Numerical Studies of Particle-Gas Two-Phase Flowing through Microshock Tubes

Abstract

Microshock tubes are always used to induce shock waves and supersonic flows in aerospace and medical engineering fields. A needle-free drug delivery device including a microshock tube and an expanded nozzle is used for delivering solid drug powders through the skin surface without any injectors or pain. Therefore, to improve the performance of needle-free drug delivery devices, it is significantly important to investigate shock waves and particle-gas flows induced by microshock tubes. Even though shock waves and multiphase flows discharged from microshock tubes have been studied for several decades, the characteristics of unsteady particle-gas flows are not well known to date. In the present studies, three microshock tube models were used for numerical simulations. One microshock tube model with closed end was used to observe the reflected shock wave and flow characteristics behind it. The other two models are designed with a supersonic nozzle and a sonic nozzle at the exit of the driven section, respectively, to investigate particle-gas flows induced by different nozzles. Discrete phase method (DPM) was used to simulate unsteady particle-gas flows and the discrete random walk model was chosen to record the unsteady particle tracking. Numerical results were obtained for comparison with those from experimental pressure measurement and particle visualization. Shock wave propagation was observed to agree well with experimental results from numerical simulations. Particles were accelerated at the exit of microshock tube due to the reservoir pressure induced by reflected shock wave. Both sonic and supersonic nozzles were underexpanded at the end of microshock tubes. Particle velocity was calculated to be smaller than gas velocity, which results from larger drag of injected particles.

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1. Introduction

During the past several decades, microshock tubes as devices to induce shock waves and supersonic flows have been widely used in mechanical, aerospace, and medical engineering fields, such as microcombustions, explosion, and needle-free drug delivery devices. Normally, a microshock tube consists of a driver section in high pressure and a driven section in low pressure which are separated by a thin diaphragm. Due to the pressure difference between two sections, shock waves are induced when the diaphragm is ruptured instantaneously [1]. If the diaphragm pressure ratio is extremely high, the diaphragm is ruptured naturally. Otherwise, it should be punctured by using a needle manually. The incident shock wave induced by a microshock tube is always a normal shock wave, while it becomes oblique shock wave after it is reflected by the end wall in the driven section in a closed-ended microshock tube. Supersonic flows are induced and accelerated behind the incident shock wave, which is always applied in experimental tests such as microcombustion and explosion. High pressure and temperature flows are generated downstream of the reflected shock wave.

With the development of microshock tubes and the innovation of medical technique, needle-free drug delivery devices are designed to inject drug powders into human body in a contactless way. The main components of the needle-free drug delivery device are a microshock tube and an expanded nozzle [2]. Gas flows with drug powders are induced and accelerated by a shock wave generated in a microshock tube and accelerated in the expanded nozzle again. Drug powders are injected into the skin tissue by obtaining enough momentum. In order to ensure the delivery without any hurts, the momentum of drug powders should be strictly controlled. Even though needle-free drug delivery devices have been studied for several decades, detailed characteristics of particle-gas two-phase flows inside the devices are not well known to date. Experimental studies are sparsely conducted on particle velocity and momentum as well. However, investigations on particle-gas flows in needle-free drug delivery devices are extremely important in improving their performance in medical engineering fields.

Flows with solid particles behave differently from single gas flows. Due to the inertia and resistance of solid particles, particles are always not able to track gas flows properly. Under the slow relaxation time of solid particles, particles follow supersonic gas flows more improperly. The above effects make the momentum of solid particles uncontrollable, which prevents the development of needle-free drug delivery devices. Due to viscous effects and the existence of boundary layers, particle dynamics are more difficultly predicted in microshock tubes. Even though researchers paid great attention to investigating shock waves and particle-gas flows in microshock tubes, detailed characteristics of shock wave and particle dynamics were not well known.

Brouillette [1] derived a new model to investigate scale effects on shock wave generation and propagation in a microshock tube and performed a comparison with experimental results. A control volume located between shock wave and contact surface was considered. Particle velocity was lower than that in microshock tubes in larger scales. The predicted numerical results agreed with experimental results well with respect to the Mach numbers of shock waves. Liu et al. [2] performed experimental and numerical investigations on shock wave propagation and particle velocity in a contoured shock tube. The momentum of drug powders was controlled by flow characteristics inside the contoured shock tube. Austin et al. [3] studied shock wave propagation and attenuation through a microscale channel of circular cross section. Shock wave velocity and pressure histories from experimental studies were compared with experimental and theoretical results. More shock wave attenuation was observed in the channels with smaller scales and lower pressures. Felling et al. [4] used pressure measurements to record pressure histories compared with existing experimental pressure results.

Effects of the pressure ratio and diaphragm location were investigated on shock waves discharged from an open-ended shock tube. Mach-disk shock, barrel shock, and reflected shock waves were observed and discussed in detail by Haselbacher et al. [5]. Labastida et al. [6] used simultaneous lateral and end-wall high-speed visualization method to observe shock wave propagation in a circular shock tube. Shock wave structure and motion were obviously visualized. A modified dissipation model was derived to discuss effects of heat transfer on shock velocity gradient by Yang et al. [7]. By considering the effects of Reynolds number and temperature difference between particle and gas phases, Henderson [8] derived mathematic models to calculate particle drag force under subsonic and supersonics gas-particle flows, respectively. Lin et al. [9–13] used CFD-EDM coupled method to calculate particle-gas flows and carried out visualization tests to validate CFD results. Particle trajectory and velocity from numerical simulations agreed with experimental results well. Sun et al. [14–19] made summaries on numerical models of calculating multiphase flows and derived some new mathematic models to calculate multiphase flows.

Numerical simulations were carried out to investigate propagation and attenuation of shock wave in micro shock tubes by using a one-dimensional approach by Ngomo et al. [20]. Attenuation of shock wave was quantitatively calculated and propagation of shock wave was obviously captured at different tube diameters. Li et al. [21] conducted time-resolved shadowgraph and transient pressure measurements to investigate shock wave structure and Mach number in a shock tube. Shock flows were discussed at the tube exit in detail. Digital particle image velocimetry was used to capture shock wave structure at the nozzle exit impinging to a plate by Henderson et al. [22]. Nozzle pressure ratio and exit diameter were investigated to have great effects on shock flows at the nozzle exit. Kendall et al. [23–25] performed experimental studies to investigate particle dynamic through contoured shock tubes by particle image velocimetry (PIV) and Schlieren visualization. The velocity and propagation of shock wave were obtained and discussed in detail.

Interaction between shock wave and solid particle was investigated by Xiong et al. [26]. Shock wave was observed to affect particle cloud clusters and small particle influenced shock wave structure as well. In addition, a controlling method for cloud cluster expansion rate was proposed. Lupoi et al. [27] conducted numerical and experimental studies to investigate particle behavior in a supersonic nozzle for cold spray system. The collisional model was derived to calculate particle dispersion in supersonic flows, which agreed with experimental results well. Wang et al. [28] derived a unified gas-kinetic scheme to calculate particle-gas flows. Collision of two particle-gas phases was discussed in detail as well as shock-driven multiphase instability. Experimental investigations were carried out to study shock wave dynamic and particle movement in microshock tubes from previous studies [29, 30]. Shock wave and particle motion were visualized and analyzed in detail.

In this article, numerical studies on shock wave propagation and particle motion were carried out in different microshock tubes. By considering solid particles behaving differently in supersonic and subsonic flows, a suitable drag coefficient model was used in present numerical simulations. Shock flows and particle motion induced by sonic and supersonic nozzles at the tube exit were obtained and discussed in detail, respectively. The comparisons were made between CFD and experimental results on pressure histories inside the microshock tube and at different nozzle exits. Shock wave propagation was visualized and compared with results got from particle tracking velocimetry. Particle velocities induced by sonic and supersonic nozzles were obtained and discussed in detail.

2. Numerical Methods

2.1. Computational Domain

A microshock tube was designed to investigate characteristics of unsteady particle-gas flows as shown in Figure 1. The shock tube was designed with the driver section in a diameter of 20 mm and a length of 41 mm, and the diameter and length of the driven section are 7.5 mm and 66 mm, respectively. All sections have circular cross sections. Sonic and supersonic nozzles were, respectively, installed at the end of the driven section in the microshock tube. The detailed schematic and size of the sonic and supersonic nozzles are shown in Figures 2(a) and 2(b). Pressure measurement and Schlieren visualization were conducted.

[figure omitted; refer to PDF]

After the diaphragm was ruptured, the incident normal shock wave was induced and moved towards the driven section. As the shock wave reached the position where pressure transducers located, the pressure increased steeply as shown in Figure 6. The instants where the shock wave met the two pressure transducers were almost similar in both experimental and numerical studies. The strength of the shock wave was observed to be stronger in CFD study compared to that in experimental test, which mainly resulted from the difference in rupture process. In experimental tests, the diaphragm was ruptured by the manual method and the rupture time was not instantaneous. However, the instantaneous rupture was simulated as the boundary condition of the diaphragm was changed from the wall to the interior. In addition, the 2D half computational domain was used in CFD study, but actually the shock wave moving in the microshock tube was not symmetrical with respect to the center line in the experimental test. The adiabatic walls were used for the numerical simulations, so the heat transfer between the shock heated air and tube walls was ignored. However, heat transfer existed in the experimental study.

As the shock wave met the end wall of the driven section, it was reflected and moved towards the direction opposite to the incident shock wave. The deviation between experimental and CFD results gradually became larger. This was mainly due to the fact that the shock wave experienced much more decay after it was reflected in the experimental test. As the shock wave collided to the solid wall, much more shock wave attenuation occurred.

3.2. Shock Wave Propagation

In previous experimental studies, Schlieren visualization was used to observe the shock wave propagation in microshock tube as shown in Figure 7. The test section was clearly observed before the normal shock wave was induced. After the diaphragm was ruptured, a normal shock wave was induced and propagated to the end wall as shown in Figures 7(b) and 7(c). When the normal shock wave met the end wall of the driven section, it was reflected and the reflected shock wave was still normal shock wave as shown in Figure 7(d). The strength of the primary shock wave was stronger than that of the reflected shock wave. This resulted from the fact that the reflected shock wave moved upstream. As reflected shock wave met the contact surface, the structure of the reflected shock wave changed as shown in Figure 7(e). Reflected shock wave was no longer a normal shock wave but the direction did not change. The dark spots and areas shown in the figures were particles seeded in the microshock tube.

[figures omitted; refer to PDF]

Numerical simulations were carried out to investigate the shock wave propagation and compared with experimental results. Temperature contours at different time are shown in Figure 8. The shock wave and the contact surface are clearly observed. Shock wave propagations from experimental and CFD studies agreed well and shock wave structures were also similar. After the shock wave was reflected, a high-temperature region occurred behind the reflected shock wave. This showed a good agreement with the previous observation that the high pressure developing behind the reflected shock wave was regarded as reservoir pressure for initializing nozzle flows. As the reflected shock wave met the contact surface, its structure changed and the shock wave was not normal any more.

[figure omitted; refer to PDF]

Particle motion and velocity contours of gas flows induced by the sonic nozzle are, respectively, shown in Figures 12(a) and 12(b). Due to the underexpanded nozzle flows, particle velocity gradually increased downstream of the sonic nozzle exit. The expansion waves were clearly observed at the exit as shown in Figure 12(b). Compared to the velocity of gas flows, the particle velocity was much lower, which resulted from the large diameter of particles. Large particles had large inertia and resistance, making particles follow gas flows improperly. As the reservoir pressure decreased in the driven section, the expansion waves became weak and particle velocity also decreased. The similar characteristics of particles and gas flows generated by the supersonic nozzle were observed compared to those induced by the sonic nozzle as shown in Figures 13(a) and 13(b). Higher particle velocity was observed at the exit of sonic nozzle compared to that at the exit of supersonic nozzle. This resulted from the fact that the flows generated by the sonic nozzle were more highly underexpanded. This is clearly shown as the flow velocity induced by the sonic nozzle was much higher at the nozzle exit.

[figures omitted; refer to PDF]

[figures omitted; refer to PDF]

In both CFD and experimental studies, particle velocity was obtained in four test sections downstream of nozzle exit as shown in Figure 14. The test section was defined in area of 1 mm × 3.36 mm with the interval of 1 mm from X = 0 mm to X = 4 mm. Similar test sections were used in the CFD simulations. Particle velocity was calculated for the numerical simulation and compared to the results obtained in the experimental studies as shown in Figure 15. As particles moved outside of the nozzle exit, particle velocity gradually increased in both CFD and experimental studies. This is due to the fact that the sonic nozzle flows were underexpanded at the nozzle exit. Particle velocity obtained from the CFD studies was higher compared to that in experimental calculations. The main reason is that the shock wave strength was observed to be stronger in the numerical simulation as mentioned previously, so the reservoir pressure generating in the driven section was also higher in the CFD study.

[figure omitted; refer to PDF][figure omitted; refer to PDF]

4. Conclusions

Numerical simulations were carried out to study shock wave propagation and particle-gas flows induced by sonic and supersonic nozzles and the comparison between numerical and experimental results was made as well. Normal shock wave and reflected shock wave were clearly observed in CFD studies and agreed with experimental results well. Pressure histories indicated that less shock wave attenuation happened in the CFD simulations compared to that in the experimental studies, which resulted from the different methods used for rupturing diaphragms. In CFD studies, the diaphragm was ruptured instantaneously instead of manual rupturing method by using a needle. In addition, much more viscous effects and friction between the shock wave front and tube walls took place in the experimental tests. After the normal shock wave met the end wall, high pressure and temperature flows were induced behind the reflected shock wave. This made the high pressure the reservoir pressure inducing nozzle flows. Particles were accelerated in the microshock tube due to the development of boundary layers behind the shock wave. Both sonic and supersonic nozzle flows were choked and shock wave structure was clearly observed. Particles were observed to be accelerated behind the exit of both sonic and supersonic nozzles due to both nozzle flows being underexpanded. Particle velocity showed large deviation from the velocity of gas flows, which resulted from large particle diameter used in the present studies. In the future studies, the small particle diameter will be considered and investigated in the microshock tube.

Acknowledgments

This research was funded by the National Natural Science Foundation of China (Grants nos. 51906222 and 51706206), Key Research and Development Program of Zhejiang Province (Grant no. 2019C03117), and Zhejiang Sci-Tech University (Grant no. 18022135-Y). The authors acknowledge the financial support.

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