# Determination of the Operating Temperature of the Gas-Insulated Transmission Line +
## 3 Numerical Example  
Based on the derived formulae, the temperature distribution in the high-current buscduct depicted in Figure 1 was calculated. The geometrical parameters were as follows: \(R_{1}\) = 12 mm, \(R_{2}\) = 15 mm, \(R_{3}\) = 23 mm, \(R_{4}\) = 25 mm and \(d\) = 60 mm. Both the phase conductors and the screen were assumed to be aluminum, which has an electric conductivity of \(\gamma\) = 33 MS m\({}^{-1}\). The frequency was 50 Hz. Currents in the phase conductors were \(I\)\({}_{1}\) = 300A, \(I\)\({}_{2}\) = 300 exp\([-\,\mathrm{j}\)\(\frac{2}{3}\)\(\pi]\)A, \(I\)\({}_{3}\) = 300 exp\([\,\mathrm{j}\)\(\frac{2}{3}\)\(\pi]\)A. The ambient temperature was \(T_{\mathrm{o}}\) = 24 \({}^{\circ}\)C. The length of the buscduct system was assumed to be \(l\) = 3900 mm. The results of the calculations are shown in Table 3. \begin{table}
\begin{tabular}{c c c} \hline
**Range of Applicability** & \(C\) & \(n\) \\ \hline \(10^{-10}<\mathrm{Ra}_{m}\leq 10^{-2}\) & 0.675 & 0.058 \\ \(10^{-2}<\mathrm{Ra}_{m}\leq 10^{2}\) & 1.020 & 0.148 \\ \(10^{2}<\mathrm{Ra}_{m}\leq 10^{4}\) & 0.850 & 0.188 \\ \(10^{4}<\mathrm{Ra}_{m}\leq 10^{7}\) & 0.480 & 0.250 \\ \(10^{7}<\mathrm{Ra}_{m}\leq 10^{12}\) & 0.125 & 0.333 \\ \hline \end{tabular}
\end{table} 
Table 1: Coefficients \(C\) and \(n\) for the horizontal cylinders [33]. \begin{table}
\begin{tabular}{c c c c c} \hline
**T** & \(\mathbf{c_{P}}\) & \(\lambda\) & \(\eta\)**-\(\mathbf{10^{-6}}\)** & \(\mathbf{\
u\)**-\(\mathbf{10^{-6}}\)** \\ \hline \({}^{\circ}\)C & \(\frac{\mathrm{J}}{\mathrm{kgK}}\) & \(\frac{\mathrm{W}}{\mathrm{mK}}\) & Pa-s & \(\frac{\mathrm{m^{2}}}{\mathrm{s}}\) \\ \hline
0 & 1005 & 0.0244 & 17.16 & 13.28 \\ \hline
10 & 1005 & 0.0251 & 17.65 & 14.16 \\ \hline
20 & 1005 & 0.0259 & 18.14 & 15.06 \\ \hline
30 & 1005 & 0.0267 & 18.63 & 16.00 \\ \hline
40 & 1005 & 0.0276 & 19.12 & 16.96 \\ \hline
50 & 1005 & 0.0283 & 19.61 & 17.95 \\ \hline
60 & 1005 & 0.0290 & 20.10 & 18.97 \\ \hline
70 & 1009 & 0.0297 & 20.59 & 20.02 \\ \hline
80 & 1009 & 0.0305 & 21.09 & 21.09 \\ \hline
90 & 1009 & 0.0313 & 21.48 & 22.10 \\ \hline
100 & 1009 & 0.0321 & 21.87 & 23.13 \\ \hline
120 & 1009 & 0.0334 & 22.85 & 25.45 \\ \hline
140 & 1013 & 0.0349 & 23.73 & 27.80 \\ \hline
160 & 1017 & 0.0364 & 24.52 & 30.09 \\ \hline
180 & 1021 & 0.0378 & 25.30 & 32.49 \\ \hline
200 & 1026 & 0.0393 & 25.99 & 34.85 \\ \hline \end{tabular}
\end{table} 
Table 2: Thermal properties of air at 1 atm [34]. Figure 4: Time variation of the phase conductor temperature: measurements (L1, L2, L3: phase conductors; \(T_{o}\): ambient temperature). Figure 5: Time variation of the enclosure temperature – measurement (E1, E2, E3: enclosure temperatures; \(T_{o}\): ambient temperature). Figure 6: Temperature distribution in the three-phase high-current b  
To carry out the simulation in the COMSOL software, the electromagnetic heating interface, using magnetic fields and heat transfer in solids modules, was used. The values of the material parameters, such as conductivity, magnetic permeability, etc., were taken from the material library available in the COMSOL software. For aluminum, they are \(\gamma\) = 33 MS\(\cdot\)m\({}^{-1}\), \(\mu_{0}=4\pi 10^{-7}\frac{H}{m}\) and \(\lambda=200\frac{W}{mK}\). The radiation was taken into account by setting the boundary condition, known as diffuse surface, on the edges of the areas and by introducing the emissivity coefficient on these surfaces (\(\varepsilon\) = 0.6). In turn, a natural convection was taken into account by applying the heat transfer in fluid condition to the areas of gas presence and setting the velocity field option at 0.01 m/s along the y axis. Physics-controlled mesh was used. Elements size was set as finer. For a more accurate analysis, the temperature distributions in phases L1 and L2 (according to Figure 7), are presented in Figures 8 and 9, respectively. The experimental tests presented in this section confirmed the correctness of the analytical method of calculating temperature in high-current busducts.