# Experimental Investigation of Thermal Contact Conductance in a Bundle of Flat Steel Bars
## 2 Materials and Methods
Due to the cooler shift, this element acts in the stand as the cold plate. The heat flux \(q\) flowing through the sample was evaluated as a quotient of heat flux rate \(Q\) generated by the main heater and its surface area \(A\). It was assumed, that the value of \(Q\) is equal to the power supply \(P\) of this heater. This assumption was possible because electric resistance heaters are 100% efficient, which results in the fact that all of the electrical energy is transferred into heat [33]. The power supply of the main heater \(P\) was measured using a 3-phase power network meter N14 [34]. An important element of the described tests is the analysis of measurement uncertainties. The total uncertainty of the measurement was estimated from an error propagation equation [35]:  
\[\frac{\delta k_{ef}}{k_{ef}}=\left(\left(\frac{\delta P}{P}\right)^{2}+\left( \frac{\delta A}{A}\right)^{2}+\left(\frac{\delta l_{sp}}{l_{sp}}\right)^{2}+ \left(\frac{\delta\Delta t}{\Delta t}\right)^{2}\right)^{0.5}, \tag{2}\]  
Figure 2: Testing stand: (**a**) general view: 1—heating chamber, 2—control unit of main and guarded heaters, 3—data logger with temperature meter, 4—autotransformer, 5—unit of cooling system; (**b**) scheme of the heating chamber: 1—retort with a hot plate, 2—investigated sample, 3—cold plate, 4—heating chamber cover with a cooler, 5—side guarded heater, 6—main heater, 7—bottom guarded heater, 8—thermal insulation, 9—support structure. The maximal measurement uncertainty of the effective thermal conductivity at the used experimental stand was 4.6% [13]. The tests encompassed the samples with three types of bar arrangement, which can be seen in Figure 3a-c. Taking into account bar arrangement in relation to the direction of heat flow, these samples have been denoted as the following: transverse TR, parallel PR and mixed ML. Five samples have been tested altogether--three samples of 5 \(\times\) 20 mm bars and two samples of 10 \(\times\) 40 mm bars. Figure 3d presents the view of one of the samples during placing in the heating chamber of the stand. Each sample, due to the individual geometry, was characterized by a different value of the \(l_{sp}\) dimension. Table 1 shows the \(l_{sp}\) values, number of layers and number of bars for individual samples. In order to prepare the samples, bars from low-carbon steel with the carbon content of 0.2% were used. The change in the thermal conductivity of such steel in the temperature function (where temperature is expressed in \({}^{\circ}\)C) is described by the following relationship [36]:  
\[k_{st}=1.24\cdot 10^{-8}t^{3}-3.26\cdot 10^{-5}t^{2}-1.19\cdot 10^{-2}t+51.35, \tag{3}\]  
The results of the measurement of effective thermal conductivity are presented in the form of diagrams. Figure 4a presents the results obtained for samples made of 5 \(\times\) 20 mm bars, whereas Figure 4b shows the results for samples made of 10 \(\times\) 40 mm bars. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Sample**} & **Number of Layers** & \multirow{2}{*}{\(l_{sp}\)} & **Number of Bars** \\  & **in the Sample** & & **in the Sample** \\ \hline
5 \(\times\) 20 TR & 12 & 60 mm = 12 \(\times\) 5 mm & 228 \\
5 \(\times\) 20 PA & 4 & 80 mm = 4 \(\times\) 20 mm & 315 \\
5 \(\times\) 20 MI & 5 & 80 mm = 4 \(\times\) 5 mm + 3 \(\times\) 20 mm & 313 \\
10 \(\times\) 40 TR & 8 & 80 mm = 8 \(\times\) 10 mm & 72 \\
10 \(\times\) 40 MI & 5 & 110 mm = 3 \(\times\) 10 mm + 2 \(\times\) 40 mm & 105 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The values of \(l_{sp}\) and number of layers and number of bars for individual samples. Figure 3: Investigated samples: (**a**) transverse sample; (**b**) parallel sample, (**c**) mixed sample; (**d**) one of the samples during placing in the heating chamber of the stand. As shown, the value of \(k_{\textit{ef}}\) coefficient depends on the bar dimensions and their arrangement. The lower the compaction of the layers of the bars on a unitary length, the bigger this parameter becomes.