Round 1

Reviewer 1 Report

Review to

Numerical Simulation of Melting Kinetics of Metal Particles during Tapping with Argon-Bottom Stirring

by

Kinnor Chattopadhyay, Rodolfo D. Morales, Alfonso Nájera-Bastida, Jafeth Rodríguez-Ávila and Carlos Rodrigo Muñiz-Valdés

The paper is about the numerical simulation of fluid dynamics of steel melt with Argon-steering and melting kinetics of added metal particles. It describes used numerical method, provides analysis of particle melting trough simulation of particle trajectory and melting rate in corresponding conditions for the different initial points of immersion. Conclusions are supported by the data. The language is clear and the paper is well organized. The title matches the content.

The figures are of good quality and prepared carefully.

Paper gives valuable information on different ways of tapping process for baths with stirring and might be useful for the design of melting furnaces. The paper can be published with minor changes.

Questions:

1) It is stated that the system is isothermal (p.2). However there a difference between melt and particle. It is more correct to say that melt is considered isothermal.

2)Physical properties in table 2 should be supplemented with the reference and value dimension.

3)Reference for properties in table 3 ([22]) is given to the paper on the melting process. It might be better to cite the direct source of properties.

4)The density is given both as ρ and ϱ symbols within the paper. Is it on purpose? Then it should be explained and ρ added to nomenclature.

5)The value T in nomenclature is explained as temperature and total stress. It is required to use separate symbols for different values.

6)First of conclusions on p.18 (lines 388-3390) is basically just a used Nusselt number definition and should be reconsidered.

Author Response

Reply to the reviewers of the paper Numerical Simulation of Melting Kinetics of Metal Particles during Tapping with Argon-Bottom Stirring by

Kinnor Chattopadhyay, Rodolfo D. Morales, Alfonso Nájera-Bastida, Jafeth Rodríguez-Ávila and Carlos Rodrigo Muñiz-Valdés

Reviewer 1

• It is stated that the system is isothermal (p.2). However there a difference between melt and particle. It is more correct to say that melt is considered isothermal.

1. Indeed, your suggestion is correct.
2. Physical properties in table 2 should be supplemented with the reference and value dimension.
3. Table 2 refers to the dimensionless numbers. Mr. Zhang and Professor Oeters, reference (23), defined the dimensionless numbers governing heat conduction in melting processes. We simply used them as they are correct and define well the thermal dynamics.
4. Reference for properties in table 3 ([22]) is given to the paper on the melting process. It might be better to cite the direct source of properties.
5. It is correct, we have added two additional original sources for these properties. However, the reviewer must be referring to reference (23). New references (25) and (26) are now in the manuscript.

4)The density is given both as ρ and ϱ symbols within the paper. Is it on purpose? Then it should be explained and ρ added to nomenclature.

1. It is misprinting. It is corrected.

5)The value T in nomenclature is explained as temperature and total stress. It is required to use separate symbols for different values.

1. Correct. The symbol of total stress has been changed.

6)First of conclusions on p.18 (lines 388-3390) is basically just a used Nusselt number definition and should be reconsidered.

1. We referrer now to the heat transfer coefficient which is a more fundamental parameter than the Nusselt number.

We acknowledge and give thanks to the reviewer for its valuable assessment and important observations.    

Reviewer 2 Report

This is a very nice paper showing numerical simulations of how particles melt under differing flow conditions that are captured at the scale of the ladel.

I have only very minor comments mostly to do with clarification:

1. Equation 7 and following text:
I think some clarification on the variables would be useful here.
The subscripts are somewhat confusing, authors are using i, j and k for directions and it is not immediately clear what l is (I assume it is for liquid, is this the same as q = 1?)
Vorticity is defined while discussing the third term, which does not contain it. It might be better to put this later.

There are also some either missing or additional commas for example v_p,i and omega_l,i and more in the rest of the text.

2. Argon Boundary Condition:
The authors state the Argon is a velocity boundary condition (inlet), but it would be useful to give a value or equation i.e. is it a fixed flow rate?

3. Equation 8:
I assume A is the cross sectional area, but it is not defined in the text or in the nomenclature.

4. Air is only mentioned twice in the paper. Properties are given in table 2 and then mentioned once in the text at line 257.
It's not clear where air comes into the simulation.

5. The authors should mention what the green iso surfaces are in the results?

6. Table 3:
Dimensionless radius and Dimensionless Shell radius is confusing.
Dimensionless radius has same definition as the shell radius and Dimensioness Shell radius uses variables defined in the nomenclature as particle trajectory (s) and ratio (r).

7. Some variables in the text or not Italic

8. Eq 16 repeated twice

Author Response

Reply to the reviewers of the paper Numerical Simulation of Melting Kinetics of Metal Particles during Tapping with Argon-Bottom Stirring by

Kinnor Chattopadhyay, Rodolfo D. Morales, Alfonso Nájera-Bastida, Jafeth Rodríguez-Ávila and Carlos Rodrigo Muñiz-Valdés

Reviewer 2

1. Equation 7 and the following text:
   I think some clarification on the variables would be useful here.
   The subscripts are somewhat confusing, authors are using i, j and k for directions and it is not immediately clear what l is (I assume it is for liquid, is this the same as q = 1?)
   Vorticity is defined while discussing the third term, which does not contain it. It might be better to put this later.

There are also some either missing or additional commas, for example, v_p,i and omega_l,i and more in the rest of the text.

463. Thank you for your observations. We added a definition of these sub-indexes in the nomenclature, line 463.

Yes, l is for liquid when we emphasize the metal phase and q, represents the index for liquid metal, air and argon. See line 112. It is mentioned in the text that, although argon and air constitute a single gas phase we, artificially, separated them using an additional dummy index in the VOF model.

You are correct, we placed, in line 175, the vorticity term which belong to the lift force.

We reviewed the comas. Thanks.

2. Argon Boundary Condition:
   The authors state the Argon is a velocity boundary condition (inlet), but it would be useful to give a value or equation i.e. is it a fixed flow rate?
3. Again you are correct. This velocity was estimated from a flow rate of 400 l/min of argon and divided by the cross-section area of the porous plug, in contact with the melt with a diameter of 60 mm, assuming uniform plug flow. See lines 150-151.

3.Equation(8):
I assume A is the cross-sectional area, but it is not defined in the text or in the nomenclature.

413. Indeed. The cross-section area of the metallic particle in the bath. It is specified in the nomenclature, line 413.
414. Air is only mentioned twice in the paper. Properties are given in table 2 and then mentioned once in the text at line 257.
     It's not clear where air comes into the simulation.

• That is correct. Although argon and air constitute a single gas phase we, artificially, separated them using an additional dummy index in the VOF model. This trick is useful to track air-argon interfaces. That is because argon is heavier than air and therefore, it would be interesting how argon segregates from the air. This is mentioned in the new text, see lines 87-89.
• The authors should mention what the green isosurfaces are in the results?

241. They represent the iso-surfaces of gas bubbles. This I mentioned in the new text, see line 241.
242. Table 3:
     The dimensionless radius and Dimensionless Shell radius is confusing.
     Dimensionless radius has same definition as the shell radius and Dimensions Shell radius uses variables defined in the nomenclature as particle trajectory (s) and ratio (r).
243. totally agree, they are ill-defined. This is corrected in the new manuscript in red.
244. Some variables in the text or not Italic
245. Reviewed
246. Eq 16 repeated twice
247. Corrected

We are glad that you liked our paper and convey to you our gratitude for your valuable time and professional work. Thanks.

Reviewer 3 Report

Issues that must be resolved.

It is worth noting that although you are interested in ensuring a homogenous chemical composition during the addition of alloying elements in the tapping process, you are not modelling the mass transport of the individual alloying elements within the melt. The proposed model provides insights into the liquation kinetics that can assess the impact of altering the spatial location in which the metal additions are introduced into the melt without modelling multi-component mass transport.

You provide a list of the assumptions made in section 2.1. You should provide a similar list of assumptions in section 2.4, including;

• The assumption that the kinetics of the particles have an insignificant impact upon the flow of the molten steel.
• The particles are assumed to instantaneously melt and form a solidified shell of steel surrounding the metal addition.
• You are ignoring any interaction between particles and only consider the trajectory of a single particle.

In the process, how many particles are added? At what point would the particles influence the flow of the melt? Would any interaction between the particles influence melting kinetics, such as amalgamation? Would their velocity be uniform? These points could be included in the discussion.

• The approximation of the temperature averaged thermophysical properties used to determine the size of the initial solid shell and the following liquation kinetics should also be in this list of assumptions. This also simplifies the complexity arising from spatial variations in chemistry altering the solidus-liquidus property diagram and thermo-physical properties.

The paper is lacking a statistical evaluation of the kinetics of the particles and should be able to answer the following questions. Does the model predict any stochastic variation in the molten flow? If so, how sensitive are the predictions if the metal particle is added moments later? How sensitive is the melting time to deviations in the velocity upon introduction to the melt? What scatter in velocity is to be expected and would it influence behaviour? The answer to these questions would determine the extent in which ignoring particle addition interactions is acceptable for modelling the process.  If the results are not sensitive to such changes, it suggests that particle interaction is not as important as the flow of the melt.

Some sentences need minor rephrasing.

The sentence starting on line 35 “Steel tapping is crucial” needs rework. It might need split into two sentences to convey the importance of the tapping process and then its complexity.

Line 37 “The efficiency of the alloys” – do you mean the efficiency of producing the alloys?

Line 38 “Steel tapping together with bottom argon …” should be rephrased to something like “Steel tapping processes with argon stirring in the ladle create a forced convection current which strongly influences the dissolution of alloying additions, and impacts the thermal and chemical homogenization of the ladle”.

Line 81 – either “There is no slag phase in the system”, or “There is not a slag phase in the system”

Line 385 – This sentence needs rephrased. Something like “The present research focusses on the melting times …” rather than “The present research works about the melting times …”.

Suggestions for the authors to consider.

The paper would benefit from discussing how the model could be adapted to other common alloying elements. What elements is the current model capable of simulating as-is? What changes are needed to capture others? What about elements with a higher melting point such as Cr described in table 4? It would be interesting to see how the optimum location of placing the metal additions varies for metal additions that are lighter than Fe.

In the Practical Implications section you could state the calculated optimum positioning of the nickel additions at all of the different tonnages of melt investigated, noting that the most problematic time for ensuring complete liquation and mixing of heavy additions such as Ni is for shallow baths.

In the paragraph starting on line 127, it would be worth mentioning details regarding the mesh sensitivity study performed to arrive at this mesh and the corresponding numerical error. This is not mandatory as many papers omit this however it would provide greater confidence in the results if the numerical error is known.

A small illustration of the ladle with an axis of origin and a cross section identifying what cross section is being viewed would be useful for the velocity vector plots.

In Table 4, you could put (s) for the Cr, Mn and Ni. I guess Ts means the solvus temperature rather than the solidus temperature. You could use Tl for the liquidus temperature or Tm for the melting temperature to make this clearer.

The layout of Figure 7 could be improved, with a title to the colour bar for Figure 7 c), so that it does not look like it is time at first glance.

In future work you could choose locations in polar coordinates and make a contour graph of the predicted melting times for different tonnages of the melt for a given superheat instead of the bar charts shown in Figure 10.

Is there any way to validate the model predictions? Can you see if the chemical homogenization in steel created with metal additions from different locations correlate to the predicted melting times? Is there any way to validate the predicted velocities of the melt? This model only needs to be qualitatively accurate to guide the decision-making process rather than predict exact liquation times. This would be worth noting in the manuscript.

Author Response

Reviewer 3

It is worth noting that although you are interested in ensuring a homogenous chemical composition during the addition of alloying elements in the tapping process, you are not modeling the mass transport of the individual alloying elements within the melt. The proposed model provides insights into the liquation kinetics that can assess the impact of altering the spatial location in which the metal additions are introduced into the melt without modeling multi-component mass transport.

1. You are right, we are not modeling mass transport here because we are more interested on fluid flow problems related with heat transfer. In the actual steelmaking shop, stirring is so intense that mass transfer is rapid an efficient. Therefore, alloying kinetics is the rate control of the process, though, we are about to deal with this topic soon. As you pointed out our interest is focused on the melting kinetics affected by local turbulence.

You provide a list of the assumptions made in section 2.1. You should provide a similar list of assumptions in section 2.4, including;

• The assumption that the kinetics of the particles have an insignificant impact upon the flow of the molten steel.
• The particles are assumed to instantaneously melt and form a solidified shell of steel surrounding the metal addition.
• You are ignoring any interaction between particles and only consider the trajectory of a single particle.

In the process, how many particles are added? At what point would the particles influence the flow of the melt? Would any interaction between the particles influence melting kinetics, such as amalgamation? Would their velocity be uniform? These points could be included in the discussion.

1. You are pointing out very important questions here that, obviously, deserve consideration. First, we are simulating single particles and this means that there is not interaction with other particles. Thereby, the field of a particle does not affect the field of the melt, then there is not existence of possible amalgamation. However, the results presented here are valid for sinking particles like nickel, niobium, chromium, etc., because the amounts of particles added are small. For example, Nb concentrations are about 0.01-0.05%. In the new manuscript, in the section of the assumptions, we specify this very clearly thanks to your sharp observation.

• The approximation of the temperature averaged thermophysical properties used to determine the size of the initial solid shell and the following liquation kinetics should also be in this list of assumptions. This also simplifies the complexity arising from spatial variations in chemistry altering the solidus-liquidus property diagram and thermo-physical properties.

1. Thanks for these observations. The model uses the physical experience related with the formation of a solid shell when we make a cold addition to a liquid metal. An outer shell forms surrounding the particle and under this condition there is not possibility for the alloy directly and having, someone, involved with the phase diagram. This is quite possible with other additions such as sponge iron, pig iron, in such cases, definitively the phase diagram must be considered since the particle will form probably eutectics accelerating its melting rate. Therefore, the physical properties of the shell will vary by few unit thermal conductivities in a relatively narrow temperature between the metal temperature and the temperature of the shell. This is implicitly expressed by equation (10) in the text.

The paper is lacking a statistical evaluation of the kinetics of the particles and should be able to answer the following questions. Does the model predict any stochastic variation in the molten flow? If so, how sensitive are the predictions if the metal particle is added moments later? How sensitive is the melting time to deviations in the velocity upon introduction to the melt? What scatter in velocity is to be expected and would it influence behavior? The answer to these questions would determine the extent in which ignoring particle addition interactions is acceptable for modelling the process.  If the results are not sensitive to such changes, it suggests that particle interaction is not as important as the flow of the melt.

93. Exactly! We did not worked statistics. Let me explain; all simulations reported in this paper correspond to an instantaneous flow field. Since the ladle is filling, there’s not ever a steady-state condition. We had a previous experience, see our reference (6) in the text, where statistics varied considerably even at a constant time, for a given ladle level in the case of light additions. In other words, sinking particles if added in the same position for a given steel level in the ladle, will yield small standard deviations regarding residence times and trajectories. This is underlined in the new text. See lines 90-93.

Some sentences need minor rephrasing.

The sentence starting on line 35 “Steel tapping is crucial” needs rework. It might need split into two sentences to convey the importance of the tapping process and then its complexity.

1. Thanks we did as suggested.

Line 37 “The efficiency of the alloys” – do you mean the efficiency of producing the alloys?

1. It is the amount of dissolved alloy divided by the total weight of the addition. The manuscript was modified in this sense.

Line 38 “Steel tapping together with bottom argon …” should be rephrased to something like “Steel tapping processes with argon stirring in the ladle create a forced convection current which strongly influences the dissolution of alloying additions, and impacts the thermal and chemical homogenization of the ladle”.

1. Excellent suggestion. Done!!!

Line 81 – either “There is no slag phase in the system”, or “There is not a slag phase in the system”

1. There is not slag.

Line 385 – This sentence needs rephrased. Something like “The present research focusses on the melting times …” rather than “The present research works about the melting times …”.

1. Excellent suggestion!! Done!!!

Suggestions for the authors to consider.

The paper would benefit from discussing how the model could be adapted to other common alloying elements. What elements is the current model capable of simulating as-is? What changes are needed to capture others? What about elements with a higher melting point such as Cr described in table 4? It would be interesting to see how the optimum location of placing the metal additions varies for metal additions that are lighter than Fe.

1. This exactly the topic of a new paper. We just left the fundamental aspects for the present paper. To scale up theses results closer to industrial conditions requires ore elaboration in our new paper we are testing additions such as Fe-Si, Fe-Mn and Fe-Nb.

In the Practical Implications section, you could state the calculated optimum positioning of the nickel additions at all of the different tonnages of melt investigated, noting that the most problematic time for ensuring complete liquation and mixing of heavy additions such as Ni is for shallow baths.

1. That is correct. We added some comments in this regard in the section headed as Practical Implications.

In the paragraph starting on line 127, it would be worth mentioning details regarding the mesh sensitivity study performed to arrive at this mesh and the corresponding numerical error. This is not mandatory as many papers omit this however it would provide greater confidence in the results if the numerical error is known.

10. I guess the reason of why most of the researchers do not elaborate on this aspect is the same as ours. To get the velocity fields took about three and half months with our current computing facilities. The mesh is fine enough and besides, the multiphase model involves the metal, the air and the argon (made artificially as different phase to track its interfaces). The system is unsteady and receives two inputs; from the impinging jet and from the ladle bottom. Such a system has not been, so far, tackled. Therefore, to carry out a study of mesh sensitivity would be very costly. Besides, we use a fine mesh, hexahedral elements and controlled aspect ratio and skewness. Our model is supported by experimental measurements and, particularly, the VOF model has been tested against the population balance models and experimental velocities at the air-liquid interface. See our paper, R.D. Morales et. al: “Physical and mathematical Modeling of Flow Structures of Liquid Steel in Ladle Stirring Operations”, Metall. and Mater. Trans. B, doi.org/10.1007/s11663-019-01759-x, 2020. Therefore, our flow field predictions are trustable.

A small illustration of the ladle with an axis of origin and a cross section identifying what cross section is being viewed would be useful for the velocity vector plots.

1. well that may be so, but Figure 2 is self-explanatory. Someone must imagine this cross-section view of the ladle at different steel levels.

In Table 4, you could put (s) for the Cr, Mn and Ni. I guess Ts means the solvus temperature rather than the solidus temperature. You could use Tl for the liquidus temperature or Tm for the melting temperature to make this clearer.

1. We are talking about pure metals. Therefore, T_s means simply, melting point.

The layout of Figure 7 could be improved, with a title to the colour bar for Figure 7 c), so that it does not look like it is time at first glance.

1. Thanks, done!!!!

In future work you could choose locations in polar coordinates and make a contour graph of the predicted melting times for different tonnages of the melt for a given superheat instead of the bar charts shown in Figure 10.

1. We, gladly, take this very nice suggestion. I guess, when you are very much involved in some task you miss that general picture. Surely we will use your idea. Thanks!!

Is there any way to validate the model predictions? Can you see if the chemical homogenization in steel created with metal additions from different locations correlate to the predicted melting times? Is there any way to validate the predicted velocities of the melt? This model only needs to be qualitatively accurate to guide the decision-making process rather than predict exact liquation times. This would be worth noting in the manuscript.

1. We must be aware of the limitations of this model to reproduce exactly what is going on the real life. Let me tell that the contribution of this model is not to tell you exactly the minute details of a real ladle. Rather, its contribution is to help you in fixing policies for additions of different metals, when and where and for that, we guarantee 100% the model as its based-on fundamentals of physics. In other words, the model predicts that the nickel addition must be in the argon plume and in the opposite side mainly at low bath levels. And, this 100% credible. Efficiency of ferroalloys, possible trajectories, effects of melt superheat can be studied through this model for process control and the establishment of tapping operations. For all that, from a qualitative standpoint, the model is quite useful.

We give you the thanks for your tough questions that made us to figure out, with great effort, reasonable answers. The amount of questions and arguments that you handled reflect the interest that you put on our work. And, for that, we are very much acknowledged.