Dynamic Modeling of an Industrial Electric Arc Furnace - American

Ind. Eng. Chem. Res. 2005, 44, 8067-8083

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Dynamic Modeling of an Industrial Electric Arc Furnace Richard D. M. MacRosty and Christopher L. E. Swartz* Department of Chemical Engineering, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4L7

This paper describes the development of a dynamic model for an industrial steelmaking electric arc furnace (EAF). The model is sufficiently detailed so as to describe the melting process and chemical reactions and account for reagent and energy additions. The lack of knowledge of reaction mechanisms due to the complex nature of the reacting system is overcome by modeling the process as equilibrium zones with mass transport limitations. An important objective of developing this model is to ensure that it can be used within an optimization framework, thus particular consideration is given to the continuity of model equations and the robustness of the model over a wide range of conditions. Parameter estimation is carried out using industrial data and the model performance illustrated through simulation studies. 1. Introduction Electric arc furnaces (EAFs) are widely used in the steel industry for recycling scrap steel. Simply stated, steel recycling involves melting down the scrap metal and adjusting the chemistry to obtain the desired product grade. The steel is melted using both chemical and electrical energy sources. In modern furnaces, the electrical power is added to the furnace via three electrodes which transfer energy to the steel in the form of an electric arc. The chemical contribution is derived from combustion reactions taking place in the furnace, fueled predominantly by coke, natural gas, and oxygen. In the steel industry EAFs are run in batches, termed heats. While processing conditions vary greatly, a typical heat takes between 1 and 3 h and consumes approximately 400 kWh/ton of steel.1 The energyintensive nature of EAFs makes these operations attractive candidates for optimization. EAFs typically involve a relatively low level of automation and rely heavily on operator involvement. As with most industrial processes, operator experience is invaluable for the operation of the process. However, this experience can be limited due to the multivariable interactions and subtle relationships that may be easily overlooked. The understanding of these complexities is confounded by the small number of useful process measurements which make it difficult to infer the current state of the process. Therefore, in most situations process operating procedures are based upon what has worked well in the past. The manner in which reagents, scrap, and electric power are added to the furnace may be carried out in many possible ways. Detailed process knowledge, in the form of a model, makes it possible to take advantage of more complex relationships to provide information such as finding the optimal balance and timing of the energy contributions from chemical and electrical sources. The main focus of this work involves the application of mathematical modeling techniques to develop a dynamic model of the EAF steel-making process. This model must be sufficiently detailed so as to describe the melting process and chemical changes and account for * To whom correspondence should be addressed. Tel: (905) 525-9140. Fax: (905) 521-1350. E-mail: [email protected].

reagent and energy additions. Further work involving applications of the model presented in this paper is under way and will be presented in subsequent communications. One such extension involves the use of the model in a mathematical optimization procedure to determine the economically optimal input profiles subject to prevailing process and operational constraints. A further extension is to develop an advanced control strategy for an industrial EAF, whereby online measurements are used together with the model to make online updates to account for disturbances and plantmodel mismatch. This paper is organized in the following manner. A brief background to the operation of the EAF is given, followed by a discussion of other published modeling work. Section 2 outlines the model of the furnace with a detailed mathematical description given in section 3. The estimation of the model parameters from industrial data is presented next. The functionality of the model is illustrated through simulation studies after which conclusions are drawn. 1.1. Process Operation. This section describes the typical operation of an EAF. The scrap charge for each heat is comprised of a range of different scrap sources. The particular mix for each heat is selected based on a number of factors such as the availability of each scrap source and the desired product grade being produced. The selection and packing of the furnace will also influence the composition of the liquid steel and slag as the scrap melts as well as the melting behavior. The charge could include lime and carbon, and/or these could be injected into the furnace during the heat. Typically, two or three buckets of scrap are charged per heat depending on the bulk density of the scrap and volume of the furnace. The capacity of a bucket is typically of the order of 100 tons. The EAF batch processing recipe involves a series of distinct stages, specifically charging, preheat, melting, and tapping (emptying of molten steel from the furnace). The preheat stage involves the combustion of natural gas to raise the temperature of the steel. Following the preheat, the electrodes are lowered into the furnace, and the power is turned on. An intermediate voltage tap is selected, while the electrodes bore into the scrap to maintain a stable arc. The voltage can be increased once

10.1021/ie050101b CCC: $30.25 © 2005 American Chemical Society Published on Web 09/14/2005

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a pool of liquid has formed at the base of the arc. During the initial stages of the meltdown, a long arc (high voltage) is typically selected. A long arc allows more energy to be transferred via radiation to the scrap surrounding the arc. This has a more global heating effect in the furnace than a shorter arc which focuses the majority of its energy to the base of the arc. As flat bath conditions are approached, a shorter arc is preferred because the furnace walls are now exposed, and energy radiated laterally is essentially lost. Toward the end of the heat, the slag layer is foamed by injecting C and O2 to form CO gas which bubbles through the slag. The foaming slag covers the arc thereby protecting the furnace walls from arc radiation and also improving the power transfer to the steel and hence the energy efficiency. The oxidation reactions occurring in the furnace also serve as a source of energy. During the heat phosphorus, sulfur, aluminum, silicon, manganese, and carbon are removed from the steel as they react with oxygen and float into the slag. These reactions can be controlled by lancing oxygen into the bath. O2 lancing is typically carried out until the carbon content is at the required level for tapping. After a certain amount of power has been added to the furnace, the bath temperature and carbon content are measured. This information indicates what further additions need to be made to reach the desired endpoint specifications. The measurements are then repeated to ensure that these specifications have been met. Once the desired composition and temperature have been obtained in the furnace, the tap-hole is opened, and the molten steel is poured into the ladle for transport to the next operation. 1.2. Modeling Approaches. The fundamental mechanisms involved in the electric arc furnace process are relatively poorly understood due to the complexity and extreme conditions of the process. These factors make the development and validation of a highly detailed model very difficult and necessitate the application of simplifying assumptions about the process. Bekker et al.2 developed a model of the furnace from fundamental thermodynamic and kinetic relationships for the purpose of closed-loop control simulation. Empirical relationships and simplifying assumptions are used to describe mechanisms which are not well understood or measurable. The model assumes that energy from the arc and heat of reaction is added to the liquid metal which in turn heats the solid component. The gas phase of the furnace is assumed to be at the same temperature as the liquid steel. The temperature of the liquid steel is increased both through the chemical and electrical energy additions, with energy losses occurring through the furnace wall. The melt rate and temperature of the solid steel are determined by the rate of heat transfer to the solid component. The ratio of the scrap temperature to the liquid steel temperature is assumed to be the fraction of energy that is available to melt the steel, with the remainder used to raise the temperature of the solid scrap. The important metallurgical reactions that are considered are those involving oxidation of Fe, C, and Si and reduction of FeO. The reaction rates of dissolved C and Si in the steel are assumed to be proportional to the difference between their concentrations and approximate equilibrium concentrations. The model assumes that all O2 fed into the furnace is consumed in the oxidation of Fe, C, and Si. Oosthuizen

et al.3 extended the model to be able to predict the offgas temperature and the slag foam depth. The next three models described consider the process as being comprised of equilibrium zones with mass transport limitations. This is motivated by arguments that reactions tend to be transport limited at steelmaking temperatures and also avoids difficulties associated with a lack of knowledge of the reaction mechanisms and estimation of kinetic parameters. Cameron et al.4 developed an EAF model for the purpose of dynamic simulation that could be used to identify improvements in EAF operating practices. The authors model the process as four phases (metal, slag, organic solid, and gas) and consider six interfaces between the metal, slag, gas, and carbon material. Chemical equilibrium is then assumed at each interface, from which the reaction products are redistributed to the bulk phases. Material flow between the bulk phases and interfaces is driven by a concentration gradient, with the chemical equilibrium state at the interface determined by minimizing the Gibbs Free Energy. The authors validated the model using off-gas chemistry data. Limited detail of the model is provided, presumably due to proprietary reasons. Matson and Ramirez5 developed a model that approximates the furnace as two separate control volumes, namely bath, slag, and gas reactors. The authors assume chemical equilibrium in the individual control volumes and transport limitations between each control volume to determine the rate of reaction. In each control volume a dynamic elemental balance is used to track the flow of material. The equilibrium algorithm minimizes the Gibbs free energy subject to atom balance constraints and considers the presence of the elements, C, H, O, N, and Fe. The quantity of Ca is assumed to remain constant. The chemical equilibrium problem is solved via a subroutine at each integration step. Mass transfer between the control volumes is modeled as diffusion across a concentration gradient. The scrap is modeled as a collection of spheres with sensible heating of the spheres determining the temperature profile of each sphere as a function of its radius. At each time step the surface temperature of the scrap is monitored to determine whether the subsequent step will be a sensible heating iteration or a melting iteration. During the melting iteration, the energy is used to overcome the latent heat of melting, and the radius of the spheres consequently changed. Small discretization steps are required to attain an acceptable level of accuracy with this method. The authors used iterative dynamic programming to determine the optimal carbon and oxygen additions and also the optimal batch time. Modigell et al.6 also developed an EAF model for use as a simulation tool. In this case the process is modeled as four distinct reaction zones that are assumed to be in a state of chemical equilibrium, with flow of material between reaction zones governed by concentration gradients and adjustable mass transfer coefficients. However, few specifics of the model are provided. The model was validated using endpoint data as this was the only data available to the authors. In contrast to the above-described models that consider the EAF operation as a whole, a number of modeling efforts have focused on individual elements of the EAF process. Examples of the latter include electrode models,7-9 detailed three-dimensional models to predict the radiative heat transfer in the furnace,10

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Figure 1. Schematic of EAF model.

models to predict slag foaming,11,12 and models for post combustion in the furnace freeboard.13,14 2. Model Overview There were five main requirements in developing the model for the furnace: 1. The model should be based primarily on fundamental principles. A key reason for this is that the intended use of the model is for optimization studies. Empirical models are reliable only within the range of the data from which they were identified which makes them less desirable for optimization applications where the model may be required to be evaluated over a relatively large decision space. 2. The model should be able to predict the nonlinear dynamic behavior of the process over the course of the batch operation. 3. The model should be robust and not susceptible to numerical difficulties. An important objective of developing this model is to ensure that it can be used within a dynamic optimization framework to determine operating conditions that would optimize a specified performance criterion such as minimizing the cost per ton of liquid steel or maximizing production rate. Therefore of particular importance are the continuity of model equations and the robustness of the model over a wide range of conditions. 4. The model should be developed in such a manner that the number of parameters to be estimated from industrial data is kept to a minimum. A particular challenge encountered was the lack of useful data for parameter estimation. The development of the model therefore involved a balance between minimizing the number of model parameters requiring estimation and ensuring sufficient model accuracy. The number of zones chosen to model the EAF (four) was felt to be an appropriate compromise.

5. A further objective was to build the model in such a way that the structure was flexible, allowing the substitution/addition of more detailed relationships into the model. 2.1. Model Structure. The EAF is modeled as a system of equilibrium zones with each zone approximating the behavior of a section of the furnace: 1. Gas zone: includes all gas in the freeboard volume, i.e., the free space in the furnace above the scrap material. 2. Slag-Metal interaction zone: includes all the slag material and the portion of iron interacting with the slag. 3. Molten steel zone: consists of all metallic elements in their liquid-state excluding that portion included in the slag-metal zone. 4. Solid scrap zone: includes the charged scrap that is still in solid form. Each zone is distinguished by its unique composition and conditions. Chemical equilibrium is assumed to exist in the slag-metal interaction and the gas zones; this equilibrium assumption is reasonable if one considers the high temperatures within the system.1,4,6 The reaction of material is limited by mass transfer between the zones, where the mass transfer coefficients are treated as adjustable parameters estimated from industrial process data. Figure 1 is a schematic diagram of the model depicting the mass flows between the above-described zones. The chemical species included in each zone and the material additions are also illustrated. The energy model considers the radiation and convective heat transfer taking place between different zones, the furnace components, and the arc. A description of each of the four zones follows. Gas Zone. The gas zone includes all material in the freeboard volume. The species considered in this zone are as follows: CO, O2, CO2, CH4, H2, H2O, N2, and C9H20. C9H20 is taken to be an average composition of

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all the volatile components that may be present in the scrap and is assumed to vaporize from the scrap in the initial minutes of charging. CH4 and O2 are added to this zone via the burners. O2, N2, and H2O are introduced from ingressed air and water-cooling of the electrodes. CO enters from the partial combustion of carbon in the slag-metal interaction zone. The components within this phase are assumed to exist in a state of chemical equilibrium. The relationships for determining the equilibrium state in the model are presented in section 3.1. Slag-Metal Interaction Zone. This zone consists of the slag material and a portion of the molten-metal phase with which it is in contact, including metal droplets in the slag. The species considered in this zone are as follows: Fe, Mn, Mg, Al, Si, FeO, Fe2O3, MnO, MgO, Al2O3, SiO2, CaO, C, CO, O2, N2. The presence of CO2 in this zone is neglected by the following argument. The reaction of carbon particles in the slag is controlled by the formation of a gaseous layer around the carbon particles as they oxidize. This gaseous layer limits transport of the O2 to the carbon particle, and thus any CO2 that is produced is quickly reduced to CO via the Boudouard equilibrium reaction

CO2 + C H 2CO

(1)

The mechanism of this reaction in the slag is discussed further by Morales et al.15 All components in the slag-metal zone are also assumed to exist in a state of chemical equilibrium. This zone is in direct contact with the gas zone and the molten-metal zone. O2 enters this zone via lancing and diffusion from the gas phase (according to its partial pressure). The presence of iron oxides in the zone also increases the availability of oxygen for components with a sufficient reducing potential. Metallic elements (Fe, Mn, Mg, and Al) and nonmetallic elements (C, Si, P, and S) enter this phase from the molten-metal phase. Carbon is also added from injection and roof additions. Lime and dolime (CaO‚MgO) added to the furnace are also included in this zone. All oxides accumulate in this zone, except for CO which leaves this zone and enters the gas phase as it is produced. Decarburization and oxidation reactions are the most important, and these are limited by the transport of C and O2 into this zone. Lancing has a dual effect on this zone. First, it provides for the addition of C and O2 to the zone, and second, it results in an increased mixing effect of this zone with the moltenmetal zone. The increased mixing effect is captured by relating the mass-transfer coefficient to the volumetric flow of O2 added through the lance. More details of this model are given in section 3.1.1. It is assumed that the majority of carbon added to the furnace via the lance will enter this zone with the remaining carbon added to the molten-metal zone. The division of lance carbon between these zones is an adjustable parameter that is estimated from industrial data. Slag foaming in the furnace is an important phenomenon; it protects the furnace elements from radiative damage and also improves the efficiency of the transfer of energy to the steel. The foaming depends on the rate of CO evolution from the bath, which is controlled by the amount of carbon and oxygen available. Oxygen is available either as FeO or it is lanced directly into the bath. The slag composition is also an important factor

in slag foaming with the correct basicity and viscosity essential for obtaining a foaming slag. Molten-Metal Zone. Material enters this zone from the solid-scrap zone as it melts and leaves to enter the slag-metal zone according to the transport rate to the slag-metal interface. It is assumed that no reactions occur in this zone because of the absence of O2. Energy from the arc is added to the molten-metal zone, and energy transfer takes place between it and the solidscrap and slag-metal zones, driven by the prevailing temperature gradient. The presence of the following components are modeled in this zone: Fe, Mn, Al, Si, C. Mass transfer of material to the slag-metal zone is driven by natural diffusion and also forced diffusion as a result of lancing. Knowledge of the initial mass of carbon in the steel allows the model to predict its mass at any time. The initial mass of carbon is estimated from the composition analysis of the scrap sources constituting the furnace charge. The mass balance equations keep track of the additions (coke additions, melting scrap, etc.) and consumption reactions (decarburization). Additions from carbon present in the scrap can be modeled simply as a fraction of the mass of steel that melts. Decarburization is modeled as the C leaving the molten-metal zone and entering the slag-metal zone where it reacts with O2 or FeO to form CO. The driving force for the mass transport of C is the concentration gradient between these phases. The equilibrium reaction in the slag-metal interaction zone thus ultimately controls the rate of decarburization. Solid-Scrap Zone. The solid-scrap zone is modeled as a mass of steel that melts according to the quantity of energy transferred from the liquid steel, gas, and arc and the proximity of the steel temperature to its melting point. As steel liquefies it is removed from the solid steel zone and added to the molten steel zone. The model predicts a homogeneous temperature in the solid-scrap zone which corresponds to the average temperature of the scrap. However, in reality the temperature is not homogeneous, and scrap material melts continuously throughout the heat. A modified version of the melting model proposed by Bekker et al.2 was implemented. The temperature ratio that divides energy between sensible heating and overcoming the heat of fusion is taken as Ts/Tmelt, where Ts and Tmelt are the scrap and steel melting point temperatures, respectively. This prevents the temperature of the solid material from exceeding its melting point temperature since the portion of energy contributing to the temperature increase diminishes to zero as the steel temperature reaches its melting point. Further detail is given in section 3.2. 3. Detailed Mathematical Model 3.1. Material Balances. The material in each zone can be tracked with an atom balance

d in out (b ) ) F k,z - F k,z dt k,z

(2)

where bk is the molar amount of element k in zone z, and Fkin and Fkout are the flow rates of element k in to and out of the zone. The chemical equilibrium for the multireaction systems can be computed by minimizing the Gibbs free energy. The method used in the model does not require reaction stoichiometry to compute equilibrium and is instead determined by solution of the


system of equations corresponding to the first-order necessary conditions for constrained minimization of the Gibbs free energy

∑i niaik ) bk ∆G°f,i + RT ln aˆ i +

∑k λkaik ) 0

(3) (4)

where ni is the number of moles of species i at equilibrium in the specified zone, aik is the number of atoms of element k in species i, bk is the number of moles of element k, ∆G°f,i is the Gibbs free-energy of formation, aˆ i is the activity of species i, and λk are Lagrange multipliers. The activity is a function of the system temperature, pressure, and composition. The thermodynamic data required were obtained from the National Institute of Standards and Technology (NIST) Chemistry Webbook.16 The activity of the nonideal slag was determined using the regular solution formalism using data for the interaction energies obtained from the literature.17 The activity for the metal phase was determined using the unified interaction parameter model using interaction parameter data also sourced from the literature.18 Subscripts indicating the zone have been omitted to simplify the notation. The benefit of using the system of equations described in eqs 3 and 4 is that a nested minimization routine is not required at each integration step, and the minimum is determined at each point by solving the necessary conditions together with the other model equations. The chemical equilibrium problem is convex when not combined with the phase equilibrium problem,19 providing added justification for this approach. A particular challenge in the development of the model was to ensure its robustness over a wide range of conditions. To achieve this, a logarithmic transformation was applied to the molar quantity at equilibrium, ni, by introducing a new variable niL defined as

niL ) ln(ni)

(5)

The term eniL now replaces ni in the formulation. The transformation was found to improve the scaling of the problem particularly when the equilibrium predicts very small concentrations of species. The remainder of this section describes the flows in to and out of the various zones. The flows are given in terms of the compounds and relate to elemental flows described in eq 2 as follows

Fk,z )

∑i aikFi,z

(6)

where k refers to the element and i to the compound. Gas zone: The net molar flows for the species into the gas zone are comprised as follows: in out - F i,gas ) F i,gas Fburner,i + Fingress,i + Fsm-gas,i + Fvolatile,i - Fo/g,i (7)

The molar flows FA-B,i indicate the flow of component i from zone A to zone B, with sm representing the slagmetal zone. Fburner,i accounts for the addition of O2 and CH4 via the burners, and Fsm-gas,i accounts for the evolution of CO from the slag as well as the flow of O2 between the slag-metal zone and the gas zone con-

trolled by the partial pressure according to Fsm-gas,O2 ) kPO2(ysm,O2 - ygas,O2). The inclusion of volatiles in the scrap is modeled by adding the C9H20 at a constant feedrate, Fvolatile,i, for a short period of time after the furnace has been charged. This is done to better approximate the dynamics of the vaporization of the volatiles since the equilibrium model would predict instantaneous vaporization. Fingress,i accounts for the gas that either leaks into the furnace or is expelled from the freeboard and is related to the furnace pressure. Fo/g is the amount of material leaving in the off-gas and has the same composition as the gas in the furnace. Slag-metal (sm) zone: The net molar flows into the slag-metal zone are comprised as in out F i,sm - F i,sm ) θL,iFlance,i + Froof,i + Fmm-sm,i + Fgas-sm,i (8)

Flance,i includes the injected carbon and lanced oxygen. The injected carbon enters both the slag-metal and the molten-metal zones. The distribution of the carbon is handled through the parameter θL,i, with θL,C estimated from operating data. θL,i is set to one for O2 and zero for all other components. Froof,i tracks the addition of fluxes through the roof, and Fmm - sm,i accounts for the movement of metallic species from the metal zone to the slag-metal zone and the flow of reduced materials back to the molten metal as determined by the concentration gradient. Molten-metal (mm) zone: The net molar flows for the molten-metal zone are comprised as in out - F i,mm ) F i,mm (1 - θL,i)Flance,i + Fsm-mm,i + Fas-mm,i (9)

where Fss-mm,i represents the addition of molten steel as a result of the solid scrap melting. Solid-steel (ss) zone: For the solid-steel zone it is more convenient to use units of mass instead of moles, and therefore the following balance is used

d (m ) ) M ˙ scrap - M ˙ melt dt ss

(10)

where M ˙ scrap is the rate of addition of the scrap to the furnace and M ˙ melt is the rate of melting. Further detail on the computation of the melt rate is given later in the discussion of the energy balance. 3.1.1. Mass Transport. The driving force for mass transport between the molten-metal and slag-metal zones is the concentration gradient across these zones. The mass transfer coefficient is expressed as the product of two parameters. The value of the first parameter is fixed and represents the mass transfer properties of the component relative to the other components, while the second parameter is the same for all components and can be considered as a base mass transfer coefficient. This latter parameter is adjustable, and its value is estimated using process data. The mass transfer is given by

Fmm-sm,i ) βikm(ymm,i - ysm,i)

(11)

where Fmm-sm,i is the molar flow of species i from the molten-metal zone to the slag-metal zone, βi is the relative mass transfer coefficient for component i, km is the base mass transfer coefficient between the molten-

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metal and the slag-metal zones, and yz,i is the molar concentration of species i in zone z. The oxidation reactions of Si, Al, Mn, Fe, P, S, and C are controlled by the presence of oxygen in the slag. The equilibrium calculation determines the preferential reaction of the components with the available oxygen according to their reduction potentials. The oxidation of these components depletes their concentration in the reaction zone, and thus the driving force for mass transport is increased. Similarly, if an oxygen deficit occurs, then a buildup of species i will inhibit the transport of that component into the reaction zone. Therefore depending on the conditions in the furnace, the oxidation reactions may be limited either by the availability of oxygen or by transport to the oxygen-rich reaction zone. The rate of decarburization is dependent on the availability of oxygen, either as O2 or as FeO, and the rate of mass transport to the oxygen interface. This is captured in the model as the transport of carbon from the molten-metal zone to the slag-metal zone. The availability of oxygen is controlled by its transport to the slag-metal zone from lancing directly into this zone and also in the form of FeO. Throughout the heat, the transport of carbon to the slag-metal zone is controlled by natural diffusion across a concentration gradient. During lancing there is an additional mixing component that is dependent on the flow rate of the O2 in the lance. Therefore an additional term is added to the diffusion relationship in eq 11 to account for the increased effect of O2 lancing

Fmm-sm,C )βCkm(ymm,C - ysm,C)+kmL(ymm,C -y*C) (12) where y*C is the equilibrium concentration of carbon in steel. Fruehan1 suggests a value of 0.03% as a practical limit in steel-making. kmL is related to the volumetric flow of O2 through the simple relationship kmL ) γFO2, where γ is an adjustable parameter determined from the process data. 3.1.2. Carbon Addition. Carbon is added to the furnace in two ways. It may be charged through the roof during the heat or injected into the slag/molten metal. The roof charging method involves adding a quantity at once, compared to the injection which is a continuous addition made over a period of time. When the carbon is charged into the furnace through the roof, a large portion initially floats on the slag surface and enters the bath as it dissolves. Ji et al.20 developed a model for the accumulation of carbon in the slag. The carbon is depleted from the slag due to reaction, and an amount floats on the slag, the mass of which is proportional to the mass of carbon in the slag. A similar approach was used in this work except from a different perspective. Here, the rate of carbon entering the slag-metal zone was modeled as being proportional to the mass floating on the slag, giving rise to the equation

d ) ) F in (m C (1 - XC) - kdcmc,float dt c,float

(13)

where mc,float is the mass of carbon floating on the slag. FCin is the rate of carbon charged into the furnace during the heat. The proportionality constant, kdc, can be estimated using industrial data. This parameter will depend on the type, quality, and method of addition of the carbon source. A portion of the carbon will be lost

as dust, some will react in the freeboard, and a portion will remain floating on the slag. Furthermore, the coke source is typically impure due to the presence of ash and volatiles. The installation on which the parameter estimation in this work is based uses metallurgical coke as the carbon source, which typically consists of 8688% fixed carbon.1 If the fraction of actual carbon entering the steel can be estimated, then it can be accounted for in the variable XC. In this work a value of 0.15 was assigned to XC to account for the presence of impurities. Injected carbon is added at a much lower rate compared to roof charging, thus it is assumed that all carbon injected into the steel will go into solution. Due to the high speed at which the carbon is injected, a portion will enter the molten metal, while the remainder will go into the slag. While in reality the division depends on a number of factors such as the angle of injection and the particle size distribution of the carbon, in this work this division is assumed constant and estimated as a parameter (θL) from the process data. 3.1.3. Slag Foaming. Slag foaming is desirable since it improves the efficiency of the electrical energy and protects the furnace from radiative damage. The viscosity, density, and surface tension of the slag have an important effect on the ability of the slag to form a stable foam, thus the composition of the slag is important. Jiang and Fruehan11 derived a relationship for the foaming factor, Σ, using dimensional analysis techniques to relate the ratio of the foam height and the superficial gas velocity to the physical properties of the slag. This relationship has been widely accepted in the literature addressing the slag foaming in steelmaking:

Σ)

115µ Hf ) s Vg xFσ

(14)

The slag viscosity, µ, was estimated using a model given in Urbain.21 The density, F, was estimated using the partial molar volumes from the data reported by Mills and Keene.22 A simple empirical model, obtained from Morales et al.,15 was used to estimate the slag surface-tension, σ. The superficial gas velocity, Vgs, was calculated from knowledge of the evolution of CO from the bath and the furnace geometry. This model, as with the majority of models in the literature, assumes that the slag depth is sufficient to not affect the foaming height of the slag. Using a model that is independent of the quantity of slag results in the prediction of unrealistically large slag foaming heights during the initial stages of the heat when CO is being produced, but the slag volume is not sufficient. To address the issue of incorporating the effect of the volume of slag (or static slag height) into the model in a simple yet robust manner, the model was adapted such that the foam height is essentially as predicted by eq 14 if the static slag depth is greater than a critical height. For smaller static slag depths, the foam height predicted from eq 14 is scaled by a fraction that is approximately proportional to the static slag height. The modified relationship takes the form

Hf ) Φ(ΣVgs)

(15)

1 1 Φ ) tanh(R(hs) + β) + 2 2

(16)

where


where Qgas-ss is the convective heat exchange between the gas and the solid scrap given by the following equation

Qgas-ss ) (hA)(Tgas - Tss)

Figure 2. Hyperbolic tangent function used to incorporate the effect of slag depth.

in which R and β are adjustable parameters. The function Φ is illustrated in Figure 2 together with a piecewise linear function that it can be thought of as approximating. The advantage of eq 16 over the more direct piecewise linear function is that the former is differentiable, making it more suitable for dynamic optimization. It should be noted that the relationship, given by Φ, between slag depth and foaming is not developed from fundamental principles since to the authors’ knowledge there have been no published studies on the relationship between the slag volume and the foaming index. In general, the volume of slag is not considered a limiting factor in the foam height, since slag foaming is of interest toward the end of the heat when there is ample slag material available. However, in this work, the entire heat is of interest, and therefore the model must be able to predict the foam height for the duration of the heat. Thus it is important to compensate for the fact that there may be insufficient slag volume during the initial stages. In the absence of literature or process data, the parameters in eq 16 were fitted using a value of 20 cm as the critical height for the static depth. 3.2. Energy Balance. The following energy balance was implemented for the gas, slag-metal, and moltenmetal zones: n

d dt

(Ez) ) Qz +

n

Fi,zHi,z|in - ∑Fi,zHi,z|out ∑ i)1 i)1

(17)

Here, Qz is the heat flow added to zone z; Fi,z is the molar flow of component i to/from zone z, and Hi,z is the corresponding enthalpy. The energy holdup at any time is computed as n

Ez )

ni,zHi,z ∑ i)1

(18)

where ni,z corresponds to the number of moles of species i in zone z. The heat flow term, Qz, is now developed for each zone in the following text. Gas zone: The gas exchanges energy with the solid scrap and the furnace roof and walls. The energy transferred to the gas is given by the following equation

Qgas ) -Qgas-ss - Qwater

(19)

(20)

The energy exchange between the gas and the solid scrap is most significant during the initial stage of the heat. To capture this behavior, the convective heat transfer coefficient is made proportional to the flowrate of material entering the furnace through the burners, Fburner. The prediction of the effective surface area of the scrap material is confounded by its random nature, and thus a simplification is made whereby the effective surface area is assumed proportional to the mass of steel divided by its bulk density. The combined area and convective heat transfer coefficient are thus computed as

hA ) kT3Fburner

( ) mss Fbulk

where the constant, kT3, is estimated from the data. Qwater is the heat extracted by the cooling water from the roof and wall panels

Qwater ) m ˘ H2OCp,H2O(Tcw,out - Tcw,in) where m ˘ H2O is the mass flowrate of water, Cp,H2O is its heat capacity, and Tcw,in and Tcw,out are the inlet and outlet cooling water temperatures, respectively. Slag-metal (sm) zone: The slag-metal zone has limited contact with the solid scrap, and thus heat exchange between these zones is assumed negligible. The amount of energy added to the slag via the arc is very complex since it will depend on the volume of slag and also the force that the arc strikes the bath, since the arc action will tend to displace the slag and expose the molten metal below. The model assumes that arc energy is not directly added to the slag, but instead it receives this energy indirectly through contact with the molten metal. The exchange between the slag and gas is assumed to be negligible compared to the other sources of energy transfer and therefore not explicitly considered. Thus the only source of heat exchange of the slag material, considered in the model, is with the molten metal

Qsm ) kT2(Tmm - Tsm)

(21)

where kT2 is the heat transfer coefficient that is estimated from process data. Molten-metal (mm) zone: As with the slag and gas, the heat exchange between the molten metal and gas is assumed negligible compared to the other sources of energy transfer. The energy flows into the molten-metal zone are given by the following equation

Qmm ) Qpower-mm - Qmm-ss - Qmm-sm - qrad 4

(22)

is the net loss of energy via radiation from where qrad 4 the bath and Qpower-mm is the energy entering the bath from the arc energy. The molten metal has contact with both the solid-scrap and slag-metal zones. The heat transfer from the molten metal to the solid scrap, Qmm-ss, is made proportional to the mass of liquid to capture the increasing heat transfer area as more

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molten metal is formed

Qmm-ss ) kT1mmm(Tmm - Tss)

(23)

where mmm is the mass of molten metal and kT1 is the heat transfer coefficient between the molten metal and solid steel. The effective heat transfer area between the slag-metal and molten metal is assumed constant, and the heat transfer is proportional to the temperature difference between the molten-metal and slag-metal zones

Qmm-sm ) kT2(Tmm - Tsm)

(24)

where kT2 is the same heat transfer coefficient as in eq 21. Solid-scrap (ss) zone: A methodology similar to that in Bekker et al.2 was used for the energy balance and the melt rate of the solid scrap. For the solid scrap there is no reaction, and an energy balance yields

d (T ) ) dt s

˙ in Qss(1 - Tss/Tmelt) - M scrap

∫TT Cp,FedT s

o

[mssCp,Fe]kdT

(25)

where Tss and Tmelt are the solid scrap and melting point temperatures. Qss(1 - Tss/Tmelt) is the fraction of energy entering the steel that contributes to sensible heating, with the remaining fraction of energy contributing to in ∫TTos Cp,FedT accounts for the melting of the scrap. M ˙ scrap the energy required to heat scrap as it is charged into the furnace. mss is the mass of solid steel computed from the mass balance in eq 10. kdT is an adjustable parameter to compensate for variations in the bulk density and composition, both of which influence the rate of melting. The bulk density will effect the heat transfer to the steel, and the composition will impact the sensible heating. The rate at which the solid scrap melts, M ˙ melt, can be determined by dividing the rate of energy available to the scrap for melting by the energy per unit mass required to melt the scrap at its current temperature

M ˙ melt )

Qss(Tss/Tmelt) [∆Hf,Fe +

∫TT

melt

s

(26)

Cp,Fe(s)dT]kdm

where (Tss/Tmelt) is the fraction of the energy that contributes to the melting of the scrap. ∆Hf,Fe is the heat of fusion of Fe, and kdm is an adjustable parameter to compensate for variations in the bulk density and composition in the scrap. The energy transferred to the solid scrap, Qss, is given by the following equation

(27) Qss )Qpower-ss + Qmm-ss + Qgas-ss - Qvolatile - qrad 3 where Qpower-ss is the portion of the electrical energy from the arc that is transferred to the solid material, Qmm-ss is as given by eq 23, Qgas-ss is as given by eq 20, is the net loss of energy from the scrap and qrad 3 material via radiation. Qvolatile ) Fvolatile(∆Hvap), which accounts for the energy required to vaporize the volatile components present in the scrap. 3.2.1. Radiative Heat Transfer. Radiation is an important mechanism of heat transfer due to the high temperatures in the furnace. It is thus necessary to be

Figure 3. Stages of the melting used in the radiation model.

able to predict the contribution of radiation as a mode of heat exchange in the furnace. This component of the model determines the radiative heat transfer between the different surfaces based on their surface temperature, emissivity, and surface area. An important characteristic that must be captured is that as the scrap melts or if more scrap is charged, the surface areas also change. The dynamics of the changing conditions in the furnace are modeled by relating the void volume in the furnace to the exposed surface area of the various elements (roof, walls, scrap, and bath) in the furnace. The dynamic behavior of melting scrap in the furnace is extremely complex and varies from heat to heat. A simple geometrical model is therefore proposed in order to approximate dynamic behavior as the steel melts. The melting model assumes an initial cone-frustum shaped void is melted into the scrap by the electrodes, which increases in volume as more material is melted. Figure 3 illustrates the furnace geometry as the heat progresses from the initial stage to the intermediate and final stages. Figure 3(a) shows the initial exposure of the roof, followed by the progressive exposure of the walls in Figure 3(b), until flat bath conditions are approached in Figure 3(c). Initially, the furnace roof and walls will be shielded from the radiation by the scrap material. As the scrap melts the roof will be progressively exposed, followed by the walls. The cone-frustum shaped void increases in height and radius, while maintaining a constant critical angle of repose, until the cone base radius is equal to that of the furnace. In reality the angle will change continually with the competing effect of the electrodes boring down and the scrap collapsing. A constant angle of repose provides a simple mechanism for averaging this apparent random behavior. More details of the changing geometry are given later in this section. The surfaces within the furnace are treated as gray bodies and thus the net radiative heat transfer from each surface is

qrad ) i

Ebi - Ji (1 - i)/iAi

(28)

where Ebi is the blackbody emissive power of surface i determined from the Stefan-Boltzmann law, Ebi ) σTi,4 σ ) 6.676 × 10-8W/(m2‚K4), and Ti is the temperature of surface i. Ji is the radiosity, which is the rate of radiation leaving a unit area of surface i; Ai is the surface area, and i is the emissivity of surface i. The destination of the radiative heat transfer to surfaces in an enclosure is described by the following equation N

) qrad i

AiFij(Ji - Jj) ∑ j)1

(29)

where Fij is the view factor, the fraction of radiation leaving surface i that is intercepted by surface j.


For an enclosure the view factors are related as follows: N

Fij ) 1 ∑ i)1

(30)

The following reciprocity relationship is always true

AiFij ) AjFji

(31)

Three-dimensional models using differential elements to compute the view factors10,23 are too computationally intensive to be used in this work. Therefore simplifying assumptions are made with respect to the internal geometry of the furnace to avoid the large computational expense in computing the view factors by integration. Thus the following simplifying assumptions are made with regard to the four surfaces considered in the model: the roof is modeled as a dome; the bath as a circular disk; the scrap surface as a cone-frustum; and the walls of the furnace as a cylinder. Reynolds23 showed that CO rich atmospheres, typical of smelting furnace freeboards, contribute less than 5% toward the radiative energy exchange within the furnace. Absorption and emission of radiative energy from the gaseous freeboard is therefore not explicitly included in the radiation model. Discrepancies as a result of this exclusion from the model would be compensated for in the estimation of the heat transfer coefficients for the gas phase, effectively converting these parameters to overall heat transfer coefficients from purely convective heat transfer parameters. The system of equations given by eqs 28-31 requires a further N(N-1)/2 equations to completely specify the radiation model for given surface temperatures, emissivities, and areas, where N is the number of surfaces in the enclosure. Here we consider five surfaces in the furnace: the furnace roof, furnace walls, scrap material, the bath, and the arc, which for convenience are numbered as follows: 1-roof, 2-wall, 3-scrap, 4-bath, 5-arc. The presence of the arc in the furnace will be addressed shortly. Analytical solutions for the following view factors, based on the assumed furnace geometry, were obtained from literature:24 F1,1, F1,2, F2,2, F2,4, F4,1, F4,4. Next, the presence of the electric arc in the furnace is considered. The arc is assumed to emit radiative energy as a blackbody and to be perfectly transparent in receiving radiative energy, as was assumed by Guo and Irons.10 Due to the transparency of the arc we do not need to consider the radiation received by the arc from the other surfaces in the furnace. Furthermore, the model makes the same assumptions about energy usage from the arc as were made by Guo and Irons.10 Specifically, 18% of the energy from the arc is delivered directly to the steel, 2% is absorbed by the electrode, and 80% is delivered in the form of radiation. Thus the energy radiated by the arc is modeled as

qrad 5 ) (0.80)Qarc

(32)

is the net radiative heat transfer from the where qrad 5 arc. Qarc is the total energy released from the arc and is related to the active power (Pr) in the primary circuit through a proportionality constant that is estimated from data, Qarc ) kPPr. A percentage of the radiated energy will be transmitted to the scrap and the steel

bath, while the remainder is “lost” to the furnace walls and roof. The fraction of radiation from the arc that by reaches surface i is determined by multiplying qrad 5 the appropriate view factor, F5i, and including its contribution in the energy balance in eq 29 N

qrad i

)

rad AiFij(Ji - Jj) - q5-i ∑ j)1

(33)

rad where q5-i ) F5,iqrad and is the radiation received by 5 surface i from the arc. The negative sign preceding the rad term indicates that surface i receives energy from q5-i is defined as the amount of radiation the arc, since qrad i leaving surface i. The view factors from the arc, F5,i, are determined based on process knowledge. As the scrap melts the most important change is that the initial shield of scrap protecting the walls from the arc melts away exposing the wall to arc radiation. During the meltdown the arc is buried in the scrap, and therefore the view factor, F5,3, between the arc and scrap is close to unity. The fraction of radiation leaving the arc that is intercepted by the scrap decreases slowly until the end of the heat is approached when the layer of scrap protecting the wall disappears very quickly. This behavior is approximated as an exponential decay with respect to the radius of the base (rb) of the cone-frustum, used to approximate the void left by molten material. Due to the constant angle of repose assumed in the model, the radius of the base will change with the changing mass of solid scrap in the furnace. The dynamics of the arc to wall view factor was approximated as

F5,3 ) 0.9 - erb/42

(34)

Note that the initial and endpoint conditions ensure that

0 < rb e rR

(35)

where rR is the furnace radius and is approximately 3.5 m. F5,1 and F5,4 are determined by treating the arc as a cylinder and using analytical expressions from literature.24 F5,5 ) 0; therefore, F5,2 can be computed using the enclosure equation for view factors, given in eq 30. Effect of Foaming on Power Transfer. Slag foaming has a significant effect on the amount of power that is transferred to the steel. As the foam covers the arc, less energy is lost to the roof and walls and is instead transported directly to the steel. Reports in the literature1 indicate energy transfer efficiency improvements from 40% efficiency without a foaming slag to 60-90% efficiency when the slag is foamed. The amount of radiative energy transferred to the scrap, as computed in the radiation component of the model, is for an ideal case where the effects of a foaming slag are not considered. This section discusses the use of an efficiency factor, E/f , to account for the effect of the foaming slag. The steel receives 18% of the arc energy (Qarc), radiative energy from the arc, and also a portion of energy recovered as a result of slag foaming. The radiative energy to the solid scrap and molten metal is determined explicitly in the radiation model. The quantity of energy transferred directly from the arc and the portion recovered due to the foaming slag is divided

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between the solid scrap and the molten metal in proportion to their relative mass. The energy transfer to the molten metal due to the arc is given by

(

)

mmm rad rad Qpower-mm ) (0.18Qarc + E/f (q5-1 + q5-2 )) mmm + mss (36) where E/f is the fraction of radiative energy recovered rad rad due to foaming; q5-1 and q5-2 are the net radiative transmission from the arc to the furnace roof and walls, respectively. Similarly, the energy to the solid scrap is given by

Qpower-ss )

(

1-

)

mmm rad rad (0.18Qarc + E/f (q5-1 + q5-2 )) (37) mmm + mss

The net radiative energy from the roof and walls, given in eq 33, is also altered to compensate for the effect of foaming N

) qrad i

rad AiFij(Ji - Jj) - (1 - E/f )q5-i ∑ j)1

(38)

The efficiency E/f is comprised of two efficiency factors expressed as fractions between 0 and 1, that together determine the impact of the foaming slag on the energy transfer in the furnace

E/f ) E1E2

(39)

The height of the foaming slag was considered previously in eq 15. The first efficiency factor, E1, relates the fraction of the arc that is covered by the foam to the fraction of radiation that is prevented from reaching the wall and instead transmitted to the steel. Using the data in the literature10 as a guideline, it was assumed that if the arc is fully covered by the foaming slag, at most 70% of the arc energy will be blocked from reaching the wall and instead transported to the steel. Furthermore, it is assumed that the arc length will be constant at approximately 0.5 m, and therefore at a slag foam height of 0.5 m or greater E1 will be at its maximum value of 0.7 or 70%. Below 0.5 m, E1 is approximately proportional to the slag foam height, decreasing toward zero as the foam height approaches zero. This relationship is modeled with the following hyperbolic tangent function

1 1 E1 ) 0.7 tanh(R1Hf + β1) + 2 2

(

)

(40)

with R1 ) 5.0 and β1 ) -1.25. The shape of the function is similar to that in Figure 2. The second factor, E2, considers that flat bath conditions are required before foaming can occur unhindered. The model used to determine the slag height does not take into account that a significant amount of solid material in the furnace will impact the foaming. The presence of solid scrap will limit the amount of foaming, and also foaming may occur in the void spaces between the scrap and thus be unable to cover the arc; subsequently the benefits in terms of efficiency are reduced. It is assumed that when the scrap material is less than 20% of the initial charge mass of scrap material

Table 1. Adjustable Model Parameters description γ km kT1 kT2 kT3 kP kdm kdT kPO2 kdc θL

parameter for mass transfer coefficient due to mixing effect of lancing mass transfer coefficient between the slag-metal and molten-metal zones heat transfer coefficient between the molten metal and solid scrap heat transfer coefficient between the molten metal and slag-metal heat transfer coefficient between the gas and the solid scrap relationship between active power and arc energy solid melt rate tuning parameter solid temperature tuning parameter mass transfer coefficient for transfer to the slag dissolution parameter for carbon addition fraction of lance coke entering the molten-metal phase

(0.01%mscrap < 0.2), E2 will be close to 1, and hence the presence of solid has a negligible effect on foaming. However, when the mass of scrap exceeds 20% of its initial charge mass, then E2 decreases with an increasing fraction of solid material. This is implemented using the relationship

1 1 E2 ) tanh(R2(1 - 0.01%mscrap) + β2) + (41) 2 2 with R2 ) 3.2 and β2 ) -1.29. 4. Parameter Estimation As has been discussed previously, an objective of the modeling work was to limit the number of parameters in the model. This goal has to some extent been achieved in that the model requires that only a relatively modest number of parameters be estimated. The parameters considered for estimation are listed in Table 1. In their review of the Dow parameter estimation problem, Biegler and Damiano25 discuss the importance of using good starting values and eliminating as many unnecessary model parameters as possible, both of which lead to better conditioned optimization problems. To this end, initial estimates of the model parameters were obtained using information from published literature sources and visually comparing the predictions against the industrial data. This was followed by a sensitivity analysis, the purpose of which was to eliminate insensitive parameters from the estimation problem. Dynamic models require specification of the initial condition of the system. The initial conditions used for the estimation problem are discussed here. The mass of each scrap charge is recorded for the individual heats. The charge is composed from different scrap sources such as pig iron, re-bar, and directly reduced iron. The mass and average chemical compositions of each source are known, which enables the overall composition of the charge to be determined. The quantity of volatile material in the scrap is computed in the same manner. The bulk density of the scrap may vary significantly depending on the scrap mix and the degree of compactness of the scrap. The bulk density will affect the ability of the burners to heat the scrap and also the rate at which the scrap melts. At the end of each heat a portion of the molten metal, known as the heel, is left in the furnace to aid the next heat. The initial heel (volume of liquid steel) is not directly measured but computed through a mass balance and recorded for each heat. The


Figure 4. Coefficients for regression model from the sensitivity analysis.

composition of the heel was assumed to be that of the endpoint target tap chemistry from the previous heat. A small mass of slag was also assumed to remain in the furnace with the heel, the composition of which was taken to be the average composition of the slag for the furnace. The initial composition of the freeboard was assumed to be that of the ambient air. 4.1. Sensitivity Analysis. The purpose of the sensitivity analysis was to isolate the least sensitive parameters in the model; these could then be set as constant values and removed from the rigorous estimation problem. The sensitivity analysis was conducted using a design of experiments approach. A resolution IV, fractional factorial design was carried out to determine the main effects of the parameters on the model predictions. This design was sufficient to provide an indication of the effect of the parameters and also specify the variables that are most strongly influenced by each parameter. The nominal case was determined using parameter values obtained from published literature together with manual adjustments based on visual comparisons to the data. Each parameter was then perturbed above and below its nominal value by 20% according to the designed experiment. To quantify the effect of the parameter changes, a combined measure of model performance was constructed from a selection of the predicted states. The states whose values are measured on the actual process were selected for the analysis since these are the only measurements which would be available for the rigorous estimation work. In addition, variables important to the performance and accuracy of the model were included, such as the temperatures of each zone and the mass of solid and liquid steel. For the analysis, it was necessary to construct a single metric corresponding to the trajectory of each state. The integral-square-error (ISE) of the predictions from the mean state predictions was calculated to summarize the time-dependent predictions for the entire heat into a single value. Centering and scaling the data ensures that all variables have the same weighting. It was therefore possible to construct a combined performance measure for the model using all the selected states. This combined measure of performance summarized the overall model sensitivity to each of the parameters. The ISE of the individual model states provided information on the parameters by which they were most strongly influenced.

To analyze the information, a linear regression model was built, and the coefficients of the regression model examined to determine the importance of the parameters. Figure 4(a),(b) shows the coefficient values for the regression model that was constructed from the sensitivity analysis data. There are 13 parameters shown in these figures; 11 correspond to the parameters described in Table 1, and the final two are, respectively, the initial concentration of carbon and the mass of the heel in the furnace. The coefficients whose confidence intervals do not include zero are deemed to be significant to the model. The coefficients for the combined measurement, as shown in Figure 4(a), indicate that only six of the 13 parameters are shown to be significant. The CO2 measurement is strongly affected by only four of the 13 parameters. By studying the model coefficients of both the combined and individual measurements a good understanding of the model sensitivity was obtained. This study enabled a number of parameters to be fixed and hence eliminated from the model estimation problem. This study is not necessarily conclusive since the results depend on the states that were chosen to evaluate the model and is based on a linear analysis. It is however a useful tool that can be implemented to provide a good starting point for difficult estimation problems. The study suggested the following parameters to be the most important for the model: km, kT1, kT3, kP, kdm, and kdc. 4.2. Available Measurements. The following measurements were available for the estimation of the model: 1. Electrical and material inputs: power; scrap additions; carbon, lime, and dolime additions; and carbon, oxygen, and lime injection 2. Off-gas chemistry: CO, CO2, O2, H2 3. Endpoint carbon concentration and temperature measurements of the steel 4. Average endpoint slag chemistry There are a number of factors which complicate the estimation problem, the most severe of which is the limited amount of data available. The only direct measurements of the process which are available for the duration of the heat are the off-gas composition. Four components, namely CO, CO2, H2, and O2, are measured. However, there is a degree of correlation between these measurements due to the O2 dependence of the other components in the system. This means that there is less actual information in these measurements than

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if they were uncorrelated. Furthermore, as is typical of industrial data, there is noise in the data. The data were processed through a low-pass filter with a data-window of 3 measurements; this was deemed sufficient to reduce the most significant effects of noise without adversely effecting the dynamics of the signal. The absence of data for the following variables makes the estimation problem particularly challenging: 1. Off-gas flowrate: The industrial system being studied has a single off-gas system serving a pair of furnaces. The off-gas system also has a number of vents which entrain the ambient air to aid cooling of the gas. The combination of these factors makes the prediction of the off-gas flowrate using the suction fan very difficult. This information would enable better predictions of the amount of ingressed air to be made and also allow a considerably more accurate mass and energy balance to be carried out. 2. Slag composition: The available data for the slag chemistry is the endpoint composition. Availability of the composition profiles over the heat duration would significantly improve the observability of the process. 3. Bath temperature: The single temperature measurement at the end of the heat makes it difficult to accurately calibrate the melting model. The above measurements would provide a much clearer understanding of the behavior within the furnace and allow the model to be better tuned to the process. Furthermore, it would be possible to add more detail to the model since there would be more information available for estimation and validation of modeling work. While technology exists for all the above measurements, there is an associated capital cost, reliability issues of equipment, and maintenance costs. Clearly, there is a tradeoff, and it is necessary to evaluate whether the benefits of such equipment would justify the cost. Evaluating the benefit of improved instrumentation is one of the planned extensions to this work. 4.3. Handling the Raw Data. There were two types of anomalies present in the raw off-gas data. The first is associated with the opening of the furnace roof during the heat to add the second charge. When the roof is swung open, the off-gas analyzer measures the surrounding air composition instead of the furnace conditions. This can be quite easily identified because of the timing and the composition of O2 spikes to approximately 20%, while all other components drop to zero (N2 is not measured). The second aberration was a result of the off-gas analyzer probes becoming blocked. The analyzer is able to sense the blockage and purges the probes with air. However, there is a short period of unreliable measurement preceding and following the purge that must also be eliminated from the data. The status of the analyzer is a recorded measurement, and thus these data can be easily removed. These anomalies need to be accounted for so that they do not adversely affect the estimation of the model parameters. 4.4. Rigorous Parameter Estimation. The differential-algebraic furnace model and dynamic data require the parameter estimation problem to be solved as a dynamic optimization problem. In the estimation problem, the model parameters are the decision variables, and the deviation of the model predictions from the data is minimized. Cervantes and Biegler26 discuss several methods for the solution of dynamic optimization problems. The two most successful techniques address the infinite-dimensional nature of the problem by

implementing some level of discretization. The first method involves partial discretization, where the control vector is parametrized, but the states remain continuous. The alternative method requires both the control inputs and the states to be discretized and reduce the problem to a standard nonlinear programming problem. The partial discretization approach is favored for this work because of the large dimensionality of the model. The estimation of the model parameters was implemented using the gPROMS/gEST27 software package which utilizes this solution technique. The particular implementation of the maximum-likelihood function used is given by

Φ)

N ln(2π) + 2 1

min 2 θ

{

NE NVi NMij

∑ ∑∑ i)1 j)1 k)1

(

2

ln(σijk ) +

)}

(z˜ ijk - zijk)2 σijk2

(42)

where N is the total number of measurements for all the experiments; θ is the set of model parameters to be estimated subject to upper and lower bounds; NE is the number of experiments performed; NVi is the number of variables measured in the ith experiment; NMij is the number of measurements of the jth variable in the ith experiment; σijk is the variance of the kth measurement of variable j in experiment i; z˜ ijk is the kth measured value of variable j in experiment i; and zijk is the model prediction corresponding to the kth measurement of variable j in experiment i. The estimation problem as given in eq 42 allows the flexibility of including several types of variance models. The variance may be assumed constant or related to the magnitude of the measured or predicted values. A drawback of increasing the flexibility in the variance model in the case of limited data is that it is equivalent to adding more parameters to the model and therefore may affect the conditioning of the estimation problem. If the variance model is completely specified, eq 42 reduces to a weighted least-squares problem. This formulation allows multiple heats to be considered simultaneously to estimate model parameters. 4.4.1. Parameter Scaling. The condition of the parameter estimation problem may vary significantly during optimization and correct scaling can help offset this effect. Parameters must be scaled so that they do not vary over many orders of magnitude. The method of scaling used in this work is of the general form

θ˜ j )

θj - 1/2(θmax - θmin ) j j 1/2(θmax - θmin ) j j

(43)

and where θ˜ j is the scaled parameter value, and θmax j are the bounds of the unscaled parameter θj. θmin j 4.4.2. Parametrization of Input Data. The controlvector parametrization approach used in the solution of the differential-algebraic equation system requires parametrization of the process inputs. A satisfactory level of discretization must be selected such that the number of intervals is reasonable and also that the inputs are accurately represented. Fortunately, in the case of the furnace the nature of the input variables makes them suitable for the piecewise-constant approximation. The process inputs are typically held constant for quite long intervals during normal opera-


Figure 5. Off-gas predictions. Table 2. Parameter Estimates parameter

estimate

parameter

estimate

γ km kT1 kT2 kT3 kP

8.50E-03 5.53E+01 1.35E-02 1.00E-04 1.07E-03 1.11E-03

kdm kdT kPO2 kdc θL

4.43E-01 1.21E-01 5.99E-01 6.94E-02 7.50E-01

Table 3. Model Prediction Data CO CO2 O2 H2

selected heat MSPE

average MSPE

validation MSPE

7.62E-03 2.67E-03 1.78E-04 4.14E-03

6.59E-03 2.54E-03 4.50E-04 4.29E-03

6.30E-03 3.70E-03 1.66E-04 6.26E-03

tion of the plant. In this study the inputs were parametrized using 3-minute intervals. Parametrization of the inputs at 3 and 5 min intervals did not show any appreciable difference. 4.4.3. Analysis of Results. Table 2 presents the final values of the model parameters listed in Table 1. The values of the parameters identified in section 4.1 as being significant were generated by gPROMS/gEST. These parameters were obtained by carrying out the estimation on 8 batches simultaneously. The only data available for this study that provide information on the progression of the heat is the offgas composition data. Figure 5 shows the model prediction (solid line) of the off-gas composition data for one of the heats used for the parameter estimation. From these predictions, the model shows reasonable agreement with the process data. The sharp changes approximately midway through the batch coincide with the introduction of the second charge. Furthermore, the burner flows are adjusted in anticipation of the second charge which also affects the off-gas composition. Table 3 summarizes the quality of model predictions in terms of the mean square prediction error for (i) the heat shown in Figure 5, (ii) the average for all the heats used in the estimation process, and (iii) validation data sets not used in the parameter estimation. These values

Figure 6. Endpoint slag composition prediction.

are computed from composition data, with a typical value of 0.2 for CO, CO2, and H2; O2 typically has a value less than 0.05. Comparing the data for the selected heat shown in Figure 5 against the average of all the heats used in the parameter estimation indicates that model performance for this heat is fairly typical. The model was validated in a straightforward manner. The model predictions, with the parameters fixed at the values in Table 2, were compared to data from two heats not used in the estimation problem. The mean square predicted error results for these new data sets compare favorably with those from the data used to carry out the estimation, providing some confidence in the model’s predictive capability. Discrepancies between the model predictions and the plant data are expected due to a number of factors. This includes the inevitably inaccurate predictions of the ingressed air and off-gas flow rates which are not measured. During the first and second scrap charges the model predictions tend to be poorest. Possible reasons are that an accurate estimate of the initial conditions related to the scrap is difficult to obtain based on the scrap charge. When material is charged, the

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Figure 7. Scenario 1. Furnace input profiles.

furnace is also cooled significantly and takes several minutes to reach the point where the assumption of equilibrium is appropriate. Another possible reason for the observed discrepancy is that the model assumption of homogeneous scrap results in carbon being released in proportion to the melt rate. During the meltdown period the system is very erratic, due for example to collapsing of a section of the scrap pile. Furthermore, the conditions in the off-gas are affected by the properties of the particular piece of scrap beneath the arcs at that time. However, as the amount of liquid steel increases these higher frequency disturbances are removed. It is also very difficult to evaluate the contribution of the various sources of energy in the absence of data from a designed experiment and temperature measurements for the duration of the heat. Figure 6 compares the model predictions of the endpoint slag chemistry with the compositions obtained from the process. The model predicted composition presented here is from the same heat as the data presented in Figure 5. The slag chemistry is not determined for each heat, and so the average of the composition data for a number of batches together with the standard deviation is presented. The slag composition in the model is determined by calculating the composition in the slag-metal zone while ignoring the presence of Fe. It is difficult to assess the model performance based upon a single data point; however, the model shows a good fit with the industrial data. 5. Simulation Studies The model comprises 85 differential variables and 1050 algebraic variables; a simulation takes approximately 30 CPU s on an Intel Pentium IV 3.0 GHz processor. Several case studies are presented in this section to illustrate the potential uses of the model and its functionality. 5.1. Scenario 1. Base Case. Figure 7 shows the normalized input profiles for a typical heat. Initially the burners are fired-up (FO2 and FCH4) to preheat the scrap.

Figure 8. Scenario 1. Solid scrap and liquid steel profiles.

Following the preheat, the power (Parc) is turned on, and the scrap will begin to melt. When sufficient space has been created in the furnace, a second charge is added; this occurs at approximately t ) 28 min in the case shown here. Toward the latter stages of the heat, carbon is injected (Cinj) into the bath to reduce the FeO and produce a foaming slag. At the same time O2 is lanced (O2,lnc) into the bath to prevent a buildup of carbon in the steel. During the course of the heat carbon, lime, and dolime are charged through the furnace roof at specific times, typically just before the second charge and then again once there is sufficient liquid steel. Figure 8 presents the normalized profiles of the mass of solid and liquid steel, indicating the melting progression. Initially, there is very little melting as one would expect during the preheat. However, once the power is turned on melting proceeds rapidly. The addition of the second charge is evident at approximately 30 min, illustrated by the sharp increase in the mass of solid steel. The initial mass of liquid steel is due to the heel


Figure 9. Scenario 1. Offgas composition profiles.

Figure 10. Scenario 1. Slag composition profiles.

left in the furnace from the previous heat in order to aid melting. Figure 9 shows the off-gas composition profile for this heat. It is evident from the graph that all O2 and CH4 are completely consumed. The initial increases in the H2 and CO concentration are due to the combustion of volatiles that vaporize from the scrap within the first few minutes of it being charged. The increase seen at the time of the second charge is sharper due to hotter furnace conditions at this time, and therefore the volatiles will vaporize faster. When a large amount of volatiles are present or CO is coming from the bath, an O2 deficit results, and the equilibrium favors the production of CO over CO2 as the products of combustion; this trend is also favored by higher temperatures in the furnace. Toward the end of the heat a large amount of CO is given off from the bath due to increased C injection and O2 lancing; evidence of this is seen in the rising CO and CO2 concentrations in the off-gas after t ) 55 min. The slag compositions shown in Figure 10 were determined by excluding the mass of Fe in the slagmetal zone. Initially the volume of slag is very small and hence subject to large variations with the addition of the initial carbon charge. Once the lime and dolime are charged into the furnace (at t ) 22 min) the volume of the slag is sufficient such that the composition is less

Figure 11. Scenario 1. Radiative heat transfer in the furnace.

prone to extreme fluctuations. The presence of C in the slag prevents the concentration of FeO from increasing. However, when the C concentration gets low (at t ) 67 min), the FeO concentration climbs rapidly. The presence of SiO2 and Al2O3 can also be seen. The functionality of the radiation model is illustrated in Figure 11, where the net radiative heat transfer from each surface is illustrated. For simplicity the net radiative heat to the scrap and the bath are combined into a single variable, steel. In the model, a negative value indicates a net gain of energy onto the surface. During the heat, the walls and roof do not heat up at the same rate as the steel due to cooling water pumped through the panels. For the first t ) 15 min, the steel is heated via the burners and increasingly radiates heat to the walls and roof. After the power is turned on, the net radiative transfer is dominated by the radiation from the arc to the steel and other furnace elements. When the arc is boring into the scrap, the arc is shielded by the scrap, and therefore the majority of the energy from the arc is radiated to the scrap. From time t ) 53 min, the ability of the scrap to shield the walls decreases quite rapidly, and the incident radiation to the walls increases accordingly. The increase is a result of the walls being exposed as the steel melts, and the dead-time in the wall exposure is a result of the cone shaped void which results when the electrodes bore into the steel; thus the walls are largely protected until most of the scrap is melted. The presence of a foaming slag, which shields the walls from the arc, prevents the radiation to the walls from increasing even further. Finally, toward the end of the heat when the power is turned off, there is a net loss of radiative energy from the bath to the roof and walls. 5.2. Scenario 2. Effect of Preheat Duration on Final Melting Time. In this scenario the preheat is reduced from 15 min in the base case to just 3 min, and the subsequent additional electrical power and time necessary to compensate for the reduced preheat are illustrated. For the case where the preheat is reduced to 3 min, the first charge will require more electrical energy before there is sufficient space in the furnace to add the second charge. In this scenario we apply the same initial power trajectory as in the base case but shifted in time to coincide with the end of the preheat. The power is also maintained at its maximum value until there is sufficient space in the furnace to add the

8082


Figure 12. Scenario 2. Scrap melting. Figure 14. Scenario 4. Slag foaming.

Figure 13. Scenario 3. Carbon injection strategy. Table 4. Comparison of Preheat Strategies variable

total change (%)

electrical power burner O2 burner CH4 melt time

17.5 -8.9 -9.3 -5.8

second charge. This is illustrated in Figure 12 over the time interval, t ) 12-23 min. Figure 12 and Table 4 show the results of this case study. In Figure 12 it is evident that the overall batch time is longer in the base case. However, studying the results in Table 4, it is evident that the savings in time and burner fuel consumption result in the use of a larger amount of electrical power. 5.3. Scenario 3. Effect of Carbon Lancing on the Slag Composition. Figure 13 shows two different carbon injection strategies. In the test case, a much higher rate of injection is maintained until the end of the batch. The consequence of keeping the injection rate higher toward the end of the heat is shown in the slag composition where slag FeO composition is reduced by 9%. Thus there is less iron lost to its oxidized state, which is desirable. 5.4. Scenario 4. Effect of Lancing Strategies on Slag Foaming Performance. For the appropriate slag composition, the C injection and O2 lancing rates can

be used to manipulate the slag foam height. In Figure 14, the solid line represents the predicted foaming height, with the base case and test case shown as (a) and (b), respectively. The dashed line corresponds to the foaming efficiency E/f , given by eq 39, which represents the percent of radiative energy recovered as a result of foaming. E/f has a maximum value of 70% since not all energy can be recovered; this is discussed earlier with the development of the foaming efficiency relationships. Comparing the foam height in (a) and (b) shows that the slag is foamed to a greater degree, in case (b) through the injection of more carbon and lancing of more oxygen. However, comparing the foaming efficiency in (a) and (b) shows that there is little difference between these scenarios. This illustrates that the greater degree of foaming in case (b) has no benefit in terms of recovered energy since E/f was already at its maximum in (a) during that time period. In fact, excessive foaming could result in the slag foaming through the electrode ports in the roof, an occurrence which is both possible and undesirable. Furthermore, there is a financial cost for the additional C and O2 consumed in increasing the foam height in (b). Determining the correct C and O2 addition policies would be an interesting optimization problem, whereby the maximum efficiency is desired for the small addition of C and O. 6. Conclusions A detailed model of the electric arc furnace has been developed. This model is based on fundamental principles, although a degree of empiricism has been introduced to model relationships where the real mechanisms are either too complex to be modeled or insufficient information is available. Key model parameters have been estimated using available industrial data. However, further measurements during the progression of the heat would be useful. It is hoped that this work and the planned optimization studies to follow will provide an incentive for further instrumentation of industrial EAF operations so that advantage may be taken of these tools. The model framework presented allows for the inclusion of further detail. Potential enhancements include detailed models for predicting the melting of scrap in


the furnace and improved prediction of decarburization and slag foaming, all of which are areas of ongoing research at the McMaster Steel Research Centre. However, the generally limited amount of data available for parameter estimation in an industrial setting should be carefully considered during model refinements. A number of case studies have been presented where operational tradeoffs to improve the profitability or production rate of the process have been illustrated. Future work will focus on incorporating the model into a mathematical optimization framework to determine optimal operation policies by explicitly considering the cost of these tradeoffs. Acknowledgment The authors gratefully acknowledge personnel at Dofasco for their valuable input, Air Liquide for making an off-gas analyzer available for this study, and Drs. Gordon Irons and Diancai Guo at McMaster University for useful discussions. This work was funded by the McMaster Steel Research Centre, Materials and Manufacturing Ontario and the McMaster Advanced Control Consortium. Literature Cited (1) Fruehan, R. J. The Making, Shaping and Treating of Steel, 11th ed.; AISE Steel Foundation: Pittsburgh, PA, 1998. (2) Bekker, J. G.; Craig, I. K.; Pistorius, P. C. Modelling and Simulation of an Electric Arc Furnace Process. ISIJ Int. 1999, 39(1), 23-32. (3) Oosthuizen, D. J.; Viljoen, J. H.; Craig, I. K.; Pistorius, P. C. Modelling of the Off-gas Exit Temperature and Slag Foam Depth of an Electric Arc Furnace. ISIJ Int. 2001, 41(4), 399-401. (4) Cameron, A.; Saxena, N.; Broome, K. Optimizing EAF Operations by Dynamic Process Simulation. In Proceedings of the 56th Electric Furnace Conference; ISS Publishers: Warrendale, PA, 1998; pp 689-696. (5) Matson, S.; Ramirez, W. F. Optimal Operation of an Electric Arc Furnace. In Proceedings of the 57th Electric Furnace Conference; ISS Publishers: Warrendale, PA, 1999; pp 719-728. (6) Modigell, M.; Trabert, A.; Monheim, P. A Modelling Technique for Metallurgical Processes and Its Applications. AISE Steel Technol. 2001, 28, 45-47. (7) Meng, M.; Irons, G. A. Comparison of Electric Arc Models with Industrial Data. In Proceedings of the 58th Electric Furnace Conference and 17th Technology Conference; ISS Publishers: Warrendale, PA, 2000; pp 183-194. (8) Billings, S. A.; Boland, F. M.; Nicholson, H. Electric Arc Furnace Modelling and Control. Automatica 1979, 15, 137-184. (9) Collantes-Bellido, R.; Gomez, T. Identification and Modeling of a Three Phase Arc Furnace for Voltage Disturbance Simulation. IEEE Trans. Power Delivery 1997, 12(4), 1812-1817.

(10) Guo, D.; Irons, G. A. Modeling of Radiation Intensity in an EAF. In Proceedings of the Third International Conference on CFD in the Minerals Process Industries; CSIRO: Melbourne, Australia, 2003; pp 651-659. (11) Jiang, R.; Fruehan, R. J. Slag Foaming in Bath Smelting. Metall. Trans. B 1991, 22, 481-489. (12) Gou, H.; Irons, G. A.; Lu, W.-K. A Multiphase Fluid Mechanics Approach to Gas Holdup in Bath Smelting Processes. Metall. Mater. Trans. B 1996, 27, 195-201. (13) Kleimt, B.; Kohle, S. Power Consumption of Electric Arc Furnaces with Postcombustion. MPT Metall. Plant Technol. Int. 1997, 20(3), 56-57. (14) Tang, X.; Kirschen, M.; Abel, M.; Pfeifer, H. Modelling of EAF Off-Gas Post Combustion in Dedusting Systems using CFD Methods. Steel Res. 2003, 74(4), 201-210. (15) Morales, R. D.; Rodrı´guez-Herna´ndez, H.; Garnica-Gonza´lez, P.; Romero-Serrano, J. A. A Mathematical Model for the Reduction Kinetics of Iron Oxide in Electric Furnace Slags by Graphite Injection. ISIJ Int. 1997, 37(11), 1072-1080. (16) National Institite of Standards and Technology (NIST) Chemistry Webbook, http://webbook.nist.gov/chemistry/. (17) Ban-ya, S. Mathematical Expression of Slag-Metal Reactions in Steelmaking Process by Quadratic Formalism Based on the Regular Solution Model. ISIJ Int. 1993, 33(1), 2-11. (18) Sigworth, G. K.; Elliott, J. F. The Thermodynamics of Liquid Dilute Iron Alloys. Metal Sci. 1974, 8(9), 289-310. (19) Smith, W. R.; Missen, R. W. Chemical Reaction Equilibrium Analysis; Wiley: New York, 1982. (20) Ji, F.; Barati, M.; Coley, K.; Irons, G. A. Some Experimental Studies of Coal Injection into Slags. In Proceedings of the 60th Electric Furnace Conference; ISS Publishers: Warrendale, PA, 2002; pp 511-524. (21) Urbain, G. Viscosity Estimation of Slags. Steel Res. 1987, 58(3), 111-116. (22) Mills, K. C.; Keene, B. J. Physical Properties of BOS Slags. Int. Mater. Rev. 1987, 32, 1-114. (23) Reynolds, Q. Thermal Radiation Modelling of DC Smelting Furnace Freeboards. Miner. Eng. 2002, 15, 993-1000. (24) Siegel, R.; Howell, J. R. Thermal Radiation Heat Transfer, 4th ed.; Taylor and Francis-Hemisphere: New York, 2002. (25) Biegler, L. T.; Damiano, J. J.; Blau, G. E. Nonlinear Parameter Estimation: A Case Study Comparison. AIChE J. 1986, 32(1), 29-45. (26) Cervantes, A.; Biegler, L. T. Optimization Strategies for Dynamic Systems. In Encylcopedia of Optimization; Floudas, C. A., Pardalos P. M., Eds.; Kluwer Academic Publishers: Dordrecht, 2001; Vol. 4,pp 216-227. (27) Process Systems Enterprise Ltd. gPROMS Advanced User Guide. Version 2.3, 2004.

Received for review January 25, 2005 Revised manuscript received July 28, 2005 Accepted July 29, 2005 IE050101B

Dynamic Modeling of an Industrial Electric Arc Furnace - American

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