Читать книгу Ice Adhesion - Группа авторов - Страница 49
3.2.1 Scenario I: Droplet in an Environment without Airflow
ОглавлениеIn this scenario, tip singularity formation is a common phenomenon that occurs during ice growth. In this phenomenon, when a droplet is placed on a cold plate, it freezes and turns to an ice drop with a pointy tip (Figure 3.5). Tip singularity formation is mainly due to the water expansion after freezing and is governed by the quasi-steady heat transfer at the later stages of ice formation. Marín et al. [21] stated that the freezing front is convex at earlier stages of ice growth and at the final stages it becomes concave. They also reported that the freezing front is almost perpendicular to the ice-air interface, i.e. γ = ϕ + θ ≈ 90° (see Figure 3.5), due to the fact that latent heat cannot transfer across the solid-air interface due to low thermal conductivity of air. The shape of solid-liquid front is obtained through the assumption of constant front temperature at the equilibrium melting temperature, Tf , i.e. neglecting Gibbs-Thomson effect.
Figure 3.5 Geometry of droplet at later stages of ice formation and after complete ice formation. θ denotes the angle between liquid-air interface and horizontal, ϕ denotes the angle between freezing front and horizontal and α denotes the final tip angle [21].
For obtaining geometric theory for tip formation, the first step is to write mass conservation with respect to z, as temporal dynamics is not significant.
Where Vl and Vs denote liquid and solid volumes, respectively, v is density ratio, and z is height of trijunction (Figure 3.5).
The liquid at the top of freezing front is divided into two parts. The upper part is like a spherical cap with angle of θ and the lower part has a volume of Vd. Thus,
(3.17)
(3.18)
Considering the geometries of upper and lower parts of the liquid, one finds,
(3.19)
Based on (Eqs. 3.16–3.20) and the fact that can be obtained and a sharp tip is formed when r→0. Thus, at this singularity point, one can find θ as follows:
Eq. (3.21) can give volumes of liquid part before and after freezing. If this equation is multiplied by r3, the left side gives the volume of liquid. Also, the right side gives the volume of this liquid when it is frozen where expansion factor is considered. According to Eq. (3.21), we have α = π–2θ regardless of v value and as mentioned before γ ≈ 90°. Therefore, from Eq. (3.21), a constant value of α = 131° for the tip angle is obtained which is in a great harmony with the experimental results [21].
Now, we determine the growth rate of ice in scenario one (Figure 3.6). In this case, as the thermal conductivity of air is low, convective heat transfer is low and the generated enthalpy of phase-change is transferred through the ice by thermalconduction mechanism and subsequently, through the substrate, Isothermal condition is assumed forthe liquid at the ice-water interface and heat flux through the liquid, is negligible. the velocity of freezing front is defined as:
Where l denotes temporal height and r denotes the radius of freezingfront. The heat transfer away from the interface to the substrate, is obtained through the energy balance at the interface for a quasi-steady process as follows:
Where ρi is the density of ice and Hm is the enthalpy of ice formation. Also, using heat conduction equation, is obtained as:
Figure 3.6 Ice growth on a sub-zero substrate when the droplet is in an environment without airflow. In this case, ice growth is controlled mainly by heat transfer through the substrate [5].
Where δT denotes the temperature difference between the substrate and ice-water interface (Ts–Tf), l0 denotes initial height of droplet, lm denotes the thickness of substrate, ki denotes thermal conductivity of ice and km denotes thermal conductivity of substrate. From equations 3.22, 3.23 and 3.24 one finds:
From Eq. (3.25) the height (l) or radius (r) of droplet as a function of time, t, can be obtained in two different conditions. The first one is for the condition where thermal conductivity of substrate is high or thickness of the coating is low. Thus, lm/km ≪ l0/ki and Eq. (3.25) can be written as:
The second condition is when the thermal conductivity of coating is low or the thickness of coating is high. In this case, lm/km ≪ l0/ki and Eq. (3.27) is obtained:
The important assumption in the aforementioned analysis is quasi-steady heat transfer. In quasi-steady heat transfer it is assumedthat time-scale for ice growth, is more than thermal diffusion, in which Di is thermal diffusivity of the ice. This assumption is correct in the case of a water droplet. For example, for a water droplet with 1mm diameter, the time-scale for growth is around 10 s and time-scale for diffusion is around 1 s.
In order to validate the model developed for ice growth rate (Eqs. 3.26 and 3.27), Irajizad et al. [5] collected some experimental data on ice growth rate on different substrates. e.g. PDMS1 and glass. Furthermore, the reported ice growth rate in [22] is included in this comparison. They plotted collected data in Figure 3.7 along with ice growth rate obtained from the theoretical model (Eqs. 3.26 and 3.27).
Figure 3.7 The experimental data for ice growth rate are compared to theoretical model obtained from Eq. (3.26) and Eq. (3.27) which shows that experimental data for ice growth match theoretical model well. l and l0 are height of liquid at the top of the ice and total height of the droplet, respectively [5].
As shown in Figure 3.7, the predicted model obtained by heat transfer analysis matches experimental data well [5]. As an example, freezing times of a water droplet on stainless-steel at -20°C and -30°C are 9.6 and 7 s, respectively, which are obtained from the experiment [23]. The freezing times obtained from predicted model are 10 and 6 s for -20°C and -30°C, respectively, which are in a great harmony with the experimental data. In order to obtain isotherms in the ice, the heat equation should be solved in the ice domain (∇2T = 0). The boundary conditions in this case are that ice-water interface temperature is constant and ice-icephobic substrate interface temperature is prescribed.
