# Model of Complex Heat Transfer in the Package of Rectangular Steel Sections
## 3 Results and Discussion
These results are presented in Figure 10. The total thermal resistance value as well as its changes in the temperature function depend significantly on the section geometry. The larger the sections size, the greater the changes of \(R_{to}\). The values of \(R_{to-\text{min}}\) and \(R_{to-\text{max}}\) obtained for each package for 20 and 700 \({}^{\circ}\)C are presented in Table 2. The minimal and maximal values for all packages were 0.0051 and 0.0238 (m\({}^{2}\cdot\)K)/W, respectively. This means that the resistance \(R_{to}\) varies more than four and a half times. \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Type of the Section**} & \multicolumn{5}{c}{\(\Delta R_{st}\) \%} \\ \cline{2-5}  & **100**\({}^{\circ}\)**C** & **300**\({}^{\circ}\)**C** & **500**\({}^{\circ}\)**C** & **700**\({}^{\circ}\)**C** \\ \hline \(20\times 40\) mm & 1.6 & 2.8 & 8.2 & 16.7 \\ \hline \(40\times 40\) mm & 5.3 & 10.2 & 26.2 & 50.1 \\ \hline \(60\times 60\) mm & 14.1 & 28.0 & 69.6 & 130.5 \\ \hline \(80\times 80\) mm & 33.2 & 67.3 & 157.5 & 293.7 \\ \hline \end{tabular}
\end{table} 
Table 1: The values of the \(\Delta R_{st}\) parameter for individual sections for chosen temperatures. Figure 9: The results of the calculation results of the \(R_{ct}\)/\(R_{st}\) quotient: (**a**) values obtained for \(R_{ct-\text{min}}\) and (**b**) values obtained for \(R_{ct-\text{max}}\). To perform a more detailed analysis, one additional parameter was calculated. This parameter, denoted as \(dR_{to}\), describes percentage changes in the resistance \(R_{to}\) in relation to its mean value and is expressed as:  
\[dR_{to}=\frac{R_{to-\max}-R_{to-\min}}{0.5(R_{to-\max}+R_{to-\min})}\cdot 100\%. \tag{25}\]  
The values of the parameter \(dR_{to}\) are shown in Figure 11. The larger the section's dimensions, the smaller the values of the considered parameter. At the same time, with the increase in dimensions of the sections in the package, we observe smaller changes in \(dR_{to}\) in the temperature function. The values of the parameter \(dR_{to}\) obtained for the extreme temperatures are summarized in Table 3. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Type of the Package**} & \multicolumn{2}{c}{\(R_{to\min}\)**, **(m\({}^{2}\)**-K)/W**} & \multicolumn{2}{c}{\(R_{to\max}\)**, **(m\({}^{2}\)**-K)/W**} \\ \cline{2-5}  & **20 \({}^{\circ}\)C** & **700 \({}^{\circ}\)C** & **20 \({}^{\circ}\)C** & **700 \({}^{\circ}\)C** \\ \hline
20 \(\times\) 40 mm & 0.00583 & 0.00510 & 0.00831 & 0.00646 \\ \hline
40 \(\times\) 40 mm & 0.00787 & 0.00651 & 0.01035 & 0.00787 \\ \hline
60 \(\times\) 60 mm & 0.01369 & 0.00869 & 0.01616 & 0.01004 \\ \hline
80 \(\times\) 80 mm & 0.02129 & 0.00983 & 0.02376 & 0.01119 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Minimal and maximal values of resistance \(R_{to}\) obtained for 20 and 700 \({}^{\circ}\)C.  
Figure 10: The results of model calculations and the measurements of the total thermal resistance of packages of rectangular steel sections: (**a**) sections 20 \(\times\) 40 mm, (**b**) sections 40 \(\times\) 40 mm, (**c**) sections 60 \(\times\) 60 mm, (**d**) sections 80 \(\times\) 80 mm. To evaluate the correctness of the presented model, the computed values of \(R_{to}\) were compared with the experimental results. The experimental values of \(R_{to}\) were measured using a guarded hot plate apparatus [19]. The device works according to the principles described in the standards [15,16]. The measurement data also apply to the same sections that were analyzed in the calculations. The tests consisted of heating flat beds of sections (see Figure 2) with a steady, one-dimensional heat flux rate. The moment the fixed state was reached, the temperature difference \(\Delta t_{s}\) between the outer layers of the sample perpendicular to the direction of heat flow was measured.