# Discrete element method-based models for the consolidation of particle packings in paper-coating applications.

код для вставкиСкачатьASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING Asia-Pac. J. Chem. Eng. 2011; 6: 44–54 Published online 20 May 2010 in Wiley Online Library (wileyonlinelibrary.com) DOI:10.1002/apj.447 Special Theme Research Article Discrete element method-based models for the consolidation of particle packings in paper-coating applications† G. Pianet,1 F. Bertrand,1 * D. Vidal1,2 and B. Mallet1 1 2 École Polytechnique de Montréal, P.O. Box 6079, Stn. CV, Montréal, Quebec, Canada H3C 3A7 FPInnovations – Paprican, Pointe-Claire, Quebec, Canada H9R 3J9 Received 2 October 2009; Revised 15 February 2010; Accepted 23 March 2010 ABSTRACT: This paper concerns the use of the discrete element method (DEM) for the three-dimensional simulation of the consolidation of particle packings in a surrounding liquid such as those in paper-coating applications. The accuracy of DEM is first assessed using X-ray microtomography experiments in the case of polydisperse particle distributions. It is shown that simulations that only account for gravitational and contact forces are in excellent agreement with literature data. However, paper-coating applications involve more complex mechanisms governing the transient particle/pigment consolidation: compression effects during metering and calendering operations, and drag forces during the drainage of the liquid into the basesheet. To simulate the deposition of particles on the basesheet under drainage conditions more realistically, three modelling strategies are considered in this work, which respectively take into account: (1) gravity, (2) mechanical compression and (3) uniform drag. The results show that the first two strategies yield stratified structures and similar bulk porosities, and that the use of a uniform drag leads to significant changes in the particle dynamics and packing properties. Finally, exploratory results with a model for the two-way coupling between the dynamics of the liquid and solid phases reveal its potential for predicting the compression of wet granular media. 2010 Curtin University of Technology and John Wiley & Sons, Ltd. KEYWORDS: particle packings; liquid; compression; discrete element method; lattice Boltzmann method; two-way coupling INTRODUCTION The numerical investigation of particulate/granular flow is a challenging research field in both engineering and computer science. The static and dynamic characterisation of powders, slurries and pastes is a key step in many industrial applications, including papermaking and paper-coating processes. The numerical approaches to simulate this type of flow are commonly classified into continuum and discrete models. Continuum models, which have been used for industrial scale systems, have proven inaccurate in the case of dense particulate flows owing to their inability to account for particle segregation and particle collisions. On the other hand, the discrete or particulate models, which can be of deterministic or probabilistic nature, simulate the motion of *Correspondence to: F. Bertrand, École Polytechnique de Montréal, P.O. Box 6079, Stn. CV, Montréal, Quebec, Canada H3C 3A7. E-mail: francois.bertrand@polymtl.ca † Presented at the 2008 TAPPI Advanced Coating Fundamentals Symposium. 2010 Curtin University of Technology and John Wiley & Sons, Ltd. Curtin University is a trademark of Curtin University of Technology each particle individually and are more appropriate for investigating collisional phenomena such as avalanches or segregation mechanisms.[1,2] Among these methods, the discrete element method (DEM) has proven to be both accurate and flexible when explicit dependency to time and non-elastic conservative particle contacts are required. In this work, a variant of the DEM model of Zhou et al .[3] is used, which is based on a soft-particle springand-dashpot model to account for energy dissipation due to inter-particle contacts, and the use of the Coulomb law of friction, to simulate the deposition of particles on the basesheet of paper-coating applications. In such a process, the time scale of buoyancy effects inherent to gravitational forces is usually much larger than that related to the compression during the metering and calendering operations, and to the drag forces during the drainage of the liquid into the basesheet, which are the two main competing mechanisms governing the transient particle/pigment consolidation. As a result, the effect of gravity is assumed to be negligible here, and the particle deposition is considered to be driven by two Asia-Pacific Journal of Chemical Engineering DEM-BASED MODELS FOR THE CONSOLIDATION OF PARTICLE PACKINGS other mechanisms related to the drag-driven settling of particles and their mechanically driven compression. It is the first time that such models are being used to simulate the settling and the compression of particle packings within the context of paper coating. The second mechanism is modelled by a downwardmoving solid wall so that the particles are progressively packed against the basesheet, while periodic conditions are imposed on the side boundaries. The integration of forces due to the contact between the top layer of particles and this moving solid wall is done at each time step. More specifically, at each of these iterations, the resulting pressure is compared to a reference pressure and the plane is moved downwards when this pressure is too low or upwards when it is too high. This mechanism was designed so as to reproduce the mechanical constraints of the calendering process. The first mechanism is much more intricate to model since the viscous drag is related to solid–liquid friction, the extent of which can be deduced from the relative local velocity of the liquid phase. As a first approximation, one may consider a uniform velocity field. More accurate strategies exist, all of which account explicitly for the complex solid–liquid interactions. For a comprehensive account on discrete particle models and the most popular solid–liquid coupling techniques, the reader is referred to the thorough review of Zhu et al .[4] In the current work, the local mean velocity of the liquid phase is solved numerically with the lattice–Boltzmann method (LBM). To this end, a two-way coupling approach is introduced to handle fluid–particle interactions and unusually high solids volume fractions (∼60%). More precisely, instead of using traditional point-force models to consider the effect of suspended particles on fluid flow, a force term is introduced, which is based on multi-scale porous media modelling, and whereby the time- and space-dependent porosity and permeability values are updated according to the instantaneous local particle distribution. In contrast to twoway coupling approaches for dilute suspensions, this method is well suited to dense particulate flow, and designed to capture the main mechanisms of momentum transfer for both particle–particle and fluid–particle interactions. First, a brief overview of DEM and underlying collisional models is presented. Second, a DEM-based model is introduced and its accuracy is assessed by simulating the packing of polydisperse spherical particles under gravity and comparing the results with X-ray microtomography (XMT) experimental data. Next, a model that explicitly takes into account fluid and solid dynamics is presented. It is based on the DEM-based model for the motion of the pigments and the LBM for the flow of the liquid phase. Results of various packing simulations obtained with both the uncoupled and coupled models are then presented to investigate global packing properties and, in an exploratory manner, the 2010 Curtin University of Technology and John Wiley & Sons, Ltd. role of the liquid phase on particle migration. Finally, concluding remarks discuss some of the current limitations of the hybrid LBM–DEM modelling strategy proposed in this work. THE DISCRETE ELEMENT METHOD Model description DEM is a time-dependent Lagrangian soft-particle method (SPM)[5] for which particle overlapping is allowed. In this method, it is possible to constrain overlaps with appropriate physical models in such a way as to recover elastic collision and particle deformation behaviour (Fig. 1). The characteristic deformation parameters are established by closure relationships that depend on particle structural properties. This additional set of physical parameters opens a whole spectrum of applications that simpler hard-sphere models cannot deal with. In DEM, the position xi = (xi , yi , zi ) of the centre of a particle i of mass mi and its velocity at given time t is computed by numerically integrating Newton’s second law of motion: mi d2 xi dt 2 = Ftotal,i (1) The rotation of each particle is obtained in a similar fashion through an angular momentum balance. In Eqn (1), Ftotal,i is the total force acting on particle i . In a general framework, it comprises three distinct components: Ftotal,i = Fhydrodynamic,i + Fnon−hydrodynamic,i + Fcontact,ij (2) j =i The first term gathers all hydrodynamic forces such as buoyant, drag, pressure gradient, lubrication and lift forces (see Zhu et al .[4] for more details). Particular attention is given to hydrodynamic forces in the following section. The second term is related to nonhydrodynamic forces as those prevailing, for example, in colloidal systems. These hydrodynamic forces, which are rather straightforward to include but rather intricate to validate when coupled with hydrodynamic forces, are not considered in the current work. The last term stands for the contact forces between particle i and neighbouring particles j involved in an (i , j ) collision. They are evaluated here by resorting to the so-called generalised spring-and-dashpot model, which is a variant of the classical linear spring-and-dashpot model introduced by Cundall and Strack.[5] Contact forces are projected on both normal and tangential directions of a local Asia-Pac. J. Chem. Eng. 2011; 6: 44–54 DOI: 10.1002/apj 45 46 G. PIANET ET AL. Asia-Pacific Journal of Chemical Engineering → t particle j at time (t + 1) → n dt particle i dn particle j at time (t) DEM representation of two colliding particles. This figure is available in colour online at www.apjChemEng.com. Figure 1. Figure 2. Sketch of the springand-dashpot model. referential (Fig. 2) where the magnitude and direction of particle overlap, δ, rules the behaviour of repulsive forces. The model used in the present work is similar to that of Zhou et al .,[3] in which shear and normal forces are connected via the Coulomb law of friction. Equations reported in Table 1 detail how stiffness coefficients kn and kt and damping coefficients Cn and Ct are brought into play in the collision rules. In this model, they are expressed as functions of Young’s modulus E , Poisson ratio υ and friction coefficient µs . More details about this model and the setting of the coefficients can be found in Bertrand et al .[6] It is sufficient to say that it has proven efficient for the simulation of many granular flow problems by our research group, mainly in the field of mixing[7] and coating.[1] Values of the coefficients used in this work are similar to those used in Ref. [7], and have led to overlaps smaller than 1% of the particle diameter. Eqn (1) was integrated using the half-step leapfrog Verlet integration scheme coupled with an appropriate time-stepping strategy. Assessment of the DEM-based model through XMT measurements Considering the large number of physical and numerical parameters to be set in a DEM-based model, proper validation is an important task. For instance, the assessment of our DEM code in a dynamical context has been Table 1. DEM model of Zhou et al.[3] (α = 3/2, β = 1, and δn,ij and δt,ij are the normal and tangential components of the overlap between particles i and j). Fcontact,ij = Fcontact,n,ij + Fcontact,t ,ij α + Cn δ̇n,ij Fcontact,n,ij = kn δn,ij β Fcontact,t ,ij = kt δt ,ij + Ct δ̇t ,ij kn = kn (E , υ) Cn = cn (E , υ) 3/2 kt = kt µs kn δn,ij 3/2 Ct = ct µs kn δn,ij 2010 Curtin University of Technology and John Wiley & Sons, Ltd. done with the flow through an orifice and the Beverloo correlation.[6] Also, in the field of powder mixing in pharmaceutical blenders, our code has proven accurate by means of comparisons with experimental data.[7] In the context of particle-settling simulations, the final packing state can be easily analysed. For instance, this can be done by means of the so-called radial distribution function (RDF) that can be readily obtained from simulation results or from experimental samples scanned with XMT.[8] The RDF provides a way to quantitatively characterise the structure that is present in a collection of particles. A comparison of numerical and experimental RDF functions is given in Fig. 3, in the case of a gravity-driven packing of 2000 polydisperse glass particles whose particle size distribution (PSD) is given in Fig. 4. For instance, the peak of the RDF g(r) at normalised inter-particle spacing r/ < D >= 1, where < D > denotes the average particle diameter, represents the nearest-neighbour contacts, whereas that at r/ < D >= 2 denotes the contacts involving particles separated by two particle diameters. This clearly shows an excellent agreement between the simulation results obtained with our DEM-based model and the experimental data from Ref. [8]. These results demonstrate the adequacy of our DEM-based model for the simulation of the gravity-dominated packing of polydisperse particles. Nevertheless, as was mentioned in Ref. [1], interstitial fluid flow may play a significant role in the packing of pigments in paper-coating applications. The next section addresses this important issue by introducing a realistic model for the simulation of the flow of wet granular media. FLUID–SOLID COUPLING MODEL In this work, the influence of the fluid phase on particles is considered from two different points of view. The first and simplistic approach[1] supposes that the fluid velocity u = UF is uniform in the whole domain. In this case, each particle i experiences a drag force Fd = 6π µai (UF − dxi /dt), where µ is the fluid Asia-Pac. J. Chem. Eng. 2011; 6: 44–54 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering DEM-BASED MODELS FOR THE CONSOLIDATION OF PARTICLE PACKINGS Figure 3. (a) Radial distribution function (RDF) corresponding to (b) 2000 gravity-packed polydisperse particles. Color indicates particle size, from blue (small) to red (large). This figure is available in colour online at www.apjChemEng.com. 14 10 5 7 11 2 0 4 3 1 12 8 Z 6 Y 9 13 X Figure 4. Particle size distribution for the gravity-driven packing of 2000 particles. Figure 5. D3Q15 lattice. This figure is available in colour online at www.apjChemEng.com. dynamic viscosity and ai the particle radius. This is a crude approximation since the Stokes law for the drag force is valid for isolated spherical particles. The second approach is elaborated in this section, and relies on the use of a Navier–Stokes solver for calculating the velocity field accurately. The main difficulty then lies in devising an adequate method for coupling the fluid and solid phases. The lattice Boltzmann method Conventional computational fluid dynamics (CFD) methods (e.g. finite element, finite volume and finite 2010 Curtin University of Technology and John Wiley & Sons, Ltd. difference methods) have been proven to have limitations in solving the Navier–Stokes equations in porous media. On the contrary, the LBM, developed in the early 1990s, is now considered by many researchers[9] as a method of choice for the simulation of single-phase or multiphase flows in complex geometries. LBM can be viewed as a particulate method inasmuch as it ‘simulates’ the macroscopic behaviour of fluid molecules in motion. More precisely, it is based on the discretisation of the Boltzmann equation from kinetic gas theory and the propagation and collision of populations of particles on the nodes of a lattice representing the computational domain. In practice, these populations of particles propagate and collide at every time step δt on a lattice with Asia-Pac. J. Chem. Eng. 2011; 6: 44–54 DOI: 10.1002/apj 47 48 G. PIANET ET AL. Asia-Pacific Journal of Chemical Engineering spacing δx and along ei velocity directions, where the number of directions i depends on the lattice type. A 15-velocity-direction D3Q15 lattice is used in this work, as described in Fig. 5. The collision step is given by can be solved with an explicit scheme, which makes it a good candidate to be combined with DEM for the development of a fully coupled fluid–solid model. The reader is referred to the book of Succi[9] for more details on LBM and its implementation. eq fi (x, t) − fi (x, t) fi (x, t) = fi (x, t) − τ∗ ∗ (3) and is followed by the propagation step ∗ fi (x + ei δt , t + δt ) = fi (x, t) (4) where fi (x, t) is the particle probability distribution function (the so-called population) in the direction of the velocity ei at position x and time t, and τ ∗ is the dimensionless relaxation time. The second term of the right-hand side of Eqn (4) approximates the collision process through the Bhatnagar, Gross and Krook (BGK) relaxation procedure, towards a local equilibrium population, which for the D3Q15 lattice is given by 2 δt eq fi (x, t) = wi ρ 1 + 3(ei · u) δx 2 4 δt 9 3 δt 2 (5) + (ei · u) − (u · u) 2 δx 2 δx 1 for where w0 = 92 , wi = 91 for i = 1 : 6 and wi = 72 i = 7 : 14. In this equation, fi (6) ρ = ρ(x, t) = i and u = u(x, t) = 1 fi ei ρ i One of the objectives of this ongoing work is to develop a simulation tool for investigating the consolidation of pigments in coating/calendering operations. More generally, we also intend to use this computer model to investigate other solid–liquid flow problems, such as the ones related to the mixing of dispersions. Coupling techniques based on direct numerical simulation (DNS) are a priori attractive because they rely on an explicit discretisation of the particles. Unfortunately, these techniques are known to be excessively expensive both in time and memory, and are still limited to the simulation of a few hundred monodisperse particles.[4] In the present context, the so-called two-way coupling techniques represent the best alternative owing to their capability of resolving the impact of the particles on fluid motion. The novel technique introduced in this paper is restricted to the case of concentrated suspensions, the behaviour of which is that of a timeand space-dependent porous medium. To this end, a porous medium model was implemented into our LBM code, which brings into play a forcing term, Fp , following along the lines of Guo and Zhao.[10] Note that this time- and space-dependent forcing technique was proposed and validated by Guo et al .[11] It takes into account the drag due to the flow of particles making up the porous medium by means of the Forcheimerextended Darcy’s drag force: (7) Fp = − represent the local fluid density and the local fluid velocity, respectively, and the dimensionless relaxation time τ ∗ is related to the fluid kinematic viscosity ν through ν 1 (8) + τ∗ = 2 2 δt cs where cs is the speed of sound defined as 3(1 − w0 ) δx cs = 7 δt Fluid–particle coupling (9) LBM is used in this work for two reasons: first, in addition to being straightforward to implement, it has already proven to be most efficient for solving a large variety of fluid mechanics problems in significantly smaller computational times as opposed to classical CFD methods and, second, the Boltzmann equation 2010 Curtin University of Technology and John Wiley & Sons, Ltd. 1.75 µε |u|u + εG u−√ ρK 150K ε (10) where ε is the porosity, K is the permeability, ρ is the fluid density and G is an additional body force if needed. Note that an extrapolation method was also used for an accurate imposition of the boundary conditions[12] delimiting the porous medium. With this technique, second-order accuracy in space can be achieved for a large set of boundary conditions, including sliding boundaries and curved walls, which are often encountered in complex applications. The validity of this LBM-based model for the solution of the generalised Navier–Stokes equations describing (isothermal) incompressible fluid flow in porous media was assessed by simulating the porous plane Couette and porous Poiseuille flows. In the first case, the forcing term is induced by a vertical velocity gradient originating from the top sliding wall. In the second case, the forcing term is induced by a uniform body force, Asia-Pac. J. Chem. Eng. 2011; 6: 44–54 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering DEM-BASED MODELS FOR THE CONSOLIDATION OF PARTICLE PACKINGS (a) (b) Figure 6. (a) Porous plane Couette flow. The plain lines represent the finite reference difference solutions, and the squares the solutions obtained with the LBM-based model and its forcing term; (b) porous Poiseuille flow. The plain lines represent the finite difference reference solutions, the triangles the solutions obtained with the LBM-based model and its forcing term. Note that Re = ρHuref /µ is the Reynolds number, where H is the width of the channel and uref the characteristic velocity (that of the sliding wall for the Couette flow of that at the centre of the channel for the Poiseuille flow), and Da = K/H2 is the Darcy number. whereas the top and bottom walls are fixed. As seen in Fig. 6, the reference solutions obtained with a secondorder finite difference solver are precisely predicted for both problems and different values of the Darcy number Da (different levels of permeability), which shows the accuracy of the proposed approach for simulating fluid flow in porous media. Two-way coupling algorithm The two-way coupling fluid–solid model proposed in this work combines the DEM-based and LBMbased models described above. The overall algorithm is transient and requires initial conditions for both the particles and the fluid. At every time step, local porosity values are first obtained for each lattice cell, and local permeability values are estimated by means of the following Carman–Kozeny correlation for particles of diameter d: K (x) = ε3 (x)d 2 /150(1 − ε(x))2 (11) The resulting porosity and permeability distributions are then introduced into the forcing term [Eqn (10)] of the LBM model. Next, the velocity and pressure distributions are computed with LBM, in which each lattice cell is considered as a macroscopic representative elementary volume (REV; see for instance Ref. [10]) of the porous medium. The following step consists of performing a number of DEM iterations, typically from 100 to 1000, to move the particles. For each particle i , the hydrodynamic forces, that is the term Fhydrodynamic,i in Eqn (2), are evaluated locally from the fluid velocity, pressure gradient, porosity and distance to the nearest 2010 Curtin University of Technology and John Wiley & Sons, Ltd. particles, all of which were previously computed. The resulting particle distribution is then used to move on to the next time step. Two remarks are in order. First, numerical experiments (not described here) have revealed that performing many DEM iterations within one single LBM time iteration has shown to be a good strategy to reduce the computational cost while yielding results that do not depend significantly on this number of iterations. Moreover, Fhydrodynamic,i corresponds in this work to the viscous drag force, which is the most important force related to the fluid–particle interactions involved in coating applications. Note that the viscous drag is evaluated here from classical correlations due to Ergun[13] and Wen and Yu.[14] Alternatively, one could resort to recent correlations based on averaged results from the DNS of randomly dispersed particles.[4] RESULTS In this section, three different packing techniques described in Fig. 7 are investigated, compared and discussed within the context of pigment consolidation in paper-coating applications. The first case is the gravity-driven packing technique. A typical DEM simulation is typically rather fast in the first steps owing to the uniform acceleration of gravity that leads to the bulk motion of the particles with very few collisions until the lowermost particles reach the bottom wall. A quick compression phase then occurs, which leads to a loose packing porosity lying within the range delimited by the loose random packing limit, εLRP ∼ 44%, and the close random packing limit, εCRP ∼ 36%, depending on both the friction parameter and the weight of the bed of particles. We revert to Asia-Pac. J. Chem. Eng. 2011; 6: 44–54 DOI: 10.1002/apj 49 50 G. PIANET ET AL. Asia-Pacific Journal of Chemical Engineering (a) (b) (c) Figure 7. Schematics of the three particle packing techniques: (a) gravity driven, (b) mechanically driven (compression) and (c) drag driven. this point below. As shown in Fig. 3 of the section on Assessment of the DEM-Based Model Through XMT Measurements, gravity-driven packings are well predicted by DEM, although these results are of limited use for paper-coating applications where gravity effects are known to be small. The second case concerns the mechanical compression of particles. As already discussed in the introduction, the particles are compressed by a downwardmoving solid wall so that they progressively pack against the bottom plane. At each time iteration, the pressure resulting from the contact between the top layer of particles and this moving plane is compared to a reference pressure, and the plane is moved downwards when this pressure is too low or upwards when it is too high. This mechanism in a way mimics the mechanical constraints of the calendering process. Figure 8 displays the initial and final states of the packing of 46 339 polydisperse particles as predicted with the DEM-based model and mechanical compression only, that is, without gravitational and drag forces. The initial mean porosity ε = 45% (∼1.02 × εLRP ), which is close to that for a monodisperse loose random packing, εLRP , reduces to 38% (∼1.06 × εCRP ), which is slightly above εCRP , the porosity of a monodisperse close random packing. In this case, the percolation of the particles are driven by the contact forces and, as a result, the porosity levels off near but not lower than the monodisperse close random packing value (εCRP ), due to the barrier effect of the largest particles. From a computational point of view, the first DEM iterations of this mechanical compression packing simulation are stable and rather similar to the ones for a gravity-driven packing technique. Things are, however, much different in the last iterations, which may become unstable if the total pressure on the moving plane is not carefully chosen and controlled. Note that, in the context of the calendering process, it is likely that these mechanically driven particle packings 2010 Curtin University of Technology and John Wiley & Sons, Ltd. are more realistic than those obtained with gravity only. In most paper-coating processes, buoyancy is often negligible due to the high fluid viscosity and small density differences between the particles and the fluid. However, we believe that an enhanced percolation of small particles and a further decrease in porosity can be obtained by bringing into play liquid drainage and the corresponding viscous drag. For comparison purposes, 2014 monodisperse particles were packed using the gravity-driven, mechanically driven and drag-driven techniques. Initially, the particles of diameter d = 800 µm were randomly dispersed in a rectangular domain of size (Lx , Ly , Lz ) = (30 000, 3000, 3000 µm), with periodic boundary conditions along the y and z directions and solid walls at x = 0 and 30 000 µm. Five initial particle configurations corresponding to random dispersions with a prescribed 75% porosity were generated. Each time, the overlapping particles were subjected to a fictitious repulsive potential to displace them towards nearest free spaces. The RDFs of these initial packings revealed that peaks due to reorganisation were almost undetectable, so that this dispersion technique can be considered as quasi-stochastic. Note also that the simulation parameters were tuned so that the relevant driving forces were equivalent in magnitude for all three packing techniques. This was done a posteriori by verifying the time-dependent force exerted on a few probe particles. All simulations were stopped when the packing heights levelled off to some constant value. Figure 9 shows the initial particle configuration and the final packings for all three techniques. Figure 10 contains a graph of the porosity vs the vertical position. The porosity values are in fact averaged values coming from the five different initial particle configurations described above. Note, however, that these configurations had a small impact on the resulting porosity curves. As seen in Figs 9 and 10, the gravity-driven Asia-Pac. J. Chem. Eng. 2011; 6: 44–54 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering DEM-BASED MODELS FOR THE CONSOLIDATION OF PARTICLE PACKINGS Figure 8. Mechanical compression of 46 339 polydisperse particles: (a) initial state and (b) final state. Colour indicates particle size, from blue (small) to red (large). This figure is available in colour online at www.apjChemEng.com. technique yields the loosest packing. The porosity oscillates around 40%, and thus falls right between the CRP and LRP bounds as has already been observed above for a different gravity-driven packing. The mechanically driven packing technique is observed to yield a global decrease in porosity, close but slightly above on average to the CRP bound. No concentration gradients are found across the height of such packings, mostly because of the monodisperse nature of the particles, so that the compression effect is mechanically transmitted across the whole packing. In the case of the drag-driven packing technique, a uniform drag was imposed, and significant differences with respect to the other two cases can be readily noticed as regard the global porosity structure. First, the packings are much tighter, the final porosity ranging between the close packing bound (εCP ∼ 26%) and the CRP bound. Second, the slope of the corresponding porosity curve in Fig. 10 reveals a noteworthy concentration gradient, which is likely due to the effect of drag and induced compression. Once again, different initial 2010 Curtin University of Technology and John Wiley & Sons, Ltd. packings were tested with no major impact on the global porosity profiles and microstructure. Increasing (respectively decreasing) the value of the friction parameters resulted in a global shift of the porosity profiles towards larger (respectively smaller) values. A tool was developed for characterising the instantaneous pore size distribution. It is based on an erosion–dilatation method proposed by Toivakka and Nyfors[15] to detect the real pores of a porous medium and determine the position of their centres. Lateral periodic boundary conditions were considered and the pore size distribution was calculated between the centres of the lowest and highest particles in order to limit vertical end effects. Figure 11 shows a graph of the pore size distribution with respect to deq /d, where the pore volumes are expressed as spheres of equivalent diameters deq . As can be seen, the gravity-driven and mechanically driven packing results exhibit similar distributions, both in terms of shape and magnitude, with one single peak in the small pore size region at around deq /d = 0.2. The drag-driven packing profile is significantly different, Asia-Pac. J. Chem. Eng. 2011; 6: 44–54 DOI: 10.1002/apj 51 52 G. PIANET ET AL. Asia-Pacific Journal of Chemical Engineering (a) (b) (c) (d) Figure 9. Packing of 2014 monodisperse particles: (a) initial particle configuration, and final packings for (b) gravity-driven, (c) mechanically driven and (d) drag-driven techniques. Colour indicates the initial vertical position of the particles. This figure is available in colour online at www.apjChemEng.com. since many of the largest pores observed at deq /d > 0.4 with the other two packing techniques are converted to smaller ones. Moreover, there is one large peak at deq /d = 0.2 and a smaller one at 0.3 and, as a consequence, the relatively important number of pores with similar properties, reveal that important particle reorganisation occurs during the drag-driven packing of particles. Another comment concerning the mechanically driven packing results is of interest. The fact that, over the course of the simulation, the total pressure applied by the particles on the compression plane increased monotonically to a user-defined maximum value (see section on Results) indicates that the particle consolidation and the apparent structural reorganisation of the bed of particles may be affected in some way by the interparticle penetration inherent to DEM. Note that this may also be attributed to the speed at which the compression plane is pushed downwards because, if this speed is too much, the particles will not have sufficient time to reorganise. Future work will address this issue. As discussed in the section on Fluid–Solid Coupling Model, we are developing an LBM–DEM two-way coupling algorithm for simulating the consolidation of packings of particles under nonuniform drag forces. One first exploratory result obtained with this hybrid model is presented here. A pressure drop was imposed as the driving force for the flow. The same initial configuration of 2014 monodisperse particles was considered and the apparent viscosity of the solid–liquid suspension was 2010 Curtin University of Technology and John Wiley & Sons, Ltd. Figure 10. Graph of the porosity vs the vertical position for all three packing techniques. calculated so as to match as closely as possible the viscosity used in the uniform drag packing simulation discussed above. The local porosities were evaluated analytically in spheres of volumes equal to the ones in the lattice Boltzmann cell. Resorting to a larger investigation zone would yield increased porous matrix homogeneity and coarser flow details. Figure 12 shows the fluid velocity and porosity fields at different times during the particle consolidation. It is first noticed Asia-Pac. J. Chem. Eng. 2011; 6: 44–54 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering DEM-BASED MODELS FOR THE CONSOLIDATION OF PARTICLE PACKINGS Figure 11. Pore size distribution. Pore volumes are expressed as spheres of equivalent diameter deq . that the fluid velocity is strongly correlated to the local packing structure characteristics, with a significant velocity decay in the consolidated zone near the bottom plane. These results are more consistent with drainage conditions than those obtained in the case of a uniform drag packing technique for which the very large fluid velocities at the bottom induce a very quick and very strong compression. Fluid momentum in the lateral directions is such that the particle kinematics is much more complex than that related to the quasi-onedimensional and subsequently stratified bulk behaviour observed with the previous packing methods. It can also be seen that the homogeneous initial configuration rapidly changes to a three-part flow comprising a free fluid zone, a fluidised particle bed zone and a fixed bed zone in which little structural changes occur. It is also interesting to note that the two interfaces between these three zones are characterised by different propagation speeds, in analogy with other settling processes. However, it must be mentioned that the local recirculations with particle resuspension occurring at the top are probably overestimated in this simulation. Experimental data are required, which would help assess the accuracy of these results and help improve the hybrid LBM–DEM model proposed in this work for simulating the compression of particle packings. CONCLUSION The objective of this work was to compare different ways of using the DEM for consolidating and compressing particle packings in the context of paper-coating applications. It was shown that simple methods, namely the gravity-driven and mechanically driven techniques, yield stratified structures all along the deposition process and similar final bulk porosities. The mechanically 2010 Curtin University of Technology and John Wiley & Sons, Ltd. Figure 12. Results obtained with the LBM–DEM twoway coupling algorithm. Particle packing consolidation under nonuniform drag forces at different times (from left to right). Fluid velocity field (arrows) and local porosity distribution. This figure is available in colour online at www.apjChemEng.com. driven packing technique was designed so as to reproduce the mechanical constraints of calendering processes in a simple way, but no significant changes in the resulting microstructures were observed compared to gravity-driven packings. Simulations with a uniform viscous drag led to significant changes in the particle dynamics and packing properties. This has provided the impetus to develop an LBM–DEM two-way coupling algorithm in which particle motion is coupled to fluid dynamics. The advantage of such a complex model lies in the fact that it gives access to heretofore inaccessible information such as the local hydrodynamic constraints on the particles. Independent assessments of both fluid (LBM) and particle (DEM) models were performed, and the results obtained were shown to be in very good agreement with experimental data or theory. In the context of high solid volume fractions, the coupling principle considers the solid phase as a time- and spacedependent porous matrix in LBM, the porosity and permeability of which are recovered from DEM results. One first exploratory result was presented and showed Asia-Pac. J. Chem. Eng. 2011; 6: 44–54 DOI: 10.1002/apj 53 54 G. PIANET ET AL. Asia-Pacific Journal of Chemical Engineering the potential of this hybrid LBM–DEM model for predicting the behaviour of wet granular media such as the ones in coating applications. Future work will focus on the validation of this LBM–DEM model against experimental data and on its extension to the case of polydisperse systems of nonspherical particles. 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