Using Discrete Element Method to Analyse the Drop Ball Test

Round 1

Reviewer 1 Report

In this paper, the author proposed a formula for predicting the probability of particle breakage according to the data from DBT experiments. Moreover, the author used the DEM to analyze the collision behavior of different ball sizes and energy during ball collisions in the DBT. The energy of each ball was recorded. The formula proposed by the author corresponds well with reality but lacks theoretical analysis, and it is still unclear whether this formula is applicable in other conditions. The data obtained from DEM are different from the focus of DBT experiments, and there is no prediction related to fragmentation in DEM simulation, which is a big problem.

Comments:

1. Please add graphs corresponding to the fracture probability of particles of different sizes in Figure 5 to prove the correctness of the formula.

2. There seems to be a mistake in Line 258, as the particle sizes in the two images should be different.

3. In Line 184, the author states that the breakage behavior of steel particles is significantly different from that of ore particles, but no corresponding data can be found in other parts of the article. Please add relevant explanations.

4. In Figure 3, it seems that more drops result in fewer fractures. This is unreasonable, please provide an explanation.

5. In Figure 4, the failure frequency of 115mm and 100mm particles is almost the same. In Figure 6, the predicted fracture probability of the model is almost size-independent, which is consistent with Figure 4. However, there are significant differences in the breakage behavior of particles of different sizes in Figure 3. Please provide an explanation.

6. In sections 3.1 and 3.2, the author recorded the energy of particles in DEM but did not predict the probability of particle breakage based on the energy. This is not related to the work in section 3.1, so you need to find the connection between energy and fragmentation.

7. According to Figure 7, particles rebound in the simulation, is the rebound height also the same in reality? Please add relevant data.

8. In line 331, the author says the larger the balls, the greater the impact, and thus local spalling is more likely with the 125 mm balls than it is on the 115 mm and 100 mm balls. This is obvious, it is not necessary to emphasize in the conclusion. Please modify the relevant information.

9. Is c the same value in Eq.7 and Eq. 8? Please provide a detailed explanation of each parameter in these two formulas.

10. As is well known, the force on a particle when it breaks must be greater than the elastic limit. Therefore, when using the linear Spring-Dashpot Model to statistically analyze the elastic energy, maybe elastic energy is useless for predicting particle breakage. Please discuss this problem in the article.

In this paper, the author proposed a formula for predicting the probability of particle breakage according to the data from DBT experiments. Moreover, the author used the DEM to analyze the collision behavior of different ball sizes and energy during ball collisions in the DBT. The energy of each ball was recorded. The formula proposed by the author corresponds well with reality but lacks theoretical analysis, and it is still unclear whether this formula is applicable in other conditions. The data obtained from DEM are different from the focus of DBT experiments, and there is no prediction related to fragmentation in DEM simulation, which is a big problem.

Comments:

1. Please add graphs corresponding to the fracture probability of particles of different sizes in Figure 5 to prove the correctness of the formula.

2. There seems to be a mistake in Line 258, as the particle sizes in the two images should be different.

3. In Line 184, the author states that the breakage behavior of steel particles is significantly different from that of ore particles, but no corresponding data can be found in other parts of the article. Please add relevant explanations.

4. In Figure 3, it seems that more drops result in fewer fractures. This is unreasonable, please provide an explanation.

5. In Figure 4, the failure frequency of 115mm and 100mm particles is almost the same. In Figure 6, the predicted fracture probability of the model is almost size-independent, which is consistent with Figure 4. However, there are significant differences in the breakage behavior of particles of different sizes in Figure 3. Please provide an explanation.

6. In sections 3.1 and 3.2, the author recorded the energy of particles in DEM but did not predict the probability of particle breakage based on the energy. This is not related to the work in section 3.1, so you need to find the connection between energy and fragmentation.

7. According to Figure 7, particles rebound in the simulation, is the rebound height also the same in reality? Please add relevant data.

8. In line 331, the author says the larger the balls, the greater the impact, and thus local spalling is more likely with the 125 mm balls than it is on the 115 mm and 100 mm balls. This is obvious, it is not necessary to emphasize in the conclusion. Please modify the relevant information.

9. Is c the same value in Eq.7 and Eq. 8? Please provide a detailed explanation of each parameter in these two formulas.

10. As is well known, the force on a particle when it breaks must be greater than the elastic limit. Therefore, when using the linear Spring-Dashpot Model to statistically analyze the elastic energy, maybe elastic energy is useless for predicting particle breakage. Please discuss this problem in the article.

Author Response

Dear Reviewer

Thank you for your useful technical comments and suggestions on our manuscript. We have modified the manuscript accordingly.

Detailed corrections are listed below point by point. Points raised by the reviewers are in italics and changes made or our response to those points in normal text.

Reviewers' comments:

Reviewer #1

In this paper, the author proposed a formula for predicting the probability of particle breakage according to the data from DBT experiments. Moreover, the author used the DEM to analyze the collision behaviour of different ball sizes and energy during ball collisions in the DBT. The energy of each ball was recorded. The formula proposed by the author corresponds well with reality but lacks theoretical analysis, and it is still unclear whether this formula is applicable in other conditions. The data obtained from DEM are different from the focus of DBT experiments, and there is no prediction related to fragmentation in DEM simulation, which is a big problem.

• From the DBT, data was collected for about a year on ball failure for the three sizes 125mm, 115mm and 100mm and these data exhibit failure behaviour that can be correlated to ball size and impact drops. Thus, it was possible to propose a mechanistic model that describes this data. The second part involves DEM simulation to get a deeper insight into the DBT and it is hereby confirmed that theoretical prediction of PE is adequate. However, the DEM is used as this can also be used in actual mill simulations and apply this data accordingly.

1. Please add graphs corresponding to the fracture probability of particles of different sizes in Figure 5 to prove the correctness of the formula.

• The scope of the work was limited to assessing ball breakage in the SAG mill, which is why a formula (Equation 7) was developed. The fracture probability of particles was studied in our previous work (Bwalya and Chimwani 2020) with useful insights drawn. We used the fracture probability of particles as a starting point however it is noted that particles because of being brittle only require a few impacts to break while steel balls require thousands of attempts.

2. There seems to be a mistake in Line 258, as the particle sizes in the two images should be different.

• Thank you for identifying this mistake, (b) should be 100 mm balls. The authors corrected it.

3. In Line 184, the author states that the breakage behavior of steel particles is significantly different from that of ore particles, but no corresponding data can be found in other parts of the article. Please add relevant explanations.

• There is no corresponding data presented in the paper about particle breakage because the paper was focusing on ball breakage as mentioned in the foregoing. The statement relays known information already in the literature which has been cited from the authors’ previous work. We now refer to Figures 5 and 6 specifically, which can be compared to the paper that we have referenced.

4. In Figure 3, it seems that more drops result in fewer fractures. This is unreasonable, please provide an explanation.

• It is assumed that if balls are not flawed, they will survive an infinite number of drops, those that fail as shown in Figure 3 are the defective ones. The earlier peak for the 125 mm balls indicates bigger ball flaws and the higher relative stress levels that these balls are subjected to. Smaller balls peak later because the cumulative effect is less intense.

5. In Figure 4, the failure frequency of 115mm and 100mm particles is almost the same. In Figure 6, the predicted fracture probability of the model is almost size-independent, which is consistent with Figure 4. However, there are significant differences in the breakage behavior of particles of different sizes in Figure 3. Please provide an explanation.

• The authors still reiterate that the results in all the figures are consistent. In Figure 3, the plots of the number of failures against the number of drops for 115- and 100-mm balls are close and following each other except at 9000 drops where they are separated by 10 failures which is relatively insignificant.

6. In sections 3.1 and 3.2, the author recorded the energy of particles in DEM but did not predict the probability of particle breakage based on the energy. This is not related to the work in section 3.1, so you need to find the connection between energy and fragmentation.

• The energy predicted in the mentioned sections was not meant to predict the probability of ball damage. The DEM was meant to confirm what was measured using DBT tests and also to provide greater insights into the conversions and transmissions of energy within the DBT equipment.

7. According to Figure 7, particles rebound in the simulation, is the rebound height also the same in reality? Please add relevant data.

• The rebound height is more or less the same as discussed in Lines 281 – 284 and presented in Table 5.

8. In line 331, the author says the larger the balls, the greater the impact, and thus local spalling is more likely with the 125 mm balls than it is on the 115 mm and 100 mm balls. This is obvious, it is not necessary to emphasize in the conclusion. Please modify the relevant information.

• Thank you, the statement was removed from the conclusion.

9. Is c the same value in Eq.7 and Eq. 8? Please provide a detailed explanation of each parameter in these two formulas.

• The c has been changed to e in Equation 7.

10. As is well known, the force on a particle when it breaks must be greater than the elastic limit. Therefore, when using the linear Spring-Dashpot Model to statistically analyze the elastic energy, maybe elastic energy is useless for predicting particle breakage. Please discuss this problem in the article.

• Here we are being more descriptive of the process, and we conclude that simple theoretical prediction is enough without requiring simulation, but simulation becomes important when the survivability of the balls is being predicted in an actual mill environment.

The authors would like to thank the reviewer once more for the constructive comments that helped to improve the quality of our work.

Kind regards,

Ngonidzashe Chimwani

Reviewer 2 Report

The paper introduces a DEM modelling of the DBT used to quality-control in the steel ball production process. In general terms, the paper concept, the state of the art revision, the methodology and the analysis of results are well written and justified. However, some lacks must be clarified before recommending its publication:

1- Line 26: “major cost component”; according to the context it should be “major operating cost component”

2-Line 54: POB, please include a reference of this test

3-Line 55: DRAWT, same as the previous one

4-Line 57: DBT, same as the previous one.

5-Line 97: Use DBTs instead of “Drop ball tests”

6-Line 121: in the description of DBT, the parameter or indicator which characterises the behaviour of the balls is missing (number of failure balls each 1000 drops). Also, a reference is needed for Figure 109a.

7-Line 153: “Spring-dashpot Model”, please insert reference.

8-Line 252: “150 and 100”, should be “115 and 100”, please confirm and correct.

9-Line 258: Figure 6 caption, 150 mm twice, shouldn’t be 115 in (a) and 100 in (b)?

10-Line 282: Please use KE instead of “kinetic energy”, as in previous paragraphs

11-Line 284: “all three balls”, please correct with “all three sizes”

12-Line 334: It should be included here the importance of including some characterization tests on the balls to validate the methodology.

13- References section: Reference of MSc Thesis from Mr. Shwarzkopf Oliver Samukute, which is available online, should be included in the references. Also, in Reference 3, please write correctly the surname “Sepulveda”.

Author Response

Dear Reviewer

Thank you for your useful technical comments and suggestions on our manuscript. We have modified the manuscript accordingly.

Detailed corrections are listed below point by point. Points raised by the reviewers are in italics and changes made or our response to those points in normal text.

 Reviewer's comments:

The paper introduces a DEM modelling of the DBT used to quality-control in the steel ball production process. In general terms, the paper concept, the state of the art revision, the methodology and the analysis of results are well written and justified. However, some lacks must be clarified before recommending its publication:

• Thank you very much.

Line 26: “major cost component”; according to the context it should be “major operating cost component”

• The correction was done as per the review’s suggestion.

2-Line 54: POB, please include a reference of this test

• The reference was included.

3-Line 55: DRAWT, same as the previous one

• The reference was included.

4-Line 57: DBT, same as the previous one.

• The reference was included.

5-Line 97: Use DBTs instead of “Drop ball tests”

• The correction was done as per the review’s suggestion.

6-Line 121: in the description of DBT, the parameter or indicator which characterises the behaviour of the balls is missing (number of failure balls each 1000 drops). Also, a reference is needed for Figure 109a.

• The authors are not sure if they got the review’s question right. Their response though is that the balls were not assessed per number of drops but were rather dropped until the ball damage was observed.

7-Line 153: “Spring-dashpot Model”, please insert reference.

• Reference included.

8-Line 252: “150 and 100”, should be “115 and 100”, please confirm and correct.

• Thank you for observing the mistake, it was corrected.

9-Line 258: Figure 6 caption, 150 mm twice, shouldn’t be 115 in (a) and 100 in (b)?

• Thank you for observing the mistake, it was corrected.

10-Line 282: Please use KE instead of “kinetic energy”, as in previous paragraphs

• The correction was done as per the review’s suggestion.

11-Line 284: “all three balls”, please correct with “all three sizes”

• The correction was done as per the review’s suggestion.

12-Line 334: It should be included here the importance of including some characterization tests on the balls to validate the methodology.

• These have been alluded to, but in this paper, the focus is on the DBT which can be used to evaluate the quality of manufactured batches of balls. If failure exceeds 15% of the 40 balls that are circulated in the test, then the batch is normally rejected.

13- References section: Reference of MSc Thesis from Mr. Shwarzkopf Oliver Samukute, which is available online, should be included in the references. Also, in Reference 3, please write correctly the surname “Sepulveda”.

• The reference to Mr Samukute’s thesis was included in the reference list and the correction was made.

The authors would like to thank the reviewer once more for the constructive comments that helped to improve the quality of our work.

Kind regards,

Ngonidzashe Chimwani

Round 2

Reviewer 1 Report

The authors have satisfactorily addressed the issues I raised in my review. I recommend the manuscript to be published on Minerals.

Reviewer 2 Report

Authors addressed all comments and the manuscript has been greatly improved, so my recommendation is to accept it at present form.