## 2 Modelling
### Thermal Network Model  
The loop heat pipe thermal network consists of two distinct elements: thermal conductance (\(G\)) and thermal capacitance (\(C_{th}\)). Thermal conductance can be either _axial_ or emphradial as shown in (1) and (2):  
\[G=\frac{kA}{L},axial, \tag{1}\]  
\[G=\frac{2\pi Lk}{\ln\left(\frac{d_{sat}}{d_{in}}\right)},radial, \tag{2}\]  
Figure 3: Equivalent hydraulic network with nodes and control volumes. Figure 2: Equivalent thermal network with nodes and control volumes. where \(A\) is the cross-sectional area through which heat transfer occurs, \(k\) is the material thermal conductivity, \(L\) is length, and \(d_{out}\) and \(d_{in}\) are the outside and inside diameters, respectively, for the case of the evaporator being a radial element. The thermal conductance values shown in Figure 2, based on the previous assumptions, are as follows:  
* \(G_{(8-1)}\): Represents the thermal conductance between Node 8 and Node 1 relative to the heat source and evaporator, respectively. * \(G_{(1-CC)}\): Represents the thermal conductance between Node 1 and the compensation chamber node. * \(G_{(1-ext)}\): Represents the thermal conductance between Node 1 and the external node, relative to the evaporator and the external environment temperature node. This includes the transport pipeline for the liquid phase. * \(G_{(CC-ext)}\): Represents the thermal conductance between the compensation chamber node and the external environment temperature node. This includes the transport pipeline for the vapour phase. The thermal conductance associated with convection between fluids and surfaces, i.e., \(G_{CC-(6-7)}\) and \(G_{(3-4)-ext}\), is as in (3):  
\[G=hA, \tag{3}\]  
where \(h\) is the convective heat transfer coefficient of liquid or vapour, and \(A_{h}\) is the contact surface area between the fluids. The \(h\) coefficient is an empirical value usually obtained through thermal tests. Finally, we can address the last two thermal conductances that are associated with the Wick and evaporator components, i.e., \(G_{(1-(1-2))}\) and \(G_{(1-(6-7))}\), respectively. In both Gs, it is necessary to understand whether the loop heat pipe is an axial (1) or radial (2) design, and also include convective coefficient \(h\) (3) every time there is a thermal contact between fluid and solid. However, a different approach is required for the Wick effective thermal conductivity, \(k_{w}\), as shown in [31]:  
\[k_{w}=k_{s}\bigg{[}\frac{2+k_{l}/k_{s}-2\varepsilon(1-k_{l}/k_{s})}{2+k_{l}/k _{s}+\varepsilon(1-k_{l}/k_{s})}\bigg{]}. \tag{4}\]  
In (4), \(k_{s}\) is the solid material thermal conductivity, \(k_{l}\) is the thermal conductivity of the working fluid in a liquid phase, and \(\varepsilon\) represents Wick porosity. The thermal capacitances (C\({}_{th}\)) considered for the thermal network from Figure 1 are  
* \(C_{th_{3}}\): Represents the thermal capacitance of evaporator saddle. * \(C_{th_{1}}\): Represents the thermal capacitance of evaporator grooves that are in contact with the Wick. * \(C_{th_{CC}}\): Represents the thermal capacitance of compensation chamber cannister. * \(C_{th_{(6-7)}}\): Represents the thermal capacitance of fluid inside the compensation chamber. These thermal capacitances are then evaluated by  
\[C_{th}=V\rho c_{p}=Mc_{p}, \tag{5}\]  
where \(M\) is the mass, \(V\) is the volume of the given node, \(\rho\) is material density and \(c_{p}\) is material heat capacitance of the associated node. Similarly, in Wick thermal conductance, there is also an effective thermal capacity to be estimated:  
\[\big{(}\rho c_{p}\big{)}_{w}=\varepsilon\big{(}\rho c_{p}\big{)}_{l}+(1- \varepsilon)\big{(}\rho c_{p}\big{)}_{s}. \tag{6}\]The general transient heat conduction equation governing each solid node (\(T_{8}\), \(T_{1}\), \(T_{\text{CC}}\), \(T_{(6-7)}\)) is given by  
\[C_{i}\frac{dT_{i}}{dt}=\pm\sum_{j=1,j\
eq i}^{j(adjacent\ nodes)}G_{ij}|T_{i}-T_{j}| \begin{cases}+\text{\emph{Incoming heat power to node }}i\\ -\text{\emph{Outgoing heat power from node }}i\end{cases}