# A Review of Research on Dropwise Condensation Heat Transfer
## 3 Dropwise Condensation Heat Transfer Model
### Surface Integral Heat Transfer Model  
In the previous mathematical modeling process of the dropwise condensation, most refer to the nucleation sites theory. They make assumptions of the surface between droplets, think that it is a bare surface, and regard the heat transfer process as the convective heat transfer process of the plate. According to the Nusselt theory [18], \(Nu_{x}=0.664 Re_{x}^{1/2}Pr^{1/3}\) (\(Re<5\times 10^{5}\)) and formula \(Nu_{x}=h_{F}x/\lambda\), the surface convective heat transfer coefficient \(h_{F}\) can be obtained. The heat transfer rate of the bare surface can be calculated as  
\[\Phi_{F}=A_{F}h_{F}\Delta\mathrm{T} \tag{1}\]  
where \(A_{F}\) is the exposed area of condensation surface, m\({}^{2}\), and \(\Delta\mathrm{T}\) is the temperature drop between steam and surface, K.  
For the calculation of the heat transfer rate of the surface covered by the droplets, a one-dimensional heat transfer model of the arbitrary radius droplet can be established. The statistical method (population balance) is introduced to predict the droplet size distribution of the surface, that is, the number of droplets of radius r on the surface, and then the heat transfer rate through all droplets on the surface can be expressed as  
\[\Phi_{D}=A_{D}(\int_{r_{min}}^{r_{e}}q_{d}(r)n(r)dr+\int_{r_{e}}^{r_{max}}q_{d }(r)N(r)dr) \tag{2}\]  
where \(n(r)\), \(N(r)\) are the population density of small drops and large drops, and as shown by Equations (8) and (9) respectively, m\({}^{-3}\). \(q_{d}(r)\) is the heat transfer rate of a droplet and is derived by building a heat transfer model through a single droplet as shown by Equation (7), W. \(A_{d}\) is the area covered by droplets on the condensing surface, m\({}^{2}\). For the limits of integration, \(r_{min}\), \(r_{e}\), \(r_{max}\) are the minimum radius, the effective radius, and the maximum radius of the droplet, respectively. These three radii play an important role in the calculation of dropwise condensation heat transfer. It is affected by the condensation temperature, the physical properties of the condensate, and the properties of the condensing surface. 1. Minimum Radius  
For a given surface undercooling, there is an ideal minimum droplet size of the condensing surface. In the nucleation stage, condensation and evaporation of vapor molecules form on the surface of the embryo. Only when the drops reach a certain size and the condensation rate is higher than the evaporation rate, can it finally grow [19]. The minimum droplet radius can be expressed by the following equation [20]. \[r_{min}=\left(\frac{2T_{S}\sigma}{H_{fg}\rho}\right)\frac{1}{\Delta\mathrm{T}} \tag{3}\]  
where \(T_{S}\) is vapor saturation temperature, K. \(\sigma\) is the surface tension of the drop, N/m. \(H_{fg}\) is the latent heat of condensation, J/kg. \(\rho\) is the density of the liquid, kg/m\({}^{3}\). 2. Effective Radius  
According to the nucleation sites theory of dropwise condensation, two kinds of drop growth form on the surface. One is small drops that are directly condensed by vapor; the other is large drops that grow by coalescing small drops. The effective droplet radius for determining the droplet size can be obtained by the following [21]:  
\[r_{e}=\sqrt{1/4N_{s}} \tag{4}\]  
where \(N_{s}\) is the number of nucleation sites per unit area on the condensing surface. 3. Maximum Radius  
[PERSON] [22] analyzed the force on droplets on the vertical surface, comprehensively considered the influence of contact angle, contact angle hysteresis, and the drop physical properties, and then obtained the droplet departure radius as follows:  
\[r_{max}=\left(\frac{6(cos\theta_{r}-cos\theta_{a})sin\theta}{(2-3 cos\theta+ cos^{3}\theta)}\frac{\sigma}{\rho g}\right)^{1/2} \tag{5}\]  
where \(\theta\) is the contact angle, deg, \(\theta_{r}\), \(\theta_{a}\) are the receding contact angle and advancing contact angle of the droplet, respectively, and \(g\) is the acceleration of gravity, m/s\({}^{2}\).