Forward and Backward Reaction Rate Constants of the Reaction of C O CO (g) on Fe C Melts

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1 ISIJ International, Vol. 50 (010), No. 5, pp Forward and Backward Reaction Rate Constants of the Reaction of CC (g) on Fe C Melts Hyun-Soo KIM, Jang Gyu KIM and Yasushi SASAKI Graduate Institute of Ferrous Technology (GIFT), Pohang University of Science and Technology (PSTECH), Pohang Korea (South). (Received on November 16, 009; accepted on March, 010) It is known that the molten iron oxide reduction by Fe C melt shows quite different behavior depending on whether the surface of Fe C melts is fully covered with molten iron oxide or not. There have been only two studies of the molten iron oxide reduction by Fe C melt with the existence of free surface. In both studies, the reduction appears to be controlled by the reaction of C C (g). The rate of this reaction has not been fully established. In the present study, based on the reexamination of overall reaction rates of previous two studies by Dancy and Lloyd et al., the forward and backward reaction rate constants of the reaction C C (g) at K have been evaluated by applying the order of magnitude evaluation method. The forward rate constant, in the unit of mol/m s, is given by: k exp(50 000/RT), and the backward rate constant, in the unit of mol/m s, is deduced to: k C exp(77 000/RT). KEY WRDS: molten iron oxide; free surface; C dissociation; Marangoni flow; direct reduction. 1. Introduction Due to the concerns about environmental and energy issues, new ironmaking processes of higher smelting intensity than a blast furnace have been developed. To meet the demand of high production and high quality of products in these processes, understanding of the details of smelting phenomena is critical. Although many investigations of the reduction of iron oxides with liquid iron carbon alloys 1 7) have been carried out, the details are not yet firmly established. In most of these investigations, the surface of Fe C melts was fully covered with molten iron oxide or iron oxide bearing slag. Under this condition, molten iron oxide reduction proceeds with the formation of bubbles at the interface between Fe C melt and molten iron oxide, and the apparent reduction reactions are expressed as followings: Fe (l)c (g) Fe (l)c (g)...(1) C (g)c C (g)...() The reaction () can be separated into several steps: C (g) C (ad)...(3) C (ad) C (ad) (ad)...(4) (ad)c (ad) C (ad)...(5) C (ad) C (g)...(6) C C (ad)...(7) and the rate-limiting process is found to be C dissociation reaction (4). Belton and Fruehan 1) examined the studies of Dancy ) and Lloyd et al. 3) on the reduction of iron oxides in iron carbon melts. In both cases, the surface of Fe C melt was only partially covered with liquid iron oxide and its free surface area of the metallic melt exceeded the contact area with the liquid oxide by more than an order of magnitude. The reactions were completed within a few seconds.,3) The reaction rates of their results were found to be significantly high at K, compared with the maximum rates for the C reaction with Fe C sat melt and for the reaction of liquid iron oxide under 1 atmosphere of C gas as shown in Fig. 1. 1) Very intriguingly, Lloyd et al. observed that there was no bubbling through the molten oxide and no Fig. 1. Rates of reactions of molten iron oxides on the surface of carbon-saturated iron melt. 1) 010 ISIJ 678

2 ISIJ International, Vol. 50 (010), No. 5 significant subsurface nucleation of C bubbles for the melts with more than 1.5 mass% of carbon. Their findings were totally different from the case that the surface of Fe C melt was fully covered with molten iron oxide. The reduction process appears to be the reaction between supersaturated oxygen in the melt supplied from liquid oxide and dissolved carbon, followed by desorption of C at the free surface. The apparent surface reaction can be expressed as: C C (g)...(8) The possible reaction mechanism is schematically shown in Fig.. It is a reliable estimate of the reaction area that is the most important thing in the determination of the rate constant. Lloyd et al. 3) assumed that the contacting area of liquid iron oxide with Fe C melt represented the reaction area. Belton and Fruehan 1) estimated the reaction constant as shown in Fig. 1 by using the same reaction area. This assumption leads that C molecules are produced at the interface between the iron oxide and the melt without C bubble formation. It is quite unreal and the assumption cannot be acceptable. As a plausible reaction mechanism, the following reaction path can be considered. Dissolved oxygen and carbon move to the surface and react each other at the free surface. C C (ad)...(7) (ad)...(9) C (ad) (ad) C (ad)...(5) The absorbed C molecule desorbs from the melt surface. C (ad) C (g)...(6) In this case, the reaction does not occur in the melt but on the free surface of the melt. The dissolved oxygen atoms must be rapidly transferred to the surface. (The rapid movement of the dissolved oxygen atoms will be discussed later.) The rate of C molecule desorption has been known from the decarburization studies to be very high. The rates of the reactions (7) and (9) are also known to be very fast and thus the reactions can be under the equilibrium condition. As a result, the rate-controlling step of the reaction (8) will be the reaction (5). In other words, the overall rate of the reaction (5) can be in the same order as that of the reaction (8). Thus, the rate of the reaction (8) is plausibly expressed by measurable dissolved oxygen and carbon concentrations: RatekSC C C...(10) where S is the reaction area and k is the rate constant of the reaction (8). C C and C are the concentrations of dissolved carbon and oxygen, respectively. Concerning with the reaction (5), the adsorbed oxygen atoms must migrate from the molten oxide to the free surface to react the adsorbed carbon atoms. Atomic diffusion cannot manifest the fast movement of the dissolved oxygen atoms because of the low diffusion coefficient of oxygen, m /s. 8) The oxygen atoms at the surface are consumed by the reaction (5) so that the oxygen concentration at the surface should decrease gradually while receding from the vicinity of molten iron oxide. If so, the evaluation of the oxygen concentration distribution on the free surface is important to determine the actual reaction area. The purpose of the present study is to find out the driving force of the fast migration of oxygen atoms in the melt and the distribution of oxygen concentration at the free surface during the direct reduction of liquid iron oxide with Fe C melt.. Mathematical Model for Molten Iron xide Reduction with Fe C Melt According to the experiments by Dancy ) and Lloyd et al., 3) the size of the free surface of Fe C melt is larger than that of molten iron oxides by more than one order of magnitude. Therefore the geometrical condition of their experiments could be simplified as shown in Fig. 3. The molten iron oxide locates in the center of the free surface of Fe C melt, where the free surface size of the melt, W is much larger than that of the iron oxide, w. Thus, we may reasonably neglect the size of the molten iron oxide, w. Both of carbon and oxygen atoms control the reduction rate of iron oxide on Fe C melt. The reaction rate slightly increased with carbon concentration in the melt above 1 mass% C although the values were widely dispersed. 3) If the initial carbon content is relatively large, say, about 4 mass%, the consumption of carbon in the melt by the reduction reaction can be negligible compared with the total amount carbon, and it can be assumed constant during the reaction as a first approximation. Then the molten iron Fig.. Possible mechanism for C evolution during reduction of molten iron oxide by Fe C melts. Fig. 3. Geometrical illustration diagram for molten iron oxide on Fe C melt ISIJ

3 ISIJ International, Vol. 50 (010), No. 5 oxide reduction could be considered as a pseudo-unimolecular reaction and the reaction rate is controlled by solely the oxygen concentration. Since the oxygen is consumed at the free surface, there must be the gradient of oxygen concentration on the surface. The lowest oxygen concentration is negligibly small at the place far away from the iron oxide because the melt contains high amount of carbon. n the other hand, the oxygen concentration would be highest at the vicinity of the floating molten iron oxide. The surface tension of Fe C melt is strongly dependent on the oxygen content in the melt. It is a function of the oxygen content, which is expressed as 9) : s ln(1170c ) (N/m) (at K)...(11) where C is mass% of oxygen. If there is the gradient of the oxygen concentration on the free surface, the gradient of the surface tension induces the solutal Marangoni flow. Yin et al. found that solutal Marangoni flow could be generated due to the solubility difference of oxygen during solidification of molten steel. 10) In our previous research, 11) we observed directly via a confocal laser-scanning microscope that there was a solutal Marangoni flow on the free surface of Fe C melt in contact with wüstite. 11) Based on these results, it can be said that the solutal Marangoni flow must have been developed in the experiments of molten iron oxide reduction with Fe C melt conducted by Dancy ) and Lloyd et al. 3) This Marangoni flow can effectively transfer the oxygen near the molten iron oxide to the free surface. The strength of the Marangoni flow is determined by the oxygen concentration distribution on the free surface of the melt. It is impossible to directly measure the oxygen distribution at the surface of melts, but the value can be deduced from thermodynamic data. Since the amount of subsurface evolution of C for Fe C melt is negligible, 3) the oxygen concentration at the interface should be approximately equal to the solubility of oxygen in liquid iron at the experimental temperature (details are discussed in Appendix A). The solubility of oxygen in liquid iron is expressed by the equation 1) : log(c S )6 30/T (1) From Eq. (1), the oxygen concentration equilibrated with liquid iron oxide is 0.9 mass% at K. It is much higher than the value expected from the equilibrium reaction (8) under the C partial pressure of 1 atm and the carbon concentration of 4 mass%. Namely, oxygen is supposed to be supersaturated at the interface. In the Fe melt, the oxygen concentration distribution is described by using a diffusive convective mass transfer equation. n the free surface of the melt, the radial directional convective flow is dominant. Hence, neglecting convection in z-direction for simplification, the following equation in cylindrical coordinate is used in this analysis by defining the free surface where. urz (, ) C(, r z) D C (, r z ) r...(13) where C (r, z) is a function of r and z, and D is oxygen diffusivity in Fe C melt. The effect of carbon on D is neglected. When the steady state is assumed at the free surface,, the flux of oxygen can be expressed as D C (, r z ) kk m C C...(15) where C C is mass% of carbon which is assumed to be constant of 4.0 mass% over the whole free surface in the present study, and k m is reaction rate constant with the unit of [m/s]. The reaction rate constant, k [mol/m s] can be converted to k m [m/s] by multiplication of 100M /r [m 3 /mol]. M is oxygen atomic weight, kg/mol. r is the density of Fe 4mass%C melt at K, 6900 kg/m 3. 13) The Laplacian of the oxygen concentration at the free surface in Eq. (13) can be simplified by using Eq (14). The oxygen concentration gradient at the surface should change below the surface and quickly go to zero because the concentration should be zero inside the melt. Therefore the gradient change can be made into the simple form in an approximate way. C(, r z) 0kC (, r0)/ D Δz...(16) And it can be conjectured that the concentration change with infinitesimal Dz is the order of the concentration at the surface because there is an abrupt change in concentration at the surface, that is, C (r, 0)/Dz C (r, z)/. Finally the Laplacian of the oxygen concentration at the free surface can be expressed as follows: C(, r z) Combining Eq (13) with Eq (17), dc r k ur (, ) (, 0) 0 dr D C k 0...(17) D C (, r )...(18) If u(r, z) becomes closed to zero at zd V, r-directional velocity, u(r, 0) can be obtained roughly by linearization of the shear stress balance on the free surface of the melt,, u μ dσ dc d dc (, r 0) ur (, 0) σ δ V dc dr μ kc (, r0)...(14) dc( r, 0) dr...(19)...(19) where m is the viscosity of Fe C melt and ds/dc (0.8)(170)/(1170C ) based on Eq. (11). Combining Eq. (19) with Eq. (18), the equation for oxygen concentration gradient will be expressed in the following from considering the oxygen concentration should decrease in the radial direction. (, r 0) 010 ISIJ 680

4 ISIJ International, Vol. 50 (010), No. 5 dc() r k μc()( r C()) r (0) dr Dδ ( 0. 8)( 170) If oxygen concentration is much larger than 0.01 mass%, 1170C 170C (r)...(1) Since oxygen concentration is 0.9 mass% at the interface with iron oxide, Eq. (1) is acceptable as a first approximation. As a result, Eq. (1) can be modified as dc() r dr Integrating Eq. () from 0 to r, the equation for carbon concentration can be obtained as follows: r C( r) C S exp β β k μ Dδ (. 08) 08. Dδ k μ...(3)...(4) where C (0) is C S in this calculation because r0 means the interface between iron oxide and Fe C melt on the free surface in this calculation. The order of d V is assumed to be close to that of b (proved in Appendix). Then b is expressed by 08. D β...(4) k μ Substituting numeric values of D ( m /s) 8) and m (610 3 Pa s) 13) at K into (4), Fig. 4. V V V C () r...() xygen concentration distribution at the surface of Fe C melt with various reaction rate constants. b /(k)...(5) From Eqs. (3) and (5), the oxygen distribution depends on the reaction rate constant k, where kk/c C /(100M /r), referring to Eq (15). The oxygen concentration distribution with various values of k is shown in Fig. 4. The widely spread oxygen distribution shrinks with increase of k. Thus, to evaluate the oxygen concentration distribution, k must be determined at first. The total reaction rate, I, over the reaction area S can be expressed as IkSC (r)c C...(6) The equilibrium oxygen concentration C eq for the reaction (8) with the carbon concentration of 4.0 mass% at K under pc of about 0.07 MPa 3) is about mass%. Therefore if the oxygen concentration decreases to C eq, the reaction (8) reaches the equilibrium condition and subsequently the reaction ceases. Namely, in order to calculate the total reaction rate over the free surface area, we must integrate the concentration from 0 to r eq (the distance where the oxygen concentration becomes C eq ), and multiplied it with k and carbon concentration. As a first approximation, however, r eq is assumed to be infinite since the oxygen concentration C decreases exponentially so that the contribution to I in this area is essentially insignificant. (The reliability of this assumption will be discussed later.) Then, the integration could be easily carried out as I kc C r rdr kc C...(7) C C S ()π π β 0 From Eq. (7), total reaction rate is proportional to b, and b depends on the diffusivity, viscosity and ds/(d ln C ), here, 0.8 of Eq. (11). Consequently, the apparent total reaction rate also depends on these parameters. The existence of the free surface is a key factor to the direct reduction of iron oxides on Fe C melt. If the area of the free surface is not large enough, C bubbles may form in the melt. Then, the rate limiting step of the molten iron oxide reduction process changes to the reaction of C (ad) C (ad) (ad) from the reaction of C (ad) (ad) C (g), and the overall reaction rate of the molten iron oxide reduction could be decreased. 3. Evaluation of Reaction Rate Constant of the Reaction of C C The observed total reaction rates by Dancy ) and Lloyd et al. 3) range mol/s. Each estimated reaction rate constant by Belton et al. 1) was almost the same and was about mol/cm s if the reacting areas were assumed to be the contact area between the molten iron oxide and Fe C melts of about 0.7 cm and 0.5 cm, 1) respectively. As already mentioned, however, their assumed reaction area would be incorrect. The calculated total reaction rate based on Eq. (7) is shown as a function of k in the logarithmic scale from 5.0 to mol/m s in Fig. 5. The total reaction rate is proportional to 1/k 3 based on Eqs. (5) and (7). When the value of k is 10.0 mol/m s, the corresponding total reaction rate is mol/s and it is very close to that reported by Dancy ) and Lloyd et al. 3) Namely, the reaction rate constant of the reaction (5) can be 10.0 mol/m s. b can be calculated from Eq. (5) and the determined k. It is about 0.63 cm. nce b is obtained, the oxygen concentration distribution can be obtained using Eqs. (3) and (4) as shown in Fig. 6. Actually, the evaluation of the total reaction rate should include the area with the oxygen concentration less than mass%, C eq. However, from Fig. 6, when r is very large about cm compared with b, C is ISIJ

5 ISIJ International, Vol. 50 (010), No. 5 I.0 is negligibly small so that the calculated total reaction rate by the integration from 0 to infinite distance can be acceptable. Thus, the reaction rate constant of the reaction (5) at K is confirmed to be 10.0 mol/m s. Although this deduced value must be regarded as approximate, the order of magnitude is physically reasonable. The activation energy of the molten iron oxide reduction measured by Lloyd et al. 3) is reasonably assumed the same as that of the reaction (8), C C, and was found to be about 50.0 kj/mol. 3) Using this activation energy, the rate constant of the reaction (8), k, in the unit of mol/m s, can be given by k exp(50 000/RT)...(8) where R is the gas constant and T is the absolute temperature. It is noted that the rate constant k is obtained by using the dissolved C and mass% but not the fractional coverage. The rate of the reaction per unit area can be expressed by using the fractional coverage as well as C and mass%, RatekC C C k q q C q...(10) where q C and q are the fraction of surface coverage of adsorbed carbon and oxygen atoms, respectively. k q is the rate constant of the reaction (5). Unfortunately, the fractional coverage is not the linear function of dissolved atoms. Moreover, the behavior of C (ad) or q C should be affected by the (ad). Further work to correlate two reaction rate constants of k q and k is needed to provide quantitative information of the reaction between adsorbed species on the surface of liquid Fe. Fig. 5. Fig. 6. Apparent total reaction rate of the reaction of C C (g) as a function of the reaction rate constant, k. Confirmed oxygen concentration distribution at the surface of Fe C melt as a function of distance from the rim of molten iron oxide. negligibly small about mass%. Tentatively, the contribution of reaction on the area with the radius of more than.0 cm, I.0, was calculated, r I0. πk C S exp rdr 0. β πkc β 0 S 4. Evaluation of C Dissociation Rate The C dissociation reaction is the reverse reaction of the reaction (8) and is expressed as C (g) C...(9) The rate of the C dissociation reaction on the liquid iron melts has not yet been experimentally established at all. The rate measurements by using isotope exchange reaction have been carried out. 15,16) However, the C dissociation rate on liquid Fe C was too fast to conquer the gas phase mass transfer step. The apparent rate of the reaction (9) can be the same as that of the reaction (30), since the rates of reactions (3), (7) and (9) are significantly fast. C (ad) C (ad) (ad)...(30) The rate of the reaction (9) per unit area can be expressed as Ratek C pckc C C...(31) where k C is the forward rate constant of C dissociation reaction (9), and pc is the partial pressure of C gas. Assuming microscopic reversibility, k C kc C C /pck/k eq...(3) where K eq is the equilibrium constant of the reaction (9) and is at K. 16) Substituting the values of K eq and k into the Eq. (3), the reaction rate constant of C dissociation reaction, k C at K is found to be about mol/m s. Sasaki and Ishii 15) estimated the value of k C at 1 53 K by assuming the uncovered reacting area of only 1.15% and was about 47.0 mol/m s. The activation energy of the reaction (30), E C is correlated to DH eq and E k through van t Hoff equation and the correlation is DH eq E C E k...(33) where DH eq and E k are the activation energies of K eq and the reaction rate constant of k, respectively. DH eq is found to be.0 kj/mol. 17) As already mentioned, E k is about 50.0 kj/mol. 3) Substituting these values to Eq. (33), E C is calculated to be 77.0 kj/mol. By using the estimated activation energy value, k C at K is calculated to be mol/m s. It is reasonably agreed with the value evaluated from the iron oxide reduction rate, mol/m s. Using the estimated activation energy, the rate constant of C dissociation reaction (9), k C, in the unit of mol/m s, 010 ISIJ 68

6 ISIJ International, Vol. 50 (010), No. 5 is given by k C exp(77 000/RT)...(34) The rate constant, k C over the range K is shown in Fig. 7 along with the rate constant of C dissociation reaction on Fe C melt, k C. 18) It is shown that k C exceeds k C by more than two order of magnitude. The rate controlling step of the gas phase carburization reaction of molten iron by C H gas mixture has not been fully confirmed. The carburization reaction by C H gas mixture is apparently expressed by CH CH...(35) In the carburization reaction of (35), the reaction rate of C dissociation (30) is believed to be significantly fast (confirmed by the present study), then the rate-limiting process is supposed to be the removal step of the absorbed oxygen (ad) of the reaction (35). (ad)h (ad) H (g)...(36) (ad)c (ad) C (ad)...(37) Sasaki and Belton 19) evaluated the relative ratio of the rate constants of C and H dissociation reactions. The calculated ratio R D at K is ) The relative ratio of the forward reaction rate constants of the reaction (36) and (37) is able to correlate to R D and is given by k H /k C R D (K /K H )...(38) where k C and k H are the forward reaction rate constants of the reactions (36) and (37), respectively. K is the adsorption coefficient of oxygen on Fe melt, and K H is the equilibrium constant of the reaction, H H (g) (g)/. The rate constant of the reaction (37) at K is 11.9 mol/m s. 18) Thus, k H is calculated to be 31.0 mol/m s. The evaluated rate constant of the reaction (36) at K is marked by a solid circle as shown in Fig. 7. It is shown that k C is about -order larger than k H. Based on these rate constants, it can be said that the rate controlling step of the gas phase carburization reaction of molten iron by C H gas mixture is the removal reaction of the absorbed oxygen Fig. 7. Comparison of the rate constants of forward and backward reaction of C C (g) together with that of C dissociation reaction. (ad) on Fe melt. 5. Conclusions (1) The reaction rate constant of the reaction C C (g) at K was evaluated by applying the order of magnitude estimation method and it was found to be about 10.0 mol/m s. Adapting the activation energy based on the work of Lloyd et al., the rate constant, in the unit of mol/m s, is given by: k exp(50 000/ RT). () Assuming microscopic reversibility of the reaction of C C (g), the rate constant of C dissociation reaction of C (g) C at K was found to be about mol/m s. Using the activation energies of k and the equilibrium constant of the reaction C (g) C, the rate constant was expressed in the unit of mol/m s as: k C exp(77 000/RT). (3) When the surface of Fe C melt is partially covered with molten iron oxide, the dissolved oxygen near the molten iron oxide can be effectively transferred to the free surface of Fe C melt by strong Marangoni flow induced by the oxygen concentration gradient. (4) The rate controlling step of the gas phase carburization reaction of molten iron by C H gas mixture was confirmed to be the removal step of the absorbed oxygen (ad) in the reaction (ad)h (ad) H (ad). Nomenclature C C : Bulk carbon concentration in Fe C melt, 4.0 [mass%] C S : Solubility of oxygen in liquid iron, 0.9 [mass%] C : xygen concentration in the melt [mass%] D : xygen atomic diffusivity in liquid iron, [m /s] m : Viscosity of Fe 4mass%C melt, [Pa s] r : Density of Fe 4mass%C melt, 6900 [kg/m 3 ] M : xygen atomic weight, [kg/mol] R : Ideal gas constant [J/mol K] C eq : Equilibrium oxygen concentration for CC with 4.0 mass% C at K under p C 0.07 MPa [mass%] r eq : eq Distance where the oxygen concentration is C q C : Fraction of surface coverage of adsorbed carbon q : Fraction of surface coverage of adsorbed oxygen k : Reaction rate constant for C C [mol/m s] k m : 100kM /r [m/s] k : k m C C [m/s] k C : Reaction rate constant for C C [mol/m s] k C : Rate constant of C dissociation reaction on Fe C melt [mol/m s] k H : Reaction rate constant for (ad)h (ad) H (g) [mol/m s] k C : Reaction rate constant for (ad)c (ad) C (ad) [mol/m s] K eq : Equilibrium constant for CC at K K : Adsorption coefficient of oxygen on Fe melt K H : DH eq : Equilibrium constant for H H (g) (g)/ Activation energy of K eq [kj/mol] E k : Activation energy of k [kj/mol] E C : Activation energy of k C [kj/mol] ISIJ

7 ISIJ International, Vol. 50 (010), No. 5 R D : Relative ratio of the rate constants of C and H dissociation reactions at K REFERENCES 1) G. R. Belton and R. J. Fruehan: Turkdgan Sympo. Proc., Fundamentals and Analysis of New and Emerging Steelmaking Technologies, Iron and Steel Society of AIME, Pittsburgh, (1994), 3. ) T. E. Dancy: J. Iron Steel Inst., 169 (1951), 17. 3) G. W. Lloyd, D. R. Young and L. A. Baker: Ironmaking Steelmaking, (1975), 49. 4) A. Sato, R. Nakagawa, S. Yoshimatsu, A. Fukuzawa and T. zaki: Tetsu-to-Hagané, 67 (1981), ) A. Sato, G. Aragane, K. Kamihira and S. Yoshimatsu: Tetsu-to- Hagané, 73 (1987), 81. 6) D. R. MacRae: J. Met., 17 (1965), ) K. Harashima, S. Mizoguchi, M. Matsuo and A. Kiyose: ISIJ Int., 3 (199), ) T. Iida and R. I. L. Guthrie: The Physical Properties of Liquid Metals, Claredon Press, xford, (1975), ) D. R. Poirier, H. Yin, M. Suzuki and T. Emi: ISIJ Int., 38 (1998), 9. 10) H. Yin and T. Emi: Metall. Mater. Trans. B, 34B (003), ) H. S. Kim, S. H. Lee and Y. Sasaki: ISIJ Int. in press. 1) C. R. Taylor and J. Chipman: Trans. AIME, 154 (1943), 8. 13) E. T. Turkdogan: Fundamentals of Steelmaking, Institute of Materials, London, (1996), ) G. R. Belton: Can. Met. Q., 1 (198), ) Y. Sasaki and K. Ishii: ISIJ Int., 44 (004), ) R. J. Fruehan and S. Antolin: Metall. Mater. Trans. B, 18B (1987), ) Steelmaking Data Sourcebook: the 19th Committee on Steelmaking. JSPS, Gordon and Breach Sci. Pub., Montreux, (1988), ) D. R. Sain and G. R. Belton: Metall. Mater. Trans B, 7B (1976), ) Y. Sasaki and G. R. Belton: Metall. Mater. Trans B, 9B (1998), 89. 0) R. Bird, W. Stewart and E. Lightfoot: Transport Phenomena, John Wiley & Sons, NY, (1960), 83. 1) M. Volmer and A. Weber: Z. Phys. Chem., Stoechiom. Verwandschaftsl., 119 (196), 7. ) R. Becker and W. Doring: Ann. Phys., 4 (1935), ) E. T. Turkdogan: Physical Chemistry of High Temperature Technology, Academic Press, New York, (1980), 46, 83. 4) P. A. Distin, G. D. Hallet and F. D. Richardson: J. Iron Steel Inst., 06 (1968), 81. 5) D. G. C. Robertson and A. E. Jenkins: Heterogeneous Kinetics at Elevated Temperatures, eds. by G. R. Belton and W. L. Worrel, Plenum Press, New York, (1970), 393. where K eq is the equilibrium constant of the reaction (8) and is 419 at K. 17) a C% and a % are the activity of carbon and oxygen in mass%, respectively. The reference state is that of unit activity coefficient at infinite dilution and the standard state is the hypothetical 1 mass% solution. The calculated pc is about MPa at K. This value is quite small compared with that of supersaturation of MPa required for the homogeneous C gas nucleation in liquid Fe C melts. In other words, if heterogeneous nucleation sites are not existed in the melt, the C gas bubble formation never occurs even under the high concentrations of C of 4.0 mass% and of 0.5 mass% in the Fe C melt, or a large amount of dissolved oxygen can exist in the high carbon Fe C melt without C bubble formation at least near the interface. The heterogeneous nucleation sites are generally supposed to be inclusions in the melt and crevices at the crucible wall. In the experiments by Darcy ) and Lloyd et al., 3) the molten iron oxide covered quite small area of less than 10% of the free surface. Therefore, the chance of finding inclusions near the interface can be small. Furthermore, in the Darcy s experiment, crevices were not existed since the molten iron oxide was floated at the bottom of parabolic plane of molten Fe C, or no existence of crevices. The lack of the heterogeneous nucleation sites can be the reason that C gas bubble formation was not observed in their experiments.,3) If C bubble formation is not allowed in the melt, the oxygen can dissolve even in the high carbon Fe C melts. Thus, we may reasonably assume that the oxygen concentration at the interface between molten iron oxide and Fe C melts can be close to that of the oxygen saturation in liquid Fe in the experiments carried out by Darcy ) and Lloyd et al. 3) The supersaturation of C gas in Fe C melts is already confirmed by several researchers. 4,5) In their experiments done with levitated liquid drops of Fe C alloys not in contact with refractory surface and in an oxidizing atmosphere, the C gas bubbles were found to be initiated with supersaturation in the range MPa. 4,5) Appendix A C Gas Nucleation in Fe Melts with High Concentration of Carbon and xygen The theory of homogeneous nucleation was developed by Volmer and Weber 1) and Becker and Doring. ) Based on the theory of homogeneous nucleation, 1,) Turkdogan 3) predicts a supersaturation of MPa for nucleation of gas bubbles in pure iron melts at K with surface tension in the range N/m 1. Since carbon content in Fe C melts has negligible effect on the surface tension, the same discussion can be applicable to liquid Fe C melts. The equilibrium pressure of C gas in the liquid Fe with C of 4.0 mass% and of 0.5 mass% (assumed in the present study) is calculated by assuming the reaction equilibrium of the reaction (8) CC (g)...(8) pck eq a C% a %...(A-1) Appendix B The continuity equation in cylindrical coordinates is given by, 0) 1 ru 0...(B-1) r ( ) r v 1 v ru 0...(B-) r ( ) r where [ ] has the meaning of the order of. The radial surface flow will be quickly diminished in the area beyond the radius of b, since ds/dc 0. Thus, as a characteristic length, r can be replaced by b. If d V is defined as the boundary-layer thickness of the surface flow, Eq. (B-) can be modified to, 010 ISIJ 684

8 ISIJ International, Vol. 50 (010), No. 5 1 βus v 0 us v0...(b-3) β β 0 δv β δv where u s is the surface flow and v 0 is the flow velocity in z- component in the center. The momentum equation for r-component in cylindrical coordinates neglecting gravitational force under the steadystate condition is as following 0) : u u u μ 1 u v ru...(b-4) r r ( ) r r ρ z Likewise, using the characteristic scales, (B-3) can be expressed The left-hand side of (B-5) turns to be zero by the continuity Eq. (B-3). Therefore the right-hand side of (B-5) should become zero. us us...(b-6) β δv Finally, bd V The assumption that the order of b is close to that of d V is proved. s s s s u u u μ s v u u 0 β δv ρ β δv...(b-5) ISIJ