# Thermophysical Model for Online Optimization and Control of the Electric Arc Furnace

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Control Volumes and State-Space Variables

#### 2.2. Chemical Reactions

_{2}O

_{3}.

#### 2.3. Heat Transfer

#### 2.4. Solid–Liquid Phase Change

#### 2.5. Arc Efficiency

## 3. Results and Discussion

#### 3.1. Model Behavior

#### 3.2. Recursive Estimation of Arc Efficiency

#### 3.3. Industrial Use and Application

^{TM}Cenit software. The goal of these MPC simulations is to optimize the electrical power input in order to increase the overall efficiency of the arc power. In order to adapt the process data for the MPC study, the basket contents and schedule of charges are preserved according to logged data without the exact schedule being preemptively revealed to the MPC. Logged power input is overwritten by the closed-loop simulation. Because the operation of the gas burners should be in sync with the accumulated electrical energy added to the furnace, logged LNG and oxygen flows are replaced with the gas burner recipe used in plant operation. Logged time delays and pauses in electrical power supply are preserved in the MPC simulations.

^{TM}Cenit implementation of MPC, the optimization takes the form of minimizing an objective function. Because the EAF is operated as a batch process, the process outputs that contribute to the objective function are evaluated at the end of the batch (the time at which the model predicts the furnace contents are fully melted). The MPC algorithm seeks to simultaneously minimize the total batch time and maximize the efficiency of the electric arc based on the objective function J:

- ${U}_{1-15}$ or MV${}_{1-15}$: Shift from the nominal power profile during optimization interval $\left(\right)$ (MW).

- ${Z}_{1}$ or CV${}_{1}$: Batch time (seconds)
- ${Z}_{2}$ or CV${}_{2}$: Energy losses from the electric arc (kWh).

- One-basket heats: 15.75 kWh/tonne per heat (2315 kWh per heat, based on an average charged weight of 147 tonne);
- Two-basket heats: 13.32 kWh/tonne per heat (1945 kWh per heat, based on an average charged weight of 146 tonne);
- Three-basket heats: 6.78 kWh/tonne per heat (983 kWh per heat, based on an average charged weight of 145 tonne).

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations and Nomenclature

## Abbreviations

EAF | Electric Arc Furnace | |

MPC | Model Predictive Control | |

NMPC | Non-linear Model Predictive Control | |

SEC | Specific Energy Consumption | |

KF | Kalman Filter | |

LNG | Liquefied Natural Gas | |

VF | View Factor | |

CV | Controlled Variable | |

MV | Manipulated Variable | |

DRI | Direct Reduced Iron |

## Nomenclature

${A}_{ij}$ | Area for heat transfer between mass i and mass j | m${}^{2}$ |

${d}_{i}$ | Diameter of i | m |

$\frac{\mathrm{d}}{\mathrm{d}t}$ | Derivative operator | |

${\u03f5}_{j}$ | Radiation emissivity of surface j | |

${F}_{i}$ | Molar rate of change of component i | $\frac{\mathrm{mol}}{\mathrm{s}}$ |

${C}_{\mathrm{p},\phantom{\rule{4pt}{0ex}}i}$ | Heat capacity of i | $\frac{\mathrm{J}}{\mathrm{kg}\phantom{\rule{4.pt}{0ex}}\mathrm{K}}$ |

${h}_{k}$ | Height of mass k | m |

${H}_{i}$ | Enthalpy of component i | $\frac{\mathrm{J}}{\mathrm{kg}}$ |

${k}_{ij}$ | Heat transfer coefficient between type i and type j | $\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ |

${m}_{i}^{j}$ | Mass of component i in phase j | kg |

${M}_{i}$ | Molar mass of component i | $\frac{\mathrm{g}}{\mathrm{mol}}$ |

${Q}_{ij}$ | Heat flowing from mass i to mass j | W |

${P}_{k}$ | Power from source k | MW |

${p}_{i}$ | Pressure of phase i | Pa |

${r}_{i}$ | Reaction rate of reaction i | $\frac{\mathrm{kg}}{\mathrm{s}}$ |

${r}_{\mathrm{particle}}$ | Radius of particle | m |

${\rho}_{i}$ | Density of i | $\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}$ |

${\sigma}_{SB}$ | Stefan-Boltzmann constant for radiation | $5.670374\times {10}^{-8}$$\frac{\mathrm{W}}{{\mathrm{m}}^{2}{\mathrm{K}}^{4}}$ |

t | Time | s |

${T}_{j}$ | Temperature of phase j | K |

${v}_{i}$ | Stoichiometric coefficient of component i in a chemical reaction | |

${V}_{k}$ | Volume of mass k | m${}^{3}$ |

${W}_{i}$ | Mass rate of change of component i | $\frac{\mathrm{kg}}{\mathrm{s}}$ |

${x}_{i}^{j}$ | Mass fraction of component i in phase j | |

${x}_{j}$ | Area fraction of control volume j | |

subscript: b | Liquid slag | |

subscript: c | Solid slag | |

subscript: g | Gas | |

subscript: l | Liquid steel | |

subscript: r | Roof | |

subscript: s | Solid steel | |

subscript: v | Vessel |

## Appendix A. Extended Model Details

#### Appendix A.1. Electrical and Chemical Power

**X**O, where

**X**O is the oxide species listed for each reaction in Table A3 and Table A4.

#### Appendix A.2. Reaction Kinetics

**Table A1.**Oxygen consumption and species rate of change for each steel reaction with oxygen. F denotes molar rates of change, v denotes stoichiometric coefficients and x denotes mass fractions.

Reaction | O${}_{2}$ Consumption Rate | Species i Rate of Change |
---|---|---|

$\mathrm{Fe}+\frac{1}{2}\left\{{\mathrm{O}}_{2}\right\}\to \left(\mathrm{FeO}\right)$ | ${F}_{{\mathrm{O}}_{2}}^{\mathrm{Fe}\to \mathrm{FeO}}={x}_{\mathrm{Fe}}^{\mathrm{steel}}{F}_{{\mathrm{O}}_{2}}^{\mathrm{refine}}$ | ${F}_{i}^{\mathrm{Fe}\to \mathrm{FeO}}=\frac{{v}_{i}^{\mathrm{Fe}\to \mathrm{FeO}}}{{v}_{{\mathrm{O}}_{2}}^{\mathrm{Fe}\to \mathrm{FeO}}}{F}_{{\mathrm{O}}_{2}}^{\mathrm{Fe}\to \mathrm{FeO}}$ |

$\left[\mathrm{C}\right]+\frac{1}{2}\left\{{\mathrm{O}}_{2}\right\}\to \left\{\mathrm{CO}\right\}$ | ${F}_{{\mathrm{O}}_{2}}^{\mathrm{C}\to \mathrm{CO}}=\frac{1}{2}{x}_{\mathrm{C}}^{\mathrm{steel}}{F}_{{\mathrm{O}}_{2}}^{\mathrm{refine}}$ | ${F}_{i}^{\mathrm{C}\to \mathrm{CO}}=\frac{{v}_{i}^{\mathrm{C}\to \mathrm{CO}}}{{v}_{{\mathrm{O}}_{2}}^{\mathrm{C}\to \mathrm{CO}}}{F}_{{\mathrm{O}}_{2}}^{\mathrm{C}\to \mathrm{CO}}$ |

$\left[\mathrm{C}\right]+\left\{{\mathrm{O}}_{2}\right\}\to \left\{{\mathrm{CO}}_{2}\right\}$ | ${F}_{{\mathrm{O}}_{2}}^{\mathrm{C}\to {\mathrm{CO}}_{2}}=\frac{1}{2}{x}_{\mathrm{C}}^{\mathrm{steel}}{F}_{{\mathrm{O}}_{2}}^{\mathrm{refine}}$ | ${F}_{i}^{\mathrm{C}\to {\mathrm{CO}}_{2}}=\frac{{v}_{i}^{\mathrm{C}\to {\mathrm{CO}}_{2}}}{{v}_{{\mathrm{O}}_{2}}^{\mathrm{C}\to {\mathrm{CO}}_{2}}}{F}_{{\mathrm{O}}_{2}}^{\mathrm{C}\to {\mathrm{CO}}_{2}}$ |

$\left[\mathrm{Si}\right]+\left\{{\mathrm{O}}_{2}\right\}\to \left({\mathrm{SiO}}_{2}\right)$ | ${F}_{{\mathrm{O}}_{2}}^{\mathrm{Si}\to {\mathrm{SiO}}_{2}}={x}_{\mathrm{Si}}^{\mathrm{steel}}{F}_{{\mathrm{O}}_{2}}^{\mathrm{refine}}$ | ${F}_{i}^{\mathrm{Si}\to {\mathrm{SiO}}_{2}}=\frac{{v}_{i}^{\mathrm{Si}\to {\mathrm{SiO}}_{2}}}{{v}_{{\mathrm{O}}_{2}}^{\mathrm{Si}\to {\mathrm{SiO}}_{2}}}{F}_{{\mathrm{O}}_{2}}^{\mathrm{Si}\to {\mathrm{SiO}}_{2}}$ |

$2\left[\mathrm{Cr}\right]+\frac{3}{2}\left\{{\mathrm{O}}_{2}\right\}\to \left({\mathrm{Cr}}_{2}{\mathrm{O}}_{3}\right)$ | ${F}_{{\mathrm{O}}_{2}}^{\mathrm{Cr}\to {\mathrm{Cr}}_{2}{\mathrm{O}}_{3}}={x}_{\mathrm{Cr}}^{\mathrm{steel}}{F}_{{\mathrm{O}}_{2}}^{\mathrm{refine}}$ | ${F}_{i}^{\mathrm{Cr}\to {\mathrm{Cr}}_{2}{\mathrm{O}}_{3}}=\frac{{v}_{i}^{\mathrm{Cr}\to {\mathrm{Cr}}_{2}{\mathrm{O}}_{3}}}{{v}_{{\mathrm{O}}_{2}}^{\mathrm{Cr}\to {\mathrm{Cr}}_{2}{\mathrm{O}}_{3}}}{F}_{{\mathrm{O}}_{2}}^{\mathrm{Cr}\to {\mathrm{Cr}}_{2}{\mathrm{O}}_{3}}$ |

$2\left[\mathrm{Al}\right]+\frac{3}{2}\left\{{\mathrm{O}}_{2}\right\}\to \left({\mathrm{Al}}_{2}{\mathrm{O}}_{3}\right)$ | ${F}_{{\mathrm{O}}_{2}}^{\mathrm{Al}\to {\mathrm{Al}}_{2}{\mathrm{O}}_{3}}={x}_{\mathrm{Al}}^{\mathrm{steel}}{F}_{{\mathrm{O}}_{2}}^{\mathrm{refine}}$ | ${F}_{i}^{\mathrm{Al}\to {\mathrm{Al}}_{2}{\mathrm{O}}_{3}}=\frac{{v}_{i}^{\mathrm{Al}\to {\mathrm{Al}}_{2}{\mathrm{O}}_{3}}}{{v}_{{\mathrm{O}}_{2}}^{\mathrm{Al}\to {\mathrm{Al}}_{2}{\mathrm{O}}_{3}}}{F}_{{\mathrm{O}}_{2}}^{\mathrm{Al}\to {\mathrm{Al}}_{2}{\mathrm{O}}_{3}}$ |

**Table A2.**Equilibrium constants for reactions. ${T}_{\mathrm{l}}$ denotes liquid phase temperatures.

Reaction | Equilibrium Constant |
---|---|

$\left(\mathrm{FeO}\right)+\left[\mathrm{C}\right]\leftrightarrow \mathrm{Fe}+\{\mathrm{CO}$} | ${log}_{10}\left(\right)open="("\; close=")">{K}_{\mathrm{eq}}^{\mathrm{FeO}\leftrightarrow \mathrm{CO}}+5.096$ |

$\left(\mathrm{FeO}\right)+\left[\mathrm{Mn}\right]\leftrightarrow \mathrm{Fe}+\left(\mathrm{MnO}\right)$ | ${log}_{10}\left(\right)open="("\; close=")">{K}_{\mathrm{eq}}^{\mathrm{FeO}\leftrightarrow \mathrm{MnO}}$ |

$\left(\mathrm{MnO}\right)+\left[\mathrm{C}\right]\leftrightarrow \left[\mathrm{Mn}\right]+\left\{\mathrm{CO}\right\}$ | ${log}_{10}\left(\right)open="("\; close=")">{K}_{\mathrm{eq}}^{\mathrm{MnO}\leftrightarrow \mathrm{CO}}+8.574$ |

$2\left(\mathrm{FeO}\right)+\left[\mathrm{Si}\right]\leftrightarrow 2\mathrm{Fe}+\left({\mathrm{SiO}}_{2}\right)$ | ${log}_{10}\left(\right)open="("\; close=")">{K}_{\mathrm{eq}}^{\mathrm{FeO}\leftrightarrow {\mathrm{SiO}}_{2}}+1.72$ |

$2\left(\mathrm{MnO}\right)+\left[\mathrm{Si}\right]\leftrightarrow 2\left[\mathrm{Mn}\right]+\left({\mathrm{SiO}}_{2}\right)$ | ${log}_{10}\left(\right)open="("\; close=")">{K}_{\mathrm{eq}}^{\mathrm{MnO}\leftrightarrow {\mathrm{SiO}}_{2}}+1.27$ |

$3\left(\mathrm{FeO}\right)+2\left[\mathrm{Cr}\right]\leftrightarrow 3\mathrm{Fe}+\left({\mathrm{Cr}}_{2}{\mathrm{O}}_{3}\right)$ | ${log}_{10}\left(\right)open="("\; close=")">{K}_{\mathrm{eq}}^{\mathrm{FeO}\leftrightarrow {\mathrm{Cr}}_{2}{\mathrm{O}}_{3}}$ |

$3\left(\right)open="("\; close=")">{\mathrm{SiO}}_{2}$ | ${log}_{10}\left(\right)open="("\; close=")">{K}_{\mathrm{eq}}^{{\mathrm{SiO}}_{2}\leftrightarrow {\mathrm{Al}}_{2}{\mathrm{O}}_{3}}-14.465$ |

**Table A3.**Forward reaction rates for equilibrium reactions. x denotes mass fractions and ${k}_{\mathrm{f}}$ refers to kinetic model constants for each reaction.

Reaction | Forward Reaction Rate | Units |
---|---|---|

$\left(\mathrm{FeO}\right)+\left[\mathrm{C}\right]\leftrightarrow \mathrm{Fe}+\left(\mathrm{CO}\right)$ | ${F}_{\mathrm{f}}^{\mathrm{FeO}\leftrightarrow \mathrm{CO}}=\left(\right)open="("\; close=")">0.11\xb7{10}^{4}$ | $\frac{\mathrm{mol}\phantom{\rule{0.166667em}{0ex}}\mathrm{FeO}}{\mathrm{s}}$ |

$\left(\mathrm{FeO}\right)+\left[\mathrm{Mn}\right]\leftrightarrow \mathrm{Fe}+\left(\mathrm{MnO}\right)$ | ${F}_{\mathrm{f}}^{\mathrm{FeO}\leftrightarrow \mathrm{MnO}}={10}^{4}\xb7{K}_{\mathrm{f}}^{\mathrm{FeO}\leftrightarrow \mathrm{MnO}}{x}_{\mathrm{FeO}}^{\mathrm{slag}}{x}_{\mathrm{Mn}}^{\mathrm{steel}}$ | $\frac{\mathrm{mol}\phantom{\rule{0.166667em}{0ex}}\mathrm{FeO}}{\mathrm{s}}$ |

$\left(\mathrm{MnO}\right)+\left[\mathrm{C}\right]\leftrightarrow \left[\mathrm{Mn}\right]+\left\{\mathrm{CO}\right\}$ | ${F}_{\mathrm{f}}^{\mathrm{MnO}\leftrightarrow \mathrm{CO}}=\left(\right)open="("\; close=")">0.017\xb7{10}^{4}$ | $\frac{\mathrm{mol}\phantom{\rule{0.166667em}{0ex}}\mathrm{MnO}}{\mathrm{s}}$ |

$2\left(\mathrm{FeO}\right)+\left[\mathrm{Si}\right]\leftrightarrow 2\mathrm{Fe}+\left({\mathrm{SiO}}_{2}\right)$ | ${F}_{\mathrm{f}}^{\mathrm{FeO}\leftrightarrow {\mathrm{SiO}}_{2}}=\left(\right)open="("\; close=")">2\xb7{10}^{4}{x}_{\mathrm{Si}}^{\mathrm{steel}}$ | $\frac{\mathrm{mol}\phantom{\rule{0.166667em}{0ex}}\mathrm{FeO}}{\mathrm{s}}$ |

$2\left(\mathrm{MnO}\right)+\left[\mathrm{Si}\right]\leftrightarrow 2\left[\mathrm{Mn}\right]+\left({\mathrm{SiO}}_{2}\right)$ | ${F}_{\mathrm{f}}^{\mathrm{MnO}\leftrightarrow {\mathrm{SiO}}_{2}}=\left(\right)open="("\; close=")">2\xb7{10}^{4}{x}_{\mathrm{Si}}^{\mathrm{steel}}$ | $\frac{\mathrm{mol}\phantom{\rule{0.166667em}{0ex}}\mathrm{MnO}}{\mathrm{s}}$ |

$3\left(\mathrm{FeO}\right)+2\left[\mathrm{Cr}\right]\leftrightarrow 3\mathrm{Fe}+\left({\mathrm{Cr}}_{2}{\mathrm{O}}_{3}\right)$ | ${F}_{\mathrm{f}}^{\mathrm{FeO}\leftrightarrow {\mathrm{Cr}}_{2}{\mathrm{O}}_{3}}=\left(\right)open="("\; close=")">3\xb7{10}^{4}$ | $\frac{\mathrm{mol}\phantom{\rule{0.166667em}{0ex}}\mathrm{FeO}}{\mathrm{s}}$ |

$3\left(\right)open="("\; close=")">{\mathrm{SiO}}_{2}$ | ${F}_{\mathrm{f}}^{{\mathrm{SiO}}_{2}\leftrightarrow {\mathrm{Al}}_{2}{\mathrm{O}}_{3}}=\left(\right)open="("\; close=")">1.5\xb7{10}^{4}$ | $\frac{\mathrm{mol}\phantom{\rule{0.166667em}{0ex}}{\mathrm{SiO}}_{2}}{\mathrm{s}}$ |

**Table A4.**Backward reaction rates for equilibrium reactions. x denotes mass fractions, ${k}_{\mathrm{f}}$ refers to kinetic model constants for each reaction, ${p}_{\mathrm{CO}}$ is the partial pressure of CO and M denotes molar masses.

Reaction | Backward Reaction Rate | Units |
---|---|---|

$\left(\mathrm{FeO}\right)+\left[\mathrm{C}\right]\leftrightarrow \mathrm{Fe}+\left\{\mathrm{CO}\right\}$ | ${F}_{\mathrm{b}}^{\mathrm{FeO}\leftrightarrow \mathrm{CO}}=\frac{{K}_{\mathrm{f}}^{\mathrm{FeO}\leftrightarrow \mathrm{CO}}}{{K}_{\mathrm{eq}}^{\mathrm{FeO}\leftrightarrow \mathrm{CO}}}{p}_{\mathrm{CO}}$ | $\frac{\mathrm{mol}\phantom{\rule{0.166667em}{0ex}}\mathrm{Fe}\phantom{\rule{4pt}{0ex}}\left(\mathrm{FeO}\right)}{\mathrm{s}}$ |

$\left(\mathrm{FeO}\right)+\left[\mathrm{Mn}\right]\leftrightarrow \mathrm{Fe}+\left(\mathrm{MnO}\right)$ | ${F}_{\mathrm{b}}^{\mathrm{FeO}\leftrightarrow \mathrm{MnO}}={10}^{2}\xb7\frac{{K}_{\mathrm{f}}^{\mathrm{FeO}\leftrightarrow \mathrm{MnO}}}{{K}_{\mathrm{eq}}^{\mathrm{FeO}\leftrightarrow \mathrm{MnO}}}{x}_{\mathrm{MnO}}^{\mathrm{slag}}$ | $\frac{\mathrm{mol}\phantom{\rule{0.166667em}{0ex}}\mathrm{Fe}\phantom{\rule{4pt}{0ex}}\left(\mathrm{FeO}\right)}{\mathrm{s}}$ |

$\left(\mathrm{MnO}\right)+\left[\mathrm{C}\right]\leftrightarrow \left[\mathrm{Mn}\right]+\left\{\mathrm{CO}\right\}$ | ${F}_{\mathrm{b}}^{\mathrm{MnO}\leftrightarrow \mathrm{CO}}={10}^{2}\xb7\frac{{K}_{\mathrm{f}}^{\mathrm{MnO}\leftrightarrow \mathrm{CO}}}{{K}_{\mathrm{eq}}^{\mathrm{MnO}\leftrightarrow \mathrm{CO}}}{p}_{\mathrm{CO}}{x}_{\mathrm{Mn}}^{\mathrm{steel}}$ | $\frac{\mathrm{mol}\phantom{\rule{0.166667em}{0ex}}\mathrm{Mn}\phantom{\rule{4pt}{0ex}}\left(\mathrm{MnO}\right)}{\mathrm{s}}$ |

$2\left(\mathrm{FeO}\right)+\left[\mathrm{Si}\right]\leftrightarrow 2\mathrm{Fe}+\left({\mathrm{SiO}}_{2}\right)$ | ${F}_{\mathrm{b}}^{\mathrm{FeO}\leftrightarrow {\mathrm{SiO}}_{2}}=\left(\right)open="("\; close=")">2\xb7{10}^{2}$ | $\frac{\mathrm{mol}\phantom{\rule{0.166667em}{0ex}}\mathrm{Fe}\phantom{\rule{4pt}{0ex}}\left(\mathrm{FeO}\right)}{\mathrm{s}}$ |

$2\left(\mathrm{MnO}\right)+\left[\mathrm{Si}\right]\leftrightarrow 2\left[\mathrm{Mn}\right]+\left({\mathrm{SiO}}_{2}\right)$ | ${F}_{\mathrm{b}}^{\mathrm{MnO}\leftrightarrow {\mathrm{SiO}}_{2}}=\left(\right)open="("\; close=")">2\xb7{10}^{4}{x}_{{\mathrm{SiO}}_{2}}^{\mathrm{slag}}$ | $\frac{\mathrm{mol}\phantom{\rule{0.166667em}{0ex}}\mathrm{Mn}\phantom{\rule{4pt}{0ex}}\left(\mathrm{MnO}\right)}{\mathrm{s}}$ |

$3\left(\mathrm{FeO}\right)+2\left[\mathrm{Cr}\right]\leftrightarrow 3\mathrm{Fe}+\left({\mathrm{Cr}}_{2}{\mathrm{O}}_{3}\right)$ | ${F}_{\mathrm{b}}^{\mathrm{FeO}\leftrightarrow {\mathrm{SiO}}_{2}}=\left(\right)open="("\; close=")">\frac{6\xb7{10}^{2}\xb7{M}_{\mathrm{Cr}}}{{M}_{{\mathrm{Cr}}_{2}{\mathrm{O}}_{3}}}$ | |

$3\left(\right)open="("\; close=")">{\mathrm{SiO}}_{2}$ | ${F}_{\mathrm{b}}^{{\mathrm{SiO}}_{2}\leftrightarrow {\mathrm{Al}}_{2}{\mathrm{O}}_{3}}=\left(\right)open="("\; close=")">1.5\xb7{10}^{4}$ | $\frac{\mathrm{mol}\phantom{\rule{0.166667em}{0ex}}\mathrm{Si}\phantom{\rule{4pt}{0ex}}\left(\right)open="("\; close=")">{\mathrm{SiO}}_{2}}{}$ |

Reaction | Species i Rate of Change |
---|---|

$\left(\mathrm{FeO}\right)+\left[\mathrm{C}\right]\leftrightarrow \mathrm{Fe}+\left\{\mathrm{CO}\right\}$ | ${F}_{i}^{\mathrm{FeO}\leftrightarrow \mathrm{CO}}=\frac{{v}_{i}^{\mathrm{FeO}\leftrightarrow \mathrm{CO}}}{{v}_{\mathrm{FeO}}^{\mathrm{FeO}\leftrightarrow \mathrm{CO}}}\left(\right)open="("\; close=")">{F}_{f}^{\mathrm{FeO}\leftrightarrow \mathrm{CO}}-{F}_{b}^{\mathrm{FeO}\leftrightarrow \mathrm{CO}}$ |

${F}_{i}^{\mathrm{FeO}\leftrightarrow \mathrm{MnO}}=\frac{{v}_{i}^{\mathrm{FeO}\leftrightarrow \mathrm{MnO}}}{{v}_{\mathrm{FeO}}^{\mathrm{FeO}\leftrightarrow \mathrm{MnO}}}\left(\right)open="("\; close=")">{F}_{f}^{\mathrm{FeO}\leftrightarrow \mathrm{MnO}}-{F}_{b}^{\mathrm{FeO}\leftrightarrow \mathrm{MnO}}$ | |

${F}_{i}^{\mathrm{MnO}\leftrightarrow \mathrm{CO}}=\frac{{v}_{i}^{\mathrm{MnO}\leftrightarrow \mathrm{CO}}}{{v}_{\mathrm{MnO}}^{\mathrm{MnO}\leftrightarrow \mathrm{CO}}}\left(\right)open="("\; close=")">{F}_{f}^{\mathrm{MnO}\leftrightarrow \mathrm{CO}}-{F}_{b}^{\mathrm{MnO}\leftrightarrow \mathrm{CO}}$ | |

${F}_{i}^{\mathrm{FeO}\leftrightarrow {\mathrm{SiO}}_{2}}=\frac{{v}_{i}^{\mathrm{FeO}\leftrightarrow {\mathrm{SiO}}_{2}}}{{v}_{\mathrm{FeO}}^{\mathrm{FeO}\leftrightarrow {\mathrm{SiO}}_{2}}}\left(\right)open="("\; close=")">{F}_{f}^{\mathrm{FeO}\leftrightarrow {\mathrm{SiO}}_{2}}-{F}_{b}^{\mathrm{FeO}\leftrightarrow {\mathrm{SiO}}_{2}}$ | |

${F}_{i}^{\mathrm{MnO}\leftrightarrow {\mathrm{SiO}}_{2}}=\frac{{v}_{i}^{\mathrm{MnO}\leftrightarrow {\mathrm{SiO}}_{2}}}{{v}_{\mathrm{MnO}}^{\mathrm{MnO}\leftrightarrow {\mathrm{SiO}}_{2}}}\left(\right)open="("\; close=")">{F}_{f}^{\mathrm{MnO}\leftrightarrow {\mathrm{SiO}}_{2}}-{F}_{b}^{\mathrm{MnO}\leftrightarrow {\mathrm{SiO}}_{2}}$ | |

${F}_{i}^{\mathrm{FeO}\leftrightarrow {\mathrm{Cr}}_{2}{\mathrm{O}}_{3}}=\frac{{v}_{i}^{\mathrm{FeO}\leftrightarrow {\mathrm{Cr}}_{2}}}{{v}_{\mathrm{FeO}}^{\mathrm{FeO}\leftrightarrow {\mathrm{Cr}}_{2}}{\mathrm{O}}_{3}}\left(\right)open="("\; close=")">{F}_{f}^{\mathrm{FeO}\leftrightarrow {\mathrm{Cr}}_{2}{\mathrm{O}}_{3}}-{F}_{b}^{\mathrm{FeO}\leftrightarrow {\mathrm{Cr}}_{2}{\mathrm{O}}_{3}}$ |

#### Appendix A.3. Overall Heat and Mass Balances

#### Appendix A.4. Model Constants and Dimensions

**Table A6.**Model constants and dimensions. An empty entry in the

**Units**column indicates a unitless quantity.

Constant | Description | Value | Units |
---|---|---|---|

${d}_{\mathrm{furnace}}$ | Furnace diameter | 8.1 | m |

${d}_{\mathrm{inner}}$ | Inner control volume diameter | 4.65 | m |

${h}_{\mathrm{furnace}}$ | Furnace height | 5.2 | m |

${h}_{\mathrm{panel}}$ | Cooling water panel height | 2.89 | m |

${C}_{\mathrm{p},\phantom{\rule{4.pt}{0ex}}\mathrm{solid}}$ | Heat capacity of solid steel | 39 | $\frac{\mathrm{J}}{\mathrm{mol}\phantom{\rule{4.pt}{0ex}}\mathrm{K}}$ |

${C}_{\mathrm{p},\phantom{\rule{4.pt}{0ex}}\mathrm{liquid}}$ | Heat capacity of liquid steel | 46 | $\frac{\mathrm{J}}{\mathrm{mol}\phantom{\rule{4.pt}{0ex}}\mathrm{K}}$ |

${C}_{\mathrm{p},\phantom{\rule{4.pt}{0ex}}\mathrm{slag}}$ | Heat capacity of slag | 50 | $\frac{\mathrm{J}}{\mathrm{mol}\phantom{\rule{4.pt}{0ex}}\mathrm{K}}$ |

${\rho}_{\mathrm{solid}}$ | Density of solid steel | 2000 | $\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}$ |

${\rho}_{\mathrm{liquid}}$ | Density of liquid steel | 7000 | $\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}$ |

${k}_{\mathrm{phase}}$ | Phase change constant | 0.005 | $\frac{1}{\mathrm{s}}$ |

${k}_{\mathrm{CO}}$ | Limiting constant for CO combustion | 0.25 | $\frac{1}{\mathrm{s}}$ |

${x}_{\mathrm{arc}}^{\mathrm{gas}}$ | Fraction of arc power used to heat gas | 0.05 | |

${x}_{\mathrm{vessel}}^{\mathrm{loss}}$ | Fraction of arc losses used to heat vessel | 0.3 | |

${k}_{\mathrm{ss}}$ | Heat transfer coefficient: solid steel–solid steel | 400 | $\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ |

${k}_{\mathrm{sl}}$ | Heat transfer coefficient: solid steel–liquid steel | 12,000 | $\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ |

${k}_{\mathrm{ll}}$ | Heat transfer coefficient: liquid steel–liquid steel | 60,000 | $\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ |

${k}_{\mathrm{cs}}$ | Heat transfer coefficient: solid slag–solid steel | 2000 | $\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ |

${k}_{\mathrm{cl}}$ | Heat transfer coefficient: solid slag–liquid steel | 2000 | $\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ |

${k}_{\mathrm{bs}}$ | Heat transfer coefficient: liquid slag–solid steel | 5 | $\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ |

${k}_{\mathrm{bl}}$ | Heat transfer coefficient: liquid slag–liquid steel | 5 | $\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ |

${k}_{\mathrm{sg}}$ | Heat transfer coefficient: solid steel–gas | 20 | $\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ |

${k}_{\mathrm{lg}}$ | Heat transfer coefficient: liquid steel–gas | 10 | $\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ |

${k}_{\mathrm{gr}}$ | Heat transfer coefficient: gas–roof | 25 | $\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ |

${k}_{\mathrm{gv}}$ | Heat transfer coefficient: gas–vessel | 25 | $\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ |

${k}_{\mathrm{rw}}$ | Heat transfer coefficient: roof–water | 300 | $\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ |

${k}_{\mathrm{vw}}$ | Heat transfer coefficient: vessel–water | 300 | $\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ |

${\epsilon}_{\mathrm{s}}$ | Emissivity of solid steel | 0.4 | |

${\epsilon}_{\mathrm{l}}$ | Emissivity of liquid steel | 0.6 | |

${\epsilon}_{\mathrm{r}}$ | Emissivity of furnace roof | 0.7 | |

${\epsilon}_{\mathrm{v}}$ | Emissivity of side panels | 0.5 |

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**Figure 1.**State variables in the EAF process model. T${}_{\mathrm{roof}}^{\mathrm{water}}$, T${}_{\mathrm{vessel}}^{\mathrm{water}}$ and T${}_{\mathrm{offgas}}$ can be compared to real-time process data.

**Figure 2.**Illustration of (

**left**) melting and (

**right**) freezing. ${T}_{\mathrm{m}}$ indicates the steel melting temperature and d${T}_{\mathrm{m}}$ is a constant used for calculating the rates of melting and freezing.

**Figure 3.**Variable and static dimensions used to calculate arc visibility for the arc efficiency model.

**Figure 4.**Steel meltdown dynamics and change in (

**A**) solid mass and (

**B**) liquid mass during five consecutive two-basket heats.

**Figure 5.**Agreement of model predictions with process measurements from 250 heats for (

**A**) liquid steel temperature and (

**B**) liquid steel weight after tapping. Model biases for temperature and weight are negligible while residual standard deviations reflect scatter.

**Figure 6.**Comparison of arc efficiency estimation and model for (

**A**) one-basket heats, (

**B**) two-basket heats and (

**C**) three-basket heats. The x-axis is the normalized progress of each heat as measured by the percentage of total electric energy added to the furnace. Low %kWh${}_{\mathrm{input}}$-results are not meaningful as the efficiency estimation requires several samples to change from the initial guess factor of 0.02.

**Figure 7.**Example of arc loss coefficient ${k}_{\mathrm{loss}}$ estimation during five consecutive two-basket heats. (

**left**) A constant ${k}_{\mathrm{loss}}$ produces model results that follow the cooling water dynamics but sometimes overpredict the outlet temperature. (

**right**) Estimation of ${k}_{\mathrm{loss}}$ produces more accurate model results.

**Figure 8.**An illustration of MPC optimization for batch time and electric arc efficiency. Historical MV and model CV predictions are shown as solid black lines. Within the prediction horizon, optimized MV (MPC output) and CV are shown in orange, while nominal MV and the resulting CV are shown in blue.

**Figure 9.**Change in specific energy consumption (SEC) due to MPC for (

**A**) one-basket heats, (

**B**) two-basket heats and (

**C**) three-basket heats. ΔSEC is calculated by subtracting the logged SEC up to the point where the model predicts steel meltdown from the MPC simulation result.

**Figure 10.**Logged specific energy consumption (SEC) up to the point of melting vs. change in SEC due to MPC for (

**A**) one-basket heats, (

**B**) two-basket heats and (

**C**) three-basket heats. ΔSEC is calculated as described in Figure 9.

Dissolved Component | Phase(s) | Reactive with O${}_{2}$ in Model? | Equilibrium Reaction in Model? |
---|---|---|---|

Fe | Liquid, Solid | Yes | Yes |

C | Liquid, Solid | Yes | Yes |

Cr | Liquid, Solid | Yes | Yes |

Si | Liquid, Solid | Yes | Yes |

Al | Liquid, Solid | Yes | Yes |

Mn | Liquid, Solid | No | Yes |

FeO | Slag | No | Yes |

SiO${}_{2}$ | Slag | No | Yes |

Cr_{2}O_{3} | Slag | No | Yes |

Al_{2}O_{3} | Slag | No | Yes |

MnO | Slag | No | Yes |

**Table 2.**Prefactors for linear heat transfer between different masses in the furnace. The subscript letters (s, l, c, b, r, v, g) refer to (solid steel, liquid steel, solid slag, liquid slag, roof, vessel, gas), respectively. An empty table entry indicates that linear heat transfer between the two masses is omitted from the model.

Inner Solid | Outer Solid | Inner Liquid | Outer Liquid | Solid Slag | Liquid Slag | Roof | Vessel | Gas | |
---|---|---|---|---|---|---|---|---|---|

Inner Solid | - | ${k}_{\mathrm{ss}}$${A}_{\mathrm{ss}}^{\mathrm{cross}}$ | ${k}_{\mathrm{sl}}$${A}_{\mathrm{sl}}^{\mathrm{inner}}$ | ${k}_{\mathrm{sl}}$${A}_{\mathrm{sl}}^{\mathrm{cross}}$ | ${k}_{\mathrm{cs}}$${A}_{\mathrm{cs}}^{\mathrm{inner}}$ | ${k}_{\mathrm{bs}}$${A}_{\mathrm{bs}}^{\mathrm{inner}}$ | - | - | ${k}_{\mathrm{sg}}$${A}_{\mathrm{bs}}^{\mathrm{inner}}$ |

Outer Solid | ${k}_{\mathrm{ss}}$${A}_{\mathrm{ss}}^{\mathrm{cross}}$ | - | ${k}_{\mathrm{sl}}$${A}_{\mathrm{ls}}^{\mathrm{cross}}$ | ${k}_{\mathrm{sl}}$${A}_{\mathrm{sl}}^{\mathrm{outer}}$ | ${k}_{\mathrm{cs}}$${A}_{\mathrm{cs}}^{\mathrm{outer}}$ | ${k}_{\mathrm{bs}}$${A}_{\mathrm{bs}}^{\mathrm{outer}}$ | - | - | ${k}_{\mathrm{sg}}$${A}_{\mathrm{bs}}^{\mathrm{outer}}$ |

Inner Liquid | ${k}_{\mathrm{sl}}$${A}_{\mathrm{sl}}^{\mathrm{inner}}$ | ${k}_{\mathrm{sl}}$${A}_{\mathrm{ls}}^{\mathrm{cross}}$ | - | ${k}_{\mathrm{ll}}$${A}_{\mathrm{ll}}^{\mathrm{cross}}$ | ${k}_{\mathrm{cl}}$${A}_{\mathrm{cl}}^{\mathrm{inner}}$ | ${k}_{\mathrm{bl}}$${A}_{\mathrm{bl}}^{\mathrm{inner}}$ | - | - | ${k}_{\mathrm{lg}}$${A}_{\mathrm{bl}}^{\mathrm{inner}}$ |

Outer Liquid | ${k}_{\mathrm{sl}}$${A}_{\mathrm{sl}}^{\mathrm{cross}}$ | ${k}_{\mathrm{sl}}$${A}_{\mathrm{sl}}^{\mathrm{outer}}$ | ${k}_{\mathrm{ll}}$${A}_{\mathrm{ll}}^{\mathrm{cross}}$ | - | ${k}_{\mathrm{cl}}$${A}_{\mathrm{cl}}^{\mathrm{outer}}$ | ${k}_{\mathrm{bl}}$${A}_{\mathrm{bl}}^{\mathrm{outer}}$ | - | - | ${k}_{\mathrm{lg}}$${A}_{\mathrm{bl}}^{\mathrm{outer}}$ |

Solid Slag | ${k}_{\mathrm{cs}}$${A}_{\mathrm{cs}}^{\mathrm{inner}}$ | ${k}_{\mathrm{cs}}$${A}_{\mathrm{cs}}^{\mathrm{outer}}$ | ${k}_{\mathrm{cl}}$${A}_{\mathrm{cl}}^{\mathrm{inner}}$ | ${k}_{\mathrm{cl}}$${A}_{\mathrm{cl}}^{\mathrm{outer}}$ | - | - | - | - | - |

Liquid Slag | ${k}_{\mathrm{bs}}$${A}_{\mathrm{bs}}^{\mathrm{inner}}$ | ${k}_{\mathrm{bs}}$${A}_{\mathrm{bs}}^{\mathrm{outer}}$ | ${k}_{\mathrm{bl}}$${A}_{\mathrm{bl}}^{\mathrm{inner}}$ | ${k}_{\mathrm{bl}}$${A}_{\mathrm{bl}}^{\mathrm{outer}}$ | - | - | - | - | - |

Roof | - | - | - | - | - | - | - | - | ${k}_{\mathrm{gr}}$${A}_{r}$ |

Vessel | - | - | - | - | - | - | - | - | ${k}_{\mathrm{gv}}$${A}_{v}$ |

Gas | ${k}_{\mathrm{sg}}$${A}_{\mathrm{bs}}^{\mathrm{inner}}$ | ${k}_{\mathrm{sg}}$${A}_{\mathrm{bs}}^{\mathrm{outer}}$ | ${k}_{\mathrm{lg}}$${A}_{\mathrm{bl}}^{\mathrm{inner}}$ | ${k}_{\mathrm{lg}}$${A}_{\mathrm{bl}}^{\mathrm{outer}}$ | - | - | ${k}_{\mathrm{gr}}$${A}_{r}$ | ${k}_{\mathrm{gv}}$${A}_{v}$ | - |

**Table 3.**Radiative heat transfer between different masses in the furnace. The subscript letters (s, l, b, r, v) refer to (solid steel, liquid steel, liquid slag, roof, vessel), respectively.

Roof | Vessel | |
---|---|---|

Inner Solid | ${\sigma}_{\mathrm{SB}}$${A}_{\mathrm{bs}}^{\mathrm{inner}}$$V{F}_{\mathrm{r}}^{\mathrm{inner}}$$\left(\right)$ | ${\sigma}_{\mathrm{SB}}$${A}_{\mathrm{bs}}^{\mathrm{inner}}$$V{F}_{v}^{\mathrm{inner}}$$\left(\right)$ |

Outer Solid | ${\sigma}_{\mathrm{SB}}$${A}_{\mathrm{bs}}^{\mathrm{outer}}$$V{F}_{\mathrm{r}}^{\mathrm{outer}}$$\left(\right)$ | ${\sigma}_{\mathrm{SB}}$${A}_{\mathrm{bs}}^{\mathrm{outer}}$$V{F}_{\mathrm{v}}^{\mathrm{outer}}$$\left(\right)$ |

Inner Liquid | ${\sigma}_{\mathrm{SB}}$${A}_{\mathrm{bl}}^{\mathrm{inner}}$$V{F}_{\mathrm{r}}^{\mathrm{inner}}$$\left(\right)$ | ${\sigma}_{\mathrm{SB}}$${A}_{\mathrm{bl}}^{\mathrm{inner}}$$V{F}_{\mathrm{v}}^{\mathrm{inner}}$$\left(\right)$ |

Outer Liquid | ${\sigma}_{\mathrm{SB}}$${A}_{\mathrm{bl}}^{\mathrm{outer}}$$V{F}_{\mathrm{r}}^{\mathrm{outer}}$$\left(\right)$ | ${\sigma}_{\mathrm{SB}}$${A}_{\mathrm{bl}}^{\mathrm{outer}}$$V{F}_{\mathrm{v}}^{\mathrm{outer}}$$\left(\right)$ |

**Table 4.**Temperature of fully-liquefied steel prior to tapping as predicted by the model when following logged process data and MPC simulations. Both means and standard deviations are presented.

T${}_{\mathbf{Log}}^{\mathbf{Full}}$ | T${}_{\mathbf{Log}}^{\mathbf{Melt}}$ | T${}_{\mathbf{MPC}}$ | |
---|---|---|---|

1-Basket | 1688.5 °C ± 27.8 °C | 1684.5 °C ± 27.9 °C | 1693.0 °C ± 13.0 °C |

2-Basket | 1693.3 °C ± 21.6 °C | 1687.6 °C ± 23.4 °C | 1703.7 °C ± 17.1 °C |

3-Basket | 1717.9 °C ± 2.1 °C | 1717.2 °C ± 2.2 °C | 1702.5 °C ± 0.9 °C |

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**MDPI and ACS Style**

Jawahery, S.; Visuri, V.-V.; Wasbø, S.O.; Hammervold, A.; Hyttinen, N.; Schlautmann, M.
Thermophysical Model for Online Optimization and Control of the Electric Arc Furnace. *Metals* **2021**, *11*, 1587.
https://doi.org/10.3390/met11101587

**AMA Style**

Jawahery S, Visuri V-V, Wasbø SO, Hammervold A, Hyttinen N, Schlautmann M.
Thermophysical Model for Online Optimization and Control of the Electric Arc Furnace. *Metals*. 2021; 11(10):1587.
https://doi.org/10.3390/met11101587

**Chicago/Turabian Style**

Jawahery, Sudi, Ville-Valtteri Visuri, Stein O. Wasbø, Andreas Hammervold, Niko Hyttinen, and Martin Schlautmann.
2021. "Thermophysical Model for Online Optimization and Control of the Electric Arc Furnace" *Metals* 11, no. 10: 1587.
https://doi.org/10.3390/met11101587