Developments in the modelling of oxygen steelmaking

Transcription

1 Unversty of Wollongong Research Onlne Faculty of Engneerng - Papers (Archve) Faculty of Engneerng and Informaton Scences 2011 Developments n the modellng of oxygen steelmakng G A. Brooks Swnburne Unversty of Technology N Dogan Unversty of Wollongong, ndogan@uow.edu.au M Alam J Naser M A. Rhamdhan Publcaton Detals Brooks, G. A., Dogan, N., Alam, M., Naser, J. & Rhamdhan, M. A. (2011). Developments n the modellng of oxygen steelmakng. Guthre Symposum Montreal: McGll Unversty. Research Onlne s the open access nsttutonal repostory for the Unversty of Wollongong. For further nformaton contact the UOW Lbrary: research-pubs@uow.edu.au

2 Developments n the Modellng of Oxygen Steelmakng G. A. Brooks, N. Dogan, M. Alam, J. Naser and M. A. Rhamdhan Faculty of Engneerng and Industral Scence Swnburne Unversty of Technology Hawthorn, VIC 3122 Australa Emal: ndogan@swn.edu.au Phone: (+61) (03) Fax: (+61) (03) Abstract Steelmakng s a complex process nvolvng smultaneous mult-phase nteractons, chemcal reactons, heat transfer and turbulent flow patterns at hgh temperatures. The severe operatng condtons make t dffcult to make measurements and drectly observe the process. Mathematcal modellng has been wdely used to evaluate the process and mprove understandng of the system and optmse process control. Three approaches to modellng the process, equlbrum modellng to predct the equlbrum chemstry of the process, knetc modellng to analyse rate of chemcal changes throughout the process, and computatonal flud modellng to understand the flud flow n the reactor, are dscussed n ths paper. The governng equatons and the forms of numercal computatonal soluton technques for these models are descrbed, and crtcally examned. Introducton Steel s the most produced metal n the world, wth over 1400 mllon metrc tons (mmt) p.a made around the world, compared to approxmately 30 mmt p.a. of alumnum. There are two major technques to produce steel, oxygen steelmakng and Electrc Arc Furnace steelmakng (EAF). These processes are llustrated schematcally n FIGURE 1. Oxygen steelmakng uses hgh speed pure oxygen to oxdze mpurtes from molten ron produced from a blast furnace. In the case of an EAF, scrap steel and other ron sources are heated up usng gas burners and melted by hgh powered electrc arcs. In EAF steelmakng, oxygen and carbon njecton s used to oxdze mpurtes and form a foamng slag (molten oxde mpurtes) above the molten metal bath. Whle there are major dfferences n these producton technques, they both have hgh rates of producton (>100 tonne/hour) and can produce hgh qualty steel, though several other processes steps after steelmakng are requred before castng.[1] Table 1 lsts the major reactons occurrng n steelmakng processes. The refnng reactons take place smultaneously between 1600 and 1700 ±C. The majorty of these reactons are exothermc reactons and subsequently no external heat s requred n oxygen steelmakng, n fact, scrap s added to cool the process. In the case of electrc arc furnace, electrc energy s used for heatng up the scrap from ambent temperatures.

3 understandng or optmsng the process beyond current operatng regmes.[3] Neural networks,[4, 5] fuzzy logc[6, 7] and multvarate statstcs[8-11] have also been used as process control tools n steelmakng operatons but these black box models[12] suffer from the same lmtatons as smple emprcal models, though they are lkely to mprove control and provde a bass for process optmsaton wthn current operatng Table 1. Major reactons n oxygen steelmakng [13] (a) (b) FIGURE 1: Schematc representaton of top-blowng oxygen steelmakng (a) and electrc arc furnace (b) Steelmakng processes are complex snce they nvolve smultaneous mult-phase (sold-gaslqud) nteractons, chemcal reactons, heat and mass transfer and complex flow patterns at hgh temperatures. The process s also hghly transent, meanng that steady state assumptons commonly used by chemcal engneers are often nvald.[2] The severe operatng condtons make t dffcult to make measurements and drectly observe the process. Ths dffculty can be addressed by developng models, whch make t possble to descrbe the complcated nature of the process tself and to understand the nterconnecton of mportant process varables. Smple emprcal models have been developed n ndustry for the purposes of process control but they are not sutable for ncreasng Oxygen pck up by the metal: O 2(g) = [O] (1) CO 2(g) = CO (g) + [O] (2) (FeO) = Fe + [O] (3) (Fe 2 O 3 ) = 2(FeO) + [O] (4) Oxdaton of elements n the metal: [C] + [O] = CO (g) (5) Fe + [O] = (FeO) (6) [S] + [O] = (SO 2 ) (7) [Mn] + [O] = (MnO) (8) 2[P] + 5[O] = (P 2 O 5 ) (9) Oxdaton of compounds n the slag: 2(FeO) + ½O 2(g) = (Fe 2 O 3 ) (10) 2(FeO) + CO 2(g) = (Fe 2 O 3 ) + CO (g) (11) Flux reactons: MgO (s) = (MgO) (12) CaO (s) = (CaO) (13) Gas reactons: CO (g) + ½O 2(g) = CO 2(g) (14) regmes. Unfortunately, the physcal complexty of the process means that a completely rgorous mathematcal descrpton of the process based on fundamental physcs and chemstry s currently not possble. In many cases, only sem-emprcal relatonshps are avalable to descrbe physcal phenomena n processes lke oxygen steelmakng.[14] Whlst, a fully scentfc model ( whte box model) of oxygen steelmakng maybe mpractcal, more scentfcally based modellng technques can be appled to mprove understandng, provdng a bass to desgn more effcent furnaces and optmze the current technology.[3] However, n developng models for complex processes lke oxygen steelmakng, t s mportant to compromse between the detal requred, the 2

4 nformaton avalable on emprcal parameters, the nherent lmts of the avalable mathematcal tools and the computatonal tme requred to fnd solutons.[15] There are varous modellng technques used to mprove the understandng of the process. The common modellng technques can be classfed nto three groups, shown schematcally n FIGURE 2: Computatonal thermodynamcs models are used to understand the lmts of the system and portonng of the mpurtes among the phases; computatonal flud models are employed to understand the flud flow patterns and nteracton between the phases; and computatonal knetcs evaluate the concentraton changes of each phase wth tme. As mpled by the schematc representaton n FIGURE 2, these technques are nterrelated and results from each model nform the development of other models. For example, computatonal flud dynamcs provdes mportant parameters for understandng convectve mass transfer. Smlarly, thermodynamcs models provde the lmts for knetc modellng. In ths paper, we revew these modellng technques, usng selected aspects of steelmakng operatons to llustrate the general approach and hghlghtng lmtatons of the technques developed to date. Computatonal Chemcal Thermodynamcs Computatonal Knetcs - Global process models - Fundamental models - Sem-emprcal models FIGURE 2: Modellng technques used n steelmakng processes Modellng technques Computatonal Flud Dynamcs Computatonal Chemcal Thermodynamcs Computatonal chemcal thermodynamcs s a well establshed technque for understandng hgh temperature processes, many aspects of materals scence, the geology of the earth and even the chemstry of the stars.[16-20] Thermodynamc models provde nformaton on the equlbrum dstrbuton of speces wthn the phases, the heat generated or consumed by these chemcal reactons and behavour of speces n soluton. Thermodynamc modellng s based on the 2 nd law of Thermodynamcs and equlbrum calculatons are made usng the Gbbs free energy mnmzaton technque, whch s developed from ths law. The bass of the method s to mnmze Gbbs free energy of the system at constant temperature and pressure usng;[19] ( µ + RT ln a ) mn G = n µ = n = 0 (15) Here varables R, T, n refer to gas constant, temperature and mole of speces, respectvely. µ 0 denotes the standard chemcal potental and a refers to actvty. Actvty can be understood as representng effectve concentraton of a partcular speces and s an mportant thermodynamc quantty for quantfyng the way speces dssolve nto each other. Chemcal potental s the ncrease n Gbbs free energy of the system for an addtonal mole of speces,, and represents how a change n composton of the system effects the amount of energy ntroduced to the system at constant temperature and pressure.[21] Two constrants should be satsfed; n must be non-negatve and total moles of the elements n the system must be equal to summaton of moles of the speces n each phase as gven n Eq. (16). k =1 n a j = bj (16) where a j s the number of g atoms of element j and b j s the total atom of element j. Ths s a standard optmzaton problem that can be solved usng the Lagrange method of unknown multplers. Eq. (15) becomes;[20] θ = G + k λ m = 1 j= 1 a j n b 0 j = 0 (17) 3

5 where the λ are the Lagrange multplers, one for each element. Ths allows each of the n j to be treated ndependently, and t can be shown usng multvarate calculus that the equlbrum condton s gven by; θ n j and θ λ = 0 = 0 (18) (19) These results n a set of (m+k) equatons wth (m+k) unknowns ( all the n and λ terms) and may, therefore, be solved for each equlbrum concentratons of a partcular speces (n ) as long as the chemcal potentals are known as functons of the concentratons at the gven temperature and pressure. The equatons that requred to be solved to obtan the composton of the system are not all lnear n regards to the composton varables. A Newton-Raphson method s used to solve the equatons usng ntal estmates of compostons n j, and Lagrangan multplers λ, for a gven quantty of speces n and temperature. [20, 22] In developng thermodynamc models, the phases and possble speces wthn the system need to be defned carefully. All the speces defned n the system requre thermochemcal data such as heat capacty, enthalpy and entropy, as functons of temperature for pure speces before Gbbs energy mnmzaton technques can be appled. Fortunately, large databases of crtcally evaluated thermodynamc data has been compled by nternatonal recognsed groups such as NASA and NPL. The approprate soluton models for each phase also need to be evaluated based on soluton structure and behavour. The ntal quanttes of speces wthn the phases and operatng condtons are then entered. The equlbrum calculatons are evaluated based on the Gbbs energy mnmzaton technque whch s readly avalable n commercal thermochemcal software. The most wdely used packages are Thermo-Calc, Factsage, MTData, Thermodata, HSC, Chemx and Gemn2.[19, 23] These software combne computatonal codes for databases and equlbrum calculatons to extend the applcatons n dfferent metallurgcal systems.[23, 24] For example, Chemx, Thermo- Calc and MTData offers flexblty to chose a soluton model such as polynomal, Redlch- Kster and Margules dependng on the user s system. HSC currently has a lmtaton on the soluton behavour, as actvty coeffcents can only be entered as constant number or as a polynomal functon of composton and temperature. Nevertheless, HSC has been wdely used n ndustry because of ts smplcty.[19] Once agan, the balance between scentfc rgor and fndng practcal solutons s evdent n modellng of complex ndustral systems such as steelmakng. One of the major challenges n a thermodynamc model of steelmakng s the determnaton of an approprate soluton model for each phase n the steelmakng processes. Models of soluton behavour allow expermental results for partcular systems to be generalsed and make predctons of soluton behavour away from the condtons of the orgnal emprcal data set. However, the nteractons between dfferent speces n a phase at dfferent concentratons and temperatures are complex, and partcularly n the case of phases where the basc molecular forms n the phase vary wth concentraton and temperature, t s hard to fnd a soluton model that work well for all condtons. It remans one of the great scentfc challenges to determne soluton behavour a pror.e. calculate from basc physcs wthout resortng to the current hgh level of emprcsm. The major phases n the system are lqud ron, slag (molten oxdes), gas, sold refractory, sold fluxng agents (e.g. lmestone) and sold steel (scrap). In the case of lqud ron, a dlute soluton model usng emprcally determned nteracton parameters s well establshed and can be readly appled to these problems. The dlute soluton model s based on Henry s Law, whch assumes that the actvty of solute has a lnear relatonshp wth concentraton.[25] However, for slag, the soluton behavour of the phase s challengng to model because t s dffcult to descrbe the complex molecular structure of slag tself, 4

6 makes the nterpretaton of expermental data troublesome. Moreover, due to the lack of knowledge about the nteracton between dfferent metal oxdes n slag results n furthers dffcultes n developng robust thermodynamc models of slags.[24] However, the mportance of molten oxde soluton behavour to materals scence, geology and norganc chemstry, has meant that that a large of works body exsts addressng these ssues.[26, 27] Several models have been developed for mult-component oxde systems, these nclude onc two sub-lattce, regular soluton, cell, assocated soluton and modfed quaschemcal models.[28-36] Although the regular soluton model s smple to employ, the cell and modfed quaschemcal models have been found more approprate for studyng steelmakng slags.[24] Whlst, the modellng of molten oxdes soluton behavour s complex and the relance on emprcal measurements s dsappontng, computatonal chemcal thermodynamc modellng s regarded as a great trumph, resultng n many useful and mportant predctons over many felds, ncludng steelmakng.[37] For example, thermodynamc modellng has been successfully appled to the predcton of the dstrbuton of sulphur n steelmakng reactons[38] usng the technques descrbed above. Computatonal Knetcs Whlst, computatonal thermodynamcs can predct the equlbrum composton of steelmakng reactons, these technquee can not predct how fast these reactons takes place and what controls ther rate. As a feld of knowledge, reacton knetcs attempts to answer such questons. It s mportant to note that the term reacton knetcs s sometmes used to only descrbe the speed of molecular nteractons durng reacton, where as, we use the term to descrbe the overall process of the reacton, whch may nclude mass transfer and other mechansms; we reserve the term nherent knetcs to descrbe the actual molecular re-arrangement step. In many systems, the overall knetcs s controlled by one or two steps of the process, whch may not be the nherent knetcs. For example, the dffuson of sugar n a tea cup s clearly mportant to the rate of sugar dssoluton, and strrng the cup speeds up the process not because strrng helps the nherent knetcs (t can t) but because the dffuson step s accelerated by the ncreased mxng ( mass transfer s ncreased) Reacton knetcs provdes nformaton on the condtons that govern the rate at whch a reacton wll occur and can descrbe the mechansm by whch the reacton proceeds. In general, computatonal knetcs s less developed than computatonal thermodynamcs, wth less commercal software and large databases avalable. In steelmakng reactons, the nherent knetcs are not usually the rate lmtng step as the hgh temperatures of steelmakng reactons favour very hgh nherent rates, thus the reactons knetcs are more lkely to be controlled by transport of elements from/to the phase nterphase. Generally speakng, reacton rates n steelmakng operatons can be predcted usng Fck s frst law, whch states that the rate of dffuson s proportonal to the concentraton gradent.[39, 40] The rate equaton s wrtten n terms of molar concentraton: C = D x J (20) where J s molar flux n x drecton (mol/m 2 sec). C refers to molar concentraton (mol/cm 3 ) and D refers to dffuson coeffcent (cm 2 /sec). To reduce the process varables a manageable level, tme s manly chosen as the dfferental ndependent varable and spatal gradents are gnored. Wth ths smplfcaton, the dependent varable, bulk concentraton, s not functon of poston and equaton can be reduced to; dc dt ( C ) A = k C V b (21) Here varables k, A and V are mass transfer constant, area and volume, respectvely. The formulatons of rate equaton dffer wth the 5

7 physcal propertes and type of flud flow over the surface of the reactng condensed phase. Ths equaton wth approprate ntal and boundary condtons can be solved to analyse the process tself or alternatvely, varous aspects of the process. In realty, the reducton of Fck 1 st Law to the smple 1 st order dfferental of Eq. (20) nvolves many gross smplfcatons. Recently, the authors of ths paper have tred to address these ssues through the development of transent knetc theores, whch deal wth changes n nterfacal area, mass transfer coeffcents and concentraton gradents.[41] For example, global model of oxygen steelmakng ncludng the knetcs of scrap meltng, flux dssoluton, slag chemstry, temperature profle of the system, formaton and resdence of metal droplets n the emulson, knetcs of decarburzaton reacton n dfferent reacton zones has been developed by the present authors.[42] In ths model, total decarburzaton rate n the emulson zone s obtaned by the summaton of decarburzaton rates of ndvdual metal droplets. The generated droplets, whose resdence tme s smaller than gven tme-step, are returnng from the emulson zone. Based on ths boundary condton, decarburzaton rate can be calculated usng; M e dc dt = n = 1 m 100 t + t t ( C C ) t (22) Here the number of the droplets n the emulson zone s represented by n, m s weght of a sngle droplet (kg) and C s the carbon content of droplet (mass%). The global model has been valdated aganst ndustral data. 66) Process models [43-49] are developed to am to express what steps a process conssts of, or how they are to be performed. These models enable us to evaluate the key process varables such as concentraton of mpurtes and temperature of lqud bath throughout the blow to understand the process better, desgn new technques and optmze the process. Numerous of knetc models have been proposed to predct process varables that nfluencng the system such as scrap meltng,[50-57] flux dssoluton[58] and carbon removal reacton n the gas-slag-metal emulson.[59-62] Few of these models are fully based on the laws of physcs formed by lmted set of equatons. Normally, a level of emprcsm s ntroduced to provde a practcal soluton. For example, when lqud metal droplets are ejected to the emulson phase due to the oxygen mpngement onto the lqud bath at hgh speeds (> Mach 1), these droplets spend some tme n emulson and fall back to metal bath. Droplet generaton enhances the reacton area sgnfcantly that ncreases the reacton rates of the process. Based on the prevous studes,[62-65] t has been found that the resdence tme of a droplet vares from 0.2 to 120 seconds. It has been observed exprmentally that ths phenomenon s strongly related to the carbon removal reacton n the emulson phase. The overall reacton s; (FeO) + [C] = Fe + CO (g) (23) Carbon s oxdzed va FeO reducton that forms carbon monoxde gas. If the rate of carbon removal s weak, the reacton product, CO gas escapes easly and detaches from the droplet as a result, the droplet mantans ts orgnal shape. After a crtcal pont, defned as threshold decarburzaton rate, gas product s not able to escape from the droplet and stays n the metal droplet that the droplet becomes bloated. Brooks et al.[62] developed a mathematcal model to predct the resdence tme of the droplets generated n the slag-gas-metal emulson system under varous operatng condtons. In ths model, a force balance was used to analyse the ballstc moton of a sngle droplet at vertcal and horzontal coordnates. The moton of a sngle droplet s llustrated n FIGURE 3. 6

8 decarburzaton rate s related to the FeO concentraton n the emulson phase usng; = ( mass%feo) (26) 4 r C FIGURE 3: The schematc llustraton of ballstc moton of a metal droplet n slag[62] The relatonshp between the forces can be represented for horzontal and vertcal coordnates n the followng equatons.[62] for z drecton: duz ρdvd = FB F dt for r drecton: ρ d V d du dt r = F D,r G F F A,r D,z F A,z (24) (25) where u z and u r are the velocty of droplet for z and r drectons, respectvely. Varables ρ d and V d refer to densty and volume of droplet. F B, F G, F D and F A are buoyancy, gravtaton, drag and added mass forces, respectvely. The numercal calculaton s undertaken usng the explct forward dfferencng method. After dfferentatng, the trajectory of metal droplet for z and r drectons can be obtaned. As the trajectory at z drecton approaches zero, the lapse tme s calculated and defned as a resdence tme of a droplet.[62] Up to ths pont, ths model s based on basc laws of Newtonan physcs. However, there s a lack of understandng on how the bloatng behavour of a droplet occurs n the emulson. Ths phenomenon cannot has not yet been fully explaned, so emprcal relatonshps were ntroduced based on the expermental study by Molloseau and Fruehan.[60] It was observed that bloatng behavour occurs f FeO concentraton n the slag s hgher than 10 mass% n the emulson phase. The threshold Ths model s able to predct the resdence tme of bloated and dense droplets usng ths smple correlaton. The detal of ths model was been explaned elsewhere.[62] Ths approach has been successfully appled to expermental results[60] and now beng ncorporated nto ndustral models.[66] In tme, t may be possble to remove the current level of emprcsm assocated wth the threshold decarburzaton rate. Ths example s typcal of modellng of complex processes, where a mxture of basc physcs and emprcsm s used produce useful results and dentfy areas requrng more research. In ths sense, model developments help us understand the gaps n our knowledge and pont towards useful research drectons. Computatonal Flud Dynamcs Computatonal flud dynamcs (CFD) s ncreasngly utlsed n modellng the flud flow behavour of dfferent ndustral processes. In ths approach, the equatons that govern a process of nterest are solved by means of computer based smulaton. The advent of powerful dgtal computers has reduced the tme requred for the smulaton process as well as opened up a new era n CFD applcaton enablng modellng of many complex processes. In steelmakng, oxygen s njected under supersonc condton of around Mach 2 nsde the furnace.[71] Supersonc gas jets are preferred over subsonc jets because of hgh dynamc pressure assocated wth t whch results n hgher depth of penetraton onto the lqud bath and better mxng of oxygen gas wth lqud melt. Also as dscussed earler, the hgh speed oxygen jet produces droplets upon mpngement on the lqud.[72] Snce the envronment nsde the furnace s very harsh to make any measurement, a number of expermental [73-82] and some numercal studes [83-87] have been performed to help 7

9 understandng of flud flow behavour nsde the furnace. In our numercal studes of gas jet mpngng on lqud bath, the unsteady contnuty, momentum and energy equatons are solved for pressure, velocty and temperature of the flud flow. The equatons are gven below[88]; Contnuty equaton: ρ ρu + = 0 t x where ρ s the densty and flud flow n th drecton. (27) U s the velocty of Momentum equaton: ρ U ( ρu U j ) P ( j uu j ) τ ρ (28) t + x j = + x x j + F where P s the pressure of flud, τ j s the molecular shear stress, ρ u u j s the turbulent shear stress and F s the body force. For Newtonan flud, the molecular shear stress s proportonal to the rate of deformaton. There are dfferent turbulent models avalable for the calculaton of turbulent shear stresses.[89, 90] Energy equaton: ρe + t ( ρeu + PU ) ( q ) ( j j j j ) + C U u u U Pρut τ ρ x x + = x (29) where E s the total energy, q s the conducton heat flux and s calculated by Fourer s law of heat conducton. The term ρ u t s the turbulent heat flux.[89] deformaton of the gas lqud nterface due to the transfer of momentum of gas jet to the lqud surface s modelled by the followng equaton.[91] α P l Pg = σdv (30) α where P l, P g are the lqud and gas sde pressure and σ s the surface tenson of lqud. Brackbll et al.[91] developed a contnuum surface force model where ths equaton s ncluded as a body force n the momentum equaton. In order to determne the sharp gas lqud nterface, the volume of flud (VOF) technque[92] s used whch solves an addtonal transport equaton for the volume fracton α of tracked phase: α + u t j α x j = 0 α = 1, cell s flled wth tracked phase α =0, no tracked phase n ths cell. 0< α <1 cell shares both phase. (31) After calculatng the volume fracton n each cell, the densty ρ and vscosty µ of the flud n each cell are determned by a weghted average, as shown n followng equatons: ρ = ρlα l + ρg ( 1 α l ) (32) µ = µ α + µ ( 1 ) (33) l l g µ l These are partal dfferental equatons and are solved by usng fnte volume method applyng approprate boundary condtons.[88] FIGURE 4 shows a computatonal doman wth boundary condtons for the smulaton of gas jet mpngng on lqud surface. Generally, velocty boundary condton s used at the nlet and statc pressure boundary condton s used at the outlet. For the sold wall, the wall boundary condton was used n our model. The FIGURE 4: Computatonal doman wth boundary condtons 8

10 where the subscrpts l and g correspond to the lqud and gas phase, respectvely. These propertes are then used to solve the transport equatons. In fnte volume method, the soluton doman s subdvded nto a number of cells and the partal dfferental equatons are ntegrated over the volume of each cell n computatonal doman. The non-lnear equatons are dscretsed to make lnear algebrac equatons. Then the algebrac equatons are solved teratvely usng a sutable teratve technque. Prevous studes [85-87] have calculated the depth and dameter of depresson created by the gas jet upon mpngement on lqud bath. The depth of depresson ncreases wth ncreasng gas jet velocty and decreasng lance heght as shown n FIGURE 5 [87]. The hgher the depth of penetraton, the better the mxng of oxygen gas and lqud metals whch, n turn, ncreases the overall rate of reacton. All of the prevous numercal studes have assumed the gas phase as ncompressble whch s not the case n the real process. As a result, prevous numercal solutons have under predcted the depth of depresson when used for modellng of oxygen steelmakng[87]. Ths s one of the lmtatons of the numercal technques descrbed, t that t does not droplet generaton rate because n order to smulate droplet generaton, a very fne mesh s requred whch wll result n a excessve computatonal tme. Smulaton of complex flud dynamcs [93] nsde the furnace at sothermal condton has been carred out n some studes but the smulaton of reactons s stll a bg challenge for CFD although models can smulate some smple reactons between gas phases. Among the advantages of numercal studes s that we can study non-sothermal condtons n the process whereas t s very dffcult to do expermental studes at nonsothermal condtons. Some numercal studes at non-sothermal condton [83, 86] have been performed. The present authors have numercally smulated the supersonc oxygen jet behavour at steelmakng temperatures [94]. It was found that the potental core length (the length up to whch the axal gas velocty s equal to nozzle ext velocty) of the supersonc jet s hgher at hgh ambent temperature and at steelmakng condton (1800K), t s 2.5 tmes hgher then the potental core length at normal room temperature condton (285K), as shown n FIGURE 6. Ths was the frst step towards developng non-sothermal modellng gas-jet mpngng phenomenon n steelmakng. Studes are gong on to smulate the flud flow phenomenon such as depresson depth, splashng at non-sothermal condton. FIGURE 5: Depth of depresson varaton wth changng flow rate and lance heght gve stable solutons when one phase s hghly compressble n multphase flows. No numercal studes have been performed on FIGURE 6: Velocty dstrbuton of supersonc jet at dfferent ambent temperatures. These examples llustrate the advances n CFD modellng of complex processes such as Oxygen 9

11 steelmakng but also llustrate how sgnfcant work s stll requred to lnk the results of CFD models to chemcal thermodynamcs and knetcs. Conclusons In ths paper, we revewed how mathematcal modellng has been used to analyse steelmakng processes. The modellng concepts were llustrated wth examples drawn from the lterature and studes by the authors. Steelmakng reactons occur under nonequlbrum condtons, thus predctons based on thermodynamcs alone wll be nsuffcent to understand and control the system. However, thermodynamcs predctons can be used to predct the lmts and drecton of reactons occurrng n the system. Models based on fundamental physcs, for example, computatonal flud dynamcs, can be used to ncrease understandng of knetc factors n the process but the use of emprcal relatonshps s stll requred n many cases, ether due to a lack of theory or dffculty n obtanng more fundamental data. In summary, models provde powerful tools for makng useful predctons, developng theoretcal understandng of the system and provde a framework to advance our understandng. The complexty of steelmakng and problems assocated wth measurng and vsualzng the phenomenon beng studed necesstates the use of sem-emprcal models and compromses between mathematcal/scentfc rgor and practcal solutons beng found. References 1. R.J. Fruehan, Overvew of Steelmakng Processes and Ther Development, n The Makng, Shapng and Treatng of Steel, R.J. Fruehan, Edtor. 1998, The AISE Steel Foundaton: Pttsburg. p G.A. Brooks, M.A. Rhamdhan, K.S. Coley, Subagyo, and P. Y., Transent Knetcs of Slag Metal Reactons. Metallurgcal and Materals Transactons B, Vol.40B No. 3, 2009, pp G.A. Irons. Developments n electrc arc furnace steelmakng. n AISTech - Iron and Steel Technology Conference Proceedngs Charlotte, NC, vol. 1, pp I.J. Cox, R.W. Lews, R.S. Ransng, H. Laszczewsk, and G. Bern, Applcaton of neural computng n basc oxygen steelmakng. Journal of Materals Processng Technology, Vol.120, No. 1-3, 2002, pp J. Falkus and P. Petrzkewcz. Neural Networks n Statstcal Controllng of Oxygen Converter Process n Proc. Conf. Hgh Technologes n Advanced Metal Scence and Engneerng St. Petersburg, Russa 6. C. Kubat, H. Tasķn, R. Artr, and A. Ylmaz, Bofy-fuzzy logc control for the basc oxygen furnace (BOF) Robotcs and Autonomous Systems 49 (3-4), pp Vol.49, No. 3-4, 2004, pp V. Sahajwalla and R. Khanna, A Monte Carlo smulaton study of dssoluton of graphte n ron-carbon melts Metallurgcal and Materals Transactons B: Process Metallurgy and Materals Processng Scence Vol.31, No. 6, 2000, pp A.A. Greenfeld. Statstcal Approach to Oxygen Steelmakng. n Proceedngs of the Conference on Mathematcal Process Models n Iron- and Steelmakng Amsterdam. The Metals Socety, pp K. Katsura, K. Isobe, and T. Itaoka, Computer control of the basc oxygen process. J Metals, Vol.16, No. 1964, pp A. Wyatt, D. Kundrat, H. Fuchs, and H. Schaefer. Reducton n energy consumpton through optmzaton of oxygen and carbon njecton n AISTech - Iron and Steel Technology Conference Proceedngs 2009, vol. pp E. Sandberg, Energy and Scrap Optmsaton of Electrc Arc Furnaces 10

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