## 2 Mathematical Modeling  
### Drop Size Distribution  
The growth process of condensing drops is divided into two phases. The first phase is when the growth of small drops is dominated by direct condensation, and it is defined by the drop size range from the minimum radius to the effective radius. The minimum droplet radius which is the smallest thermodynamically viable drop is given by [41]  
\[r_{min}=\frac{2\ T_{sat}\sigma}{H_{fg}\rho\Delta T} \tag{1}\]  
where \(T_{sat}\) is the water-saturated temperature, \(\sigma\) is the surface tension, \(H_{fg}\) is the latent heat of vaporization, and \(\rho\) is the density of the condensate. \(\Delta T\) is the total temperature drop in the path of heat transfer due to thermal resistance. The effective radius is when small droplets reach a certain size and the growth process is dominated by coalescence. The effective radius is provided as follows [16; 42]:  
\[r_{\epsilon}=\left(\frac{1}{N_{s}}\right)^{0.5} \tag{2}\]  
The nucleation sites per unit area, \(N_{S}\), are derived analytically for hydrophobic surfaces and given as [37]  
\[N_{s}=\frac{0.037}{r_{min}^{2}} \tag{3}\]  
Drops that mainly grow by coalescence may reach a certain size and start to depart from their position. The maximum radius \(r_{max}\) of a departing drop is obtained by balancing the gravitational and interfacial forces. The expression with respect to contact angle hysteresis \(\theta_{hys}\) and inclination angle \(\alpha\) is given as [17; 43]  
\[r_{max}=\left[\frac{3\sigma(\text{cos}\theta_{r}-\theta_{a})\text{sin}\theta_{ a}}{2p\beta\pi\ \text{sin}\alpha\ (2-3\text{cos}\theta_{a}+\text{cos}^{3}\theta_{a})}\right]^{0.5} \tag{4}\]  
where \(\theta_{a}\) and \(\theta_{r}\) are advanced and receding contact angles and they define the contact angle hysteresis. In the special case when the drop maximum size is limited to DWC region width within a hybrid surface, the maximum radius becomes  
\[r_{max}=\frac{w_{DWC}}{2} \tag{5}\]The empirical formula of drop size distribution for large drops, which is bounded between the effective and maximum radius, is provided in the following form [12]:  
\[N(r)=\frac{1}{3\pi\ r^{2}\ r_{\max}}\bigg{(}\frac{r}{r_{\max}}\bigg{)}^{n-1} \tag{6}\]  
The exponent number \(n\) in the above expression is originally \(1/3\), and sometimes the power number is shown as \(-2/3\) in the literature. In this case, Equation (6) is applicable for a full-DWC surface when contact angle is \(\theta\ \leq 90^{\circ}\) without the consideration of surface renewal. It is in good agreement with the simulation work provided by [PERSON] and [PERSON] [5], and it is used for validation with experimental work. Despite its limitations, it is widely used for modeling and evaluating the thermal performance of full-DWC and DWC within hybrid surfaces [14; 16; 17; 18; 19; 20; 21; 32; 37; 39]. Drop size distributions, as given by Equation (6) with \(n=1/3\), for various applications are shown in Figure 1 from some sources in the literature. The original work has a size range defined by an effective and maximum radius of 0.001 mm and 0.45 mm, respectively, for \(\theta\ \leq 90^{\circ}\)[12]. The same drop size distribution was applied for \(\theta\) of 90 and 120 degrees [16]. Both cases are for full-DWC surfaces without considering the effect of surface renewal. For DWC within a hybrid surface in [21], Equation (6) is used and \(r_{max}\) is limited to DWC region width. The simulation work in [19] is performed in the presence of coalescence and re-nucleation, and data of drop size distribution for contact angles of 90 and 120 degrees are plotted for a 1 mm maximum radius. The discrepancy between [16] and [19] may be due to the consideration of coalescence and re-nucleation in [19]. The drop size distribution of small drops, which is bounded between the minimum and effective radius, is derived by applying the population balance model.