# Experimental Investigation of Thermal Contact Conductance in a Bundle of Flat Steel Bars
## 3 Calculations and Results
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sample** & \(\mathbf{B_{5}}\) & \(\mathbf{B_{6}}\) & \(\mathbf{B_{7}}\) & \(\mathbf{R^{2}}\) \\ \hline
5 \(\times\) 20 TR & \(-2.18\times 10^{-4}\) & 0.280 & 383.3 & 0.999 \\
5 \(\times\) 20 PA & \(-2.33\times 10^{-4}\) & 0.184 & 168.7 & 0.996 \\
5 \(\times\) 20 MI & \(-2.18\times 10^{-4}\) & 0.151 & 159.2 & 0.996 \\
10 \(\times\) 40 TR & \(-2.78\times 10^{-4}\) & 0.172 & 237.2 & 0.993 \\
10 \(\times\) 40 MI & \(-2.15\times 10^{-4}\) & 0.150 & 126.0 & 0.996 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The values of coefficients \(B_{5}\), \(B_{6}\), \(B_{7}\) and \(R^{2}\) from Equation (22) obtained for individual samples. Figure 9: Calculated values of contact conductance: (**a**) results obtained for samples made of 5 \(\times\) 20 mm bars; (**b**) results obtained for samples made of 10 \(\times\) 40 mm bars. unambiguously indicate the factors which determine the obtained character of changes of the \(h_{ct}\) coefficient. Another factor which determines thermal contact conductance is contact pressure \(p\). The influence of this parameter on the value of the \(h_{ct}\) coefficient is described by the equation proposed by [PERSON] [38]:  
\[h_{ct}=\frac{1.13k_{m}\sigma^{\prime}}{\sigma_{p}}\bigg{(}\frac{p}{H_{c}} \bigg{)}^{0.94}, \tag{23}\]  
where \(k_{m}\) is the harmonic mean thermal conductivity:  
\[k_{m}=\frac{2k_{1}k_{2}}{k_{1}+k_{2}}, \tag{24}\]  
In case of the tested samples the elements which make up joints have the same thermal conductivity \(k_{st}\), thus:  
\[k_{m}=\frac{2k_{st}^{2}}{k_{st}+k_{st}}=k_{st}, \tag{25}\]  
Using Equation (23) the influence of contact pressure on the value of \(h_{ct}\) has been investigated. Since the values of \(\sigma^{\prime}\) and \(\sigma\) for the tested bars were unknown, the value of the following expression has been determined indirectly:  
\[G_{ct}=\frac{1.13k_{m}\sigma^{\prime}}{\sigma}, \tag{26}\]  
Namely it has been assumed that:  
\[G_{ct}=\frac{h_{ct}}{\bigg{(}\frac{p}{H_{c}}\bigg{)}^{0.94}}, \tag{27}\]  
The tested samples were 0.1 m high. The unit pressure generated by the layer of steel of this height is 7.8 kPa. It has been assumed that the microhardness of steel \(H_{c}\) equals 1130 MPa [43]. Taking into account the above-mentioned values of \(p\) and \(H_{c}\) for each sample the value of \(G_{ct}\) has been calculated, which corresponds to the thermal contact conductance in the temperature of 20 \({}^{\circ}\)C. The values of \(G_{ct}\), which have been determined this way are collated in Table 8.  
Using the obtained values of \(G_{ct}\) parameter the changes of the value of \(h_{ct}\) coefficient in the contact pressure p function have been calculated according to the relationship:  
\[h_{ct}=G_{ct}\bigg{(}\frac{p}{H_{c}}\bigg{)}^{0.94}, \tag{28}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Sample**} & \(\mathbf{h_{ct-min}}\) & \(\mathbf{h_{ct-avg}}\) & \(\mathbf{h_{ct-max}}\) & \\ \cline{2-5}  & **W/(m2 K)** & **W/(m2 K)** & **W/(m2 K)** & \(\delta h_{ct}\), \% \\ \hline
5 \(\times\) 20 TR & 390.1 & 446.3 & 472.5 & 18.4 \\
5 \(\times\) 20 PA & 159.8 & 176.8 & 185.4 & 14.1 \\
5 \(\times\) 20 MI & 173.7 & 195.5 & 205.3 & 15.9 \\
10 \(\times\) 40 TR & 223.7 & 252.7 & 263.5 & 15.4 \\
10 \(\times\) 40 MI & 128.1 & 144.0 & 152.4 & 16.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The minimum, average and maximum values of \(h_{ct}\) obtained for individual samples. The maximum value of \(p\) (amounting to 77.5 kPa) taken into account in the calculations corresponds to the unit pressure generated by a layer of bars with the height of 1 m. The changes of \(h_{ct}\) parameter in the p function for the chosen three samples are presented in Figure 10. As can be seen contact conductance for all the samples is rising linearly in the contact pressure function. However, the dynamics of such a rise for individual samples is highly diversified.