# The Stationary Thermal Field in a Multilayer Elliptic Cylinder
## 5 Conclusions  
* An analytical-numerical method for the determination of the thermal field in a multilayer elliptical system was presented in the paper. An example of application is the analysis of the thermal field in a double-insulation conductor. The eigenfunctions (6) were computed analytically by the superposition of the general and particular solutions of the Poisson Equation (2) [29] and by the variable-separation method. In turn, the coefficients of the eigenfunctions were computed by the iterative solution of the system of Equations (22) and (23) after numerical computation of the integrals in (24)-(30). * no need for area discretization;
- no need for area discretization;
- the possibility of calculating field values at any point in the system and not only at grid nodes;
- Formulas (8), (15) and (16) support the physical interpretation and discussion of the influence of individual parameters. Moreover, they facilitate the determination of scaling laws and the fast estimation of the field at selected points;
- Formulas (8), (15) and (16) enable easy calculation of the sum of series with the necessary limitation of the number of summed terms (due to the calculation time). * The thermal field in the central part of the system (\(0\leq\eta\leq\eta_{1}\)) is almost uniform. This is observed both in the conductor core (Figures 2-4) and in the test model (Figures 5 and 6). The maximum temperature drop in the conductor core is negligible: \(T_{1}(\eta=0,\,\psi=\pi/2)\)\(-\)\(T_{1}(\eta=\eta_{1},\,\psi=0)\)\(=0.034\)\({}^{\circ}\)C. In the test system, the respective temperature drop is \(0.033\)\({}^{\circ}\)C. The uniformity of the field is justified by a very large thermal conductivity in the central part of the system (\(\lambda_{1}=180\)\(W/(\)mK\()\)). * Clear temperature drops occur in regions of lower heat conductivity, compared to the central part (\(\lambda_{2}<<\lambda_{1},\,\lambda_{3}<<\lambda_{1}\)). The drop increases with the increase in insulation thickness. For example, for the conductor, the following occurs: (\(|b_{3}-b_{1}\approx 8.99\) mm \(>a_{3}-a_{1}\approx 6.08\) mm) \(\Rightarrow\)\(\{T_{1}(\eta=\eta_{1},\,\psi=\pi/2)-T_{3}(\eta=\eta_{3},\,\psi=\pi/2)=22.09\)\({}^{\circ}\)C \(>T_{1}(\eta=\eta_{1},\psi=0)-T_{3}(\eta=\eta_{3},\,\psi=0)\)\(=19.99\)\({}^{\circ}\)C})--Figure 1. The above is due to the increase in the thermal resistance of two layers of insulation with their total thickness and to thermal Ohm's law. * The increase in the thermal conductivity of the middle layer \(\lambda_{2}\) (e.g., tenfold) substantially changes the thermal field distribution. For the conductor, the tempera  
Figure 8: The relative differences (34) on the minor semi-axis (\(x=0\)). ture drop in the double insulation is almost linear (Figures 2 and 3). In the test model, the temperature drop is continuous by sections (Figures 5 and 6). In addition, the maximum temperature decreases from 70.01 \({}^{\circ}\)C (Figures 2 and 3) to 58.62 \({}^{\circ}\)C (Figures 5 and 6). This is due to better heat dissipation with the increase in \(\lambda_{2}\). * On the system's outer perimeter (\(\eta=\eta_{3}\)), the temperature decreases from \(T_{3}\)(\(\eta=\eta_{3}\), \(\psi=0\)) = 49.99 \({}^{\circ}\)C to \(T_{3}\)(\(\eta=\eta_{3}\), \(\psi=\pi/2\)) = 47.91 \({}^{\circ}\)C for the conductor, and from 49.61 \({}^{\circ}\)C to 48.22 \({}^{\circ}\)C for the test system. The above proves that the layer boundaries (ellipses \(\eta_{1}\), \(\eta_{2}\), \(\eta_{3}\)) are not isotherms. This is due to the variable distance of the points on the ellipses \(\eta_{1}\), \(\eta_{2}\), \(\eta_{3}\) from the system's center (\(\eta=0\), \(\psi=\pi/2\)). * Figures 7 and 8 show that the relative differences (34) between the temperature distributions (as calculated, respectively, by the method presented in the paper and numerical method) are very small. Therefore, the above-mentioned methods give very similar results. The module of difference (34) is the smallest in the region with a uniform temperature field (\(0\leq\eta\leq\eta_{1}\)).