Modelling mineral slurreis using coupled discrete element method and smoothed particle hydrodynamics

Journal Pre-proof Modelling mineral slurreis using coupled discrete element method and smoothed particle hydrodynamics

Wei Chen, Damian Glowinski, Craig Wheeler PII:

S0032-5910(20)30115-7

DOI:

https://doi.org/10.1016/j.powtec.2020.02.011

Reference:

PTEC 15170

To appear in:

Powder Technology

Received date:

12 November 2019

Revised date:

19 January 2020

Accepted date:

4 February 2020

Please cite this article as: W. Chen, D. Glowinski and C. Wheeler, Modelling mineral slurreis using coupled discrete element method and smoothed particle hydrodynamics, Powder Technology(2020), https://doi.org/10.1016/j.powtec.2020.02.011

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© 2020 Published by Elsevier.

Journal Pre-proof MODELLING MINERAL SLURREIS USING COUPLED DISCRETE ELEMENT METHOD AND SMOOTHED PARTICLE HYDRODYNAMICS 1,3

2

Wei Chen , Damian Glowinski and Craig Wheeler

3

1

School of Energy Science and Power Engineering, Central South University, Changsha, Hunan, China

410083 2

Bradken Resources Pty Limited, Bassendean, Australia, 6054

3

School of Engineering, The University of Newcastle, Callaghan, Australia 2308

ABSTRACT: Slurry in wet grinding mills is critical for transporting fine progenies out of the system to downstream floatation process. It is commonly modelled as Newtonian fluids when simulating grinding mills with numerical tools. However, rheology of the slurry exhibits shear thinning non-Newtonian behaviours. This study aims to

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investigate the non-Newtonian characteristics of mineral slurries both experimentally and numerically. NonNewtonian smoothed particle hydrodynamics (SPH) and its coupling to discrete element modelling (DEM)

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framework was initially described. Non-Newtonian rheology of a copper slurry with various solids concentrations was determined experimentally by a rotary viscometer. SPH-DEM modelling of the viscometry

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test was conducted with both Newtonian and Power-law non-Newtonian settings in fluid phase, and comparisons were performed. Results suggested that the non-Newtonian SPH based method better reflects

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actual rheological behaviours of the slurry. In addition, DEM parameters exhibited limited impacts on rheology of the solids-liquid mixture, particularly at low solids concertation and high shear rate states. The findings

slurry flows within grinding mills.

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suggested that the non-Newtonian based SPH-DEM method should be used to more accurately model the

1.

INTRODUCTION

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KEYWORDS: Slurry, Non-Newtonian Fluid, SPH, SPH-DEM Coupling

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Mineral processing is the most important stage of copper and gold production, during which grinding mills are utilised to reduce size of the ore materials for further mineral enrichment processing. Grinding mills often account for 30% ~ 50% of the total production cost [1]. Current engineering design and optimisation of grinding mills have benefited significantly from numerical modelling tools, such as Discrete Element Modelling (DEM). The ability to predict and visualise the interactions of the rock, grinding media and mechanical geometries provides some extents of design validations. Additional contact mechanics, including rolling friction or particle shape modelling [2], cohesive/adhesive contact modelling [3], particle breakage modelling [4] have also been devised to more closely represent the dynamic process within a grinding mill. Mineral slurry is produced during grinding mill operation. It is well known that the slurry phase plays several important roles, such as the transportation of finer particles after particle breakage and in the modification of the coarse particle dynamics. This is particularly important for predicting flow through discharge grates and pebble ports and in the pulp chamber. Presence of the water in the grinding chamber also alters the overall rock and ball charge behaviour which leads to varied grinding performance (Figure 1) [5], [6]. Therefore, inclusion of the water phase is essential to accurate grinding mill modelling.

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Figure 1. The presence of slurry in a grinding mill. Smoothed particle hydrodynamic (SPH) [7]–[9] is often utilised to be coupled to DEM to model the overall grinding mechanism [10]–[12]. Comparing to mesh based computational fluid dynamics (CFD), its Lagrangian numerical method ensures convergences of the computation. Cleary [13]–[15] has extensively applied coupled DEM-SPH method to model the grinding mechanism. The development enhanced overall accuracy of the grinding mill modelling by accounting for interactive force between the fluid and particle phases. Nevertheless, rheological properties of the mineral slurry may not be accurately reflected in current DEM-SPH numerical framework. As shown in Figure 2, in SPH phase, viscosity term of the fluid particle is selected as a Newtonian characteristic (water viscosity). In DEM phase, fine particles suspended in the fluid phase cannot

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be properly modelled due to restricted computational resources. This essentially manifests into Newtonian rheological behaviours of the mineral slurry; whereas, slurry exhibits non-Newtonian rheological behaviours,

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such as Power-law, in the reality [16].

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Figure 2. (a) The methodology of the smoothed particle hydrodynamics; (b) Comparison between the

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Newtonian and non-Newtonian rheological responses in a shear diagram.

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Based on the forgoing comments, the purpose of this paper is to utilise a non-Newtonian fluid-based SPHDEM numerical framework for modelling mineral slurries. A suite of mineral slurries is selected to experimentally characterise the rheological characteristics using a rotary viscometer device. Numerical

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modelling of the viscometry tests is to be performed with both Newtonian fluid-based method and the modified Non-Newtonian fluid-based DEM-SPH framework, and results are to be compared. NUMERICAL METHOD

2.1

Discrete Element Modelling

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2.

The Hertz-Mindlin model (Cundall & Strack, 1979) is often used to compute the particle-particle and particlewall contacts. The contact force between two particles includes a normal force ( 𝑭𝑛 ) component and a tangential force (𝑭𝑡 ) component, 𝑭 = 𝑭𝑛 + 𝑭𝑡

(1)

𝑭 = (𝑘𝑛 𝛿𝒏𝑖𝑗 − 𝛾𝑛 𝒗𝒏𝑖𝑗 ) + (𝑘𝑡 𝛿𝒕𝑖𝑗 − 𝛾𝑡 𝒗𝒕𝑖𝑗 ) where 

𝑭𝑛 is the normal contact force,



𝑭𝑡 is the tangential contact force,



𝑘𝑛 is the elastic stiffness for normal contact,



δ𝐧𝑖𝑗 is the normal overlap,



𝛾𝑛 is the viscoelastic damping constant for normal contact,

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Journal Pre-proof 

𝒗𝒏𝑖𝑗 is the normal relative velocity (normal component of the relative velocity of the two particles),



𝑘𝑡 is the elastic stiffness for tangential contact,



𝛿𝒕𝑖𝑗 is the tangential overlap,



𝛾𝑡 is the viscoelastic damping constant for tangential contact,



𝒗𝒕𝑖𝑗 is the tangential relative velocity (tangential component of the relative velocity of the two particles).

Static friction is obtained by tracking the elastic shear displacement throughout the period of the contact. The tangential overlap 𝛿𝒕𝑖𝑗 is truncated when necessary to fulfil a local Coulomb yield criterion: 𝑭𝑡 ≤ 𝜇𝑭𝑛 , where 𝜇 is the particle-particle friction coefficient. Therefore, there is assumed to be no relative movements between contact surfaces when 𝑭𝑡 > 𝜇𝑭𝑛 , and when the Coulomb yield criterion is satisfied the contact surfaces slip

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relatively to each other. The normal force has two terms, a spring force and a damping force. The tangential force also has two terms:

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a shear force and a damping force. The shear force is a “history” effect that accounts for the tangential displacement (tangential overlap) between the particles for the duration of the time they are in contact.

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Considering that the shear modulus (G) can be calculated from Young’s modulus and Poisson’s ratio, the Hertz-Mindlin contact model depends on the following material parameters: Coefficient of restitution, e



Young’s modulus, Y



Poisson ratio, 𝜈



Coefficient of static friction, 𝜇𝑠



Coefficient of rolling friction, 𝜇𝑟

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

(2) (3) (4) (5)

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The contact model coefficients in Eq. (8) were calculated as follows, 4

𝑘𝑛 = 𝑌 ∗ √𝑅∗ 𝛿𝑛 3

5

𝛾𝑛 = −2√ 𝛽 √𝑆𝑛 𝑚∗ ≥ 0 6

𝑘𝑡 = 8𝐺 ∗ √𝑅∗ 𝛿𝑛 5

𝛾𝑡 = −2√ 𝛽 √𝑆𝑡 𝑚∗ ≥ 0 6

where (6)

𝑆𝑛 = 2𝑌 ∗ √𝑅∗ 𝛿𝑛

(7)

𝑆𝑡 = 8𝐺 ∗ √𝑅∗ 𝛿𝑛

(8)

𝛽=

(9) (10)

1 𝑌∗ 1 𝐺∗

= =

𝑙𝑛(𝑒) √𝑙𝑛2 (𝑒)+𝜋2 (1−𝜈1 2 ) 𝑌1

+

(1−𝜈2 2 ) 𝑌2

2(2+𝜈1 )(1−𝜈1 ) 𝑌1

+

2(2+𝜈2 )(1−𝜈2 ) 𝑌2

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Journal Pre-proof 1

(11)

𝑅∗ 1

(12)

1

=

𝑚∗

𝑅1

=

+

1 𝑚1

1 𝑅2

+

1 𝑚2

where e is the coefficient of restitution, m is the mass and R is the radius of a particle. The subscripts 1 and 2 denote to the two particles in contact. Additionally, when the rolling resistance is selected for the contacts, the elastic-plastic spring-dashpot (EPSD) model [18], [19] is utilised. This model adds an additional torque contribution in an incremental way by the following;

(13)

={

𝑘 𝑀𝑟,𝑡 − 𝑘𝑟 𝜔𝑟 ∆𝑡;

𝑘 if |𝑀𝑟,𝑡 − 𝑘𝑟 𝜔𝑟 ∆𝑡| < 𝜇𝑟 𝑅 ∗ |𝐹𝑛 |

𝑘 𝑀𝑟,𝑡 −𝑘𝑟 𝜔𝑟 ∆𝑡 𝑛 |𝑀𝑘 −𝑘 𝜔 ∆𝑡| 𝑟 𝑟 𝑟,𝑡

𝜇𝑟 𝑅 ∗ |𝐹 |

;

otherwise

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𝑘 𝑀𝑟,𝑡+∆𝑡

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where 𝑘𝑟 = 𝑘𝑡 𝑅∗ 2 is the rolling stiffness [20],



𝜔𝑟 is the relative angular velocity of the two particles in contact.

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

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The EPSD rolling friction model also has a viscous damping component [18]. However, it is neglected in this study since the rolling stiffness model selected above can already provide an ideal case for well damped

2.2

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rolling resistance behaviour without introducing a hard-to-define viscous damping coefficient [21]. Smoothed Particle Hydrodynamic

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Smoothed Particle Hydrodynamics (SPH) is a powerful particle-based method for modelling complex, splashing, free surface fluid flows. This method is particularly well suited for modelling industrial fluid flows, such as the fine slurry component of the multiphase flow. Once coupled to DEM, it may be applied to various

(14)

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industrial applications which involve solids and liquid flows. The continuity equation in SPH is defined as: 𝑑𝜌𝑎 𝑑𝑡

= ∑𝑏 𝑚𝑏 (𝑣𝑎 − 𝑣𝑏 ) ∙ 𝛻𝑊𝑎𝑏

where 𝜌𝑎 is the density of particle a with velocity 𝑣𝑎 and 𝑚𝑏 is the mass of particle b. The relative position vector from particle b to particle a by 𝑟𝑎𝑏 = 𝑟𝑎 − 𝑟𝑏 . The fluid relative velocity between particles is defined 𝑣𝑎𝑏 = 𝑣𝑎 − 𝑣𝑏 . A spatial size h is used as the smoothing kernel for all interpolation calculation. The smoothing kernel (𝑊𝑎𝑏 = 𝑊(𝑟𝑎𝑏 , ℎ)) evaluated for the distance |𝑟𝑎𝑏 | between particles a and b. Utilising this continuity equation can ensure numerical conservation and validity in free surface flow and density discontinuities applications. The acceleration for each particle a in the momentum equation of the SPH form is: (15)

𝑑𝑣𝑎 𝑑𝑡

= 𝑔 − ∑𝑏 𝑚𝑏 [(

𝑃𝑏 𝜌𝑏 2

+

𝑃𝑎 𝜌𝑎 2

)−

𝜉

4𝜇𝑎 𝜇𝑏

𝑣𝑎𝑏 𝑟𝑎𝑏

𝜌𝑎 𝜌𝑏 (𝜇𝑎 +𝜇𝑏 ) 𝑟𝑎𝑏 2 +𝜂 2

] ∙ 𝛻𝑎 𝑊𝑎𝑏

where 𝑃𝑎 is the pressure and 𝜇𝑎 is the viscosity at particle a. ξ is a calibration factor for the viscous term. g is the gravity vector. The parameter η is small and is added to regularize the singularity at 𝑟𝑎𝑏 = 0. In Newtonian

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Journal Pre-proof fluid based SPH, the viscosity 𝜇 of the particle is simply defined as the water viscosity. However, for nonNewtonian fluid, providing the mineral slurry exhibits Power-law rheological properties, the viscosity 𝜇𝑎 of particle a is defined as 𝜇𝑎,𝑏 = 𝐾 (

(16)

𝜕𝑣𝑎 𝑛−1 𝜕𝑦

)

where 

𝐾 is the flow consistency index [22];



𝜕𝑣𝑎



𝑛 is the dimensionless flow behaviour index [22].

𝜕𝑦

is the shear rate;

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Non-Newtonian SPH modelling development has been discussed in previously studies which include the Bingham model as well as the Herschel-Bulkley model [23]–[26]. To directly calculate the pressure from the

𝛾

𝑃 = 𝑃0 [( ) − 1]

(17)

𝜌0

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where 𝑃0 is the reference density. Here we use γ = 7.

𝛾𝑃0 𝜌0

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The pressure scale 𝑃0 is defined by: (18)

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𝜌

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density based on an equation of state, a quasi-compressible approach is used:

= 100𝑉 2 = 𝑐𝑠 2

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where 𝑉 is a characteristic fluid speed. Coupled to the equation of state can ensure that the density variations

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are <1% and that the flow is incompressible.

Solid surfaces in simulations are modelled using particles by enforcing a normal repulsive force. The equations of motion are utilised if the boundary objects are in motion, in which their rigid body motion is resolved based on the summation of the stress terms.

2.3

Coupling of DEM-SPH

Both resolved and unresolved methods can be used to compute the interactions between the solids phase and the liquid phase. For the resolved method, the full pressure-continuity equation solution is calculated around each solid particle. This requires the resolution of the fluid phase that is much finer than the particle size. It is a preferred method when it is allowed by the computational resources, in which the phase averaging process is eliminated [27]. In terms of the unresolved method, it is used when the solid particle size is small or in large population. The DEM phase is averaged to calculate the continuum solid fraction distribution. Then, it is used in a continuum multiphase formulation of the fluid equations [28], [29]. The one-way SPH-DEM coupling methodology is defined in [13]. The approach has been used to model industrial applications, such as slurry flow in mills. The full two-way SPH-DEM coupling methodology is given

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Journal Pre-proof in [15], which is used in applications where particulate solids exhibit significant impacts on the fluid flow. The drag force exerted on a particle in such a multi-particle fluid system with solid fraction corrections is: 1

𝐹𝐷 = 𝐶𝑑 𝜌|𝑢𝑟 |2 𝜀 −𝜒 𝐴⊥ 𝑢𝑟

(19)

2

where 1

𝜒 = 3.7 − 0.65𝑒𝑥𝑝 {− (1.5 − 𝑙𝑜𝑔𝑅𝑒)2 }

(20)

2

𝑢𝑟 is the local relative flow velocity between phases and ε is the local fluid fraction. The particle Reynolds number is 𝑅𝑒 =

2𝑟𝜌|𝑢𝑟| 𝜇

where 𝑟 is the equivalent spherical radius of the particle and μ is the fluid dynamic

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viscosity. 𝐶𝑑 is the drag coefficient which is based on correlations to experimental data. The drag is dependent on 𝐴⊥ is the projected area of the spherical particles. EXPERIMENTAL SCHEME

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3.

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The experimental apparatus used to conduct the viscometry test is shown schematically in Figure 3. The rotary viscometer used in this research was a coaxial cylinder type and the measurement principle was

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explicitly discussed in [30]. The measurement system is comprised of a cup and a bob. The geometry of the systems follows the strictly to the testing standard [31]. The measuring bob inside the specimen cup is driven

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by an electromotor through a gear. The motor’s rotating speed is strictly proportional to a frequency which is produced by the control unit. The frequency can be pre-set by a 30-step knob. The 30 rotating speed steps are subdivided into geometrical progression within a range of 0.0478–350 rpm. Potential viscosity measuring 7

0

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range for such a system is from 0.001 to 1.7×10 Pa s with testing temperature range in−20–50 C. Both the motor and the gear are bolted to the casing. During operation, the rotation is transmitted to the measuring

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system though a cardan chuck, thus preventing the horizontal forces from affecting the driving mechanism. This also eliminates the influence caused by the mechanical friction of the instrument.

Figure 3. Experimental setup of the rotary viscometer test to determine the fluid rheology.

A copper slurry sample collected from commercial operation in Australia was selected in this study. The particle size distribution and material properties of the specimen are shown in Figure 4. Four solids concentrations (mass based) were selected to determine the rheological properties, 30%, 50%, 65% and 80%. The 80% solids concentration slurry closely represents the operational conditions during production. 0

At the beginning of a test, the ambient temperature is maintained at around 20 C. Once the temperature is stabilised, the slurry sample is continuously poured into the cup until the bob is fully submerged. The rotary viscometer is then switched on and the rotational speed was continuously incremented from step 1 to 30. The corresponding torque reading from the gauge at each rotational step was then logged into the computer. Each test is repeated three times and the averaged results were reported.

Figure 4. Particle size distribution and particle density of the copper slurry.

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Journal Pre-proof 4.

NUMERCIAL MODELLING PROGRAM

To perform the coupled SPH-DEM modelling of the viscometry test, the aforementioned Newtonian and nonNewtonian fluid based numerical framework were both used. The code used in the modelling suite is LAMMPS [32], [33]. The computational procedure of the coupling framework is shown in Figure 5. At the beginning of the computation, a number of DEM calculations is initially performed, after which the coordinates and the momentum of the particles are updated and stored. The DEM phase data is then transferred to SPH computation, and the fluid-fluid particle interaction and fluid-solids particles drag force are calculated. Results are subsequently transferred back to DEM phase to form a close loop. The actual number of calculations

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steps before transferring the drag force to solids particles depend on the numerical settings of each phase.

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Figure 5. Methodology of the one-way SPH-DEM coupling numerical modelling framework. In terms of the numerical modelling program, similar to the experimental scheme, four different solids

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concentrations are modelled under both the Newtonian and non-Newtonian SPH modes, which leads to a total of 8 modelling cases. In each case, the slurry material of a pre-defined solids concentration is initialled loaded into the cup to a fixed volume which enables the bob to be fully immersed. Then, rotational speed of the bob is

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slowly ramped up from zero to pre-defined levels. 8 rotational speeds from 3.5 rpm to 350 rpm were selected in the numerical program. The stress response on surface of the bob is collected under each testing condition.

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During the rotation, once the shear stress response measured from the bob reached a relatively steady state, the shear stress was recorded and rotation of the bob was then stopped.

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It is critical to select appropriate fluid particle and solids particle size and distribution in the modelling. The actual particle size and distribution of the slurry sample is too small to achieve realistic computational

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turnaround time. Thus, scale up of the particles is required in the numerical modelling. However, the followings should be considered during the scale up process: 

Gap between bob and cup is at least five folds of the maximum diameter of the fluid particle/solids particle

To fulfil the requirements above as well as achieving realistic computational time, a mono-sized fluid and solids particle size of 400 μm were selected in the modelling. Solids and fluid particles were inserted into the viscometer device to a fixed volume. The corresponding total particle count in each solids concentration is: 

30% solids concentration: 396,329 solids particles; 2,589,339 fluid particles.



50% solids concentration: 785,702 solids particles; 2,199,966 fluid particles.



65% solids concentration: 1,190,605 solids particles; 1,795,064 fluid particles.



80% solids concentration: 1,756,276 solids particles; 1,229,393 fluid particles.

This essentially led to an unresolved method utilised in all modelling cases. Detailed parameters used in SPH and DEM phases are discussed below.

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RESULT AND DISCUSSION

Following the experimental and numerical programs above, the rheology of the slurry samples is obtained and shown below. Comparisons between the two different SPH-DEM modelling modes are made. 5.1

Experimental Slurry Rheology

Following the experimental procedure outlined above, shear stress-shear rate correlations of the slurry samples are obtained and shown in Figure 6 (a) and (b). It was indicated that all slurry samples exhibited shear thinning rheological characteristics. Therefore, viscosity of the tested samples is continuously decreasing along with an increasing shear rate. The shear thinning behaviours may be approximated as Power-law correlations. At higher solids concentrations, the power-law rheological effect was observed to be

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more evident. Power-law rheology function fitting results for each solids concentration are shown in Table 1. Correlations of

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the flow consistency index 𝐾 and the flow behaviour index 𝑛 to the solids concentration of the slurry samples were investigated and shown in Figure 7. A pseudo-linear relationship between the solids concentration and

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the flow consistency index as well as the flow behaviour index was indicated. From this result, the rheology of a slurry sample at an untested solids concentration may be approximated based on the empirical correlations

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formulated.

Figure 6. Rheology testing results of the selected copper slurry;(a) Shear diagram; (b) Viscosity

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diagram.

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Table 1. Flow consistency and flow behaviour index results from the rheological experiments.

Figure 7. Correlations between the Power-law rheology model parameters and the solids concentration of the copper slurry. 5.2

Non-Newtonian SPH-DEM Rheology

SPH-DEM modelling of the viscometry testing was conducted with parameters shown below in Table 2. Both Newtonian and non-Newtonian settings of the numerical modelling were performed. Figure 8 shows flow of the solids particles during viscometry testing. It was indicated from the figure that solids particles generally suspended uniformly within the slurry during rotation. Liquid-solids mixture between cup and rotating bob exhibited highest velocity during the test. Table 2. Numerical modelling parameters used in the simulations.

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Journal Pre-proof Figure 8. Non-Newtonian SPH-DEM modelling of the viscometry testing at 30% solids concentration under a rotational speed of 75.4 rpm; (a)&(c) water and solids particles distribution; (b) velocity of the slurry during rotation. Results are shown in Figure 9 for the Newtonian setting and Figure 10 for the non-Newtonian setting for the SPH phase. It was observed that the rheology of the solids-liquid mixture exhibited Newtonian characteristics in Figure 9. Viscosity of the modelled slurry was close to water viscosity across all solids concentrations. In contrary to the results from Newtonian SPH-DEM modelling, shear thinning behaviours was indicated in the non-Newtonian SPH-DEM modelling as shown in Figure 10. Analogous Power-law correlations may also be formulated across all solids concentrations for the numerical modelling results in such cases. Nevertheless,

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the shear stress obtained at a specific shear rate was observed to be larger than experimental results, in which the DEM particles suspended in the SPH phase may play a role in elevating the shear stress. Such a

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phenomenon is discussed below.

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Figure 9. Newtonian SPH-DEM modelling results of the viscometry testing at all solids concentrations.

Figure 10. Non-Newtonian SPH-DEM modelling results of the viscometry testing at all solids

5.3

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concentrations.

Effect of DEM Parameters on Slurry Rheology

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Effect of the contact parameters in the DEM phase on the rheology of the slurry was also investigated. This was achieved by varying the particle-particle frictional coefficient (µs) and the particle rolling frictional coefficient (µr); however, maintaining the SPH setting of the non-Newtonian fluid. Both µs the µr are increased from 0.1 to 0.9 at a step of 0.1, and corresponding rheology results of the slurry were compared. Minimum and maximum shear stresses at a specific shear rate were recorded when varying DEM contact parameters. Results for each solids concentration are shown in Figure 11. It was indicated that increasing the DEM contact parameters resulted in elevation of the shear stress, and subsequently the viscosity. Such a phenomenon was more evident at relatively higher solids concentrations. In addition, considering the increase of shear stress due to DEM contact parameters in a specific solids concentration, the elevation of the shear stress is more evident at lower shear rates. This is due to that comparatively more contacts between the solids particles and the measurement bob occurred during modelling at lower shear rates. When increasing the shear rates, the centrifugal force induced particle segregation, which effectively reduced the contacts between the DEM particles and the measurement bob. The distribution of the solids particles suspended in the SPH phase for the 80% solids concentration under various shear rates was shown in Figure 12. It was indicated that the less particles were observed in the

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Journal Pre-proof annulus between the measurement cup and bob at highest shear rate, suggesting less impacts of the DEM particles on the rheology of the mixture.

Figure 11. Effect of the DEM parameters on the rheology results in non-Newtonian SPH-DEM modelling at all solids concentrations.

Figure 12. Distribution of solids particles suspended in the SPH phase at increasing shear rates under the 80% solids concentration. CONLUSION

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6.

Non-Newtonian smoothed particle hydrodynamics and its coupling to the discrete element modelling was

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studied in this paper. A numerical framework of the SPH-DEM coupling was initially described. A suite of experimental study on the non-Newtonian slurry samples was conducted using a rotary viscometer. Numerical

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modelling of the viscometry testing was perform to compare ] rheological performance from experiments and



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simulation. Results yielded the following major conclusions:

Selected copper slurry sample exhibited shear thinning behaviours which can be described using a

solids concentration. 

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Power-law correlation. The shear shinning characteristics became more evident when increasing the

The rheology of the solid-liquid mixture predominantly depends on the rheology settings of the liquid



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phase in the DEM-SPH modelling.

DEM parameters exhibited more significant impacts on the rheology at lower shear rates and/or

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higher solids concentration. At higher shear rates, segregation behaviour of the DEM particles was observed due to the relatively high centrifugal force. Consequently, it is suggested that non-Newtonian based rheology setting should be used when modelling slurries in industrial flows to accurately represent its shearing characteristics. REFERENCE [1]

G. R. Ballantyne and M. S. Powell, “Benchmarking comminution energy consumption for the processing of copper and gold ores,” Miner. Eng., vol. 65, pp. 109–114, 2014.

[2]

W. Chen, T. Donohue, A. Katterfeld, and K. Williams, “Comparative discrete element modelling of a vibratory sieving process with spherical and rounded polyhedron particles,” Granul. Matter, vol. 19, no. 4, 2017.

[3]

W. Chen, T. Donohue, K. C. Williams, A. Katterfeld, and T. Roessler, “Modelling Cohesion and Adhesion of Wet, Sticky Iron Ore in Discrete Element Modeling for Material Handling Process,” in Iron Ore 2015 Conference Proceedings, 2015.

[4]

P. W. Cleary and M. D. Sinnott, “Simulation of particle flows and breakage in crushers using DEM: Part

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Journal Pre-proof 1–Compression crushers,” Miner. Eng., vol. 74, pp. 178–197, 2015. [5]

M. D. Sinnott, P. W. Cleary, and R. D. Morrison, “Combined DEM and SPH simulation of overflow ball mill discharge and trommel flow,” Miner. Eng., vol. 108, pp. 93–108, 2017.

[6]

P. W. Cleary and R. D. Morrison, “Prediction of 3D slurry flow within the grinding chamber and discharge from a pilot scale SAG mill,” Miner. Eng., vol. 39, pp. 184–195, 2012.

[7]

J. J. Monaghan, “Smoothed particle hydrodynamics,” Annu. Rev. Astron. Astrophys., vol. 30, no. 1, pp. 543–574, 1992.

[8]

R. A. Gingold and J. J. Monaghan, “Smoothed particle hydrodynamics: theory and application to nonspherical stars,” Mon. Not. R. Astron. Soc., vol. 181, no. 3, pp. 375–389, 1977. J. J. Monaghan, “Smoothed particle hydrodynamics,” Reports Prog. Phys., vol. 68, no. 8, p. 1703,

of

[9]

[10]

ro

2005.

P. W. Cleary, M. D. Sinnott, and G. G. Pereira, “Computational prediction of performance for a full

-p

scale Isamill: Part 1–Media motion and energy utilisation in a dry mill,” Miner. Eng., vol. 79, pp. 220– 238, 2015.

P. W. Cleary and M. D. Sinnott, “Computational prediction of performance for a full scale Isamill: Part

re

[11]

[12]

lP

2–Wet models of charge and slurry transport,” Miner. Eng., vol. 79, pp. 239–260, 2015. P. W. Cleary and R. D. Morrison, “Predicting charge motion, slurry flow and rock breakage within and

[13]

na

discharge from a 3D pilot SAG mill using a coupled DEM-SPH model,” 2014. P. W. Cleary, M. Sinnott, and R. Morrison, “Prediction of slurry transport in SAG mills using SPH fluid

[14]

Jo ur

flow in a dynamic DEM based porous media,” Miner. Eng., vol. 19, no. 15, pp. 1517–1527, 2006. P. W. Cleary, M. Prakash, J. Ha, N. Stokes, and C. Scott, “Smooth particle hydrodynamics: status and future potential,” Prog. Comput. Fluid Dyn. an Int. J., vol. 7, no. 2–4, pp. 70–90, 2007. [15]

P. W. Cleary, “Prediction of coupled particle and fluid flows using DEM and SPH,” Miner. Eng., vol. 73, pp. 85–99, 2015.

[16]

W. Chen, K. C. Williams, T. F. Bunn, C. Liang, and M. G. Jones, “Measurement and modelling of the apparent shear resistance for aerated fine powders,” Granul. Matter, vol. 17, no. 5, 2015.

[17]

P. A. Cundall and O. D. L. Strack, “A discrete numerical model for granular assemblies,” Geotechnique, vol. 29, no. 1, pp. 47–65, 1979.

[18]

J. Ai, J.-F. Chen, J. M. Rotter, and J. Y. Ooi, “Assessment of rolling resistance models in discrete element simulations,” Powder Technol., vol. 206, no. 3, pp. 269–282, 2011.

[19]

W. Chen, A. Roberts, A. Katterfeld, and C. Wheeler, “Modelling the stability of iron ore bulk cargoes during marine transport,” Powder Technol., vol. 326, 2018.

[20]

K. Iwashita and M. Oda, “Rolling resistance at contacts in simulation of shear band development by

11

Journal Pre-proof DEM,” J. Eng. Mech., vol. 124, no. 3, pp. 285–292, 1998. [21]

C. M. Wensrich and A. Katterfeld, “Rolling friction as a technique for modelling particle shape in DEM,” Powder Technol., vol. 217, pp. 409–417, 2012.

[22]

W. H. Herschel and R. Bulkley, “Measurement of consistency as applied to rubber-benzene solutions,” in Proceedings of the American Society for the Testing of Materials, 1926, vol. 26, pp. 621–633.

[23]

N. S. Martys, W. L. George, B. W. Chun, and D. Lootens, “A smoothed particle hydrodynamics-based fluid model with a spatially dependent viscosity: Application to flow of a suspension with a nonNewtonian fluid matrix,” Rheol. Acta, vol. 49, no. 10, pp. 1059–1069, 2010.

[24]

S. Shao and E. Y. M. Lo, “Incompressible SPH method for simulating Newtonian and non-Newtonian

A. Peer, M. Ihmsen, J. Cornelis, and M. Teschner, “An implicit viscosity formulation for SPH fluids,” ACM Trans. Graph., vol. 34, no. 4, p. 114, 2015.

S. M. Hosseini, M. T. Manzari, and S. K. Hannani, “A fully explicit three-step SPH algorithm for

-p

[26]

ro

[25]

of

flows with a free surface,” Adv. Water Resour., vol. 26, no. 7, pp. 787–800, 2003.

simulation of non-Newtonian fluid flow,” Int. J. Numer. Methods Heat Fluid Flow, vol. 17, no. 7, pp.

[27]

re

715–735, 2007.

W. Chen, K. Williams, and M. G. Jones, “Applications of Numerical Modelling in Pneumatic Conveying,”

[28]

lP

in Pneumatic Conveying Design Guide, Oxford: Elsevier, 2016, pp. 521–552. Y. Tsuji, T. Tanaka, and T. Ishida, “Lagrangian numerical simulation of plug flow of cohesionless

[29]

na

particles in a horizontal pipe,” Powder Technol., vol. 71, no. 3, pp. 239–250, 1992. Y. Tsuji, T. Kawaguchi, and T. Tanaka, “Discrete particle simulation of two-dimensional fluidized bed,”

[30]

Jo ur

Powder Technol., vol. 77, no. 1, pp. 79–87, 1993. M. C. Wendl, “General solution for the Couette flow profile,” Phys. Rev. E, vol. 60, no. 5, pp. 6192– 6194, Nov. 1999. [31]

A. D2196, “Standard Test Methods for Rheological Properties of Non-Newtonian Materials by Rotational Viscometer.” ASTM International, West Conshohocken, PA USA, 2015.

[32]

S. Plimpton, “Fast parallel algorithms for short-range molecular dynamics,” J. Comput. Phys., vol. 117, no. 1, pp. 1–19, 1995.

[33]

C. Kloss and C. Goniva, “Liggghts–a new open source discrete Element simulation software,” in Proc. of The 5th International Conference on Discrete Element Methods, 2010, pp. 25–26.

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Journal Pre-proof Table 3. Flow consistency and flow behaviour index results from the rheological experiments. 65% Solids Concentration

80% Solids Concentration

1.49

6.22

10.87

11.87

0.132

0.142

0.244

0.298

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50% Solids Concentration

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Flow Consistency Index Flow Behaviour Index

30% Solids Concentration

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Parameters

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Journal Pre-proof Table 4. Flow consistency and flow behaviour index results from the rheological experiments. Particle Density – kg/m

3

2800

Interparticle Friction Coefficient

0.5

Wall Friction Coefficient

0.3

DEM

Restitution Coefficient

0.3

Settings

Poisson’s Ratio

0.3

Young’s Modulus – Pa

1 × 10

Particle Size and Distribution:

400 μm

Solids Concentration (mass based)

30%, 50%, 65% and 80%

Newtonian

Fluid Particle Size – mm

400 μm

SPH

Fluid Particle Viscosity (Water Viscosity) @25°C – Pa·s

8.9 × 10

Settings

Point Density – kg/m

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7

3

Flow Consistency Index - 𝐾



Flow Behaviour Index - 𝑛

Fluid Particle Density – kg/m Time step (DEM) - s

Control

Time step (SPH) - s

-4

1000 10 mm

As per Table 3

1000 ‒7

1 × 10

‒6

1 × 10

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Simulation

3

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Settings

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SPH

Fluid Particle Viscosity:

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Newtonian

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Point Size – mm Non-

100%

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Journal Pre-proof Non-Newtonian SPH-DEM coupling method incorporates the fluid rheology model



Mineral slurries exhibit the shear thinning non-Newtonian behaviour



Newtonian SPH-DEM incorrectly models the slurry rheology



DEM parameters may elevate the slurry rheology in SPH-DEM modelling

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Journal Pre-proof Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Figure 10

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Figure 12