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.
Steel bars, due to their mechanical properties and relatively low cost, have a wide range of applications in various branches of industry and the economy. They are used both as finished products and semi-finished products intended for further processing. Subjecting the bars to heat treatment improves the desired mechanical properties, which increases their durability, strength, and resistance to various operating factors. This process allows the steel to be tailored to specific applications in construction, industry, and other branches of engineering. During heat treatment, in many cases, steel bars are heated not individually but in bundles, i.e., groups arranged parallel (Fig. 1). The use of this method of heating has several important technological, economic, and quality-related reasons. Specifically, heating a larger number of bars simultaneously allows for more efficient use of the furnace's thermal energy which ultimately contributes to the reduction of fuel or electricity consumption. Additionally, this approach increases production efficiency, improves the uniformity of the structure, and minimizes deformation and oxidation.
Because heat treatment processes of steel products are characterized by significant energy consumption and pollutant emissions, researchers in the metallurgy industry are continuously conducting studies to improve these technologies. Among other things, this includes the optimization of heating processes. This is related to the development of algorithms that allow for the most precise prediction of the current temperature of the heated components. This problem is particularly complex when analyzing the heating of a bar bundle, because heat transfer in this medium being a combination of: (a) conduction across the bars, (b) conduction in the gas gaps (pores), (c) thermal contact conduction between adjacent bars, and (d) thermal radiation between surfaces of bars. The simultaneous occurrence of four completely different modes of heat transfer makes the mathematical description of this process much more difficult.
The analysis of this problem is significantly facilitated by the use of the concept of effective thermal conductivity (k or ETC). The most dependable method for determining ETC is through experimental testing. This is usually done using the steady-state method with a guarded hot plate apparatus. A significant drawback of these measurements is that the results are not universally applicable, as they apply only to particular material samples. Moreover, this technique requires specialized equipment, is time-intensive, and requires the preparation of specific test samples. Consequently, model calculations are often employed as an effective alternative for determining the ETC. However, as demonstrated by the authors based on their own research, the models of ETC commonly presented in the literature, do not yield accurate results when analyzing bundles of steel bars. As a result, an original model of ETC was developed, specifically designed to determine the thermal properties of the discussed charge. The basis for developing this model was the so-called unit cell, identified as a repeating periodic fragment of a flat arrangement of bars with a staggered configuration (Fig. 2a). Such an approach is commonly used in the analysis of heat transfer processes in porous media. The unit cell considered in this situation contains three bar fragments from two adjacent layers. In the vertical plane the cell is divided into three sections I-III. Thanks to such division, eight elements are distinguished in the cell (Fig. 2b).
By using the so-called thermal-electrical analogy, which arises from the similarity of the mathematical notation of Ohm and Fourier lows, the intensity of thermal energy flow within the unit cell is expressed through the total thermal resistance, R. Using the value of this parameter, the ETC is calculated based on the relationship describing the conduction resistance through plane wall:
where δ is the height of the cell.
For the adopted division of the unit cell, the resistance R is calculated as a combination of series and parallel connections of thirteen distinct thermal resistances related to: conduction within the bars (five resistances), conduction in the gas area (three resistances), contact conduction (two resistances), and thermal radiation (three resistances). The equivalent network of these resistances is shown in the Fig. 3 and their explanation is presented in Table 1.
In this paper, a simplified version of the heat transfer model for a bundle of round bars is presented. In this approach, the total thermal resistance of the medium is calculated as a combination of only four thermal resistances. The reason for eliminating the number of resistances is obvious. This reduction significantly simplifies the heat transfer model. In addition to describing the model itself, an analysis will also be provided to demonstrate the extent to which the introduced simplifications affect the accuracy of the computational results. The model was also validated using the results of experimental investigations.
The values of the analyzed resistances for two cases of a 20 mm bar bundle for four selected temperatures: 100; 300; 500 and 700 °C are presented in Tables 2 and 3. Table 2 presents the results for a bundle with porosity of 0.15, Table 3 the results for a bundle with porosity of 0.21.