Experimental facility

The experimental facility used in this study is illustrated in Fig. 1, was designed with reference to previous experimental setups [39-41] and adapted to meet the specific objectives and technical requirements of the present research. The system consists of the following major components: a vacuum graphite electrode furnace, a melt-release mechanism, a transparent water tank, temperature and pressure acquisition instruments, a high-speed camera, and an external disturbance device.

Fig. 1Full-screen View

(Color online) The overall experimental apparatus

The vacuum graphite electrode furnace features a double-layer annular structure heated by five graphite electrodes [42], with a maximum design temperature of 2200 °C. A graphite crucible placed inside the furnace is used to melt the metal, and the temperature at the center of the heating zone is monitored using an infrared pyrometer. The exterior of the furnace is lined with graphite wool insulation to minimize heat loss. In addition, a vacuum system was employed to prevent the oxidation of the graphite electrodes and molten material.

The melt-release device consists of a graphite support rod and an electric lift, allowing for precise and automated control of the melt release rate. During the metal melting process, the graphite rod is lowered via a control cabinet so that its conical tip tightly seals against the corresponding conical outlet at the bottom of the graphite crucible, preventing leakage. Once the metal reaches the target temperature, the graphite rod is retracted upward by the control system, enabling the melt to be released and fall freely under the effect of gravity.

The water tank is a rectangular stainless-steel frame fitted with transparent viewing panels. To facilitate observation and data recording, the front and rear sides are equipped with pressure-resistant acrylic windows. The left-side wall features two rows of ports for mounting pressure transmitters and thermocouples. Figure 2 shows the vertical arrangement of the temperature and pressure measurement points. The internal dimensions of the tank are 250 mm (length) × 250 mm (width) × 1800 mm (height). The specifications of the measurement equipment used in this study are listed in Table 2. The bottom of the tank is equipped with a fragment collector and an electric heating unit, which are used for retrieving debris to analyze particle size distribution and heating the coolant water, respectively.

Fig. 2Full-screen View

Location of temperature and pressure measuring points

The temperature and pressure acquisition system comprises thermocouples, pressure transmitters, and a corresponding data acquisition platform, enabling the comprehensive recording of temperature and pressure variations during the melt settling in water. Additionally, a high-speed camera capable of frame rates of up to 200,000 fps, along with its supporting software, meets the requirements for the dynamic recording and image processing of the melt descent process.

The external disturbance device is shown in Fig. 3, comprises a gas cylinder (1), a pressure reducing valve (2), a trigger switch (3), a pressure relief switch (4), piston components (5), pressure transmitters (6), and other associated parts. Its operating principle involves regulating high-pressure gas from the cylinder via a pressure reducing valve. The regulated gas was then delivered through the trigger switch to the piston assembly located at the bottom of the water tank, generating a controlled external pressure pulse. This system enables the precise generation of millisecond-scale pressure disturbances of up to 3 MPa within a water environment, offering a safer and more stable alternative to traditional methods of introducing explosive disturbances [43, 44].

Fig. 3Full-screen View

Schematic diagram of external disturbance device. Gas cylinder (1), pressure reducing valve (2), trigger switch (3), pressure relief switch (4), piston components (5), pressure transmitters (6)

Case details and data processing

This study focused on the influence of external disturbances on the evolution characteristics of droplets during the triggering stage. A comparative experimental analysis of the transient velocity and expansion rate of droplets with and without external disturbance provides supporting evidence for understanding the droplet triggering mechanism. The materials and experimental conditions used are listed in Tables 3 and 4, respectively. The experimental procedure was as follows: Prior to each test, a predetermined amount of cooling water was added to the tank to maintain a consistent droplet falling height across all cases. The initial droplet temperature, water temperature, and external disturbance pressure were then set according to the requirements of each test condition. Once the target temperatures were reached, the melt-release device was activated to discharge the droplet. Data recording was simultaneously initiated using the temperature and pressure acquisition system and a high-speed camera.

Controlling uncertainties in FCI experiments remains challenging because of the complex coupled hydrodynamic and thermal effects occurring within extremely short timeframes. Therefore, strict pre-test protocols are essential. During the pre-test preparation, an infrared pyrometer was used to accurately monitor the melt temperature in the furnace. The molten droplet was released only after the temperature remained stable for at least 15 min, limiting the maximum relative error in the droplet temperature to within 5%. To minimize impurities, all tests employed high-purity materials: Sn-99.99%, Pb-99.999%, and a Pb-Sn alloy with a composition of 50:50 wt%. These high-purity materials ensure consistent thermal properties of the melt. Although the droplet size (particularly droplet shape) cannot be precisely controlled and may influence local heat transfer and related processes, certain measures were taken to improve repeatability. As with other internationally recognized FCI experiments (e.g., MISTEE by KTH, KROTOS by CEA, TROI by KAERI, and SIGMA by UCSB), perfect reproducibility is not always achievable. Nevertheless, to promote droplet uniformity, an adjustment rod was used to regulate both the release frequency and droplet diameter. During preliminary tests, the rod was rotated until a stable stream of freely falling droplets was achieved—set here at 100 drops per minute. The rod position corresponding to this stable condition was marked and used in all subsequent tests to ensure essentially consistent droplet size.

Figure 4(a) shows two consecutive frames captured during droplet entry into water, with a time interval of 0.1 ms between frames.

Fig. 4Full-screen View

Processing of the images. (a) Instantaneous droplet motion; (b) Instantaneous interaction between a droplet and water

The settling velocity was determined from close-range experimental observations. The transient velocity of the droplet was estimated by calculating the average velocity between two consecutive images, which is defined as the ratio of the relative displacement to the time interval. The leading-edge position of each droplet was tracked using a high-speed video, and the displacement (denoted as ) over a 0.1 ms interval (corresponding to the system response time) was recorded. Therefore, the corresponding velocity is given by Eq. (1).(1)Figure 4(b) shows the droplet shape at a specific instant during water entry. Although the shape change of the droplet within the mixing region could not be precisely quantified from the image, the boundary of the interaction zone was clearly distinguishable. Therefore, the instantaneous droplet diameter was approximated based on an equivalent circular area, and the droplet expansion rate was taken as the average expansion rate between two consecutive images. The equivalent diameter of the molten droplet was estimated using the following Eq. (2), where Dx and Dy represent the horizontal and vertical diameters of the droplet, respectively, and D_d denotes the equivalent diameter [1] used to characterize the size of the irregularly shaped droplet.(2)The droplet expansion rate was determined by analyzing high-speed video footage of the vapor film-water interface evolution. Using the equivalent diameter method, the instantaneous droplet diameter was measured over a 0.1 ms interval (system response time). Therefore, the corresponding expansion rate is given by Eq. (3).(3)

Experimental measurements are inevitably subject to discrepancies from true values owing to factors such as instrument accuracy, measurement techniques, and test conditions. In this study, measurement errors were categorized as direct or indirect. For parameters with direct measurement errors, the true value is expressed as Eq. (4):(4)where X_real is the true value of the parameter, X is the measured value of the parameter, φ is the absolute error of this parameter.

The physical variables directly measured are X₁, X₂, X₃, …, Xn, because X₁, X₂, X₃, …, Xn are independent variables. The indirect measurement of the physical variable is G, which is given by the following relationship:(5)For parameters with indirect measurement errors, based on the error transfer theory, the error of the physical variable by indirect measurement can be calculated using the following Eq. (6):(6)where φG is the error of the indirectly measured physical variable, is the error of the direct measurement value Xi, is the error transfer coefficient.

Therefore, the relative error of the indirectly measured physical variable G during the experiment is given by Eq. (4):(7)Based on the error calculation methodology described above, the maximum relative errors for the key experimental parameters, including the droplet mass, droplet temperature, water temperature, pressure variations during the interaction, external disturbance pressure, droplet settling velocity, and droplet expansion rate, are presented in Table 5.

Typical phenomena

The A1 condition, characterized by a lead-tin alloy temperature of 300 °C, water temperature of 20 °C, and absence of external disturbance pressure, is illustrated in Fig. 5(a). The penetration event was of short duration. Prior to water contact, the droplet fell in a regular shape. The immense temperature difference upon impact induces violent heat transfer, resulting in the formation of a vapor film that encapsulates the droplet. Subsequent expansion of the vapor was observed in both the horizontal and vertical directions, yielding a vapor pocket. A pronounced separation of this pocket from the droplet occured in the vertical direction, in contrast to the more limited horizontal development. Air entrainment, which is attributed to the droplet-water velocity difference, was also observed. As presented in Fig. 5(b), the vapor pocket subsequently undergoes constriction at its center, with the majority of the entrained air migrating backwards under aqueous surface tension until atmospheric release.

Fig. 5Full-screen View

Image processing. (a) Diagram of droplet motion; (b) Air entrainment phenomenon

Figure 6(a) shows the transient velocity variation of a droplet. In the plot, line a-b corresponds to the droplet velocity change in air. The water surface is indicated by a solid black line, and the moment at which the droplet crosses this line is defined as t = 0 s for simplicity. Line b-c reflects the velocity variation during the initial penetration into the water. Upon contacting the water, the droplet experiences deceleration owing to both hydrodynamic drag and an upward force caused by localized steam generation at the leading edge, resulting in an increase in pressure. As steam generation increases over time, the pressure at the front of the droplet increases, forcing the vapor to move rearward and form an airbag along with the entrained air. As demonstrated in our previous study [45], this vapor envelope reduces drag, resulting in a velocity recovery, as shown by line c-d. When the vapor shifted backward, the droplet’s leading edge reestablished contact with the liquid water. Although some steam continues to form, its quantity is reduced, and the stability is diminished. Consequently, the hydrodynamic resistance increased again, and the droplet velocity decreased, as shown in line d-e. Finally, as the droplet detaches from the airbag and undergoes further deformation, its velocity continues to decline until it stabilizes, as represented by the line e-f.

Fig. 6Full-screen View

Data processing

Figure 6(b) shows the expansion rate history of the same droplet. Line a-b corresponds to the period from the initial water contact to full submersion. The vapor generated by intensive droplet-water heat transfer causes droplet expansion. However, because the immersion process initially occurs at a low velocity and the wetted area of the droplet increases only gradually, a stable interaction state is not yet reached. Thus, the expansion rate increased only slowly during this stage. Line b-c corresponds to a faster immersion phase, during which the airbag at the droplet leading edge began to collapse. The droplet may also undergo expansion and fragmentation upon contact with cold water, leading to a sharp rise in the expansion rate. However, over time, the droplet cooled and the airbag detached, causing the expansion rate in line c-d to gradually decline toward zero.

No external disturbance pressure

Figure 7(a) shows the transient velocity of a single droplet at different temperatures upon contact with water. As the droplet temperature increased, the duration of acceleration and peak velocity were extended and elevated, respectively, thereby raising the overall velocity level. At a constant water temperature, a hotter droplet induces faster and more extensive water evaporation, producing more vapor. This influences the process in two ways. First, enhanced evaporation accelerates cavity formation, prolonging the droplet’s fall within the cavity. As previously analyzed, the cavity reduces the direct contact between the droplet and water, leading to lower resistance and thus a higher droplet velocity. Second, more vapor accumulates in the droplet’s wake region [46], providing additional downward propulsion. Figure 7(b) presents the droplet expansion rate across different droplet temperatures. As the temperature rises, the growth duration of the expansion rate after water entry was prolonged, whereas the peak expansion rate initially increased and then decreased. At 300 °C, the droplet was rapidly cooled upon contact, resulting in only a minor increase in the expansion rate. At 400 °C, 500 °C, and 600 °C, the droplet enters an unstable film boiling regime, where intermittent vapor film collapse causes repeated water contact and fine-scale fragmentation, leading to large fluctuations in the expansion rate. At 700 °C and 800 °C, stable film boiling occured, enveloping the droplet in a persistent vapor film that protected it during descent until cooling. Here, the expansion rate increased smoothly and exhibited the longest duration among all cases. Figure 7(c) illustrates transient thermal-hydraulic behavior and resulting debris fragments at droplet temperatures of 300 °C, 500 °C, and 700 °C. At 300 °C, the droplet’s leading edge develops a conical protrusion, and the resulting debris fragment is smooth and elliptical. At 500 °C, the leading edge becomes flared and spiky, and the debris body appears rough and curled. At 700 °C, the leading edge reverts to a smooth elliptical shape, while the debris fragments form slender, flake-like structures with spiky surfaces. In addition, the peak pressures measured by a high-frequency dynamic pressure transducer further supported these observations, showing a trend of initial increase followed by a decrease with rising droplet temperature. The maximum pressure, 0.022 MPa, occurs at 500 °C, as summarized in Table 6. Additionally, by comparing the timing of the transient velocity and expansion rate, it is noteworthy that their peak values occur simultaneously.

Fig. 7Full-screen View

(Color online) Different droplet temperatures. (a) Transient velocity; (b) Expansion rate; (c) Experimental phenomena and products

Figure 8(a) shows the transient velocity of a single droplet at different coolant temperatures. As the coolant temperature increased, the acceleration period of the droplet lengthened, and its peak velocity also rised. This behavior is analogous to the effect of the droplet temperature described earlier. When the droplet temperature was fixed, a higher coolant temperature allowed the water to reach boiling more readily, leading to a prolonged evaporation time, higher evaporation rate, and greater total vapor volume. This promoted the formation of a protective cavity that propelled the droplet downward. Figure 8(b) presents the droplet expansion rate under different coolant temperatures. Although the duration of the expansion rate growth lengthens with increasing coolant temperature, the peak expansion rate decreases. At 20 °C, the droplet experiences unstable film boiling, where the vapor film collapses intermittently, causing repeated contact with water and fine-fragmentation. As the water temperature rises, the heat transfer shifts toward stable film boiling, forming a persistent cavity that shields the droplet during its descent until cooling. Consequently, the expansion rate increased gradually, and the growth period became the longest of all treatments. Figure 8(c) shows representative phenomena and resulting debris at coolant temperatures of 20 °C, 50 °C, and 80 °C. With increasing coolant temperature, the droplet’s leading edge became more rounded, and the resulting debris fragments became smoother. This occurs because more vapor is generated over the same period as the coolant temperature rises, creating a more stable cavity that envelops the droplet, reduces direct water contact, and limits deformation. Peak pressure measurements obtained using a dynamic high-frequency pressure transducer further supported these observations. As listed in Table 7, the peak pressure decreased with increasing coolant temperature. Finally, consistent with earlier findings, the moment of the peak transient velocity coincided with that of the expansion rate.

Fig. 8Full-screen View

(Color online) Different water temperatures. (a) Transient velocity; (b) Expansion rate; (c) Iconic phenomena and product fragments

Figure 9(a) shows the transient velocity of the droplets for different materials. The lead-tin alloy droplet reached the highest peak velocity upon entering the water. This occurs because of the lower melting point of the alloy, which allows it to store more internal energy at the same temperature. Consequently, water evaporated more readily, and a larger volume of vapor was generated, increasing the peak transient velocity of the droplet. Figure 9(b) presents the expansion rate for each material. The lead-tin alloy and tin droplets exhibited significant fluctuations in the expansion rate, whereas the lead droplet remained largely stable. This further suggests that under identical conditions, lead is less susceptible to vapor explosions. Figure 9(c) shows the typical experimental phenomena and resulting debris for the lead-tin alloy, tin, and lead. Owing to its higher melting point, lead has a lower degree of superheating at the same temperature, making it less prone to fine-fragmentation. Peak pressure measurements support this observation, with values decreasing in the order of lead-tin alloy >tin >lead, as summarized in Table 8. Consistent with earlier results, the moment of the peak transient velocity coincided with that of the peak expansion rate.

Fig. 9Full-screen View

(Color online) Different droplet materials. (a) Transient velocity; (b) Expansion rate; (c) Experimental phenomena and product debris

External disturbance pressure

The preceding analysis examined the laws governing the transient velocity and expansion rate of the droplets under self-triggering conditions. However, in severe accidents, various disturbances in the external environment can affect the triggering process of droplets. This section introduces external disturbances by setting a disturbance plate in motion at the bottom of the water tank, focusing on the impact of external disturbance pressure on the process, with specific conditions listed in Table 4. Figure 10 illustrates the motion of one single droplet under Test D6, where the lead-tin alloy is initialized at 300 °C, and the coolant is maintained at 20 °C, with the external disturbance pressure of 3 MPa. Before the lead-tin alloy droplet contacts the water surface, the water has already been initialized from the external disturbance pressure, as reflected by the bubbles moving upwards in Fig. 10. Compared with Test A1, after the lead-tin alloy droplet entered the water surface, its penetration depth at the same moment was significantly reduced owing to the action of the external disturbance, and the gas-liquid interface changes became more chaotic upon detachment from the water surface.

Fig. 10Full-screen View

Droplet motion process under external disturbance pressure

Figure 11 shows the transient velocity of the droplet under different external disturbance pressures. With the increase of the external disturbance pressure, the duration of the velocity increase of the droplet after penetrating the water decreased, and the peak velocity of the droplet also decreased accordingly. This is because after the droplet penetrated the water pool, the drag force can rapidly boost the decrease in velocity. Meanwhile, with the enhancement of heat transfer at the interface, the hot droplet turns to be surrounded by a thick layer of steam and air mixture, thus promoting the drag reduction in the subsequent droplets. This competition leads to an increase in the velocity of the hot droplet until the peak velocity is reached. Moreover, when an external disturbance pressure is introduced, it may weaken the stability of the gas layer, thus increasing the drag force. Therefore, as the external disturbance pressure increased, the peak pressure owing to the drag reduction decreased. Figure 12 shows the transient velocity comparison of lead-tin alloy droplet with or without external disturbance pressure at different droplet temperatures and water temperatures. The results also show that the external disturbance pressure can reduce the peak velocity of the droplets. Moreover, under the external disturbance conditions, the initial temperature of the droplets may have less effect on the drag reduction of the droplets, and the external pressure dominates the settling process.

Fig. 11Full-screen View

(Color online) Droplet transient velocity under different external disturbance pressures. (a) Lead-tin alloy; (b) Tin; (c) Lead

Fig. 12Full-screen View

(Color online) Comparison of transient droplet velocity of lead-tin alloy with or without external disturbance pressure. (a) Different droplet temperatures; (b) Different water temperature

Figure 13 shows the droplet expansion rate under different external disturbance pressures. As the external disturbance pressure increased, the duration of the expansion rate after the droplet entered the water decreased; however, the peak expansion rate increased. Even for the case of the lead droplet, the expansion rate can reach up to 9.46 m/s, as shown in Fig. 13(c). This phenomenon occurs because as the droplet descends through the water, the initiation of the trigger at the interface leads to the fine-fragmentation process in some local regions. The resulting fine particles dispersed around the parent droplet, forming a gaseous film that contributed to the expansion of the hot droplets. When the external disturbance pressure propagates, it readily induces an increase in the instability of the droplet surface, thereby triggering further fine-fragmentation. Compared to the self-trigger scenario, this localized fine-fragmentation process was more pronounced. Consequently, as the external disturbance pressure increased, the droplet expansion rate correspondingly escalated. Moreover, the cavity is more likely to collapse under the action of external disturbance pressure, which accelerates the contact time between the droplet and water, thereby increasing the degree of fine-fragmentation of the droplet, as shown in Fig. 14. Figure 15 shows a comparison of the expansion rate of lead-tin alloy droplets with and without external disturbance pressure at different droplet temperatures and water temperatures. The results also show that the external disturbance pressure can reduce the rise time of the droplet expansion rate and increase the peak value of the expansion rate. In addition, with an increase in the initial temperature of the droplet or the coolant temperature, the external disturbance pressure can further promote the triggering process. The highest growth rate can be achieved by 108% when the coolant temperature is approximately 80°C. Therefore, attention should be paid, although it is very close to the saturation condition for real severe accidents, when an external trigger may occur. Figure 16 shows the mass ratio of product fragments with diameters of less than 0.5 mm under different working conditions. With an increase in the external disturbance pressure, the mass ratio of product fragments with diameters less than 0.5 mm also increased. Simultaneously, the peak pressure measured by the dynamic high-frequency pressure transmitter in the experiment also illustrated this phenomenon, as shown in Table 8. As the external disturbance pressure increased, the peak pressure values exhibited an ascending trend. Conversely, as the coolant temperature increased, the peak pressure values exhibited a descending trend. Furthermore, an increase in the droplet temperature was associated with a biphasic trend in the peak pressure values, which initially increased and subsequently decreased. Ultimately, a comparative analysis of the transient velocity and expansion rate moments revealed that the peak transient velocity moment was synchronized with the peak expansion rate moment.

Fig. 13Full-screen View

(Color online) The droplet expansion rate under different external disturbance pressures. (a) Lead-tin alloy; (b) Tin; (c) Lead

Fig. 14Full-screen View

The steam film at the leading edge of the droplet is collapsing under the action of external disturbance pressure

Fig. 15Full-screen View

(Color online) Comparison of droplet expansion rate of lead-tin alloy with and without external disturbance pressure. (a) Different droplet temperatures; (b) Different water temperature

Fig. 16Full-screen View

(Color online) The mass ratio of product fragments with a diameter of less than 0.5 mm under different working conditions. (a) Type of material; (b) Water and droplet temperature

Another important point to address is that the primary factor influencing the measured pressure variation is the distance between the measurement point and molten droplet. As pressure propagates from the local triggering region, it is attenuated by water resistance. The pressure decay can be estimated using the following Eq. (8), where P and P₀ refer to the triggering pressure at the measuring point and the surface of the molten droplet, respectively, r refers to the distance between the measuring point and the location of the molten droplet at the triggering time, b refers to the empirical coefficient, which is usually regarded as 1.03 in water. Using this equation, the corresponding peak pressure at the droplet surface can be effectively reconstructed. For instance, in Test A3 and Test D3, the measured triggering pressures were 0.022 MPa and 0.081 MPa, respectively. After applying the correction, the estimated pressures at the droplet surface were approximately 0.57 MPa and 2.1 MPa. These results demonstrate that the applied external disturbance significantly enhanced the trigger pressure compared to the spontaneous triggering conditions.(8)

Comparison with previous work and limitations of the present work

A recent study by KTH in Sweden [47] examined steam explosion characteristics involving multiple molten tin droplets falling through a coolant pool. This configuration is highly similar to that in the present study, in which streams of molten Sn, Pb, and Sn-Pb alloy droplets descend through a coolant pool. The authors reported distinct and complex steam explosion behaviors not observed in earlier single-droplet experiments, an important finding given the limited available data on multi-droplet configurations and the clear need for deeper investigation into such conditions.

In their tests, a spontaneous steam explosion was initiated, and triggered explosions were speculated to occur subsequently; however, the underlying mechanism remains unclear. In the present study, external triggering was reliably achieved by applying an external disturbance pressure in direct contact with the molten droplets. Notably, even lead, a material known for its resistance to explosions, could be triggered under a 3 MPa external disturbance, generating a peak pressure of 64 kPa.

It should also be noted that, owing to the relatively similar mass of Sn droplets used in the experiments, the peak pressures ranged from 1 to 33 kPa (as shown in Fig. 9) were obtained for different initial melt masses. In the present study, Sn was used in cases C1, D8, D9, and D10. For the internally triggered case, a peak pressure of 19 kPa was recorded, whereas the externally triggered cases yielded peak pressures of 26 kPa, 49 kPa, and 47 kPa, as shown in Table 9, which align reasonably well with earlier experimental results from KTH (Fig. 17).

Fig. 17Full-screen View

Peak pressure of steam explosion against melt sample initial mass [47]

In summary, this study focused primarily on the triggering behavior of small-scale molten droplets at temperatures up to 800 °C, a range compatible with high-speed visualization. However, evidence [40] suggests that higher-melting-point materials may undergo significantly different fuel-coolant interactions, potentially altering the steam explosion strength. Moreover, the explosion behavior of these materials under external triggering remains complex and inadequately understood. Additionally, because of obscuration by the vapor envelope, the droplet-gas interface could not be clearly resolved with high-speed imaging. Consequently, the expansion rates could only be estimated approximately from the extent of the gas phase. X-ray imaging is a promising alternative for tracking internal structures during FCI, particularly for high-melting-point materials. As such materials cool, rapid crust formation [12] may influence the interaction more strongly than external triggers. Understanding how these mechanisms compete and interact will be the key focus of our subsequent research.