The possibility of simplified modelling of radiation heat transfer within a steel porous charge

The article refers to the problem of calculating the effective thermal conductivity kef of a steel porous charge. In proposed approach for each heat transfer mechanism, which occurs in the considered medium, the corresponding thermal resistance is assigned. The model values of the kef coefficient were determined twice for the same input data (geometric dimensions, emissivity, temperature, thermal conductivity of steel and gas). The difference between successive calculations depended on the use of two different methods to determine the thermal radiation resistance Rrd. In the first approach the resistance Rrd was calculated using the exact method and in the second approach the simplified, approximate method was used. The discrepancy between the obtained results of kef in both approaches provides the evidence to use the approximate method to determine the resistance Rrd. A bundle of square steel sections were used to demonstrate the challenge. In the exact method, data regarding the temperature distribution within a single profile was used. These data were obtained based on experimental research using a guarded hot plate apparatus. The calculations were performed using a temperature range between 200C and 800C for two sections: 6060mm and 8080mm and three emissivities: 0.5, 0.7 and 0.9.


Introduction
Heat treatment operations of steel products have significant bearing on all key performance metrics of the plant, i.e.: productivity, energy consumption, product quality and emission of pollutants. This causes that heat treatment parameters should be selected very carefully. In the metallurgical industry for control and design of heat treatment parameters special numerical models have been used for more than three decades [1][2][3]. Such models predict spatial and temporal changes in the temperature of the charge. One of the challenges of this prediction is having required knowledge about the thermal properties of the heated charge. This issue becomes particularly complicated when the charge with a porous structure (e.g bundles of long elements or wire coils) is treated. Such kind of charge is a granular medium with the gas phase filling the gaps in between the steel elements. Therefore, during the heat treatment the following complex processes take place simultaneously: conduction in steel, conduction in gas, contact conduction, free convection and thermal radiation. As a result, the effective thermal conductivity kef becomes the key thermal property of the porous charge. This parameter is commonly used in the theory of porous media [4,5].
The effective thermal conductivity of the porous material can be calculated by applying the model which is based on the analysis of thermal resistance for individual modes of heat transfer [6,7]. If the hightemperature process is applied to the above scenario, one of the heat transfer modes to be considered in the kef model is thermal radiation. The thermal resistance of radiation Rrd in this model can be calculated in two ways: exact and approximate. In the exact approach, resistance Rrd is determined using the radiosity method. The main challenge comes down to solving the system of equations. Obtaining a solution by this method requires knowledge on the temperature values of all the surfaces that close the space of radiative heat transfer. To obtain information on changes in the temperature distribution within the heated porous charge, experimental investigations are necessary. Therefore, this is a significant drawback of this method. This problem is eliminated in the approximate method as this utilises one simple equation in which only the average temperature is applied.
This paper compares the two methods of calculating the effective thermal conductivity of a porous charge. The first, thermal radiation resistance is determined using the exact method, the second the approximated method. The analysis was carried out for the porous charge of a steel square section bundle which is illustrated in Fig. 1. Due to the high porosity of this charge, the fraction of thermal radiation in the total heat transfer is particularly high in this case.

Analysis and modelling
When analyzing the thermal radiation in the area of the bundle, the process within each profile needs to be considered. This is radiation heat transfer in a threesurface enclosure as illustrated in Fig 2. It is assumed that the temperature of the bottom surface A1 is T1, the temperature of the top surface A2 is T2 and the temperature of the lateral surface A3 is T3 and T1 > T3 > T2. Since the enclosure is square the following relationship takes place: A1 = A2 = 0.5 A3. It is also assumed that the surfaces of the enclosure are opaque, diffuse, grey and its radiative properties are the same and expressed by the emissivity  (1 = 2 = 3). The net heat flux of radiation qrd in this system is [8]: were J1 and J2 represents radiosity of the surfaces A1 and A2 respectively. The radiation resistance Rrd for this system is expressed by the following relationship: Therefore, the challenge in calculating the resistance Rrd in the exact method is to determine the radiosities Ji of all surfaces in the considered enclosure, which are described by the following equations: is surface reflectivity. The emissive power per unit area of each surface Ei is described by the equation [8]: The view factors Fi-j used in equations (3) for the surface layout under consideration have the following values: In order to determine each radiosity, the system of equations (3) comes down to the matrix form: where Xrd is a dimensionless coefficient with the value depending on the emissivity and the shape, as well as the relative position of the surfaces that represent the boundaries for the space [9]. For square enclosure Xrd = 1/. The effective thermal conductivity of the analyzed charge is calculated using the definition of conduction resistance for the flat layer with dimension l [10]: where Rto is the total thermal resistance of the considered medium.
For the section bundle the l represents the sum of section dimension in the direction of heat flow and the width of the gap between the individual sections. While resistance Rto is the sum of the section thermal resistance Rst and thermal resistance of the gap Rgp: The methodology for calculating the section thermal resistance Rst is described in the publication [11]. While calculating this parameter, the Rrd resistance needs to be taken into account. Whereas the gap thermal resistance Rgp is calculated by the polynomial [12]: The values of coefficients Bi from this polynomial depend on the surface physical state of the adjacent sections.

Results and discussion
The calculations presented below were performed for two square sections: 60 mm and 80 mm with the wall thickness of 3 mm and three emissivities: 0.5, 0.7 and 0.9. The temperatures of the individual section surfaces required for the calculation of the resistance Rrd in the exact method were obtained from experimental research. These measurements were taken while determining the effective thermal conductivity of the section bundles using the guarded hot plate apparatus [13]. During these experiments, temperatures t1 and t2 were captured using thin jacked thermocouples within the selected sections. Measurements points were located on opposite surfaces of the section, perpendicular to the direction of the heat flow. The results of the measurements were used to calculate the temperature differences t = t1 -t2, which corresponds to the mean temperature tm = 0.5(t1 + t2). The values of the parameter t in the function of the temperature tm obtained for the sections 60 and 80mm are shown in Fig. 3. The data presented in Fig. 3 was then approximated by linear regression equations. The following relationships were obtained for each section: The equations (11) were used to determine the temperatures for each surfaces of the section (T1, T2 and T3) which are necessary for calculating the Rrd resistance in the exact method: Fig. 4 presents the results of radiation resistance calculations for the considered scenarios, which were obtained by both methods (exact and approximate). It was observed that the results for both sections using the same method (Figs. 4a and 4b) were very similar with only a very minor difference of 0.1% However, there are significant differences between the exact and the approximate methods for the same sections, with the exact method producing higher values. These differences in the results of both methods depend on the emissivity. When the emissivity decreases greater differences are observed.
In order to analyze the deviations in detail, the percentage excess of radiation resistance Rrd were calculated: where Rrd-ex is the resistance obtained using the exact method, whereas Rrd-ap is the resistance obtained in the approximate method. The results of the Rrd parameter obtained for the 80mm section are presented in Fig. 5. It can be noted that this parameter is almost constant for each emissivity in a temperature function. Very similar results were obtained for the 60 mm section. This implies that for a given emissivity the average value of this parameter can be applied. Thus, the difference in Rrd resistance between the exact and the approximate method is reduced linearly with the increase in emissivity, reaching zero for the black body surface. However, for a square enclosure with an emissivity of 0.5, the Rrd-ex value is about 20% greater than the Rrd-ap value. In line with the purpose of this article, it was decided to evaluate how the observed difference in Rrd-ex and Rrd-ap values affects the value of the effective thermal conductivity of the square section bundle. In these calculations it was assumed that the thermal conductivity of steel ks and the thermal conductivity of gas phase kg change with temperature in the following relationships:  (15) Equation (14) describes changes in thermal conductivity of low-alloy steel with carbon content of 0.2%, while equation (15) describes changes in thermal conductivity of air. These equations were determined through approximation of the literature data [14,15].
The calculations of the coefficient kef were performed for the same scenarios as for the radiation resistance. To determine the gap thermal resistance Rgp, in equation (10), the following values of Bi coefficients were used:  B1 = 2.3110 -5 ;  B2 = -0.0534;  B3 = 58.16.
Calculations results for the effective thermal conductivity for the analyzed bundles are presented in Fig. 7. The obtained kef values for the 60 mm sections are within the 3.98.8 W/(m 2 K) range, whereas for the 80 mm sections the kef values are within the 3.511.3 W/(m 2 K) range. This coefficient increases as a function of temperature, and the dynamic of it increases with the increase in the emissivity. At the same time, higher values were obtained when the radiation resistance Rrd-ap was taken into account in the calculations. In order to demonstrate the influence of the radiation resistance value on the effective thermal conductivity, the percentage difference between the kef values obtained for both scenarios was calculated. This parameter is defined as follows: where kef-ap denote the kef value calculated taking into account the resistance Rrd-ap, and kef-ex denote the kef value calculated taking into account the resistance Rrd-ex.
The results of the calculations for the kef for each bundle are presented in Fig. 8. The value of the kef for the 60mm section bundle in 200C temperature was not greater than 1.5%. However, when the temperature increased to 800C the obtained values were between 1.7% and 8.7%. The highest values were observed for the emissivity of 0.5. For the emissivity of 0.7 the maximum value was 5.3%. A similar trend was observed for the 80mm sections. For 200 the maximum value obtained was 2.6%, whilst for 800C and the successive emissivities obtained values were 10.6%, 6.1% and 1.9% respectively. The values of the kef parameter averaged for the whole temperature range obtained for all the analyzed scenarios are summarized in Table 1. As it can be seen, the use of the approximate method for calculations the radiation resistance in the model of the effective thermal conductivity, in relation to the entire temperature range, exceeds the maximum value of the kef coefficient by approximately 7%. However, this applies to the emissivity of 0.5, while the surface of steel profiles subjected to heat treatment is usually characterized by the emissivity above 0.7. Therefore, the overestimation of the value of kef using the approximate method will be less than 5%. It can be stated that for industrial needs, the accuracy of calculations at this level is sufficient. However, in a situation where more accurate calculations of the kef coefficient are required, the radiation resistance can be calculated using the approximate method. In order to apply this approach, the use of an appropriate correction term is required, which takes into account the linear impact of the emissivity:   Due to the use of equation (17), the mathematical model of the effective thermal conductivity avoids solving the system of equations (6a), however, the results of these calculations will not be overstated as observed with equation (7).

Conclusions
The problem presented in this article is related to the optimization of heat treatment of the square steel profiles heated in the form of bundles. Due to the porous structure of this charge, its basic thermal property is effective thermal conductivity. This parameter quantifies the ability of the bundle to transmit heat as a result of the complex processes of conduction in steel and air, contact conduction, thermal radiation and free convection. The calculations of the kef coefficient were performed for two scenarios, which differ in the way of their modeling (exact and approximate) for radiation resistance. It has been shown that with the approximate method of determining resistance Rrd, the results of the kef calculations are overstated by an average of about 5%. In order to eliminate this discrepancy an updated version of the approximate equation was proposed, in which a corrective term was added. This term takes into account the linear impact of emissivity on the value of radiation resistance Rrd.