Evaluation of Approximate Expressions to Calculate the Area of the Intersection between a Sphere and a Cylindrical Plane
Round 1
Reviewer 1 Report
This review concerns the manuscript of du Toit "Evaluation of Approximate Expressions to Calculate the Area of the Intersection Between a Sphere and a Cylindrical Plane". In examination of random packed beds, radial distribution of porosity is one of the most important charactersitics. Calculation of local porosity requires information about the area of intersection between the particles in the packed bed and a cylindrical surface of given radius. Exact calculation of the area is difficult even for particles as simple as spheres, as typically elliptic integrals are involved. Thus, approximate formulas are of certain interest to the researchers in the field.
The submitted paper is devoted to evaluation of some approximate formulas for the calculation of the intersection area. The validity and accuracy of the formulas is verified for a few representative configurations. In my opinion, this is a solid scientific work and I recommend its publication in Powders.
I can only suggest to integrate Section 3.4.7 (Summary) with Conclusions, as currently the paper ends with two concluding sections, mostly overlapping in the content.
Author Response
This review concerns the manuscript of du Toit "Evaluation of Approximate Expressions to Calculate the Area of the Intersection Between a Sphere and a Cylindrical Plane". In examination of random packed beds, radial distribution of porosity is one of the most important characteristics. Calculation of local porosity requires information about the area of intersection between the particles in the packed bed and a cylindrical surface of given radius. Exact calculation of the area is difficult even for particles as simple as spheres, as typically elliptic integrals are involved. Thus, approximate formulas are of certain interest to the researchers in the field.
The submitted paper is devoted to evaluation of some approximate formulas for the calculation of the intersection area. The validity and accuracy of the formulas is verified for a few representative configurations. In my opinion, this is a solid scientific work and I recommend its publication in Powders.
I can only suggest to integrate Section 3.4.7 (Summary) with Conclusions, as currently the paper ends with two concluding sections, mostly overlapping in the content.
The suggestion was implemented in part. The last paragraph of Section 3.4.7 Summary was deleted and used to replace the second last paragraph of Section 5 Conclusions.
Reviewer 2 Report
In this manuscript, the author studies the porous structure of packed cylindrical beds of spherical particles and evaluates the precision of approximations to estimate the radial distribution of porosity. The analysis of the porous structure of packed beds is essential not only in powder or chemical technologies but also in materials science and engineering.
The comments related to the present manuscript are summarized below.
1. Comment on the applicability of developed approximations to the analysis of the porous structure of powder beds composed of mixtures of particles of different sizes.
2. Lines 164-167: Elaborate the sentence starting “Du Toit [34] has demonstrated …”.
3. Lines 490-492: Confirm/revise the sentence starting “In a previous study [34] it was shown the …”
Author Response
In this manuscript, the author studies the porous structure of packed cylindrical beds of spherical particles and evaluates the precision of approximations to estimate the radial distribution of porosity. The analysis of the porous structure of packed beds is essential not only in powder or chemical technologies but also in materials science and engineering.
The comments related to the present manuscript are summarized below.
- Comment on the applicability of developed approximations to the analysis of the porous structure of powder beds composed of mixtures of particles of different sizes.
The developed methods can be used to analyse packed beds composed of mixtures of spheres of different sizes. The following sentence has been inserted in the last paragraph of the Conclusions: “Due to the fact that the approaches of Mariani et al [19] and Du Toit [20,21] and the approximate expressions of Mueller [22] consider individual particles when calculating the intersection areas, the packed beds being analysed may consist of mixtures of spheres of different sizes.”
- Lines 164-167: Elaborate the sentence starting “Du Toit [34] has demonstrated …”.
The paragraph has been rephrased / rewritten to become: “Du Toit [34] considered four representative sphere-cylindrical plane test configurations to demonstrate that the numerical integration of the analytical integral formulations developed by Mariani et al. [19], Du Toit [20,21] and Mueller [22] obtain the correct areas for the intersections between the spheres and cylindrical planes. The integral formulations of Mariani et al. [19], Du Toit [20,21] and Mueller [22] can thus be used to evaluate the validity of other proposed methods to calculate the area of the intersection between a sphere and a cylindrical plane.”
- Lines 490-492: Confirm/revise the sentence starting “In a previous study [34] it was shown the …”.
The sentence was revised to read: “In a previous study by Du Toit [34] it was shown that when the integral expressions of Mariani et al. [19], Du Toit [20,21] and Mueller [22] are integrated numerically they provide the correct results”.
Reviewer 3 Report
The submitted manuscript is well written and presents study of evaluation the performance of the approximate expressions that had been derived to calculate the intersection areas. Firstly, the ability of the approximate expressions to calculate the intersection area is evaluated by considering several typical sphere‐cylindrical plane configurations that can be encountered in packed beds. Secondly, the application of the approximate expressions to obtain the radial variation in porosity for a selection of cylindrical packed beds is evaluated.
The authors should define the practical applicability of the presented model. How can it be used, for example, for heat transfer or storage processes? Is it valid only for monodisperse particle sizes? In practice, few materials have the shape of an ideal sphere. What can be the variation in particle sphericity to make the results still applicable?
For example, in the paper https://doi.org/10.1002/ceat.201700018 the authors investigate the thermal conductivity coefficient, porosity and number of contacts as a function of particle size. Can you give an example of applying your model to this study?
Perhaps a graphical representation of the cases shown in Tables 1 and 3 would be useful.
Author Response
The submitted manuscript is well written and presents study of evaluation the performance of the approximate expressions that had been derived to calculate the intersection areas. Firstly, the ability of the approximate expressions to calculate the intersection area is evaluated by considering several typical sphere‐cylindrical plane configurations that can be encountered in packed beds. Secondly, the application of the approximate expressions to obtain the radial variation in porosity for a selection of cylindrical packed beds is evaluated.
The authors should define the practical applicability of the presented model. How can it be used, for example, for heat transfer or storage processes? Is it valid only for monodisperse particle sizes? In practice, few materials have the shape of an ideal sphere. What can be the variation in particle sphericity to make the results still applicable?
In the Introduction a brief overview is given of some of the practical applications where the porosity of a packed bed plays an important role. A sentence was added in the last paragraph of the Conclusions of the paper to explain that the methodologies can be applied to packed beds composed of mixtures of spheres of different sizes. Examples of that can be found in Wang et al (2022) and Du Toit [21]. The variation of the porosity of packed beds consisting of non-spherical particles can be investigated using the current methodology when the non-spherical particles are represented by clumps or clusters of spherical particles of varying sizes.
For example, in the paper https://doi.org/10.1002/ceat.201700018 the authors investigate the thermal conductivity coefficient, porosity and number of contacts as a function of particle size. Can you give an example of applying your model to this study?
The author could not download the referenced paper. However, examples of where the analysis of the porous structure play a role in various aspects of packed beds can be found in Van Antwerpen et al (2010), Van Antwerpen et al (2012), Wang et al (2022), De Beer et al (2017) and De Beer et al. (2018).
Perhaps a graphical representation of the cases shown in Tables 1 and 3 would be useful.
The author agrees with the remark. It was considered to include the graphical representations of the four primary configurations. However, it was decided that since the graphical representations of the four primary configurations can be found in Du Toit [34] that they will not be repeated in the current study. It was also decided that sufficient information is provided in Table 3 for the reader to form visual images of the various configurations.
Wang et al. (2022) Investigation into the packing structure of binary pebble beds using X-ray tomography. Powder Technology 406, 117589. https://doi.org/10.1016/j.powtec.2022.117589
Van Antwerpen et al. (2010) A review of correlations to model the packing structure and effective thermal conductivity in packed beds of mono-sized spherical particles. Nuclear Engnineering and Design 240, 1803-1818. https://doi:10.1016/j.nucengdes.2010.03.009
Van Antwerpen et al. (2012) Multi-sphere Unit Cell model to calculate the effective thermal conductivity in packed pebble beds of monosized spheres. Nuclear Engineering and Design 247, 183-201. https://doi:10.1016/j.nucengdes.2012.03.012
De Beer et al. (2017) A methodology to investigate the contribution of conduction and radiation heat transfer to the effective thermal conductivity of packed graphite pebble beds, including the wall effect. Nuclear Engineering and Design 314, 67-81. http://dx.doi.org/10.1016/j.nucengdes.2017.01.010
De Beer e al. (2018) Experimental study of the effective thermal conductivity in the near-wall region of a packed pebble bed. Nuclear Engineering and Design 339, 253-268. https://doi.org/10.1016/j.nucengdes.2018.09.014
Round 2
Reviewer 1 Report
The paper can be accepted in the present form.
Reviewer 2 Report
I am satisfied with the author's revision of the manuscript.
Reviewer 3 Report
Manuscript can be published in present form.
