## 3 FEM Model of the Rotating Target
### Boundary Conditions  
The rotating target, during its operation, is subject to different types of actions: centrifugal force, pressure gradient, and variable thermal load.  
However, due to the presence of the ion beam, the following temperature field was analytically and numerically estimated on the surface of the target [6] taking into account:  
* presence of boiling water inside the device with a saturation temperature of 50 \({}^{\circ}\)C;
* working pressure inside the component of 0.1 bar;
* boiling water heat transfer coefficient (analytically estimated with Rohsenow correlation [18]) of about 18,165 W/m\({}^{2}\)/K;
* target thickness of 10 mm of aluminum alloy in the evaporator region;  
Figure 7: Target Mesh details.  
the density of the aluminum is to 2.7 g/cm\({}^{3}\), the thermal conductivity of the aluminum alloy is to 170 W/m/K, and the heat capacity of the material is equal to 900 J/kg/K;  
Thanks to all these assumptions, a transient RANS thermal-hydraulic simulation was performed on the heated region of the rotating target [6] to highlight the periodic temperature profile on the external surface of the evaporator to adopt as BC for the mechanical simulations. The time step selected for the simulation grants a Courant number lower than 1 (0.82) for the stability of the numerical simulation. The contour plot of the thermal field on the external surface of the rotating target is reported in Figure 8; the maximum temperature, about 135 \({}^{\circ}\)C, was located in the area exposed to the beam spot while a kind of \"comet effect\" was highlighted downstream of it thanks to the cooling effect of the boiling water inside. The cold spot of about 98 \({}^{\circ}\)C was located at the opposite side just upstream of the beam footprint.  
Thanks to these assumptions, the following boundary conditions were fixed for the FEM analyses:  
* the temperature in the zone of incidence of the ion beam T = 135 \({}^{\circ}\)C;
* the temperature on the rotating disc T = 100 \({}^{\circ}\)C;
* the evaporator's internal temperature T = 50 \({}^{\circ}\)C;
* the internal condenser's temperature T = 20 \({}^{\circ}\)C.  
The rotating target during its operation rotates at a constant angular speed of 33.15 rad/s. Further assumptions were (1) the absence of pressure inside, (2) the presence of standard atmospheric pressure outside, and (3) the self-weight of the structure.  
Once the model, nominal loads, and boundary conditions were defined, the mechanical performance of the device was investigated.  
A couple of temperature-displacement procedures were implemented. The thermomechanical analysis allows a nonlinear calculation in which the mechanical stress and temperatures are solved iteratively accounting for the reciprocal action of the temperature on the stress and vice-versa.  
Steady-state thermal analysis for evaluating the temperature at the thermal equilibrium of the system on the surface where subject to constant heat loads and environmental conditions over time was set.  
The impinging ion beam source was modelled as a surface heat flux and was applied on the external surface of the evaporator.  
Figure 8: Temperature distribution on the lateral surface of the rotating target [6].  
A cylindrical coordinate system was adopted to consider the rotation of the target concerning the ion beam and to define the variable impact area:  
\[x_{cord}=R\cos(\vartheta) \tag{4a}\] \[y_{cord}=R\sin(\vartheta)\] (4b) \[Z_{cord}=Z_{center} \tag{4c}\]  
where \(R\) is the radius of the evaporator, \(\vartheta\) the angle defined by the angular velocity and time, and \(Z_{center}\) is the coordinate along the vertical axis of the center of the circular impact area.  
A subroutine was implemented in the FEM software to define the dimension of the area where the thermal power is deposited by the ion beam source.