# Engineering of a FGM Interlayer to Reduce the Thermal Stresses Inside the PFCs
## 3 Conceptual Design of a Steel-Based PFC with a Engineered W/Steel FGM Interlayer  
In this section, we apply the methodology just described to a more realistic conceptual design of a PFC having a steel plate with grooved coolant channels as heat sink. The main differences from the simple ideal case lie in the fact that now the body will be multiply connected and the temperature field \(T\) will not depend only on the plate thickness. These hypothesis would result in a total strain tensor that has not anymore equal components along the \(x\) and \(z\) directions. Another issue that will be solved is the discretization of the ideal FGM in a multi-layer FGM having discrete concentrations.  
The chosen layout for the PFC is the flat-tile configuration, with the heat sink being a steel plate with squared conduits for the water. The geometry of the component is sketched in Figure 3. The choice of the squared shape of the coolant channel is to lower the spread between the water bulk temperature, which is a good approximation of the mean temperature of the heat sink, and the maximum temperature of the steel domain. In such a way, one minimizes the thermal stresses due to the gradient since they depend on the difference between the local temperature and the average temperature of the body \(\sim(T-T_{m})\), as we discussed previously.  
The materials used for the design are AISI316L for the heat sink and Plasma Sprayed tungsten for the armor. This austenitic steel, which has a very low thermal conductivity, is characterized by low costs and high availability. Moreover, the choice of this material for the heat sink is also for testing the robustness of the proposed design methodology and assessing its capability of designing a PFC with such a thermally insulating material (\(\lambda\sim 15\) Wm\({}^{-1}\)K\({}^{-1}\)). Plasma Spraying is a coating technique able to deposit millimeters-thick coatings with high density [23]. The material properties of AISI316L and of W will be taken from [6]. The PFM properties are chosen to be the one of bulk W, with the exception of the thermal conductivity, which must be modified to reproduce the performance of the plasma sprayed coating. In fact, it has been observed experimentally that the conductivity of plasma sprayed W is far below the one of the hot-rolled material. In particular, measurements show [8; 10] that this thermophysical property is, for the coating, very similar to the one of steel (\(\lambda\sim 15\) Wm\({}^{-1}\)K\({}^{-1}\)). Therefore, due to the lack of a specific database, one will consider the thermal conductivity of the armor to be equal to that of the heat sink. Such a hypothesis is also useful to decouple the thermal problem from the thermomechanical one. In this way, the temperature field will not depend on the FGM concentration function. As mentioned before, the coupled problem can be easily solved with an iterative algorithm, converging on both the FGM thickness and concentration. However the implementation of such numerical scheme is outside the scope of this paper.  
The design inputs we chose for this study are the following:  
* Heat load coming from the plasma \(q=0.5\) MWm\({}^{-2}\)
* Operative coolant pressure \(p=4\) MPa
* Operative water bulk temperature \(T_{water}=60\)\({}^{\circ}\)C
* Heat transfer coefficient of convection \(h=50\) kWm\({}^{-2}\)K\({}^{-1}\)
* Stress free temperature \(T_{0}=20\)\({}^{\circ}\)C
* Thickness of the tungsten armor \(\delta_{1}=2\) mm  
Which correspond to possible boundary conditions for a first wall module where the impinging load is mainly due to radiation and little to no charged particle irradiation is present. The heat load \(q\) is considered constant on the whole plasma-facing surface.