Chuan-Yu (Charley) Wu is a senior lecturer at the School of Chemical Engineering at University of Birmingham and has research interests in understanding and modelling the behaviour of particulate materials during the manufacturing processes for pharmaceutical and other particulate products at microscopic and macroscopic levels. In particular, he is interested in developing models for predicting the properties of products based on the properties of particles and individual constituents.

Discrete Element Modelling of Particulate Media

By Chuan-Yu Wu

The Royal Society of Chemistry

Copyright © 2012 The Royal Society of Chemistry
All rights reserved.
ISBN: 978-1-84973-360-1

Contents

Two-Phase Systems,
FROM SINGLE PARTICLE DRAG FORCE TO SEGREGATION IN FLUIDISED BEDS A. Di Renzo and F. P. Di Maio, 3,
ENHANCING THE CAPACITY OF DEM/CFD WITH AN IMMERSED BOUNDARY METHOD C.-Y. Wu and Y. Guo, 10,
EFFECT OF SOLID AND LIQUID HEAT CONDUCTIVITIES ON TWO-PHASE HEAT AND FLUID FLOWS T. Tsutsumi, S. Takeuchi and T. Kajishima, 21,
GRAVITATIONAL SEDIMENTATION AND SEPARATION OF PARTICLES IN A LIQUID: A 3D DEM/CFD STUDY L. Qiu and C.-Y. Wu, 30,
DEM SIMULATION OF MIGRATION PHENOMENA IN SLOW, DENSE SLURRY FLOW WITH BROWNIAN MOTION EFFECTS M.A. Koenders, M. Ibrahim and S. Vahid, 39,
FORCE EVALUATION FOR BINGHAM FLUIDS USING MULTIPLE-RELAXATION-TIME LATTICE BOLTZMANN MODEL S. Chen, Q. Sun and F. Jin, 46,
THE EFFECT OF INITIAL BED HEIGHT ON THE BEHAVIOUR OF A SOIL BED DUE TO PIPE LEAKAGE USING THE COUPLED DEM-LBM TECHNIQUE X. Cui, J. Li, A.H.C. Chan and D. Chapman, 51,
GRANULAR FLOWS IN FLUID K. Kumar, K. Soga and J.-Y. Delenne, 59,
Cohesive Systems,
A STUDY OF THE INFLUENCE OF SURF ACE ENERGY ON THE MECHANICAL PROPERTIES OF LUNAR SOIL USING DEM C. Modenese, S. Utili and G.T. Houlsby, 69,
MODELLING OF THE CONT ACT BEHAVIOUR BETWEEN FINE ADHESIVE PARTICLES WITH VISCOUS DAMPING K. Mader and J. Tomas, 76,
REBOUND OF A PARTICLE FROM A SOLID SURFACE WITH A VISCOUS OR NONLINEAR VISCOELASTIC LIQUID FILM IN THE CONTACT ZONE J. Bowen, D. Cheneler, J.W. Andrews, C-Y. Wu, M.C.L. Ward and M.J. Adams, 86,
EFFECT OF THE PENDULAR STATE ON THE COLLAPSE OF GRANULAR COLUMNS R. Artoni, F. Gabrieli, A. Santomaso and S. Cola, 95,
INVESTIGATION OF DYNAMIC BEHAVIOUR OF A PARTICLE-LOADED SINGLE FIBRE USING DISCRETE ELEMENT METHODS M. Yang, S.Q. Li, G. Liu and J. S. Marshall, 103,
MODELLING OF THE FILTRATION BEHAVIOUR USING COUPLED DEM AND CFD S. Stein and J. Tomas, 113,
Granular Flows,
DEM MODELLING OF SUBSIDENCE OF A SOLID PARTICLE IN GRANULAR MEDIA C.H. Goey, C. Pei and C.-Y. Wu, 123,
NUMERICAL SIMULATION Of THE COLLAPSE OF GRANULAR COLUMNS USING DEM T. Zhao, G.T. Houlsby and S. Utili, 133,
DEM MODELLING OF THE DIGGING PROCESS OF GRAVEL: INFLUENCE OF PARTICLE ROUNDNESS S. Miyai, T. Katsuo, T. Tsuji, T. Takayama and T. Tanaka, 141,
DEM MODELLING OF HIGH SPEED DIE FILLING PROCESSES C.-Y. Wu, F. Ogbuagu and C. Pei, 149,
DEM ANALYSIS OF LOADS ON DISC INSERTS IMMERSED IN GRAIN DURING SILO FILLING AND DISCHARGE R. Kobylka and M. Molenda, 158,
THREE DIMENSIONAL DEM/CFD ANALYSIS OF SEGREGATION DURING SILO FILLING WITH BINARY MIXTURES OF DIFFERENT PARTICLE SIZES C.-Y. Wu and Y. Guo, 165,
MODELING PACKING OF SPHERICAL FUEL ELEMENTS IN PEBBLE BED REACTORS USING DEM H. Suikkanen, J. Ritvanen, P. Jalali and R. Kyrki-Rajamaki, 175,
Quasi-Static Deformation,
A NUMERICAL INVESTIGATION OF QUASI-STATIC CONDITIONS FOR GRANULAR MEDIA C. Modenese, S. Utili and G.T. Houlsby, 187,
EXPLORING THE CONTROLLING PARAMETERS AFFECTING SPECIMENS GENERA TED IN A PLUVIATOR USING DEM L. Cui, 196,
DEM TRIAXIAL TESTS OF A SEABED SAND G. Macaro and S. Utili, 203,
THE STEADY STATE SOLUTION OF GRANULAR SOLID HYDRODYNAMICS FOR TRIAXIAL COMPRESSIONS S. Song, Q. Sun and F. Jin, 212,
3D DEM SIMULATIONS OF UNDRAINED TRIAXIAL BEHAVIOUR WITH PRESHEARING HISTORY G. Gong and A.H.C. Chan, 219,
STRONG FORCE NETWORK OF GRANULAR MIXTURES UNDER ONE-DIMENSIONAL COMPRESSION N.H. Minh and Y.P. Cheng, 227,
VERIFICATION OF THE DOUBLE SLIP AND ROTATION RATE MODEL FOR ELLIPTICAL GRANULAR FLOW USING THE DISTINCT ELEMENT METHOD L.Q. Li, M.J. Jiang and Z.F. Shen, 236,
MICROMECHANICS OF SEISMIC WAVE PROPAGATION IN GRANULAR MATERIALS J. O’Donovan, C. O’Sullivan and G. Marketos, 245,
MICROMECHANICAL STUDY ON SHEAR WAVE VELOCITY OF GRANULAR MATERIALS USING DISCRETE ELEMENT METHODS X. Xu, D. Ling, Y. P. Cheng and Y. Chen, 255,
MECHANICAL BEHAVIOUR OF METHANE HYDRA TE SOIL SEDIMENTS USING DISCRETE ELEMENT METHOD: PORE-FILLING HYDRATE DISTRIBUTION Y. Yu, Y. P. Cheng and K. Soga, 264,
ON THE EFFECT OF SOIL MODIFICATION WITH LIME USING GRADING ENTROPY E. Imre, J. Szendefy, J. Lorincz, P.Q. Trang and Vijay P. Singh, 271,
SUBJECT INDEX, 280,

CHAPTER 1

FROM SINGLE PARTICLE DRAG FORCE TO SEGREGATION IN FLUIDISED BEDS

A. Di Renzo and F. P. Di Maio

Dipartimento di Ingegneria Chimica e dei Materiali, Università della Calabria Via P. Bucci, Cubo 44A, 1-87036 Rende (CS), Italy

1 INTRODUCTION

In numerical simulations of dense two-phase flow involving particulate materials the Discrete Element Method (DEM) has proved particularly effective in capturing the complex hydrodynamics of the solid phase. DEM-based granular solid dynamics, including collisions and persistent contact with elaborate force-displacement laws, friction and cohesion have shown to be superior to traditional fluid-like, continuum approaches, which typically require coarse approximations and the introduction of artificial variables like solids pressure and viscosity. However, computational limitations of DEM models do not allow adding also the burden of flow simulations resolved at the level of particle-particle interstices, so that typically an averaged scale approach, with computational cell sizes of the order of a few particle diameters, is used. As a consequence, formulations of the drag force acting on individual particles are required to close the set of equations to solve for the solid and fluid phases. While many drag force models for monodisperse systems have been proposed in the literature, as discussed below, expressions for such force on a particle in a multi-particle system is currently the subject of extensive research work.

2 DRAG FORCE AND CLOSURE IN DEM-CFD MODELS

2.1 Momentum exchange and two-way coupling

Characterisation of the relative motion between a fluid and dense particle system by a DEM-CFD approach requires the solution of the averaged equations of motion of the fluid phase and the classical Newton’s second law of dynamics for each particle, where the drag force appears explicitly. The fluid flow field is obtained from the solution of the discretised locally averaged continuity and Navier-Stokes equations, which in differential terms are expressed, respectively, as:

[MATHEMATICAL EXPRESSION OMITTED] (1)

[MATHEMATICAL EXPRESSION OMITTED] (2)

where ρf U and P are the fluid density, fluid velocity and pressure, respectively, ε is the volumetric fraction of the fluid (or voidage), τ is the deviatoric stress tensor, S is the fluid-particle inter-phase momentum exchange density and g the acceleration of gravity.

The corresponding equations for each particle of the solid phase follow the conventional DEM approach, i.e.:

[MATHEMATICAL EXPRESSION OMITTED] (3)

[MATHEMATICAL EXPRESSION OMITTED] (4)

where m, V, I, a and a are the particle mass, volume, moment of inertia, linear and angular acceleration, respectively. The forces considered are gravity, contact forces fc, pressure gradient and drag force fd, in the order of appearance in Equation (3). Note that the last two terms arise from the interaction with the fluid. In the rotational direction only torques arising from contact forces are considered.

Interphase coupling is achieved by connecting the momentum exchange density source term S in Equation (2) with the drag force acting on individual particles, i.e.:

[MATHEMATICAL EXPRESSION OMITTED] (5)

where the wj coefficient plays the role of distance weighting function per unit volume.

2.2 Drag force

Expressions accounting for the influence of velocity and voidage on the drag force exerted on individual particles have been often derived based on established correlations for the pressure drop across fixed beds of a single material of diameter D. In general terms, the modulus of the dissipative pressure gradient is related to the modulus of the drag force by:

[nabla]p = 1 – ε/ε 6/π fd/D3 (6)

Extensive studies in the literature led to a number of common, relatively accurate expressions valid for monodisperse suspensions that cover many orders of magnitude of the Reynolds number and from dense packing to highly dilute systems, like the combination of Ergun and Wen and Yu or Di Felice’s formula.

In the case of disperse systems or when multiple particulate solids are present simultaneously, the drag force acting on a particle becomes much more difficult to evaluate. This is also related to the fact that experimental accessibility to such datum is very limited. The first theoretical advancements indeed appeared as a result of fully resolved simulations of fluid flow through static arrays of spheres. In particular, van der Hoef et al., based on lattice-Boltzmann simulations of the flow through random arrays of spherical particles, were able to propose the first theoretical approach in the characterisation of the phenomenon. The most significant result of their work, later used also in other papers, is the formulation of the drag force acting on a generic particle as proportional to the average drag force in the system, and to express the coefficient as a function of a poly-dispersion index and bed voidage, as detailed in the next Section.

3 DRAG IN MULTIPARTICLE TWO-PHASE FLOW

In analogy with the relationship between individual drag force and pressure gradient across the bed, van der Hoef et al. proposed the following starting point:

[MATHEMATICAL EXPRESSION OMITTED] (7)

where xi is the volumetric fraction of species i in the multi-particle mixture. To keep the derivation simple, only two solids will be considered and, without loss of generality, species 1 will be assumed to be the smaller one. The two key steps are (i) the definition of an average drag force [bar.fd] and average diameter [bar.D] for the system, related to the overall pressure drop by:

[nabla]p = 1 – ε/ε 6/π [bar.fd]/[bar.D]3 (8)

and (ii) the definition of the drag force on an individual species as proportional to average drag force in the system, i.e.:

fdi = αi[bar.fd] (9)

Then, by introducing the average diameter, as defined by Sauter’s mean, and a polydispersion index given, respectively, by:

[bar.D] = (x1/D1 + x2/D2)-1 (10)

yi = Di/D (11)

van der Hoef et al. proposed the proportionality coefficient αi to be derived based on considerations in the viscous flow regime, giving the individual drag force by:

fdi = y2i[bar.fd] (12)

It is the case to mention that the derivation presented here is formally different, though conceptually equal, to the original treatment in two aspects. Firstly, a dimensional notation is used here. Secondly, all previous considerations involve the drag force intended as the net of the pressure gradient interaction term. Additional details can be found in papers elaborating further on the presented approach, e.g. by Cello et al.

4 A MODEL FOR SEGREGATION IN FLUIDISED BEDS

The capability to compute the drag force at the individual particle scale is particularly useful in dealing with fluidised beds of binary mixtures of solids and the related segregation problems. An initially mixed binary bed upon fluidisation shows a tendency for the component with the smaller size and lower density to accumulate to the bed surface, acting as flotsam, the reverse occurring for the other component, the jetsam. The matter becomes problematic when smaller and denser particles are mixed with larger but less dense particle, a case in which it is difficult even to attribute the roles of flotsam and jetsam to the mixture components.

Despite the severe consequences in process performances, the complexity of the two-phase flow and the substantial previous inability to set the force balance over individual solids species has prevented a theoretical treatment of segregation problems. We have recently proposed to include the result of Equation (12) into a force balance on a particle of, by convention, species 2 immersed in a homogeneous mixture of the two solids.

Under the hypothesis of viscous flow regime, the drag force exerted by the fluid over a particle of diameter D is:

fd = 30 1 – ε/ε2 πμuD (13)

and the corresponding minimum fluidisation velocity is:

umf = gε3/180(1 – ε)μρD2 (14)

Equations (13) and (14) are intended here applicable to both particle species, provided the appropriate diameter and density are used, as well as the mixture, for which the average size is defined by Equation (10) and the average density is:

ρ = ρ1×1+ρ2(1-x1) (15)

It is useful to recall that the pressure gradient developed as a result of fluidisation of a binary bed is:

[nabla]p = [bar.ρ](1 – ε)g (16)

The force balance on a particle of species 2 can be set, by comparing the ratio of the hydrodynamic action of the fluid on the particle weight to the value of one, to establish whether it will be pushed upwards or downwards, i.e. it will act as flotsam or jetsam. In formula, this reads:

[MATHEMATICAL EXPRESSION OMITTED] (17)

where the numerator results from the sum of the pressure gradient term (neglecting Archimedean buoyancy) and the drag force evaluated at the minimum fluidisation velocity of the mixture and the denominator is the particle weight. Introducing the dimensionless ratio of the particle-to-average density [bar.s] = ρ2√ρ and of the average-to-particle diameter [bar.d] = [bar.D] / D2, Equation (17) can be rewritten as:

[bar.s] = 1 – ε + ε[bar.d] (18)

Alternatively, using the corresponding version based on the particle-to-particle density and diameter ratios s = ρ2 / ρ1 and d = D1 / D2, we have:

[MATHEMATICAL EXPRESSION OMITTED] (19)

The attention will be focussed on binary mixtures composed of smaller, denser and larger, less dense materials so that values of s and d lie in the range 0 to 1. The possibility for either solid to become the flotsam component determines the occurrence of two possible segregation directions of an initially mixed bed. Equilibrium lines that allow discriminating between the two directions can be prescribed by Equation (19) and are shown in Figure 1 at various bed compositions. The chart with the discriminating lines can be used to predict the tendency for a given solids pair at a given composition to segregate one way or the other. This is achieved by computing s and d to locate the point on the chart and compare its position with the equilibrium line at the corresponding composition.

Similar considerations apply to the comparison of data of a given system in terms of [bar.s] and [bar.d] with the unique equilibrium line prescribed by Equation (18). Comparison of the predicted segregation direction with experimental data available in the literature is shown in Figure 2. Details of the examined systems and further comments are reported elsewhere. However, it is evident that agreement is found for the great majority of the data, without adjustable parameters in the model, confirming the soundness of the approach and, particularly, the realistic predictions of the drag force of Equation (12). Discrete Element simulations are then expected to benefit from the adoption of Equation (12) in modelling fluid-particle flows involving multi-particle mixtures.

5 CONCLUSION

Simulations based on Discrete Element Method and averaged CFD approaches require a model for the drag force. In systems involving multi-particle mixtures such drag force is shown to require specific treatment and a recently proposed model is discussed. The same is shown to allow establishing a force balance at the particle scale that proves very useful in addressing problems related to segregation in fluidised beds. Quantitative model validation is shown for a large set of systems available in the literature.

CHAPTER 2

ENHANCING THE CAPACITY OF DEM/CFD WITH AN IMMERSED BOUNDARY METHOD

C.-Y. Wu and Y. Guo

INTRODUCTION

Discrete Element Methods (DEM) have been coupled with Computational Fluid Dynamics (CFD) for analysing fluid-solid particle flows. In the coupled DEM and CFD (i.e. DEM/CFD), DEM is used to model the motion of particles and CFD is employed to analyse the fluid flow, while empirical correlations for the drag forces are generally introduced to analyse the interaction between the fluid and particles and two-way coupling of fluid-particle interaction is considered. DEM/CFD is a computationally efficient technique that has been widely used in modelling two-phase flows, in which the fluid domain is generally discretised into fluid cells using fixed and rectangular grids and all quantities such as pressure, density and velocity are volume-averaged in the fluid cells. In order to simulate the evolution of bubbles and the detailed fluid flow inside the bubbles, the size of the fluid cell should be smaller than the macroscopic bubbles. On the other hand, it has to be larger than the particle size so that the void fraction (the ratio of the volume of void, excluding that occupied by the solid particles, to the total volume of the cell) will not become zero. Typically, the size of the fluid cell is 3~5 times the particle diameter.

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