Simplified model of the effective thermal conductivity of a bundle of round steel bars

The paper presents a simplified model of the effective thermal conductivity (ETC) of a bundle of circular steel bars. This model is a modification of a previously introduced detailed model, in which the ETC value is determined based on an analysis of thirteen thermal resistances. In the simplified model, the number of resistances was reduced to four, and some of the equations describing individual resistances were modified. In the computational section, the results of both models were compared. Calculations were performed for twelve cases, determined by bar diameters (10, 20, 30, and 40 mm) and three bundle porosities (0.09, 0.14, and 0.21). The bars were assumed to be made of steel with 0.2% carbon content and a surface emissivity of 0.7. Air was considered as the gas filling the gaps in the bundle. The calculations were carried out for a temperature range of 25-800 °C. It was found that, compared to the detailed model, the simplified model yields values that are on average about 4.8% higher. For this reason, a correction factor of 0.954 was introduced into the simplified model. After applying this correction, the results of the simplified model deviate from those of the detailed model by only about 1%. The simplified model was also used for a qualitative analysis of heat transfer within the bundle. It was determined that, depending on temperature and bar diameter, the gap resistance is higher than the bar layer resistance by a factor of approximately 3 to 49. Moreover, it was shown that contact conduction is the key thermal phenomenon during the heating of the bundle. In this way, more than 60% of the thermal energy is transferred in the bundle gaps. The results of the presented model can serve as a valuable source of information for optimizing the heat treatment processes of steel bars.