propagation in supercritical CO₂.

In Figure 6(d), further analysis of the cross-sectional sound pressure distribution in the high-pressure chamber at Z = 185 mm reveals that the end face has seven maximum values, and four of these values of sound pressure amplitude reach the 10⁶ Pa level, with a maximum of 1.51 × 10⁶ Pa.

According to the actual sound power of the transducer, the effective value of the sound pressure radiated by the transducer into supercritical CO₂ is 1.22 × 10⁵ Pa. Figure 7 shows the simulated distribution of sound pressure amplitude along the axis of the ultrasonic transducer. The maximum at 191.2 mm is approximately 3.6 × 10⁵ Pa, which is still far below the cavitation threshold for supercritical CO₂.

Figure 7

Axial pressure amplitude distribution.

Figure 8 shows that the maximum velocity of carbon dioxide particle vibration on the axis of the ultrasonic transducer is 2.95 m/s, the average speed is 1.79 m/s, and the root-mean-square speed is 1.99 m/s, which is one order of magnitude higher than the vibration velocity of particles in water under similar conditions [12]. Figure 9 shows that the axis acceleration is in the order of 10⁵m/s². The particle vibration velocity and acceleration exhibit a periodic distribution. The mechanical effect in supercritical CO₂ is far stronger than that in water under similar conditions due to the minimal viscosity coefficient and surface tension of supercritical CO₂, which is why ultrasound enhances metal corrosion and polymer formation in supercritical CO₂. High-pressure liquid carbon dioxide has physical characteristics that are nearly supercritical [10]. The outcomes of the simulation also match the properties of supercritical carbon dioxide. This makes it very easy to explain the metal corrosion and polymer production processes in hyperbaric carbon dioxide liquids.

Figure 8

Velocity of carbon dioxide particle vibration.

Figure 9

Acceleration of carbon dioxide particle vibration.

5 Conclusion

The acoustic process is typically considered an isentropic process, where the vapor pressure inside a bubble changes as the bubble grows or collapses. Specifically, the vapor pressure inside the bubble decreases as its size increases and increases as the bubble shrinks. In this study, the contribution of the vapor pressure to the overall internal pressure of the bubble is considered equivalent to that of other gases within it.

This dynamic behavior of vapor pressure has direct implications for cavitation in both liquid and supercritical fluids. Cavitation occurs when the amplitude of ultrasonic pressure reaches or exceeds the Blake pressure threshold, which is a key factor for bubble formation. However, the challenge in supercritical fluid technology arises from the difficulty of obtaining the high-power ultrasound required to induce cavitation effectively. The high-pressure conditions that define supercritical fluids further complicate the process, as they impose physical limitations.

Despite these challenges, the unique properties of liquid CO₂ and supercritical carbon dioxide, such as minimal surface tension and low viscosity, allow for the mechanical action of ultrasound to produce effects similar to cavitation. In this context, while true cavitation may not occur under the given conditions, ultrasound can still induce a cavitation-like effect, providing some potential for applications in supercritical fluid technology. Therefore, while cavitation in supercritical fluids is difficult to achieve, understanding the relationships between vapor pressure, bubble dynamics, and the properties of supercritical CO₂ is crucial for advancing the use of ultrasound in these systems.