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 non-linearity 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 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].
3. Results
In the field of research, considering modelling (MLR) as a first step, an attempt was made to apply and subsequently modify the Köhle equation [
22]. Based on the analyses carried out on a furnace operating under the industrial conditions, one should state the lack of universality for the form of the model proposed by Köhle. The obtained values differ from the measured ones by an average of 50% for the 1992 version and by an average of 10% for the 2002 version of the equation [
23]. As part of the work performed, the model was modified, and one of the changes made was the replacement of the M
L coefficient relating to oxygen consumption by an equation taking into account the heat gain from the oxidation of individual elements. This is due to the previously mentioned large number of scrap classes, as well as the large share of the iron oxidation reaction as a source of chemical energy. Therefore, the replacement of the coefficient that represents the average heat gain by a separate function is justified. This function is written in the form:
where M
sl is the mass of slag, Mg; M
AM is the atomic weight of considered metal, u; avg%
MO.sl is the average considered metal oxide content of slag, %; M
AMO is the atomic weight of considered metal oxide, u; and ΔH
M is the the enthalpy of the oxidation reaction of a considered element, converted to kWh/kg; (where M = Cr, Mn, Fe, Al, Si).
Figure 1 shows the comparison of the real values of electricity consumption with the model values obtained based on two versions of the Köhle model: the 2002 version and its modification. It can be noted that for the modified version a much higher model fit was obtained (average difference 1%, maximum 20%).
In the realities of the furnace characterised, the model did not adequately provide quantitative answers as to what steps and extent of change should be taken to optimise the process.
Therefore, mass and energy balances were calculated and a linear multiple regression (MLR) model was undertaken.
For the calculation of mass and energy balances, it was necessary to identify the chemical composition of the liquid products (slag and steel). For this purpose, slag and metal samples were taken. A study of the chemical composition of slags was carried out for 200 melts in the same group of steel grades. The obtained average chemical compositions of slag and steel are presented in
Table 2 and
Table 3.
In the case of the unit under consideration, the insufficient metering, the high diversification of the steel grades, and the high variability of the charge (based on more than 40 scrap grades) led to some simplifications. For example, assuming a constant hot heel level; ignoring the change in furnace geometry resulting from erosion of the lining and, thus, assuming that the oxygen stream always reaches the bath and is fully consumed; and assuming that the chemical composition of the scrap corresponds to the average of the ranges described in PN 85/H 15000.
The results of the mass and energy balances obtained, and their analysis, clearly indicate the areas which should be investigated in order to improve the scrap melting process, e.g., appropriate use of natural gas, problems with furnace leak-tightness, and use of oxygen. However, the predominance of chemical energy in the melting process suggests that its optimisation will have a significant impact on the overall energy balance of the melt.
The parameters used to create the MLR were time, which is related to the amount of heat loss in the process; oxygen and gas consumption, as sources of chemical energy; the amount of slag-forming materials used, whose melting requires a significant amount of energy; and the scrap yield, which gives information on the amount of iron oxidised.
Calculations and multiple regression analyses were carried out to obtain the equation identifying the demand for electric energy.
Table 4 presents the results for the selected backward MLR analysis, which gave the selected statistical model—the form of Equation (17). Statistical calculations were performed for the total sample size, which was N = 1959; the significance level of α = 0.05 was assumed. Statistical correlation between the considered independent variables was checked; the relationships between the variables present in the equation were statistically insignificant.
Statistical results were obtained: the regression coefficient at the level of R = 0.497; comparing for Rtable = 0.1946 (for N > 1000), one can state the significance of the calculated linear correlation. Fisher–Snedecor’s F-value, at the α = 0.05 level, F(5.1953) = 128; for Ho’s hypothesis, comparing it with the table critical value of Ftable = 2.108, Ho’s hypothesis of statistical significance R2 of the equation—can be accepted at the significance level of α = 0.05, which implies the possibility of rejecting the alternative hypothesis. It can also be seen that for all selected variables, the probability p is less than the significance level of 5% (p < α), and the independent variables present in the equation are individually statistically significant.
Following the above, a linear regression model was created (3), resulting in:
where t
ttt is the tap to tap time, min.; m
scr is the mass of scrap in charge, kg; m
ox is the mass of oxygen, kg; m
CH4 is the mass of natural gas, kg; m
sf is the mass of slag formers, kg; and m
hm is the mass of hot metal, kg
In
Figure 2, the normality analysis of the distribution of the residuals was shown. It seems, Shapiro–Wilk test: SW–W = 0.9971,
p = 0.0009. Variance constancy: fulfilled, the uniformity of the scatter of the residuals is constant across the width of the interval.
Simulations using the created MLR model confirmed the conclusions of the analysis of the balances performed, i.e., the predominance of chemical energy and the inverse proportionality of methane consumption to electrical energy consumption.
Taking into account the analysis results, which confirmed the conclusions of the mass and energy balance, the description of the process efficiency (defined as the ratio of the supplied energy to the amount of physical heat of the metal bath) as a function of the share of chemical energy was undertaken.
In
Figure 3, the analysis of the dependence of process efficiency on the share of chemical energy was shown. A decrease in process efficiency (η
EAF) is visible with an increase in the share of chemical energy in the process. This trend in melt efficiency prompted an analysis of the efficiency of chemical energy use in the process under study.
As an example, the amount and timing of CH4 addition was analysed as a function of the efficiency of the heat generated by its combustion.
It was assumed that the efficiency will change with the progress of scrap melting. This is due to phenomena such as a change in the scrap surface with the progress of melting, but most importantly due to a decrease in the temperature difference between the flame and the charge, which heats up during the process. The change in burner efficiency was re-calculated on the basis of the approach presented in [
21,
24,
25] and is presented in
Figure 4.
The energy efficiency of the burners was considered first. This varies with the progress of melting. This efficiency is, on average, between 70 and 40%, and can drop as low as 20%. To identify the energy transfer efficiency, the Gottardi methodology was used [
25], which made the energy efficiency of the burner dependent on, among other things, the temperature difference between the flame and the charge, which decreases with the progress of melting and, additionally, with a decrease in the scrap surface area. This dependence is described by the function (18):
where
is the energy efficiency of the burner [%]; and p
t is the melting progression [%].
Based on the mathematical description of the change in burner efficiency as a function of melting progress, three patterns of energy gain as a function of the average gas flow possible under the conditions of the unit analysed were simulated. Boundary conditions were established: minimum gas flow in burner mode, 90 Nm3/h; the currently used average gas flow rate, 110 Nm3/h; and near-maximum flow rate, 200 Nm3/h. The maximum burner operation time—resulting from the melting stage characterised by the presence of scrap in the “cold regions” of the walls—is 30 min in total for all the baskets.
The change in energy contributed by the burners over time (
Figure 4), the cumulative energy per 30 min (
Figure 5a), and the cumulative energy as a function of the maximum possible amount of gas supplied (
Figure 5b) were considered.
From the simulations obtained, it can be concluded that it is beneficial to intensify the gas flow in the initial stage of the melt, as this allows the burner to use its moment of greatest efficiency. The calculated maximum achievable gas consumption is, respectively, 135, 165, and 300 Nm3, and the energy contributed is 826 kWh, 1009 kWh, and 1835 kWh.
Finally, the equation was created to simulate production costs (in the sense of direct manufacturing costs), CO2 emissions, and process efficiency (Mg of hot metal/h)—it is assumed that such a construction of the model will allow to control the technology in a manner depending also on economic objectives.
The attempt to consolidate the results in terms of obtaining the unit melt cost function contributed to the creation of Equation (19).
where P
c is the production cost; X
1 is the cost per unit of electrical energy, PLN/kWh; X
2 is the cost per unit of natural gas, PLN/Nm
3; X
3 is the cos per unit of technical oxygen, PLN/Nm
3; X
4 is the average unit cost of scrap purchase, PLN/Mg; X
5 is the cost per emission of CO
2 unit PLN/Mg; and X
6 is the cost per unit of slag formers, PLN/kg.
Equation (5) was used to calculate production costs (in terms of direct production costs of liquid steel), CO
2 emissions, and process efficiency (Mg of hot metal/h)—assuming that such a model construction would allow to control the technology in a manner depending also on economic objectives. After calculating the production costs for the smelters under consideration (forming the database), a correlation analysis was carried out between the production cost (less the price of scrap metal forming the metal bath) and the amount of chemical energy input; the results are presented in
Figure 6.
The correlation coefficient is R = 0.513 and the
t-Student’s
t-value reaches
t = 41.51 with t
tabular = 1.960. From the existence of this relationship (
Figure 6) it should be concluded that it is advantageous to increase the share of chemical energy in the process, but taking into account the efficiency of its transfer by the oxygen treating bath stage.