KEY WORDS: surface tension; prediction; surface thermodynamic model; Mukai-value; bubble entrapment.

Transcription

1 ISIJ Internatonal, Vol. 53 (013), No. 1, pp Correspondence between Surface Tenson Estmated by a Surface Thermodynamc Model and Number of Bubbles n the Vcnty of the Surface of Steel Products n Contnuous Castng Process Tash MATSUSHITA, 1) * Kusuhro MUKAI ) and Masafum ZEZE 3) 1) Department of Materals and Manufacturng-Castng, School of Engneerng, Jönköpng Unversty, P.O. Box 106, SE Jönköpng, Sweden. ) Professor emertus, Department of Materals Scence and Engneerng, Kyushu Insttute of Technology, 1-1, Sensu-cho, Tobata, Ktakyushu, Japan. 3) Yawata R&D Laboratory, Nppon Steel Corporaton, 1-1, Tobhata-cho, Tobata, Ktakyushu, Japan. Now at Nppon Steel & Sumtomo Metal Corporaton. (Receved on July 1, 01; accepted on September 0, 01) Surface tensons of low carbon slabs and 16 mass%cr stanless steel were estmated usng a surface thermodynamc model proposed by Muka et al. As an applcaton of the model, an ndex to evaluate the drvng force for the fne bubble entrapment by the soldfyng shell, the Muka-value, M, was calculated from the surface tenson values. The relatonshp between Muka-value and number of entrapped bubbles was dscussed. A lnear relatonshp was found between the number of captured bubbles and Mukavalue. In the prevous work, the Muka-value was used as a relatve scale to evaluate the drvng force for the movement of bubbles. However, by calculatng the M from the surface tenson values by the surface thermodynamc model, physcally reasonable Muka-values could be obtaned. KEY WORDS: surface tenson; predcton; surface thermodynamc model; Muka-value; bubble entrapment. 1. Introducton * Correspondng author: E-mal: tash.matsushta@th.h.se DOI: In the steelmakng process, surface tenson of multcomponent dlute soluton system s regarded as sgnfcant. However, to the knowledge of the present authors, effcent and reasonable methods to estmate or predct the surface tenson have not been establshed so far. Hence, the measurements have to be done n each case. A goal of the present paper s to predct the surface tenson of multcomponent dlute metallc solutons. In ths paper, an attempt was made to predct the surface tenson of low carbon steel and 16 mass%cr stanless steel (16Cr SUS), by a recently developed surface tenson model by some of the present authors. 1) The surface tenson of multcomponent system has been predcted usng parameters derved from surface tenson values n lterature. Another goal of the present paper s to suggest a gudelne to suppress the entrapment of the fne gas bubbles and/or nonmetallc nclusons by the soldfyng shell durng contnuous castng. The entrapment of the fne partcles at soldfyng nterface s a serous problem snce t becomes a cause of defects. Such defects wll lower the mechancal propertes and t wll become surface defects after the machnng. Hence, t s desred to suppress the entrapment of such fne partcles to mprove the qualty of steel product. A mechansm of the fne partcle entrapment was proposed by some of the present authors.,3) In the prevous works, t has become apparent that fne partcles n the molten steel such as bubble are entrapped by soldfyng nterface durng the soldfcaton and remaned n the fnal products. It s concluded that the surface tenson gradent n the vcnty of soldfcaton nterface s a controllng factor of the bubble entrapment. In the present study, the relatonshp between the number of entrapped bubbles durng the castng process and an ndex to evaluate the drvng force of the entrapment, Muka-value, M, was nvestgated. Feasblty of the suppresson of the fne partcle entrapment at soldfyng nterface was dscussed based on the above-mentoned mechansm Predcton of Surface Tenson of Multcomponent Dlute Soluton (Surface Thermodynamc Model) 1) In the present work, attempt was made to estmate the surface tenson of commercal steels based on a surface thermodynamc model proposed by Muka et al. 1) The calculated surface tenson values are used to calculate the Muka-value whch s mentoned n the next secton. An outlne of the surface thermodynamc model for the estmaton of surface tenson s brefly summarzed below: A thermodynamc formula has been derved to descrbe (or predct) the surface tenson of bnary and mult-component systems by some of the present authors. 1) The startng pont of the present model for the estmaton of surface tenson s the total Helmholtz energy of the system, F l l FTV (,, An, ) = PV σa μ n = 1... (1) where P l s the pressure of lqud phase, V l s the volume of r 013 ISIJ 18

2 ISIJ Internatonal, Vol. 53 (013), No. 1 lqud phase, σ s the surface tenson, μ s the chemcal potental of component, n s the number of moles of component n the system, vz., n = n n ( n b s b : the number of s moles of component n the bulk phase, n the number of moles of component n (or at) the surface (area)). It was found that the surface tenson could be descrbed by the dfference of logarthm of the actvty coeffcent of component of the lqud phase ncludng surface and logarthm of the actvty coeffcent of component of the lqud bulk phase. It was attempted to descrbe the surface tenson for bnary and ternary dlute ron alloys as a functon of concentraton at a certan temperature. In the case of ternary dlute ron alloy, the actvty coeffcent of solute could be estmated by the polynomal equaton as a functon of concentraton wth nteracton parameters. In the same way, the surface tenson of ternary dlute ron alloys was descrbed by the polynomal equaton as a functon of concentraton. For example, the surface tenson of Fe-- system could be descrbed as follows: 1) μ MFe μ Fe σ = σ ( Fe k. 303M w ) 100M T M e e w R 100 ( ) (. 303M e e e e ) Fe ww 100 M M MFe μ μ ( ). 303MFe w ( ) 100M RT 100M e e w... () where k=(rt/a 0) 10 7, A 0 s the surface area of the system wth one mole soluton, σ Fe s the surface tenson of pure ron at temperature T, w s the mass percent of the component, M s the atomc weght of component, μ s the chemcal potental of n the bulk for the standard state at 1 mass% w for a dlute, deal soluton, R s the gas constant, e s the nteracton parameter and prme denotes the property of the phase ncludng surface phase. For the smplcty, all coeffcents may be replaced by a parameter P as follows: σ = σ Pw P w Pw P ww P w Fe... (3) As shown n the equaton, the surface tenson can be descrbed by surface tenson of pure element (σ Fe), frstorder terms, second-order terms and cross terms of mass percent of each element. By a crtcal survey of the data on the parameters, P, P and P, for ternary system and by extendng Eq. (), t s also possble to predct the surface tenson of hgher order multcomponent dlute ron alloys. In ths paper, an attempt was made to predct the surface tenson of low carbon steel and 16 mass%cr stanless steel (16Cr SUS) by usng ths model. The parameters, P, can be calculated usng thermodynamc parameters as shown n Eq. (). However t s practcally more dffcult to obtan such thermodynamc parameters by experments or calculaton than measure the surface tenson. Hence these parameters were obtaned by fttng the expermental surface tenson values aganst concentraton n the present paper. 1.. Entrapment Behavor of Fne Bubbles n Contnuous Castng Process As an applcaton of above mentoned surface thermodynamc model 1) for surface tenson, the entrapment behavor of the fne bubbles by the soldfyng shell n the contnuous castng slabs could be dscussed snce the Muka-value, M, could be calculated usng the surface tenson values. The mechansm of the fne partcles entrapment by soldfyng nterface has been elucdated by Muka and hs group. They have concluded that a mechansm of the entrapment of fne partcles by soldfyng nterface s due to the surface (nterfacal) tenson gradent whch s generated around the surface of the bubble (or around the nterface of the partcle) caused by a solute concentraton gradent.,3) The partcle should be moved toward soldfyng nterface when the surface (nterfacal) tenson of bulk sde of the partcle s hgher than that of soldfyng nterface sde (Fg. 1). In the contnuous castng, the bubbles n the melts wll be entrapped at soldfyng nterface durng the castng. Therefore t mght be possble to estmate when and where the bubbles are captured by nvestgatng the locaton of the pores after the castng. The bubbles whch entrapped at an earler stage of the soldfcaton wll be remaned near the surface and t mght become a cause of surface defects. On the other hand, the bubbles whch captured at a later stage wll be remaned far from the surface. It mght lead the lower mechancal propertes and surface defects after the machnng, whch are not desred. Accordng to the mechansm, the entrapment of the bubbles can be suppressed by ntroducng postve surface tenson gradent toward the soldfcaton nterface. In the present work, an attempt was made to fnd out the feasblty of the suppresson of bubble capturng based on the mechansm. The ndex to estmate the drvng force for the fne bubble entrapment, Muka-value, can be descrbed as follows:.,3) or M =Σdσ/dw (1 1/k 0 )w... (4) M =σ (w ) σ (w /k o )... (5) where σ s the surface tenson. w s the concentraton of component n the bulk. k 0 s the equlbrum partton coeffcent of component. w /k 0 s the solute concentraton at the sold-lqud nterface. Fg. 1. Moton of a fne partcle at sold-lqud nterface ISIJ

3 ISIJ Internatonal, Vol. 53 (013), No. 1 σ (w ) s the surface (nterfacal) tenson wth the steel composton at the bulk phase. σ (w /k o ) s the surface (nterfacal) tenson wth the steel composton at the soldfyng nterface. As can be seen from the Eq. (5), Muka-value corresponds to the surface tenson dfference due to the composton dfference between bulk and sold-lqud nterface. The surface tenson gradent n the boundary layer could be obtaned by dvdng the Muka-value by the concentraton boundary layer thckness. Even f the boundary layer thckness s unknown, t s possble to compare the drvng forces for the fne bubble entrapment by composton dfference wth Muka-value f t s assumed that the concentraton boundary layer thckness s same. The boundary layer thckness can be regarded as same under the same castng condtons. In the present study, the surface thermodynamc model for surface tenson whch s descrbed n the prevous secton s used to estmate the σ (w ) and σ (w /k o ) n the Eq. (5) for the calculaton of Muka-value, M The Number of the Captured Bubbles n Contnuous Castng Strand To study the relatonshp between Muka-value and the number of captured bubbles, the number of captured bubbles were counted for sx dfferent stanless steels (16Cr SUS) and three dfferent low carbon steels whch cast n contnuous castng by Muka et al. 4) The composton of each steel grade are shown n Tables 1 and. The 16Cr SUS shown n Table 1 were cast usng the No. 1 vertcal-bendng type contnuous caster of Yawata works, Nppon Steel Corporaton under about the same castng condton. 4) Then, the number of bubbles wth radus larger than 0.5 mm n the stanless steel slab (0 15 mm from the surface) was counted from the X-ray mages ) snce such bubbles have the potental to become harmful defects. All low carbon steels shown n Table were cast usng the No. vertcal-bendng type contnuous caster of Yawata works, Nppon Steel Corporaton under about the same castng condton ncludng flow rate of argon, castng speed and Table 1. Composton of each stanless steel grades (mass%). wdth of the slab. Then, the number of bubbles, whch radus s larger than 0.1 mm, n the sub-surface layer of low carbon steel slab (0 15 mm) was counted. 3) The number of the captured bubbles for low carbon steel and stanless steel are summarzed n Tables 3 and 4. The number of bubbles per m was counted for the low carbon steel and the number of bubble per cm 3 was counted from the X-ray mages for the 16Cr SUS. The number of bubbles were counted n a dfferent way. However the Muka-value s a parameter to evaluate the drvng force of the movement of fne partcle toward sold lqud nterface due to the surface tenson dfference. Therefore the dfference of the measurement method s out of queston so far as the comparson s made wth the number of bubbles whch s counted by the same method Parameters for the Surface Tenson of Multcomponent Dlute Soluton (Low Carbon Steel) An equaton to descrbe the surface tenson was derved usng the surface thermodynamc model 1) to calculate surface tenson and Muka-value of present steel grades. For the predcton of surface tenson, the nfluence of strong sur- Table. Composton of each low carbon steel grades (mass%). Element Low carbon (1) Low carbon () Low carbon (3) Fe Balance Balance Balance C S Mn P S Soluble Al * T N Free O * * The concentraton of free oxygen and soluble alumnum was calculated from the alumnum deoxdaton equlbrum n the steel. Interacton of the other elements was also consdered. Element Low Al (1) Low Al () Low Al (3) Medum Al (1) Medum Al () Hgh Al Fe Balance Balance Balance Balance Balance Balance C S Mn P S Soluble Al * Cr Mo N Free O * * The concentraton of free oxygen and soluble alumnum was calculated from the alumnum deoxdaton equlbrum n the steel. Interacton of the other elements was also consdered. Table 3. Table 4. Number of captured bubbles per m (Low carbon steel). Number of bubbles LC(1) 4 LC() 100 LC(3) 108 Number of captured bubbles per cm 3 (Stanless steel). Number of bubbles Low Al (1) Low Al () 0.30 Low Al (3) Medum Al (1) Medum Al () 0.04 Hgh Al ISIJ 0

4 ISIJ Internatonal, Vol. 53 (013), No. 1 face actve elements, sulfur, oxygen and ntrogen, were consdered. In addton, nfluence of ttanum, nobum and boron were taken nto account n the present paper. Accordng to the model for surface tenson, the surface tenson of Fe N O S T Nb B system at a certan temperature, σ, could be expressed as follows by extendng Eq. (3): σ = σ Pw Pw Pw P w P w PSSwS PNOwNwO PNSwNwS POSwOwS Pw T T PNbwNb Pw B B... (6) In the above equaton, the second-order term and cross term for ttanum, nobum and boron were gnored snce these elements are not strong surface actve elements. For the same reason the nteracton on the surface tenson between T O or T N also can be gnored although the ttanum has strong nteracton wth oxygen and ntrogen n bulk phase n chemcal thermodynamcs pont of vew. The parameters, P, P, P, could be obtaned from the surface tenson of Fe-- ternary system. For example, parameters, P N, P O, P NN, P OO and P NO could be obtaned by determne the parameters that make Eq. (3) gve the best ft to the expermental surface tenson for Fe N O system 5) as a functon of concentraton. However, there are some dffcultes n assgnng values to all the parameters from only the surface tenson of ternary system due to the lack of expermental data. In the present paper, t has been attempted to derve the parameters from multcomponent system as well n addton to ternary (or bnary) system. By usng the surface tenson data for the multcomponent system, the all parameters for the elements whch s ncluded n the system can be determned at same tme by fttng. In the followng calculatons, a commercal software (Mathematca ver. 7.0, Wolfram Research) was used to determne the parameter values that make the equaton for the surface tenson (Eq. (6)) gve the best ft to the surface tenson data as a functon of concentraton. Frstly, the parameters P O and P OO were determned from the surface tenson of Fe O bnary system at 1 83 K 5,6) by Table 5. Fe N N O O S S NN N Interacton parameters for multcomponent dlute soluton (low carbon steel) at K. Parameter Values at K σ Fe P N 7.09 P O P S.89 P NN P OO 10.1 P SS 1.41 P NO 16.6 P NS 0 P OS P T P Nb P B OO O fttng the expermental values to the followng equaton whch corresponds to Eq. (3). σ = σ Pw P w Fe O O OO O... (7) The surface tenson of pure ron, σ Fe, at 1 83 K was determned n advance from the lterature 5,7) by lnear fttng of surface tenson aganst temperature. Secondly, the parameters P S, P SS and P OS as well as P N, P NN and P NO were determned from the surface tenson for Fe O S ternary system at K 8,9) and Fe N O ternary system at 1 83 K, 5) respectvely, by fttng the expermental values wth the Eq. (3). The surface tenson of pure ron at K and 1 83 K were calculated from the lterature data 5,7) and used for the fttng. After that, the other parameters, P T, P Nb and P B, were determned from the surface tenson at K for syntheszed multcomponent alloy 3) by fttng the expermental values to the Eq. (3). All the above parameters were regarded vald at K. The temperature dfference was neglected due to the lack of expermental data. The measurement error of the surface tenson by sessle drop method s about ±3% 10) whch s correspondng to about ± N m 1 n the present system. On the other hand, accordng to the lterature data for the surface tenson of pure ron whch s used n the present paper, 5,7) the temperature coeffcent for the surface tenson of pure ron s N m 1 K 1. If t s assumed that ths value s vald for the present systems as well, the temperature dfference n the present calculaton (±5 K) s correspondng to ± N m 1 n surface tenson dfference. Hence, consderng the expermental error, the temperature dfference could be neglgble n the present system. Regardng the parameter P NS, t was found that the value was not relable due to the fact that the P NS value became postve when the surface tenson of Fe O S system by Ogno et al. 8) was used for the calculaton. On the other hand, t became negatve value when the data by Gupt et al. 9) was used. At ths stage, t s not adequate to evaluate the P NS value accurately. Eventually, data sets of both Ogno et al. 8) and Gupt et al. 9) are used for the calculaton and P NS value was put as zero (nfluence of the nteracton between ntrogen and sulfur on the surface tenson was gnored). Then the parameters, P T, P Nb and P B were determned that make the equaton for the surface tenson gve the best ft to the surface tenson data for the multcomponent system whch contan ttanum, nobum and boron 3) as a functon of concentraton. Snce the parameters P T, P Nb and P B were fxed n the fnal step, the nfluence of neglect for the nteracton between ntrogen and sulfur (P NS=0), temperature dfference and measurement errors were dumped nto these parameters. The values of the parameter at K are summarzed n Table 5. The unt of surface tenson s N m Parameters for the Surface Tenson of 16 mass%cr SUS The surface tenson of 16Cr SUS was also estmated usng the surface thermodynamc model. 1) Due to the lack of expermental data to determne the parameters for surface tenson model, only four elements, ron, chromum, oxygen and sulfur, were taken nto account for the calculaton. Regardng the 16Cr SUS, the calculaton by the model amed to estmate the surface tenson wth only strong sur ISIJ

5 ISIJ Internatonal, Vol. 53 (013), No. 1 Table 6. Interacton parameters for Fe 16 mass% Cr O S system at 1 83 K. Parameter Values at 1 83 K σ Fe16Cr P O P S P OO 17.0 P SS 0.97 P OS 1.50 face actve elements. The nfluence of the other elements on the surface tenson wll not be assessed n the present calculaton due to the smplfcaton. The nfluence of the other elements on the surface tenson are dumped nto the parameters for oxygen and sulfur n the present calculaton. In ths system, the Cr cannot be regarded as a dlute solute. Therefore the Fe 16 mass% Cr was treated as a pure element. Then the surface tenson of 16 mass%cr SUS can be descrbed by Eq. (8) whch corresponds to Eq. (3). σ = σ Pw Pw P w P w P ww Fe16Cr O O S S OO O SS S OS... (8) where σ Fe16Cr: the surface tenson of Fe 16 mass%cr, P : Parameter (Coeffcent) for the equaton on surface tenson of Fe 16 mass%cr O S system by the surface thermodynamc model and w : the concentraton of component (mass%). New nteracton parameters, P, n the Eq. (8) were determned by fttng the surface tenson values for Fe 16 mass%cr O S system at 1 83 K 11,1) aganst concentraton usng Eq. (8). The obtaned nteracton parameters are shown n Table 6. The unt of surface tenson s N m 1.. Dscusson.1. Calculated Surface Tenson The expermental surface tenson values could be descrbed by sngle equaton. It mples also that the contrbuton of weak surfactants such as ttanum, boron and nobum to the surface tenson can be descrbed by consderng only frst order terms. The calculated surface tenson values by Eq. (6) are plotted aganst expermental data whch was used for the determnaton of parameters n Eq. (6) (Fg. ). The data except for Ref. 13) was used for the determnaton of parameters. As shown n the Table 5, the parameters for Eq. (6) are vald at K and the surface tenson of pure ron, σ Fe, at K s used. Therefore the calculaton results on Fe O S system whch measured at hgher temperature (1 873 K) are overestmated as can be seen from Fg.. Further, The data by Ogno et al. 8) are devated from the calc.=exp. lne snce the hgh number of data ponts by Gupt et al. 9) were used together wth the data by Ogno et al. to determne the parameters by fttng. The dsagreement between the expermental and calculated values may become less by ncreasng the order of the equaton for the surface tenson snce t s stll possble to descrbe the parameters, P, by thermodynamc parameters O S Fg.. Expermental surface tenson value vs. Calculated surface tenson value. even f the order of the equaton s ncreased. However, an approprate order should be second order for the relablty of the expermental data. It s to be noted that the degree of agreement between the expermental and calculated values can be mproved f accurate expermental data for ternary system could be generated n the future. It s especally mportant to obtan the relable measurement values of Fe O S and Fe N S systems (and Fe O N system, f possble) for the surface tenson model snce the parameters for these strong surface actve elements s senstve aganst surface tenson. As mentoned, the measurement error of surface tenson s about ± N m 1. Hence, t s desred to determne the parameter values to make the average error of calculated surface tenson value become less than ± N m 1. In order to verfy the valdty of the surface tenson model, surface tenson of two dfferent practcal steels, 13) whch are not used for the determnaton of parameters, were calculated from the composton by usng the surface tenson model. The data s plotted n the Fg. as open crcles. The calculated values show reasonable agreement wth the expermental data. As descrbed before, the P NS =0 s appled and the nfluence of neglect for the nteracton between ntrogen and sulfur s corrected by the parameters for ttanum, nobum, and boron. In the present case, these practcal steels contan hgh sulfur (0.0 mass%) and ntrogen (0.01 mass%), and the amount of ttanum, nobum and boron are not analyzed except for the ttanum amount for one of them whch has lower calculated and hgher expermental value. Therefore the correcton of the nfluence of neglect for the nteracton between ntrogen and sulfur by P T, P Nb and P B terms s not suffcent. As a results, the calculated values became hgher than the expermental values. However the surface tenson wll not be ncreased n total unless the amount of nobum become same level wth boron or ten tmes of ttanum as can be seen from parameter values, P T, P Nb, P B n Table 5. The estmaton of the surface tenson by the model s stll rough. However, there are some fndngs and advantages of the model. Generally speakng, t s dffcult to know the 013 ISIJ

6 ISIJ Internatonal, Vol. 53 (013), No. 1 nfluence of a specfc speces on the surface tenson from the expermental results n the case of multcomponent systems when the concentratons of two or more speces are changed smultaneously. However, t s possble to dstngush the contrbuton of each element to the surface tenson by usng the model. For example, n the experment, the surface tenson of a commercal steel seems to be decreased wth ncreasng of nobum. 3) However the concentraton of a surface actve element, oxygen, was also ncreased. Therefore, t was mpossble to evaluate the net effect of nobum. However, t s possble to evaluate the net effect of nobum on the surface tenson by applyng the surface tenson model. As can be seen from the parameters n Table 5 and Eq. (6), the parameter for nobum, P Nb, has a postve value. Therefore, t can be concluded that the surface tenson s ncreased wth ncreasng of nobum... Valdaton of the Parameters In the prevous paper, 1) t has been revealed that the surface tenson can be descrbed as a functon of concentraton and the coeffcents (parameters) have thermodynamcal meanng. At the present stage, t may dffcult to calculate the parameters P, P and P ndvdually wth thermodynamcs or nterfacal chemstry. Therefore the parameters must be determned by the measurements of surface tenson. However, t s stll possble to dscuss the thermodynamcal meanng of obtaned parameter values as follows qualtatvely. As can be seen from Eqs. () and (3),... (9)... (11) where a s actvty of component and prme denotes the correspondng propertes of the phase ncludng surface phase. At equlbrum state, μ = μ. Therefore, μ μ = RTln( a ) RTln a... (1) In the case of postve adsorpton, a < a. Hence sgn of μ μ, n other words, sgn of P must be negatve. Consequently, t s reasonable that the parameters P whch obtaned by curve fttng based on the expermental values of surface tenson (Tables 5 and 6) have negatve values. Regardng the parameters P and P, t can be descrbed as follows as can be seen from Eqs. () and (3). and MFe μ μ P = k 100MRT The chemcal potental of bulk phase, μ and the chemcal potental of the phase ncludng surface, μ, n the above equaton can be expressed as follows, μ = μ RTln( a )... (10) μ = μ RTln a P. 303MFe k M e = e 100 M e e e. 303 e Fe P = k 100 M M... (13) ( )... (14) e e e e M M snce M e ( e ) e n the Eq. (14) may be arranged as 30Me M M = 30M 30Me M M and e. = 30M Hence, Eq. (14) may be arranged as follows: M e e. 303 Fe P = k... (15) 100 M Now, log f log f = e w e w e w e w ( e e ) w =... (16) where f s the actvty coeffcent of component. In the case of postve adsorpton, logf logf > 0. Hence t could be concluded that the postve P ( e ) and e >0 P ( e ) values whch obtaned by curve fttng e >0 based on the expermental values of surface tenson are reasonable. The P NN whch has slghtly postve value s only excepton. Ths ought to be expermental error of the surface tenson measurements..3. Muka-value and the Number of Entrapped Bubbles The drvng force for the fne bubble entrapment by the soldfyng shell can be assessed by the Muka-value, M as mentoned n the begnnng of ths paper. In the prevous paper, ) the Muka-value was calculated usng Eq. (4) wth the dσ/dw value at w =0. It s assumed that the concentraton of element s decreased lnearly n the boundary layer and the surface tenson s ncreased lnearly toward the bulk. As a results, the dσ/dw value on the bulk sde was overestmated and the Muka-value was also overestmated. Especally, the Muka-value for Low Al stanless steel was hgher n the prevous calculaton and the value became more than N m 1. Muka-value, whch corresponds to the surface tenson dfference, cannot have such hgh value snce the surface tenson value tself s less than N m 1 n the present system. The Muka values derved n the prevous paper was useful as a relatve scale to evaluate the dfference of the drvng force for the movement of bubbles by composton dfference. In ths paper, an attempt was made to evaluate the Muka-value quanttatvely. In ths paper, the Muka values were calculated usng Eq. (5) wth the estmated surface tenson values by the surface tenson model. The surface tenson at bulk (σ (w )) n the Eq. (5) was estmated from the composton shown n Tables 1 and by usng Eq. (6) for low carbon steels and Eq. (8) for 16 Cr SUS stanless steels. To estmate the surface tenson at the soldfyng nterface n the Eq. (5) from the steel composton, the composton at the soldfyng nterface was estmated by usng followng equlbrum partton coeffcent, k 0 : k 0 S =0.05, k 0 T =0.40, k 0 N =0.8 and k 0 O = ) for low carbon steel. The same ( ) e e w ISIJ

7 ISIJ Internatonal, Vol. 53 (013), No. 1 Table 7. The composton at bulk and soldfyng nterface (low carbon steel). Element Low carbon (1) Bulk Low carbon (1) Soldfyng nterface Low carbon () Bulk Low carbon () Soldfyng nterface Low carbon (3) Bulk Low carbon (3) Soldfyng nterface S T N Free O Table 8. The composton at bulk and soldfyng nterface (low Al stanless steel). Element Low Al (1) Bulk Low Al (1) Soldfyng nterface Low Al () Bulk Low Al () Soldfyng nterface Low Al (3) Bulk Low Al (3) Soldfyng nterface S N Free O Table 9. The composton at bulk and soldfyng nterface (medum and hgh Al stanless steel). Element Medum Al (1) Bulk Medum Al (1) Soldfyng nterface Medum Al () Bulk Medum Al () Soldfyng nterface Hgh Al Bulk Hgh Al Soldfyng nterface S N Free O Table 10. Calculated surface tenson and Muka-values. Table 11. Calculated surface tenson and Muka value for stanless steel. Steel grade Surface tenson at bulk [N m 1 ] Surface tenson at soldfyng nterface [N m 1 ] Muka-value [N m 1 ] (σ (w ) σ (w /k o )) LC (1) LC () LC (3) equlbrum partton coeffcents were used to estmate the concentraton of each element at soldfyng nterface of stanless steels as well due to the lack of the data on equlbrum partton coeffcents for 16Cr SUS. Regardng the equlbrum partton coeffcents for oxygen, k O 0 =0.0~0.1 have been reported. 16 ) In the present system, alumnum oxde wll be thermodynamcally formed at soldfyng nterface when k O 0 =0.0 s appled snce the oxygen concentraton at soldfyng nterface becomes hgh. However the alumnum oxde was not formed n the practcal operaton. Hence k O 0 =0.1, whch does not form alumnum oxde thermodynamcally, seems to be reasonable value n the present case. The maxmum oxygen concentraton at soldfyng nterface becomes mass% wth k O 0 =0.1 and alumnum oxde wll not be formed snce the oxygen can be supersaturated. Accordng to Hlty and Crafts, 3) and Novokhatsky and Belov, 4) the oxygen could be supersaturated at least mass%. The composton at bulk and soldfyng nterface whch used to estmate the surface tenson s summarzed n Tables 7 9. The surface tenson at 1 83 K and the Muka values calculated from the surface tenson dfference between bulk and soldfyng nterface (Eq. (5)) are shown n Tables 10 and 11. The Muka-value calculated usng Eq. (5) wth the surface Steel grade Surface tenson at bulk [N m 1 ] Surface tenson at soldfyng nterface [N m 1 ] Muka-value [N m 1 ] (σ (w ) σ (w /k o )) Low Al (1) Low Al () Low Al (3) Medum Al (1) Medum Al () Hgh Al tenson model became smaller compared wth prevous values ) calculated usng Eq. (4) wth k 0 O =0.1 and physcally reasonable Muka-values could be obtaned. In the present calculaton. the surface tenson dfference (σ (w ) σ (w /k o )) becomes between 0.01 and 0.81 N m 1. In the prevous work, the Muka-value was not calculated from the surface tenson dfference drectly but calculated by usng surface tenson gradent (Eq. (4)). Therefore the Muka-value became larger than reasonable surface tenson dfference for some cases. By calculatng the Muka-value from the surface tenson dfference drectly (Eq. (5)) by usng the surface tenson model, Muka-value could be determned quanttatvely. In order to verfy the valdty of the Muka-value, the number of entrapped bubbles was plotted aganst Mukavalue. Fgures 3 and 4 show the results for low carbon steel and stanless steel, respectvely. As can be seen from these fgures, the number of bubbles s ncreased wth ncreasng of Muka-value and t shows lnear relatonshp. The force act on the bubbles (ex. buoyancy, force by convecton) can be regarded as same except for the force caused by the surface tenson gradent snce the castng condton s same. 013 ISIJ 4

8 ISIJ Internatonal, Vol. 53 (013), No. 1 Fg. 3. Relatonshp between Muka-value, M and number of bubbles n the low-carbon slabs. Fg. 4. Relatonshp between Muka-value, M and number of bubbles n the 16Cr SUS steels. Hence t s possble to correlate the movement of bubbles drectly to the Muka-value. The results ndcate that the Muka-value (drvng force for the fne bubble entrapment by the soldfyng shell) are proportonal to the number of captured bubbles, whch shows the Muka value s a vald parameter to evaluate the bubble entrapment. The number of bubbles, whch radus s larger than 0.1 mm, becomes zero below a certan crtcal Muka-value, M cr =0.4 under the present castng condtons n the low carbon steel case. The smaller bubbles can be regarded as harmless bubbles from the castng qualty pont of vew and practcally the nfluence of such small bubbles can be neglgble. In the case of 16Cr SUS, the crtcal Muka-value becames 0.1 under the present castng condtons. The dfference of the crtcal value s due to the dfference of concentraton boundary layer thckness n front of the soldfyng nterface caused by the castng condton dfference such as coolng speed and the dfference of the counted bubble sze. The crcal Muka-value wll be vared by the followng castng condtons as well: ) amount of argon, ) castng speed, ) condton of electromagnetc strrng, v) shape of nozzle, v) wdth of slab, etc. snce such operatng condtons are not taken nto account n the Muka-value. The number of entrapped bubbles at a certan Muka-value wll be vared by changng above mentoned condtons. It s, however, possble to propose gudng prncples to suppress the number of defects caused by entrapped bubbles based on the Muka-value whch s derved for the same castng condtons to evaluate the nfluence of composton. As mentoned prevously, the entrapment of the bubbles wll be suppressed by ntroducng postve surface tenson gradent toward the soldfcaton nterface or mnmze the Mukavalue. By combnng t wth the surface tenson model for surface tenson, a gudelne to decrease the Muka-value can be found theoretcaly wth hgher accuracy compared wth prevous calculaton method. It could be possble to fnd out the elements whch gve smaller or negatve Muka-values by usng the surface tenson model. For example, as can be seen from Table 5, the nobum has postve surface tenson gradent toward the soldfcaton nterface (P Nb > 0). Therefore, f the nobum s ntroduced n the system so that the concentraton of nobum becomes hgher toward the soldfcaton nterface, the Muka-value wll become smaller and the fne bubble capturng by the nterface wll be suppressed. As shown n the relatonshp between the number of bubbles and the Muka-value whch obtaned under the same castng condton, a lnear relatonshp were found between these two factors. Hence, t s also possble to fnd out the nfluence of the castng condton on the bubble entrapment by changng the castng condton and nvestgate the relatonshp of these two factors n the same way. Such results wll gve useful nformaton for the optmzaton of the process. 3. Concluson The surface tenson of the multcomponent practcal steels could be predcted by usng the surface thermodynamc model wth parameters derved from lterature data. It was found that the nfluence of weak surfactants such as ttanum, nobum and boron on the surface tenson could be descrbed by consderng only frst order terms. Further, the nfluence of a specfc element on the surface tenson was clarfed based on the model. The precse surface tenson measurements for Fe O S and Fe N S systems (and Fe O N system, f possble) are essental to determne the parameters related to these elements more precsely for the model. The Muka-value, M was calculated from the surface tenson whch s predcted by the surface tenson model and appled for the evaluaton of the entrapment behavor of the fne bubbles by soldfyng shell. A gudelne to suppress the fne bubble entrapment has been proposed based on the mechansm of the fne partcle entrapment and the surface tenson model. The bubble entrapment wll be suppressed by ntroducng elements such as nobum whch gves hgher surface tenson toward soldfyng shell. It can be concluded that the Muka-value calculated by usng the surface tenson model s a useful parameter for the optmzaton of the process to suppress the number of defects caused by the bubble entrapment, manly from the aspect of optmzaton of composton. Nomenclature A: Area A 0: Surface area of the system wth one mole soluton a : Actvty of component e : Interacton parameter n the bulk e : Interacton parameter n the phase whch ncludng surface F: Helmholz energy ISIJ

9 ISIJ Internatonal, Vol. 53 (013), No. 1 f : Actvty coeffcent of component k: k=(rt/a 0) 10 7 k 0 : Equlbrum partton coeffcent of component M: Muka value M cr: Crtcal Muka-value M : Atomc weght of component n : Number of moles of component : Number of moles of component n the bulk n b n s phase : Number of moles of component n (or at) the surface (area). P l : Pressure of lqud phase P : Parameter (Coeffcent) for the equaton on surface tenson by the surface thermodynamc model P : Parameter (Coeffcent) for the equaton on surface tenson of Fe 16 mass%cr O S system by the surface thermodynamc model R: Gas constant T: Temperature V: Volume V l : Volume of lqud phase w : Mass percent of the component μ : Chemcal potental of component μ μ μ : Chemcal potental of component n the bulk phase : Chemcal potental of component n the phase ncludng surface : Chemcal potental of n the bulk for the standard state at 1 mass% for a dlute, deal soluton μ : Chemcal potental of n the phase whch ncludng surface for the standard state at 1 mass% for a dlute, deal soluton σ : Surface tenson σ Fe: Surface tenson of pure ron σ Fe16Cr: Surface tenson of Fe16 mass%cr REFERENCES 1) K. Muka, T. Matsushta, K. C. Mlls, S. Seetharaman and T. Furuzono: Metall. Mater. Trans. B, 39 (008), 561. ) K. Muka and M. Zeze: Steel Res., 74 (003), ) K. Muka, L. Zhong and M. Zeze: ISIJ Int., 46 (006), ) M. Zeze and K. Muka: Proc. of 3rd Int. Cong. on the Scence and Technology of Steelmakng (ICS005), Warrendale, PA, USA, (005), ) J. Zhu and K. Muka: ISIJ Int., 38 (1998), ) N. Takuch, T. Tanguch, N. Shnozak and K. Muka: J. Jpn. Inst. Met., 55 (1991), 44. 7) N. Takuch, T. Tanguch, Y. Tanaka, N. Shnozak and K. Muka: J. Jpn. Inst. Met., 55 (1991), ) K. Ogno, K. Nog and C. Hoso: Tetsu-to-Hagané, 69 (1983), ) K. M. Gupt, V. I. Vshkaryov and S. A. Blznukov: Trans. Ind. Inst. Met., 9 (1976), ) K. Muka: Kōon yūta no kamen butsur kagaku, Agne, Tokyo, (007), ) K. Muka, Z. L and M. Zeze: Mater. Trans., 43 (00), ) Z. L, M. Zeze and K. Muka: Mater. Trans., 44 (003), ) R. F. Brooks and P. N. Quested: J. Mater. Sc., 40 (005), ) Handbook of Iron and Steel - I -, 3rd ed., ISIJ, Tokyo, (1981), ) Z. Morta and T. Tanaka: Tetsu-to-Hagané, 74 (1988), ) A. Hays and J. Chpman: Trans. Am. Inst. Mn. Metall. Eng., 135 (1939), ) W. A. Tller: J. Iron Steel Inst., 19 (1959), ) J. Chpman: Basc Open Hearth Steelmakng, the Amercan Insttute of Mnng and Metallurgcal Engneerng, New York, (1951), ) W. A. Fscher, H. Sptzer and M. Hshnuma: Arch. Esenhüttenwes., 31 (1960), ) M. T. Hepworth, R. P. Smth and E. T. Turkdogan: Trans. Metall. Soc. AIME, 36 (1966), ) A. Kusano, K. Ito and K. Sano: Tetsu-to-Hagané, 54 (1968), 553. ) F. Oeters and K. Rüttger: Arch. Esenhüttenswes., 40 (1969), ) D. C. Hlty and W. Crafts: J. Met., Trans. AIME, 188 (1950), ) I. A. Novokhatsky and B. F. Belov: Izv. Akad. Nauk SSSR, 3 (1969), ISIJ 6