Packing fraction of particles with a Weibull size distribution

This paper addresses the void fraction of polydisperse particles with a Weibull (or Rosin-Rammler) size distribution. It is demonstrated that the governing parameters of this distribution can be uniquely related to those of the lognormal distribution. Hence, an existing closed-form expression that predicts the void fraction of particles with a lognormal size distribution can be transformed into an expression for Weibull distributions. Both expressions contain the contraction coefficient β. Likewise the monosized void fraction ${\phi }_1$, it is a physical parameter which depends on the particles’ shape and their state of compaction only. Based on a consideration of the scaled binary void contraction, a linear relation for $(1-{\phi }_1)\beta$ as function of ${\phi }_1$ is proposed, with proportionality constant B, depending on the state of compaction only. This is validated using computational and experimental packing data concerning random close and random loose packing arrangements. Finally, using this β, the closed-form analytical expression governing the void fraction of Weibull distributions is thoroughly compared with empirical data reported in the literature, and good agreement is found. Furthermore, the present analysis yields an algebraic equation relating the void fraction of monosized particles at different compaction states. This expression appears to be in good agreement with a broad collection of random close and random loose packing data.

Figure 1

The cumulative finer distribution F of two lognormal and Weibull distributions as a function of the particle size $d$: (a) $\mathrm{m}=\sqrt{2},\ln {\sigma }_g=1,\delta =0.1,\ln d_g=-2.56$, and (b) $\mathrm{m}=4\sqrt{2},\ln {\sigma }_g=0.25,\delta =10,\ln d_g=2.24$, whereby $m$ and $\ln {\sigma }_g$ on the one hand, and lnδ and $\ln d_g$ on the other, are coupled by Eqs. (8) and (10), respectively (with $\alpha =2)$.

Figure 2

Void fraction reduction quotient $(h/{\phi }_1)$ of random close packing (RCP) of binary spheres (shaded area) as a function of the large sphere volume fraction $c_L(0\le c_L\le 1)$ and sphere size ratio $u(1\le u<\infty )$. The boundaries for $u\to \infty$ follow from Eqs. (13) and (14) with ${\phi }_1=0.36$ [indicated by (a) and (b), respectively]. The approximation [Eq. (15) with $\beta =0.20,{\phi }_1=0.36$] and the simulation results of [29] for $u=2$ are included as well. For $u\to \infty$ the composition $[c_L={(1+{\phi }_1)}^{-1}]$ and the scaled void fraction $(h/{\phi }_1={\phi }_1)$ pertaining to maximum void reduction are indicated.

Figure 3

Values of $\left(1-{\phi }_1^{\mathrm{RCP}}\right){\beta }^{\mathrm{RCP}}$ versus ${\phi }_1^{\mathrm{RCP}}$ for a number of particle shapes (Table 2), and Eq. (18) with $B^{\mathrm{RCP}}=0.358$.

Figure 4

(a) Experimentally measured [35] void fraction of glass beads and crushed glass particles with various lognormal size distributions, using tapping and prodding, versus the standard deviation $\ln {\sigma }_g$. (a) Equation (11) with ${\phi }_1=0.395$ and $\beta =0.182$ (glass beads), and (b) Eq. (21) with ${\phi }_1=0.505$ and $B=0.279$ (crushed glass). (b) Experimentally measured [35] void fraction of glass beads and crushed glass particles with various Weibull size distributions, using tapping and prodding, versus the distribution shape parameter $m$. (a) Equation (12) with ${\phi }_1=0.395$ and $\beta =0.182$ (glass beads), and (b) Eq. (22) with ${\phi }_1=0.505$ and $B=0.279$ (crushed glass).

Figure 5

Void fraction range of monosized particles as a function of ${\phi }_1^{\mathrm{RCP}}$ (shaded area). The curve pertaining to the maximum void fraction, ${\phi }_1^{\mathrm{RLP}}$, and the line pertaining to the minimum void fraction, ${\phi }_1^{\mathrm{RCP}}$, are shown using Eq. (A2) with $\lambda =2$ and $\lambda =1$, respectively, as well as known combinations of ${\phi }_1^{\mathrm{RLP}}$ and ${\phi }_1^{\mathrm{RCP}}$ values for a number of particle shapes.