Inclusion Behavior in a Curved Bloom Continuous Caster with Mold Electromagnetic Stirring
Round 1
Reviewer 1 Report
Page 5, Table 2: “Liquid steel density = 7810 kg/m3”
Please add reference from which was obtained this value of liquid steel density.
I think that it is density of solid steel. Usually, the density of liquid steel is significantly smaller (~7000 kg/m3). If all numerical simulations were done by using 7810 kg/m3 value as the liquid steel density the obtained results and conclusions can be wrong.
Pages 5-6, Table 2: Please describe clear for which steel grade and specific non-metallic inclusions (composition, liquid or solid, etc.) were done the presented numerical simulations. According to the given "Inclusion density = 3900 kg/m3", it is pure Al2O3 inclusions. However, in practice, these pure Al2O3 inclusions are solid in the liquid steel at the given temperature and have regular or irregular (but not spherical) shape, if radius of Al2O3 inclusions is larger than 1.5 µm.
Page 7, lines 173-175: “In order to confirm the validity of the numerical results, fluid flow and inclusion number density was validated with the experimental data of the tundish[20]. These results will not be discussed in detail.”
The given reference [20] has not numerical quantitative validation of numerical simulation model for fluid flow and inclusion number density based on the experimental data of the tundish. Please add quantitative validation of your numerical simulation by obtained experimental results of real casting process.
Page 9, lines 214-216 and Figure 8: “Turbulent collision is the most important mechanism of collision among inclusions. It is about one order greater than Archimedes collision and about two orders greater than Stokes collision.”
According to results given in Figure 8, the turbulent collision of inclusions having diameter <10 µm is about two (not one) orders (~100 times) greater than Archimedes collision. It means that the effect of the Archimedes collision is too small.
Page 11, Figure 11b: The oxygen content in the 27SiMn steel given in this figure (6-9 ppm) does not correspond to parameters of inclusions given in Table 2. Based on values given in Table 2 and on assumption that present inclusions are pure Al2O3 inclusions (because “Inclusion density = 3900 kg/m3"), the O content in steel can be calculated around 40 ppm, which is much larger compared to that given in Figure 11b. Therefore, it is very important to use correct data for specific NMIs for numerical simulation and experimental data for comparison.
Author Response
Thanks for the suggestion and I agree with the comment. All comments are very helpful for us to improve our manuscript. We have addressed all comments by the reviewers in detail. The modifications are highlighted in the revised manuscript. 1. Page 5, Table 2: "Liquid steel density = 7810 kg/m3". Please add reference from which was obtained this value of liquid steel density. I think that it is density of solid steel. Usually, the density of liquid steel is significantly smaller (~7000 kg/m3). If all numerical simulations were done by using 7810 kg/m3 value as the liquid steel density the obtained results and conclusions can be wrong. Solution: Thanks for the suggestion. The steel is alloy steel (bearing steel) in our simulation, so the density is greater. Line 154-155 in the revised manuscript as follows. "Table 2 gives geometric and physical parameters during the calculation, which are obtained from the industrial experiment of bearing steel in steel company." 2. Pages 5-6, Table 2: Please describe clear for which steel grade and specific non-metallic inclusions (composition, liquid or solid, etc.) were done the presented numerical simulations. According to the given "Inclusion density = 3900 kg/m3", it is pure Al2O3 inclusions. However, in practice, these pure Al2O3 inclusions are solid in the liquid steel at the given temperature and have regular or irregular (but not spherical) shape, if radius of Al2O3 inclusions is larger than 1.5 µm. Solution: Thanks for the suggestion and I agree with the reviewer's comment. The current work is focus on the inclusion behavior of pure Al2O3 in bearing steel during continuous casting process. The initial characteristic radius is 1.7 µm as solid. In order to simplify calculation, the inclusion are modeled as sphere. Line 81 and line 154-156 in the revised manuscript as follows. "(2) Inclusions are assumed as spherical particles. Table 2 gives geometric and physical parameters during the calculation, which are obtained from the industrial experiment of bearing steel in steel company. The inclusions are pure solid Al2O3." 3. Page 7, lines 173-175: "In order to confirm the validity of the numerical results, fluid flow and inclusion number density was validated with the experimental data of the tundish[20]. These results will not be discussed in detail." The given reference [20] has not numerical quantitative validation of numerical simulation model for fluid flow and inclusion number density based on the experimental data of the tundish. Please add quantitative validation of your numerical simulation by obtained experimental results of real casting process. Solution: Thanks for the suggestion and I agree with the reviewer's comment. We apologize for the carelessness in citation of reference, and this is modified in revised manuscript. There are two reasons about the validation. (1) In this case, we do not consider the solidification in the mathematical model. Thus, the mechanisms of fluid flow and inclusion collision-aggregation in the mold is similar to that in the tundish. The difference between mold and tundish is only the geometric structure. (2) We do not fine the reference about the experimental data of inclusion size distribution in the mold. Line 194-197 in the revised manuscript as follows. "In curved bloom continuous caster, the mechanisms of fluid flow and inclusion behavior are similar with that in tundish. In order to confirm the validity of the numerical results, fluid flow and inclusion number density was validated with the experimental data of the tundish[20,24]. These results will not be discussed in detail." 4. Page 9, lines 214-216 and Figure 8: "Turbulent collision is the most important mechanism of collision among inclusions. It is about one order greater than Archimedes collision and about two orders greater than Stokes collision." According to results given in Figure 8, the turbulent collision of inclusions having diameter
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Despite a few points needing clarification before publication, the paper shows interesting results. The quality of the writing is not perfect but good enough to be understood. However, the comments on the modeling and on the physics are often not very developed. The text of the article is not that long, so there is room for deeper analysis of the presented results. Also the formatting does not help in many places, the authors have used in-line numbered list of short sentences, which breaks the text with disturbing numbers, I suggest changing these lists to bullet lists or into paragraphs for long items. But first, there is a question of vocabulary, right from the abstract: Archimedes’ force, usually refers to buoyancy, while in this submitted paper it refers to an electromagnetic force. A quick search has not revealed extensive usage of this naming convention is the scientific literature, hence naming this force Archimedes’ force is misleading. If I am not mistaken, what the authors name Archimedes’ force is more of a Lorentz force. Also, the paper is in the direct continuation of references [11], [12] and [19], where more information can be found. However, it would be nice to discuss in more details the main assumptions of the model described in the paper. Assumption (3) (line 78) should be discussed more than just providing a reference to another paper (where the assumption is not really justified either). It may be acceptable to justify the usage of such a particle size distribution in order to lower the computational cost of the simulations as it makes the resolution of the whole population balance equation reduce to three variables, N, C, and r*. However, this also introduces significant biases in the inclusion aggregation kinetics since all size classes evolve with completely correlated behavior. Literature on metallurgical reactor modeling using population balances have shown that this assumption is far from being true in general. Ref. [12] cites Zhang et al. (2000) to justify this modeling, but in this reference the initial particle size distribution of inclusions is only piecewise exponential, which does not allow the mathematical derivation presented in ref. [12] and in the article under current review. This makes it a strong limitation of the approach in the current paper that is only acceptable if clearly discussed and justified. Related to the previous point, undefined notations, such as f(r) and r* in equations (10) and (11) must be made explicit. While r* can intuitively be interpreted as some kind of average radius, the details in f(r) hide significant modeling of the inclusion distribution that are not discussed in the paper. f(r) should be replaced by the actual function the authors have apparently used, that is f(r) = A exp(-B r). r* is defined in [12] as “characteristic inclusion radius” but is not much more discussed there either. It should, since it is required to understand Figures 9c and 10c. The turbulent collision model by Saffman and Turner (1956) is very commonly used, yet it should be stated clearly that it is used in equation (15). In the same way, a reference would be welcome for the Brownian collision in equation (12), since only equations (13) and (14) derive from equation (9). In general, the list of references does not give a good overview or the multiple works in related subjects, despite population balances being regularly applied to metallurgical reactors in many published papers. No information is given on the mesh for the finite-volume simulation. It should be provided, at least the number of cells in each direction and their size. It would be nice to explain the derivation of equations (19) and (20) since these are not presented in the same form in Ref. [12] where everything else is detailed. Moreover, the value C₂ in equation (21) must be supported by some reference and explained, τ₀ is not even defined. Apart from Figure 6c that is ugly and hard to read, Figures 6 to 10 show interesting simulations results. Figure 8 is a very interesting piece of information. It is disappointing that the results regarding particle size (r*) are not discussed further. The physical explanation how bigger inclusions appear due to EMS is well detailed, but the consequences in terms of process operation should be commented shortly. Getting less but bigger inclusions in the final product can be very detrimental to mechanical performances of the material, so this may be a significant drawback of EMS. This comment is all the more necessary since the meaning of r* is not discussed, so it not easy read from the figures how much bigger the inclusions are with EMS. About 2µm may seem small enough, but r* is actually an exponent in an exponential distribution (if I understood Ref. [12] correctly), not just an average particle size. Line 260 “As we all know” would better be replaced by a reference to support that statement. Moreover, the conclusion in lines 261 and 262 is worded in a very strong way while it is a very qualitative comparison. It is a very good sign that supports the findings in the paper, but it is not a validation. The conclusion summarizes the important points from the paper in a clear and concise way. It would be interesting to extend it with concluding remarks regarding the process and the role played by EMS on inclusions since, as stated in the introduction, EMS is used in order to promote inclusion removal. As such, it seems effective, but at the cost of getting bigger inclusions, which must be stressed in the conclusion and put into perspective with the overall goal of controlling mechanical performances of the product.Author Response
Thanks for the suggestion and I agree with the comment. All comments are very helpful for us to improve our manuscript. We have addressed all comments by the reviewers in detail. The modifications are highlighted in the revised manuscript. Despite a few points needing clarification before publication, the paper shows interesting results. The quality of the writing is not perfect but good enough to be understood. However, the comments on the modeling and on the physics are often not very developed. The text of the article is not that long, so there is room for deeper analysis of the presented results. Also the formatting does not help in many places, the authors have used in-line numbered list of short sentences, which breaks the text with disturbing numbers, I suggest changing these lists to bullet lists or into paragraphs for long items. But first, there is a question of vocabulary, right from the abstract: Archimedes' force, usually refers to buoyancy, while in this submitted paper it refers to an electromagnetic force. A quick search has not revealed extensive usage of this naming convention is the scientific literature, hence naming this force Archimedes' force is misleading. If I am not mistaken, what the authors name Archimedes' force is more of a Lorentz force. Also, the paper is in the direct continuation of references [11], [12] and [19], where more information can be found. However, it would be nice to discuss in more details the main assumptions of the model described in the paper. Assumption (3) (line 78) should be discussed more than just providing a reference to another paper (where the assumption is not really justified either). It may be acceptable to justify the usage of such a particle size distribution in order to lower the computational cost of the simulations as it makes the resolution of the whole population balance equation reduce to three variables, N, C, and r*. However, this also introduces significant biases in the inclusion aggregation kinetics since all size classes evolve with completely correlated behavior. Literature on metallurgical reactor modeling using population balances have shown that this assumption is far from being true in general. Ref. [12] cites Zhang et al. (2000) to justify this modeling, but in this reference the initial particle size distribution of inclusions is only piecewise exponential, which does not allow the mathematical derivation presented in ref. [12] and in the article under current review. This makes it a strong limitation of the approach in the current paper that is only acceptable if clearly discussed and justified. Related to the previous point, undefined notations, such as f(r) and r* in equations (10) and (11) must be made explicit. While r* can intuitively be interpreted as some kind of average radius, the details in f(r) hide significant modeling of the inclusion distribution that are not discussed in the paper. f(r) should be replaced by the actual function the authors have apparently used, that is f(r) = A exp(-B r). r* is defined in [12] as "characteristic inclusion radius" but is not much more discussed there either. It should, since it is required to understand Figures 9c and 10c. The turbulent collision model by Saffman and Turner (1956) is very commonly used, yet it should be stated clearly that it is used in equation (15). In the same way, a reference would be welcome for the Brownian collision in equation (12), since only equations (13) and (14) derive from equation (9). In general, the list of references does not give a good overview or the multiple works in related subjects, despite population balances being regularly applied to metallurgical reactors in many published papers. No information is given on the mesh for the finite-volume simulation. It should be provided, at least the number of cells in each direction and their size. It would be nice to explain the derivation of equations (19) and (20) since these are not presented in the same form in Ref. [12] where everything else is detailed. Moreover, the value C₂ in equation (21) must be supported by some reference and explained, τ₀ is not even defined. Apart from Figure 6c that is ugly and hard to read, Figures 6 to 10 show interesting simulations results. Figure 8 is a very interesting piece of information. It is disappointing that the results regarding particle size (r*) are not discussed further. The physical explanation how bigger inclusions appear due to EMS is well detailed, but the consequences in terms of process operation should be commented shortly. Getting less but bigger inclusions in the final product can be very detrimental to mechanical performances of the material, so this may be a significant drawback of EMS. This comment is all the more necessary since the meaning of r* is not discussed, so it not easy read from the figures how much bigger the inclusions are with EMS. About 2µm may seem small enough, but r* is actually an exponent in an exponential distribution (if I understood Ref. [12] correctly), not just an average particle size. Line 260 "As we all know" would better be replaced by a reference to support that statement. Moreover, the conclusion in lines 261 and 262 is worded in a very strong way while it is a very qualitative comparison. It is a very good sign that supports the findings in the paper, but it is not a validation. The conclusion summarizes the important points from the paper in a clear and concise way. It would be interesting to extend it with concluding remarks regarding the process and the role played by EMS on inclusions since, as stated in the introduction, EMS is used in order to promote inclusion removal. As such, it seems effective, but at the cost of getting bigger inclusions, which must be stressed in the conclusion and put into perspective with the overall goal of controlling mechanical performances of the product. Solution: Thanks for the detailed suggestion. We have studied the comments carefully and listed the response in the following order. (1) According to reference [4] (revised manuscript), the vocabulary of "Archimedes force" is modified as "Archimedes electromagnetic force". Besides, "Archimedes slipping velocity" is modified as "Archimedes electromagnetic slipping velocity", and "Archimedes collision" is modified as Archimedes electromagnetic collision. These revisions are adopted throughout the manuscript. Line 41-42 in the revised manuscript as follows. "In the electromagnetic field, the inclusion movement is affected by Archimedes electromagnetic force[4]." (2) Assumption (3) is further explained in the revised manuscript. Line 82-84 in the revised manuscript as follows. "(3) According to industrial experiment, the fractional inclusion number density increases exponentially as the inclusion size decreases, , and A and B only depend on time and coordinates[3,12,13]." (3) Our previous work in Reference [20] (revised manuscript) shows that the predicted inclusion number density agrees with the experimental data when the inclusion diameter is less than 15.5 μm, but there are obvious differences between the predicted number density and the experimental data when the inclusion diameter was greater than 15.5μm. Several reasons lead to such difference: (a) The inclusion number density distribution function and inclusion size satisfy the exponential function approximately. (b) In experiment, the number density of small inclusions increases and the number density of large inclusions decreases because one cluster inclusion is regarded as several smaller inclusions during two dimensional (2D) inclusion analysis. (c) In experiment, the increase of the number density of large inclusions may result from refractory erosion, while the current mathematical model ignores this factor. (4) The equation of is added in assumption (3), and equation (10) and (11) is further described in the revised manuscript. Line 82-84 and line 123-129 in the revised manuscript as follows. "(3) According to industrial experiment, the fractional inclusion number density increases exponentially as the inclusion size decreases, , and A and B only depend on time and coordinates[3,12,13]. The inclusion slipping velocity in equation (7) and equation (8) is the function of Archimedes electromagnetic force and density difference between molten steel and inclusion, as shown in equation 9. Thus the inclusion slipping velocity also include two parts: Stokes floating velocity and Archimedes electromagnetic slipping velocity. In this way, on the base of in assumption (3), the inclusion slipping velocity can be expressed as follows. (10) (11)" (5) Line 103-106 in the revised manuscript as follows. "On the base of the mass conservation, the inclusion characteristic radius (or volume average radius) is the function of inclusion volume concentration (C) and the inclusion number density (N). (6)" (6) Reference [21] and reference [22] are added in the revised manuscript to further describe equation (12) and (15). Line 138 and line 141 in the revised manuscript as follows. "Brownian collision[21] (12) Turbulent collision[22] (15)" (7) The detail of mesh configuration is given in revised manuscript. Line 153-154 and line 160-164 in the revised manuscript as follows. "Figure 3 shows the schematic diagram of M-EMS, and the computational domain is covered by about 350 000 structured grids. (a) Schematic diagram (b) Local grids Figure 3. Schematic diagram and local grids of M-EMS with curved continuous casting bloom" (8) Equation (19), (20) and (21) are further explained in the revised manuscript. Line 172-180 in the revised manuscript as follows. "For the inclusion in liquid steel, there are two transport mechanisms: the convection and the diffusion. The inclusion flux at the walls includes the convection flux , ) and the diffusion flux ( , ): (19) (20) where is the normal vector, and the coefficient of can be expressed as: (21) where is the turbulent wall friction, Pa, and is the empirical constant in the standard turbulent model." (9) Figure 6c describes the liquid steel flow in curved bloom continuous caster without M-EMS. The length arrow represents the magnitude of fluid velocity. In general, the fluid velocity in the central of the cross-section is less than the surrounding zones in Figure 6c and 7c. Line 212-213 in the revised manuscript as follows. "(3) The velocity of liquid steel near the central of cross-section is less than surrounding zones." (10) Line 287-291 in the revised manuscript as follows. "The total oxygen mass fraction is the sum of dissolved oxygen and insoluble oxygen in the oxide inclusion, and the dissolved oxygen is saturated in the steel. Thus, the distribution of total oxygen mass fraction is similar to the distribution of the inclusion volume concentration. In this way, we can come to this conclusion: the predicted inclusion distribution in Figure 10 conforms to the industrial experiment quantitatively."
Author Response File: Author Response.pdf
Reviewer 3 Report
This is an interesting and well-organized paper concerning behavior of inclusions in a continuous caster with electromagnetic stirring. The major shortcoming of this paper that the authors omitted important explanations and references. Below is the main points which need to be properly improved and clarified before publication.
- In the introduction of this paper the authors mentioned the importance of Archimedes force when considering the behavious of inclusions. It seems this was the main motivation to undertake this study. However, in conclusion, the authors summarized that the Archimedes collisions are much less important as compared to the turbulent collisions. By the way, in one of the previous studies of the authors (ISIJ International, Vol. 59 (2019), No. 10), where they investigated behaviour of inclusions in tundish using a very similar approach and similar numerical model, the conclusion was that Archimedes collisions are of prime importance in the inclusion collison and coalecsence. It is the reviwer’s oppinion that, in general, the turbulence befaviour of particles including their collision in a tandish is even more important than that in a continuous caster. Proper comments and explanations should be added to the maniscript
- If turbulent collisions are considered as the main factor influencing the inclusion behavior, the values of turbulence dissipation rate, e (epsilon) are of crucial importance in their model. However, the authors did not give any detais on how they calculated the turbulence dissipation rate and what turbulent model was used in their calculation. Appropriate data and their explanations should be added to the manuscript.
- What is the chemical composition of inclusions used in the study ? A question to be answered is whether or not the electromagnetic migration force and migration-driven collisions between inclusion particles can be ignored ? At least this issue should be discussed in the manuscript.
- A lot of relationships are given without explanations or/and references. Examples are Eqs.(10),(11) , (20) and some others. It makes reading the paper difficult.
Author Response
Thanks for the suggestion and I agree with the comment. All comments are very helpful for us to improve our manuscript. We have addressed all comments by the reviewers in detail. The modifications are highlighted in the revised manuscript. This is an interesting and well-organized paper concerning behavior of inclusions in a continuous caster with electromagnetic stirring. The major shortcoming of this paper that the authors omitted important explanations and references. Below is the main points which need to be properly improved and clarified before publication. 1. In the introduction of this paper the authors mentioned the importance of Archimedes force when considering the behaviors of inclusions. It seems this was the main motivation to undertake this study. However, in conclusion, the authors summarized that the Archimedes collisions are much less important as compared to the turbulent collisions. By the way, in one of the previous studies of the authors (ISIJ International, Vol. 59 (2019), No. 10), where they investigated behavior of inclusions in tundish using a very similar approach and similar numerical model, the conclusion was that Archimedes collisions are of prime importance in the inclusion collision and coalescence. It is the reviewer's opinion that, in general, the turbulence behavior of particles including their collision in a tundish is even more important than that in a continuous caster. Proper comments and explanations should be added to the manuscript Solution: Thanks for the suggestion. In reference [17] (revised manuscript) (ISIJ International, Vol. 59 (2019), No. 10), we elaborated that Archimedes electromagnetic collision is only one of the important collision mechanisms for inclusion coalescence, turbulent collision is of prime important. The effect of Archimedes electromagnetic force on the inclusion behavior consist of Archimedes electromagnetic slipping velocity and Archimedes electromagnetic collision. According to Figure 8 in current manuscript, it can be concluded that turbulent collision rate is greater than Archimedes electromagnetic collision. And with inclusion diameter increases, the difference between turbulent collision rate and Archimedes electromagnetic collision decreases gradually. Line 236-243 in the revised manuscript as follows. "The effect of Archimedes electromagnetic force on the inclusion behavior consists of Archimedes electromagnetic slipping velocity and Archimedes electromagnetic collision. Figure 8 gives the contribution of different collision mechanism to the inclusion collision. Turbulent collision is the most important mechanism of collision among inclusions. It is about one order greater than Archimedes electromagnetic collision and about two orders greater than Stokes collision. And with inclusion diameter increases, the difference between turbulent collision rate and Archimedes electromagnetic collision decreases gradually." 2. If turbulent collisions are considered as the main factor influencing the inclusion behavior, the values of turbulence dissipation rate, e (epsilon) are of crucial importance in their model. However, the authors did not give any details on how they calculated the turbulence dissipation rate and what turbulent model was used in their calculation. Appropriate data and their explanations should be added to the manuscript. Solution: Thanks for the suggestion and I agree with the reviewer's comment. Line 50-51 and line 74-75 in the revised manuscript as follows. "In the curved bloom continuous caster, the liquid steel flow is calculated by the standard turbulence model. is the effective viscosity coefficient, Pa•s, which is determined by standard turbulence model;" 3. What is the chemical composition of inclusions used in the study? A question to be answered is whether or not the electromagnetic migration force and migration-driven collisions between inclusion particles can be ignored? At least this issue should be discussed in the manuscript. Solution: Thanks for the suggestion and I agree with the reviewer's comment. The inclusions are pure solid Al2O3. The buoyancy of particle with the same size depends on the function of the density difference between steel and inclusion, it is a constant. But the Archimedes electromagnetic force is the function of electromagnetic force. It will change with the local electromagnetic condition. Thus, Archimedes electromagnetic migration and Archimedes electromagnetic collision cannot be ignored. Line 155-156 and line 111-117 in the revised manuscript as follows. "The inclusions are pure solid Al2O3. The forces acting on the inclusion particle include the gravity, the buoyancy, the viscous drag and Archimedes electromagnetic force. The buoyancy of particle with the same size depends on the function of the density difference between steel and inclusion, it is a constant. But the Archimedes electromagnetic force is the function of electromagnetic force. It will change with the local electromagnetic condition. Thus, the terminal velocity of inclusion can be obtained under the condition of force balance of inclusion particle: (9)" 4. A lot of relationships are given without explanations or/and references. Examples are Eqs.(10), (11) , (20) and some others. It makes reading the paper difficult. Solution: Thanks for the suggestion and I agree with the reviewer's comment. Line 103-106, line 123-129, line 137-141 and line 172-180 in the revised manuscript as follows. "On the base of the mass conservation, the inclusion characteristic radius (or volume average radius) is the function of inclusion volume concentration (C) and the inclusion number density (N). (6) The inclusion slipping velocity in equation (7) and equation (8) is the function of Archimedes electromagnetic force and density difference between molten steel and inclusion, as shown in equation 9. Thus the inclusion slipping velocity also include two parts: Stokes floating velocity and Archimedes electromagnetic slipping velocity. In this way, on the base of in assumption (3), the inclusion slipping velocity can be expressed as follows. (10) (11) Their related collision rates can be expressed as follows: Brownian collision[21] (12) Stokes collision (13) Archimedes electromagnetic collision (14) Turbulent collision[22] (15) For the inclusion in liquid steel, there are two transport mechanisms: the convection and the diffusion. The inclusion flux at the walls includes the convection flux , ) and the diffusion flux ( , ): (19) (20) where is the normal vector, and the coefficient of can be expressed as[23]: (21) where is the turbulent wall friction, Pa, and is the empirical constant in the standard turbulent model."
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
1. It is well known that the density of solid iron (or low alloyed steel) at 25oC is 7800-7860 kg/m3 and the density of the iron melt or liquid steel at metallurgical temperatures is significantly lower (~7000-7100 kg/m3). The density value (= 7810 kg/m3) given in Table 2 of this paper as “Liquid steel density” corresponds to the solid bearing steel measured at room temperature (https://www.astmsteel.com/product/52100-bearing-steel-aisi/). However, the liquid bearing steel at 1768 K temperature should be significantly lower. Please check and correct this density (or add a reference, in which this value was obtained by measurement of density of liquid bearing steel).
4. According to Figure 8, the following values of collision rate for inclusions having diameter 5 µm can be obtained: Turbulent collision (for 0.001 m2s-3) ~ 3x10-14, Archimedes electromagnetic collision ~ 2x10-16, Stokes collision ~ 1x10-16. It follows from these values that the ratio between collision rates for Turbulent collision and Archimedes electromagnetic collision is >100 (3x10-14 / 2x10-16 =150). It means that the Turbulent collision is about two (not one) orders greater than Archimedes collision.
5. If the numerical simulation was done for bearing steel, it is strange that the validation of obtained results forbearing steel was done based on experimental results for 27SiMn steel. Characteristics of these steels, inclusions and casting conditions can be significantly different. Therefore, this explanation and validation are very doubtful (not convincing) for me.
Author Response
- It is well known that the density of solid iron (or low alloyed steel) at 25oC is 7800-7860 kg/m3 and the density of the iron melt or liquid steel at metallurgical temperatures is significantly lower (~7000-7100 kg/m3). The density value (=7810 kg/m3) given in Table 2 of this paper as "Liquid steel density" corresponds to the solid bearing steel measured at room temperature (https://www.astmsteel.com/product/52100-bearing-steel-aisi/). However, the liquid bearing steel at 1768 K temperature should be significantly lower. Please check and correct this density (or add a reference, in which this value was obtained by measurement of density of liquid bearing steel).
Solution: Thanks for the suggestion. According to the reference of bearing steel (https://www.astmsteel.com/product/52100-bearing-steel-aisi/), the melting point is 1697K, and the thermal expansion coefficient of solid phase is 11.9*10-6/K. The thermal expansion coefficient of liquid phase is an estimated value of 1*10-4/K(Aboutalebi, M.R.; Hasan, M.; Guthrie, R.I.L. Coupled turbulent flow, heat, and solute transport in continuous casting processes. Metall. Mater. Trans. B 1995, 26(4), 731-744), and the degree of superheat is 30K. Thus, the density from solid phase to liquid phase of bearing steel can be approximated expressed by:
(11.9*10-6/K)*(1697K-298K)+(1*10-4/K)*30K=0.0196481=2%
The density difference of bearing steel between solid phase and liquid phase can be ignored in the current work, so the density of liquid bearing steel is 7810 kg/m3 in the manuscript.
- According to Figure 8, the following values of collision rate for inclusions having diameter 5 µm can be obtained: Turbulent collision (for 0.001 m2s-3) ~ 3x10-14, Archimedes electromagnetic collision ~ 2x10-16, Stokes collision ~ 1x10-16. It follows from these values that the ratio between collision rates for Turbulent collision and Archimedes electromagnetic collision is >100 (3x10-14 / 2x10-16 =150). It means that the Turbulent collision is about two (not one) orders greater than Archimedes collision.
Solution: Thanks for the suggestion and I agree with the reviewer's comment.
Line 239-243 in the revised manuscript as follows.
"Turbulent collision is the most important mechanism of collision among inclusions. It is about two orders greater than Archimedes electromagnetic collision and Stokes collision if the inclusion size is less than 10 μm, and it is about one order greater than Archimedes electromagnetic collision and Stokes collision if the inclusion size is greater than 10 μm."
- If the numerical simulation was done for bearing steel, it is strange that the validation of obtained results for bearing steel was done based on experimental results for 27SiMn steel. Characteristics of these steels, inclusions and casting conditions can be significantly different. Therefore, this explanation and validation are very doubtful (not convincing) for me.
Solution: Thanks for the suggestion. Bearing steel continuous casting technology involves in M-EMS and soft-reduction. At the final solidification zone, soft-reduction affects the fluid flow in the liquid pool. But the current numerical simulation doesn't consider the soft reduction. Thus, the numerical result cannot be validated by the total oxygen distribution of bearing bloom. Fortunately, 27SiMn steel continuous casting technology only involves in M-EMS, so our numerical result is validated by the total oxygen distribution of 27SiMn bloom quantitatively.
Author Response File: Author Response.pdf
Reviewer 2 Report
The authors have addressed my comments. They should proofread the parts that they have amended to answer the reviewers’ remarks, because their English is much less understandable there than in the rest of the text. Some sentences are quite hard to understand, which is why I recommend acceptance after minor revision. Beside that, the comments hereinafter are of secondary importance. I stand by my remark that “Archimedes electromagnetic force” is an unusual naming convention. The authors have justified such a name by referencing a paper where the name appears only once and that is itself based on a book form 1984. It confirms that this naming convention is not very common. On the other hand, the expression given in reference [4] is the one that commonly appears as “Lorentz force” (https://en.wikipedia.org/wiki/Lorentz_force). This is a detail and if the authors are much more comfortable with the naming as “Archimedes electromagnetic force”, they can keep it, this force does result from the action of a field on particle’s volume, and hence it is some kind of Archimedes force. I find the discussion about the exponential law for inclusion PSD to be disappointing, especially since the authors provided interesting complementary information in their answer to the reviewers. The revised wording of the list of hypotheses is much more clear than it was in the original submission and it adds citations that make it relatively easy for the reader to find more complete information on that assumption. As such, it meets the minimal requirement to address my remark, and may be published as is, but it is disappointing that the authors did not comment more. What the authors have included in their answers to reviewers (point (3)) is the kind of comment that would have been beneficial to the article. The reason why I insist on that point is that Reference [3] presents actual measurements, approximates inclusion PSD as a piecewise exponential distribution, and solves the population balance with a class method where classes are defined as particle size ranges. Reference [12] shows that if inclusion are distributed in size following an exponential law, the population balance equation can be simplified and resolved with a single apparent radius for the whole size distribution, but their derivation does not apply to piecewise exponentials. Reference [13] shows an example of an exponential law for inclusion distribution, but it makes clear that this is an assumption: “Assuming the density distribution, n₁⁰…”. So, in the end, when the authors of the current draft state “According to industrial experiment, the fractional inclusion number density increases exponentially as the inclusion size decreases”, this is false and (purposely ?) misleading. They should have made it clear that this is an assumption, and have found a justification for this assumption. The interest of simplifying the inclusion PSD to an exponential is totally understandable, in regard to the drastic simplification of the model that it leads to. However, it makes the simulations a bit less robust, so it should have been stated clearly. The current (even revised) wording of this part looks suspicious, like the authors try to hide this approximation, while a slightly more careful wording would make it clear that such a level of approximation is worth doing, since it makes it possible to computationally track inclusion evolution over time in the whole casting bloom.Author Response
Solution: Thanks for the suggestion and I agree with the reviewer's comments. These comments are very helpful for us to improve our manuscript. We have studied the comments carefully and listed the response in the following order.
(1) We ask a professor in our lab to review this paper and make some revisions.
Line 28-30, 163-164, 216-217, 219-220, 232-233 and 290-291 in the revised manuscript as follows.
"In recent years, the quality requirements of steel products are increasingly strict for the excellent drawing property, low-temperature toughness and fatigue resistance."
"For the electromagnetic field, the software ANSYS is employed to obtain the electromagnetic force."
"A strong jet flow comes from the nozzle to the mold, and then spreads outward gradually with the fluid flow."
"In the M-EMS central cross-section, the fluid velocity near the mold wall is greater than that near the center."
"The strong electromagnetic force also results in the more contribution of Archimedes electromagnetic collision on the inclusions."
(2) Line 41-46 in the revised manuscript as follows.
"In the electromagnetic field, the inclusion movement is affected by the electromagnetic force (Archimedes electromagnetic force[4]), which has the similar characteristics to the buoyancy (Archimedes force) of inclusion particle in molten steel. The difference between Archimedes force and gravity of inclusion causes Stokes collision. Such a rule also applies to Archimedes electromagnetic force. Consequently, the collision among inclusions caused by Archimedes electromagnetic force is called as Archimedes electromagnetic collision."
(3) Line 88-90 in the revised manuscript as follows.
"In order to simplify the inclusion collision-aggregation model, it is assumed that the fractional inclusion number density increases exponentially as the inclusion size decreases, f(r)=Ae-Br , and A and B only depend on time and coordinates[3,12,13]."
Author Response File: Author Response.pdf
Reviewer 3 Report
Thanks for revision
Author Response
Solution: Thanks for the suggestion and I agree with the comment. All comments are very helpful for us to improve our manuscript. All revisions are highlighted in the revised manuscript. are adopted throughout the manuscript.
Round 3
Reviewer 1 Report
Please add the given explanations for questions 1 and 5 in the article text for clear understanding.
Author Response
- Please add the given explanations for questions 1 and 5 in the article text for clear understanding.
Solution: Thanks for the suggestion and I agree with the reviewer's comment. The comment is very helpful for us to improve our manuscript. We have addressed all comments by the reviewer in detail. The explanations for questions 1 and 5 are highlighted in the revised manuscript.
Line 70-72 and line 286-290 and line in the revised manuscript as follows.
"The density of bearing steel at liquid state is equal to that at solid state because the density difference of bearing steel is about 2% when the temperature changes from 298 K to 1768 K."
"In steelmaking process, bearing steel continuous casting technology involves in M-EMS and soft-reduction. At the final solidification zone, soft-reduction affects the fluid flow in the liquid pool. But the effect of soft-reduction is not considered in the current numerical simulation. Fortunately, 27SiMn steel continuous casting technology only involves in M-EMS, so our numerical result is validated by the total oxygen distribution of 27SiMn bloom quantitatively."
