Thermodynamic Assessment of Liquid Fe Si C System by Unified Interaction Parameter Model
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1 ISIJ Internatnal, Vl. 54 (014,. 4, pp Thermdynamc Assessment f Lqud Fe System by Unfed Interactn Parameter Mdel Seung Hwan AH, Yang Hng KIM, Jng-Pl SHI and Yung Eun LEE* Dngbu Metal mpany, Dnghae, Gangwn-d, 40-10, Republc f Krea. (Receved n September 5, 01; accepted n December 1, 01 The slutn behavrs f Fe system are characterzed by the frmatn f, and the prpertes are defned by the value f the standard Gbbs free energy f frmatn f, ΔG. Hwever, a revew shws that reprted values f ΔG d nt agree amng the nvestgatrs and are shwn a sgnfcant dscrepancy amng them as hgh as 16 kj/ml. In rder t reslve ths uncertanty, new value f ΔG s assessed by reprducng expermentally determned tw-fld saturatn data wth the UIP mdelng Fe melt. The prpsed value f s gven by fllwng equatn. ΔG ( + ( gr = ( β ΔG = T ( J / ml, K ( errr ± 100J ΔG Wth the prpsed, the UIP mdel fr Fe system was develped frm the expermentally determned slublty f and tw-fld saturatns f and taken frm the lterature reprted. The resultng actvty ceffcents are expressed as fllws: ln γ = ε ( x + ε ( x x + ε ( x + ε ( x x ( ( Fe + ε x x + ε x + ε ( x x + ε ( x lnγ = ln γ + ε ( x x + ε ( x x x + ε ( x + ε 4 + ε ( x + ε ( x x x + ε ( x 4 4 ( x x x x + ε ( x x x lnγ = ln γ + ε( x + ε( x x x + ε( x x + ε( x x x + ε( x x x x ε ( x + x x x x x x + ε ( + ε ( 4 x where lnγ = /T, lnγ = /T, ε = /T, ε = /T, ε = /T, ε = /T, ε = /T, ε = /T, ε = /T, and ε = /T. The mdel reprduces well the expermentally determned slublty as well as the tw-fld saturatn f and. It als descrbes well the slutn behavrs n a wde range f cmpstns and temperatures, permttng ts use fr varus applcatns such as rnmakng, steelmakng, cast-rn, -based ferrallys, and the lw temperature lqud phase grwth f electrnc grade slcn carbde. KEY WORDS: actvty ceffcents f Fe,, and ; standard Gbbs free energy f frmatn f ; unfed nteractn parameter mdel. 1. Intrductn Irn, slcn, and carbn serve as cmmn ngredents fr prcesses f rnmakng, steelmakng, cast-rn, -based ferrallys, and the lw temperature lqud phase grwth f electrnc grade slcn carbde (. The nfrmatn f slutn prpertes f Fe system s essental n develpment and mprvement f these prcesses. Many nvestgatrs examned the slutn prpertes f Fe melt. 1 7 Because f ther specfc prcessng nterests, ther results are lmted n cmpstns and temperatures and nt suffcent enugh t descrbe the slutn prpertes fr the * rrespndng authr: E-mal: yung.lee@dngbu.cm DOI: entre cmpstn range f Fe system. The slutn behavrs f Fe system are characterzed by decreasng carbn slublty as the slcn cntent ncreases, and the degree f ts slublty s determned by the frmatn f. Therefre, the accurate nfrmatn f the standard Gbbs free energy f frmatn f, ΔG, s necessary t crrectly descrbe the slutn behavrs f Fe system. Hwever, the values reprted by hpman et al., 1 Turkdgan, 8 Barn, 9 Pankratz, 10 Lacaze et al. 11 and JAAF 1 d nt agree amng themselves, and, mre mprtantly, d nt reprduce the expermentally determned twfld saturatns f and. 1,,5,6 Ths study s t determne the crrect value fr the ΔG and t develp a slutn mdel n Unfed Interactn Parameter frmalsm, 1,14 whch reprduces cnfdently the 014 ISIJ 750
2 ISIJ Internatnal, Vl. 54 (014,. 4 expermental results f Fe melt.. Unfed Interactn Parameter (UIP Frmalsm The present study adpts the Unfed Interactn Parameter (UIP frmalsm prpsed by Bale and Peltn 1,14 as t s thermdynamcally cnsstent at all cmpstn ranges and satsfes Gbbs-Duhem relatnshp. Accrdng t the cnventn by UIP frmalsm, ne f the elements s chsen as slvent and the thers as slutes. The actvty ceffcents f slvent and slutes are expressed n terms f cmpstns as fllws, 1 lnγslvent = ε jk xx j k ε 4 jk, = 1 jkl,, = 1 jklm,,, = 1 ε jklm xxxx j k l m lnγ = lnγ + lnγ + ε x ( slute + ε xx + jk j k jk, = 1 xxx jkl j k l... (1... ( where ε jk, ε jkl, ε jklm, ε j, ε jk, ε jkl are nteractn parameters. In ths study, Fe s taken as slvent, and and as slutes. Ther actvty ceffcents are presented n expanded frms as the Eqs. (, (4, and (5. ε + slvent j j j= 1 jkl,, = 1 xxx+ jkl j k l ln γ = ε ( x + ε ( x x + ε ( x Fe + ε ( x + ε ( x x + ε ( x x + ε ( x + ε ( 4 x + ε ( x x + 9 ε ( x x + ε ( xx + ε ( x ( lnγ = ln γ + ε ( x + ε ( x x x + ε ( x x + ε ( x + ε ( x x x + ε ( x x x x + εs ( x x ( + ε 4 x + ε ( x x x + ( x x 9 x x + ε ( x x x x + εs ε x x 4 4 ( +... (4 ε jk=a+b/t. The nteractn parameters were determned by the multple lnear regressn analyss usng avalable expermental data. They were btaned frst fr the bnary systems and, subsequently, fr the hgher rder systems.. Interactn Parameters fr the Fe Bnary System As shwn n the Eq. (5, the value f the actvty ceffcent f at nfnte dlute slutn, lnγ, s necessary t be determned frst. In the present study, t s taken as assessed by gwrth et al. 15 as shwn by the Eq. (6. ln γ = / T... (6 The slublty n Fe melt was studed by many nvestgatrs.,16 19 As the recent assessment by Japan Scety fr the Prmtn f Scence (Steelmakng Data Surcebk 19 represents ther results well, ths study adpts ths assessment as gven by the fllwng Eq. (7. lg x =... ( / T lg T. 049 Ths Eq. (7 s used t calculate the slublty f at 50 K ncrement f temperature n the range f 1 4 t 1 97 K. The actvty ceffcents at saturatn and at nfnte dlutn becme the set f data fr the present analyss. Fr the multple regressn analyss, the Eq. (5, whch s the representatn f the actvty ceffcent f n UIP frmalsm, s rearranged by transpsng lnγ t the left-hand sde f the equatn n rder t relate the expermentally determned terms n cmpstns and nteractn parameters. The result s gven by the Eq. (8. lnγ ln γ = ε ( x x + ε ( x x +... (8 Multple regressn analyss s perated n the rearranged Eq. (8 n rder t determne the nteractn parameters, whch reprduce best the expermental data. The result shws that the slublty f Fe system s descrbed reasnably well by ne nteractn parameter, ε, whch s gven by Eq. (9. ε = / T ( K... (9 lnγ = ln γ + ε ( x x + ε ( x x x + ε ( x + ε ( x x + ε ( x x x x + ε ( x x x + ε ( x + ε ( x 4 x + ε ( x x x x + ε ( x x 9 x x + ε ( x x x + ε ( x (5 where the standard states fr Fe, and are taken as pure lqud Fe and and graphte, respectvely. The temperature dependency f nteractn parameters s expressed as Fg. 1. Slublty f carbn n Fe melt ISIJ
3 ISIJ Internatnal, Vl. 54 (014,. 4 The calculated slublty by the UIP mdel s cmpared wth the wrk by hpman et al., Ktchener et al., 16 Matba et al. 17 and Turkdgan et al. 18 n Fg. 1. The present assessment reprduces well the expermental data. 4. Interactn Parameters fr the Fe Bnary System The slutn prpertes f Fe bnary system were assessed by Lacaze et al., 11 hart, 0 and Schmd. 1 Agreement amng them s excellent, but the assessment by hart 0 s shwn t apprach medan values f actvty ceffcents f by the ther tw. Therefre, ths study adpts the assessment f hart 0 t prduce a set f actvty ceffcents f Fe and at varus cmpstns and temperatures. hart 0 rganzed the slutn prpertes f Fe melt n terms f enthalpes and excess entrpes, and ths study calculates the actvty ceffcents frm hs assessed excess prpertes as shwn by the fllwng Eqs. (10 and (11. XS XS G = ΔH TΔS = RT lnγ... (10 XS lnγ = G RT... (11 The actvty ceffcents f Fe and are calculated n the entre cmpstn range at 0.05 mles ncrement and n temperature range frm 1 5 t 1 97 K at 50 K ncrement. Frm ths set f data, the actvty ceffcent f at nfnte dlutn, lnγ, s determned by ts values at x =0. As the smrphc nature f the set f data assures the cnsstency n the behavr f slutn prpertes n Fe system, ths study adpts the values fr lnγ develped frm hart. 0 It s apprxmated by Eq. (1 and shwn n Fg.. ln = / T ( K γ... (1 Such develped set f data fr actvty ceffcents f Fe and s used t determne nteractn parameters by perfrmng multple regressn analyss n the Eqs. (a and (14. ln γfe = ε ( x + ε ( x + ε ( x +...(a lnγ ln γ = ε ( x x + ε ( x x + ε ( x x (14 The results shw that the slutn prpertes f Fe system are descrbed best by three nteractn parameters as gven by the Eqs. (15, (16, and (17. ε ε = / T ( K... (15 = / T ( K... (16 ε = / T ( K... (17 Fgure shws that the assessed UIP mdel reprduces well the wrk f hart Expermental Data fr the Fe System The avalable expermental data fr ths study are the slublty f n and Fe systems The Slublty f Bnary System The phase dagram fr system by Massalsk shws that melt s n saturatn wth. Hwever, ts slublty s reprted t be n the range f x c=0.009 t 0.0 fr temperatures frm t K, whch s substantally hgher than the recent nvestgatn by Bakke et al., Iguch et al. 4 and Tuset et al. 5 (x c= t These recent measurements are cnsstent wth the peratn parameters fr the refnng prcess f hgh purty metal, and ths study adpts ther results. Fgure 4 cmpares ther wrks. The best fttng curve fr the slublty f was develped frm ther data, and t s apprxmated by the Eq. (18. ln x / T ( K... (18 = Fgure 4 shws the lqudus n equlbrum wth develped by the Eq. (18. The eutectc cmpstn s determned t be x c= by extraplatn. Ths s far less than that by Massalsk at x c= Ths places the assessed slublty f at cmpstn lwer than that Fg.. The actvty ceffcents f at nfnte dlutn f Fe. Fg.. The actvty ceffcents f Fe and. 014 ISIJ 75
4 ISIJ Internatnal, Vl. 54 (014,. 4 (a Fg. 4. Slublty f carbn n system. reprted by Massalsk. As the melt alng the slublty curve s n equlbrum wth, the actvtes f and were calculated wth the use f Eq. (19 n the temperature range f t 100 K. a ( + = ( β, ΔG = RT ln... (19 a a As the ml fractn f s less than abut x c=0.001 n ths temperature range, the actvty f s cnsdered t be the same as the ml fractn f, a =x, and the actvty f s determned frm the value f ΔG wth the use f Eq. (19. A set f data fr actvty ceffcents f and s calculated frm the abve assessed slublty f and used n the multple regressn analyss fr ternary nteractn parameters f UIP mdel. 5.. The Slublty f Fe Ternary System The slublty f n the Fe melt was determned expermentally by hpman et al., 1 Schurmann et al., 4 Kawansh et al., 6 and Sanbng et al. 7 Ther results are presented n Fg. 5. The Fe melt s n equlbrum wth graphte when cntent s lw. The slublty f n Fe melt decreases as the cntent ncreases. When the cntent s suffcently hgh, the Fe melt s n equlbrum wth. In-between, the Fe melt exsts tgether wth graphte and. These tw-fld saturatns are un-varant cmpstns and vary wth temperature. At the cmpstns f tw-fld saturatn, the actvtes f and are unty,.e., a =a =1. Actvty ceffcents f are determned at varus cmpstns and temperatures up t tw-fld saturatns frm the expermentally determned slublty. At tw-fld saturatns, the actvty ceffcents f are determned frm the value f ΔG. Such develped sets f data fr actvty ceffcents f and are served t determne ternary nteractn parameters f the UIP mdel. As mentned abve, the tw-fld saturatn s the manfestatn f un-varant reactn n Fe system. Its behavr s characterzed by ΔG, and the crrect value f ΔG defnes unquely the temperature dependency f cmpstn fr tw-fld saturatn. hpman et al. 1, nvestgated extensvely the tw-fld (b Fg. 5. Slublty f carbn n Fe melt. (a 1 56, 1 698, 1 81 and 1 96 K. (b 1 6, 1 77 and 1 87 K. Fg. 6. Tw-fld saturatn f and n Fe system. saturatns n Fe system, and Fg. 6 reprduces ther expermental results. The cmpstn f fr tw-fld saturatn s shwn t ncrease wth ncreasng cmpstn f, and they ncrease wth ncreasng temperature. Ths relatnshp s als supprted by Ohtan et al. 5 and Kawansh ISIJ
5 ISIJ Internatnal, Vl. 54 (014,. 4 et al. 6 As they are the manfestatn f un-varant reactn n the Fe system, the nfrmatn f these tw-fld saturatns s used t valdate the value f ΔG n ths study. 5.. Apprasal f the Standard Gbbs Free Energy f Frmatn f, ΔG Many nvestgatrs 1,8 1 studed the standard Gbbs free energy f frmatn f, ΔG. Fgure 7 reprduces them as they reprted. It shws that ther values d nt agree amng themselves. The values f ΔG quted by hpman et al. 1, are mst pstve amng them. Thse by Pankratz 10 are negatve than thse by hpman et al. Thse by Barn, 9 JAAF, 1 Lacaze et al., 11 and Turkdgan 8 are practcally dentcal but mst negatve. The dscrepancy amng them s as hgh as abut 16 kj/ml. Obvusly, ther valdty needs t be examned. The valdty f a gven ΔG s examned by develpng a UIP mdel wth the use f t. The develped mdel s used t reprduce the expermental data fr tw-fld saturatns. If tw sets f nfrmatn are n agreement, the gven ΔG s taken t be crrect. (The peratn f UIP mdel s dscussed n detal n the next sectn. The values f ΔG by Barn 9 are mre negatve than thse by Pankratz, 10 and ther effects are dscussed n the reprductn f tw-fld saturatns f and n the fllwngs. ΔG The UIP mdel wth by Barn 9 reprduces prly the expermental data fr tw-fld saturatns f and. As shwn n Fg. 6, the tw-fld saturatns calculated by the UIP mdel devate sgnfcantly frm the expermental data fr tw-fld saturatns f and tward lwer ΔG cntent f. A smlar exercse wth by Pankratz 10 shws that the tw-fld saturatns by the UIP mdel apprach clser t the expermental data fr tw-fld saturatns f and, but the reprductn by the UIP mdel s nt satsfactry yet. Fgure 6 llustrates ths effect well. It ΔG appears that by Pankratz 10 are clser t the crrect ΔG. The pr reprductn wth f Barn 9 ΔG s due t the fact that the excessvely negatve values f ΔG lwer the actvty ceffcent f at the expermental cmpstn f fr tw-fld saturatns and that the UIP mdel can prduce such lw actvty ceffcent f nly at lwer cmpstn f than the expermental ne. The assessment fr the crrect value f ΔG was made by tral and errr untl a cnfrmty s reached n the cmpstns f tw-fld saturatns between UIP mdel and expermental data. The resulted ΔG s prpsed by the Eq. (0. ( + ( gr = ( β ΔG = T( J / ml,... ( K errr ± 100J ( The ranges f errr and effectve temperature are based n expermental temperatures by hpman et al. 1, The prpsed ΔG cmpares wth the prevusly reprted values n Fg. 7. It s a lttle pstve than that by Pankratz. 10 The cmpstns f tw-fld saturatns were reprduced by the UIP mdel wth the prpsed ΔG. Fgure 6 shws that they reprduce very well the expermental cmpstns f tw-fld saturatns. The bserved gd agreement lends a credence t the prpsed value f. ΔG 6. Interactn Parameters fr the Fe System The develped set f data fr actvty ceffcents f and n Fe system are used t determne ternary nteractn parameters. After a necessary rearrangement, a multple regressn analyss was perfrmed n the Eqs. (1 and (. lnγ ln γ ε ( x ε ( x x + ε ( x x x + ε ( 4 x = ε ( x x x + ε ( x x x... (1 + ε ( x x x x + ε ( x x x x + lnγ ln γ ε ( x x ε ( x + ε ( x + ε ( 4 x... ( = ε ( x x x + ε ( x x x x + ( x x + ε ( x x x + ε It was fund that fur ternary nteractn parameters are suffcent t reprduce the expermentally bserved slutn behavrs. They are gven by the Eqs. (, (4, (5, and (6 and Table 1 as well. ε = / T ( K... ( ε = / T ( K... (4 Fg. 7. Standard Gbbs free energy f frmatn f. ε ε = / T ( K... (5 = / T ( K... (6 The slublty f n Fe melt was reprduced by the present UIP mdel and cmpared wth expermental data n Fg. 5. The mdel s shwn t reprduce well the expermental data. Tw-fld saturatns f and by the present UIP mdel are cmpared wth the expermental data n Fg. 6. A gd agreement between them s bserved. 014 ISIJ 754
6 ISIJ Internatnal, Vl. 54 (014,. 4 Table nclusn Interactn parameters f UIP mdel fr the Fe system. Systems Interactn parameters Temperatures, K * Fe (slvent Fe Fe Fe lnγ = /T ε = /T lnγ = /T ε = /T ε = /T ε = /T ε = /T ε = /T ε = /T ε = /T The thermdynamc prperty f plays an mprtant rle n defnng slutn behavrs f Fe system, but reprted values fr ΔG are fund t be ncrrect. The crrect value f ΔG was determned by reprducng exper- mental tw-fld saturatns wth the use f UIP mdel as fllws: ( + ( gr = ( β ΔG = T J / ml, ( ( ± K errr J ΔG Wth the prpsed n hand, a UIP mdel was assessed fr the Fe system. The UIP mdel was fund t descrbe well the slutn behavrs f Fe melt n a wde range f cmpstns and temperatures. It ensures a wde applcatn n the prcesses f rnmakng, steelmakng, cast-rn, -based ferrallys, and the lw temperature lqud phase grwth f electrnc grade slcn carbde. REFEREES 1 J. hpman, J.. Fultn,. Gkcen and G. R. askey: Acta Metall., (1954, 49. J.. Fultn and J. hpman: J. Met., 6 (1954, 56. J. hpman, R. M. Alfred, L. W. Gtt, R. B. Small, D. M. Wlsn,. Thmsn, D. L. Guernsey and J.. Fultn: Trans. Am. Sc. Met., 44 (195, E. Schurmann and D. Kramer: Gesserefrschung, 1 (1969, 9. 5 M. Ohtan: Sc. Rep. RITU, A7 (1955, S. Kawansh, T. Yshkawa and T. Tanaka: Mater. Trans., 50 (009, K. Sanbng, M. Ohtan and K. Tta: Sc. Rep. RITU, A9 (1957, E. T. Turkdgan: Physcal hemstry f Hgh Temperature Technlgy, Academc Press, ew Yrk, (1980, 0. 9 I. Barn: Thermchemcal Data f Pure Substance, rd ed., VH, Germany, ( L. B. Pankratz: Thermdynamc Prpertes f arbdes, trdes, and Other Selected Substances, US Bureau f Mnes Bulletn 696, US Dept. f the Interr, Washngtn D, ( J. Lacaze and B. Sundman: Metall. Trans., A (1991, M. W. hase, Jr.: IST-JAAF Thermchemcal Tables, 4th ed., Am. hem. Sc. & Am. Inst. Phys., Wdbury, ew yrk, ( A. D. Peltn and. W. Bale: Metall. Mater. Trans., A17 (1886, W. Bale and A. D. Peltn: Metall. Mater. Trans., A1 (1990, G. K. gwrth and J. F. Elltt: Met. Sc., 8 (1974, J. A. Ktchener, J. O. Bckrs and D. A. Sparat: Trans. Faraday Sc., 48 (195, S. Matba and S. Banya: Technl. Rep. Thku Unv., 0 (1955, E. T. Turkdgan and L. E. Leake: J. Irn Steel Inst., 180 (1955, Steelmakng Data Surcebk, ed. by Japan Scety fr the Prmtn f Scence, the 19th mmttee n Steelmakng, Grdn and Breach Scence Publshers S. A., Mntreux, Swtzerland, (1988, T. G. hart: Hgh Temperature - Hgh Pressures, (1970, R. Schmd: alphad, 4 (1980, 101. T. B. Massalsk, J. L. Murray, L. H. Bennett and H. Baker: Bnary Ally Phase Dagrams, Vl. 1, Amercan Scety fr Metals, Materals Park, OH, (1986, 589. P. Bakke and O. S. Klevan: Prcess Technlgy nf. Prc., Vl. 14, ISS, Warrendale, PA, (1995, Y. Iguch and T. arushma: Prc. f 1st Int. nf. n Prcessng Materals fr Prpertes, TMS, Warrendale, PA, (199, J. Tuset, L. Ottem, R. Lvk: lcn fr hemcal Industry, Prc., ed. by H. A. Oye, H. Rng, Insttute f Inrganc hemstry, TH, Geranger, rway, ( ISIJ
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