**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 ML 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:

$$\mathrm{pH} = \sum\_{\mathrm{M}} \left( \mathrm{M}\_{\mathrm{sl}} \cdot \frac{\mathrm{M}\_{\mathrm{AM}} \cdot \mathrm{avg} \, \%\_{\mathrm{MO}, \mathrm{sl}} \cdot 1000 \, \frac{\mathrm{kg}}{\mathrm{Mg}}}{100 \, \% \cdot \mathrm{M}\_{\mathrm{AMO}}} \cdot \Delta \mathrm{H}\_{\mathrm{M}} \right) \tag{16}$$

where Msl is the mass of slag, Mg; MAM is the atomic weight of considered metal, u; avg%MO.sl is the average considered metal oxide content of slag, %; MAMO is the atomic weight of considered metal oxide, u; and Δ H 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%).

Real energy consumption on the furnace, [kWh/Mg]

**Figure 1.** Comparison of Köhle model results with real electricity consumption [23].

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 Tables 2 and 3.

**Table 2.** Average chemical composition of EAF slags before tap.



**Table 3.** Average chemical composition of liquid bath before tap.

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.


**Table 4.** Statistical parameters of calculated MLR model.

b\* is a structural parameters of the regression function; B is an estimators of obtained regression equation; t is a t-Student test probability distribution; and *p*-value is a probability level of variable.

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 R<sup>2</sup> 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:

$$\mathbf{E}\_{\rm el} = 169.94 + 0.26 \cdot \mathbf{t\_{TT}} + 3.27 \cdot \frac{\mathbf{m\_{ox}}}{\mathbf{m\_{scr}}} - 1.98 \cdot \frac{\mathbf{m\_{CH\_4}}}{\mathbf{m\_{scr}}} + 0.11 \cdot \frac{\mathbf{m\_{sf}}}{\mathbf{m\_{scr}}} + 48.79 \cdot \frac{\mathbf{m\_{hml}}}{\mathbf{m\_{scr}}} \tag{17}$$

where tttt is the tap to tap time, min.; mscr is the mass of scrap in charge, kg; mox is the mass of oxygen, kg; mCH4 is the mass of natural gas, kg; msf is the mass of slag formers, kg; and mhm 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.

**Figure 2.** Normal distribution of residuals.

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 (ηEAF) 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.

**Figure 3.** Dependence of process efficiency on the share of chemical energy.

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.

**Figure 4.** Burner efficiency as a function of melting time progression [%] (own study based on data from [21]).

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):

$$
\eta\_{\rm CH\_4} = 70.272 - 8 \cdot 10^{-3} \cdot \text{p}\_{\rm t}^2 + 5 \cdot 10^{-5} \cdot \text{p}\_{\rm t}^3 \tag{18}
$$

where ηCH4is the energy efficiency of the burner [%]; and pt 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.

**Figure 5.** (**a**) Variation of burner energy input as a function of burner operating time (**b**) cumulative energy as a function of burner operating time

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).

$$\mathbf{P\_{c}} = \mathbf{E\_{el}} \cdot \mathbf{X\_{l}} + \mathbf{m\_{CHi}} \cdot \mathbf{X\_{2}} + (\mathbf{m\_{6T}} - \mathbf{m\_{hm}}) \cdot \mathbf{X\_{4}} + \mathbf{m\_{0i}} \cdot \mathbf{X\_{3}} + \left(\mathbf{m\_{C}} + \mathbf{m\_{CHi}}\right) \cdot \frac{44}{16} \cdot \mathbf{X\_{5}} + \mathbf{m\_{st}} \cdot \mathbf{X\_{6}} \tag{19}$$

where Pc is the production cost; X1 is the cost per unit of electrical energy, PLN/kWh; X2 is the cost per unit of natural gas, PLN/Nm3; X3 is the cos per unit of technical oxygen, PLN/Nm3; X4 is the average unit cost of scrap purchase, PLN/Mg; X5 is the cost per emission of CO2 unit PLN/Mg; and X6 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), CO2 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.

**Figure 6.** The amount of chemical energy in relation to the direct cost of producing a tonne of metal bath.

The correlation coefficient is R = 0.513 and the *t*-Student's *t*-value reaches *t* = 41.51 with ttabular = 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.

## **4. Discussion**

Modelling of the melting process in an electric arc furnace is possible under industrial conditions, even with incomplete metering. However, the individual characteristics of the unit under consideration have to be taken into account as existing models do not show universality. The use of models developed for other units requires their modification and taking into account the characteristics resulting from the proportion and share of chemical and electric energy as well as the input and technological conditions of the furnace.

Increasing the chemical energy input into the melt lowers the required total amount of electrical energy—methane consumption is inversely proportional to electrical energy consumption. However, due to the technological effect of the fact that oxygen is fed most intensively at the time of charge reheating—throughout this period—its correlation with electrical energy is positive.

Based on model calculations for the analysed furnace unit using a modified Köhle model from 2002, the model and actual data converged at the level of R = 0.36. The calculated electricity demand differed from the actual demand at the maximum level of 20%. However, it did not give sufficient guidance for process control.

Thanks to the mass and energy balance calculations performed, results were obtained from which it can be concluded that there is a predominance of chemical energy (52%) in the process compared to electrical energy.

It can be concluded that it is advantageous to intensify the gas flow at the initial stage of melting, as this makes it possible to use the moment of maximum efficiency of the burners (presented on Figure 5a,b). These conclusions require a complementary analysis; the real conditions of the process should be taken into account, which limit the maximum flow on the oxy–gas burners in the initial stages of the process (risk of burner damage by flame reflection from the scrap metal near the walls) and the fact that a higher energy input will lead to a faster melting of the deposited scrap metal

The statistical calculations carried out and the analyses of the statistical indices obtained made it possible to select the multiple linear regression model Equation (17)—as a method giving the possibility of an easy interpretation with an appropriate representation

of the process in these conditions. The regression coefficient for the calculated model is R = 0.497, the difference between the calculated value and the real one is, at maximum, 17% (with mean 0). The obtained model was extended by the analysis of natural gas efficiency and consolidated with the cost calculation equation.

**Author Contributions:** Conceptualization, P.M. and M.K.; methodology, M.M.; resources, M.M.; data curation, M.M.; writing—original draft preparation, M.M.; writing—review and editing, P.M. and M.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The present work was write as part of the second edition of "Implementation Doctorate" programme. The authors gratefully acknowledge their support.

**Conflicts of Interest:** The authors declare no conflict of interest.
