A Model for Dissolution of Lime in Steelmaking Slags

A Model fo Dissolution of Lime in Steelmaking Slags RAHUL SARKAR, USHASI ROY, and DINABANDHU GHOSH In a pevious study by Saka et al. (Metall. Mate. Tans. B 46B:96 05), a dynamic model of te LD steelmaking was developed. Te pediction of te pevious model (Saka et al. in Metall. Mate. Tans. B 46B:96 05) fo te bat (metal) composition matced well wit te plant data (Cicutti et al. in Poceedings of 6t Intenational Confeence on Molten Slags, Fluxes and Salts, Stockolm City, 000). Howeve, wit espect to te slag composition, te pediction was not satisfactoy. Te cuent study aims to impove upon te pevious model Saka et al. (Metall. Mate. Tans. B 46B:96 05) by incopoating a lime dissolution submodel into te ealie one. Fom te industial point of view, te undestanding of te lime dissolution kinetics is impotant to meet te eve-inceasing demand of poducing low-p steel at a low basicity. In te cuent study, tee-step kinetics fo te lime dissolution is ypotesized on te assumption tat a solid laye of Æ sould fom aound te uneacted coe of te lime. Fom te available expeimental data, it seems impobable tat te obseved kinetics sould be contolled singly by any one kinetic step. Accodingly, a geneal, mixed contol model as been poposed to calculate te dissolution ate of te lime unde vaying slag compositions and tempeatues. Fist, te ate equation fo eac of te tee ate-contolling steps as been deived, fo tee diffeent lime geometies. Next, te ate equation fo te mixed contol kinetics as been deived and solved to find te dissolution ate. Te model pedictions ave been validated by means of te expeimental data available in te liteatue. In addition, te effects of te pocess conditions on te dissolution ate ave been studied, and compaed wit te expeimental esults weeve possible. Incopoation of tis submodel into te ealie global model (Saka et al. in Metall. Mate. Tans. B 46B:96 05) enables te pediction of te lime dissolution ate in te dynamic system of LD steelmaking. In addition, wit te inclusion of tis submodel, significant impovement in te pediction of te slag composition duing te main blow peiod as been obseved. DOI: 0.007/s663-06-0659-0 Te Mineals, Metals & Mateials Society and ASM Intenational 06 I. INTRODUCTION IN tei ecent publication, [] te autos developed a dynamic model fo pedicting te bat and slag compositions duing blowing in a 60-ton LD convete. Wile model pedictions fo bat compositions appeaed to be in elatively good ageement wit measuements in an actual LD convete by Cicutti et al., [] tose fo te slag compositions did not coespond well to te actual values. Pedictions fo pecent in slag wee too ig and consequently tose fo pecent FeO wee too low. Among te assumptions made in te model, [] one was te unifom dissolution of lime duing te blowing peiod. Te model tus lacked a pope submodel fo lime dissolution wic migt ave esulted in te obseved inconguity wit te eal values. Te cuent study aims to develop a submodel fo lime dissolution in steelmaking slags and study te effects of diffeent pocess vaiables on dissolution ates. Finally, tis submodel would be RAHUL SARKAR and USHASI ROY, Reseaces, ae wit Reseac and Development, Tata Steel Limited, Jamsedpu, India. DINABANDHU GHOSH, Pofesso, is wit te Depatment of Metallugical and Mateials Engineeing, Jadavpu Univesity, Kolkata, 70003, India. Conatct e-mail: aulsaka.jumet@gmail.com Manuscipt submitted August 0, 05. Aticle publised online Apil, 06. incopoated into te global model fo dynamically calculating lime dissolution ates wit te aim of impoving upon te pedictions fo slag compositions. In te context of LD steelmaking, undestanding dissolution kinetics of lime is impeative fo seveal easons. Lime constitutes a significant pat of te total expenses fo aw mateials, and cutting down on lime consumption would consideably educe te oveall cost of steelmaking. A fa moe impotant eason is to get id of fee lime wic is a peennial poblem fo opeating steel plants. Fo conditions petinent to Tata Steel, te amount of fee lime in slags vaies in te ange fom 5 to 0 pct, [3] and tis limits its usage as a suitable building mateial. Anote key motivation to study lime dissolution is to poduce low-p steels wit low-basicity slags. LD sops in Tata Steel typically opeate in te basicity ange fom 3. to 3.8, and ecently, tee as been an inceasing dive to educe te basicity of LD slags witout compomising on tundown pospous. Pocess conditions, sequence of lime addition, lime paticle size, etc. may be contolled witin pemissible limits to acieve tis goal, but fist, te effects of tese paametes on tundown slag conditions must be undestood. A fundamental study on lime dissolution in steelmaking slags is necessay because it plays a vey impotant ole in acieving te desied slag composition at tundown. METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 47B, AUGUST 06 65

II. MODEL FORMULATION 4 Reaction between solid and (as SiO 4 ions) in slags esults in te fomation of calcium silicates, te expeimental evidence fo wic may be found in te studies of seveal eseaces. [4 7] Te type of silicates fomed depends on slag conditions (pimaily slag basicity and FeO content). In te cuent study, it is assumed tat te eaction between solid and in slags esults only in te fomation of solid Æ wic foms a laye aound te uneacted coe of. Unde suc conditions, fute eaction between and would equie te diffusion of fom te slag toug te solid Æ laye to te inteface between te uneacted and Æ. Te kinetic steps involved in te dissolution pocess may ten be summaized as:. Slag-film diffusion Tis step involves te diffusion of fom te bulk slag toug te slag-film bounday laye fomed aound solid Æ. Poduct-laye diffusion Tis involves te diffusion of toug te Æ laye fomed aound te uneacted coe of. 3. Intefacial eaction It involves te eaction between solid and at te inteface between uneacted and te solid Æ. A scematic epesentation of tese steps fo a speical paticle is sown in Figue. Expeimental study on lime dissolution in steelmaking slags ave been done by seveal eseaces. [4 0] Modeling studies on lime dissolution ave been elatively less but still available. [ 3] Wile seveal insigts about te dissolution of lime in slags ae available fom tese studies, none of tem povide a definite conclusion about te ate-contolling step. A majoity of te expeimental and modeling studies on cylindical lime paticles assumes diffusion toug te slag film as ate-contolling, efeing to te study by Matsusima et al. [4] Howeve, Matsusima et al. s [4] conclusion tat slag-film diffusion is ate-contolling is not cooboated by tei expeimental data. Tey agued tat an incease in dissolution ates wit te inceasing speeds of evolution (of te otating lime cylinde) confims tat slag-film diffusion is ate-contolling but even in case of mixed contol ates would incease wit incease in te speed of evolution. Futemoe, plots of f (X) vs t fo tei expeimental data do not confim tat slag-film diffusion is ate-contolling. Guo et al. s [0] expeimental esults fo dissolution of speical lime paticles in blast funace slags clealy sow tee egimes in wic tee diffeent mecanisms contol te ate. Fo ectangula specimens, Deng et al. s [7] expeiments indicate tat diffusion toug te poduct laye may be ate-contolling. Howeve, te numbe of data points epoted by Deng et al. [7] is too less to conclude anyting decisive about te ate-contolling step. In view of te above, te cuent autos debunk te ypotesis made in seveal pevious investigations (especially tose on cylindical lime specimens) tat slag-film diffusion contols te oveall kinetics of lime dissolution in steelmaking slags. It is fute agued tat in a eal steelmaking pocess wee slag conditions like basicity, FeO content, tempeatue etc. ae continually vaying duing te blow it seems unlikely tat a single mecanism will be ate-contolling tougout. Rate, a mixed contol model tat takes into account contibutions fom te all te tee (ypotetical) kinetic steps would be moe appopiate. Tus in te cuent study, mixed contol models ave been developed fo calculating dissolution ates of lime. A. Model Assumptions Apat fom te basic assumption about te fomation of solid Æ, te following ae te main assumptions of te model:. At te inteface between te uneacted lime and te solid Æ laye, equilibium conditions ae attained.. Steady-state conditions ae assumed to pevail, except in some cases wee a pseudo-steady-state appoximation as been made. 3. Diffusion of only is consideed. Diffusion of ote slag constituents like FeO, P O 5, etc. is not consideed. 4. Diffusion is pimaily consideed to be taking placing along one pincipal diection. 5. Diffeent geometies of lime specimen ave been consideed, but it as been assumed tat lime paticles of a paticula geomety etain tei oiginal sape. B. Govening Equations Te initial pat of te model development involves te deivation of ate equations fo eac of te tee ate contolling steps fo tee diffeent lime geometies. Afte tat ate equations fo mixed contol kinetics ave been deived fo eac of tese geometies.. Cylindical specimen a. Slag-film diffusion contol. Figue gives a scematic epesentation of a cylindical lime specimen in slag and te concentation pofiles fo slag-film diffusion contol. A sell mass balance fo te species ove a cylindical sell in te slag-film of adius, tickness D, and lengt L gives (wen D! 0): d d N ðþ 0: ½ Integating Eq. [] and using te elationsip between J ðþ and N ðþ as [4] J ðþ ð x SiO Þ N ðþ: ½ we get J ðþ ð x SiO Þ C : ½3 Using Fick s fist law, [8] te elation between x SiO and C SiO and substituting in Eq. [3], we get 65 VOLUME 47B, AUGUST 06 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig. Scematic epesentation of a paticle in slag and te ypotetical kinetic steps in te dissolution pocess. Fig. Sell mass balance fo a cylindical paticle in slag and scematic concentation pofiles fo slag-film diffusion contol. ðq s C SiO Þ dc C d : Integating Eq. [4] wit te bounday conditions C SiO C e at 0 C SiO C b at 0 þ d C ½4 ) : ½5 we obtain te concentation pofile fo as ln q s C e ln q! s C e ln q s C SiO q s C b 0 : ½6 ln 0 þd C 0 Te mola ate of diffusion W SiO ðþ is given as METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 47B, AUGUST 06 653

W SiO ðþ plq s D s ln q sc e q s C pl N SiO b SiO ðþ : ln 0 þd C 0 ½7 Since all te kinetic steps ae in seies, and using te stoiciomety of te eaction we get W SiO ð Þ ðsþþð Þ ðsþ W SiO ðþ dn dt Substituting W SiO ð in tems of we get d dt Þ dn dt : ½8 in Eq. [8] and ewiting n q s D s ln q sc e q s C b SiO ½9 q ln 0 þd C 0 Integating Eq. [9] witin pope limits and eplacing in tems of X we get 4q s D s ln q sc e q s C b SiO X t: ½0 q 0 ln 0 þd C 0 Equation [0] is te integated ate equation fo slag-film diffusion fo cylindical geomety. b. Poduct-laye diffusion contol. Figue 3 gives a scematic epesentation of te concentation pofile fo a cylindical lime specimen wen te diffusion toug te poduct (Æ (s)) laye contols te oveall ate. Poceeding in te same way as above and integating using te pseudo steady-state assumption [5] fo te bounday conditions, C SiO C e at C SiO C b at 0 : ½ Te concentation pofile fo C SiO is obtained as ln q s C e ln q! s C e ln q s C SiO q s C b : : ½ ln 0 Following te same pocedue, te diffeential equation fo te vaiation of wit t comes as q s D p ln q sc e ln 0 d q s C b SiO : ½3 dt q Te integated ate equation fo poduct-laye diffusion contol is ten obtained as X þ ð XÞlnð XÞ 8q s D p ln q sc e q s C b q 0 t: ½4 c. Intefacial eaction contol. Fo te eaction, ðsþþð Þ ðsþ assuming tat te concentation of solids emain uncanged, te ate of eaction of pe unit aea (v ) is given by v Cb k 0 b ðwee k0 f k fc and k 0 b k bc SiO Þ: ½5 At equilibium, v 0 wic gives k 0 b C e : ½6 Substituting tis in Eq. [5], we get i v C b C e : ½7 Substituting v wit n and ewiting n in tems of ; we get i d k 0 f C b C e : ½8 dt q Te integated ate equation fo intefacial eaction contol is ten obtained as i k 0 ð XÞ f C b C e q 0. Speical specimen t: ½9 a. Slag-film diffusion contol. Figue 4 gives a scematic epesentation of a speical lime specimen in slag and te concentation pofiles fo slag-film diffusion contol. A sell mass balance fo ove a speical sell of adius and tickness D yields (wen D! 0): d d N SiO ðþ 0: ½0 Integating Eq. [0] and following te same pocedue as fo a cylinde, we get ðq s C SiO Þ : dc C 3 d : ½ Integating Eq. [] wit te same bounday conditions as Eq. [5], we get ln q s C e ln q! s C e q s C SiO q s C b 0 ln@ 0 0 0 þd C A: ½ 654 VOLUME 47B, AUGUST 06 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig. 3 Sell mass balance fo a cylindical paticle in slag and scematic concentation pofiles fo poduct-laye diffusion contol. Fig. 4 Sell mass balance fo a speical paticle in slag and scematic concentation pofiles fo slag-film diffusion contol. W SiO ðþ 4pq s D s ln q sc e 4p q s C N SiO b SiO ðþ : ½3 0 0 þd C Te equation fo te vaiation of wit t as is obtained using te same metod as d dt q s D s ln q sc e q s C b SiO : ½4 q 0 0 þd C Te integated fom of ate equation fo slag-film diffusion contol is ten obtained as METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 47B, AUGUST 06 655

X 6q s D s ln q sc e q s C b SiO t: ½5 0 3 q 0 0 þd C b. Poduct-laye diffusion contol. Figue 5 gives a scematic epesentation of te concentation pofile fo a speical lime specimen wen te diffusion toug te poduct (Æ ) laye is ate contolling. Pefoming a sell mass balance in te poduct laye and integating wit te bounday conditions of Eq. [], te concentation pofile fo C SiO is obtained as ln q s C e ln q! s C e q s C SiO q s C b 0 ln@ 0 A: ½6 Te diffeential equation fo te vaiation of wit t and te integated ate equation ae ten obtained, espectively, as q d s D s ln q sc e q s C b SiO : ½7 dt q 0 3 X ð X 4q s D s ln Þ 3 q s C e q s C b q 0 t: ½8 c. Intefacial eaction contol. To obtain te ate equations fo intefacial eaction contol, we make use of Eq. [7] and ten eplace v in tems of fo a speical geomety. Using te same metod as befoe, te diffeential fom of ate equation obtained is te same as Eq. [8], and te integated ate equation is deived as i k 0 ð XÞ f C b C e 3 q 0 3. Rectangula specimen t: ½9 a. Slag-film diffusion contol. Consideing a ectangula sell aving dimensions Dl, w and b (Figue 6) and a pefoming sell mass balance fo te species ove tis ectangula sell, we get d N SiO ðlþ 0: ½30 dl Te diffeential equation fo te vaiation of C SiO wit l is ten obtained as ðq s C SiO Þ dc C 6 : dl ½3 Integating Eq. [3] wit te bounday conditions ) C SiO C e at l l 0 C SiO C b at l l 0 þ d : ½3 C Te concentation pofile fo C SiO is obtained as ln q s C e ln q! s C e l l 0 q s C SiO q s C b : ½33 d C Te mola ate of diffusion at is given by W SiO ðlþ l wbn SiO ðlþ l wbq sd s ln q! s C e d C q s C b : ½34 Te diffeential fom of te ate equation is obtained likewise as dl dt q s D s ln q sc e q s C b ; ½35 q d C and te integated ate equation fo slag-film diffusion contol is given by q s D s ln q sc e q s C b SiO X q d C l 0 t: ½36 b. Poduct-laye diffusion contol. Consideing a sell aving dimensions Dl, w and b in te poduct laye (Figue 7) and following te same pocedue as above, an equation simila to Eq. [3] is obtained. Now integating wit te bounday conditions C SiO C e at l l C SiO C b at l l 0 : ½37 Te concentation pofile fo is obtained as ln q s C e ln q! s C e l l q s C SiO q s C b l 0 l : ½38 Following te same pocedue as befoe, te diffeential and integated foms of ate equations ae obtained, espectively, as q l 0 l dl s D p ln q sc e q s C b SiO : ½39 dt q X 4q s D p ln q sc e q s C b q l 0 t: ½40 656 VOLUME 47B, AUGUST 06 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig. 5 Sell mass balance fo a speical paticle in slag and scematic concentation pofiles fo poduct-laye diffusion contol. Fig. 6 Sell mass balance fo a ectangula paticle in slag and scematic concentation pofiles fo slag-film diffusion contol. c. Intefacial contol. Using Eq. [7] and substituting v in tems of w, b, l fo ectangula geomety, diffeential and integated ate laws fo intefacial eaction contol ae obtained, espectively, as i dl dt X C b C e : ½4 q i C b C e q l 0 t: ½4 A summay of integated ate equations fo diffeent geometies of lime specimen and fo diffeent contolling mecanisms is given in Table I. 4. Mixed contol model fo lime dissolution a. Cylindical specimen. Using te diffeential and integated foms of ate equations fo te tee diffeent contols and following te geneal pocedue used fo deiving mixed contol equations, [6] te diffeential and integated ate equations fo mixed contol kinetics ae obtained, espectively, as METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 47B, AUGUST 06 657

0 d ln 0 þd C @ 0 þ dt q s D s þ ia C p C e ln 0 q s D p þ ln q sc e q s C b q : ½43 0 0 ln 0 þd C @ 0 X þ 0 ðx þ ð XÞlnð XÞÞ 4q s D s 8q s D p þ ð þ ln q sc e XÞ q s C ia b SiO C p C e q 0 t: ½44 SiO b. Speical specimen. Diffeential and integated ate laws fo a speical lime specimen ae obtained likewise as 0 d 0 0 þd C @ þ 0 dt q s D s q s D p þ ln q ½45 sc e q s C þ ia b SiO : C p C e q d C X þ 0 3 3 ð XÞ 3 X 6q s D s q s D p X þ ð þ ln q sc e Þ 3 q s C ia b SiO C p C e q 0 t: SiO ½46 c. Rectangula specimen. Diffeential and integated ate laws fo a cube lime specimen ae obtained likewise as 0 dl d C þ l0 l @ þ ia dt q s D s q s D p C p C e þ ln q sc e q s C b SiO : ½47 q 0 @ d C X þ l0 X X þ ia q s D s 4q s D p C p C e þ ln q sc e q s C b SiO q l 0 t: ½48 5. Calculation of concentation bounday laye tickness (d C ) Rate equations developed in te peceding sections necessitate te estimation of d C. Kosaka et al. [7] empiically expessed d C as a function of slag popeties, and te speed of evolution fo te dissolution of a otating steel cylinde into liquid zinc o aluminum as d C 4:76 d :5 Re0:6 Sc 0:35 : ½49 In te absence of any suitable coelation fo calculating d C ; in te case of lime dissolution in steelmaking slags, Eq. [49] is used to estimate d C. A coection facto a is ten used so tat easonable ageement between te model pedictions and expeimental esults is acieved. 6. Tempeatue dependence of diffusivity (D s ) and viscosity (g) In te poposed model, effects of tempeatue on D s (and D p ) and g ae modeled by assuming tat tey exibit an Aenius type of elationsip wit tempeatue, espectively, as follows: D s D 0 exp E S RT : ½50 g g 0 exp E g : ½5 RT 7. Estimation of activity coefficient of (c SiO ) To estimate c SiO as a function of slag composition and tempeatue, a lage numbe of temodynamic calculations ae caied out using Equilib module of FactSage Vesion 6.4 [8] at vaious slag compositions and tempeatues (elevant to steelmaking opeations). Finally, te temodynamic data ave been cuve-fitted to obtain an equation fo pedicting c SiO as a function of slag composition and tempeatue. In te anges studied, analyses yield log c SiO :44 4:059x þ 0:8373x SiO þ 4:5569x FeO 543:4 ðr 0:99Þ: T ½5 C. Solution Stategy Computational study fo te pesent model as been caied out using MATLAB Vesion R009a.Algebaic and tanscendental equations ae solved using te Newton-Rapson metod. An unde-elaxation paamete (b) is used weeve necessay, and its value as been cosen by eacing a compomise between accuacy in calculations and computational speed. Fo validating te model wit expeimental data of ote eseaces, [4,7,0] ate equations fo diffeent geometies ae solved wit input conditions simila to tose mentioned in te elevant liteatue. Wen tis model is applied as a submodel to te global model fo 658 VOLUME 47B, AUGUST 06 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig. 7 Sell mass balance fo a ectangula paticle in slag and scematic concentation pofiles fo poduct-laye diffusion contol. Table I. Summay of Integated Foms of Rate Equations fo Diffeent Geometies and Rate-Contolling Mecanisms Slag-Film Diffusion Poduct-Laye Diffusion Intefacial Reaction Cylindical X k t Xþ ð XÞlnð XÞ k t ð XÞ k3 t Speical X k 4 t 3 X X 3 k5 t ð XÞ 3 k6 t Rectangula X k 7 t X k 8 t X k 9 t LD steelmaking, te petinent equations ae solved wit input conditions aleady publised in te pevious study [] and ence not epeated ee. Only, te list of new model vaiables used in te submodel fo lime dissolution is mentioned in Table II. A new lance eigt pactice is applied in te cuent study fo bette ageement wit plant measuements [] and as also been mentioned in Table II. Te temodynamic data equied fo model calculations ae obtained fom one of te standad data-souces. [9] Diffusivity of silica in slags is obtained fom typical values epoted by Dolan et al. [0] and te values of E S and E g ae obtained fom Matsusima et al. [4] In te absence of expeimental data on D p, simulation studies ae caied out fo diffeent values of u (te atio between D p and D s ). A value of u 0:5 gives a easonably good ageement between model pedictions and te obseved values. III. RESULTS AND DISCUSSIONS In tis section, te esults of te solutions of ate equations fo mixed contol (Eqs. [43] toug [48]) ae discussed and compaed wit expeimental data publised elsewee. [4,7,0] In addition, te effects of pocess vaiables on lime dissolution ate ave been identified and also compaed wit expeimental esults, watsoeve available. [4,9] Finally, tis model is incopoated as a submodel in te global model [] fo calculating lime dissolution ates duing actual blowing conditions. Wit te lime dissolution submodel incopoated, te pedictions fo slag compositions ave been discussed, and impovements vis-a-vis te pevious model ave been analyzed. A. Dissolution Rates fo Diffeent Lime Geometies. Cylindical specimen To calculate te ate of dissolution fo a cylindical lime specimen, Eq. [44] is solved to obtain te factional convesion of lime (X) at vaious times. Te esults ae plotted in Figue 8 and compaed wit te expeimental esults publised by Matsusima et al. [4] fo diffeent speeds of evolution of te otating lime specimen. As evident fom Figue 8, model pedictions fo sow good ageement wit te expeimental values. Te model pedicts te igt tend in Xvstplots fo all diffeent speeds of evolutions, and te ageement wit expeimental data is inceasingly pominent fo ige speeds of evolution. At 300 and 400 pm, model pedictions matc excellently wit te expeimental esults. At 00 and 00 pm, pedictions ae elatively poo altoug te model still pedicts te igt tends. A possible eason fo te elatively poo ageement at low pms (especially fo pm = 00) migt be te eo involved in calculating d C using Eq. [49]. At low pms, values of d C ae expected to be vey ig esulting in muc slowe ates of diffusion toug te slag-film. Unde suc conditions slag-film diffusion may be ate contolling. Howeve, wen d C is calculated using Eq. [49], it does not become sufficiently lage at low METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 47B, AUGUST 06 659

Table II. List of Input Paametes used in te Lime Dissolution Submodel [4,0] Quantity Unit Value Lance eigt pactice (as input to global model) m.6-.45-.30-.05-.75-.5 Initial adius of cylindical lime paticle m 0.05 Lengt of cylindical lime paticle m 0.03 Modified fowad eaction ate constant fo te eaction of m/s.0 9 0 4 wit at -Æ inteface Activation enegy fo diffusion in slags [4] KJ 94 Activation enegy fo viscous flow of slags [4] KJ 0 Diffusivity of in slag at 773 K (500 C) [0] m /s.0 9 0 9 Numbe of evolutions of te otating lime cylindes (used fo min 400 calculating bounday laye tickness) Ratio of te diffusivities of toug Æ and slag 0.5 Unde elaxation paamete 0.0 Coection facto fo bounday laye calculation 5 pms, and ence te model ove-pedicts te values of X at lowe speeds of evolutions.. Speical specimen Rates of dissolution fo a speical lime specimen ave been calculated by solving Eq. [46] to obtain te factional convesion of lime (X) at vaious times. Te esults ae plotted in Figue 9 and compaed wit te expeimental data epoted by Guo et al. [0] An analysis of te expeimental data publised by Guo et al. [0] clealy sows tee distinct egimes in Xvstplot. Te autos ague tat te fomation of multisolid laye (assumed to Æ in te cuent study) stats afte some peiod, eaces a paticula tickness and ten te laye gets completely dissolved. Suc penomena, oweve, ae not consideed in te model (wic assumes tat a laye of Æ is pesent always) and tus pedictions do not necessay cooboate wit te expeimental esults, especially duing te initial egime. Pedicted tend of Xvstplot sows only two distinct egimes: an initial peiod wen d C is ig (since is lage) and slag-film diffusion is ate-contolling; and a final egime wen d C is low (because of smalle )and a mixed contol mecanism of slag-film diffusion and poduct-laye diffusion goven te ate. In te late stages, tee is a good coespondence between pedicted and expeimental values, altoug pedictions fo te initial peiods ave a geate deviation. 3. Rectangula specimen Fo ectangula specimen Eq. [48] is solved to obtain X at vaious times (t). Results ave been compaed wit te expeimental data on lime dissolution fo cubic specimens by Deng et al. [7] and plotted in Figue 0. As evident fom te model esults fo Xvstplot, a mixed contol mecanism of slag-film and poduct-laye diffusion detemines te ate of dissolution. Expeimental esults also point towad suc a mecanism, and a faily good ageement is acieved between te model esults and te expeimental values duing a majo pat of te time-peiod. Only towad te vey late stages, te model pedictions sow some deviation fom te expeimental data. A possible souce of eo, as aleady explained, may be te inaccuacy in calculating d C. Also, te pesent model assumes diffusion to be occuing only along one diection. Fo a ectangula geomety, suc an assumption olds tue only if eac of te ote two diections is muc lage tan te diection of diffusion. Howeve, Deng et al s. [7] expeimental data ae fo cubic specimens wee all dimensions ae equal and diffusion is expected to take place along all te tee diections. Tus evidently, tis appoximation migt ave esulted in some eos in calculation. B. Effects of Pocess Vaiables on Dissolution Rate. Tempeatue Effects of tempeatue on dissolution ate ave been modeled by taking into account its effect on te model vaiables. C e (a SiO and c SiO ),,q s,q,d s (and also D p ), g (and ence d C ) ae te model vaiables wic vay wit tempeatue. In te ange of tempeatues consideed, q s and q do not pactically vay wit tempeatue, and te effect of tempeatue on ; altoug significant, is of no pactical impotance since intefacial eaction does not ave muc effect on te oveall ate. Bot a SiO and c SiO incease wit tempeatue tus counteacting te effect of tempeatue on C e (wic depends on te atio between a SiO and c SiO ). Still C e maginally inceases wit tempeatue but tat pactically as no effect on dissolution ate wic depends on te concentation diffeence (q s C e ) and q s is at least two odes of magnitude ige tan C e. Tus D s (and also D p ) and g (and ence d C ) ae te only vaiables wic vay significantly wit tempeatue. Fom Eq. [49], we get k m / D 0:7 s g 0:3 : ½53 As evident fom Eqs. [50] and [5], D s exponentially inceases and g exponentially deceases wit tempeatue, bot of wic incease k m. D p (assumed to be 0:5D s ) also inceases exponentially wit tempeatue, and ence lime dissolution ates incease significantly wit tempeatue. In fact, lime dissolution ate also follow an Aenius elationsip wit tempeatue as illustated in Figue. Fo an incease in tempeatue by 50 K (50 C) fom 773 K to 83 K (500 C to 550 C) te dissolution ate becomes almost twice its 660 VOLUME 47B, AUGUST 06 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig. 8 Model pedictions of factional convesion of (X) fo a otating cylindical specimen compaed wit te expeimental data of Matsusima et al.: [4] (a) 00 pm, (b) 00 pm, (c) 300 pm, and (d) 400 pm. Fig. 9 Model pedictions of factional convesion of (X) fo a speical specimen compaed wit te expeimental data of Guo et al. [0] value at 773 K (500 C). Fo validation, model esults ae compaed wit expeimental data epoted by Matsusima et al. [4] and a faily good ageement between te two is obtained as evident fom Figue.. Slag composition Effects of slag composition on lime dissolution ates ae modeled by consideing its effects on c SiO and g. Te effects of slag composition on c SiO ae studied using Eq. [5] wile tose on g ae investigated using te Fig. 0 Model pedictions of factional convesion of (XÞ fo a ectangula specimen compaed wit te expeimental data of Deng et al. [7] Viscosity module of FactSage vesion 6.4. [8] Te esults ae analyzed by consideing te effects of te two most impotant paametes tat define te composition of LD slags: 3. Slag basicity Fo fixed FeO content, te effects of slag basicity on dissolution ates ae plotted in Figue and compaed wit te expeimental investigations of Matsusima et al. [4] and Hamano et al. [9] As evident fom Eq. [5], METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 47B, AUGUST 06 66

Fig. Model pedictions of te effects of tempeatue on dissolution ate of compaed wit te expeimental data of Matsusima et al. [4] Fig. Model pedictions of te effects of slag basicity on dissolution ate of compaed wit te expeimental data of Matsusima et al. [4] and Hamano et al. [9] c SiO deceases wit inceasing basicity, and ence C e inceases. Tis, oweve, does not ave a significant effect on te ate of dissolution fo easons peviously discussed. Slag basicity as a muc moe ponounced effect on g. In te basicity ange consideed (0.5 to 3.0), as te slag basicity inceases, g initially inceases, and ten afte a limiting basicity is eaced, g emains pactically uncanged wit basicity. Vaiation of k m wit basicity follows te evese tend, and tus wit inceasing basicity, dissolution ates fist deceases, and ten emains pactically constant. Model pedictions coespond well wit te expeimental esults of bot sets of investigations at low values of basicity (0.5 to.0). [4,9] At ige basicities, model esults cannot be validated wit expeimental esults since dissolution ates at ige basicities ae not epoted. [4,9] 4. Slag FeO content Figue 3 sows te effects of FeO content of slags on dissolution ates at constant basicity. As in te pevious case, model pedictions ae compaed wit te expeimental data publised by Matsuima et al. [4] and Hamano et al. [9] Wit te inceasing FeO content of slags, c SiO inceases, and C e deceases. Tis, oweve, as negligible effects on te ate of dissolution as peviously discussed. FeO content as a muc moe significant effect on g. In te ange cuently consideed (0 to 60 pct FeO), g geatly deceases wit incease in slag FeO content but afte about 50 pct FeO, g emains pactically uncanged wit fute incease in FeO. Tus k m inceases wit te inceasing FeO content till some limiting value (at aound 50 pct) ae eaced afte wic it faily emains constant. Dissolution ates follow a simila tend, as illustated in Figue 3. Expeimental esults publised by Matsusima et al. [4] and Hamano et al. [9] indicate simila tends in te vaiation of dissolution ates wit vaiation in FeO content of slags. In te case of Matsusima et al., [4] model pedictions agee vey well wit te expeimental esults in te ange 0 to 40 pct FeO. In Hamano et al. s [9] case, oweve, te expeimental studies epot a muc geate incease in ates wit te inceasing FeO in te ange 40 to 55 pct FeO, tus indicating tat model pedictions ae not vey accuate at ige FeO anges (>40 pct). Eo involved in calculating g toug FactSage calculations at ige FeO contents migt be a possible eason fo suc deviation. Fo all pactical puposes, oweve, dissolution ates at FeO content>40 pct is unimpotant because FeO content in eal steelmaking slags adly becomes >40 pct. Neveteless, fute investigations on te easons beind suc deviations ae necessay. a. Lime paticle size. To examine te effect of lime paticle size on dissolution ate, te time fo complete convesion of lime (s) is calculated fo tee diffeent geometies cuently consideed (temed s c, s s and s fo cylindical, speical, and ectangula specimens, espectively). Putting X in Eqs. [44], [46], and [48], we get 0 0 q 0 ln 0 þd C 0 B 4q s D s þ 0 8q s D p þ ic @ A s c s s s 0 q 0 @ 0 q l 0 @ þ ln q sc e q s C b C p C e : d C 6q s D s þ 0 q s D p þ k 0 C p C e f þ ln q sc e q s C b ia ½54 : ½55 d C q s D s þ l0 4q s D p þ k 0 C p C e f þ ln q sc e q s C b ia : ½56 66 VOLUME 47B, AUGUST 06 METALLURGICAL AND MATERIALS TRANSACTIONS B

Fig. 3 Model pedictions of te effects of slag FeO content on dissolution ate of compaed wit te expeimental data of Matsusima et al. [4] and Hamano et al. [9] Fig. 5 Model pedictions of vaiation in dissolution ate of duing blowing in a 60-ton LD convete. steelmaking, [] and in tis section, te model pedictions fo vaiation in lime dissolution ates (calculated using te lime dissolution submodel) ave been discussed. Finally, impovements in slag compositions (in tems of ow close tey appoac eal-time values) wit espect to te pevious model ave been analyzed. Fig. 4 Effect of paticle size on te time equied fo complete convesion of (s) fo diffeent geometies. Figue 4 illustates te effect of lime paticle size on s fo te tee geometies. Fo all tee geometies, s vaies as te squae of te lime paticle size. Tis is also evident fom Eqs. [54] toug [56] and is caacteistic of mixed contol kinetics. It is also to be noted fom Figue 4 tat fo te same paticle size, s s is te least, and s is maximum. s c is only sligtly ige tan s s and bot s c and s s ae muc smalle tan s. An explanation fo tis can be sougt fom Eqs. [54] toug [56]. Wit te substitution of l 0 0 and assuming tat te contibutions fom intefacial eaction contol is negligible, it can be seen fom Eqs. [55] and [56] tat s is appoximately twelve times s s. Using a simila agument, s can be found to be appoximately eigt times s c. C. Application to a Real Steelmaking Pocess In te peceding sections, a mixed contol model fo calculating te dissolution ate of lime in steelmaking slags as been developed. Also, te effects of pocess vaiables on dissolution ates ave been examined. Tis model is now incopoated in te global model fo LD. Vaiation in lime dissolution ate duing te blow Figue 5 sows te vaiation in lime dissolution ate duing blowing in a 60-ton LD convete. Fo a fixed paticle size of te lime paticle, dissolution ate depends on tempeatue, slag basicity, and FeO content. As aleady discussed, dissolution ates inceases wit te inceasing tempeatue and FeO content and deceases wit te inceasing basicity. Fo te fist to minutes of te blow, FeO content is vey ig (about 40 pct) and basicity is low (in te ange. to.). Tese factos esult in ig dissolution ates fo te fist to minutes even toug te slag tempeatue is not ig. As te blow poceeds, FeO content in slag stats deceasing and basicity inceases. Bot tese factos esult in a decease in dissolution ate wic eaces a minimum afte about 5 minutes. Afte tis, slag FeO content emains elatively constant fo te majo pat of te blow. Bot basicity and tempeatue, oweve, inceases continually as te blow poceeds. Tese factos ave counteacting effects on dissolution ates. As a esult, lime dissolution ates emains faily constant afte about 5 minutes tougout te majo pat of te emaining blowing peiod. Sligt incease in lime dissolution ate can be obseved in Figue 5 because tempeatue as a muc geate effect on dissolution ates as compaed to slag basicity. Towad te vey end of te blow, (afte about 4 minutes fom te stat) FeO content stats inceasing significantly. Tempeatues ae also vey ig and slag basicity sligtly deceases due to te dilution effect of vey ig FeO content. All tese factos favo te dissolution of lime in slag. Tus dissolution ates saply inceases in tis peiod (4 to 6 minutes).. Vaiation in slag compositions duing te blow As peviously discussed, te pevious model [] aleady pedicted te vaiations in slag compositions duing METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 47B, AUGUST 06 663

Fig. 6 Models pedictions of vaiations in slag compositions duing blowing in a 60-ton LD convete wit and witout te dissolution model: (a) and (b) FeO. blowing in a 60-ton LD convete. Tis model aimed at impoving te model pedictions of slag compositions by incopoating a lime dissolution submodel into te global model. Figues 6(a) and (b) sow te vaiations in slag compositions duing te blow wit and witout te lime dissolution model. As evident fom Figues 6(a) and (b), wit te lime dissolution submodel, te global model pedictions coespond muc bette wit actual plant measuements. [] Towads te beginning of te blow dissolution ate of is ig, as peviously discussed. Howeve, as te blow poceeds, it stats to decease. Consequently, lesse amount of goes into te slag, and te oveestimation in pecent is significantly educed. Since pecent FeO in slag depends on te dilution of slag by, te undeestimation in FeO pecent is also eliminated to a lage extent. Simila to pedictions of te pevious model, pecent eaces a maximum, and pecent FeO eaces a minimum afte about 5 minutes fom te stat of te blow. Howeve, wit te lime dissolution model incopoated, te maximum in pecent is eaced at a muc lowe value, and ence te minimum in pecent FeO is eaced at a muc ige value. Aftewad, te pecent compositions of and FeO emain pactically constant, and tis tend is obseved in model pedictions bot wit and witout te lime dissolution model as well as in plant measuements by Cicutti et al. [] Towad te vey end of te blow, (in te peiod 4 to 6 minutes afte te blow-stat), tee is not muc impovement in model pedictions wit te incopoation of te lime dissolution model. Tis deviation, oweve, is attibuted moe to a demeit ineent to te global model. In te end-blow egime, te amount of ot metal in te emulsion becomes too low, and ence, te consumption of FeO due to eactions is vey less, tus esulting in vey ig FeO content in te slag. [] Fute efinement wit egad to tis aspect may be necessay to obtain ealistic pedictions duing te final stages of te blow. Vey ig ates of lime dissolution in tese stages may compensate fo te ig FeO content tus binging down te pecent FeO values by dilution effects. Suc ig ates of dissolution, oweve, ae not obtained using te pesent submodel even duing te vey late stages of te blow. IV. CONCLUSION Te cuent study was aimed to develop a model fo dissolution of lime in steelmaking slags based on te assumption tat eacts wit in slags to fom solid Æ. A tee-step kinetic pocess fo te dissolution of lime as been ypotesized. Citically analyzing expeimental data publised by ote eseaces, it as been agued tat te obseved kinetics cannot be suitably explained if any one of te tee (ypotetical) eaction steps is ate contolling. Tus, mixed contol models ave been poposed in te cuent study fo calculating lime dissolution ates. Integated ate equations fo eac of te tee ate-contolling mecanisms ae fist obtained, and ten te ate laws fo mixed contol kinetics ave been deived. Poposed mixed contol models fo diffeent lime geometies ae fist validated using expeimental data publised elsewee. Ten, te effects of tempeatue, slag basicity, slag FeO content, and lime paticle size on dissolution ates ave been analyzed and validated wit expeimental esults of ote eseaces. Finally, tis model as been incopoated in te aleady existing global model fo LD steelmaking. Incopoation of te submodel into te global model enabled te calculation of lime dissolution ates dynamically duing blowing in a 60-ton LD steelmaking convete. Moe impotantly, consideable impovements in pedictions of slag compositions ave been obseved. Wit te inclusion of te lime-dissolution submodel, pedictions fo pecent and pecent FeO cooboate muc bette wit te industial esults fo a majo pat of te blow. In te end-blow egime, not muc impovement in model pedictions can be acieved, tus indicating tat fute efinement of te model may be equied. 664 VOLUME 47B, AUGUST 06 METALLURGICAL AND MATERIALS TRANSACTIONS B

NOMENCLATURES b Beadt of ectangula specimen (m) d Diamete of speical/cylindical specimen at any time t (m) k 0 b Modified backwad eaction ate constant fo te eaction ðsþþð Þ ðsþ (m/s) Modified fowad eaction ate constant fo te eaction ðsþþð Þ ðsþ (m/s) k m Mass-tansfe co-efficient of in slag (m/s) l Lengt of ectangula specimen at any time t (m) l 0 Initial lengt of ectangula specimen (m) Radius of speical/cylindical specimen at time t (m) 0 Initial adius of paticle (fo speical/cylindical specimen) (m) u Linea velocity of paticle (fo otating specimen) (m/s) v Mola ate of eaction of as pe te eaction ðsþþð Þ ðsþ (mol/ m s) w Widt of ectangula specimen (m) x j Mole-faction of component j in slag (-) CSiO b Mola concentation of in te bulk slag (mol/m 3 ) CSiO e Mola concentation of at te inteface between and Æ (mol/m 3 ) C p Mola concentation of at te inteface between Æ and slag-film (mol/ m 3 ) C SiO Mola concentation of (mol/m 3 ) D p Diffusivity of toug Æ (s) laye (m /s) D s Diffusivity of in slag (m /s) D 0 Pe-exponential facto in te Aenius elation fo D s (m /s) E g Activation enegy fo viscous flow of slag (J/mol) E S Activation enegy fo diffusion of in slag (J/mol) JSiO ðþ i Molecula flux of in te i diection (mol/ m s) L Lengt of cylindical paticle (m) N SiO ðþ i Combined mola flux of in te i diection (mol/m s) R Univesal gas constant (J/mol K) Re Reynolds numbe ( ) Sc Scidmt numbe ( ) T Tempeatue (K) W SiO ðþ i Mola ate of diffusion of in te i diection (mol/s) X Factional convesion of ( ) GREEK SYMBOLS a Coection facto fo bounday laye calculation ( ) b Unde-elaxation paamete ( ) c SiO Activity co-efficient of in slag wit espect to pue (l) at te same tempeatue ( ) d C Concentation bounday laye tickness in slag (m) g Viscosity of slag(pa s) g 0 Pe-exponential facto in te Aenius elation fo g (Pa s) q Mola density of (mol/m 3 ) q s Mola density of slag (mol/ m 3 ) s Time equied fo complete convesion of (s) u Ratio of te diffusivities of toug CaoÆ and toug te slag ( ) REFERENCES. R. Saka, P. Gupta, S. Basu, and N.B. Ballal: Metall. Mate Tans. B, 05, vol. 46B, pp. 96 76.. C. Cicutti, M. Valdez, T. Pe ez, J. Petoni, A. Go mez, R. Donayo, and L. Feo: Poc. 6 t Intenational Confeence on Molten Slags, Fluxes and Salts, Stockolm City, 000, p. 367. 3. S.K. Couday, S.N. Lenka, and A. Gos: Ionmak. Steelmak., 007, vol. 37 (4), pp. 343 49. 4. M. Matsusima, S. Yadoomau, K. Moi, and Y. Kawai: Tans. Ion Steel Inst. Jpn., 977, vol. 7, pp. 44 49. 5. T. Hamano, S. Fukagai, and F. Tsukiasi: Ion Steel Inst. Jpn. Int., 006, vol. 46 (4), pp. 490 95. 6. S. Amini, M. Bungs, and O. Ostovski: Ion Steel Inst. Jpn. Int., 007, vol. 47 (), pp. 3 37. 7. T. Deng and D. Sicen: Metall. Mate Tans. B, 0, vol. 43B, pp. 578 86. 8. Z.S. Li, M. Witwood, S. Millman, and J. van Boggelen: Ionmak. Steelmak., 04, vol. 4 (), pp. 0. 9. T. Hamano, M. Hoibe, and K. Ito: Ion Steel Inst. Jpn. Int., 004, vol. 44 (), pp. 63 67. 0. M. Guo, Z. Sun, X. Guo and B. Blanpain: Poc. 03 Int. Symp. on Liquid Metal Pocessing and Casting, TMS, Austin, TX. 03, pp. 0 08.. S.Y. Kitamua, H. Sibata, and N. Mauoka: Steel Res. Int., 008, vol. 79 (8), pp. 586 90.. A.K. Sukla, B. DeO, S. Millman, B. Snoeije, A. Ovebosc, and A. Kapilasami: Steel Res. Int., 00, vol. 8 (), pp. 940 48. 3. N. Dogan, G.A. Books, and M.A. Ramdani: Ion Steel Inst. Jpn. Int., 009, vol. 49 (0), pp. 474 8. 4. R.B. Bid, W.E. Stewat, and E.N. Ligtfoot: Tanspot Penomena, nd ed., Wiley, New Yok, 00, pp. 536 37. 5. O. Levenspiel: Cemical Reaction Engineeing, 3d ed., Wiley, New Yok, 999, p. 48. 6. A. Gos and S. Gos: A Textbook of Metallugical Kinetics, PHI Leaning Pvt. Ltd., Deli, 04, pp. 66 68. 7. M. Kosaka and S. Minowa: Tetsu-to-Hagane, 965, vol. 5 (), pp. 8 5. 8. FactSage: Vesion 6.4, Database FToxid, Temfact and GTT Tecnologies, 03. 9. O. Kubascewski and C.B. Alcock: Metallugical Temocemisty, 5t ed., Pemagon Pess, Oxfod, 979, pp. 37 4. 0. M.D. Dolan and R.F. Jonston: Metall. Mate Tans. B, 004, vol. 35B, pp. 675 84. METALLURGICAL AND MATERIALS TRANSACTIONS B VOLUME 47B, AUGUST 06 665