Area of the Intersection between a Sphere and a Cylindrical Plane

Round 1

Reviewer 1 Report

General comment: 

In this study, the intersection area was calculated by different integral expressions and verified by computer software. This study is helpful to analyze some structural characteristics of the particle system (such as porosity). However, it should be noted that some details need to be supplemented. Such as application background, comparison of several integration methods (such as calculation efficiency), etc.

Detailed comment:

1.In Abstract,” When the positions and sizes of the spheres are known, the radial variation in porosity can be determined using volume-based, area-based, or line based approaches. Here the focus is on the area-based methods which employ the intersections between the spheres and selected cylindrical planes to determine the radial variation in porosity, focusing specifically on the calculation of the area of the curved elliptic intersection between a sphere and a cylindrical plane.” The main purpose of this article is to discuss the calculation of the intersection area, but there is less description on how to obtain the radial change of porosity. I think readers will be more interested in obtaining porosity.

2. In Abstract,” The integral expressions cannot be integrated analytically and have been evaluated using approximations or numerical integration.” It is suggested that the calculation amount or calculation time of different integration methods be compared in the article.

3.In Introduction, the application background of the study should be supplemented, such as the influence of the radial distribution of porosity on the movement or heat storage of pebble bed, or the engineering background of particle system. The recommended references are as follows:

Zafar Hayat Khan, Rashid Ahmad, Licheng Sun, Effect of instantaneous change of surface temperature and density on an unsteady liquid–vapour front in a porous medium. Experimental and Computational Multiphase Flow, 2020, 2(2), 115–121. https://doi.org/10.1007/s42757-019-0027-9.

Goodling, J. S., Vachon, R. I., Stelpflug, W. S., Ying, S. J., & Khader, M. S. (1983). Radial porosity distribution in cylindrical beds packed with spheres. Powder technology, 35(1), 23-29.

4.In line 64, 2.1 Intersection of cylindrical plane and sphere, this part should be supplemented with an overview of the problems to be solved.

4.In line 164, 3.1.1. Test cases, what is the purpose of the test cases? Is this to determine the convergence of the example, or to give the geometric characteristics of the example, or for other purposes?

5.In Conclusions, “It can also be concluded that the procedure of Mariani et al. [16] and the angular integration procedure of Du Toit [17,19] are the most efficient approaches.” What is the definition of "most effective"?

6. In Conclusions, the innovation points or main characteristics in the text should be supplemented.

Author Response

General comment: 

In this study, the intersection area was calculated by different integral expressions and verified by computer software. This study is helpful to analyze some structural characteristics of the particle system (such as porosity). However, it should be noted that some details need to be supplemented. Such as application background, comparison of several integration methods (such as calculation efficiency), etc.

Detailed comment:

1.In Abstract,” When the positions and sizes of the spheres are known, the radial variation in porosity can be determined using volume-based, area-based, or line-based approaches. Here the focus is on the area-based methods which employ the intersections between the spheres and selected cylindrical planes to determine the radial variation in porosity, focusing specifically on the calculation of the area of the curved elliptic intersection between a sphere and a cylindrical plane.” The main purpose of this article is to discuss the calculation of the intersection area, but there is less description on how to obtain the radial change of porosity. I think readers will be more interested in obtaining porosity.

The author agrees with the reviewer that most readers will be more interested in how to obtain the radial variation in porosity and, therefore, how the porosity is calculated. The porosity at a given radial position is determined by the area of the voids in the cylindrical plane, at the radial position, between the spheres intersected by the cylindrical plane. This requires the calculation of the areas of the intersections between the spheres and the cylindrical plane. The calculation of the porosity is explained and demonstrated by Mariani et al. [16], Du Toit [17,19] and Mueller [18], as well as in other articles by the same authors. The current study are extractions from the articles of Mariani et al. [16], Du Toit [17,19] and Mueller [19] of the methodologies that determine the success of the calculation of the porosity, namely the evaluation of the area of the intersection between a sphere and a cylindrical. The first aim of the study is to demonstrate directly, as opposed to indirectly, that the methodologies evaluate the area of the intersection between a sphere and a cylindrical plane correctly. The second aim of the study is to compare the characteristics and performance of the methodologies considering the fact that they respectively employ the axial, angular and radial directions as independent variables.

2. In Abstract,” The integral expressions cannot be integrated analytically and have been evaluated using approximations or numerical integration.” It is suggested that the calculation amount or calculation time of different integration methods be compared in the article.

 Although the accurate integration of the relevant integrals is important, it is not the purpose of the study to compare various numerical integration techniques. Simpson’s Rule was selected to evaluate the integrals numerically due to the fact that the functions to be integrated are smooth and because Simpson’s Rule is third order accurate and simple to implement. CPU or computational time was found to be an impractical measure due to the speed of the calculations and the factors that have an influence on the CPU time. The size of the integration step (Δ) required to get an accurate result was used as the first measure to compare the performance. Because the natures (physical meaning) of the integration steps for the different methodologies are not the same, the number of integration points was also introduced as an additional measure.

3.In Introduction, the application background of the study should be supplemented, such as the influence of the radial distribution of porosity on the movement or heat storage of pebble bed, or the engineering background of particle system. The recommended references are as follows:

 The author has used the articles already cited in the manuscript under review to supplement the application background to provide a broader context for the relevance of the study.

Zafar Hayat Khan, Rashid Ahmad, Licheng Sun, Effect of instantaneous change of surface temperature and density on an unsteady liquid–vapour front in a porous medium. Experimental and Computational Multiphase Flow, 2020, 2(2), 115–121. https://doi.org/10.1007/s42757-019-0027-9.

 The author doesn’t have access to Springer to read the article. The author can’t from the abstract determine the precise nature of the porous medium considered in the paper and to what extended it is relevant to the manuscript under review.

Goodling, J. S., Vachon, R. I., Stelpflug, W. S., Ying, S. J., & Khader, M. S. (1983). Radial porosity distribution in cylindrical beds packed with spheres. Powder technology, 35(1), 23-29.

 Note that this paper is cited in the first version of the manuscript, although not in the context implied by the reviewer. The focus is rather on the fact that Goodling et al. employed a physical (destructive) experimental approach to determine the radial variation in porosity.

4.In line 64, 2.1 Intersection of cylindrical plane and sphere, this part should be supplemented with an overview of the problems to be solved.

 The author has inserted a relevant introductory paragraph at the beginning of the section and trust that it addresses the concern of the review satisfactorily. The paragraph inserted hopefully provides the necessary context for the definitions that follow.

5.In line 164, 3.1.1. Test cases, what is the purpose of the test cases? Is this to determine the convergence of the example, or to give the geometric characteristics of the example, or for other purposes?

 The author addressed this concern by changing the title of section 3.1 and adding material that explain the purpose of the test cases that were considered.

6.In Conclusions, “It can also be concluded that the procedure of Mariani et al. [16] and the angular integration procedure of Du Toit [17,19] are the most efficient approaches.” What is the definition of "most effective"?

 The author has now addressed this in more detail in section 3.2 (previously section 3.2.5).

6. In Conclusions,the innovation points or main characteristics in the text should be supplemented.

The conclusions have been extensively revised to address the concerns raised by the reviewer.

Reviewer 2 Report

This study validates the integral expressions for angular and axial integration in Du Toit [17, 19] , elliptical integration in Mariani [16, 17] and radial integration in Mueller [18] for predicting an intersection between a sphere and a cylindrical plane. COMSOL Multiphysics, SOLID WORKS, and NX were used here to reconstruct their geometry and compute their areas using their tab notations.

I think this work is suitable for publication in MDPI, but as a technical report only. This work doesn't provide enough results for a high-quality full-length article, but rather for a technical report involving detailed review and validation.  To further substantiate the computer-modelled results, comprehensive and tabular representations of several mathematical expressions and their results are also necessary.

Author Response

This study validates the integral expressions for angular and axial integration in Du Toit [17, 19], elliptical integration in Mariani [16, 17] and radial integration in Mueller [18] for predicting an intersection between a sphere and a cylindrical plane. COMSOL Multiphysics, SOLID WORKS, and NX were used here to reconstruct their geometry and compute their areas using their tab notations.

I think this work is suitable for publication in MDPI, but as a technical report only. This work doesn't provide enough results for a high-quality full-length article, but rather for a technical report involving detailed review and validation.  To further substantiate the computer-modelled results, comprehensive and tabular representations of several mathematical expressions and their results are also necessary.

It is not entirely clear to the author what comprehensive and tabular representations of mathematical expressions and their result the reviewer would like to see. Based on the comments of Reviewer 1, the author has provided more detail of the numerical integration of the integrals and validation results for the numerical integration of the Legendre complete elliptic integrals, as well as representative results showing the computational effort required to obtain the converged solutions for the intersection areas. The author trusts that the revision of the manuscript addresses the concerns expressed by the reviewer.

Reviewer 3 Report

  The study of the properties and performance of fixed beds has relatively recently been extensively carried out by means of particle-resolved computational fluid dynamics, which enables the resolution of local effects by constructing a packing of particles via discrete element method (DEM) or rigid body dynamics (RBD). Both methods provide an explicit representation of particle positions, from which properties of the packing can theoretically be generated. One of the most useful is the void fraction, especially its variation in the radial, or transverse to flow, direction. Formulas have been derived for these calculations for packings of spheres, by volume-based, area-based or line-based methods. The lower the dimension, the easier the computations but then the question arises of validation and accuracy of the approximations necessary to obtain the simpler equations. In this contribution, the author compares the area-based formulas of Mariani et al., himself (two versions) and Mueller. Most of these formulas have previously been validated by extensive comparisons to computer-generated data, and have been found to perform well. It is therefore no surprise that the present author shows that the three methods agree well with one another in predicting the area of intersection between a sphere and a cylindrical plane, which is the problem at the core of these void fraction calculations. Although this agreement was to be expected, it has not been demonstrated clearly before, so in that regard the present manuscript makes a novel contribution.

  Each method requires the numerical evaluation of an integral, which the author has chosen to perform by Simpson’s rule. It is then observed that Mueller’s method, which has an integral with singularities at the interval endpoints, does not give as accurate results and does not converge as quickly as the others when the quadrature increments are decreased.

  Several years ago, when Mueller published his 2010 and 2012 papers, I was asked to review them. Having an interest in this area, I evaluated all Mueller’s formulas, including those for subintervals of the bed using the line-based method, and including my own extension of his area-based formulas to bed subintervals, and found excellent agreement between them and in comparison to my own computer-generated sphere packings. In performing these calculations, I used a quadrature rule from the IMSL numerical subroutine library, (QDAGS). This is “a general-purpose integrator that uses a globally adaptive scheme to reduce the absolute error. It subdivides the interval [A, B] and uses a 21-point Gauss-Kronrod rule to estimate the integral over each subinterval”, to quote the description. Further, “this routine is designed to handle functions with endpoint singularities.” I found no problems with convergence or accuracy of the integrals for the area-based formulas of Mueller. I therefore suggest that before this manuscript can be accepted, the author should repeat the calculations of Mueller’s method using a quadrature method designed to handle endpoint singularities, or at the very least include some text indicating that the inferior results that were found for Mueller's method may be due to the choice of quadrature method, and the method may be as good as the other methods if a better quadrature method were to be chosen.

Author Response

The study of the properties and performance of fixed beds has relatively recently been extensively carried out by means of particle-resolved computational fluid dynamics, which enables the resolution of local effects by constructing a packing of particles via discrete element method (DEM) or rigid body dynamics (RBD). Both methods provide an explicit representation of particle positions, from which properties of the packing can theoretically be generated. One of the most useful is the void fraction, especially its variation in the radial, or transverse to flow, direction. Formulas have been derived for these calculations for packings of spheres, by volume-based, area-based or line-based methods. The lower the dimension, the easier the computations but then the question arises of validation and accuracy of the approximations necessary to obtain the simpler equations. In this contribution, the author compares the area-based formulas of Mariani et al., himself (two versions) and Mueller. Most of these formulas have previously been validated by extensive comparisons to computer-generated data, and have been found to perform well. It is therefore no surprise that the present author shows that the three methods agree well with one another in predicting the area of intersection between a sphere and a cylindrical plane, which is the problem at the core of these void fraction calculations. Although this agreement was to be expected, it has not been demonstrated clearly before, so in that regard the present manuscript makes a novel contribution.

  Each method requires the numerical evaluation of an integral, which the author has chosen to perform by Simpson’s rule. It is then observed that Mueller’s method, which has an integral with singularities at the interval endpoints, does not give as accurate results and does not converge as quickly as the others when the quadrature increments are decreased.

  Several years ago, when Mueller published his 2010 and 2012 papers, I was asked to review them. Having an interest in this area, I evaluated all Mueller’s formulas, including those for subintervals of the bed using the line-based method, and including my own extension of his area-based formulas to bed subintervals, and found excellent agreement between them and in comparison to my own computer-generated sphere packings. In performing these calculations, I used a quadrature rule from the IMSL numerical subroutine library, (QDAGS). This is “a general-purpose integrator that uses a globally adaptive scheme to reduce the absolute error. It subdivides the interval [A, B] and uses a 21-point Gauss-Kronrod rule to estimate the integral over each subinterval”, to quote the description. Further, “this routine is designed to handle functions with endpoint singularities.” I found no problems with convergence or accuracy of the integrals for the area-based formulas of Mueller. I therefore suggest that before this manuscript can be accepted, the author should repeat the calculations of Mueller’s method using a quadrature method designed to handle endpoint singularities, or at the very least include some text indicating that the inferior results that were found for Mueller's method may be due to the choice of quadrature method, and the method may be as good as the other methods if a better quadrature method were to be chosen.

The primary purpose of the study was to show directly through the numerical integration of the relevant integral expressions, that the methodologies of Mariani et al. [16], Du Toit [17,19] and Mueller [19] give the correct result for the area of the intersection between a sphere and a cylindrical plane. Although the accurate integration of the relevant integrals is important, it is not the purpose of the study to compare various numerical integration techniques. Simpson’s Rule was selected to evaluate the integrals numerically due to the fact that the functions to be integrated are smooth and because Simpson’s Rule is third order accurate and simple to implement. The computational effort, based on Simpson’s Rule, associated with the methodologies was also considered by comparing the number of integration points required to obtain a converged result for the intersection area.

It would be interesting to know what the coding for the QDAGS routine looks like, as well as how many subdivisions were required to obtain an accurate result for the Mueller integral expression, as well as the number of integrations points = number of subdivisions x 21 + number of subdivisions x 10. How many refinements of adaptations did the routine perform?

Having been involved in the development of 1D, 2D and 3D finite element CFD codes, the author appreciates the power of the 21-point Gauss-Kronrod rule to integrate the integrals under consideration. The author also understands why the routine would be able to handle the end-point singularities associated with the Mueller formulation.

Although this is not normally the place to do it, the author would be interested to collaborate with the reviewer in a study on the computational effort of more advanced numerical integration methods to solve the relevant integrals. As recommended at the end of the manuscript this be done in the context of the calculation of the radial variation in porosity for a selection of cylindrical packed beds consisting of varying numbers of spheres, including the pebble bed model consisting of 450000 spheres generated by Suikkanen et al. [13]. An additional aspect regarding the overall computational effort of the calculation of the radial variation in the porosity would be to consider the effect of not sorting the spheres prior to the calculation (as done by Mueller) compared to sorting the spheres into radial bins prior to the calculation (as done by Du Toit).

Round 2

Reviewer 1 Report

no more comments