# Experimental Investigations of Effective Thermal Conductivity of the Selected Examples of Steel Porous Charge

## Abstract

**:**

_{ef}. This paper presents the results of experimental examinations of effective thermal conductivity of the porous charge, which is composed from various types of steel long components. Due to the specific nature of the samples, a special measurement stand was constructed based on the design of a guarded hot plate apparatus. The measurements were performed for sixteen different samples across a temperature range of 70–640 °C. The porosity of the samples, depending on the type of components used, ranged from 0.03 to 0.85. Depending on these factors, the effective thermal conductivity ranged from 1.75 to 8.19 W·m

^{−1}·K

^{−1}. This accounts for 0.03 to 0.25 of the value of thermal conductivity of the solid phase of the charge, which in the described cases was low-carbon steel. It was found that the effective thermal conductivity rises linearly with temperature. The intensity of this increase and the value of coefficient k

_{ef}depend on the transverse dimension of the components that form the charge. The results may represent the basis for the validation of various models of effective thermal conductivity with respect to the evaluation of thermal properties of the porous charge.

## 1. Introduction

^{−2}[1]. This accessibility of steel properties over such a wide range is obtained mainly through heat treatment. The use of precisely controlled steel heating and cooling operations with consideration for the chemical composition of the material allows obtaining the most desired mechanical properties. The heat treatment effects are induced by microstructural changes due to solid-state phase transformation. This means that the heat treatment processes have a direct influence on the quality of steel products. These heat treatments are also essential for the entire manufacturing process, since they have a significant impact on energy consumption, production efficiency, and emission of pollution. For these reasons, the manufacturers of steel need to optimize the heat treatment processes. In modern technological lines, this is achieved by automatic systems where furnaces are operated based on the use of appropriate numerical models [2,3,4,5]. One of the key input data for such models are the thermophysical properties of the heated elements. When solid components such as billets or slabs are heated, the basic property of the charge is the thermal conductivity of steel k

_{st}[6,7]. The value of this parameter depends on the steel chemical composition, its crystal structure, and its temperature. The data on the thermal conductivity of the most popular carbon and alloy steels are generally available in the literature [1,8,9,10]. However, in many cases, the heat-treated charge is not solid. Such situations can be met in the heat treatment of the charge in the form of coils or bundles. Coils are used to heat sheets and wires, while bundles are popular in heating various types of long components i.e., bars, tubes, rectangular sections, and shapes [11,12,13]. These types of charge are two-phase structures consisting of a steel skeleton and the gas-filled voids. Due to a discrete form of the solid phase, coils and bundles are considered to be a granular porous charge [14]. Typical examples of the steel porous charge used in industrial practice are shown in Figure 1.

_{ef}. This quantity is commonly used in the theory of porous media [15]. The value of k

_{ef}coefficient is a function of complex heat transfer mechanisms related to conduction, contact conduction, free convection, and radiation, which occur within the space of the porous medium. Many different models of the effective thermal conductivity have been developed over the past several decades [16,17,18,19]. These models are very diverse and determine the value of k

_{ef}based on several parameters and divided into primary and secondary parameters [20]. Primary parameters include porosity φ and thermal conductivities of the solid phase k

_{s}and gas phase k

_{g}. Secondary parameters include thermal contact resistance, heat transfer by radiation, the Knudsen effect, and quantities, which describe the geometric configuration of the medium, the most common being the mean diameter of particles or pores. Despite this diversity, only one model relates directly to the porous charge. It has been developed to determine the radial effective thermal conductivity of steel coils annealed in bell-type hydrogen furnaces [21,22,23]. Furthermore, no experimental studies have analyzed this area of research. Experimental tests of heat transfer in metallic porous media relates mostly to steel and aluminum foams [24,25] or metallic thermal protection systems [26].

## 2. Materials and Methods

#### 2.1. Experimental Setup

_{i−h}and cold surface (top) t

_{i−c}were measured in five opposite points. One point was located in the geometrical center of the surface, whereas four other points were in the corners of the square with the side of 260 mm, and its center was overlapped with the sample center. The thermocouples that were used to measure the temperature at the lower surface of the sample were fixed to the bottom of the retort that acted as a hot plate. Furthermore, the thermocouples used for the temperature measurement on the upper surface of the sample were fixed to the steel plate that covered the sample. Due to the cooler shift, this plate performed the role of the cold plate. Its thickness was 15 mm with transverse dimensions of 390 × 390 mm. In order to determine the value of the coefficient k

_{ef}, after measurement of the temperature in the described ten points, mean temperatures were determined for each surface i.e., t

_{hot}(Equation (2)) and t

_{cold}(Equation (3)), and difference in temperature Δt along sample high (dimension L) (Equation (4)) and mean measurement temperature t

_{m}(Equation (5)):

#### 2.2. Investigated Samples

_{ef}.

#### 2.3. Measurement Procedure

_{ef}in possibly the widest range of temperature. The value of the parameter P was changed from 200 to 3200 W, which corresponded to the change in mean temperature of the samples of 70–640 °C. The upper temperature of the examination was limited by the maximal temperature of the main heater. The heater was automatically switched off by the control system after it achieved a temperature of 900 °C. This temperature was reached at a power adjustment of 3400 W.

#### 2.4. Measurement Uncertainty

## 3. Results and Discussion

^{−1}·K

^{−1}for the staggered arrangement and from 1.64 to 6.03 W·m

^{−1}·K

^{−1}for the in-line arrangement. The percentage increase in the coefficient k

_{ef}caused by the change in the arrangement and averaged for the entire temperature range was, depending on the bar diameter, 4% for 10 mm bars, 13% for 20 mm bars, and 10% for the 30 mm bars. These results support the previously proposed presumption that the effective thermal conductivity of the bar bed depends on the arrangement. The greater the contact area between the adjacent bars in the sample, the greater the value of k

_{ef}.

_{ef}range from 2.69 to 8.19 W·m

^{−1}·K

^{−1}. These are the highest values of k

_{ef}obtained among all the examined types of charge. The result can be considered as obvious, since the charge is characterized by the lowest porosity (0.03). The porosity of these samples results from narrow gaps between individual bars, which are caused by errors in the shapes of these components. Furthermore, compared to round bar beds, the contact area is greater, which reduces the resistance of thermal contact conductance between individual layers.

^{−1}·K

^{−1}. As can be seen, the arrangement of bars has an effect on the value of k

_{ef}in this case as well. Higher values were obtained for mixed samples. The percentage increase in k

_{ef}averaged for the entire temperature range between mixed and parallel samples was 19% for 5 × 20 mm bars and 42% for 10 × 40 mm bars. This shows that the effective thermal conductivity of the porous charge made of components with varied transverse dimensions depends significantly on the orientation of these components with the direction of heat flow.

^{−1}·K

^{−1}. This means that the range similar to previous cases. Interestingly, for the sample of 20 × 40 mm sections, the value of k

_{ef}did not increase with temperature, ranging from 3.63 to 3.85 W·m

^{−1}·K

^{−1}at a mean value of 3.73 W·m

^{−1}·K

^{−1}. For the samples of 40 × 40 mm and 60 × 60 mm sections, coefficient k

_{ef}was rising with temperature, as it was in previous cases.

_{0}and β and coefficient of determination R

^{2}for individual samples are listed in Table 1.

_{0}and β ranged from 1.58 to 4.18 and 0.0019 to 0.0054, respectively. Furthermore, the range of coefficient of determination R

^{2}is 0.967–0.998. Values of this parameter similar to unity indicate that the adopted equations are well adjusted to the measurement results. The values of coefficient k

_{0}confirmed unequivocally the previous conclusion that the effective thermal conductivity of a specific type of porous charge increases with the increase in dimensions of the components. For example, for the round bar samples, the coefficient k

_{ef}is greater for larger bar diameters. Furthermore, the values of coefficient β indicate that with the increase in the dimensions of the components, the effect of temperature on k

_{ef}is also more pronounced.

_{ef}of the samples made of 10 mm components. These were samples of square bars, flat bars with parallel arrangement, and round bars. As can be seen, the increase in porosity in this case from 0.03 to 0.09 leads to a substantial decline in coefficient k

_{ef}. Furthermore, for both samples of round bars, effective thermal conductivity, despite a substantial difference in porosity, is at nearly the same level. This can be explained in the following manner. Round bars with a diameter of 10 mm are unrolled from the coil and consequently are characterized by high rectilinearity errors. Consequently, the contact areas in the beds of such bars are substantially less dependent on the bars’ arrangement.

_{ef}versus temperature, a constant value of 3.73 W·m

^{−1}·K

^{−1}was adopted for the entire temperature range. The diagram shows that the increase in external porosity (the first three points for each temperature) of the charge substantially reduces the value of k

_{ef}. Furthermore, the increase in charge porosity related to internal porosity does not have such a high effect.

_{ef}for the sections (comparable to the values obtained for bars) results from the fact that gas conduction is compensated by other heat transfer mechanisms. In the free space of the section, heat is also transferred through the free convection and thermal radiation between the internal surfaces. The problem of complex heat flow in the steel rectangular sections was analyzed in the papers [40]. It has been shown that heat transfer in the section occurs with over twice lower intensity compared to heat conduction in the solid equivalent. It was also demonstrated that above 300 °C, thermal radiation is essential for the whole phenomenon. In the next article [41,42], the complex heat transfer in the packages of rectangular steel sections was analyzed in turn. It was established that the heat transfer intensity in such systems mostly depends on the thermal contact resistance R

_{ct}between adjacent layers of the package. The cited studies also show that the phenomenon of heat flow in the analyzed systems is the result of the simultaneous and interrelated mechanisms of conduction in steel and air, contact conduction, free convection, and thermal radiation. As a result, the internal porosity does not drastically reduce the value of the k

_{ef}coefficient. Further research will be carried out to obtain more precise results. Currently, due to the limited amount of the experimental data, it is impossible to draw a more precise conclusion.

_{ef}are much similar to those obtained for the samples made of 10 mm components. Figure 12, Figure 13 and Figure 14 reveal that porosity is not the only parameter that has the effect on changes in the value of coefficient k

_{ef}of the charge. It should be concluded that another important factor in this respect is the thermal contact resistance R

_{ct}between the adjacent components of individual layers. This resistance is dependent, among other things, on the actual contact area A

_{r}[43,44]. In the packed beds of the porous charge, this surface depends on the shape of the components and their work accuracy [45]. Any shape errors, and in the case of long components, especially the lack of rectilinearity, leads to a reduction in this surface. In a bed of square bars, this area should be many times greater than that of round bars. However, if the bars are characterized by substantial rectilinearity errors, numerous and randomly distributed gaps will occur between the adjacent bars. In this case, the previously mentioned disproportion in the surface A

_{r}between round and square bars will be substantially lower. The results obtained lead to the conclusion that the phenomenon of contact conduction is critical to the intensification of a global heat flow in the area of porous charge.

_{s}(Equation (8)):

_{ef}was obtained for the 10 mm round bar sample with the in-line arrangement. The highest value of k

_{ef}was observed for 30 mm square bars. Therefore, the minimum and maximum value of k

_{ef}for the charge depending on temperature can be described by the Equations (10) and (11):

_{ef}. The most common method for determining this property is to calculate it from effective thermal conductivity and effective volumetric heat capacity (Equation (13)) [49]:

## 4. Conclusions

- Depending on the temperature, size, shape, and arrangement, the effective thermal conductivity of packed bundles of various long components made of low-carbon steel changes over the range of 1.7–8.2 W·m
^{−1}·K^{−1}; - The values of coefficient k
_{ef}range from 0.03 to 0.25 of thermal conductivity of the solid phase of the charge; - The effective thermal conductivity rises linearly with temperature, whereas the intensity of this increase and the value of coefficient k
_{ef}depend on the transverse dimension of the components that form the charge; - Coefficient k
_{ef}declines with the increase in the external porosity of the charge, which depends on the transverse shape of the components and their arrangement; - A lower effect on the value of k
_{ef}is observed from internal porosity that concerns hollow components and is related to their geometry; - A substantial impact on the value of coefficient k
_{ef}is observed for thermal contact resistance, occurring between the adjacent components of the charge, with the value of this resistance depending mainly on the real contact area in the joints.

_{ef}, will represent the basis for the validation of these models.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Sample Availability

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**Figure 1.**Typical examples of the steel porous charge: (

**a**) wire coil treated in bell-type furnace, (

**b**) square bar bundles treated in soaking furnace.

**Figure 2.**A general view of the testing stand: 1—heating chamber, 2—control unit of main and guarded heaters, 3—data logger with temperature meter, 4—autotransformer, 5—unit of cooling system.

**Figure 3.**Scheme of the heating chamber: 1—retort with a hot plate, 2—investigated sample, 3—cold plate, 4—heating chamber cover with a cooler, 5—side guarded heater, 6—main heater, 7—bottom guarded heater, 8—thermal insulation, 9—support structure.

**Figure 4.**Pictorial photographs showing geometrical structure of two samples: (

**a**) sample of square bars, (

**b**) sample of rectangular sections.

Sample Type | Element Dimension | k_{0} | β | R^{2} |
---|---|---|---|---|

Staggered round bars | 10 mm | 1.57 | 0.0021 | 0.981 |

20 mm | 2.39 | 0.0042 | 0.974 | |

30 mm | 3.11 | 0.0051 | 0.998 | |

In-line round bars | 10 mm | 1.48 | 0.0022 | 0.993 |

20 mm | 2.02 | 0.0037 | 0.993 | |

30 mm | 2.54 | 0.0054 | 0.990 | |

Square bars | 10 mm | 2.54 | 0.0019 | 0.994 |

20 mm | 4.48 | 0.0031 | 0.967 | |

30 mm | 5.91 | 0.0036 | 0.982 | |

Parallel flat bars | 5 × 20 mm | 1.86 | 0.0014 | 0.993 |

10 × 40 mm | 2.43 | 0.0015 | 0.992 | |

Mixed flat bars | 5 × 20 mm | 2.10 | 0.0021 | 0.980 |

10 × 40 mm | 3.09 | 0.0033 | 0.988 | |

Sections | 40 × 40 mm | 3.95 | 0.0019 | 0.984 |

60 × 60 mm | 4.18 | 0.0031 | 0.977 |

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Wyczółkowski, R.
Experimental Investigations of Effective Thermal Conductivity of the Selected Examples of Steel Porous Charge. *Solids* **2021**, *2*, 420-436.
https://doi.org/10.3390/solids2040027

**AMA Style**

Wyczółkowski R.
Experimental Investigations of Effective Thermal Conductivity of the Selected Examples of Steel Porous Charge. *Solids*. 2021; 2(4):420-436.
https://doi.org/10.3390/solids2040027

**Chicago/Turabian Style**

Wyczółkowski, Rafał.
2021. "Experimental Investigations of Effective Thermal Conductivity of the Selected Examples of Steel Porous Charge" *Solids* 2, no. 4: 420-436.
https://doi.org/10.3390/solids2040027