## 2 Mathematical Modeling
### Proposed Drop Size Distribution
#### 2.2.1 Simulation Procedure  
Our simulation work is divided into multiple modules that are capable of illustrating the process of drop growth, generating newly formed drops, and surface renewal by sweeping and merging. All measured distances are based on a three-dimensional domain; however, simulation work is performed in two dimensions or from a top view. In the initial condition module, the effective radius and maximum radius are prescribed. All droplets are randomly distributed without overlapping on a quadrilateral domain defined by coordinates \((0,0)\) and \((a,b)\), where \(a\) and \(b\) are domain width and length, respectively. Drops size at the initial state varies from 0.2 \(r_{e}\) to 0.2 \(r_{max}\). Three distances are measured for each drop: \(x\)-axis coordinate \(x_{i}\), \(y\)-axis coordinate \(y_{i}\), and the radius \(r_{i}\), where subscript \(i\) indicates drop number, as shown in Figure 2a. These distances are updated throughout the simulation process because drop location and size are subject to change. In addition, distances from each drop center to all boundaries \(s_{ik}\) are measured and updated, where \(k\) is a boundary number. Only drops completely or partially within the domain are accounted for in estimating the area fraction, \(f\). Therefore, when the difference between distance \(s\) with upper or lower boundaries and drop radius is negative, it indicates that the drop is partially outside the boundary. \[s_{ik}-r_{i}\leq 0 \tag{13}\]  
The second module is used to represent drop growth by direct condensation. Substituting Equation (10) into (12) and solving for \(G\) gives  
\[G=\frac{dr}{dt}=\frac{\Delta T\pi r^{2}\left(1-\frac{r_{min}}{r}\right)/\rho H _{fg}2\pi r^{2}(1-cos\theta)}{\frac{\delta}{k_{cost}sin^{2}\theta}+\frac{r \theta}{4k_{c}sin\theta}+\frac{1}{2h_{i}(1-cos\theta)}} \tag{14}\]  
\(\Delta T\) is specified in an initial state, since it determines the rate of drop growth. By applying the Euler integration method, a small increment in radius \(\Delta r_{i}\) is applied to each drop for every time step \(\Delta t\). To maintain coalescence as the dominant mode for drop growth, new drops are continuously generated with a radius size equal to half of the effective radius. Therefore, the rate of generation of new drops in this module is dependent on the area available to them at every time step. New generations of drops are created without overlapping with older drops and domain boundaries. In the next module, drops are checked for coalescence by measuring two distances: the distance between each drop center to another and the sum of their radius. Drops overlapping is checked based on the three-dimensional domain when the following condition is met [19]:  
\[\sqrt{\left(x_{i}-x_{j}\right)^{2}+\left(y_{i}-y_{j}\right)^{2}}\leq\sqrt{ \left(r_{i}+r_{j}\right)^{2}-\left(r_{i}-r_{j}\right)^{2}\cos^{2}\theta} \tag{15}\]  
where \(i\
eq j\). Figure 2 simply illustrates the coalescence between two drops. Overlapped drops are deleted, and a new larger drop is formed at the center of the mass of overlapped droplets and a volume equal to the sum of their volumes. The new drop location and size are updated to check for further overlapping until no coalescence is detected. Figure 2: Demonstration of DWC process: (**a**) coalescence and merging of drops and (**b**) drop generation on renewed region. Dropwise and filmwise surfaces are integrated, and surface renewal is dependent on drops merging with neighboring films, as shown in Figure 2. It is assumed that the film region height is larger than the assigned maximum radius. Merging is detected when Equation (13) is met with either side of dropwise boundaries. When a drop size reaches the maximum radius, it departs from its origin and sweeps all drops beneath it. This is simulated by deleting any drop that reaches or exceeds the maximum radius as well as deleting all drops completely and partially located within the swept area.