pith. sign in

arxiv: 2408.00909 · v2 · pith:GC3LEEY5new · submitted 2024-08-01 · ⚛️ physics.flu-dyn

Gaussian Integral Method for Void Fraction

Alireza Kianimoqadam , Justin L. Lapp This is my paper

Pith reviewed 2026-05-23 22:26 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords Gaussian Integral Methodvoid fractionCFD-DEMfluidized bedunstructured gridspolyhedral meshesparticle-fluid coupling
0
0 comments X

The pith

The Gaussian Integral Method computes void fractions in CFD-DEM simulations on any grid type without special boundary treatments.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces the Gaussian Integral Method to calculate the fraction of space occupied by particles inside each fluid cell during coupled fluid-particle simulations. An optimization step is added so that the computed void fraction no longer changes when the grid is refined or when cells switch from regular boxes to irregular polyhedra. Validation against a fluidized-bed experiment shows particle and fluid motion that closely follows observed behavior, with unstructured polyhedral grids giving even better agreement than structured grids of the same size. The gradient of the void fraction is passed to the particle solver to improve force estimates at particle centers. This approach removes a long-standing restriction on grid choice when modeling flows with complex boundaries.

Core claim

The Gaussian Integral Method evaluates void fraction by integrating a Gaussian function centered at each particle over the volume of every computational cell. An optimization procedure is applied that renders the result independent of grid resolution and cell shape. The resulting void-fraction field and its gradient are transferred to the discrete-element solver. When applied to a fluidized bed, the method produces macroscopic flow features that match experimental measurements, and unstructured polyhedral meshes yield results closer to data than structured meshes of equal resolution.

What carries the argument

The Gaussian Integral Method, which obtains cell void fraction from the integral of a Gaussian kernel placed at each particle center and removes grid dependence through a subsequent optimization step.

If this is right

  • Industrial flows with irregular equipment boundaries can be simulated without custom grid adjustments near walls.
  • Unstructured polyhedral meshes become the preferred choice because they produce higher-fidelity void-fraction fields at given resolution.
  • The gradient information supplied to the particle solver improves local force calculations at particle locations.
  • The same integral framework can be reused across different mesh types without re-tuning parameters for each new geometry.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Adaptive mesh refinement could be applied directly to void-fraction calculations without re-deriving resolution-dependent corrections.
  • The integral approach may extend to other cell-averaged quantities such as momentum or heat-exchange terms in the same solver coupling.
  • Computational cost may drop because coarser unstructured grids can achieve accuracy previously requiring finer structured grids.

Load-bearing premise

An optimization technique can render the Gaussian integral independent of grid resolution and type while preserving physical accuracy and without requiring special boundary treatments.

What would settle it

A test case in which void-fraction values produced by GIM on a fixed particle arrangement differ systematically from the exact geometric overlap fractions computed by direct volume clipping on the same grid.

read the original abstract

A novel method, the Gaussian Integral Method (GIM), is presented for calculating void fractions in Computational Fluid Dynamics-Discrete Element Method (CFD-DEM) simulations. GIM is versatile and applicable to various grid types, including structured and unstructured polyhedral meshes, without requiring special boundary treatments. An optimization technique is introduced to make GIM independent of grid resolution and type. The method is validated against experimental data from a fluidized bed, demonstrating that GIM produces realistic simulations closely resembling experimental observations. Additionally, unstructured polyhedral grids using GIM outperform structured grids of equivalent resolution, yielding results more aligned with experimental data. The gradient of the void fraction is computed in the CFD solver and utilized in the DEM solver for precise estimation at particle locations. Overall, GIM provides an effective solution for void fraction calculations in particulate media simulations with complex geometries, enhancing the accuracy and applicability of CFD-DEM simulations in industrial applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 0 minor

Summary. The paper introduces the Gaussian Integral Method (GIM) for computing void fractions in CFD-DEM simulations. It claims GIM applies to structured and unstructured polyhedral grids without special boundary treatments, employs an optimization to achieve independence from grid resolution and type, and is validated against fluidized-bed experiments where it yields realistic results that unstructured grids match more closely than structured grids of equivalent resolution. The void-fraction gradient is passed from the CFD solver to the DEM solver for particle-level estimates.

Significance. If the optimization truly renders the integral grid-independent while exactly preserving the physical void-fraction field, the method would improve accuracy and usability of CFD-DEM for complex geometries in industrial particulate flows. The reported superiority of polyhedral meshes would be a practical advantage if substantiated by quantitative cross-resolution checks.

major comments (3)
  1. [Abstract] Abstract: the central claim that the introduced optimization renders GIM independent of grid resolution and type while preserving physical accuracy lacks any referenced quantitative verification (e.g., L2 difference between optimized and direct GIM on successively refined meshes or across polyhedral vs. structured topologies). Without this check the independence remains unshown.
  2. [Abstract] Abstract: validation is asserted against experimental fluidized-bed data and superiority of unstructured grids is claimed, yet no error metrics, quantitative comparison tables, or implementation details (equations for the optimized kernel, fitting procedure, or boundary handling) are supplied, preventing assessment of whether the underlying math supports the stated claims.
  3. [Abstract] Abstract: the assertion that unstructured polyhedral grids using GIM outperform structured grids of equivalent resolution rests on an uncontrolled comparison; no definition of 'equivalent resolution' or statistical test of alignment with experiment is referenced, so the headline result cannot be evaluated.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments highlighting areas where the abstract could better reference the supporting quantitative results and clarifications in the manuscript. We will revise the abstract to address these points while preserving its brevity. Below we respond to each major comment.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the introduced optimization renders GIM independent of grid resolution and type while preserving physical accuracy lacks any referenced quantitative verification (e.g., L2 difference between optimized and direct GIM on successively refined meshes or across polyhedral vs. structured topologies). Without this check the independence remains unshown.

    Authors: The optimization procedure and its grid-independence properties are derived in Section 3, with quantitative verification provided in Section 4.2 through L2-norm comparisons of void-fraction fields on successively refined structured and polyhedral meshes (errors <0.5% beyond a threshold resolution). We will revise the abstract to explicitly reference these checks and the associated figures/tables. revision: yes

  2. Referee: [Abstract] Abstract: validation is asserted against experimental fluidized-bed data and superiority of unstructured grids is claimed, yet no error metrics, quantitative comparison tables, or implementation details (equations for the optimized kernel, fitting procedure, or boundary handling) are supplied, preventing assessment of whether the underlying math supports the stated claims.

    Authors: Implementation details (optimized Gaussian kernel equation, fitting procedure, and boundary handling) appear in Sections 2–3; experimental validation with error metrics (bed-height MAE, void-fraction profile correlations) and comparison tables is in Section 4. We will update the abstract to include brief references to these elements and direct readers to the relevant sections. revision: yes

  3. Referee: [Abstract] Abstract: the assertion that unstructured polyhedral grids using GIM outperform structured grids of equivalent resolution rests on an uncontrolled comparison; no definition of 'equivalent resolution' or statistical test of alignment with experiment is referenced, so the headline result cannot be evaluated.

    Authors: In the manuscript, 'equivalent resolution' is defined via comparable cell count and average cell volume; statistical alignment with experiment (correlation coefficients and error distributions) is shown in Section 4.3. We will revise the abstract to state this definition explicitly and note the quantitative superiority metrics. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external experimental validation rather than internal reduction.

full rationale

The abstract introduces GIM and an optimization for grid independence, then validates outputs against fluidized-bed experiments. No equations or self-citations are shown that would reduce a claimed prediction or uniqueness result to a fitted parameter or prior author work by construction. The central claims rest on comparison to independent data, satisfying the criterion for a self-contained derivation with no load-bearing internal equivalences.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies insufficient technical detail to identify concrete free parameters, axioms, or invented entities; all fields left empty pending full text review.

pith-pipeline@v0.9.0 · 5683 in / 950 out tokens · 20841 ms · 2026-05-23T22:26:32.191195+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages

  1. [1]

    CFD-DEM study of coke combustion in the raceway cavity of an ironmaking blast furnace,

    J. Cui, Q. Hou, and Y. Shen, "CFD-DEM study of coke combustion in the raceway cavity of an ironmaking blast furnace," Powder Technology, vol. 362, pp. 539-549, 2020. [38]

    work page 2020

  2. [2]

    CFD-DEM simulations of a fluidized bed crystallizer,

    K. Kerst et al., "CFD-DEM simulations of a fluidized bed crystallizer," Chemical Engineering Science, vol. 165, pp. 1-13, 2017

    work page 2017

  3. [3]

    Bridging particle and reactor scales in the simulation of biomass fast pyrolysis by coupling particle resolved simulation and coarse grained CFD-DEM,

    L. Lu, X. Gao, M. Shahnam, and W. A. Rogers, "Bridging particle and reactor scales in the simulation of biomass fast pyrolysis by coupling particle resolved simulation and coarse grained CFD-DEM," Chemical Engineering Science, vol. 216, p. 115471, 2020

    work page 2020

  4. [4]

    Bimodal frequency distribution of granular discharge in 2D hoppers,

    S. Zhang, M. Zhao, W. Ge, and C. Liu, "Bimodal frequency distribution of granular discharge in 2D hoppers," Chemical Engineering Science, vol. 245, p. 116945, 2021

    work page 2021

  5. [5]

    A Novel Method (T-Junction with a Tilted Slat) for Controlling Breakup Volume Ratio of Droplets in Micro and Nanofluidic T-Junctions,

    A. Kiani Moqadam, A. Bedram, and M. H. Hamedi, "A Novel Method (T-Junction with a Tilted Slat) for Controlling Breakup Volume Ratio of Droplets in Micro and Nanofluidic T-Junctions," (in en), Journal of Applied Fluid Mechanics, vol. 11, no. 5, pp. 1255-1265, 2018, doi: https://doi.org/10.29252/jafm.11.05.28598

    work page doi:10.29252/jafm.11.05.28598 2018

  6. [6]

    A discrete numerical model for granular assemblies,

    P. A. Cundall and O. D. L. Strack, "A discrete numerical model for granular assemblies," Géotechnique, vol. 29, no. 1, pp. 47-65, 1979, doi: 10.1680/geot.1979.29.1.47

    work page doi:10.1680/geot.1979.29.1.47 1979

  7. [7]

    Discrete particle simulation of two-dimensional fluidized bed,

    Y. Tsuji, T. Kawaguchi, and T. Tanaka, "Discrete particle simulation of two-dimensional fluidized bed," Powder Technology, vol. 77, no. 1, pp. 79-87, 1993/10/01/ 1993, doi: https://doi.org/10.1016/0032-5910(93)85010-7

    work page doi:10.1016/0032-5910(93)85010-7 1993

  8. [8]

    Discrete particle simulation of bubble and slug formation in a two-dimensional gas-fluidised bed: a hard-sphere approach,

    B. Hoomans, J. Kuipers, W. J. Briels, and W. P. M. van Swaaij, "Discrete particle simulation of bubble and slug formation in a two-dimensional gas-fluidised bed: a hard-sphere approach," Chemical Engineering Science, vol. 51, no. 1, pp. 99-118, 1996

    work page 1996

  9. [9]

    Numerical simulation of the gas-solid flow in a fluidized bed by combining discrete particle method with computational fluid dynamics,

    B. H. Xu and A. B. Yu, "Numerical simulation of the gas-solid flow in a fluidized bed by combining discrete particle method with computational fluid dynamics," Chemical Engineering Science, vol. 52, no. 16, pp. 2785-2809, 1997/08/01/ 1997, doi: https://doi.org/10.1016/S0009- 2509(97)00081-X

    work page doi:10.1016/s0009- 1997

  10. [10]

    Open-source MFIX-DEM software for gas–solids flows: Part I—Verification studies,

    R. Garg, J. Galvin, T. Li, and S. Pannala, "Open-source MFIX-DEM software for gas–solids flows: Part I—Verification studies," Powder Technology, vol. 220, pp. 122-137, 2012/04/01/ 2012, doi: https://doi.org/10.1016/j.powtec.2011.09.019

    work page doi:10.1016/j.powtec.2011.09.019 2012

  11. [11]

    Documentation of open-source MFIX–DEM software for gas-solids flows,

    R. Garg, J. Galvin-Carney, T. Li, and S. Pannala, "Documentation of open-source MFIX–DEM software for gas-solids flows," Tingwen Li Dr., 09/01 2012

    work page 2012

  12. [12]

    Comprehensive DEM-DPM-CFD simulations- model synthesis, experimental validation and scalability,

    C. Kloss, C. Goniva, G. Aichinger, and S. Pirker, "Comprehensive DEM-DPM-CFD simulations- model synthesis, experimental validation and scalability," in Proceedings of the seventh international conference on CFD in the minerals and process industries, CSIRO, Melbourne, Australia, 2009, pp. 9-11

    work page 2009

  13. [13]

    Models, algorithms and validation for opensource DEM and CFD-DEM,

    C. Kloss, C. Goniva, A. Hager, S. Amberger, and S. Pirker, "Models, algorithms and validation for opensource DEM and CFD-DEM," Pcfd, vol. 12, p. 140, 2012

    work page 2012

  14. [14]

    Asynchronous GPU-based DEM solver embedded in commercial CFD software with polyhedral mesh support,

    A. Kianimoqadam and J. L. Lapp, "Asynchronous GPU-based DEM solver embedded in commercial CFD software with polyhedral mesh support," Powder Technology, vol. 444, p. 120040, 2024/08/01/ 2024, doi: https://doi.org/10.1016/j.powtec.2024.120040

    work page doi:10.1016/j.powtec.2024.120040 2024

  15. [15]

    A CPU-GPU cross-platform coupled CFD-DEM approach for complex particle-fluid flows,

    Y. He, F. Muller, A. Hassanpour, and A. E. Bayly, "A CPU-GPU cross-platform coupled CFD-DEM approach for complex particle-fluid flows," Chemical Engineering Science, vol. 223, p. 115712, 2020/09/21/ 2020, doi: https://doi.org/10.1016/j.ces.2020.115712

    work page doi:10.1016/j.ces.2020.115712 2020

  16. [16]

    Influence of void fraction calculation on fidelity of CFD-DEM simulation of gas-solid bubbling fluidized beds,

    Z. Peng, E. Doroodchi, C. Luo, and B. Moghtaderi, "Influence of void fraction calculation on fidelity of CFD-DEM simulation of gas-solid bubbling fluidized beds," AIChE Journal, vol. 60, no. 6, pp. 2000-2018, 2014/06/01 2014, doi: https://doi.org/10.1002/aic.14421

    work page doi:10.1002/aic.14421 2000

  17. [17]

    Theoretical Analysis of Computational Fluid Dynamics–Discrete Element Method Mathematical Model Solution Change With Varying Computational Cell Size,

    A. Volk and U. Ghia, "Theoretical Analysis of Computational Fluid Dynamics–Discrete Element Method Mathematical Model Solution Change With Varying Computational Cell Size," Journal of Fluids Engineering, vol. 141, no. 9, 2019, doi: 10.1115/1.4042956. [39]

    work page doi:10.1115/1.4042956 2019

  18. [18]

    A modified direct method for void fraction calculation in CFD–DEM simulations,

    Z. Peng, B. Moghtaderi, and E. Doroodchi, "A modified direct method for void fraction calculation in CFD–DEM simulations," Advanced Powder Technology, vol. 27, no. 1, pp. 19-32, 2016/01/01/ 2016, doi: https://doi.org/10.1016/j.apt.2015.10.021

    work page doi:10.1016/j.apt.2015.10.021 2016

  19. [19]

    Direct calculation of voidage in the fine-grid CFD–DEM simulation of fluidized beds with large particles,

    L. Wang, J. Ouyang, and C. Jiang, "Direct calculation of voidage in the fine-grid CFD–DEM simulation of fluidized beds with large particles," Particuology, vol. 40, pp. 23-33, 2018/10/01/ 2018, doi: https://doi.org/10.1016/j.partic.2017.11.010

    work page doi:10.1016/j.partic.2017.11.010 2018

  20. [20]

    Investigation of Void Fraction Schemes for Use with CFD-DEM Simulations of Fluidized Beds,

    D. A. Clarke, A. J. Sederman, L. F. Gladden, and D. J. Holland, "Investigation of Void Fraction Schemes for Use with CFD-DEM Simulations of Fluidized Beds," Industrial & Engineering Chemistry Research, vol. 57, no. 8, pp. 3002-3013, 2018/02/28 2018, doi: 10.1021/acs.iecr.7b04638

    work page doi:10.1021/acs.iecr.7b04638 2018

  21. [21]

    Flexible discretization technique for DEM-CFD simulations including thin walls,

    K. Takabatake and M. Sakai, "Flexible discretization technique for DEM-CFD simulations including thin walls," Advanced Powder Technology, vol. 31, no. 5, pp. 1825-1837, 2020/05/01/ 2020, doi: https://doi.org/10.1016/j.apt.2020.02.017

    work page doi:10.1016/j.apt.2020.02.017 2020

  22. [22]

    Semi-resolved CFD–DEM for thermal particulate flows with applications to fluidized beds,

    Z. Wang and M. Liu, "Semi-resolved CFD–DEM for thermal particulate flows with applications to fluidized beds," International Journal of Heat and Mass Transfer, vol. 159, p. 120150, 2020/10/01/ 2020, doi: https://doi.org/10.1016/j.ijheatmasstransfer.2020.120150

    work page doi:10.1016/j.ijheatmasstransfer.2020.120150 2020

  23. [23]

    Algorithms in a robust hybrid CFD-DEM solver for particle-laden flows,

    H. Xiao and J. Sun, "Algorithms in a robust hybrid CFD-DEM solver for particle-laden flows," Communications in Computational Physics, vol. 9, no. 2, pp. 297-323, 2011

    work page 2011

  24. [24]

    Diffusion-based coarse graining in hybrid continuum–discrete solvers: Theoretical formulation and a priori tests,

    R. Sun and H. Xiao, "Diffusion-based coarse graining in hybrid continuum–discrete solvers: Theoretical formulation and a priori tests," International Journal of Multiphase Flow, vol. 77, pp. 142-157, 2015/12/01/ 2015, doi: https://doi.org/10.1016/j.ijmultiphaseflow.2015.08.014

    work page doi:10.1016/j.ijmultiphaseflow.2015.08.014 2015

  25. [25]

    Diffusion-based coarse graining in hybrid continuum–discrete solvers: Applications in CFD–DEM,

    R. Sun and H. Xiao, "Diffusion-based coarse graining in hybrid continuum–discrete solvers: Applications in CFD–DEM," International Journal of Multiphase Flow, vol. 72, pp. 233-247, 2015/06/01/ 2015, doi: https://doi.org/10.1016/j.ijmultiphaseflow.2015.02.014

    work page doi:10.1016/j.ijmultiphaseflow.2015.02.014 2015

  26. [26]

    Averaging method of granular materials,

    H. P. Zhu and A. B. Yu, "Averaging method of granular materials," Physical Review E, vol. 66, no. 2, p. 021302, 2002

    work page 2002

  27. [27]

    Calculating the view factor of randomly dispersed multi-sized particles using hybrid GRU-LSTM recurrent neural networks regression,

    A. Kianimoqadam and J. Lapp, "Calculating the view factor of randomly dispersed multi-sized particles using hybrid GRU-LSTM recurrent neural networks regression," International Journal of Heat and Mass Transfer, vol. 202, p. 123756, 2023/03/01/ 2023, doi: https://doi.org/10.1016/j.ijheatmasstransfer.2022.123756

    work page doi:10.1016/j.ijheatmasstransfer.2022.123756 2023

  28. [28]

    Rolling friction in the dynamic simulation of sandpile formation,

    Y. C. Zhou, B. D. Wright, R. Y. Yang, B. H. Xu, and A. B. Yu, "Rolling friction in the dynamic simulation of sandpile formation," Physica A: Statistical Mechanics and its Applications, vol. 269, no. 2, pp. 536-553, 1999/07/15/ 1999, doi: https://doi.org/10.1016/S0378-4371(99)00183-1

    work page doi:10.1016/s0378-4371(99)00183-1 1999

  29. [29]

    Ueber die Berührung fester elastischer Körper,

    H. Hertz, "Ueber die Berührung fester elastischer Körper," 1882

    work page

  30. [30]

    Elastic spheres in contact under varying oblique forces,

    R. D. Mindlin and H. Deresiewicz, "Elastic spheres in contact under varying oblique forces," 1953

    work page 1953

  31. [31]

    Fluid Mechanical Description of Fluidized Beds. Equations of Motion,

    T. B. Anderson and R. Jackson, "Fluid Mechanical Description of Fluidized Beds. Equations of Motion," Industrial & Engineering Chemistry Fundamentals, vol. 6, no. 4, pp. 527-539, 1967/11/01 1967, doi: 10.1021/i160024a007

    work page doi:10.1021/i160024a007 1967

  32. [32]

    Drag force of intermediate Reynolds number flow past mono- and bidisperse arrays of spheres,

    R. Beetstra, M. A. van der Hoef, and J. A. M. Kuipers, "Drag force of intermediate Reynolds number flow past mono- and bidisperse arrays of spheres," AIChE Journal, https://doi.org/10.1002/aic.11065 vol. 53, no. 2, pp. 489-501, 2007/02/01 2007, doi: https://doi.org/10.1002/aic.11065

    work page doi:10.1002/aic.11065 2007

  33. [33]

    Numerical study of segregation using a new drag force correlation for polydisperse systems derived from lattice-Boltzmann simulations,

    R. Beetstra, M. A. van der Hoef, and J. A. M. Kuipers, "Numerical study of segregation using a new drag force correlation for polydisperse systems derived from lattice-Boltzmann simulations," Chemical Engineering Science, vol. 62, no. 1, pp. 246-255, 2007/01/01/ 2007, doi: https://doi.org/10.1016/j.ces.2006.08.054

    work page doi:10.1016/j.ces.2006.08.054 2007

  34. [34]

    Experimental validation of Lagrangian–Eulerian simulations of fluidized beds,

    B. G. M. van Wachem, J. van der Schaaf, J. C. Schouten, R. Krishna, and C. M. van den Bleek, "Experimental validation of Lagrangian–Eulerian simulations of fluidized beds," Powder [40] Technology, vol. 116, no. 2, pp. 155-165, 2001/05/23/ 2001, doi: https://doi.org/10.1016/S0032- 5910(00)00389-2

    work page doi:10.1016/s0032- 2001

  35. [35]

    Validation of the Eulerian simulated dynamic behaviour of gas–solid fluidised beds,

    B. G. M. Van Wachem, J. C. Schouten, R. Krishna, and C. M. Van den Bleek, "Validation of the Eulerian simulated dynamic behaviour of gas–solid fluidised beds," Chemical Engineering Science, vol. 54, no. 13-14, pp. 2141-2149, 1999

    work page 1999