Numerical Analysis on Si Deoxidation of Molten Ni and Ni Cu Alloy by Quadratic Formalism

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1 Materals ransactons, Vol. 44, No. 9 (2003) pp to 1823 #2003 he Japan Insttute of Metals Numercal Analyss on S Deoxdaton of Molten N and N Cu Alloy by Quadratc Formalsm akahro Mk, Fuo Ish and Mtsutaka Hno Department of Metallurgy, Graduate School of Engneerng, ohoku Unversty, Senda , Japan Relaton between the actvty coeffcent expressed by Darken s quadratc formalsm and the excess Gbbs energy change of mxng descrbed by Redlch-Kster type polynomal was dscussed n S deoxdaton of N alloy. he actvty coeffcents of S and O n metal expressed by quadratc formalsm have been converted nto formula usng nteracton parameters under the condton where concentraton of S and O are dlute. Numercal analyss on S deoxdaton of molten N and N Cu alloy has been carred out. It has been found to be outstandng n the agreement of equlbrum S and O contents n molten N and N Cu alloy analyzed n the present work wth the expermental results. he deoxdaton equlbrum of not only pure metal but also alloy can be analyzed numercally usng the formula determned n the present work. (Receved May 23, 2003; Accepted July 4, 2003) Keywords: quadratc formalsm, thermodynamcs, actvty, excess Gbbs energy, Redlch-Kster polynomal, deoxdaton equlbrum, slcon, nckel, copper 1. Introducton he equaton proposed by Wagner 1) has been wdely used to express the actvty coeffcents of solute n multcomponent solutons. However, second order or hgher terms must be added to represent the actvty coeffcents on the condton where solute contents are rch. Darken 2,3) has proposed the quadratc formalsm to express the actvty coeffcents that can be appled to hgher solute content. he actvty coeffcent of solvent 1 and solute 2, 3 can be expressed for ternary soluton by the followng equatons. ln 1 ¼ 12 X2 2 þ 13X3 2 þð 12 þ ÞX 2 X 3 ð1þ ln 2 ¼ 2 12 X 2 þð ÞX 3 þ 12 X 2 2 þ 13 X 2 3 þð 12 þ ÞX 2 X 3 þ ln o 2 ln 3 ¼ 2 13 X 3 þð ÞX 2 þ 12 X 2 2 þ 13 X3 2 þð 12 þ ÞX 2 X 3 þ ln 3 o ð3þ Where X denotes mole fracton of component, denotes the actvty coeffcent of component, o denotes the actvty coeffcent of component at nfnte dluton, and s a constant that characterzes the thermodynamc property of bnary soluton wthn the of valdty of equatons at constant temperature and pressure, respectvely. he followng relatons can be obtaned 2,3) by comparng the equaton proposed by Wagner 1) wth eq. (2). " 2 2 ¼ 2 12 ð4þ ð2þ " 3 2 ¼ ð5þ 2 2 ¼ 12 ð6þ 3 2 ¼ 13 ð7þ ð2;3þ 2 ¼ 12 þ ð8þ Where " and are the frst and the second order nteracton coeffcents, respectvely. Quadratc formalsm ncludes all the frst and the second order terms represented by Wagner s formalsm. Hence, the vald composton that quadratc formalsm stands s essentally wder than that where Wagner s formalsm are avalable. Numercal analyss on S deoxdaton of molten N and N Cu alloy utlzng quadratc formalsm has been carred out n the present work. 2. Preparaton of Numercal Analyss 2.1 Relaton between Darken s quadratc formalsm and Redlch-Kster type polynomal Quadratc formalsm was converted n to Redlch-Kster type polynomal 4,5) to express the excess Gbbs energy change of mxng. hs converson was done because t would be easer to expand Darken s quadratc formalsm to mult-component alloy systems. he excess Gbbs energy change of mxng for 1 2 bnary system,, and partal molar excess Gbbs energy change, 2, can be expressed as eqs. (9) and (10), respectvely. ¼ X 1 X 2 f 0 1{2 þ 1 1{2 ðx 1 X 2 Þg 2 ¼ R ln ð9þ 2 ¼ þð1 X 2 ¼ X 2 1 ð0 1{ {2 ÞþX 3 1 ð41 1{2 Þ ð10þ Where 0 and 1 are bnary nteracton parameters. On the condton of X 1 ¼ 1, the actvty coeffcent of component 2 wll be 2 o. Hence, the followng relatonshp can be obtaned n the nfnte dlute soluton of 2. R ln 2 o ¼ 0 1{2 þ 1 1{2 ð11þ he actvty coeffcent of component 2 n 1 2 bnary system can be expressed as eq. (12), usng Darken s 2,3) quadratc formalsm. ln 2 ¼ 12 X1 2 þ I ð12þ Where I s a constant on the condton of constant temperature and pressure. Equaton (12) s vald n the dlute soluton of component 2. Multplyng gas constant, R, and

2 1818. Mk, F. Ish and M. Hno absolute temperature,, to both sdes of eq. (12) and partal dfferentatng wth X1 2 gves us the followng ln 2 2Þ ¼ R 12 ð13þ On the other hand, the followng equaton can be derved from partal dfferentaton of eq. (10) wth X1 ln 2 2Þ ¼ð 0 1{ {2 Þþ6 1 1{2 X 1 ð14þ On the condton X 1 ¼ 1, eqs. (13) and (14) wll be equvalent and the followng relaton wll be obtaned for nfnte dlute soluton. R 12 ¼ 0 1{2 þ 3 1 1{2 ð15þ he bnary nteracton parameters can be obtaned by combnng eqs. (11) and (15) as follows. 0 1{2 ¼ R 3 2 ln o ð16þ 1 1{2 ¼ R 1 2 ln o 2 þ ð17þ Quadratc formalsm can be converted nto Redlch-Kster type polynomal usng 0 1{2 and 1 1{2 for dlute soluton usng eqs. (16) and (17). 2.2 Standard state of components Pure substance s chosen as a standard state (Raoultan standard state) for condensed phases n the present work. It s general to select Henran standard state for oxygen dssolved n pure melts. However, problem arses n case of alloys. he actvty coeffcents of oxygen n metal 1 and 2 are taken as unty to descrbe the actvty coeffcent of oxygen n 1 2 bnary alloy by usng Henran standard state, and the effect of alloy component s compensated by nteracton coeffcents. When the oxygen actvtes usng Henran standard state n metal 1 and 2 are dentcal, the equlbrum oxygen partal pressure dffers each other due to the dfference of the Gbbs free energy change of oxygen dssoluton nto pure 1 and 2. herefore, the Henran standard state s not a unversal reference n case of oxygen n alloy. herefore, dssolved oxygen n the melt equlbratng wth Pa (1 atm) oxygen gas has been selected as a standard state n the present work. he relaton between the oxygen actvty and oxygen partal pressure n ths standard state can be expressed as follows. a o ¼ P 1 2 O2 ð18þ he actvty coeffcent of oxygen at nfnte dluton n metal s not unty as the case of Henran standard state but O o. he oxygen actvty or equlbrum oxygen partal pressure s, therefore, ndependent of the knd of metal solvent and can be expressed by utlzng ths standard state. 3. Numercal Analyss on S Deoxdaton of Molten N S deoxdaton reacton can be expressed by eq. (19). S(l) þ 2O ¼ SO 2 (s) ð19þ hs equaton can be separated nto followng equatons. S(l) þ O 2 (g) ¼ SO 2 (s) ð20þ 2O ¼ O 2 (g) ð21þ he Gbbs free energy change of eq. (20) s dentcal to the Gbbs free energy of SO 2 formaton (G o f,so 2 ) and that of eq. (21) s zero due to the relatonshp of eq. (18). herefore, the Gbbs free energy change of eq. (19) wll be dentcal to the Gbbs free energy of SO 2 formaton (G o f,so 2 ). Equaton (19) and the Gbbs free energy of SO 2 formaton can be utlzed for S deoxdaton of any metal or alloy, and ths s a great advantage. he equlbrum constant of eq. (19), K S, can be expressed as follows. ln K S ¼ Go f,so 2 R ¼ ln a SO2 ln S 2ln O ln X S 2lnX O ¼ ln S 2ln O ln X S 2lnX O ð* a SO2 ¼ 1Þ ð22þ Where, the actvty of SO 2 s taken as unty, snce we consder the case that deoxdaton product s pure SO 2. he excess free energy change of mxng N S O ternary system can be expressed as eq. (23) by usng Redlch-Kster type polynomal. ¼ X N X S f 0 N{S þ 1 N{S ðx N X S Þg þ X N X O f 0 N{O þ 1 N{O ðx N X O Þg ð23þ þ X S X O S{O he partal molar excess free energy change of S and O can be derved as follows. S ¼ R ln S ¼ þð1 X X O ¼ 0 N{S X N ð1 X S Þþ 1 N{S X N ðx N 2X S 2X N X S þ 2XS 2 Þ 0 N{O X N X O 2 1 N{O X N X O ðx N X O Þþ S{O X O ð1 X S Þ ð24þ

3 Numercal Analyss on S Deoxdaton of Molten N and N Cu Alloy by Quadratc Formalsm 1819 O ¼ R ln O X S þð1 X O ¼ 0 N{S X N X S 2 1 N{S X N X S ðx N X S Þþ 0 N{O X N ð1 X O Þ þ 1 N{O X N ðx N 2X O 2X N X O þ 2XO 2 Þþ S{OX S ð1 X O Þ Equaton (26) can be deduced by substtutng eqs. (24) and (25) nto eq. (22). 0 N{S X N ð1 3X S Þþ 1 N{S X N ðx N 2X S 6X N X S þ 6XS 2 Þ þ 0 N{O X N ð2 3X O Þþ 1 N{O 2X N ðx N 2X O 3X N X O þ 3XO 2 Þ þ S{O ð2x S þ X O 3X S X O ÞþRðln X S þ 2lnX O Þ G o f,so 2 ¼ 0 ð25þ ð26þ Equaton (26) s the fundamental equaton for numercal analyss of S deoxdaton n molten N. he expermental results of Ish and Ban-ya 6) have been utlzed n the present work. he bnary nteracton parameters 0 N{O, 1 N{O, S{O were obtaned from the Gbb free energy change of oxygen dssoluton nto molten N, the self-nteracton coeffcent of oxygen n molten N reported by Sgworth et al. 7) and the Gbb free energy change of oxygen dssoluton nto molten S reported by Narushma et al. 8) he Gbbs free energy of SO 2 formaton has been taken from NIS-JANAF hermochemcal ables. 9) Bnary nteracton parameters have been determned from the references utlzng the results at nfnte dluton and have been shown n the followng wth the Gbbs free energy of SO 2 formaton. 0 N{O ¼ þ 44:80=J ð27þ 1 N{O ¼ :92=J ð28þ F (X N -2X S -6X N X S +6X 2 S )/(1-3X S ) S{O ¼ þ 129:2=J G o f,so 2 ¼ þ 197:7=J Equaton (31) can be obtaned by rearrangng eq. (26). 1 F ¼ X N ð1 3X S Þ f 0 N{O X N ð2 3X O Þ 1 N{O 2X N ðx N 2X O 3X N X O þ 3XO 2 Þ S{O ð2x S þ X O 3X S X O Þ Rðln X S 2lnX O ÞþG o f,so 2 g ð29þ ð30þ ð31þ ¼ 0 N{S þ 1 X N 2X S 6X N X S þ 6XS 2 N{S 1 3X S By usng the results of Ish and Ban-ya 6) and eqs. (27) (30) and takng F n eq. (31) as a vertcal axs and X N 2X S 6X N X S þ6xs 2 1 3X S as a horzontal axs, 0 N{S, 1 N{S can be derved from the ntercept and the slope of the regressed lne, respectvely. he results are shown n Fg. 1. Lner relatons have been observed at each temperature, and 0 N{S, 1 N{S have been determned as follows. Values n the parenthess are standard error. 0 N{S ¼ ð212900Þþ382:6ð115:6Þ=J ð32þ 1 N{S ¼ ð227100Þ 191:6ð123:3Þ=J ð33þ he relaton between equlbrum S and O content n molten N compared wth the results of Ish and Ban-ya s shown n Fg. 2. he present result agrees extremely well wth ther expermental results. Fg. 1 log[mass % O] Relaton between F and ðx N 2X S 6X N X S þ 6X 2 S Þ=ð1 3X SÞ pure N log[mass%s] Fg. 2 Relaton between S and O content n molten N. he actvty coeffcent of S n molten N determned n the present work s shown wth lterature s values 10 13) n Fg. 3. he bold curve was determned from eqs. (24), (32) and (33). Dashed lne shows the actvty coeffcent expressed by quadratc formalsm. Dfference between bold curve and dashed lne s lttle were S content s dlute, and t s confrmed that the present analyss satsfes quadratc formal-

4 1820. Mk, F. Ish and M. Hno R ln γ S(l) / kj.mol S (1-X S ) 2 N Fg Present work () Bowles et al. (1833K) Stukalo et al. () Sano et al. (1743K) Sano et al. (1783K) termnal Quadratc formalsm () Schwerdtfeger and Engell ( K) Schwerdtfeger and Engell ( K) Schwerdtfeger and Engell ( K) Schwerdtfeger and Engell ( K) central termnal he actvty coeffcent of S n molten N S alloy. sm wthn the composton range of mole fracton of S less than 0.042, whch s the expermental range of Ish and Ban-ya. 6) he present results agree remarkably wth Schwerdtfeger and Engell. 13) It s observed from the present work and lterature s values except Stukalo et al. 11) and Sano et al. 12) that the actvty coeffcent of S n two termnal composton s can be expressed wth quadratc formalsm. he actvty coeffcent n the central or transton appears to be complcated. Snce the present results agreed well wth lterature s values, 0 N{S and 1 N{S determned n the present work can be consdered reasonable. S deoxdaton equlbrum n molten N has been quanttatvely analyzed wth hgh accuracy by the quadratc formalsm. 4. Numercal Analyss on S Deoxdaton of Molten N Cu Alloy In case of analyzng numercally S deoxdaton of molten N Cu alloy, bnary nteracton parameters of N Cu, Cu S, and Cu O systems must be addtonally consdered wth the bnary nteracton parameters used for the numercal analyss of S deoxdaton n molten N. he excess Gbbs free energy change of mxng for N Cu bnary alloy reported by Mey, 14) the Gbbs free energy change of oxygen dssoluton nto molten Cu and self-nteracton parameter of oxygen n molten Cu reported by Sgworth and Ellott 15) were used to determne the bnary nteracton parameters of N Cu and Cu O systems, respectvely. Expermental results for S deoxdaton of molten N Cu alloys reported by Ish and Ban-ya 16) were utlzed for the analyss. Bnary nteracton parameters of N Cu system reported by Mey 14) and that for Cu O system obtaned from the results of Sgworth and Ellott 15) are shown n the followng. 0 N{Cu ¼ þ 1:291=J ð34þ Equatons (34) and (35) are vald for any composton of N Cu bnary, whle eqs. (36) and (37) are vald n the dlute of O n molten Cu. he excess Gbbs free energy change for N Cu S O quaternary system can be expressed as follows. ¼ X N X Cu f 0 N{Cu þ 1 N{Cu ðx N X Cu Þg þ X N X S f 0 N{S þ 1 N{S ðx N X S Þg þ X N X O f 0 N{O þ 1 N{O ðx N X O Þg þ X Cu X S f 0 Cu{S þ 1 Cu{S ðx Cu X S Þg þ X Cu X O f 0 Cu{O þ 1 Cu{O ðx Cu X O Þg þ X S X O S{O ð38þ he partal molar excess free energy change of S and O can be derved as follows. S ¼ R ln S ¼ Cu þð1 X S ¼ 0 N{S X N ð1 X S Þ þ 1 N{S X N ðx N 2X S 2X N X S þ 2XS 2 Þ þ 0 Cu{S X Cu ð1 X S Þ þ 1 Cu{S X Cu ðx Cu 2X S 2X Cu X S þ 2XS 2 Þ 0 N{O X N X O 2 1 N{O X N X O ðx N X O Þ 0 Cu{O X Cu X O 2 1 Cu{O X Cu X O ðx Cu X O Þ þ S{O X O ð1 X S Þ 0 N{Cu X N X Cu 2 1 N{Cu X N X Cu ðx N X Cu Þ O ¼ R ln O O X Cu X S þð1 X O ¼ 0 N{S X N X S 2 1 N{S X N X S ðx N X S Þ 0 Cu{S X Cu X S 2 1 Cu{S X Cu X S ðx Cu X S Þ þ 0 N{O X N ð1 X O Þ þ 1 N{O X N ðx N 2X O 2X N X O þ 2XO 2 Þ þ 0 Cu{O X Cu ð1 X O Þ þ 1 Cu{O X Cu ðx Cu 2X O 2X Cu X O þ 2XO 2 Þ þ S{O X S ð1 X O Þ 0 N{Cu X N X Cu 2 1 N{Cu X N X Cu ðx N X Cu Þ ð40þ he followng relaton can be obtaned by substtutng eqs. (39) and (40) nto eq. (22). 1 N{Cu ¼ 1861:6 0:9420=J ð35þ 0 Cu{O ¼ þ 134:7=J ð36þ 1 Cu{O ¼ :90=J ð37þ

5 Numercal Analyss on S Deoxdaton of Molten N and N Cu Alloy by Quadratc Formalsm N{S X N ð1 3X S Þ þ 1 N{S X N ðx N 2X S 6X N X S þ 6XS 2 Þ þ 0 N{O X N ð2 3X O Þ þ 1 N{O 2X N ðx N 2X O 3X N X O þ 3XO 2 Þ 0 Cu{S X Cu ð1 3X S Þ þ 1 Cu{S X Cu ðx Cu 2X S 6X Cu X S þ 6XS 2 Þ þ 0 Cu{O X Cu ð2 3X O Þ þ 1 Cu{O 2X Cu ðx Cu 2X O 3X Cu X O þ 3XO 2 Þ þ S{O ð2x S þ X O 3X S X O Þ 3 0 N{Cu X N X Cu 6 1 N{Cu X N X Cu ðx N X Cu Þ þ Rðln X S þ 2lnX O Þ G o f,so 2 ¼ 0 Rearrangng eq. (41) derves eq. (42). F 0 1 ¼ X Cu ð1 3X S Þ f 0 N{S X N ð1 3X S Þ 1 N{S X N ðx N 2X S 6X N X S þ 6XS 2 Þ 0 N{O X N ð2 3X O Þ 1 N{O 2X N ðx N 2X O 3X N X O þ 3XO 2 Þ 0 Cu{O X Cu ð2 3X O Þ 1 Cu{O 2X Cu ðx Cu 2X O 3X Cu X O þ 3XO 2 Þ S{O ð2x S þ X O 3X S X O Þþ3 0 N{Cu X N X Cu þ 6 1 N{Cu X N X Cu ðx N X Cu Þ Rðln X S 2lnX O ÞþG o f,so 2 g ð41þ ð42þ ¼ 0 Cu{S þ 1 X Cu 2X S 6X Cu X S þ 6XS 2 Cu{S 1 3X S By utlzng eqs. (27) (30), (32) (37) and expermental results reported by Ish and Ban-ya 16) and takng F 0 n eq. (42) as a vertcal axs and X Cu 2X S 6X Cu X S þ6x 2 S 1 3X S as a horzontal axs, 0 Cu{S and 1 Cu{S can be determned from the ntercept and the slope of the regressed lne, respectvely. he results are shown n Fg. 4. he plots seem to scatter n the where X Cu 2X S 6X Cu X S þ6x 2 S 1 3X S s small or Cu s dlute. F 0 mathematcally scatters n Cu dlute because X Cu s ncluded n the denomnator of F 0. he followng relatons has been obtaned from regresson of Fg. 4. Values n the parenthess are standard error. 0 Cu{S ¼ ð82700Þ 443:8ð45:2Þ=J ð43þ 1 Cu{S ¼ ð124000Þþ675:9ð68:3Þ=J ð44þ Apparent deoxdaton product of S n molten N Cu alloy determned n the present work are shown wth the results reported by Ish and Ban-ya n Fg. 5. he present result agrees well wth ther expermental results n the whole of the alloy. he actvty coeffcent of S n molten Cu s shown wth lterature values 10,17 19) n Fg. 6. Dfference between present work and actvty coeffcent expressed by quadratc formalsm s lttle. he values of at R ln S(l) /kjmol 1 at ð1 X S Þ 2 ¼ 0:9 and 1873 K were 153:2(present work) and 152:1(quadratc formalsm), respectvely, and t s confrmed that the present analyss satsfes quadratc formalsm F ' K 1723K (X Cu -2X S -6X Cu X S +6X 2 S )/(1-3X S ) Fg. 4 Relaton between F 0 and ðx Cu 2X S 6X Cu X S þ 6X 2 S Þ= ð1 3X S Þ. log([mass%s][mass% O] 2 ) Fg. 5 R lnγ S(l) / kj. mol [mass%s]= K [mass% Cu] Concentraton product durng S deoxdaton n molten N Cu alloy Present work () Bowles et al. (1833K) Nktn (1623K) termnal central Bergman et al. (1753K) Sano et al. (1783K) Sano et al. (1743K) termnal S (1-X S ) 2 Cu Fg. 6 he actvty coeffcentof S n molten Cu S alloy.

6 1822. Mk, F. Ish and M. Hno able 1 he bnary nteracton parameters utlzed and determned n the present work. Values (J) Regon of valdty Lterature 0 N{S þ 382:6 X S < 0:042, Present work 1 N{S :6 X S < 0:042, Present work 0 N{O þ 44:80 X O < 0:00060, Present work 1 N{O :92 X O < 0:00060, Present work 0 Cu{S :8 X S < 0:013, Present work 1 Cu{S þ 675:9 X S < 0:013, Present work 0 Cu{O þ 134:7 X O < 0:000082, Present work 1 Cu{O :90 X O < 0:000082, Present work 0 N{Cu þ 1:291 0 < X Cu < 1, Mey 14) 1 N{Cu 1861:6 0: < X Cu < 1, Mey 14) S{O þ 129:2 X S < 0:042, X O < 0: Present work able 2 he actvty coeffcent of elements nfnte dluton and frst order nteracton coeffcent n molten N at 1773 to 1923 K. S O Cu ln o þ 23: þ 3:47 þ 0:0420 S þ 46: þ 39: " O þ 39: þ 0:541 Cu þ 0: þ 0:369 Regon of X S < 0:042 X O < 0:00060 X Cu < 0:089 valdty able 3 Second order nteracton coeffcent n molten N at 1773 to 1923 K. S O Cu :1 0:356 0: S 39:0 214 ð;þ O 39: : Cu :541 Regon of X S < 0:042 X O < 0:00060 X Cu < 0:089 valdty able 4 he actvty coeffcent of elements nfnte dluton and frst order nteracton coeffcent n molten Cu at 1723 to 1923 K. S O N ln o þ 27: þ 10:8 þ 0: S " O : N : :990 Regon of X S < 0:042 X O < 0:00060 X N < 0:089 valdty able 5 Second order nteracton coeffcent n molten Cu at 1723 to 1923 K. S O N þ 191 þ 0:495 S þ ð;þ O þ 175 þ 0:851 N þ 0:851 Regon of X S < 0:042 X O < 0:00060 X N < 0:089 valdty wthn the composton range of the present work. he reported actvty coeffcents of S dffer each other among researchers. he actvty coeffcent of S at nfnte dluton determned n the present result s close to the value reported by Nktn. 16) he bnary nteracton parameters utlzed and determned n the present work are summarzed n able 1 wth the composton range of valdty. Also, the actvty coeffcents of elements n molten N and Cu at nfnte dluton and Wagner s 1) nteracton coeffcents n molten N and Cu derved from eqs. (4) (8), (11), (15) and the bnary nteracton parameter are shown n ables 2 to 5. It was confrmed from the present numercal analyss that the S deoxdaton equlbrum of molten N and N Cu alloy can be quanttatvely expressed wth hgh accuracy by utlzng quadratc formalsm. 5. Concluson Deoxdaton equlbrum n molten N and N Cu alloy can be quanttatvely expressed by bnary nteracton parameters determned on bass of the quadratc formalsm. he Gbbs energy of formaton of deoxdaton product can be utlzed by selectng dssolved oxygen n the melt equlbratng wth Pa (1 atm) oxygen gas as a standard state to analyze numercally deoxdaton equlbrum n metal or ts alloy.

7 Numercal Analyss on S Deoxdaton of Molten N and N Cu Alloy by Quadratc Formalsm 1823 REFERENCES 1) C. Wagner: hermodynamcs of Alloys, (Addson-Wesley, Cambrdge, MA., 1952). 2) L. S. Darken: rans. Metall. Soc. AIME 239 (1967), ) L. S. Darken: rans. Metall. Soc. AIME 239 (1967), ) M. Hllert and L.-I. Staffanson: Acta Chem. Scand. 24 (1970), ) N. Saunders and A. P. Modownk: Calphad(Calculaton of Phase Dagrams), A Comprehensve Gude, (Pergamon, Oxford, 1988), pp ) F. Ish and S. Ban-ya: etsu-to-hagaé 75 (1989), ) G. K. Sgworth, J. F. Ellott, G. Vaughn and G. H. Geger: rans. Metall. Soc. CIM, Annual volume (1977), ). Narushma, K. Matsuzawa, Y. Muka and Y. Iguch: Mater. rans., JIM 35 (1994), ) NIS-JANAF hermochemcal ables, 4th ed. Part II, ed. By M. W. Chase, Jr, J. Phys. Chem. Ref. Data, (1998), ) P. J. Bowles, H. F. Ramstad and F. D. Rchardson: J. Iron Steel Inst. 202 (1964), ) V. A. Stukalo, G. I. Bataln, N. Ya. Neshchmenko and V. P. Kurach: Izv. Akad. Nauk SSSR, Met. 6 (1979), ) K. Sano, K. Okama and N. Okuda: Memors Fac. Eng. Nagoya Unv. 8 (1956), ) K. Schwedrtfeger and H. J. Engell: rans. Metall. Soc. AIME 233 (1965), ) S. Mey: Calphad, 16 (1992), ) G. K. Sgworth and J. F. Ellott: Can. Metall. Quart. 13 (1974), ) F. Ish and S. Ban-ya: etsu-to-hagaé 77 (1991), ) C. Bergman, R. Chastel and J.-C. Matheu: J. Chem. hermodynamcs 18 (1986), ) K. Sano, K. Okama and N. Okuda: Memors Fac. Eng. Nagoya Unv. 8 (1956), ) Yu. P. Nktn: Izv. Vys. Uchebn. Zaved. svetn. Met. 2 (1623),