## 1 Introduction
[30; 31] and has been used to assess the effectiveness and sustainability of DWC. In the first study, the authors investigated the optimal heat flux occurring at widths of 0.6 and 0.3 mm for hydrophobic and less hydrophobic materials at a variety of subcooling temperatures. Under ideal circumstances and due to the increased draining rate, the heat transfer coefficient was higher than that for full FWC and full DWC. The second study [31] showed that with a proper region width, better drop shedding from the hydrophobic region occurred on the hybrid surface, which can be attributed to an increase in capillary force. [PERSON] et al. [32] performed an experimental study measuring heat flux on round surfaces with vertically adjacent hydrophobic and hydrophilic regions of different widths. The overall thermal performance was improved for a smaller maximum radius and a higher population density in the hydrophobic zone. The study achieved a heat transfer enhancement of 23% better than full DWC, which occurred within a hydrophobic region width of 0.55 mm at a total temperature drop of 2 K.  
Other DWC models such as the random fractal model describe drop distribution on the condensing surface and use the Le Fevre and Rose model for heat transfer through a single drop. Drop size distribution in the model agrees with that of the Le Ferve and Rose model [33; 34; 35]. Another model proposed by [PERSON] proposes two correlations for the DWC heat transfer coefficient [36]. The model, which is a modification of the Le Fevre and Rose model, considers contact angle variation with droplet height. Relevant studies and detailed explanations for DWC modeling and simulation work have also been reported [37; 38; 39; 40]. The above discussion of the literature indicates that the importance of predicting thermal performance and understanding drop size distribution is well known. Different approaches have been used in the literature to model drop size distribution; one of them is the establishment of an empirical expression of drop size distribution for a full-DWC surface [12]. The expression is applicable for hemispherical drops with a contact angle of \(\leq\)90\({}^{\circ}\) without the consideration of surface renewal. Despite its limitation, the expression is widely used in modeling heat transfer for full-DWC surfaces, DWC within hybrid surfaces, and for comparison with experimental works. Surface renewal alters drop populationsand creates regions with less thermal resistance. Surface renewal means drops depart from the hydrophobic surface by merging sideways with neighboring film regions or sweeping downward. Thus, modeling drop size distribution considering surface renewal and coalescence intensities will benefit heat transfer modeling for hybrid surfaces. Additionally, previous works on drop size distribution for DWC within a hybrid surface were limited to experimental work, and their modeling was approximated to that of full DWC. Additionally, the reported experimental work was conducted for specific cases of contact angle and effective radius and is not generalizable to the topic of drop size distribution. On the other hand, the reported numerical investigation that dealt with hybrid surfaces is based on the drop size distribution of full dropwise condensation. Additionally, the effects of surface renewal and coalescence intensity were not considered in the previous numerical models. In the present work, the surface renewal and coalescence intensities of DWC within a hybrid surface are modeled and investigated. Additionally, the validity of the current empirical expression of the drop size distribution that is developed for DWC without considering surface renewal and coalescence intensities is examined. The classical exponent (\(n\)) of the drop size distribution profiles is replaced by a polynomial expression in terms of the effective and maximum radii.