An improved contact method for quantifying the mixing of a binary granular mixture

Abstract

When two granular phases are brought into motion in a rotating drum, a competition of mixing and segregation occurs. Several image analysis methods have been used to quantify the mixing. In this work, a modification of the contact method, originally proposed by Van Puyvelde et al. (Powder Technol. 106, 183–191 (1999)), is suggested to allow evaluation of the mixing index for each separate image. A strength of this modified method lies in the removal of the case-dependent normalization of the mixing index, which has so far impaired a direct comparison to other studies. This modified method is tested on artificial and experimental images of a granular bed composed of spherical glass and polypropylene beads of equal size. The bed evolves in a rotating drum under the rolling regime. The temporal evolution of the mixing index is in excellent agreement with the commonly used variance method.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Data availability

 The image sequence used in this work is available in the supplementary material. Additional material on the minimum contact length is also available.

Code availability

 The implementation of the variance and modified contact method is accessible with the DOI number (10.14278/rodare.549) after an embargo period, that is when the manuscript is accepted for publication.

References

  1. 1.

    Sandadi, S., Pandey, P., Turton, R.: In situ, near real-time acquisition of particle motion in rotating pan coating equipment using imaging techniques. Chem. Eng. Sci. 59, 5807–5817 (2004). https://doi.org/10.1016/j.ces.2004.06.036

    Article  Google Scholar 

  2. 2.

    Silvério, B.C., Arruda, E.B., Duarte, C.R., Barrozo, M.A.S.: A novel rotary dryer for drying fertilizer: comparison of performance with conventional configurations. Powder Technol. 270, 135–140 (2015). https://doi.org/10.1016/j.powtec.2014.10.030

    Article  Google Scholar 

  3. 3.

    Boateng, A.A., Barr, P.V.: Modelling of particle mixing and segregation in the transverse plane of a rotary kiln. Chem. Eng. Sci. 51, 4167–4181 (1996). https://doi.org/10.1016/0009-2509(96)00250-3

    Article  Google Scholar 

  4. 4.

    Yazdani, E., Hashemabadi, S.H.: DEM simulation of heat transfer of binary-sized particles in a horizontal rotating drum. Granul. Matter. 21, 1–11 (2019). https://doi.org/10.1007/s10035-018-0857-3

    Article  Google Scholar 

  5. 5.

    Bui, R.T., Perron, J., Read, M.: Model-based optimization of the operation of the coke calcining kiln. Carbon. 31, 1139–1147 (1993). https://doi.org/10.1016/0008-6223(93)90067-K

    Article  Google Scholar 

  6. 6.

    Dury, C.M., Ristow, G.H.: Competition of mixing and segregation in rotating cylinders. Phys. Fluids. 11, 1387–1394 (1999). https://doi.org/10.1063/1.870003

    ADS  Article  MATH  Google Scholar 

  7. 7.

    Huang, A.N., Kuo, H.P.: A study of the three-dimensional particle size segregation structure in a rotating drum. AIChE J. 58, 1076–1083 (2012). https://doi.org/10.1002/aic.12658

    Article  Google Scholar 

  8. 8.

    Jain, N., Ottino, J.M., Lueptow, R.M.: Regimes of segregation and mixing in combined size and density granular systems: an experimental study. Granul. Matter. 7, 69–81 (2005). https://doi.org/10.1007/s10035-005-0198-x

    Article  Google Scholar 

  9. 9.

    Rong, W., Li, B., Feng, Y., Schwarz, P., Witt, P., Qi, F.: Numerical analysis of size-induced particle segregation in rotating drums based on Eulerian continuum approach. Powder Technol. 376, 80–92 (2020). https://doi.org/10.1016/j.powtec.2020.07.101

    Article  Google Scholar 

  10. 10.

    Norouzi, H.R., Zarghami, R., Mostoufi, N.: Insights into the granular flow in rotating drums. Chem. Eng. Res. Des. 102, 12–25 (2015). https://doi.org/10.1016/j.cherd.2015.06.010

    Article  Google Scholar 

  11. 11.

    Zuriguel, I., Gray, J.M.N.T., Peixinho, J., Mullin, T.: Pattern selection by a granular wave in a rotating drum. Phys. Rev. E. 73, 061302 (2006). https://doi.org/10.1103/PhysRevE.73.061302

    ADS  Article  Google Scholar 

  12. 12.

    Chou, S.H., Liao, C.C., Hsiau, S.S.: The effect of interstitial fluid viscosity on particle segregation in a slurry rotating drum. Phys. Fluids. 23 (2011). https://doi.org/10.1063/1.3623275

  13. 13.

    Khakhar, D.V., McCarthy, J.J., Ottino, J.M.: Radial segregation of granular mixtures in rotating cylinders. Phys. Fluids. 9, 3600–3614 (1997). https://doi.org/10.1007/s10035-011-0259-2

    ADS  Article  Google Scholar 

  14. 14.

    Hill, K.M., Caprihan, A., Kakalios, J.: Axial segregation of granular media rotated in a drum mixer: pattern evolution. Phys. Rev. E. 56, 4386–4393 (1997). https://doi.org/10.1103/PhysRevE.56.4386

    ADS  Article  Google Scholar 

  15. 15.

    Windows-Yule, C.R.K., Scheper, B.J., van der Horn, A.J., Hainsworth, N., Saunders, J., Parker, D.J., Thornton, A.R.: Understanding and exploiting competing segregation mechanisms in horizontally rotated granular media. New J. Phys. 18 (2016). https://doi.org/10.1088/1367-2630/18/2/023013

  16. 16.

    Santos, D.A., Duarte, C.R., Barrozo, M.A.S.: Segregation phenomenon in a rotary drum: experimental study and CFD simulation. Powder Technol. 294, 1–10 (2016). https://doi.org/10.1016/j.powtec.2016.02.015

    Article  Google Scholar 

  17. 17.

    McCarthy, J.J., Khakhar, D.V., Ottino, J.M.: Computational studies of granular mixing. Powder Technol. 109, 72–82 (2000). https://doi.org/10.1016/S0032-5910(99)00228-4

    Article  Google Scholar 

  18. 18.

    Rapaport, D.C.: Radial and axial segregation of granular matter in a rotating cylinder: a simulation study. Phys. Rev. E. 75, 031301 (2007). https://doi.org/10.1103/PhysRevE.75.031301

    ADS  Article  Google Scholar 

  19. 19.

    Yamamoto, M., Ishihara, S., Kano, J.: Evaluation of particle density effect for mixing behavior in a rotating drum mixer by DEM simulation. Adv. Powder Technol. 27, 864–870 (2016). https://doi.org/10.1016/J.APT.2015.12.013

    Article  Google Scholar 

  20. 20.

    Lacey, P.M.C.: Developments in the theory of particle mixing. J. Appl. Chem. 4, 257–268 (1954). https://doi.org/10.1002/jctb.5010040504

    Article  Google Scholar 

  21. 21.

    Fan, L.T., Chen, S.J., Watson, C.A.: Solids mixing. Ind. Eng. Chem. 62, 53–69 (1970). https://doi.org/10.1021/ie50727a009

    Article  Google Scholar 

  22. 22.

    Bridgwater, J.: Mixing of powders and granular materials by mechanical means—A perspective. Particuology. 10, 397–427 (2012). https://doi.org/10.1016/j.partic.2012.06.002

    Article  Google Scholar 

  23. 23.

    Siiriä, S., Yliruusi, J.: Determining a value for mixing: mixing degree. Powder Technol. 196, 309–317 (2009). https://doi.org/10.1016/j.powtec.2009.08.009

    Article  Google Scholar 

  24. 24.

    Chibwe, D.K., Evans, G.M., Doroodchi, E., Monaghan, B.J., Pinson, D.J., Chew, S.J.: Particle near-neighbour separation index for quantification of segregation of granular material. Powder Technol. 360, 481–492 (2020). https://doi.org/10.1016/j.powtec.2019.10.079

    Article  Google Scholar 

  25. 25.

    Chandratilleke, G.R., Yu, A.B., Bridgwater, J., Shinohara, K.: A particle-scale index in the quantification of mixing of particles. AIChE J. 58, 1099–1118 (2012). https://doi.org/10.1002/aic.12654

    Article  Google Scholar 

  26. 26.

    Asmar, B.N., Langston, P.A., Matchett, A.J.: A generalised mixing index in distinct element method simulation of vibrated particulate beds. Granul. Matter. 4, 129–138 (2002). https://doi.org/10.1007/s10035-002-0112-8

    Article  MATH  Google Scholar 

  27. 27.

    Ayeni, O.O., Wu, C.L., Joshi, J.B., Nandakumar, K.: A discrete element method study of granular segregation in non-circular rotating drums. Powder Technol. 283, 549–560 (2015). https://doi.org/10.1016/j.powtec.2015.06.038

    Article  Google Scholar 

  28. 28.

    Bhalode, P., Ierapetritou, M.: A review of existing mixing indices in solid-based continuous blending operations. Powder Technol. 373, 195–209 (2020). https://doi.org/10.1016/j.powtec.2020.06.043

    Article  Google Scholar 

  29. 29.

    Lacey, P.M.C.: The mixing of solid particles. Trans. Inst. Chem. Eng. 21, 53–59 (1943). https://doi.org/10.1016/S0263-8762(97)80004-4

    Article  Google Scholar 

  30. 30.

    Van Puyvelde, D.R., Young, B.R., Wilson, M.A., Schmidt, S.J.: Experimental determination of transverse mixing kinetics in a rolling drum by image analysis. Powder Technol. 106, 183–191 (1999). https://doi.org/10.1016/S0032-5910(99)00074-1

    Article  Google Scholar 

  31. 31.

    Chou, S.-H., Song, Y.-L., Hsiau, S.-S.: A study of the mixing index in solid particles. KONA Powder Part. J. 34, 275–281 (2017). https://doi.org/10.14356/kona.2017018

    Article  Google Scholar 

  32. 32.

    Gosselin, R., Duchesne, C., Rodrigue, D.: On the characterization of polymer powders mixing dynamics by texture analysis. Powder Technol. 183, 177–188 (2008). https://doi.org/10.1016/j.powtec.2007.07.021

    Article  Google Scholar 

  33. 33.

    Liu, X., Zhang, C., Zhan, J.: Quantitative comparison of image analysis methods for particle mixing in rotary drums. Powder Technol. 282, 32–36 (2015). https://doi.org/10.1016/j.powtec.2014.08.076

    Article  Google Scholar 

  34. 34.

    Cundall, P.A., Strack, O.D.L.: A discrete numerical model for granular assemblies. Géotechnique. 29, 47–65 (1979). https://doi.org/10.1680/geot.1979.29.1.47

    Article  Google Scholar 

  35. 35.

    Haralick, R.M., Sternberg, S.R., Zhuang, X.: Image analysis using mathematical morphology. IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 532–550 (1987). https://doi.org/10.1109/TPAMI.1987.4767941

    Article  Google Scholar 

  36. 36.

    Qi, F., Heindel, T.J., Wright, M.M.: Numerical study of particle mixing in a lab-scale screw mixer using the discrete element method. Powder Technol. 308, 334–345 (2017). https://doi.org/10.1016/j.powtec.2016.12.043

    Article  Google Scholar 

  37. 37.

    Pratt, V.: Direct least-squares fitting of algebraic surfaces. In: Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques—SIGGRAPH ’87, pp. 145–152. ACM Press, New York, New York, USA (1987)

  38. 38.

    Mellmann, J.: The transverse motion of solids in rotating cylinders—forms of motion and transition behavior. Powder Technol. 118, 251–270 (2001). https://doi.org/10.1016/S0032-5910(00)00402-2

    Article  Google Scholar 

  39. 39.

    Papapetrou, T.N., Lecrivain, G., Bieberle, M., Boudouvis, A., Hampel, U.: Code, data and supplementary material for: an improved contact method for quantifying the mixing of a binary granular mixture. https://rodare.hzdr.de/record/549

Download references

Acknowledgements

This work was supported by a “Landesstipendium” awarded by the Graduate Academy of the Technische Universität Dresden.

Funding

 Funded within the Saxon State Scholarship by the Free State of Saxony.

Author information

Affiliations

Authors

Contributions

 TNP performed the experiments, designed and implemented the method. GL, MB, and UH supervised the project. All authors, including AB, analyzed the data and wrote the manuscript.

Corresponding authors

Correspondence to Theodoros Nestor Papapetrou or Gregory Lecrivain.

Ethics declarations

Conflicts of interest

 The authors declare no conflict of interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Supplementary material 1 (AVI 48262 kb)

Supplementary material 2 (PDF 428 kb)

Online Resource 3

: Online_Resource_3 (PNG 392 kb)

Appendices

Appendix 1: Modified filling height as a function of time

The modified filling height \(\tilde{h}\), normalized with the theoretical initial value h = 53 mm, is depicted for each image of the experimental sequence in Fig. 13. The modified height \(\tilde{h}\) is higher by up to two particle diameters than its theoretical counterpart, which would be achieved if all particles were distributed inside the drum so that the free surface would form a perfect plane.

Fig. 13
figure13

The modified filling height, \(\tilde{h}\), normalized with the theoretical initial value h, as a function of time

Appendix 2: Definition of the function \({\tilde{A}}_R\left(\tilde{\ell}\right)\) used in Eq. (13)

The modified red granular area (\({\tilde{A}}_R\)) in Eq. (13) and better illustrated in Fig. 14 is expressed as a function of the horizontal length (\(\tilde{\ell }\)). Using Green’s theorem, one obtains

Fig. 14
figure14

Diagram illustrating the calculation of \(\tilde{\ell }\)

$${\tilde{A}}_R\left(\tilde{\ell}\right)=\frac{\tilde{\ell }}{2}\left(\tilde{h}-{\tilde{h}}^{\ast}\right)+\frac{\tilde{h}}{2}\left(\tilde{\ell }-{\tilde{\ell}}^{\ast}\right)+\frac{\tilde{D}}{4}\left({\tilde{h}}^{\ast }-\tilde{h}+{\tilde{\ell}}^{\ast }-\tilde{\ell}\right)+\frac{{\tilde{D}}^2}{8}\tilde{\theta },$$
(15)

where \({\tilde{h}}^{\ast }=\tilde{D}/2-\sqrt{\tilde{\ell}\left(\tilde{D}-\tilde{\ell}\right)}\), \({\tilde{\ell}}^{\ast }=\tilde{D}/2-\sqrt{\tilde{h}\left(\tilde{D}-\tilde{h}\right)}\), \(\tilde{\theta }=\pi -{\cos}^{-1}\left(2\tilde{\ell }/\tilde{D}-1\right)+{\sin}^{-1}\left(2\tilde{h}/\tilde{D}-1\right)\). \({\tilde{A}}_R\) is defined for \({\tilde{\ell}}^{\ast}\le \tilde{\ell}\le \tilde{D}-{\tilde{\ell}}^{\ast }\).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Papapetrou, T.N., Lecrivain, G., Bieberle, M. et al. An improved contact method for quantifying the mixing of a binary granular mixture. Granular Matter 23, 15 (2021). https://doi.org/10.1007/s10035-020-01073-3

Download citation

Keywords

  • Binary particle mixing
  • Rotating drum
  • Image analysis
  • Mixing index
  • Contact method
  • Variance method