# Experimental Investigation of Thermal Contact Conductance in a Bundle of Flat Steel Bars
## 3 Calculations and Results
Therefore, the calculation results have been approximated with the linear regression functions:  
\[h_{j}(t)=B_{3}t+B_{4}, \tag{14}\]  
The values of the \(B_{3}\), \(B_{4}\) and \(R^{2}\) coefficients obtained for individual samples have been collated in Table 4. In relation to all samples the values of the \(h_{j}\) conductance are within the range from 133 to 603 W/(m\({}^{2}\)-K). The transverse samples are characterized by the biggest values, whereas in case of the samples with the same geometry higher \(h_{j}\) values occur for 5 \(\times\) 10 mm bars. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sample** & \(\mathbf{B_{3}}\) & \(\mathbf{B_{4}}\) & \(\mathbf{R^{2}}\) \\ \hline
5 \(\times\) 20 TR & 0.314 & 388.8 & 0.998 \\
5 \(\times\) 20 PA & 0.207 & 170.5 & 0.998 \\
5 \(\times\) 20 MI & 0.184 & 159.8 & 0.996 \\
10 \(\times\) 40 TR & 0.164 & 242.4 & 1.000 \\
10 \(\times\) 40 MI & 0.186 & 126.3 & 0.996 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The values of coefficients \(B_{3}\), \(B_{4}\) and \(R^{2}\) from Equation (14) obtained for individual samples. Figure 6: Calculated values of the joint 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. The following part of the paper presents an attempt of a qualitative analysis, which consists in investigating the share of particular kinds of heat exchange in the joints. When two nominally flat (rough) surfaces are placed in mechanical contact, the interface (joint) is formed and consists of numerous discrete microcontact spots and a gap that separates the two adjacent surfaces [37; 38]. In such a joint, the real contact area \(A_{re}\) is much smaller than the apparent contact area \(A_{ap}\). The amount of contact area in the joint can be expressed with the use of the contact coefficient \(a_{ct}\):  
\[a_{ct}=\frac{A_{re}}{A_{ap}}, \tag{15}\]  
According to the test results the value of the \(a_{ct}\) coefficient for joints of two flat surfaces depending on the roughness and contact pressure varies from 0.005 to 0.05 [39]. If the substance which fills the gaps is transparent to radiation (for example dry gas), steady heat transfer across the joint is described by the relation [40]:  
\[q_{j}=q_{ct}+q_{g}+q_{rd}, \tag{16}\]  
where \(q_{ct}\) is the conduction via the microcontacts, \(q_{g}\) conduction through the interstitial gas, and \(q_{rd}\) heat transferred by radiation. If the conductance's are used to model the heat transfer across the joint, we can obtain:  
\[h_{j}=h_{ct}+h_{g}+h_{rd}, \tag{17}\]  
For the considered case conductance \(h_{rd}\) can be described with the use of a relationship which describes heat transfer between two parallel flat surfaces [41]:  
\[h_{rd}=4\epsilon_{ef}(1-a_{ct})\,\sigma_{c}T_{j}^{3}, \tag{18}\]  
where: \(\sigma_{c}\) Stefan-Boltzmann constant, \(T_{j}\) average absolute temperature of the joint, \(\epsilon_{ef}\) effective emissivity. The effective emissivity for a system of two parallel surfaces with the identical emissivity's \(\epsilon\) is described by the relation [10]:  
\[\epsilon_{ef}=\left(\frac{1}{\epsilon}+\frac{1}{\epsilon}-1\right)^{-1}=\frac {\epsilon}{2-\epsilon}, \tag{19}\]  
The results of the calculations of radiation conductance \(h_{rd}\) are presented in Figure 7. The calculations were made for four computational cases concerning the values of the \(a_{ct}\) parameter (extreme values of 0.005 and 0.05 were assumed) and two bar emissivity's of 0.7 and 0.8. Experimental investigations have shown that in such a range the emissivity of steel bars changes during heating to the temperature of 800 \({}^{\circ}\)C [42]. In the analysed temperature range the \(h_{j}\) value increases from 3 to 131 W/(m\({}^{2}\) K). As can be seen contact coefficient has a relatively small influence on the calculation results. The influence of the emissivity is much bigger.