**1. Introduction**

Electrical Energy (EE) can account for between 40–66% of the total energy usage during the Electric Arc Furnace (EAF) process, which is a number that highlights the importance of further improvements in modeling of the EE consumption [1]. The energy losses, which partly governs the EAF process energy dynamics, are mostly related to off-gases, slag, dust, furnace cooling, electrical and radiative losses. Most of these energy losses are closely linked to the total process time of any given heat. The process time itself is influenced by numerous impositions. One of these is the melting time of the charged raw materials, which in the scope of this study is primarily steel scrap.

Many articles have studied and proposed statistical models predicting the EE consumption of the EAF [1,2]. However, only a handful of studies have used scrap types as input variables to a statistical model predicting the EE of an EAF [2–7]. Neither of these studies analyze the contributions of each scrap type on specific predictions by the statistical model. Verifying the effects of the input variables on the complete prediction space is paramount to evaluate the practical usefulness of any statistical model let alone to make the users, i.e., process engineers, trust the model. Only when these two conditions are met can the model possibly be used to solve practical problems. An example of a practical problem where an EE prediction model can be used is when determining the EE requirement for the EAF in a Demand Side Management (DSM) system, which optimizes the processes in a steel plant with respect to the available power in the transmission line [8].

Shapley Additive Explanations (SHAP) is a recent development in the field of interpretable machine learning [9], and has previously been used to analyze a statistical model predicting the EE of an EAF producing stainless steel [10]. Tap-to-Tap time (TTT), delays, and total charged weight were found to be the three most influencing variables. TTT and total charged weight were correctly interpreted by the model with respect to what is known from process metallurgical experience. The delay variable was incorrectly interpreted by the model, which was concluded to be due to the high correlation between delays with TTT.

The aim of the current study is to investigate the effect of scrap types on a statistical model predicting the EE of an EAF. Although verifying the effects of other input variables on the model output is important, it is not the main focus of this article. For such an analysis we refer to previous studies [2,10]. To investigate the effects of scrap, three distinct representations of scrap types based on the plant scrap codes, scrap physical shape, and scrap apparent density, respectively, will be used in the models. The reasons are two-fold. First, to provide the steel plant engineers with an intuitive and simple method to categorize scrap for modeling purposes. Second, to find the optimal scrap representation for the prediction problem with respect to the accuracy and precision of the statistical model.

In addition to its scrap-oriented focus, this study further builds on the modeling methodology presented in two previous studies [2,10]. Four different data cleaning strategies will be used to investigate the effect of data cleaning on the accuracy of the statistical model and an additional non-linear statistical model framework will be employed; Random Forests (RF). Furthermore, this study uses data from an EAF producing steel for tubes, rods, and ball-bearing rings and not from an EAF producing stainless steel, thus broadening the application of the modeling methodology.

The results demonstrated that the three subsets of input variables provided by the scrap representations all increase the performance of the models. Using SHAP, it was found that heavy scrap, i.e., scrap with low surface-area-to-volume ratio, contributed to an increased EE consumption while steel sheets, a scrap type with high surface-area-to-volume ratio, contributed to a decreased EE consumption. These findings were confirmed by the steel plant engineers to agree well with previous experiences using these scrap types as raw material.

#### **2. Background**

#### *2.1. Melting of Steel Scrap in Liquid Steel*

#### 2.1.1. Driving Forces

The driving forces in scrap melting are present in any process in which scrap melting occurs. However, the driving factors vary significantly between the EAF and Basic Oxygen Furnace (BOF). This section highlights the driving factors in scrap melting with respect to the EAF in the steel plant of study. The goal is not to create an extensive review over the various research topics governing the melting of scrap, but rather to motivate the scrap representations used in the experimental part of this study. A comprehensive review in the field was compiled by Freidrich [11]. Prominent later developments in the field of scrap melting have been summarized in a recent review [12].

The melting of solid scrap in liquid steel is dominated by several factors. These are temperature gradients between solid scrap and liquid steel, concentration gradients between solid scrap and liquid steel, the freezing effect, the rate of stirring of the steel melt. These phenomena are explained further below.

**Temperature gradients** between the solid scrap and the steel melt is one of the most important driving factors in the melting of steel scrap. The melting rate, in m/s, can be determined by the following equation:

$$\frac{d\mathbf{x}}{dt} = h \cdot \frac{T\_{HM} - T\_{liq}}{\rho\_{scr} \cdot (H\_s + (T\_{HM} - T\_{liq}) \cdot \mathbf{c}\_p)}\tag{1}$$

where *THM* is the temperature of the molten steel and *Tliq* is the scrap melting temperature. Furthermore, *Hs* is the heat of melting of scrap, *cp* is the specific heat of scrap, *ρscr* is the density

of the scrap metal, and *h* is the heat transfer coefficient in the interface of the molten steel and scrap [12]. The higher the temperature gradient, *THM* − *Tliq*, the faster the steel scrap will melt.

**Alloying element gradients** also contribute to the melting of scrap in a process known as dissolution. In this case, alloying elements migrate to the solid-liquid metal interface. The most dominant alloying element in this process is carbon. However, in the BOF, the carbon concentration difference can exceed 4 wt-% while in the EAF the carbon concentration difference seldom exceeds 1 wt-%.

Assuming that the dissolution rate by carbon can be determined by the shortest length of the scrap, the following equation can be used:

$$\frac{d\mathbf{x}}{dt} = \frac{\beta \cdot (\mathbf{C}\_l - \mathbf{C}\_i)}{\rho\_s (\mathbf{C}\_l - \mathbf{C}\_0)}\tag{2}$$

where *β* is the mass transfer coefficient of carbon, *Cl* is the carbon content in the liquid steel, *C*<sup>0</sup> is the initial carbon content in the steel scrap, and *Ci* is the carbon content in the solid-liquid interface [13].

A similar equation can be determined for silicon, which can also be an alloying element to account for should the difference in silicon concentration between steel scrap and the molten steel be large.

The **freezing effect** occurs in the solid-liquid interface when the scrap first comes in contact with the liquid steel. A solidified shell is formed due to the large temperature difference between the two. This means that the volume of the scrap increases initially. The solidified shell is proportional to the surface area of the scrap that is submerged in the hot metal or molten steel. Hence, the reduction in the steel scrap size does not occur instantly, rather it decreases after the solidified shell has melted.

The **stirring** velocity is the velocity of the melt in the boundary layer between the melt and the scrap surface area. Numerous studies have related the stirring velocity to the mass transfer coefficient on scrap in liquid steel. However, there exists a wide range of reported mass transfer coefficient values for scrap in liquid steel under forced convection [12]. Nevertheless, a commonly deduced relationship between the mass transfer coefficient and the stirring velocity may be written as follows:

$$h\_{scr} = c \cdot u^p \tag{3}$$

where *hscr* is the mass transfer coefficient under forced convection, *c* and *p* are constants that are determined experimentally. *u* is the average stirring power, which is related to the average stirring velocity due to the physical relationship between energy, momentum and velocity. The stirring power is governed by, for example, oxygen blowing and carbon boil.

Furthermore, the effect of stirring on the melting rate of scrap in the EAF is low compared to the BOF since the stirring is more intense in the latter, i.e., higher stirring velocity governed by the stirring power per unit volume. Furthermore, one should not expect the stirring to be very intense in the EAF since the liquid steel depth is low and the solid-to-liquid ratio is high prior to the final stages of the process, i.e., superheating, which hampers the flow velocity of the liquid steel. The device that primarily facilitates stirring in the EAF is oxygen lancing, but other devices such as porous plugs and induction stirring enhance the stirring.

#### 2.1.2. Scrap Surface-Area-to-Volume Ratio

It is evident that the aforementioned factors are dependent on the surface-area-to-volume ratio of the scrap pieces. On the one hand, the effect from temperature and alloying element gradients influence on the complete surface area exposed to the steel melt. On the other hand, the mass of the steel scrap piece determines the melting time since more mass needs to be heated (Equation (1)) and more mass of the alloying elements have to be transported (Equation (2)). The mass is proportional to the volume of the scrap piece. Thus, the surface-area-to-volume ratio can be expressed either as a function of the surface area and volume or as a function of the surface area, apparent density of scrap, and mass of the scrap piece:

$$R\_{SV} = \frac{A}{V} = \frac{A \cdot \rho\_s}{m} \tag{4}$$

Hence, to facilitate lower melting times one should use scrap that has a high surface-areato-volume ratio. The surface-area-to-volume ratios of some elementary geometrical shapes are presented below to illuminate the effects of the surface-area-to-volume ratio in scrap melting. However, real scrap pieces are often of more complex geometric shapes.

\*\*Sphere:\*\*
 $\frac{A}{\mathcal{V}} = \frac{4\pi r^{2}}{\frac{4\pi r^{3}}{3}} = \frac{3}{r}$ 

\*\*Cylinder:\*\*
 $\frac{A}{\mathcal{V}} = \frac{2\pi rl + 2\pi r^{2}}{\pi r^{2}l} = 2\left(\frac{1}{r} + \frac{1}{l}\right)$ 

\*\*Cube:\*\*
 $\frac{A}{\mathcal{V}} = \frac{6l^{2}}{l^{3}} = \frac{6}{l}$ 

\*\*Square\*\*
 $\frac{A}{\mathcal{V}} = \frac{2l^{2} + 4lt}{l^{2}t} = 2\left(\frac{1}{t} + \frac{2}{l}\right)$ 

The thickness is defined as the thinnest dimension of the scrap piece. For the cylinder, one ought to keep *r* << *l*. For the square plate, one ought to keep *t* << *l*. The cube and sphere are equidistant from the center of the scrap piece, which means that one should keep the length and radius as small as possible, respectively.
