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


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.

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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.


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


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

Author information




 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.

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Supplementary material 1 (AVI 48262 kb)

Supplementary material 2 (PDF 428 kb)

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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

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

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

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

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 \({\tilde{\ell}}^{\ast}\le \tilde{\ell}\le \tilde{D}-{\tilde{\ell}}^{\ast }\).

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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).

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