# Experimental Investigation of Thermal Contact Conductance in a Bundle of Flat Steel Bars
## 3 Calculations and Results  
The values of contact thermal conductance \(h_{\textit{ct}}\) of the tested samples has been calculated on the basis of the analysis of thermal resistances. In order to do it the notion of the elementary cell has been used. Elementary cells of the tested samples constitute the halves of two adjacent layers of bars, which can be seen in Figure 5 (they have been marked with a broken white line). These cells are the smallest repeated parts of the considered medium. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sample** & \(\mathbf{B_{1}}\) & \(\mathbf{B_{2}}\) & \(\mathbf{R^{2}}\) \\ \hline
5 \(\times\) 20 TR & 0.0013 & 1.87 & 0.988 \\
5 \(\times\) 20 PA & 0.0027 & 3.01 & 0.987 \\
5 \(\times\) 20 MI & 0.0021 & 2.08 & 0.978 \\
10 \(\times\) 40 TR & 0.0016 & 2.42 & 0.988 \\
10 \(\times\) 40 MI & 0.0034 & 3.07 & 0.987 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The values of coefficients \(B_{1}\), \(B_{2}\) and \(R^{2}\) from Equation (4) obtained for individual samples. Figure 4: Measured effective thermal conductivity as a function of temperature: (**a**) results obtained for samples made of 5 \(\times\) 20 mm bars; (**b**) results obtained for samples made of 10 \(\times\) 40 mm bars. With the assumption that in the cell occurs a unidimensional vertical heat transfer, the total thermal resistance of the cell can be calculated as a sum of: conduction thermal resistance in the lower layer of bars, joint thermal resistance and conduction thermal resistance in the upper layer of bars:  
\[R_{tot}=R_{cdI}+R_{j}+R_{cdII}, \tag{7}\]  
where,  
\[R_{cdI}=\frac{l_{I}}{k_{st}}, \tag{8}\]  
\[R_{j}=\frac{1}{h_{j}}, \tag{9}\]  
\[R_{cdIII}=\frac{l_{II}}{k_{st}}, \tag{10}\]  
The values of the \(l_{I}\) and \(l_{II}\) dimensions corresponding to the individual samples have been collated in Table 3.  
Joint thermal conductance \(h_{j}\) which appears in Equation (9) expresses quantitatively the heat transferred in the joint between the adjacent layers of the bed. Using the definition of the heat conduction resistance for a flat layer in relation to an elementary cell it is possible to note [22]:  
\[R_{to}=\frac{l_{cl}}{k_{ef}}, \tag{11}\]  
Figure 5: Elementary cells defined for the tested samples: (**a**) parallel (PA); (**b**) mixed (MI); (**c**) transverse (TR). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Sample** & \(l_{L}\)**m** & \(l_{II}\)**m** \\ \hline
5 \(\times\) 20 TR & 0.0025 & 0.0025 \\
5 \(\times\) 20 PA & 0.0100 & 0.0100 \\
5 \(\times\) 20 MI & 0.0100 & 0.0025 \\
10 \(\times\) 40 TR & 0.0050 & 0.0050 \\
10 \(\times\) 40 MI & 0.0200 & 0.0050 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The values of \(l_{I}\) and \(l_{II}\) dimensions adopted for individual samples. where,  
\[l_{cl}=l_{I}+l_{j}+l_{II}, \tag{12}\]  
The \(l_{j}\) dimension which appears in Equation (12) indicates the mean width of the joint. Based on the measurements made with the use of a micrometer it has been established that for the tested samples the value of this parameter ranges from 0.03 to 0.1 mm. In the performed calculations it has been assumed that \(l_{j}\) = 0.07 mm. After rearranging Equation (7) and taking into account dependences (8)-(11) it can be noted:  
\[h_{j}=\left(\frac{l_{cl}}{k_{ef}}-\frac{l_{I}}{k_{st}}-\frac{l_{II}}{k_{st}} \right), \tag{13}\]  
Using Equations (3) and (4) for each sample the changes of the \(h_{j}\) conductance in the temperature function have been calculated. The results of these calculations are presented in Figure 6. The calculations take into account the fact that both the \(k_{ef}\) coefficient and the \(k_{st}\) coefficient are burdened with a 5% uncertainty. As a result of this assumption the uncertainty of the \(h_{j}\) value is also 5%, which has been marked in the diagrams. As can be seen, joint conductance as well as effective thermal conductivity for all the samples increases linearly in the temperature function.