Impact of boiling liquid droplets: Vapor entrapment suppression

Abstract

There hardly is a fluid mechanics phenomenon attracting more attention than the impact of a droplet, due to its undeniable beauty, many applications and the numerous challenges it poses. One of the crucial factors turns out to be the cushioning effect of the gas surrounding the droplet. This fact, together with the observation that almost all of the relevant literature was done in air, triggers the question what would happen when the liquid was a boiling liquid, i.e., a liquid in thermal equilibrium with its own vapor, as is the case during transport of cryogenic liquids such as liquid hydrogen. To investigate precisely this question, we experimentally generate droplets in thermodynamical equilibrium with their own vapor, even before impact, such that minute energy exchanges of the droplet with its surroundings can trigger phase change. Using a frustrated total internal reflection (TIR) setup, we make the exciting observation that depending on the impact speed and vapor conditions, the entrapment of vapor can be completely suppressed under boiling liquid conditions. We create a simplified model based on scaling arguments and perform numerical simulations considering both the compressible and condensable properties of the vapor layer that are in very good agreement with our experimental findings. Our results can be of great consequence to the pressures exerted during droplet impact and on an industrial scale may help better understand the loads experienced during sloshing wave impact inside cryogenic liquid containers.

Figures (5)

• Figure 1: Sketch of the experimental setup with as main elements the closed, temperature-controlled chamber containing HFE-7000 liquid (Res1) in equilibrium with its vapor, the droplet generated from the second reservoir (Res2), and the prism (P) used for frustrated total internal reflection imaging (TIR), the vacuum pump (VP), and the two high-speed cameras (CAM1,2) used for imaging.
• Figure 2: (A) TIR image sequence (after background subtraction) of a series of HFE-7000 droplets with radius $R_0 \approx 0.6$ mm in a vapor atmosphere at $T_{0} \approx 22$$^\circ$C; the droplets impact a sapphire substrate heated to a temperature difference $\Delta T \lesssim 2$$^\circ$C with impact velocities $U_0 = 0.24$, $0.38$, $0.51$, $0.74$ m/s, Fig. S2 and movies S1 to S5 in the SI. At low impact velocity a vapor bubble is entrapped, but at high impact velocities this entrapment is suppressed. (B) TIR image sequence (after background subtraction) of a series of droplets with radius $R_0 \approx 0.6$ mm in a vapor atmosphere at $T_0 \approx 22$$^\circ$C impacting with velocity $U_0 \approx 0.7$ m/s onto a sapphire substrate heated to an increasing temperature difference $\Delta T = 2.4$, $6.4$, $11.3$, $17.6$ K (with respect to $T_0$), Movies S6 to S10 in the SI. At this impact velocity, the entrapment of vapor is suppressed for small $\Delta T$, but entrapment can be recovered by heating the substrate to a higher temperature. In all cases, time is measured with respect to the first TIR frame in which the droplet becomes visible.
• Figure 3: (A) Azimuthally averaged inner ($r_b$, blue points) and outer ($r_d$, orange points) radius evolution for an experiment at $U_0 = 0.38$ m/s, with images available in the SI, Fig. S1B. The orange line is obtained from a linear fit to $r_d^2$ plotted against $t$ and the blue line is a linear fit for $r_b$. The intersection of the two curves provides the initial radius $r_0$ and also the initial impact time. The entrapment time $t^*$ is then measured from the inner radius fit as the time it takes for $r_b(t)$ to go from $r_0$ to $0$. (B) Initial entrapment radius $r_0$ as a function of the impact velocity $U_0$ for a set of experiments with a small temperature difference of the impact surface with respect to the ambient temperature ($\Delta T = 1.77 \pm 0.49$ K). The points represent the mean of a minimum of three repetitions and the bars display the total error. (C) entrapment time $t^*$ as a function of the impact velocity $U_0$, for the same set of experiments.
• Figure 4: (A) Time evolution of the axisymmetric and normalized profile $h/R_0$ as a function of the normalized radial coordinate $r/R_0$ for four boundary integral simulations of a HFE-7000 droplet impacting at different speeds $U_0 = 0.2$, $0.4$, $0.6$, $0.8$ m/s. The horizontal black dashed lines at $h = 455$ nm indicate the height at which a droplet would enter the evanescent wave of the light source, and would thus show up in a TIR experiment. Color indicates time progression from yellow to dark blue. (B) Time evolution of the center height $h_c$, normalized with the initial height $h_0$, for various impact velocities $U_0$ between $0.2$ m/s and $1.0$ m/s (blue to yellow). The straight dashed black line represents the time evolution of the position of the bottom of a moving sphere in the absence of vapor. Inset: Zoom into the blue dashed rectangular area. (C) The center touchdown time $\Delta t$ (blue pentagons), plotted as a function of the impact velocity $U_0$, where the continuous red curve is a hyperbolic fit to the data and the dashed red vertical line represents its asymptote. The red stars represent the entrapped vapor volume $V_\text{e}$ indicated in (A).
• Figure 5: Experimental phase diagram tracing the boundary between droplet impacts with (colored circles) and without (crosses) vapor entrapment. On the vertical axis the impact velocity $U_0$ is plotted, whereas on the horizontal axis we find the temperature difference $\Delta T$ of that of the heated substrate and the equilibrium temperature of the surrounding vapor. The color of the circles denotes the bubble lifetime $t^*$, as indicated by the color bar. The blue curve indicates the phase boundary predicted from the model.