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.
Cellular solid materials are commonly found in industrial applications. By definition, cellular solids are porous materials that are built of distinct cells. One of the groups of such materials contains metal foams. Another group of cellular metals contains bundles of steel bars, which create charges during the heat treatment of the bars. A granular structure connected by the lack of continuity of the solid phase is the main feature that distinguishes bundles from metal foams. The boundaries of the bundle cells are made of adjacent bars, with the internal region taking the form of an air cavity. In this paper, we discuss the possibility of using the Krischer model to determine the effective thermal conductivity of heat-treated bundles of steel bars based on the results of experimental tests and calculations. The model allows the kef coefficient to be precisely determined, although it requires the weighting parameter f to be carefully matched. It is shown that the value of f depends on the bar diameter, while its changes within the examined temperature range (25–800 °C) can be described using a third-degree polynomial. Determining the coefficients of such a polynomial is possible only when the effective thermal conductivity of the considered charge is known. Moreover, we analyze a simplified solution, whereby a constant value of the f coefficient is used for a given bar diameter; however, the kef values obtained thanks to this approach are encumbered with inaccuracy amounting to several dozen percentage points. The obtained results lead to the conclusion that the Krischer model cannot be used for the discussed case.
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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