## 2 Materials and Methods
The inlet and outlet of both the coolants domains are represented by the black arrows (cooling water) and red arrows (cooling helium). The developed method does not include gas phase transitions and boiling/condensation of the water inside the target, in order to keep computational requirements within standard specifications (a commercial PC). The target cavity was considered to be full of liquid water at constant atmospheric pressure with density 997 kg/m\({}^{3}\) and specific heat capacity at constant pressure of 4181.7 J/(kg.K). Gravity and density differences were considered through buoyancy, and turbulence was modeled by the k-\(\epsilon\) model. Both helium and the cooling water were implemented as a fluid domain, with inlets and outlets defined as shown in Figure 1. The external surface of the target components in contact with the surrounding air was defined with a boundary condition with a constant temperature. On the other hand, the titanium window surface located on the vacuum side was defined as an adiabatic wall. Standard initial conditions for the simulations were established as a reference to provide means of comparison for further simulation results. This latter corresponds to the liquid target system equipped with 35 \(\upmu\)m thin havar foil as an entrance window in order to match the experimental system used by [PERSON] et al. (2016), whose results are further used for validation. These standard initial conditions care presented in Table 2.  
Furthermore, several simulations were performed with distinct initial conditions for the main cooling characteristics of the target system in order to evaluate their influence and to validate the model developed. The parameters with impact on the thermodynamic behavior of the target chosen for these studies were: \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{3}{*}{**Beam Power**} & \multirow{3}{*}{\begin{tabular}{c} **Target** \\ **Window** \\ \end{tabular} } & \multicolumn{3}{c}{**Cooling Water**} & \multicolumn{3}{c}{**Cooling Helium**} & \multirow{3}{*}{**Initial Temperature of the Solid Elements**} \\ \cline{3-3} \cline{5-8}  & & **Temperature** & **Flux** & **Temperature** & **Flux** & & \\ \cline{3-3} \cline{5-8}  & & (c\({}^{\circ}\)C) & **(L/min)** & (c\({}^{\circ}\)C) & **(L/s)** & **Window Spacer** & **Insert Diffuser Support** \\ \hline \multirow{3}{*}{\
\begin{tabular}{c} 1000 W \\ (55.5 \(\upmu\)A) \\ \end{tabular} } & \multirow{3}{*}{
\begin{tabular}{c} 35 \(\upmu\)m thick \\ Havar \\ \end{tabular} } & \multirow{3}{*}{23} & \multirow{3}{*}{3} & \multirow{3}{*}{23} & \multirow{3}{*}{3} & \multirow{3}{*}{3} & \(25\ {}^{\circ}\)C with \(250\ \mathrm{W}\) \\  & & & & & & \(27\ {}^{\circ}\)C with \(500\ \mathrm{W}\) \\ \cline{1-1} \cline{6-6}  & & & & & & \(29\ {}^{\circ}\)C with \(750\ \mathrm{W}\) \\ \cline{1-1} \cline{6-6}  & & & & & & \(31\ {}^{\circ}\)C with \(1000\ \mathrm{W}\) \\ \cline{1-1} \cline{6-6}  & & & & & & (values taken from [16]) & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Standards initial conditions of the simulations. Figure 2: Stopping power of the proton beam along its trajectory (grey curve) and simulated matching power distribution used in the simulations (black curve). * the mass flow rates of the helium and the cooling water;
* the initial temperatures of the helium and the cooling water;
* the current of the proton beam (i.e., the energy deposited). Considering that both the target window and the conical insert are expected to reach significantly high temperatures, it is important to take into account the temperature dependence of the thermodynamic properties of the material involved, such as the specific heat, \(C_{p}\), and thermal conductivity, \(k\). Since the temperature dependence of \(C_{p}\) and \(k\) were not available in the literature for the specific case of havar, both of these properties were estimated based on data for similar cobalt-based alloys [20; 21].