During heat treatment of rectangular steel sections, a heated charge in the form of regularly arranged packages is placed in a furnace. The article presents a model of a complex heat transfer in such a package using the thermo-electric analogy. The model considers the following types of heat transfer: conduction in section walls, conduction and natural convection within gas, heat radiation between the walls of a section, as well as contact conduction between the adjacent sections. The results of our own experimental research were used for calculations of heat resistance applying to natural convection and contact conduction. We assumed that the material of sections was low-carbon steel and the gas was air. The result of the calculations of the presented model is total thermal resistance Rto. The calculations were performed for the temperature range 20–700 °C for four geometrical cases. Due to the variability of conditions for contact heat conduction, we assumed that total thermal resistance for a given charge is contained within a value range between Rto-min and Rto-max. We established that the value of Rto depends significantly on the section’s geometry. The larger the section sizes, the greater the changes of Rto. The minimal and maximal values of Rto for all packages were 0.0051 (m2·K)/W and 0.0238 (m2·K)/W, respectively. The correctness of model calculations was verified with the use of experimental data.

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A method to obtain a radiation exchange factor FR in the bundle of steel round bars is presented. This parameter is required for determination of the radiative thermal conductivity krd, which is one of the basic thermal properties of the bar bundles. In the presented approach, the analyzed parameter is calculated indirectly. The initial point for calculations is the geometric model of the medium defined as a unit cell. Then, for the elements present in this cell, the thermal resistance of both conduction and radiation is determined. The radiation resistance is calculated from the radiosity balance of the surfaces enclosing the analyzed system. On this basis, the radiation thermal conductivity krd is calculated. Next, taking into account the bar diameter, the value of parameter FR is also determined. The analysis is performed at the process temperature range of 200 to 800 °C for three bar diameters: 10, 20 and 30 mm, and for the three porosities of the bundle. Different emissivity of bars in the range of 0.5 to 0.9 was also taken into account. Finally, a relationship that allows calculating the FR factor correlated with the emissivity of the bars and the bundle porosity was established. The krd obtained from the methodology presented and compared with the values calculated directly do not exceed 9%; however, after averaging over the entire temperature range of the process, the difference does not exceed 0.2%.

Bundles of steel bars, besides metal foams, are an example of cellular solids. Such bundles constitute a charge during the heat treatment of bars. The paper presents a mathematical model of transient heat transfer in a bundle of rectangular steel bars based on the energy balance method. The key element of this model is the procedure of determining the effective thermal conductivity using the electrical analogy. Different mechanisms of heat transfer occurring within the analysed medium (conduction in steel and contact conduction) are assigned corresponding thermal resistances. The discussed procedure involves expressing these resistances with the use of arithmetic relationships describing their changes in the temperature function. Thermal contact resistance has been described with the use of the relationships determined experimentally. As a result of the performed calculations, the influence of contact conduction between the adjacent bars and bundle arrangement on its heating time was established. The results of the calculations show that the heating time of bundles can be lowered by 5–40% as a result of a decrease in the thermal contact resistance. This effect depends on the bar size and bundle arrangement. From the practical point of view, the analysed problem is connected with the optimization of the heat treatment processes of steel bars.

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