## 2 Numerical Method and Calculation Procedure
### The Nano-Structure Model  
In the present study, we coupled the nano-structure heat transfer to the heat transfer caused by the micro-layer. As shown in Figure 3, one can imagine that the micro-layer exists on the nano-structure, and the micro-layer is fed by the wicked fluid inside the nano-structure. The micro-layer's mass flux change rate caused by the wicked fluid can be calculated as follows:  
\[\dot{m}^{\prime\prime}_{wick}=\rho_{l}\dot{V}^{\prime\prime}_{wick} \tag{17}\]  
where \(\dot{V}^{\prime\prime}_{wick}\) is the velocity of the wicked fluid.  
As shown in Figure 3, nano-structure is divided into two parts, i.e., the heat conduction-dominated domain and the wicked fluid-dominated domain. The height of the two domains' interfaces is \(h_{c}\). The time required for heat to be transferred from the heating surface to a thickness of \(h\) can be calculated by:  
\[\tau_{cond}=\frac{h^{2}}{2\alpha_{porous}} \tag{18}\]  
where \(\tau_{cond}\) is heat transfer time under thermal conduction, \(\alpha_{porous}\) is the effective thermal diffusivity of the nano-structure, which can be calculated as follows:  
\[\alpha_{porous}=\lambda_{porous}/\rho_{porous}C_{p,porous} \tag{19}\]  
The effective thermal conductivity of the porous nano-structure can be achieved using Equation (20) [56], and the equivalent density and specific heat capacity are obtained based on the weighted average of fluid and solid according to the porosity \(\epsilon\).  
\[\lambda_{\text{porous}}=\lambda_{f}\Bigg{(}\frac{\lambda_{s}}{\lambda_{f}} \Bigg{)}^{0.28-0.757\text{log}-0.057\text{log}(\frac{3k}{\lambda_{f}})} \tag{20}\]  
Figure 3: Schematic of the heat and mass transfer through nano-structures.  
Assuming that the wicked path of the fluid is a semicircular arc with a radius of the thickness of the region (\(\delta_{nano}-h\)), the wicked fluid flow time \(\tau_{wick}\) is calculated using the following equation:  
\[\tau_{wick}=\frac{\pi(\delta_{nano}-h)}{\dot{V}^{\prime\prime}_{wick}} \tag{21}\]  
In the domain close to the heating surface, thermal conduction dominates, and the thermal transfer time through conduction is much shorter than the thermal transfer time through the wicked fluid (\(\tau_{cond}<<\tau_{wick}\)). However, the opposite is true in the domain close to the surface of the nano-structure. Considering the continuity of heat transfer, \(\tau_{cond}\) must be equal to \(\tau_{wick}\) at the interface between two domains (\(h=h_{c}\)). Thus, we can obtain the relationship shown as Equation (22):  
\[\frac{h_{c}^{2}}{2\alpha_{porous}}=\frac{\pi(\delta_{nano}-h_{c})}{\dot{V}^{ \prime\prime}_{wick}} \tag{22}\]  
The model of \(\dot{V}^{\prime\prime}_{wick}\) is shown in Equation (23) [40], which considers the fluid kinematic viscosity \(v_{porous}\), the static contact angle of the fluid on the nano-structure \(\theta\), and the nano-particle's radius \(r_{p}\). The static contact angle \(\theta\) is regarded to be zero since the nano-structure in this study is super hydrophilic. The appearance of the adopted nano-structure is introduced later.  
\[\dot{V}^{\prime\prime}_{wick}=\frac{4\rho_{0}\tau(\delta_{nano}-h_{c})v_{porous}}{\rho_{l}r_{p}^{2}}\left[-1+\sqrt{1+\frac{ \sigma\text{cos}\theta r_{p}^{3}}{\rho_{0}\pi^{2}(\delta_{nano}-h_{c})^{2}8v _{porous}^{2}}}\right] \tag{23}\]  
Substituting Equation (23) into Equation (22), the \(h_{c}\) is the only parameter to be solved. Then, the wicked fluid velocity and mass flux change rate can be finally achieved.  
After obtaining the wicked fluid mass flux change rate \(m^{w}_{wick}\), the heat flux caused by the wicked fluid can be calculated as follows:  
\[q^{\prime\prime}_{wick}=\frac{2\pi r_{f}(\delta_{nano}-h_{c})m^{w}_{wick}h_{lv} }{4r_{f}^{2}} \tag{24}\]  
where \(r_{f}\) is the radius of the bubble footprint. The \(2\pi r_{f}(\delta_{nano}-h_{c})\) is the circulation area of the wicked fluid, and the \(4r_{f}^{2}\) scales the area of influence of each bubble [57].