**1. Introduction**

Observed climatic changes in the environment are causing recycling technologies for steel production to increase their share in the total balance of crude steel production. This is particularly noticeable in the European Union. Therefore, the electric arc furnace is becoming one of the leading processing units for steel scrap used in crude steel production [1]. Electricity consumption in this process still accounts for about 50–80% of the total energy consumption during melting. Technological progress and the development of information technology create opportunities to reduce the unit demand for electrical or chemical energy.

Mathematical descriptions and characterisations of the arc furnace smelting process are complicated due to its nature resulting from, among others, high temperature and chemical composition gradients, turbulent flows, multiphase and heterogeneity of the system, significant mass flow of substrates, high speed of phase transformations and nonlinearity of phenomena. Therefore, the development of modelling of selected stages or the entire arc furnace smelting process remains justified [2].

The models used to describe the melting technology can be physical models— reproducing the processes on a smaller scale (e.g., laboratory and/or using other media than liquid metal and slag with the use of criterion numbers). An example of the application of a high-temperature model can be the studies on the performance of burners in an electric arc furnace carried out by Yonmo Sung [3]. They presented modification of oxy–gas burners, which was validated on a test bench.

Studies of scrap melting phenomena were also carried out based on water models [4] (using ice, three heating rods, and steam nozzles, which reflected scrap, electric arc, and oxy–gas burners, respectively).

Numerical methods can also be used to model the process. Mombeni [5] used a CFD model calculated in the commercial software ANSYS FLUENT to address the crack

**Citation:** Moskal, M.; Migas, P.; Karbowniczek, M. Multi-Parameter Characteristics of Electric Arc Furnace Melting. *Materials* **2022**, *15*, 1601. https://doi.org/10.3390/ ma15041601

Academic Editors: Qing Liu and Jiangshan Zhang

Received: 31 January 2022 Accepted: 17 February 2022 Published: 21 February 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

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formation of water-cooled panels installed in the furnace vault. CFD modelling can also be used to determine the temperature distribution in a metal bath in the context of burner placement and media flow rates [6].

However, the mathematical models are the most universal method, which take into account physicochemical processes, thermodynamic and theoretical equations, mass and energy balances, or only technological data from the process [7]. They can be divided into two categories: linear and non-linear.

Linear models are created as a linear combination of predictor variables [8] based on average statistical input and output data from the process, or assumed theoretical values. They describe the relationship between individual parameters and their importance for the outcome of the calculation. Linear models are static and refer to only two points in time; they are the result of input data and, as mentioned above, they can only be used with reference to average values of a given process [9].

The most popular is the Kohle model, which describes the influence of process parameters on electricity demand.

Another model based on a multivariate linear regression equation is the one developed by Haupt [10]. He determined the influence of scrap quality on the melting process in an electric arc furnace.

However, non-linear models have the ability to describe the process more accurately. R. Morales created one of the first models, which included heat and mass transfer calculations and chemical reaction kinetics in terms of use the direct reduced iron [11]. The author created a dynamic model, which took into account the relationships between individual phenomena in terms of their physical course and not just the result, which was a statistical outcome of many variables. Thus, it makes it possible to analyse single melts and to follow the course of change of the result in time.

Attempts have been made to simplify the models and organise their construction. Clear boundaries of individual zones in the furnace were set. In terms of both: furnace construction, melting stages, and division into individual phases of materials/substances present in the furnace space. Bekker [12] specified two groups of solids and liquids in the constructed model. On the other hand, Logar [13,14] created a system of dependent modules representing particular groups of phenomena (including solid scrap zone, solid slag zone, liquid slag zone, liquid metal zone, and wall and roof zone). Models using neural networks are becoming increasingly popular. The first versions were developed in the 1990s. Ledoux and Bonnard, in their work, built a model based on a multi-layer perceptron. The model described the dynamics of an electric arc during the melting process.

Due to the process complexity, it is necessary to select models in such way that sufficient representation of the processes is maintained with the best possible ability to interpret and understand the simulation results [15]. The balance between these two characteristics depends largely on the type of models and the modelling methods [16].

#### **2. Materials and Methods**

#### *2.1. Theoretical Mathematical Models*

Mathematical models are the most commonly used in EAF simulation. They can be both linear and non-linear in nature. This classification results from taking into account the above-described variability of process efficiency as well as the variability of the parameters themselves during the melt [17]. Taking into account the kinetics of the reaction or the variation in the efficiency of media use or their control transforms the model into a non-linear form. Non-linearity is understood as effects ranging from media consumption dependent on melting programmes to the dependence of heat losses on the number and height of scrap heaps and thus arc exposure. When creating mathematical models, theoretical data or available technological and statistical data are used—which, depends on their availability and quality.

#### *2.2. Linear Modelling of Steel Melting in Electric Arc Furnace*

Regression linear models are created as a linear combination of predictor variables based on statistical mean input and output data from a process, or assumed theoretical values. This means that they refer to (only) two points in time, they are the result of input data and, e.g., output data, and as mentioned above, and they can only be used with respect to the average values of a given process. Static models often do not take into account some process disturbances and micro-scale phenomena (of low weights) and, therefore, their use for the analysis of single melts becomes difficult or, even impossible, due to the high variability of the real process conditions [18].

The most widely used statistical model in the literature is the Köhle model [18,19]. Köhle's Equation (1) describing the demand for electricity:

$$\frac{\mathbf{W\_{r}}}{\mathbf{kWh/t}} = 300 + 900 \left[ \frac{\mathbf{G\_{E}}}{\mathbf{G\_{A}}} - 1 \right] + 1600 \frac{\mathbf{G\_{Z}}}{\mathbf{G\_{A}}} + 0.7 \left[ \frac{\mathbf{T\_{A}}}{^{\circ}\mathbf{C}} - 1600 \right] + 0.85 \frac{\mathbf{t\_{S}} + \mathbf{t\_{N}}}{\min} - 8 \frac{\mathbf{M\_{G}}}{\mathbf{m^{3}}/\mathrm{t}} - 4.3 \frac{\mathbf{M\_{L}}}{\mathrm{m^{3}}/\mathrm{t}} \tag{1}$$

where Wr is the electric energy demand, kWh/Mg; GE is the weight of ferrous materials, kg; GA is the tap weight (mass of the bath), kg; GZ is the weight of slag components, kg; TA is the temperature before taping, ◦C; tS is the power on time, min; tN is the power off time, min; MG is the natural gas consumption, Nm3; and ML is the oxygen consumption, Nm3.

This model characterises the electric energy demand, which in this case (of the units analysed by Köhle) is between 380 and 600 kWh/Mg with an accuracy equal to the standard deviation of 5 kWh/Mg.

The disadvantage of this solution is the low sensitivity of the model to changes in process conditions (occurrence of disturbances, change of the melting programme for the arc control system, or burners) which translates into a high error for individual melts. However, it is also an advantage in terms of long-term prediction. The Köhle model, due to its hybrid empirical–theoretical character, remains easy to interpret while maintaining sufficient accuracy.

In industrial practice, problems are encountered due to, e.g., insufficient process metering and difficult definition of boundary conditions. The amount of electrical energy supplied is easy to determine, whereas the efficiency of heat transfer through the arc depends on a large number of factors, e.g., the melting stage, the phenomena associated with arc burning (arc length, arc stability), or the plasma temperature associated with the gas atmosphere of the furnace, as well as the parameters characterising the steel scrap. Identifying the efficiency of the use of chemical energy—dependent on the reactions taking place in the reactor, which is the volume of the furnace bowl—is also a challenge. Its determination requires the identification of the reactions taking place and the distribution of the heat obtained between the phases, as well as the dynamics of the process [11–13,20].

When constructing models, it is necessary to perform mass and energy balances. Carrying out correctly defined balance calculations (of mass and energy) is essential from the point of view of the overall characteristics of the energy efficiency of the process and the determination of process boundary conditions [11].

In the course of the analyses, a database of indicators and technological parameters was created, which included 1953 records from one specific EAF. Due to a wide range of steel grades, the database was based on the most frequently manufactured group of steel grades and data was selected by rejecting melts:


The parameters included in the database are batch, material, and energy parameters.

#### *2.3. Mass Balances*

The mass balance is based on the assumption that the sum of the masses of the inputs min equals the sum of the outputs mout.

$$\mathbf{m}\_{\rm bST} + \mathbf{m}\_{\rm sf} + \mathbf{m}\_{\rm c} + \mathbf{m}\_{\rm fa} + \mathbf{m}\_{\rm ox} + \mathbf{m}\_{\rm CPI\_{\rm f}} + \mathbf{m}\_{\rm h} = \mathbf{m}\_{\rm hm} + \mathbf{m}\_{\rm sl} + \mathbf{m}\_{\rm f} + \mathbf{m}\_{\rm Wg} \tag{2}$$

where mscr is the mass of the scrap, kg; msf is the mass of the slag formers, kg; mox is the mass of the oxygen, kg; mCH4 is the mass of the natural gas, kg; mc is the mass of the carburizer, kg; mfa is the mass of the foaming agent, kg; mla is the mass of the leaked air, kg*;* mhm is the mass of the hot metal, kg; msl is the mass of the slag, kg; mwg is the mass of the waste gases, kg; and mf is the mass of the fumes, kg.

For the purposes of consideration, the assumption has been made that the mass of the hot heel does not change between heats. In reality, the mass of hot heel changes and is difficult to estimate due to the relatively small volume it occupies and the constant irregular changes in the scale changing the volume of the furnace hearth.

#### 2.3.1. Slag Weight Calculations

Slag mass was calculated on the basis of the chemical composition of the scraps used, assuming that the elements contained in them: Si, Mn, Cr, C, P, and Al are oxidised to the average of the level characterising the composition of the metal bath, and the amount of oxidised Fe constitutes 5.5% of the charge mass. These elements and the oxygen needed to burn them together with the addition of CaO and MgO make up the slag mass. The slag mass also includes ash from carbon carriers.

Chemical compositions of the slags were identified via energy dispersive X-ray fluorescence (XRF Oxford X-MET 3000TX +). In order to identify the chemical composition of the steel, a spectrophotometric method of analysis was used on a FOUNDRY-MASTER spark spectrometer.

The correctness of assumptions and calculations was checked by calculating the slag mass on the basis of CaO introduced into the process, which, as a component not dissolving in the metal bath, theoretically remains entirely in the slag.

$$\mathrm{m\_{sl}} = \frac{\left(\mathrm{m\_{CaO\\_met.}} \cdot 0.958 + \mathrm{m\_{CaO\\_inj.}} \cdot 0.864\right) \cdot 100\%}{\% \mathrm{CaO\_{slag}}}\tag{3}$$

where mCaO met. is the mass of the metallurgical calcium, kg; mCaO inj. is the mass of the injected pulverized calcium, kg; and %CaOslag is the percentage of mass weight of CaO in slag.

A calculation was then made of the mass of each element that was oxidised during bath melting and slag formation, thus reducing the metal yield.

$$\mathbf{m}\_{\rm ae} = \frac{\mathbf{M}\_{\rm A\_{\rm v}} \cdot \%t! \cdot \mathbf{m}\_{\rm sl}}{\mathbf{M}\_{\rm A\_{\rm o}}}.\tag{4}$$

where mae is the mass of the alloying element melted in slag, kg; %tl is the percentage of mass weight of given element in slag; M Ae is the atomic mass of element, u; and M Ao is the atomic mass of oxide of given element, u.

#### 2.3.2. Calculation of Waste Gas Masses

In order to calculate the mass of waste gases taking into account the course of the reaction of combustion of coal and methane, Reactions (5)–(7) were written:

$$\text{C} + \frac{1}{2}\text{O}\_2 \rightarrow \text{CO} \tag{5}$$

$$\text{C} + \text{O}\_2 \rightarrow \text{CO}\_2 \tag{6}$$

$$\text{CH}\_4 + 2\text{O}\_2 \rightarrow \text{CO}\_2 + 2\text{H}\_2\text{O} \tag{7}$$

The mass of carbon monoxide and carbon dioxide for each reaction for one kilogram of product burnt was calculated. The values obtained were, respectively, 2.33 kg, 3.67 kg, and 2.75 kg (from Reactions (5)–(7)). Due to the lack of measurement of the flue gas composition, it was assumed that all the substrates burned completely, and left entirely with the flue gases. Therefore, in the mass of waste gasses mwg, the components according to the Equation (8) are included:

$$\mathbf{m}\_{\rm W\%} = \mathbf{m}\_{\rm N\_2} + \mathbf{m}\_{\rm CO\_2} + \mathbf{m}\_{\rm H\_2O} \tag{8}$$

#### 2.3.3. Calculation of Metal Bath Weight

The theoretical weight of the metal bath was calculated as the difference between the weight of the scrap, the alloying elements contained in the bath (including iron), and the estimated amounts of fumes.

$$\mathbf{m}\_{\rm hm} = \mathbf{m}\_{\rm scr} - \boldsymbol{\Sigma}\mathbf{m}\_{\rm ae} - \mathbf{m}\_{\rm f} \tag{9}$$

where mhm is the mass of hot metal, kg; mscr is the mass of the scrap, kg; mae is the sum of masses of alloying elements that went into slag, kg; and mf is the mass of the fumes, kg.

#### *2.4. Energy Balances*

The energy balance assumes that the sum of the energy input Ein into the process is equal to the sum of the energy output from the process in the form of product heat Qout and thermal losses Qloss. The energy input to the system consists of electrical energy and chemical energy. In the other direction, the energy is transferred out of the process in the form of metal baths, liquid slag, waste gases, dusts, and heat losses.

$$\rm E\_{el.} + E\_{chem.} = Q\_{hm} + Q\_{sl} + Q\_{wg} + Q\_{f} \tag{10}$$

where Eel. is the electrical energy, MJ; Echem. is the chemical energy, MJ; Qhm is the heat of hot metal, MJ; Qsl is the heat of the slag, MJ; Qwg is the heat of waste gases, MJ; and Qf is the heat of the fumes, MJ.

The chemical energy was calculated on the basis of enthalpies of exothermic reactions— oxidation of bath components and combustion of methane. The Perry Nist Janaf method was used to calculate the enthalpies of the reactions.

#### 2.4.1. Calculation of the Physical Heat of Hot Metal

Calculation of the physical heat of the hot metal is based upon the values of the specific heat of the hot metal, its mass, and the temperature rise. It also takes into account the phase transition by using the values of latent heat, liquid temperature, and differentiation of specific heat into that occurring in the liquid and solid state.

For the calculation of the physical heat of the metal bath, the Formula (11) is used:

$$\mathbf{Q\_{hm}} = \mathbf{m\_{hm}} \cdot \left( \mathbf{c\_{sc}} \cdot \left( \mathbf{T\_m} - \mathbf{T\_a} \right) + \mathbf{Q\_{lst}} + \mathbf{c\_k} (\mathbf{T} - \mathbf{T\_m}) \right) / 1000 \tag{11}$$

where, Qhm is the physical heat of the metal bath at temperature T, MJ; mhm is the mass of hot metal, kg; css is the average specific heat of steel in the solid state, kJ/kg K; cls is the average specific heat of steel in the liquid state, kJ/kg·K; Qlst is the latent heat of fusion of steel, kJ/kg; T is the steel tap temperature, K; T m is the steel melting temperature, and K; Ta is the ambient temperature, K.

The specific heat of the liquid state was taken as 0.836 MJ/Mg·<sup>K</sup> and that of the solid state as 0.698 MJ/Mg·K. The latent heat of fusion for the steel was taken as 271.7 kJ/kg.

The melting point of steel with a hot metal composition was calculated according to the Formula (12):

$$\mathbf{T\_m} = \mathbf{T\_{Fe}} - \sum\_{i=1}^{m} \mathbf{p\_i} \mathbf{k\_i} \tag{12}$$

where Tm is the melting point, K; TFe is the melting point of pure iron (1812 K); pi is the percentage content of the element in the metal bath, %; and ki is the temperature reduction factor (according to the Table 1) [21].

**Table 1.** Temperature reduction factor for individual elements [21].


2.4.2. Calculation of the Physical Heat of Slag

The calculation of the physical heat of slag was based on the values of the slag's specific heat, mass, temperature rise, and latent heat. The following Formula (15) was used to calculate the physical heat of slag:

$$\mathbf{Q}\_{\rm sl} = \frac{\mathbf{m}\_{\rm sl} (\mathbf{c}\_{\rm sl} (\mathbf{T} - \mathbf{T}\_{\rm a}) + \mathbf{Q}\_{\rm lsl})}{1000} \tag{13}$$

where Qsl is the physical heat of slag at temperature T, MJ; msl is the slag mass, kg; csl is the specific heat of slag at temperature T*,* kJ/kg·K; Qlsl is the latent heat of melting of slag, kJ/kg; T is the slag tap temperature, K; and Ta is the ambient temperature, K.

The specific heat of slag was calculated from the Formula (14):

$$\mathbf{c}\_{\rm sl} = 0.736 + 2.93 \cdot 10^{-4} \mathbf{T} \tag{14}$$

The assumed latent heat of fusion is 209 kJ/kg.

#### 2.4.3. Calculation of the Physical Heat of Waste Gases

The calculation of the physical heat of the waste gases is based on the specific heat values of the gases of their mass and temperature rise. The formula was used to calculate the physical heat of the waste gases:

$$\mathbf{Q\_{w\_{K}}} = \mathbf{m\_{w\_{\overline{\otimes}}}c\_{w\overline{\otimes}}(T - T\_{\overline{a}})/1000} \tag{15}$$

where Qwg is the physical heat of the waste gases at temperature T, MJ; mwg is the mass of the waste gases, kg; cwg is the specific heat of the waste gases at temperature T, kJ/kg·K; T is the temperature of the gases, K; and Ta is the ambient temperature, K.

The average specific heat of the waste gases is 1.33 kJ/Nm3, whereas 1450 ◦C was taken as the value of gas temperature.
