## 2 Modelling
### Hydraulic Network Model  
Modelling fluid processes pose significant challenges in this study. To overcome this challenge, first, it is necessary to understand the fundamentals of the elements used in the hydraulic network; Figure 3. The modelling of fluid processes necessitates the use of three elements, including fluid storage Capacitance, \(C_{fl}\), used to estimate the pressure variation in \(P_{1}\) due to \(V\); and since the vapour phase is compressible, i.e., \(\frac{\partial\rho_{v}}{\partial t}\
eq 0\), it results in  
\[C_{fl}=\frac{V_{v}}{\gamma R_{gas}T_{v}}, \tag{8}\]  
where \(V_{v}\) is vapour tank volume, \(\gamma\) is vapour adiabatic index, \(R_{gas}\) is specific gas constant and \(T_{v}\) is vapour temperature. Flow inductance \(L\) is used to estimate the \"quantity of motion\", \(\frac{\partial\mu_{v}}{\partial t}\), and it is therefore expressed as  
\[L_{fl}=\frac{L_{eff}}{A_{v}}, \tag{9}\]  
where \(L_{eff}\) is the effective length of the pipe at vapour tank exit, and \(A_{v}\) is the hydraulic area of the transport lines. Lastly, we have the fluidic resistance, \(R_{fl}\), that is obtained from Hagen-Poiseuille and the Darcy equation:  
\[R_{fl}=\frac{8\mu_{v}L_{eff}}{\rho_{v}A_{v}{r_{v}}^{2}}, \tag{10}\]  
where \(L_{eff}\) is the effective length of the pipe at vapour tank exit, \(\mu_{v}\) is the vapour dynamic viscosity, \(\rho_{v}\) is the vapour density, \(A_{v}\) is the hydraulic area, and \(r_{v}\) is the internal radius of the transport lines. The foregoing three basic elements, \(C_{fl}\), \(L_{fl}\) and \(R_{fl}\), are used in the governing equations of energy, momentum and continuity in order to construct an equivalent hydraulic network as illustrated in Figure 3. For the hydraulic network, several fundamental assumptions are to be considered:  
**Assumption 1**. : _Fluid path is considered one-dimensional._  
**Assumption 2**. : _The liquid phase is assumed to be incompressible._  
**Assumption 3**. : _The vapour phase is represented using ideal gas equations._  
**Assumption 4**. : _Vapour temperature and pressure variation are derived from the Clausius-Clapeyron equation._  
**Assumption 5**. : _The flow is assumed to be laminar for both vapour and liquid phases._  
**Assumption 6**. : _Evaporation and condensation phenomena are confined to the evaporator and condenser zones, respectively._  
**Assumption 7**. : _The temperature of liquid present in the Wick section of the heat pipe is assumed to be equal to the temperature of the Wick in that region._  
The liquid and the vapour are divided into two different regions, i.e., liquid tanks and vapour tanks, respectively, as shown in Figure 3. For the vapour, we have three tanks corresponding to the vapour storage in the evaporator grooves, the adiabatic tube and the condenser. There are two vapour paths corresponding to the flow through the vapour tanks. Similarly, three liquid tanks and two liquid paths are assumed for the liquid region. The liquid region and vapour region are connected by two additional paths, which represent evaporation and condensation, using latent heat \(h_{fg}\) expression \(\dot{m}=\dot{Q}_{h}/h_{fg}\). The fluid cycle, represented in Figures 2 and 3, can be summarized in six steps:  
* The evaporator, Node 1, is heated up and the liquid tank \(M_{(7-1)}\) produces vapour; a mass of vapour \(\dot{m}_{1}\) is produced and flows into vapour tank \(M_{(1-2)}\). * Pressure, \(P_{(1-2)}\), increases inside vapour tank \(M_{(1-2)}\) and the mass of vapour, \(\dot{m}_{2}\), is pushed into vapour tank \(M_{(2-3)}\), i.e., representing the adiabatic vapour tube pipe. * Pressure, \(P_{(2-3)}\), increases inside the vapour tank \(M_{(2-3)}\) and the mass of vapour, \(\dot{m}_{3}\), is pushed into the vapour tank \(M_{(3-4)}\), i.e., representing vapour on the condenser tube. * The condenser (Node 4) cools down and a mass of fluid \(\dot{m}_{4}\) is condensed and flows into condenser liquid tank \(M_{(4-5)}\).