Ice Adhesion. Группа авторов

      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 (T_s–T_f), 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:

(3.25)

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:

(3.26)

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:

(3.27)

      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.

      3.2.2 Scenario II: Droplet in an Environment with External Airflow

      Second scenario occurs when there is airflow around the water droplet in which convective heat transfer becomes significant (Figure 3.8). In this case, instead of solid-liquid interface, ice nucleation occurs at liquid-vapor interface, as convective heat transfer reduces temperature at the surface [24]. Figure 3.8 shows ice growth pattern for the droplet exposed to an environment with airflow.

      Figure 3.8 Ice growth on a sub-zero substrate when the droplet is in an environment with external airflow. In this case, ice growth is controlled mainly by convective heat transfer through airflow [5].

      Similarly to Scenario I, one can use energy balance at phase-changing interface to obtain ice growth rate. The enthalpy released from freezing makes a heat flux at ice-water interface which is carried out with airflow. Thus,

      (3.28)

Where is convective heat transfer coefficient. In this case, the Nusselt number, written as a function of diameter which is representative dimension (NuD), by which one can obtain convective heat transfer coefficient, is written as Eq. (3.29) [25]. Note that there is no evaporation on the droplet surface, since ice is formed around the droplet.

(3.29)

Where ReD is Reynolds number for external airflow, Pr is Prandtl’s number and

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