Packing fraction of particles with lognormal size distribution

H. J. H. Brouwers

Phys. Rev. E 89, 052211 – Published 21 May 2014

Abstract

This paper addresses the packing and void fraction of polydisperse particles with a lognormal size distribution. It is demonstrated that a binomial particle size distribution can be transformed into a continuous particle-size distribution of the lognormal type. Furthermore, an original and exact expression is derived that predicts the packing fraction of mixtures of particles with a lognormal distribution, which is governed by the standard deviation, mode of packing, and particle shape only. For a number of particle shapes and their packing modes (close, loose) the applicable values are given. This closed-form analytical expression governing the packing fraction is thoroughly compared with empirical and computational data reported in the literature, and good agreement is found.

Received 1 January 2014

DOI:https://doi.org/10.1103/PhysRevE.89.052211

Authors & Affiliations

H. J. H. Brouwers

Department of the Built Environment, Eindhoven University of Technology, P. O. Box 513, 5600 MB Eindhoven, The Netherlands

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Vol. 89, Iss. 5 — May 2014

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Images

Figure 1
Binary void fraction, $h$ ( $u$ , $c_{L}$ ), of mixes with a large size ratio ( $d_{L}$ / $d_{S}$ = $u$ > $u_{b}$ ) as a function of the volume fraction of large constituent ( $c_{L}$ ), with φ $_{L}$ = 0.4 and φ $_{S}$ = 0.5 as the void fractions of the large and small constituents. The volume fraction ( $c_{L}^{sat}$ ) and binary void fraction ( $h^{sat}$ ) pertaining to a saturated mix are indicated, as well as linear interpolations [Eqs. (6) and (7)] and the curves $A$ or $B$ [Eqs. (8) and (9), respectively] for all other compositions.
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Figure 2
Cumulative finer fraction $F$ versus the logarithm of the scaled particle size for a discretely sized ternary mix (with $d_{1}$ / $d_{2}$ = $d_{2}$ / $d_{3}$ = $u$ ) having a binomial distribution ( $p$ = ½) of the particle volumes and, hence, $c_{1}$ = 0.25 $_{,}$ $c_{2}$ = 0.5, and $c_{3}$ = 0.25.
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Figure 3
Void fraction $φ_{n}^{BN}$ of binomially distributed combinations of noninteracting particle classes as a function of the void fraction φ $_{1}$ of each particle class for $n$ = 3 and $n$ = 10, taken from Table 1, and following Eq. (16) with ψ $_{n}$ , taken from Table 2 (fitted value).
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Figure 4
(Color online) Scaled ternary void fraction of spheres ( $u$ = 2) versus the probability as measured [38] and as computed by Eq. (30) with substituting $n$ = 3 and using the scaled binary void fraction ( $u$ = 2) as measured [38]. The values are summarized in Table 4.
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Figure 5
(Color online) (a) Computer-generated void fraction of random close packed spheres $φ^{LN}$ , with a lognormal size distribution, versus the standard deviation [11, 12]. Equation (31) with φ $_{1}$ = 0.37 and where β = 0.195 is also set out. (b) Experimentally measured [26, 37] and computer-generated [12] packing fraction of spheres, with a lognormal size distribution, versus the standard deviation. Equation (32) with φ $_{1}$ = 0.409 and where β = 0.176 is also set out, as well as with $\sqrt{2 π} β (1 - φ_{1}^{})$ = 0.19, which was conjectured in Ref. [26]. (c) Computer-generated void fraction of random loose packed spheres $φ^{LN}$ [11], with a lognormal size distribution, versus the standard deviation. Equation (31) with φ $_{1}$ = 0.466 and where β = 0.155 and β = 0.21 are also set out.
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Figure 6
Experimentally measured packing fraction of two sands (OS and MR), with a lognormal size distribution for different geometric mean sizes $d_{g}$ (in mm), versus the standard deviation [25]. Equation (32) with $f_{1}$ = 0.576 and β = 0.26 (MR), and with $f_{1}$ = 0.624 and β = 0.25 (OS).
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