## 3 Liquid Film Distribution Model  
### Liquid Film Thickness in the Condensation Section  
To further understand the mechanism of inclination angle, the thickness of the liquid film was calculated by a mathematic model in this study. The condensation model in heat pipes is different from the well-known Nusselt model [49]. The large shear stress induced by high speed vapour flow is considered in this heat pipe condensation model [50]. The model of heat pipe condensation liquid film distribution could be derived as follows. The derivation is based on perfect liquid distribution without considering the uneven thickness caused by entrained droplets. According to the conservation law of momentum and energy, we have the following governing equations:  
\[u_{1}\frac{\partial^{2}\mathrm{u}}{\partial\mathrm{y}^{2}}+(\rho_{1}-\rho_{ \mathrm{v}})\mathrm{g}^{\prime}=0 \tag{10}\]  
\[\frac{\partial^{2}\mathrm{T}}{\partial\mathrm{y}^{2}}=0 \tag{11}\]  
The boundary conditions are:  
\[\mathrm{u_{1}}=0,\mathrm{T}=\mathrm{T_{wc}}\;\;\mathrm{at}\;\;\mathrm{y}=0, \tag{12}\]  
\[u_{1}\bigg{(}\frac{\partial\mathrm{u_{1}}}{\partial\mathrm{y}}\bigg{)}=-\tau_ {\mathrm{i}},\;\mathrm{T}=\mathrm{T_{s}}\;\;\mathrm{at}\;\mathrm{y}=\delta, \tag{13}\]  
where \(\mathrm{u}\) is the velocity; \(\mathrm{T_{wc}}\) is the wall temperature; \(\mathrm{T_{s}}\) is the vapour temperature; \(\tau_{\mathrm{i}}\) is the shear stress; \(\delta\) is the liquid film thickness; and \(\mathrm{y}\) is the position at normal direction of the wall.  
In Equations (10), \(\mathrm{g}^{\prime}\) is the corrected gravitational acceleration, which includes the effect of inclination angle and capillary pressure. The determination of \(\mathrm{g}^{\prime}\) could be solved by the below equation:\[\rho g^{\prime}L_{eff}=\rho gsin\beta L_{eff}+P_{c} \tag{14}\]  
where L\({}_{eff}\) is the effective length of the heat pipe, \(\beta\) is the heat pipe inclination angle, and P\({}_{c}\) is the capillary pressure generated by the wick.  
From equations (10)-(13), the following relationship could be expressed:  
\[u_{l}=-\frac{(\rho_{l}-\rho_{v})g^{\prime}}{u_{l}}\frac{y^{2}}{2}+\frac{(\rho_ {l}-\rho_{v})g^{\prime}\delta-\tau_{i}}{u_{l}}y \tag{15}\]  
\[u_{i}=\frac{(\rho_{l}-\rho_{v})g^{\prime}}{u_{l}}\frac{\delta^{2}}{2}-\frac{ \tau_{i}}{u_{l}}\delta \tag{16}\]  
\[G=\frac{\rho_{l}(\rho_{l}-\rho_{v})g^{\prime}}{3u_{l}}\delta^{3}-\frac{\rho_{ l}\tau_{i}}{2u_{l}}\delta^{2} \tag{17}\]  
\[\left(\frac{3}{32}C_{1}+\frac{1}{4}C_{2}\right)C_{3}\delta^{4}-\frac{\tau_{i} }{u_{l}\
u_{i}}\left(\frac{1}{9}C_{1}+\frac{1}{3}C_{2}\right)\delta^{3}=x \tag{18}\]  
The C\({}_{1}\), C\({}_{2}\), and C\({}_{3}\) are defined as:  
\[C_{1}=\frac{\rho_{l}\gamma_{l}c_{p}}{k_{l}},\ C_{2}=\frac{\rho_{l}\gamma_{l}h _{fg}}{k_{l}(T_{s}-T_{wc})}\,\ C_{3}=\frac{(\rho_{l}-\rho_{v})g^{\prime}}{u_{l}\
u_{l}},\]  
where \(u_{i}\) is the interfacial velocity, \(\Gamma\) is the mass flow rate per unit width of liquid film, \(\
u_{l}\) is the kinematic viscosity, and \(k_{l}\) is the thermal conductivity of liquid.  
The shear stress \(\tau_{i}\) is calculated by the equation:  
\[\tau_{i}=\tau_{f}+\tau_{m}=\frac{c_{f}}{2}\rho_{v}|u_{i}-u_{v}|^{2}+\frac{k_{ l}(T_{s}-T_{wc})}{h_{fg}\delta}|u_{i}-u_{v}| \tag{19}\]  
where \(\tau_{f}\) is shear stress due to friction force, \(\tau_{m}\) is shear stress due to mass transfer, and \(c_{f}\) is the frictional coefficient.  
The velocity of vapour is calculated by:  
\[u_{v}=\frac{4G}{\rho_{v}D_{v}} \tag{20}\]  
where \(D_{v}\) is the hydraulic diameter of vapour channel.  
The heat transfer rate of condensation is expressed as:  
\[Q_{c}=(T_{s}-T_{wc})\pi D_{v}\int hdx \tag{21}\]  
where the local heat transfer coefficient determined by liquid film thickness is calculated by: h = \(\lambda_{l}/\delta\).