Full length articleSpreading and freezing of supercooled water droplets impacting an ice surface
Graphical abstract
Introduction
The accretion of ice on cold surfaces is a common phenomenon present in various fields of nature and technology which poses hazards to many applications. For example, icing can present threats to aircrafts [1], roadways [2], wind turbines [3], and power transmission lines [4], etc. The impact of supercooled drops on cold surfaces is a main factor of icing in the above examples. After forming an initial ice surface on a surface, water drops will collide with ice. Thus, understanding the mechanisms of impact and icing of a droplet on an ice surface is a fundamental endeavor for increasing knowledge of ice accretion and formulating anti-icing methods.
In general, the impact of a supercooled droplet on a cold surface involves several physical processes, including droplet deformation, heat transfer, supercooled liquid nucleation, and liquid solidification. The dynamics of droplet deformation and spreading under room temperature have been studied with various methods, including theoretical analysis [5], [6], [7], [8], [9], [10], experimental investigations [11], [12], [13], [14], as well as numerical simulations [8], [15], [16]. This kind of spreading process is isothermal, which is mainly driven by inertia, hindered by capillary and viscous forces, and affected by surface wettability [17]. These parameters are scaled by the ratio between inertial and capillary forces (Weber number) , the ratio between inertial and viscous forces (Reynolds number) , and the contact angle . Here, denotes the fluid density, the initial diameter of the droplet, the impact velocity, the liquid–vapor interfacial tension, and the dynamic viscosity. The dominant methodology of theoretical models is based on energy balance. Generally, the energy balance equation includes kinetic and surface free energy just before impact (initial stage) and surface free energy in the maximum spreading stage (final stage). A significant amount of energy is dissipated due to the viscosity of liquid during spreading. Difference in gravitation energy between the initial and final stages can be neglected, and previous models assume that the kinetic energy in the final stage is zero. Pasandideh-Fard et al. [8] suggested that the velocity gradient in the impinging direction should only be calculated only within boundary layers. The contact angle , which is exhibited by the droplet at maximum spreading when contact line stops, is reported more suitable for predicting [8]. This model yielded a good agreement with the experiment results by Ukiwe and Kwok [9], resulting a mean relative error of 6.52%. The predicted is given as follows:
Published models mainly differed in terms of the estimation of viscous dissipation and surface energy. Four kinds of contact angles are used in estimating the surface energy change. And the different assumptions will produce different coefficients of the energy terms, resulting predictions that are far from one another. Nevertheless, the dimensionless numbers derived from the models reveal the fundamental physics to some extent. In Eq. (1), is an important parameter determining the characteristics of the spreading. If and is also large, the equations will be reduced to , which is called the viscous regime where the initial energy mainly transforms into viscous dissipation. By contrast, if , , which represents the capillary regime where viscous effects are negligible. These formulations agree well with many experiments and other models [11], [16], [18]. Various studies including theoretical modeling [15], [18], [19], experiments [19] and computational simulations [15] point out that in the viscous regime, is better for scaling maximum spreading. In the limit of capillary regime, is reported to be fitted when modeling . These conclusions are close to the analysis results based on Eq. (1). Eggers et al. [15] introduced to distinguish between the two regimes, describing the transition from one limit to another quantitatively. Thus, they deduced the form that should be satisfied by maximum spreading factor as The function has the following characteristics: By using a mathematical approximation, which is a ratio of two first order polynomials, Laan et al. [12] estimated as where, is a fitting constant. The model by Laan et al. [12] is reported to perform well in modeling numerous experiment data among several models [17].
Non-isothermal impact process involving heat transfer has also been investigated experimentally [20], numerically [20], [21], and theoretically [22]. Nucleation is a solid crystal formation process, which triggers of solidification in a bulk of liquid [23]. Heterogeneous nucleation at a substrate initiates the solidification of the drops on the solid substrate [24], [25]. In a moving liquid such as impacting droplet, the mechanisms of nucleation is complicated as there are a variety of factors whose influence lacks complete clarification [26]. Nucleation typically occurs later than impact, and this stochastic freezing delay time can be different even if the impact conditions are constant [27]. However, ice surfaces act as a uniform and immediate nucleator, which enable droplet spreading and freezing to begin simultaneously. Thus, the mutual interaction between fluid flow and freezing can be taken into account without concerning about the stochastic nucleation process.
The influence of solidification on droplet spreading has not been discussed in depth. With regard to a supercooled droplet on a cold surface, solidification occurs in two distinct phases [26], [28], [29]. The first phase of solidification is a rapid process [28], beginning with nucleation [25], [30], continuing with the partial-solidification of liquid [27], [30], and ending by reaching thermodynamic equilibrium [31]. The second phase is equilibrium solidification, which occurs in non-supercooled liquid at freezing temperatures, is well understood and the propagation of its solidification front can be described as a one-phase Stefan problem [32]. Both phases of solidification are shown schematically in Fig. 1. Usually, nucleation occurs in different nucleation sites on a solid surface, depending heavily on surface properties [33] and dissolved salt [34]. After nucleation, a thin layer of ice will spread over the surface in a velocity determined by the liquid temperature and surface properties [33]. Because supercooled liquid is in a metastable equilibrium, the ice layer will trigger dendritic solidification in the bulk liquid [26], [31]. Growing dendrites change the region they cover from supercooled liquid to a mixed-phase state [32], resulting a transformation from a metastable state to a stable equilibrium state [31]. This mixed-phase region is referred to as the dendrite cloud, where the latent heat released by the portions of the liquid converted into dendrites increases the temperature from a supercooled value to the equilibrium freezing value [28]. The thermal boundary layer thickness is much smaller than that of the dendrite cloud over a wide range of conditions [29]. Surface temperature is considered to have minor influence on the propagation of the dendrite cloud [29]. The second phase of solidification begins when the drop has been filled with dendrite cloud, which is an isothermal process at equilibrium freezing temperature [28]. This process is not a concern in the study of droplet spreading, as it is much slower [29], [31].
An ice surface, which has been covered by crystal nucleus already, is unique for considering the solidification of supercooled liquid on it. Accordingly, dendrites grow immediately form the sites where the supercooled liquid contacts the ice, making impact and freezing begin simultaneously [29]. Jin et al. [35] studied the impact and freezing of a droplet at room temperature on a sooth ice surface. Schremb et al. [29] investigated the lamella thinning process of supercooled droplets impacting on an ice surface, providing a theoretical model and semi-empirical formula to predict the final center height.
However, the spreading process of a supercooled droplet on an ice surface, which can be affected by the immediate solidification of a supercooled liquid, has not been investigated before. In the present work, experiments are performed involving the impact of droplets with varying velocities, initial diameters and supercoolings on smooth ice surfaces at different temperatures. The spreading and freezing processes are documented by a high-speed camera. During the short time of droplet spreading, solidification is in the first phase. Growing dendrite cloud is considered in the analysis of the spreading process. As solidification mainly affects the fluid flow rather than the surface forces, the analysis of icing focuses on the viscous regime. Utilizing perturbation theory, approximate solutions are deduced with a newly defined dimensionless number, the icing number, which scales the influence of icing on spreading flow. A semi-empirical model is established by combining the capillary and refined viscous effects. It agrees well with the experimental results. The conditions, under which solidification is important or not, are discussed, giving a criterion for evaluating the influence of solidification.
Section snippets
Materials and methods
To gain a larger range of experimental data, two suits of experiment devices are used, which are the low-speed impact device accelerating droplets through gravity and the high-speed impact device accelerating a surface through rotating.
Impact photographs
Fig. 4 shows the impact, spread and recoiling processes of four droplets with the same impact velocity of 1.1 m/s but different supercoolings: K, K, K, and K (from right to left in Fig. 4). For the drop of K and , in the initial stage (from about 0.0 ms to 1.0 ms) the upper part of the drop remains nearly spherical, while the bottom spread out on ice surface in the shape of a disk. Then (from about 1.0 ms to 4.0 ms), as the central part of the drop turns into
Conclusions
In the present study, two apparatuses are built and used in observing the impact of a supercooled water droplet on an ice surface under controlled conditions. By varying impact velocity (1.1 16.8 m/s), initial diameter (1.5 3.5 mm), droplet supercooling (4 16 K), and ice surface temperature ( °C), the deformation process is captured for every drop. Experimental results have shown that surface temperature has negligible influence on droplet spreading, which is consistent with
CRediT authorship contribution statement
Yizhou Liu: Conceptualization, Methodology, Formal analysis, Investigation, Data curation, Writing – original draft, Visualization. Tianbao Wang: Conceptualization, Methodology, Validation, Investigation, Data curation, Writing – review & editing. Zhenyu Song: Investigation. Min Chen: Conceptualization, Writing – review & editing, Supervision, Project administration, Funding acquisition.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (No. 11972211), the National Key Basic Research Program of China (No. 2015CB755801), the National College Students’ Innovation and Entrepreneurship Training Program, China (No. 201910003024), and Tsinghua University Initiative Scientific Research Program for Undergraduates, China .
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