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.

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

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 $\tilde{\ell }$

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

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