2. Experiment set-up

A dodecahedral chamber has been constructed to investigate this problem. A schematic of the set-up is shown in figure 1(a). The chamber has 12 pentagonal faces with a side length of $30$ cm, resulting in a volume of ${\approx }210$ l. To generate a homogeneous isotropic turbulent flow field in the centre of the dodecahedron, 20 identical axial fans are mounted at the vertices and driven by the same DC power. All fans are rigged to simultaneously blow into the centre of the dodecahedron. The chamber is capable of achieving ‘ambient turbulence’ or ‘quiescent’ conditions by switching these axial fans on or off. The chamber is not hermetic and the pressure remains around 1 atm during operation. The temperature in the chamber can be regulated through the two heating/cooling plates on the side, which are connected to an external heater/cooler Julabo FP51-SL capable of achieving a temperature range of $-30\unicode{x2013}80\,^{\circ }{\rm C}$. The relative humidity (RH) can also be controlled within a range between $20\,\%$ and $100\,\%$, by either injecting water using an air humidifier or removing excess moisture by pumping the air through an in-line dryer filled with desiccants. To avoid any disturbance to the flow field, RH is adjusted beforehand and it remains relatively unchanged during each measurement thanks to the enclosed nature of the chamber. The temperature, RH and atmospheric pressure are sampled using a BME280 sensor with an in-house built Arduino data acquisition system.

Figure 1.

(a) The dodecahedral chamber with PDA measurement system. The chamber is enclosed by 12 pentagonal transparent acrylic window panels (not shown in the drawing). The droplets are injected through a nozzle chip from the top of the chamber. (b) Schematic showing a close-up view of the nozzle provided by Medspray. The nozzle chip contains 90 holes with a diameter of $4.5\ \mathrm {\mu }{\rm m}$. (c) Laser optical path, with the first-order refraction mode captured by the receiver. The droplets in all panels are not shown to scale.

The nozzle used to generate the spray is provided by Medspray. As illustrated in figure 1(b), droplets are formed at the nozzle chip driven by a syringe pump, and carried downstream by a $15\ {\rm l}\ \min ^{-1}$ (approximately $7\ {\rm m}\ {\rm s}^{-1}$ at the nozzle chip) co-flow to prevent coalescence. We choose to use Novec 7000 Engineered Fluid (chemical name hydrofluoroethers HFE-7000, hereinafter referred to as ‘HFE-7000’ in short) manufactured by 3M (3M 2021), and also characterised by Rausch et al. (Reference Rausch, Kretschmer, Will, Leipertz and Fröba2015). In essence, HFE-7000 is extremely volatile, with a very high saturation vapour pressure of $52.7$ kPa at $20\,^{\circ }{\rm C}$, and it is relatively harmless to the ozone layer and the acrylic windows of the dodecahedron.

The droplet diameters are measured using a Dantec two-component phase Doppler anemometry (PDA) system with a 112 mm FiberFlow probe. Schematics of the set-up and the optical arrangement are shown in figures 1(a) and 1(c). The receiving probe is positioned at an off-axis angle of $\theta \approx 60^{\circ }$, where the first-order refraction is the dominant mode. Details of the measurement principle can be found in Albrecht et al. (Reference Albrecht, Damaschke, Borys and Tropea2013). Essentially, light scattered by a droplet in the measurement volume, which is the intersection of two focused laser beams, is captured by two or more receivers at slightly different angles. Each detector converts the optical signal into a Doppler burst, the frequency of which is linearly proportional to the droplet velocity, and the droplet size is directly inferred from the phase difference in the signals. The PDA set-up is equipped with a 3-detector system, which can largely prevent phase ambiguity, and cover a wide particle size range without sacrificing the resolution. The measurable size range is $0.5\unicode{x2013}137.4\ \mathrm {\mu }{\rm m}$. The measurement volume is fixed at the centre of the dodecahedron, while the spray nozzle is traversed to obtain axial and radial profiles. At each measurement location, the PDA signal is sampled for 30 s. The concentration of HFE-7000 vapour in the ambient environment is considered negligible, as during each run less than 2 ml of the liquid is injected, whereas 650 ml is needed to saturate the entire volume of the dodecahedron.

3. Experimental results on droplet evaporation in a spray

We first explore the change in the average droplet diameter along the centreline ($x$ coordinate) of the spray jet in a quiescent ambient environment, where the 20 turbulence generating axial fans are off. Figures 2(a) and 2(b) show the probability density distributions (p.d.f.) of the measured droplet diameters for the ambient RH cases of ${\rm RH} = 20\,\%$ and ${\rm RH} = 100\,\%$, respectively. Close to the nozzle, the distribution exhibits a broad peak centred around $d=6\ \mathrm {\mu }{\rm m}$. Further away from the nozzle, as shown by the lines with a lighter shade, the magnitude of this peak reduces while a second peak emerges at a much smaller size, $d = 2\mbox {--}3\ \mathrm {\mu }{\rm m}$. This peak at small diameters keeps growing with increasing $x$, eventually exceeding the magnitude of the first peak. We believe that the emergence of the second peak at much smaller diameters is a result of water in the ambient air condensing on the rapidly evaporating HFE-7000 droplets. An analytical model is extended based on the work of Marié et al. (Reference Marié, Grosjean, Méès, Seifi, Fournier, Barbier and Lance2014) and Tonini & Cossali (Reference Tonini and Cossali2015), taking the immiscibility between HFE-7000 and water into consideration. Details of the model are summarised in Appendix A. Assuming that the droplets measured at $x = 20$ mm contain HFE-7000 only, we use the recorded size distribution at this location as the initial condition, and final condensed water droplet size distribution can be predicted by the model (Appendix A) and is shown by the black lines. The peak at $d \approx 2\ \mathrm {\mu }{\rm m}$ in the predicted condensed water droplet size directly results from the peak at $d_0 \approx 6\ \mathrm {\mu }{\rm m}$ in the initial size distribution. Indeed, the measured size distribution can be well captured by the model prediction. Note that, at this stage, we consider a single, stationary droplet, not affected by the vapour field of neighbouring droplets and the evaporation–condensation rate is limited by the diffusion of the vapour film around the droplet. Although many effects, including the turbulent convection (Birouk & Gökalp Reference Birouk and Gökalp2006; Méès et al. Reference Méès, Grosjean, Marié and Fournier2020; Dodd et al. Reference Dodd, Mohaddes, Ferrante and Ihme2021) and the sheltering in clusters of droplets (Villermaux et al. Reference Villermaux, Moutte, Amielh and Meunier2017; Sahu, Hardalupas & Taylor Reference Sahu, Hardalupas and Taylor2018; Dodd et al. Reference Dodd, Mohaddes, Ferrante and Ihme2021), are not taken into consideration, the modelled remaining water droplet size distribution still has a good agreement with the measurements.

Figure 2.

Droplet size distribution at (a) ${\rm RH} = 20\,\%$ and (b) ${\rm RH} = 100\,\%$ in a quiescent ambient environment. From dark to light, the line colour indicates an increasing distance $x$ from the nozzle along the centreline, ranging from $x = 20$ to $x = 60$ mm. The solid black line is the predicted condensed water droplet size distribution based on the analytical model of Appendix A, valid for large distances $x$.

We consider the HFE-7000 component fully evaporated in any measured droplet with a diameter smaller than a certain threshold $d_{{thr}}$. Thus, a ratio $\alpha$ can be defined as the ratio between the number of droplets with $d< d_{{thr}}$ and the total number of droplets measured at the same location. The threshold is chosen as $d_{{thr}}=4.7\ \mathrm {\mu }{\rm m}$, which is larger than 99.5 % of the predicted remaining water droplet size in the ${\rm RH} = 100\,\%$ case (black line in figure 2b). Note that the same value for $d_{{thr}}$ is used across all RH cases for consistency, although more droplets that still contain liquid HFE-7000 will be mistaken as remaining water droplets in low ambient RH cases. The spatial variation of $\alpha$ at a range of ambient relative humidities is depicted in figure 3. In both axial and radial directions, $\alpha$ increases with increasing distance from the nozzle, confirming the sheltering effect in the jet (Villermaux et al. Reference Villermaux, Moutte, Amielh and Meunier2017; Wang, Dalla Barba & Picano Reference Wang, Dalla Barba and Picano2021). Furthermore, we observe a strong dependence of $\alpha$ on RH: at the same location, $\alpha$ increases with RH, indicating that HFE-7000 evaporates faster in a more humid environment.

Figure 3.

The fraction $\alpha$ of remaining water droplets among the total number of measured droplets along (a) the axial direction ($r = 0$) and (b) the radial direction ($x = 40$ mm) in a quiescent ambient environment.

The observation can be explained by inspecting the modelled single droplet behaviour. As shown in figures 4(a) and 4(b), indeed, the predicted lifetime of a HFE-7000 droplet at ${\rm RH} = 100\,\%$ is around $40\,\%$ shorter than that at ${\rm RH} = 20\,\%$. The reason behind this higher evaporation rate lies in the latent heat and the resulting temperature. As shown in figure 4(e,f), when the HFE-7000 droplet is exposed to ambient air, its temperature rapidly decreases as a result of the latent heat absorbed during the liquid-to-gas phase transition. This phenomenon is also known as ‘evaporative cooling’. The cold droplet cools down its surrounding gas film, reducing the saturation vapour mass fraction of water it carries. As a consequence, the local RH will increase, and once it exceeds 1, the water vapour becomes over-saturated and will start condensing on the surface of the droplet. At a high ambient RH (figure 4d), the rate of water condensation significantly increases and the latent heat released in the process heats up the droplet to a higher temperature, therefore the HFE-7000 component will evaporate at a higher temperature and a faster rate. Furthermore, high RH environment leads to a larger water core, giving rise to a higher surface-to-volume ratio of the HFE-7000 component, which also accelerates the evaporation.

Figure 4.

Model prediction of (a,b) the equivalent droplet diameter of composing liquids, (c,d) the rate of latent heat absorbed/released by the droplet and (e,f) the droplet temperature as a function of time at (a,c,e) ${\rm RH} = 20\,\%$ and (b,d,f) ${\rm RH} = 100\,\%$. The droplet initially contains only HFE-7000 and the ambient temperature is $T_{\infty } = 20\,^{\circ }\mathrm {C}$ ($293.15$ K), and the initial diameter is set to a typical size of $d_0 = 6\ \mathrm {\mu }{\rm m}$.

4. Effect of turbulence

In this section, we discuss the effect of ambient turbulence on the observed evaporation–condensation process. The effect of turbulence is usually modelled through Ranz–Marshall correlations, and is characterised by a single non-dimensional parameter $Re_{d}\equiv d U_{r}/\nu$, which is the droplet Reynolds number based on $d$, the diameter, $U_r$, the velocity relative to the surrounding gas, and $\nu$, the kinematic viscosity of air. Since the droplet diameter $d$ changes during evaporation, we represent the level of turbulence using $Re_{d_0}\equiv d_0 U_{r}/\nu$, the droplet Reynolds number based on the initial diameter and assume the relative velocity $U_r$ remains constant. In the present experiments, the relative velocity is not measured, nevertheless, $Re_{d_0}$ is estimated to be well below 10 considering the small size of the droplets. Figure 5 compares $t_{{evap}}$, the time required for HFE-7000 to fully evaporate, and $d_{w}$, the water condensation droplet size at various $Re_{d_0}$ values. Overall, $t_{{evap}}$ reduces with increasing ambient RH. For a given RH, $t_{{evap}}$ is also found to be smaller at a higher $Re_{d_0}$, as the mass and heat exchange is enhanced by turbulent convection. In the limit of extremely large $Re_{d_0}$ which is not practical for the current configuration, a power law relationship $t_{{evap}}\propto Re_{d_0}^{-2/5}$ emerges as a consequence of (A14a). Interestingly, despite the large variation in $t_{{evap}}$, very little change is observed in the final condensed water droplet size $d_w$, as shown in figure 5(b). This suggests that as turbulent convection enhances the evaporation, the rate of condensation is also increased almost proportionally: similar to its effect in reducing the local HFE-7000 vapour concentration by advecting the HFE-7000 vapour away from the droplet, the turbulence also serves to replenish the droplet surface with over-saturated humid air. This result also explains why even the diffusion-limited model can produce a remaining water droplet size distribution with a high level of agreement with the experimental data.

Figure 5.

Model prediction at a range of droplet Reynolds numbers, $Re_{d_0}$. (a) The time required for the HFE-7000 component to fully evaporate $t_{{evap}}$, and (b) the diameter of condensed water droplets $d_w$. All results are for a HFE-7000 droplet with an initial diameter of $d_0 = 6\ \mathrm {\mu }{\rm m}$ at an ambient temperature of $T_{\infty } = 20\,^{\circ }{\rm C}$ ($293.15$ K).

Experimentally, the ambient turbulence is generated by the 20 axial fans mounted in the dodecahedral chamber. We would like to point out that the spray is already at a turbulent state even though the surrounding air in the dodecahedral chamber is under quiescent conditions. The ambient turbulence is, in fact, acting on an already turbulent jet spray. Figure 6 compares $\alpha$, the proportion of the condensed water droplets at a number of measurement locations along the radial direction with quiescent and turbulent ambient conditions. The data with ambient turbulence display a similar trend of increasing $\alpha$ with increasing $x$ and RH, and there is no major difference from the quiescent case.

Figure 6.

The fraction of remaining water droplets among the total number of measured droplets with (dashed line) and without (solid line) the presence of ambient turbulence. The error bars show the standard deviation of the mean estimated using a bootstrapping approach.

5. Observations of a single droplet

In this section, we provide further experimental evidence of the water condensation that accompanies the rapid vaporisation process by directly imaging a single, millimetre-scale HFE-7000 droplet in an acoustic levitator. The droplet is imaged using a Photron Nova S12 high-speed camera with a Navitar $12\times$ lens at 500 Hz.

Snapshots of the multi-component droplet are shown in figure 7. The moment the droplet is injected in the acoustic levitator is denoted as $t = 0$. At $t = 2.0$ s (figure 7b), small water droplets with a diameter around $20\ \mathrm {\mu }{\rm m}$ distribute within the HFE-7000 drop. Carried by the internal flow, which is caused by the internal circulation as well as the acoustic field, these water droplets move in the HFE-7000 drop, collide and coalesce into larger clusters as clearly visible in figures 7(c) and 7(d). These clusters have complex, non-spherical shapes. When the amount of remaining liquid HFE-7000 is not sufficient to wet the entire surface of the water clusters and fill the grooves, these water clusters are exposed and the drop exhibits an irregular shape with protrusions (figure 7e). After this transient phase, only a single-composition water droplet remains (figure 7f). It has a spherical shape, as a result of the increase in the surface tension with HFE-7000 being replaced by water at the droplet surface. These observations clearly validate the modelling assumption that the water condensation forms an inclusion within a HFE-7000 shell (see figure 8b), and the droplet surface contains predominantly only HFE-7000. Supplementary movie 1 available at https://doi.org/10.1017/jfm.2023.25 shows the entire evaporation–condensation process. Note that for a typical droplet in the spray, the diameter is less than 1 % of the acoustic levitated drop diameter here, and it shape is more spherical even in the multi-component stage because of the much smaller Weber number.

Figure 7.

Temporal evolution of the droplet equivalent diameter $d_e$. The solid green line is measured from droplet images, and the dotted black line is computed from the model, with $Re_{d_0} = 10$ selected to best replicate the experimental observation. Insets (a–f) are droplet images at time instances marked by the arrows. The scale bar in (a) holds for all images.

Figure 8.

(a) Low-surface-tension component (HFE-7000) wraps around the high-surface-tension water core, driven by the difference in surface tension. (b) The wrapping process is complete and HFE-7000 forms a shell around the water core.

The droplet has an oblate ellipsoidal shape as a result of levitation by the acoustic field (Yarin et al. Reference Yarin, Brenn, Kastner, Rensink and Tropea1999; Al Zaitone Reference Al Zaitone2018). The major and minor axis lengths $d_a$ and $d_b$ are measured from the image, and the equivalent diameter $d_e$ is calculated as $d_e = (d_a^2d_b)^{1/3}$. The ratio $d_b/d_a$ is approximately 0.6 until HFE-7000 is fully evaporated. The evaporation time of a spherical droplet with the same equivalent diameter gives a reasonable approximation of the ellipsoidal case (Tonini & Cossali Reference Tonini and Cossali2016, Reference Tonini and Cossali2019). The measured diameter variation matches well with the model prediction at $Re_{d_0} = 10$, which corresponds to a relative velocity of $U_r \approx 0.2\ {\rm m}\ {\rm s}^{-1}$.