# Model of Complex Heat Transfer in the Package of Rectangular Steel Sections
## 2 Materials and Methods
Hence, the final value of the section thermal resistance \(R_{st}\) is calculated using [7]:  
\[R_{st}=\frac{R_{st-d}+2R_{st-b}}{3}. \tag{4}\]  
The resistances occurring in the Equations (2) and (3), which apply to conduction in section walls (they are marked grey in Figure 4), are described with the following relationships:  
\[R_{I}=R_{III}=\frac{l_{y}\cdot l_{x}}{s\cdot k_{s}}, \tag{5}\]  
\[R_{II-1}=R_{II-3}=\frac{s\cdot l_{x}}{(l_{x}-2s)\cdot k_{s}}, \tag{6}\]  
Figure 1:0  
Figure 1:1  
where \(k_{s}\) is the thermal conductivity of steel. The resistance \(R_{gs}\) applies to the heat flow in the gas area, which is connected with the conduction and the natural convection in a horizontal enclosure occurring at the same time. Both phenomena can be treated jointly as an intensified heat conduction expressed quantitatively by equivalent gas thermal conductivity \(k_{cg}\)[7]:  
Figure 1:2  
where \(k_{g}\) is the thermal conductivity of gas and \(Nu\) is the Nusselt number. The following correlations are recommended for horizontal enclosures that contain air [6]:  
For \(10^{4}<Ra<4\times 10^{5}\):  
Figure 1:3  
Figure 1:4 Thermal resistance networks of a rectangular section: (**a**) network for resistance \(R_{st-d}\) and (**b**) network for resistance \(R_{st-b}\). For \(4\times 10^{5}<Ra<10^{7}\):  
Figure 1:5  
where \(Ra\) is the Rayleigh number. For an enclosure, this number is defined as:  
Figure 1:6  
where \(g\) is gravitational acceleration, \(\beta\) is the coefficient of volume expansion, \(\Delta t\) is the temperature difference between hot and cold surfaces of the enclosure, \(L_{c}\) is the characteristic length of the geometry (in the section case, this is the distance between the hot and the cold surface), \(v\) is the kinematic viscosity of the fluid, and \(Pr\) is the Prandtl number. For air, \(Pr=0.71\)[6]. Based on the research described in [8, 9], the values of the Rayleigh number for square 60 mm and 80 mm sections were calculated. The research consisted of heating the sections with a one-dimensional (oriented vertically from the bottom to the top), steady heat flux and measuring the temperature difference \(\Delta t\) between their hot (lower--with \(t_{h}\) temperature) surface and their cold (upper--with temperature \(t_{c}\)) surface with the use of jacked thermocouples. The research was conducted for the temperatures ranging from room temperature to 750 \({}^{\circ}\)C. The values of the \(\Delta t\) parameter in the function of the mean temperature of a section \(t_{\rm m}=0.5(t_{\rm h}+t_{\rm c})\) obtained this way are presented in Figure 5a. Then, the values of the Nusselt number (Figure 5b) for the above-mentioned sections were calculated using Equations (10)-(12). Depending on the size of a section, natural convection in relation to pure conduction can increase the intensity of heat transfer four to seven times. However, this effect decreases significantly with the temperature increase. The highest values of the \(Nu\) occur for the temperature of 100 \({}^{\circ}\)C. This is mainly caused by changes in the temperature difference \(\Delta t\) observed in the area of sections and changes in kinematic viscosity of air. The increase in the \(Nu\) value at the initial stage is attributable to changes in the parameter \(\Delta t\). After exceeding 100 \({}^{\circ}\)C, the main effect on convection is from changes in air viscosity. Thus, with a further increase in temperature, the Nusselt number starts to decline rapidly despite a further rise in the difference in temperature. At the same time, the value of parameter \(\Delta t\) depends on the other heat transfer mechanisms that are present in the section. All these mechanisms are interrelated and, as a result, the course of convection and the Nusselt value depend on the section geometry and the thermal properties of the steel [10]. Based on the curves in Figure 5b for calculating the change in the Nusselt number in relation to temperature, the following relationships were derived.