## 3 Results and Discussion
### Drop Size Distribution for Hybrid Surfaces  
From the simulation work, values of \(f\) lie between 0.280 and 0.296 for \(R_{D}\) from 0.125 to 1, and an average \(f\) of 0.288 is selected to reflect \(R_{D}\) variation only. Data fitting \(f\) and \(\gamma\) in the form of Equation (26) is performed to outline new sets of exponents \(n\) for different \(r_{e}\) and \(r_{max}\) values.  
The results of drop size distribution \(N(r)\) for DWC within the hybrid surface for \(r_{max}\) of 0.5 and 2.0 mm are illustrated in Figure 10. Results are also compared with Equation (6) for \(n\) equaling \(1/3\) for full DWC when \(r_{e}\) is set at \(1\times 10^{-6}\) m. The main dissimilarity is that hybrid \(N(r)\) profiles show lower logarithmic slopes \(n\) of 0.167 and 0.134 at \(r_{max}\) values of 0.5 and 2.0 mm, respectively. Steeper slopes in hybrid \(N(r)\) reveal not only the intensity of drops' coalescence but also the disappearance of large drops due to surface renewal. More specifically, merging reduces the large drop population, allowing drops of smaller size to  
Figure 9: Total surface renewal rate based on domain area showing the contribution of merging.  
grow without disruption, which is indicated by hybrid profiles of effective radius \(N(r_{e})\), as shown in Figure 10. The change in \(N(r_{e})\) is 73% and 41% for \(r_{max}\) values of 0.5 and 2.0 mm, respectively, when compared with full DWC.  
From Equation (2), the effective radius \(r_{e}\) is dependent on the nucleation number \(N_{s}\) that ranges from \(10^{9}\) to \(10^{13}\) sites/m\({}^{2}\) and determines the population of small drops. Obtained values for \(n\) exponent are used in (9) to define the drop size distribution of small drops \(n(r)\). The impact of \(r_{e}\) variation on hybrid \(N(r)\) and \(n(r)\) is illustrated in Figure 11 for \(r_{max}\) at 0.5 mm. The lower \(r_{e}\) value reflects a steeper logarithmic slope and a lower \(n\) value, which is presented in Figure 11 at \(N(r_{max})\). When coalescence starts at lower \(r_{e}\), it creates more area for more drops to nucleate and accelerates the process of coalescence. The percent change at \(N(r_{max})\) is up to 85% between \(N_{s}\) at \(10^{9}\) and \(10^{13}\) sites/m\({}^{2}\). This is also because surface renewal due to sweeping and merging will result in more dry areas for more nucleation.  
Data regression is performed to produce values for exponent \(n\) in terms of \(r_{max}\) from \(0.5\times 10^{-3}\) to \(2.0\times 10^{-3}\) m and \(r_{e}\) from \(1.6\times 10^{-7}\) to \(5.0\times 10^{-5}\) m, corresponding to \(N_{s}\) from \(1.0\times 10^{13}\) to \(1.0\times 10^{8}\) sites/m\({}^{2}\). The final expression \(n\) for DWC within the hybrid surface is given in the following polynomial form:  
\[n=0.1475+2.135\times 10^{4}\;r_{e}-21.98\;r_{max}-7.375\times 10^{8}-1.418 \times 10^{6}\;r_{e}\;r_{max} \tag{26}\]  
The expression has a sum of the squared estimate of \(3.918\times 10^{-4}\) and a coefficient of determination of 0.9725. In addition to Equation (26), the model results for exponent \(n\) are also given in Figure 12 for different \(r_{e}\) and \(r_{max}\). It generally has lower values than  
Figure 11: Drop size distribution for different \(N_{s}\) and corresponding \(r_{e}\) for given \(r_{max}\) of 0.5 mm.  
Figure 10: Drop size distribution model for different \(r_{max}\) values with comparison when \(n=1/3\)[12] at a given \(r_{e}\) value of \(1\times 10^{-6}\) m.  
\(1/3\) due to the consideration of surface renewal by sweeping and merging. It also reflects the coalescence intensity, which is based on drop nucleation. The simple expression can be utilized in Equations (23)-(25) to provide a better understanding of the drop population and estimate the thermal performance of DWC within hybrid surfaces.