# Model of Complex Heat Transfer in the Package of Rectangular Steel Sections
## 3 Results and Discussion  
In the presented model, the following parameters are used as input data: section dimensions (width \(l_{x}\), height \(l_{y}\), and wall thickness \(s\)), emissivity of the section surface \(\varepsilon\), Nusselt number \(Nu\), thermal conductivities of steel \(k_{s}\), and gas \(k_{g}\). The last two parameters with respect to temperature are described by the following equations [17]:  
\[k_{s}=1.2\cdot 10^{-8}t^{3}-3.2\cdot 10^{-5}t^{2}-1.2\cdot 10^{-2}t+51.3, \tag{22}\]  
\[k_{g}=-2.88\cdot 10^{-8}t^{2}+8.05\cdot 10^{-5}t+0.024. \tag{23}\]  
Equation (22) refers to steel with a carbon content of 0.2%, whereas Equation (23) refers to the air. For the emissivity of section surface, a value of 0.5 was assumed, which relates to the rolled carbon steel [18]. The presented model enables the analysis of the process of heat flow in the area of the package of sections, both in a quantitative and qualitative manner. The quantitative analysis relates to determining the value of individual resistances, whereas the quantitative analysis consists of determining the share of the individual mechanisms in the heat transfer that occurs while heating the package. The results of the calculations presented below relate to four packages of the following sections: \(20\times 40\times(2.5)\) mm, \(40\times 40\times(3)\) mm, \(60\times 60\times(3)\) mm, and \(80\times 80\times(3)\) mm, where the number in parentheses is the thickness of a section wall. First, the results of the calculations of the \(R_{st}\) resistance of the sections are presented. This parameter was calculated for two cases. Case I applies to the situations when all the types of heat transfer (conduction, radiation and convection) occur within a section; the results of these calculations are presented in Figure 8a. Case II applies to the situations when only heat conduction occurs within a section (gas convection and heat radiation between the inner surfaces of a section were ignored); the results of these calculations are presented in Figure 8b. By comparing the diagrams, it is possible to notice that the difference between the values of \(R_{st}\) resistance for both cases increases with temperature; the bigger the size of a section, the bigger this difference. This effect is caused by the intensity of heat radiation increasing with the temperature function (proportional to the third power of \(T_{\text{m}}\)) and the distance between the surfaces that exchange heat. Figure 8: The results of the calculations of \(R_{st}\) resistance of sections: (**a**) the values obtained for all kinds of heat transfer and (**b**) values obtained while considering conduction only. To quantitatively express the differences in the value of \(R_{st}\) resistance between case I and case II, a relative percentage difference \(\Delta R_{st}\) was calculated, which was defined by the following relationship:  
\[\Delta R_{st}=\frac{R_{st-II}-R_{st-I}}{R_{st-I}}\cdot 100\%, \tag{24}\]  
where \(R_{st-I}\) is the value of resistance of a section in case I and \(R_{st-II}\) is the value of resistance of a section in case II. The values of the \(\Delta R_{st}\) parameter obtained for individual sections for chosen temperature values are collated in Table 1. The maximum value of \(\Delta R_{st}\) was nearly 300%. In the case of the performed analysis, the shape of the relationship between the \(R_{st}\) resistance of individual sections and the \(R_{ct}\) resistance of contact conduction is interesting. To show the problem from the quantitative perspective, the quotient of resistances \(R_{ct}\)/\(R_{st}\) was calculated for each section. The calculations were run both for the maximal and the minimal value of \(R_{ct}\). The results of these calculations are collated in Figure 9. In most cases, the \(R_{st}\) resistance was characterized by a higher value. At the end of the performed analysis, the results of calculations of \(R_{to}\) resistance of packages will be discussed.