Capturing the dynamics of a two orifice silo with the I model and extensions_2

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Capturing the dynamics of a two oriﬁce silo with the

µ(I)model and extensions

Samuel K Irvinea, Luke A Fullard, Daniel J Hollandb, Daniel A Clarkec,

Thomasin A Lynch 1a, Pierre-Yves Lagréed

aSchool of Mathematical and Computational Sciences, Massey University, Palmerston

North, New Zealand

bDepartment of Chemical and Process Engineering, University of

Canterbury, Christchurch, New Zealand

cSchool of Chemical and Physical Sciences, Victoria University of Wellington, PO Box

600, Wellington, 6140, New Zealand

dSorbonne Université, Institut Jean Le Rond d’Alembert, CNRS,

UMR7190, Paris, 75005, France

Abstract

Granular material in a silo with two openings can display a ‘ﬂow rate dip’,

where a non-monotonic relationship between ﬂow rate and oriﬁce separation

occurs. In this paper we study continuum modelling of the silo with two

openings. We ﬁnd that the µ(I)rheology can capture the ﬂow rate dip if

physically relevant friction parameters are used. We also extend the model

by accounting for wall friction, dilatancy, and non-local eﬀects. We ﬁnd that

accounting for the wall friction using a Hele-Shaw model better replicates

the qualitative characteristics of the ﬂow rate dip seen in experimental data,

while dilatancy and non-local eﬀects have very little eﬀect on the qualitative

characteristics of the mass ﬂow rate dip. However, we ﬁnd that all three of

these factors have a signiﬁcant impact on the mass ﬂow rate, indicating that

a continuum model which accurately predicts ﬂow rate will need to account

for these eﬀects.

Keywords: Granular ﬂow, Silo, non-local, dilatancy

PACS: 47.57.Gc

2000 MSC: 74E20

1Corresponding author t.a.lynch@massey.ac.nz

Preprint submitted to Advanced Powder Technology February 27, 2023

arXiv:2210.01992v2 [cond-mat.soft] 24 Feb 2023

1. Introduction

Whether it is in small-scale situations such as a salt shaker, or large

industrial-scale situations such as blasted ore being mined from a draw-point,

granular material is often stored in silos or silo-like domains. These domains

can be challenging to model as they provide conditions for many diﬀerent

granular phenomena. In a ﬂowing granular silo the ﬂow behaviour can vary

from the quasi-static regime where the material is static or nearly static, the

dense regime where the granular material ﬂows analogously to a ﬂuid, and the

dilute regime at the oriﬁce where the material is in near free-fall. Developing

models which can capture the ﬂow of behaviour in such a complex domain is

valuable to inform industrial silo design, as well as understanding granular

ﬂows in general.

One interesting ﬂow rate phenomena is the ﬂow rate ‘dip’, which can arise

when a silo has multiple oriﬁces. Previous experiments done with spherical

steel beads in a two opening silo have shown a monotonic decrease in ﬂow

rate as the oriﬁce distance increases [1]. However, experiments done using

coarser, more industrially relevant materials result in a ﬂow rate dip, where

the ﬂow rate for a silo with two openings in close proximity to each other

is lower than the ﬂow rate for a silo with larger separations between the

openings [2]. A multiple oriﬁce silo has been modelled using the kinematic

and plasticity models [3, 4], however due to the ﬂow rate being prescribed

by the choice of parameters, the ﬂow rate dip could not be analysed.

One method of modelling these ﬂows is using Discrete Element modelling

(DEM) [5]. This is a powerful method capable of predicting granular dynam-

ics by considering interactions of pairs of particles, and can replicate some

of the dynamics of the two oriﬁce silo [6]. However, because DEM requires

modelling each particle individually it is computationally expensive, with the

feasible number of particles that can be simulated being orders of magnitude

smaller than the number of particles that are seen in an industrial context.

Alternatively, granular material may be modelled as a continuous pseudo-

ﬂuid. Such a continuum model could capture the desired macro-behaviour

of granular ﬂows while bypassing the computational overhead involved with

modelling the micro-behaviour of granular material. As such a continuum

model capable of accurately replicating the behaviour of granular material

in a silo is relevant to many industries, however such a model is diﬃcult

to develop, with granular materials exhibiting many phenomena that are

diﬃcult to describe.

Some continuum models already exist, most notably the µ(I)model [7, 8].

This model captures the transition between quasi-static and dense ﬂows (with

dilute ﬂows being predicted inaccurately [9]) using the dimensionless inertial

number I. The inertial number represents a ratio between two timescales:

how long it takes for granular material to move due to shear, and how long

it takes for conﬁning pressure to return dilated material to a resting state.

As such, high Irepresents the dilute regime where the material ﬂows in

a ‘gas-like’ manner with shear rate being more important than conﬁning

pressure, while low Irepresents the dense regime with a more ‘liquid-like’

ﬂow and longer lasting particle contacts, with the ﬂow approaching a ‘solid-

like’ quasi-static regime as I→0. The inertial number is used to modify the

frictional behaviour of the continuum model, which can be used to model

ﬂows in multiple diﬀerent conﬁgurations.

However, while the µ(I)model can give good qualitative predictions for

a silo [10, 11, 12], the quantitative ﬂow rate predictions are not accurate [13].

The µ(I)rheology does not account for several key phenomena that can occur

in a granular ﬂow, which may explain this discrepancy. The µ(I)model

applied to a pseudo-2Dsilo does not account for the friction of the front and

back walls, which other works have modelled as a Hele-Shaw like friction [14].

Another eﬀect which is not modelled is dilatancy, where a ﬂowing mass of

granular material will be less densely packed than a stationary mass [15].

This packing density likely signiﬁcantly aﬀects the mass ﬂow rate for a silo,

and as such will need to be accounted for in a continuum model which is

expected to predict the mass ﬂow rate. It also does not account for non-local

eﬀects [16, 17, 18, 19], which are where the properties of ﬂow at a point

are determined by the ﬂow behaviour of nearby material and not simply by

the forces at that point. The µ(I)model is also not well behaved for all

parameters and domains [20, 21].

Other continuum models exist and have been applied to silos. Sev-

eral of these models, including plasticity models [22, 23], the kinematic

model [24, 25], and the stochastic model [26], can give good descriptions for

the qualitative behaviour of ﬂow within the silo. However, each of these mod-

els have the ﬂow rate determined by the choice of a parameter. This means

that while these models can be useful for determining mixing behaviour and

other such phenomena, they have little use when trying to determine quan-

titative ﬂow rate behaviour.

In this paper we examine the two opening silo, as depicted in Figure 1

with parameters given in Table 1. Although this system is 3D, we assume

Figure 1: Schematic of the system modelled. Two openings of diameter Wseparated by a

distance Lallow the granular material to drain. While the simulations are 2D, experiments

are done with some thickness Wdwith the assumption that Wd<< Wsilo so that the silo

can be considered 2Dfor the purpose of simulations, with 3Deﬀects being accounted for

in the Hele-Shaw extension.

that the thickness is small so that it can be treated as a 2Dsystem. We focus

on using the µ(I)rheology, extending it to capture Hele-Shaw wall friction,

dilatancy, and non-local eﬀects. We additionally use the kinematic model for

a simple comparison model for veriﬁcation, while also demonstrating that it

is unsuitable for double opening silos. We examine the ﬂow rate phenomena

using the µ(I)model, determining the eﬀects each of the extensions have on

mass ﬂow rate magnitude and the ﬂow rate dip. We use data from other

experiments [2] for a comparison for the ﬂow rate dip.

2. Model and Implementation

2.1. The µ(I)model

The µ(I)model, as described by [7] and extended to 3Dby [8], is the

baseline model that we use to describe the granular ﬂows we are investigating.

The model uses the incompressible Navier-Stokes Equations (1)

∂tu+u· ∇u=1

ρ[−∇p+∇ · (2ηD)] −g,

∇ · u= 0,

(1)

where uis the velocity vector, ρ=φρgranular + (1 −φ)ρair is the density

derived from the packing fraction φand the mix of material and gas density

(ρgranular and ρair respectively), Dis the strain rate tensor given by D=

[∇u+ (∇u)T]/2, and gis the acceleration due to gravity. The relevant

parameters are given in Table 1.

These equations are combined with the µ(I)rheology, which assumes that

the granular friction coeﬃcient varies only on the inertial number I, which

is given by

I=|˙γ|d

pp/ρ,(2)

where ˙γij is the shear rate tensor given by

˙γij =∂ui

∂xj

+∂uj

∂xi

= 2Dij ,(3)

and |˙γ|is the second invariant of the shear rate tensor given by

|˙γ|=r˙γij ˙γij

2=p2Dij Dij ,(4)

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Capturingthedynamicsofatwooricesilowiththe(I)modelandextensionsSamuelKIrvinea,LukeAFullard,DanielJHollandb,DanielAClarkec,ThomasinALynch1a,Pierre-YvesLagréedaSchoolofMathematicalandComputationalSciences,MasseyUniversity,PalmerstonNorth,NewZealandbDepartmentofChemicalandProcessEngineering,Universit...

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