Modelling Complex Particle–Fluid Flow with a Discrete Element Method Coupled with Lattice Boltzmann Methods (DEM-LBM)

chemengineering

Article Modelling Complex Particle–Fluid Flow with a Discrete Element Method Coupled with Lattice Boltzmann Methods (DEM-LBM)

Wenwei Liu and Chuan-Yu Wu *

Department of Chemical and Process Engineering, University of Surrey, Guildford GU2 7XH, UK; [email protected] * Correspondence: [email protected]

Received: 1 July 2020; Accepted: 29 September 2020; Published: 7 October 2020

Abstract: Particle–fluid flows are ubiquitous in nature and industry. Understanding the dynamic behaviour of these complex flows becomes a rapidly developing interdisciplinary research focus. In this work, a numerical modelling approach for complex particle–fluid flows using the discrete element method coupled with the lattice Boltzmann method (DEM-LBM) is presented. The discrete element method and the lattice Boltzmann method, as well as the coupling techniques, are discussed in detail. The DEM-LBM is thoroughly validated for typical benchmark cases: the single-phase Poiseuille flow, the gravitational settling and the drag force on a fixed particle. In order to demonstrate the potential and applicability of DEM-LBM, three case studies are performed, which include the inertial migration of dense particle suspensions, the agglomeration of adhesive particle flows in channel flow and the sedimentation of particles in cavity flow. It is shown that DEM-LBM is a robust numerical approach for analysing complex particle–fluid flows.

Keywords: particle–ﬂuid ﬂow; discrete element method; lattice Boltzmann method

1. Introduction Particulate flows are extensively encountered in nature and industrial processes, attracting tremendous engineering research interests in almost all areas of sciences [1–3], including astrophysics, chemical engineering, biology, life science and so on. Due to the complex particle–particle interactions and their interactions with the surrounding media, i.e., gas/liquid environment or electrostatic/magnetic field, the dynamic behaviour of particulate flow is very complicated. Therefore, it is of significant importance to understand the particle dynamics, in order to promote relevant manufacture and production processes. With the rapid development of the modern modelling technique, abundant numerical studies on the particulate flows have sprung up in recent years. As the name suggests, the particulate flow can behave like a continuous fluid or a rigid solid. Therefore, studies of particulate flows can be classified into three categories based on the types of approach: (1) continuum modelling, which extends the continuum mechanics of single-phase fluid to describe the particle transportation, leading to a representative population balance method [4,5]; (2) developing the kinetic theory of particulate flow [6] based on the averaged equations for multi-particle systems, which generalise the dynamics of particle–particle collision processes; (3) discrete particle modelling, which solves the particle’s motion individually based on certain interaction laws. The third approach is also classified as Lagrangian particle method and has many different variations, including the discrete element method (DEM), dissipative particle dynamics (DPD), molecular dynamics (MD) and Brownian dynamics (BD). These discrete particle methods differentiate themselves with different particle–particle interaction laws and their corresponding length and time scales [3]. DEM is widely used for modelling of granular

ChemEngineering 2020, 4, 55; doi:10.3390/chemengineering4040055 www.mdpi.com/journal/chemengineering ChemEngineering 2020, 4, 55 2 of 33 materials and adhesive particulate flows [7–17], and was first introduced by Cundall and Strack [18] to study the rock mechanics. In DEM, particles are treated as individual entities, and interact with each other through contact and non-contact forces. The translational and rotational motions of the particles are described using the Newton’s second law. The greatest advantage of DEM is that a large amount of dynamic information, which is almost unattainable experimentally, such as the transient forces and torques, can be obtained in great detail. Apart from the particle–particle interaction, the hydrodynamic interaction associated with the surrounding fluid is also an important issue in the study of particulate flow. Various computational fluid dynamics (CFD) methods at different scales of time and length are developed to model the single-phase fluid flow, ranging from discrete models, e.g., the lattice Boltzmann method (LBM) [19–21], to continuum models, e.g., the direct numerical simulation (DNS) [22], the large eddy simulation (LES) [23] and the two-fluid model (TFM) [24]. Hence, a large number of hybrid models are developed for modelling particle–fluid flow [20,25–33]. A comprehensive comparison of various hybrid models was discussed in the literature [1,34]. Comparing with other conventional CFD approaches, LBM arises in the last two decades and has become a promising numerical method due to several advantages [19,20]. First, the primary computation procedure, i.e., the so-called collision operator, is performed locally with information only from where the computation takes place, facilitating straightforward implementation in a parallel way with high computational efficiency. Second, the LBM shows significant flexibility in handling complex boundary conditions, as they are treated with a local bounce-back fashion instead of the conventional boundary treatment. Third, the lattice grid adopted in LBM is traditionally orthogonal and smaller than the particle size, providing a high space resolution on the evaluation of the hydrodynamic interactions. As a result, the LBM was coupled with DEM and widely applied in analysing a large range of particle–fluid problems, such as particle suspensions [35–38], flow through porous media [39,40], multicomponent flows [20,41,42], particle-laden turbulent flows [31,43,44] and magnetohydrodynamics [45]. In this paper, a numerical approach to model complex particle–fluid flow using a coupled DEM-LBM is introduced. The fundamental equations of DEM and LBM, as well as various coupling techniques are presented in detail. The validity and accuracy of the numerical approach are demonstrated with some benchmark tests. Finally, case studies, including the migration, the agglomeration and the sedimentation of particle suspensions, are presented to illustrate the potential and applicability of the numerical approach. It should be noted that the interactions between particles and the surrounding environment, such as electrostatic interactions, liquid bridge effect and so on, are not within the scope of this work, but can be found in the literature [46–54].

2. Discrete Element Method (DEM) In DEM, particle motion is solved individually using the Newton’s equation of motion,

dUp P m = G + F, dt (1) dΩp P I dt = M, where Up and Ωp are the translational and rotational velocities of a particle, respectively. m and I are the mass and rotational inertia of the particle. G is the gravitational force. F and M represent the force and torque acting on the particle, which could come from various sources, such as interparticle collision, hydrodynamic interactions with the ﬂuid, electrostatic interactions with charge and so on. As DEM was designed to deal with the numerous collision problems between the distinct particles, ChemEngineering 2020, 4, 55 3 of 33 the collision forces and torques are the most important aspect. When two particles get in contact with each other, the collision force and torque can be decomposed as

F = F n + F t , collision n s s (2) M = rpFs(n ts) + Mr(tr n) + Mtn, collision × × where Fn and Fs denote the contact forces in the normal and tangential direction, respectively. Mr and Mt are the rolling and twisting resistance torques, respectively. rp represents the particle radius n, ts and tr are the unit vectors in the normal, tangential and rolling direction at the contact point, respectively.

2.1. Normal Contact Force The most common model for the normal contact force is the Hertz model [55], which can be expressed by a non-linear Hook’s law. The Hertz model was validated and widely applied in the contact of spherical particles with size above millimeters. However, as the particle approaches the micrometer size range, the adhesion eﬀect due to the van der Waals force must be taken into consideration. Extensions of the Hertz model, including the well-known Johnson–Kendall–Roberts (JKR) theory [56] and Derjaguin–Muller–Toporov (DMT) theory [57], were also proposed to describe the contact force in presence of adhesion. Let us consider two contacting particles at centroid positions xi and xj with radii rp,i and rp,j, respectively. The normal overlap δ between these two particles is derived as δ = r + r |x x |. N N p,i p,j − j− i Then the normal forces between two particles are given as:

= 1.5 Fne Hertz kNδN , − − 3 1.5 (3) Fne JKR = 4FC(aˆ aˆ ), − − − where kN is normal stiffness coefficient in the Hertz model, aˆ = a/a0 is the normalized contact area radius and FC represents the critical pull-off force obtained from the JKR theory. In the Hertz model, the normal stiffness is analytically derived as kN = 4/3E √R, where R is the effective particle radius and E is the effective elastic moduli, defined as:

1 = 1 + 1 , R rp,i rp,j 2 1 σ2 1 σ (4) 1 = − i + − j , E Ei Ej with Ei and Ej being elastic moduli and σi and σj being the Poisson’s ratios, respectively. Note that the contact area radius a, the effective radius R and the normal overlap δN are related to each other via a simple geometrical relation, a = √RδN. In the JKR theory, the particles will keep in contact due to the presence of van der Waals adhesion even if the external load is zero [56], which gives rise to an equilibrium contact radius a0, 1/3 9πγR2 a = ( ) , (5) 0 E where γ is the particle’s surface energy. The critical pull-off force in the JKR theory is then derived as FC = 3πγR [56]. However, both the Hertz model and the JKR theory are energy conserved, i.e., there is no energy dissipation during the process of particle collision. To consider the energy dissipation during a collision, various dissipation mechanisms were proposed, including the viscoelastic response, plastic deformation as well as the fluid-phase dissipation [58]. For the low-speed impact regime, the viscoelastic dissipation is dominant. The dissipation force Fnd is approximated with a dashpot model, which is proportional to the approaching velocity of the particles, i.e.,

F = ηNvR n, (6) nd − · where ηN is the normal dissipation coeﬃcient, and vR is the relative velocity at the contact point. ChemEngineering 2020, 4, 55 4 of 33

2.2. Sliding, Rolling and Twisting Resistance The tangential contact force results from the relative sliding motion between two particles. Similar to the normal force, the standard sliding model is approximated by a linear spring-dashpot,

Fs = kTξT ηTvs ts, (7) − − · where kT is the spring stiffness coefficient in the tangential direction, vs is relative sliding velocity, R t ξT = vs(t) ts dt is the cumulative displacement in the tangential direction and ηT is the sliding t0 · dissipation coefficient. According to the Amonton’s friction law, two contacting particles start to slide relative to each other when the tangential force reaches a critical value Fs,crit and the dynamic friction force remains constant as Fs = F . For non-adhesive particles, the critical value is F = µs |Fne|, − s,crit s,crit where µs is sliding friction coefficient. Due to the presence of adhesion, the critical value for adhesive particles is larger, Fs,crit = µs |Fne + 2FC| [3,58]. The rolling and twisting resistances are similarly postulated in the form:

Mr = kRξR ηRvL tr, − − · (8) Mt = k ξ η ΩT, − Q Q − Q R t R t where ξR = vL(t) tr dt and ξQ = ΩT(t)dt are the rolling and twisting displacements, vL= t0 · t0 R(Ω Ω ) n and Ω = (Ω Ω ) n are the relative rolling and twisting velocity, respectively. − p,i − p,j × T p,i − p,j · The rolling direction is tr = vL/|vL|. kR, kQ are the rolling and twisting stiﬀness, respectively, and ηR, ηQ are the rolling and twisting dissipation coeﬃcients, respectively. Correspondingly, the critical values for rolling and twisting resistances are given as:

M = k θ R, r,crit R crit (9) Mt,crit = 3πaFs,crit/16, where θcrit is the critical rolling angle in the relative rolling motion between two contacting particles.

3. Lattice Boltzmann Method (LBM)

3.1. Lattice Boltzmann Equation Fundamentally, the LBM aims to establish a microscopic kinetic model involving essential physics, so as to make the average physical quantities in macroscopic scale follow the expected equations, which is completely different from the conventional numerical methods that directly discretise the macroscopic continuum equations. Hence an important premise lies in that the dynamic behaviour of a fluid in the macroscopic scale is the outcome from the collective behavior of numerous constitutions in the microscopic scale [19]. The LBM originated from lattice gas automata, which utilises a discrete lattice and discrete time. In each lattice, the fluid is described by a packet of microscopic particles, which only move in certain directions with certain discrete velocities. A number of lattice speed models are available, depending on the dimension of the problem. For example, the D2Q9 and D3Q19 are frequently used in the 2D and 3D problems, respectively, as illustrated in Figure1. The individual particle motion is neglected in the LBM and a single-particle distribution function fi(x,t) is adopted instead. ChemEngineering 2020, 4, 55 5 of 33

ChemEngineering 2020, 4, 55 5 of 33

FigureFigure 1.1. TheThe D2Q9D2Q9 ((aa)) andand D3Q19D3Q19 ((bb)) latticelattice speedspeed model.model.

TheThe evolutionevolution ofof thethe particleparticle distributiondistribution functionsfunctionsf fi(i(xx,t),t) followsfollows thethe latticelattice BoltzmannBoltzmann equationequation (LBE)(LBE) [[19]19] withwith anan externalexternal forceforce term, term, +Δ +Δ= +Ω +Δ fii(xe tttft , ) i ( x ,) iii [ ftFt ( x ,)] , (10) fi(x + ei∆t, t + ∆t) = fi(x, t) + Ωi[ fi(x, t)] + Fi∆t, (10) where x is the position of the node, ei is the unit vector of the lattice speed in the direction i as denoted wherein Figurex is the1, Δ positiont is the ofexplicit the node, timee istep,is the F uniti is the vector external of the body lattice force speed and in theΩi[ directionfi(x,t)] is thei as denotedcollision inoperator. Figure1 Equation, ∆t is the (10) explicit can be time evaluated step, F ini is two the in externaldependent body processes: force and collisionΩi[fi(x,t)] and is streaming, the collision as operator.shown below: Equation (10) can be evaluated in two independent processes: collision and streaming, as shown below: fttft(xxx ,+Δ ) = ( ,) +Ω [ ftFt ( ,)] + Δ , (11) fiiiiii(x, t + ∆t) = fi(x, t) + Ωi[ fi(x, t)] + Fi∆t, (11) f (x ++Δe ∆t, t + +Δ=∆t) = f (x, t + +Δ∆t).(12) (12) fiii(,)(,).xei tttftti i x EquationEquation (11)(11) represents represents the the collision collision process, process, which which updates updates the distribution the distribution functions functions at position at xpositionfrom t to xt +from∆t based t to t on+Δ thet based relaxation on the operator relaxationΩi[f ioperator(x,t)]. Then Ωi the[fi(x streaming,t)]. Then processthe streaming (Equation process (12)) takes(Equation place, (12)) which takes simply place, propagates which simply the updated propagates distribution the updated functions distribution from position functionsx to theirfrom nearestposition neighbor x to their nodes nearestx +neighborei∆t according nodes x to + theeiΔtlattice according speed to model.the lattice The speed sequence model. of The collision sequence and streamingof collision becomes and streaming irrelevant becomes after a irrelevant few time steps.after a few time steps.

3.1.1.3.1.1. Single-Relaxation-TimeSingle-Relaxation-Time ModelModel TheThe single-relaxation-timesingle-relaxation-time (SRT) (SRT) model model refers torefers the Bhatnagar-Gross-Krook to the Bhatnagar-Gross-Krook (BGK) approximation (BGK) ofapproximation the collision operator of the collision [59]. The operator collision [59]. operator The coll isision simplified operator to ais linear simplified form [to59 a– 62linear]: form [59– 62]: ∆t = [ ( ) eq( )] Ωi Δ fi x, t fi x, t , (13) Ω=−− τt − − eq iii[f (xx ,tf ) ( , t )], (13) τ eq where τ is the dimensionless relaxation parameter and fi (x,t) is the equilibrium distribution function thatwhere is definedτ is the dimensionless as: relaxation parameter and fieq(x,t) is the equilibrium distribution function 2 2 that is defined as: eq e u (ei u) u f = ρω [1 + i + · ]. (14) i i ·2 4 2 cs⋅⋅2cs 2− 2cs 2 eq eu() eu u f =++ρω [1ii − ]. In this equation, the weight coeiifficient ωi depends242 on the lattice speed model. For example, ω0 =(14)4/9, cccsss22 ω1,2,3,4 = 1/9 and ω5,6,7,8 = 1/36 for D2Q9 model, while ω0 = 1/3, ω1, ... ,6 = 1/18 and ω7, ... ,18 = 1/36 for D3Q19In this model.equation,cs the= 1 /weight√3 is thecoefficient lattice sound ωi depends speed. on the lattice speed model. For example, ω0 = 4/9, ω1,2,3,4 = 1/9 and ω5,6,7,8 = 1/36 for D2Q9 model, while ω0 = 1/3, ω1,…,6 = 1/18 and ω7,…,18 = 1/36 for D3Q19 = model. cs 1/ 3 is the lattice sound speed. The discrete body force term Fi in Equation (10) is given as [63]: ChemEngineering 2020, 4, 55 6 of 33

The discrete body force term Fi in Equation (10) is given as [63]:

1 e u (ei u) F = (1 )ω [ i + · e ] F, (15) i i −2 4 i − 2τ cs cs · where F represents the macroscopic body force. The macroscopic ﬂuid properties, including the density ρ, velocity u and pressure p are related to the microscopic particle distribution function, which are determined as: P ρ = fi, i P ∆t ρu = fiei + 2 F, (16) i 2 p = cs ρ. The kinematic viscosity of the ﬂuid ν is related to the relaxation parameter as:

2τ 1 ν = − . (17) 6

3.1.2. Multi-Relaxation-Time Model It was reported that the SRT LBE does not show good stability with small relaxation time τ or with high Reynolds number. To improve the performance of the LBM, the multi-relaxation-time (MRT) LBE was proposed, which can be expressed as [20]:

X eq X fi(x + ei∆t, t + ∆t) = fi(x, t) Ωiα[ fα(x, t) fα (x, t)] + [Si(x, t) 0.5 ΩiαSα(x, t)], (18) − α − − α where S is the force term. From Equation (15), we have:

e u (ei u) S = ω [ i + · e ] F. (19) i i −2 4 i cs cs ·

Equation (18) can be transformed into the matrix form as:

1 eq 1 Λ f(x + e ∆t, t + ∆t) = f(x, t) M− Λ[m(x, t) m (x, t)] + M− [I ]S(x, t), (20) i − − − 2 eq where M is the transformation matrix, m and m are the moment spaces of the distribution function fi eq eq eq and its equilibrium distribution fi , which can be derived as m = Mfi and m = Mfi , respectively. The collision matrix Λ is a diagonal matrix in the moment space. The matrixes M and Λ vary for diﬀerent speed models. For D2Q9 model, the matrix M is:    1 1 1 1 1 1 1 1 1     4 1 1 1 1 2 2 2 2    − − − − −   4 2 2 2 2 1 1 1 1   − − − −   0 1 0 1 0 1 1 1 1     − − −  M =  0 2 0 2 0 1 1 1 1 . (21)  − − −   0 0 1 0 1 1 1 1 1    − − −   0 0 2 0 2 1 1 1 1  − − −   0 1 1 1 1 0 0 0 0   − −   0 0 0 0 0 1 1 1 1 − − ChemEngineering 2020, 4, 55 7 of 33

The equilibrium distribution functions meq in the moment space are related to that in the velocity space as follows:      ρ   ρ       eeq   2ρ + 3(u2 + u2 )     x y   eq   − 2 2   ε   ρ 3(ux + uy)   eq   −   j     x   ρux  eq eq  eq    m = Mf =  qx  =  ux . (22) i  eq   −   j     y   ρuy   eq     qy   uy   eq   −   p   u2 u2   xx   x y   eq   −  pxy uxuy

The diagonal matrix Λ is Λ = diag(s1, s2, s3, s4, s5, s6, s7, s8, s9). In order to ensure the mass and momentum conservation after collision, it is easily derived that s1 = s4 = s6 = 0. s8 and s9 are set as s8 = s9 = 1/τ in order to reproduce the same viscosity of SRT model [64]. The other relaxation parameters s2, s3, s5 and s7 can be decided with more ﬂexibility according to the physical model. For D3Q19 model, the matrix M is:

 1111111111111111111       30 11 11 11 11 11 11 8 8 8 8 8 8 8 8 8 8 8 8   − − − − − − −   12 4 4 4 4 4 4111111111111     − − − − − −   0 1 0 0 1 0 0 1 1 0 1 1 0 1 1 0 1 1 0   − − − − −   0 4 0 0 4 0 0 1 1 0 1 1 0 1 1 0 1 1 0   − − − − −     0 0 1 0 0 1 0 1 0 1 1 0 1 1 0 1 1 0 1   − − − − −   0 0 4 0 0 4 0 1 0 1 1 0 1 1 0 1 1 0 1   − − − − −     0 0 0 1 0 0 1 0 1 1 0 1 1 0 1 1 0 1 1   − − − − −   0 0 0 4 0 0 4 0 1 1 0 1 1 0 1 1 0 1 1   − − − − −  =   M  0 2 1 1 2 1 1 1 1 2 1 1 2 1 1 2 1 1 2 . (23)  − − − − − − − −   0 4 2 2 4 2 2 1 1 2 1 1 2 1 1 2 1 1 2   − − − − − −   0 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 1 1 0     − − − − − −   0 0 2 2 0 2 2 1 1 0 1 1 0 1 1 0 1 1 0   − − − − − −   0000000100100 1 0 0 1 0 0     − −   000000000100100 1 0 0 1   − −   00000000100100 1 0 0 1 0     − −   0 0 0 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0   − − − −   0 0 0 0 0 0 0 1 0 1 1 0 1 1 0 1 1 0 1   − − − −   0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1  − − − − ChemEngineering 2020, 4, 55 8 of 33

The equilibrium distribution functions meq in the moment space is:    ρ   2 2 2   11ρ + 19ρ(ux + uy + uz )   −   11 (u2 + u2 + u2)   3ρ 2 ρ x y z   −   ρux     2 u   3 ρ x   −   ρuy     2 u   3 ρ y   −   ρuz     2 ρu   3 z  eq eq  2− 2 2  m = Mf =  ρ(2ux uy uz ) . (24) i  − −   1 (2u2 u2 u2)   2 ρ x y z   − 2 − 2 −   ρ(uy uz )   −   1 ( 2 2)   2 ρ uy uz   − −   ρuxuy       ρuyuz     ρuxuz       0     0     0 

The diagonal matrix Λ is Λ = diag(s1, s2, ...... , s18, s19), where s1 = s4 = s6 = s8 = 1.0, s2 = s5 = s7 = s9 = 1.2, s3 = s11 = s13 = 1.4, s10 = s12 = s14 = s15 = s16 = 1/τ, s17 = s18 = s19 = 1.98.

3.2. Boundary Conditions In the LBM, the conventional velocity and pressure boundary conditions cannot be applied directly, since these quantities are not primary variables in the LBE. Instead, the boundary conditions are imposed with the distribution functions. In this section, only the ‘no-slip’ wall and the periodic boundary conditions are introduced. It should be noted that particles are treated as moving boundaries, which play important roles in the coupling between LBM and DEM. The interactions between the moving particles and the ﬂuid will be discussed separately later. The ‘no-slip’ boundary condition is implemented with the so-called bounce-back rule [65,66]. As shown in Figure2, if fi is a distribution function entering into the solid stationary wall, the undetermined distribution function that comes out of the wall is simply a reﬂection of fi, i.e.,

+ f i(x, t + ∆t) = f (x, t), (25) − i + where f i and fi denote the distribution functions in the opposite direction of i and after collision, − respectively. This simple bounce-back rule ensures a strict ‘no-slip’ condition at the wall position, as the tangential velocity is zero. It should be noted that the accuracy of the simple bounce-back rule is only ﬁrst order, except that the wall is just in the middle of the link between a solid node and a ﬂuid node, where the accuracy is second order. However, the bounce-back rule is easy to implement and can be extended to obstacles with arbitrary shape, which makes the LBM quite suitable for problems with complex geometries. ChemEngineering 2020, 4, 55 9 of 33

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Figure 2. Schematic of the bounce-back rules and periodic boundary conditions for D2Q9 model. Figure 2. Schematic of the bounce-back rules and periodic boundary conditions for D2Q9 model. Periodic boundary conditions are also straightforward to impose in the LBM. As shown in Periodic boundary conditions are also straightforward to impose in the LBM. As shown in Figure2, Figure 2, distribution functions propagating out of the domain stream into the corresponding distributionboundary functions node at the propagating opposite boundary out of the in the domain same direction. stream into Take the Figure corresponding 2 as an example, boundary the inlet node at the oppositeand outlet boundary planes are inperiodic. the same As a direction. result, the Takedistribution Figure functions2 as an example, f1, f5 and f8 the at the inlet outlet and boundary outlet planes are periodic.will stream As into a result, the corresponding the distribution node functions at the inlet,f 1 ,whilef 5 and f3, ff68 andat the f7 at outlet the inlet boundary propagate will into stream the into the correspondingcorresponding node node at at the the outlet. inlet, while f 3, f 6 and f 7 at the inlet propagate into the corresponding node at the outlet. 3.3. Unit Conversion in LBM 3.3. UnitIn Conversion the LBM, inthe LBM physical parameters in the computation are commonly dimensionless, which necessitatesIn the LBM, a unit the conversion. physical In parameters principle, one in on thely needs computation to determine are the commonly conversion coefficients dimensionless, whichof three necessitates fundamental a unit units, conversion. i.e., length, time In principle, and mass, and one then only the needs conversion to determine coefficients the of conversionall the other physical parameters can be sequentially decided. coefficients of three fundamental units, i.e., length, time and mass, and then the conversion coefficients Normally, the lattice space Δx, the time step Δt, and the fluid density ρ are set to be one in the of allLBM the othercomputation. physical How parameters to perform can the be unit sequentially conversion decided.will be explained using an example of pipe flowNormally, in 3D. Suppose the lattice that space the pipe∆x ,size the is time L × D step = 8 ∆mmt, and × 4 mm, the fluidwhere density L is the ρpipeare length set to and be one D is in the LBMthe computation. pipe diameter, How and water to perform flows through the unit this conversion pipe with an will average be explained velocity of using V = 0.0125 an example m/s. First, of pipe flowset in 3D.the number Suppose of thatlattice the as, pipe for instance, size is L nx ×D n=y ×8 n mmz = 100 ×4 50 mm, × 50, where whichL resultsis the pipein a lattice length length and Dof is the × × pipedx diameter, = L/nx = D and/ny = D water/nz = 0.08 flows mm. through Note that this the pipelattice with should an be average square in velocity 2D or cube of V in= 3D,0.0125 indicating m/s. First, set thethat number the lattice of latticelength as,is identical for instance, in everynx dimensny nion.z = Then100 the50 conversion50, which coefficient results in of a length lattice is length decided by Cl = dx/Δx = 8 × 10−5 m. Second,× given× that the density× × of water is ρf = 1000 kg/m3, the of dx = L/nx = D/ny = D/nz = 0.08 mm. Note that the lattice should be square in 2D or cube in 3D, 3 indicatingconversion that coefficient the lattice of length density is is identical Cρ = ρf/ρ = in 1000 every kg/m dimension.. Then the conversion Then the conversioncoefficient of coe massffi cientis of determined by Cm = CρCl3 = 5.12 × 10−10 kg. For better numerical stability, the relaxation time τ shall be length is decided by C = dx/∆x = 8 10 5 m. Second, given that the density of water is ρ = 1000 kg/m3, at least 0.55, implyingl that the kinematic× − viscosity ν shall be at least 0.017. Here let us setf ν = 0.05. the conversion coefficient of density is C = ρ ρ = 1000 kg/m3. Then the conversion coefficient of mass Given that the kinematic viscosity of ρwaterf / is νwater = 1.0 × 10−6 m2/s, the conversion coefficient of 3 10 is determined by Cm = CρC = 5.12−5 102 kg. For better numerical stability, the relaxation time τ shall viscosity is Cν = νwater/ν = l2.0 × 10 ×m /s.− Then the conversion coefficient of time is determined as Ct = be atC leastl2/Cν = 0.55,3.2 × 10 implying−4 s. So far, that the conversion the kinematic coefficients viscosity of theν shall three be fundam at leastental 0.017. units Here are decided. let us setOtherν = 0.05. 6 2 Givenparameters that the kinematicthat need to viscosity be nondimensionalized of water is νwater can =be1.0 converted10− accordingly.m /s, the conversion As a summary, coeffi thecient of 5 2 × viscosityunit conversion is Cν = νwater of this/ν = pipe2.0 flow10 problem− m /s. is Then listed the in Table conversion 1. coefficient of time is determined as 2 4 × Ct = C /Cν = 3.2 10 s. So far, the conversion coefficients of the three fundamental units are decided. l × − Other parameters that need to be nondimensionalized can be converted accordingly. As a summary, the unit conversion of this pipe flow problem is listed in Table1. ChemEngineering 2020, 4, 55 10 of 33

Table 1. Unit conversion of 3D pipe flow.

ChemEngineering 2020, 4, 55 10 of 33 Physical Parameters Unit Real Value Lattice Value Conversion Coefficient

Pipe length (L) m 0.008 100 8.0 × 10−5 Table 1. Unit conversion of 3D pipe flow. Pipe diameter (D) m 0.004 50 8.0 × 10−5 Conversion Physical Parameters Unit Real Value Lattice Value Fluid density (ρf) kg/m3 1000 1 1000Coe fficient 5 Pipe length (L) m 0.008 100 8.0 10− Fluid kinematic viscosity (νf) m2/s 1.0 × 10−6 0.05 2.0 × 10−5 × 5 Pipe diameter (D) m 0.004 50 8.0 10− 3 × Fluid density (ρf) kg/m 1000 1 1000 Average fluid velocity (V) m/s2 0.0125 6 0.05 0.025 5 Fluid kinematic viscosity (νf) m /s 1.0 10 0.05 2.0 10 × − × − AveragePressure fluid velocity drop ( (ΔVP)) m Pa/s 0.2 0.0125 3.2 × 10−3 0.0562.5 0.025 Pressure drop (∆P) Pa 0.2 3.2 10 3 62.5 × − TimeTime step step (∆t) (Δt) s s 3.2 × 103.2−4 10 4 1 1 3.2 × 103.2−4 10 4 × − × −

4.4. LBM-DEM LBM-DEM Coupling Coupling ForFor coupling coupling LBM LBM with with DEM, DEM, it it is is of of significant significant importance importance to to properly properly describe describe the the fluid-solid fluid-solid interactions.interactions. The The first first step step is to is establish to establish a lattice a lattice discretisation discretisation of the of solid the solid particle particle on the on lattice the lattice grid. Asgrid. illustrated As illustrated in Figure in Figure3, four3 types, four of types lattice of no latticedes are nodes obtained are obtained as: (1) the as: pure (1) the fluid pure node fluid without node connectionwithout connection to any pure to any solid pure node; solid (2) node; the pure (2) the solid pure node solid that node is not that adjacent is not adjacent to any pure to any fluid pure node; fluid (3)node; the (3)fluid the boundary fluid boundary node that node is thatlinked is linked with bo withth pure both fluid pure node fluid nodeand solid and solidboundary boundary node; node; and (4)and the (4) solid the solid boundary boundary node node that that is connected is connected with with both both fluid fluid boundary boundary node node and and pure pure solid solid node. node. Obviously,Obviously, it becomes becomes very very easy easy and and straightforward straightforward to to handle handle any any shaped shaped particles particles with with the the discrete discrete latticelattice representation. representation. Hence, Hence, there there was was a a variety variety of of techniques techniques with with different different levels levels of of complexity complexity andand accuracy accuracy to to compute compute the the fluid-solid fluid-solid interact interaction,ion, including including the the modified modified bounce-back bounce-back (MBB) (MBB) schemescheme [65,66], [65,66], interpolated interpolated bounce-back bounce-back (IBB) (IBB) scheme scheme [67–72] [67–72] and and immersed immersed boundary boundary methods methods (IBM)(IBM) [73–81]. [73–81].

FigureFigure 3. 3. TheThe mapping mapping of of a a circular circular solid solid particle particle on on the the lattice lattice grid, grid, showing showing solid solid boundary boundary nodes nodes ((orangeorange),), fluid ﬂuid boundary boundary nodes nodes ( (blueblue),), internal internal solid solid nodes nodes ( (grey)) and and pure pure fluid ﬂuid node ( white).

4.1.4.1. Modified Modified Bounce-Back Bounce-Back Scheme Scheme TheThe MBB MBB scheme scheme was was proposed proposed by by Ladd Ladd [65,66] [65,66] in in order order to to improve improve the the simple simple bounce-back bounce-back rulesrules to to incorporate incorporate the the influence influence of of moving moving fluid-solid fluid-solid interfaces. interfaces. As As shown shown in in Figure Figure 33,, this method assumes that the exact solid surface lies halfway between fluid boundary node and solid boundary ChemEngineering 2020, 4, 55 11 of 33 node, where the bounce-back condition is enforced. The particle is treated as a shell full of fluid. Therefore, the bounce-back takes place on both side of the fluid-solid interface. The first step is to estimate the velocity at the fluid-solid interface based on the solid particle’s velocity, which is given as: 1 v = Up + Ωp (x + e ∆t Xp), (26) b × 2 i − where Xp is centre of the solid particle. Then the undetermined distribution functions at the fluid boundary node and solid boundary node are computed respectively, according to the modified bounce-back rule, + f i(x, t + ∆t) = f (x, t) 2ωiρvb ei, − i − · (27) ( + + ) = +( + ) + fi x ei∆t, t ∆t f i x ei∆t, t 2ωiρvb ei. − · It can be seen that Equation (27) reduces to the simple bounce-back scheme (Equation (25)) when vb = 0. Once the bounce-back procedure is accomplished, the hydrodynamic force on the solid particle at this particular boundary node is calculated with the momentum exchange method,

1 1 ( + + ) = [ +( ) +( + ) ] F f , i x ei∆t, t ∆t 2 fi x, t f i x ei∆t, t 2ωiρvb ei ei. (28) − 2 2 − − − · · The total hydrodynamic force and torque are computed by summing up the contributions from every lattice speed direction and boundary node, P P F f = F f , i, link i − P P 1 (29) M f = (x + 2 ei∆t Xp) F f , i. link i − × −

Apparently, the discrete representation of the solid particle surface on the lattice grid is in a stepwise fashion. It is reported that the accuracy of MBB is of first order only due to the zig-zag staircases of a curved surface. Large fluctuations of the hydrodynamic interactions are also observed. Using a sufficiently high lattice resolution could provide much more smooth and accurate force evaluations, which, however, will lead to a huge increase of the computational cost. To overcome these drawbacks of MBB, several improved techniques were proposed, which will be introduced below.

4.2. Interpolated Bounce-Back Scheme To precisely describe the actual boundary of the solid particle, several bounce-back schemes using the spatial interpolation were also proposed [67–72]. In all these techniques, the relative location of the exact ﬂuid-solid interface is interpolated with a location parameter q. Hence more accurate and smoother evaluations of the hydrodynamic force and torque are obtained, which is demonstrated to maintain the second-order accuracy. Here a double interpolation scheme proposed by Yu et al. [71,72] is introduced to show how these schemes work. As shown in Figure4, the undetermined distribution function is the one coming out of the solid boundary f i(xf, t + ∆t). In most of time, the solid boundary − is not exactly located on a lattice node. Then the relative location of the solid boundary can be described by a weighting parameter q by means of spatial interpolation,

q = x x / x xs , (30) f − b f − where xs, xb and xf represent the positions of the solid node, the solid boundary and the first fluid node next to the solid boundary, respectively. The weighting parameter q varies between 0 and 1. When q = 0, the solid boundary is exactly located on the nearest fluid node, which is turned into a solid node. When q = 1, the solid boundary lies just on the nearest solid node. Then the density function at the temporary location xb that propagates exactly to the solid boundary during the streaming process ChemEngineering 2020, 4, 55 12 of 33 can be interpolated with the existing density functions f (x , t + ∆t) and f (x , t + ∆t), through a first ChemEngineering 2020, 4, 55 i f i ff 12 of 33 order form, f (x , t + ∆t) = q f (x , t) + (1 q) f (x , t). (31) i b +=Δ i f +−− i f f ftib(,xxt ) qft if (,)(1)(,). qft iff x (31) Next, a bounce-back operation takes place instantaneously at the solid boundary as: Next, a bounce-back operation takes place instantaneously at the solid boundary as: f i(xb, t+Δ+ ∆t) = = fi(xb, t + +Δ∆t) −2ωωρiρub e ⋅i, (32) f−ib−(,xx ttf ) ib (, tt )2− i ue· bi , (32) which isis the the same same as as the the MBB MBB described described in Equation in Equati (27).on In(27). the In last the step, last the step, unknown the unknown density functiondensity ffunctioni(xf, t + f∆−it()xf is, t determined + Δt) is determined from f i(xb ,fromt + ∆ tf)−i and(xb, ft i+( xΔff,tt) +and∆t) f through−i(xff, t + aΔ firstt) through order interpolation, a first order − − − interpolation, 1 q f i(x f , t + ∆t) = f i(xb, t + ∆t) + f i(x f f , t + ∆t). (33) − +=1 +1 q − +Δ+1 +qq − +Δ ft−−−(,xxxΔt ) ftt (, ) f ( , tt ). (33) if11++qq ib iff It should be noted that the interpolation can also be performed with second order form, where the densityIt should function be noted from that further the interpolation fluid node xcanfff is also needed. be performed The force andwith torque second on order the solidform, particle where canthe bedensity evaluated function similarly from further with a momentumfluid node x exchangefff is needed. method, The force and torque on the solid particle can be evaluated similarly with a momentum exchange method, P P + F f = [ f+ (x f , t) + f i(x f , t + ∆t)]ei, Fxxe=++Δ[(,)ftftti − (, )], fififiall x f i − allx f P i P + (34) M f = (xb Xp) [ f (x f , t) + f i(x f , t + ∆t)]ei. i + − (34) =−×++Δall x f i − × MxXxxefbpififi()[(,)(,)].ftftt− allx f i

Figure 4. Schematic of the double interpolation scheme.

4.3. Immersed Moving Boundary Method The immersedimmersed boundary boundary method method (IBM) (IBM) was proposed was proposed by Peskin by [73 Peskin,74] and [73,74] further implementedand further inimplemented LBM by other in researchersLBM by other [75 –researchers80]. In this method,[75–80]. extraIn this moving method, Lagrangian extra moving nodes Lagrangian are introduced nodes to representare introduced the solid to represent particle shape, the solid which particle are assumed shape, which to be deformable are assumed with to be a large deformable stiffness. with The a e fflargeects ofstiffness. the immersed The effects particle of the boundary immersed on particle the fluid boun are modelleddary on the by fluid restoring are modelled forces based by restoring on the ‘no-slip’ forces boundarybased on the condition, ‘no-slip’ whichboundary tend condition, to keep the which particle tend to to its keep original the particle shape. to Then its original the restoring shape. forces Then the restoring forces are distributed to their surrounding fluid nodes and considered as the external force terms in the governing equations. The drawbacks of the IBM include the introduction of additional free parameters, the problem-sensitive determination of the spring stiffness and the damping constant, as well as the severely restricted computational time step. ChemEngineering 2020, 4, 55 13 of 33 are distributed to their surrounding fluid nodes and considered as the external force terms in the governing equations. The drawbacks of the IBM include the introduction of additional free parameters, the problem-sensitive determination of the spring stiffness and the damping constant, as well as the severely restricted computational time step. As an alternative, Noble and Torczynski [81] proposed an immersed moving boundary technique based on the local solid fraction in each lattice cell, which is also known as the partially-solid scheme. This method not only overcomes the momentum discontinuity of MBB-based techniques and provides an adequate representation of complex boundaries at relative lower lattice resolutions, but also retains the prominent advantages of the LBM, i.e., the simple linear collision operator and its locality in computation. In this method, the lattice Boltzmann equation is modified to include an additional collision term, which depends on the local volume fraction of solid (see Figure3 right part),

∆t X ( + + ) = ( ) ( )[ ( ( ) eq( ))] + s + ( ) fi x ei∆t, t ∆t fi x, t 1 Bn fi x, t fi x, t BsΩi 1 Bn Fi∆t, (35) − − τ − s − where Bn is a total weighting function in each lattice cell, Bs is the weighting function from each solid s particle in the same lattice cell, and Ωi is the additional collision term. The total weighting function is calculated by summing up the weighting function of each solid particle that intersect with the same P lattice cell such that Bn = s Bs. Each particle’s weighting function is determined by its own solid fraction and the relaxation parameter as:

εs(τ/∆t 0.5) Bs(εs, τ) = − , (36) (1 εn) + (τ/∆t 0.5) − − where εs is the corresponding particle’s solid fraction and εn is total solid fraction in the same lattice P cell that is the sum of each particle’s contribution, i.e., εn = s εs. It can be seen that when the solid fraction εs varies from 0 (a completely fluid cell) to 1 (a completely solid cell), Bs varies from 0 to 1. Equation (35) returns to the original SRT-LBE for pure fluid when Bs = 0, and returns the new collision s operator Ωi plus the distribution from the previous time step when Bs = 1. The new collision operator is given by s = ( ) eq( ) + eq( ) ( ) Ωi f i x, t f i ρ, u fi ρ, Up fi x, t . (37) − − − − The total hydrodynamic force and torque acting on the solid particle can be evaluated using the momentum exchange method with the additional collision operator over all lattice directions at each node and then over all fluid boundary, solid boundary and internal solid nodes, which are expressed as:

P P s F f = Bs( Ωi ei), − n i P P s (38) T f = [(xn Xp) Bs( Ωi ei)]. − n − × i

Note that the implementation of the immersed moving boundary method is straightforward. Only quantities already available in the lattice cell or easily derived are used. No additional data storage or organization is needed, which is a crucial advantage over other moving boundary techniques. However, it is also noticed that the critical step in this method is the calculation of the solid fraction in each lattice cell, which is primarily a geometry problem. Many methods are proposed to estimate the solid coverage ratio of a geometry with a square or a cube, including analytically solution, Monte Carlo sampling technique, cell decomposition method, polygonal approximation, edge-intersection averaging method, linear approximation and so on [82,83]. In the current LBM-DEM numerical framework, the analytical solution is used for 2D problems, while for 3D a linear approximation method is adopted, in order to save computational cost but still retain relatively accurate results. ChemEngineering 2020, 4, 55 14 of 33

4.4. Time Steps in the LBM-DEM Coupling The time step coupling is also a crucial issue in the LBM-DEM coupling [31,82]. As we discussed previously, the time step of LBM is generally set as a dimensionless value of one in the computation. The real physical time is implicitly determined by computational parameters as well as the unit conversion scheme. Based on Li and Marshall’s work [3,58], the DEM time step depends on the interparticle collision time scale, 2 1/5 ρp t R( ) , (39) c 2 ≈ E vR 6 9 which typically varies around 10− ~ 10− s. Therefore, it is believed that the LBM time step is generally larger than the DEM time step. A time step ratio λ is then introduced as ∆tLBM = λ∆tDEM, which can be decided after all the parameters are settled. It should be noted that the time step ratio λ could be either greater or lower than one. If λ < 1, the critical time step can be simply set as ∆tLBM, and the DEM time step can be equal to ∆tLBM. When λ > 1, a number of DEM computation steps are performed within one LBM time step, during which the ﬂuid forces and torques are kept unchanged.

5. Validation In this section, a few validation studies are performed to demonstrate the accuracy of our LBM-DEM numerical approach, including single-phase Poiseuille ﬂow, gravitational settling of a particle, and the drag force on a stationary particle.

5.1. Poiseuille Flow Single-phase Poiseuille flows in both 2D and 3D are first analysed using the LBM-DEM. The validation is performed with both SRT and MRT models. As shown in Figure5, the boundaries in are periodic, and y-direction are bounded by two ‘no-slip’ walls. A constant pressure gradient is imposed in the channel in x-direction to drive the flow. The channel size is H H for the 2D case and × 10 H H for the 3D case. A few lattice resolutions are selected in the range H = 11 ~ 101 to examine × × the convergence of the numerical model. The relaxation parameter is fixed at 0.65. Different channel Reynolds numbers are achieved by changing the magnitude of the body force. Figures6 and7 show the normalised velocity profiles for 2D and 3D Poiseuille flow, respectively. The L-2 norm calculation is used to compute the relative error: v tu 2 X (U Utheory) relative error = − , (40) 2 n Utheory where U is the flow velocity obtained from the simulation and Utheory is the analytical solution of the Poiseuille flow [84]. It can be inferred that the numerical results agree very well with the theoretical prediction. The accuracy is second order for the 2D flow, while it is slightly lower than second order for the 3D duct flow. ChemEngineeringChemEngineering 20202020,, 44,, 5555 1515 ofof 3333 ChemEngineering 2020, 4, 55 15 of 33 ChemEngineering 2020, 4, 55 15 of 33

FigureFigure 5. Schematic5. Schematic of of single-phase single-phase Poiseuille Poiseuille flow ﬂow in in2D. 2D. FigureFigure 5.5. SchematicSchematic ofof single-phasesingle-phase PoiseuillePoiseuille flowflow inin 2D.2D.

Figure 6. (a) Normalised velocity profile for 2D Poiseuille flow with channel size 101 × 101. (b) The relative error of the fluid velocity as a function of the lattice resolution. FigureFigureFigure 6.6. 6. ((a(aa)) ) NormalisedNormalised Normalised velocityvelocity velocity profileprofile profile forfor for 2D2D 2D PoiseuPoiseu Poiseuilleilleille flowflow flow withwith with channelchannel channel sizesize size 101101 ×× 101.101.101. ((b (bb)) ) TheThe The × relativerelativerelative errorerror error ofof of thethe the fluiflui fluiddd velocityvelocity velocity asas as aa a function function ofof of the the lattice lattice resolution. resolution.

Figure 7. (a) Normalised velocity profile for 3D duct flow with channel size 10 × 51 × 51. The velocity profile is along the vertical direction in the mid-plane in y-direction. (b) The relative error of the fluid velocity as a function of the lattice resolution.

FigureFigureFigure 7.7. 7. ((a(aa)) ) NormalisedNormalised Normalised velocityvelocity velocity profileprofile profile forfor for 3D3D 3D ductduct duct flowflow flow withwith channelchannel size 10 ×× 5151 ×× 51.51.51. TheThe The velocityvelocity velocity × × profileprofileprofile isis is alongalong along thethe the verticalvertical vertical directiondirection direction inin in thethe the mid-planemid-plane mid-plane inin in yyy-direction.-direction.-direction. ((b (bb)) ) TheThe The relativerelative relative errorerror error ofof of thethe the fluidfluid fluid velocityvelocityvelocity asas as aa a functionfunction function ofof of thethe the latticelattice lattice resolution.resolution. resolution. ChemEngineering 2020, 4, 55 16 of 33 ChemEngineering 2020, 4, 55 16 of 33

5.2. Gravitational Se Settlingttling of a Particle For gravitationalgravitational settling settling of of a particle,a particle, both both 2D and2D 3Dand cases 3D cases are simulated are simulated to evaluate to evaluate the validity the ofvalidity our numerical of our numerical approach approach in a dynamic in a dynamic system. system. This validation This validation is performed is performed with SRT-LBM with SRT-LBM along withalong the with Hertz the model Hertz inmodel DEM. in For DEM. the 2D For case, the the2D computationalcase, the computational setup is identical setup is to identical that in Wen to that et al.’s in workWen et [85 al.’s]. As work shown [85]. in As Figure shown8, ain cylinder Figure 8, with a cylinder diameter withd pdiameter= 0.1 cm d isp = initially0.1 cm is located initially 0.076 located cm away0.076 cm from away the leftfrom wall the of left a verticalwall of channela vertical with channel width with 0.4 cm.width The 0.4 fluid cm. densityThe fluid and density kinematic and viscositykinematic are viscosity 1000 kg are/m3 1000and 1kg/m103 and6m2 /1s, × respectively. 10−6m2/s, respectively. The mass density The mass of the density cylinder of the is1030 cylinder kg/m is3. × − Therefore,1030 kg/m the3. Therefore, cylinder willthe settlecylinder under will the settle gravitational under the force gravitational and finally force reaches and a finally steady reaches state that a movessteady alongstate that the centrelinemoves along at a constantthe centreline velocity. at a The constant computational velocity. setupThe computational in the lattice unit setup scheme in the is aslattice follows. unit Thescheme size ofis as the follows. channel The is n xsizen yof= the121 channel1201 and is n ‘no-slip’x × ny = 121 wall × boundaries1201 and ‘no-slip’ are imposed wall × × onboundaries all the four are faces. imposed The cylinder’son all the diameterfour faces. and The mass cylinder’s density diameter are dp = 30and and massρp = density1.03, respectively. are dp = 30 Theand dimensionlessρp = 1.03, respectively. relaxation The parameter dimensionless is set relaxation as 0.6. Note parameter that the IBMis set is as applied 0.6. Note in thethat solid–fluid the IBM is couplingapplied in in the this solid–fluid particular coupling problem. in this particular problem. A detailed comparison between the present numericalnumerical results and that obtained by the arbitrary Lagrangian–Eulerian technique (ALE) [85] [85] is presen presentedted in Figure 99,, wherewhere veryvery goodgood agreementagreement isis reached. Furthermore, the force profilesprofiles areare quitequite smoothsmooth withoutwithout anyany largelarge fluctuations.fluctuations.

Figure 8. Schematic of a single particle settling under gravity. ChemEngineering 2020, 4, 55 17 of 33

ChemEngineering 2020, 4, 55 17 of 33

FigureFigure 9.9.Time-dependent Time-dependent (a )(a particle) particle trajectory, trajectory, (b) angular(b) angular velocity, velocity, (c) horizontal (c) horizontal velocity, velocity, (d) vertical (d) velocity,vertical velocity, (e) horizontal (e) horizontal force and force (f) vertical and (f) force. vertical force.

ForFor thethe 3D 3D case, case, the the single single particle particle settling settling experiment experiment reported reported by Ten Cateby Ten et al. Cate [86] is et numerically al. [86] is 3 3 reproduced.numerically reproduced. A spherical particleA spherical with particle diameter withdp =diameter15 mm anddp = 15 density mm andρp = density1,120 kg ρ/pm = 1,120is initially kg/m releasedis initially from released a height fromh a= height120 mm h = in120 a mm cuboid in a box cuboid with box a size with of a 100size of100 100 × 160100 mm× 1603. mm The3. boxThe × × isbox full is offull silicon of silicon oil, oil, and and four four diff differenterent types types of oilof oil are are used used in in the the experiment. experiment. The The densitiesdensities andand 3 dynamicdynamic viscositiesviscosities ofof thethe siliconsilicon oiloil areare ρρf == 970, 965,965, 962,962, 960960 kgkg/m/m 3 and ννf = 373, 212, 113, 58 PaPa·s,s, f f · respectively.respectively. TheThe terminalterminal settlingsettling velocityvelocity ofof thethe particleparticle inin thethe fourfour oilsoils are areu uterter == 0.038, 0.060, 0.091, 0.1280.128 mm/s,/s, respectively,respectively, whichwhich resultsresults inin fourfour didifferentfferent particleparticle ReynoldsReynolds numbernumberRe Repp == 1.5, 4.1, 11.6, 31.9,31.9, respectively.respectively. The computational parameters in the dimensionless lattice unit unit are are given below. TheThe domaindomain sizesize isis 5151 × 5151 × 81 and the particleparticle diameterdiameter isisd dpp = 7.5. The The density density of the fluid fluid is fixedfixed atat × × ρρff == 1.0, so that the correspondingcorresponding particleparticle densitydensity isis ρρpp = 1.155, 1.161, 1.164, and 1.167 for didifferentfferent fluidsfluids considered,considered, respectively.respectively. The relaxation parameterparameter is setset asas 0.65.0.65. FigureFigure 10 showsshows thethe timetime evolutionevolution ofof particleparticle trajectorytrajectory andand velocityvelocity profilesprofiles forfor didifferentfferent ReynoldsReynolds numbers.numbers. ItIt isis apparentapparent thatthat thethe numericalnumerical resultsresults areare inin excellentexcellent agreementagreement withwith thethe experimentalexperimental results,results, whichwhich furtherfurther demonstratesdemonstrates thethe validityvalidity andand accuracyaccuracy ofof thethe presentpresent numericalnumerical approach.approach. ChemEngineering 2020, 4, 55 18 of 33

ChemEngineering 2020, 4, 55 18 of 33

Figure 10. (a) Particle trajectory and (b) velocity profiles as a function of time.

5.3. Drag Force on a Stationary Particle FigureFigure 10. 10. ((aa)) Particle Particle trajectory trajectory and and ( (bb)) velocity velocity profiles profiles as as a a function function of of time. time. The classical drag force problem is also simulated to verify the accuracy of the force evaluation in5.3.5.3. our Drag Drag LBM-DEM Force Force on on a a coupling,Stationary Stationary whichParticle Particle is carried out with the SRT-LBM. As depicted in Figure 11, a circular (spherical) particle is placed in the centre of a rectangular (cuboid) domain and kept TheThe classical drag forceforce problemproblem isis alsoalso simulated simulated to to verify verify the the accuracy accuracy of of the the force force evaluation evaluation in stationary. The domain size is L × H = 50dp × 50dp in 2D and L × H × H = 20dp × 10dp × 10dp in 3D. The inour our LBM-DEM LBM-DEM coupling, coupling, which which is carried is carried out without with the SRT-LBM. the SRT-LBM. As depicted As depicted in Figure in 11Figure, a circular 11, a inlet boundary is constant flow with velocity U0 and the outlet is set as constant pressure boundary. (spherical) particle is placed in the centre of a rectangular (cuboid) domain and kept stationary. Allcircular the other (spherical) boundaries particle are setis asplaced open boundaryin the centre (zero of gradient). a rectangular The particle (cuboid) Reynolds domain number and kept and stationary.The domain The size domain is L Hsize= is50 Ld p× H50 = d50p indp × 2D 50 anddp in L2D Hand LH ×= H20 ×d Hp = 1020dp × 1010ddp p× in10d 3D.p in The 3D. inletThe the drag coefficient are× calculated ×as × × × × inletboundary boundary is constant is constant flow withflow velocitywith velocityU0 and U the0 and outlet the outlet is set asis constantset as constant pressure pressure boundary. boundary. All the other boundaries are set as open boundary (zeroUd gradient).0 p The particle Reynolds number and the All the other boundaries are set as open boundaryRe= (zero ,gradient). The particle Reynolds number and thedrag drag coe fficoefficientcient are calculatedare calculated as as p ν U0dp (41) Rep = , Ud8νFd = 08Fdp (41) CdC = 22, , Repd ρπρ U2πd ,2 fpUdfν 00 p (41) where Fd is the fluid drag force on the particle. Note8F that both the IBB and IBM are used in the drag where Fd is the fluid drag force on the particle.C = Note thatd both, the IBB and IBM are used in the drag forceforce validation. validation. d ρπ22 fpUd0 where Fd is the fluid drag force on the particle. Note that both the IBB and IBM are used in the drag force validation.

FigureFigure 11. 11. SchematicSchematic of of flow ﬂow past past a a stationary stationary particle. particle.

FigureFigure 12 showsshows thethe drag drag coe coefficientfficient C dCasd as a functiona function of theof the particle particle Reynolds Reynolds number numberRep for Rep both for both2D and 2D 3Dand cases. 3D cases. For the ForFigure 2D the case 2D11. shown Schematiccase shown in Figureof flowin Figure 12pasta, thea stationary12a, experimental the experimentalparticle. results results reported reported by Tritton by Trittonare also are superimposed also superimposed for comparison for comparison [87]. It is[87]. clear It thatis clear both that the both results the obtained results obtained by IBB and by IBMIBB andagree IBMFigure well agree with12 showswell the experiments.with the the drag experi coefficient Secondments. orderSecondCd as accuracya orderfunction accuracy of of the the force ofparticle the evaluation force Reynolds evaluation is achieved number is achieved for Re IBM,p for forbothwhile IBM, 2D the whileand accuracy 3D the cases. accuracy for IBBFor isthefor lower IBB2D iscase than lower shown second than in se orderFigurecond butorder 12a, higher butthe higherexperimental than firstthan order. first results order. For reported the For 3D the case 3Dby Tritton are also superimposed for comparison [87]. It is clear that both the results obtained by IBB and IBM agree well with the experiments. Second order accuracy of the force evaluation is achieved for IBM, while the accuracy for IBB is lower than second order but higher than first order. For the 3D ChemEngineering 2020,, 4,, 55 55 1919 of of 33 33 case shown in Figure 12a,b widely accepted empirical law of the drag coefficient is introduced to serveshown as in the Figure theoretical 12a,b widelyprediction accepted [88] empirical law of the drag coefficient is introduced to serve as the theoretical prediction [88] 2424 =+= ( + 0.6870.687) CCdpd (11 0.15Re0.15Rep . ). (42) (42) Rep Rep We can see that the simulation results agree well with the above theoretical prediction, We can see that the simulation results agree well with the above theoretical prediction, which which demonstrates the validity and accuracy of our coupled LBM-DEM. Furthermore, it is noticed demonstrates the validity and accuracy of our coupled LBM-DEM. Furthermore, it is noticed that that accurate drag coefficients are obtained using both IBB and IBM even with a relatively low size accurate drag coefficients are obtained using both IBB and IBM even with a relatively low size resolution dp = 6, implying that acceptable accuracy can still be retained when the computational cost resolution dp = 6, implying that acceptable accuracy can still be retained when the computational cost is reduced. is reduced.

FigureFigure 12. 12. DragDrag coefficient coeﬃcient as as a a function function of of particle particle Reynolds Reynolds number number for for ( (a)) 2D 2D and and ( (b)) 3D 3D cases. cases.

6.6. LBM-DEM LBM-DEM Applications Applications InIn this section,section, aa few few numerical numerical examples examples are are presented presented to demonstrateto demonstrate the capabilitythe capability of LBM-DEM, of LBM- DEM,which which include include inertial inertial migration migration of dense of particle dense suspensions, particle suspensions, agglomeration agglomeration of adhesive of particles adhesive in particleschannel flow,in channel and sedimentation flow, and sedimentation of particle suspensions of particle suspensions in a cavity flow. in a cavity flow. 6.1. Inertial Migration of Dense Particle Suspensions 6.1. Inertial Migration of Dense Particle Suspensions In an experimental study of pipe flow of a dilute suspension consisting of neutrally buoyant In an experimental study of pipe flow of a dilute suspension consisting of neutrally buoyant particles, Segré and Silberberg [89,90] first observed that the particle suspensions tend to migrate laterally particles, Segré and Silberberg [89,90] first observed that the particle suspensions tend to migrate and focus in an annulus at radial position r 0.6R, with R being the pipe radius. This phenomenon laterally and focus in an annulus at radial≈ position r ≈ 0.6R, with R being the pipe radius. This is named as the tubular pinch effect (or the Segré–Silberberg effect). Due to the lack of theoretical phenomenon is named as the tubular pinch effect (or the Segré–Silberberg effect). Due to the lack of explanation at the time when it was discovered, this interesting phenomenon prompted a great deal of theoretical explanation at the time when it was discovered, this interesting phenomenon prompted a interest to further explore the underlying mechanism. Since then, many studies were carried out on this great deal of interest to further explore the underlying mechanism. Since then, many studies were fascinating phenomenon experimentally [91–95], theoretically [96–99] and computationally [100–106]. carried out on this fascinating phenomenon experimentally [91–95], theoretically [96–99] and The lateral focusing of the particle suspensions is found to be closely related to the fluid inertia. computationally [100–106]. The lateral focusing of the particle suspensions is found to be closely It is well recognised that the lateral migration exists in both 2D and 3D flows, and the equilibrium related to the fluid inertia. It is well recognised that the lateral migration exists in both 2D and 3D position moves closer to the wall as the channel Reynolds number increases, which is successfully flows, and the equilibrium position moves closer to the wall as the channel Reynolds number described with the theoretical solution based on the perturbation theory and the asymptotic expansion increases, which is successfully described with the theoretical solution based on the perturbation method. However, most of the previous works are limited to very dilute suspensions with a particle theory and the asymptotic expansion method. However, most of the previous works are limited to concentration lower than 1%. When the particle concentration is increased, whether the lateral focusing very dilute suspensions with a particle concentration lower than 1%. When the particle concentration phenomenon still exists remains less explored. Therefore, the coupled LBM-DEM is employed in the is increased, whether the lateral focusing phenomenon still exists remains less explored. Therefore, current study to explore this problem. the coupled LBM-DEM is employed in the current study to explore this problem. As shown in Figure 13, let us consider a pressure-driven flow of non-Brownian particle suspensions As shown in Figure 13, let us consider a pressure-driven flow of non-Brownian particle in a 2D channel. Periodic boundary conditions are applied in x-direction and the top and bottom suspensions in a 2D channel. Periodic boundary conditions are applied in x-direction and the top and planes in y-direction are set as ‘no-slip’ walls. The particles are neutrally buoyant, i.e., the fluid and the bottom planes in y-direction are set as ‘no-slip’ walls. The particles are neutrally buoyant, i.e., the particle have the same mass density. The particles’ initial positions are randomly generated inside the fluid and the particle have the same mass density. The particles’ initial positions are randomly generated inside the channel at the beginning of the simulation. Driven by the pressure gradient, the ChemEngineering 2020, 4, 55 20 of 33 channel at the beginning of the simulation. Driven by the pressure gradient, the particle suspensions ChemEngineering 2020, 4, 55 20 of 33 will transport and migrate to their equilibrium positions. A steady particle–fluid flow is expected after a suffiparticleChemEngineeringcient longsuspensions time 2020, of4, 55will computation. transport and The migrate parameters to their usedequilibrium in the positions. simulation A steady are as particle–fluid follows.20 Theof 33 size of theflow channel is expected is L afterH = a501 sufficient101. long The particletime of computation. diameter is dThep = parameters6 and 12. Theused particle in the simulation concentration × × are as follows. The size of the channel is L × H = 501 × 101. The particle diameter is dp = 6 and 12. The φ rangesparticle between suspensions 1% and will 50%. transport The and channel migrate Reynolds to their equilibrium number Re positions.0 is tuned A steady in the particle–fluid range 4 ~ 100 by varyingparticleflow the is pressureexpectedconcentration gradient.after ϕa rangessufficient Note between thatlong thetime 1% channel andof computation. 50%. Reynolds The channel The number parameters Reynolds refers number used to the in Re onethe0 issimulation under tuned ain single the range 4 ~ 100 by varying the pressure gradient. Note that the channel Reynolds number refers to phaseare flow as follows. with no The particles. size of the The channel SRT model is L × H and = 501 the × 101. Hertz The contact particle model diameter are is used dp = 6 inand the 12. LBM The and theparticle one under concentration a single phaseϕ ranges flow between with no 1% particles. and 50%. The The SRT channel model Reynoldsand the Hertz number contact Re0 ismodel tuned are in DEM for this particular problem, respectively. usedthe range in the 4 LBM ~ 100 and by varyingDEM for the this pressure particular gradient. problem, Note respectively. that the channel Reynolds number refers to the one under a single phase flow with no particles. The SRT model and the Hertz contact model are used in the LBM and DEM for this particular problem, respectively.

FigureFigure 13. 13.Schematic Schematic of of pressure-driven pressure-driven flow ﬂow of of dense dense particle particle suspensions. suspensions.

FigureFigure 14 displays14 displaysFigure the 13. the Schematic snapshots snapshots of pressure-driven ofof thethe particleparticle flow suspension suspension of dense particle with with different suspensions. different channel channel Reynolds Reynolds numbers,numbers, where where the the particle particle concentration concentration is is fixedfixed at ϕφ == 10%.10%. It Itis isobserved observed that that only only a small a small part of part of the particlesFigure 14undergoes displays thethe lateralsnapshots migration of the at particle a relatively suspension small Reynoldswith different number channel Re0 = 4.Reynolds As Re0 the particles undergoes the lateral migration at a relatively small Reynolds number Re0 = 4. As Re0 increases,numbers, awhere complete the particle migration concentration takes place isat fixed both atRe ϕ0 = = 4010%. and It Reis 0observed = 100, where that allonly the a particlessmall part are of increases, a complete migration takes place at both Re0 = 40 and Re0 = 100, where all the particles are focusedthe particles at a certain undergoes lateral the position. lateral Furthemigrationrmore, at ait relativelyis found out small that Reynolds the larger number Re0 is, the Re shorter0 = 4. As time Re0 focused at a certain lateral position. Furthermore, it is found out that the larger Re4 is, the shorter time itincreases, takes to developa complete into migration a full migration. takes place For example,at both Re it0 =occurs 40 and around Re0 = 100, t = 5.5 where × 10 all for0 the Re 0particles = 40, while are 4 4 it takesitfocused is to much develop at aearlier certain into around alateral full migration. position. t = 4 × Furthe10 For for example,rmore, Re0 = it100. is it found occursFigure out around15 that illustrates thet =larger5.5 the Re10 0 effectsis,for theRe shorterof0 =particle40, time while it 4 × is muchconcentrationit takes earlier to develop around on the intot =particle4 a full10 migration.migration.for Re0 = ItFor100. is clearexample, Figure that 15withit occurs illustrates relatively around the low t e= ff particle5.5ects × 10 of 4concentrationparticle for Re0 = concentration40, while (ϕ = × on the1%it particle isand much 10%), migration. earlier the lateral around It migration is cleart = 4 that× is 10 still with4 for notable. relativelyRe0 = However,100. lowFigure particleas 15ϕ increasesillustrates concentration to the40%, effects the (φ migration= of1% particle and is 10%), the lateraldramaticallyconcentration migration suppressed on the is stillparticle and notable. themigration. particle However, It suspensions is clear as thatφ increases appear with relatively to tojam 40%, near low thethe particle wall. migration concentration is dramatically (ϕ = suppressed1% and and 10%), the the particle lateral migration suspensions is still appear notable. to jamHowever, near theas ϕ wall. increases to 40%, the migration is dramatically suppressed and the particle suspensions appear to jam near the wall.

Figure 14. Snapshots of the migration process of particle suspensions with different channel Reynolds

numbers, (a) Re0 = 4, (b) Re0 = 40, and (c) Re0 = 100. The particle diameter is dp = 6 and the concentration FigureisFigure ϕ 14. = 10%.Snapshots 14. Snapshots of theof the migration migration process process ofof particle suspensions suspensions with with different diﬀerent channel channel Reynolds Reynolds numbers, (a) Re0 = 4, (b) Re0 = 40, and (c) Re0 = 100. The particle diameter is dp = 6 and the concentration numbers, (a) Re0 = 4, (b) Re0 = 40, and (c) Re0 = 100. The particle diameter is dp = 6 and the concentration is ϕ = 10%. is φ = 10%.

To quantify the influence of the particle concentration, the degree of inertial migration can be defined as [94], X 1 P = PDF( y H), (43) f ≥ 2 which estimates the total probability distribution function (PDF) of the particles in the upper and lower quarter of the channel. Since the lateral equilibrium position is around 0.6 according to the Segré and Silberberg effect, the degree of the migration must be P = 1 if all the particles are laterally f ChemEngineering 2020, 4, 55 20 of 33

particle suspensions will transport and migrate to their equilibrium positions. A steady particle–fluid flow is expected after a sufficient long time of computation. The parameters used in the simulation are as follows. The size of the channel is L × H = 501 × 101. The particle diameter is dp = 6 and 12. The particle concentration ϕ ranges between 1% and 50%. The channel Reynolds number Re0 is tuned in the range 4 ~ 100 by varying the pressure gradient. Note that the channel Reynolds number refers to the one under a single phase flow with no particles. The SRT model and the Hertz contact model are used in the LBM and DEM for this particular problem, respectively.

Figure 13. Schematic of pressure-driven flow of dense particle suspensions.

Figure 14 displays the snapshots of the particle suspension with different channel Reynolds numbers, where the particle concentration is fixed at ϕ = 10%. It is observed that only a small part of the particles undergoes the lateral migration at a relatively small Reynolds number Re0 = 4. As Re0 increases, a complete migration takes place at both Re0 = 40 and Re0 = 100, where all the particles are focused at a certain lateral position. Furthermore, it is found out that the larger Re0 is, the shorter time it takes to develop into a full migration. For example, it occurs around t = 5.5 × 104 for Re0 = 40, while it is much earlier around t = 4 × 104 for Re0 = 100. Figure 15 illustrates the effects of particle concentration on the particle migration. It is clear that with relatively low particle concentration (ϕ = 1% and 10%), the lateral migration is still notable. However, as ϕ increases to 40%, the migration is dramatically suppressed and the particle suspensions appear to jam near the wall.

ChemEngineering 2020, 4, 55 21 of 33

focused. On the other hand, if the particle suspension remains randomly distributed in the channel, ChemEngineeringPf is expected 2020 to be, 4, 0.5.55 Figure 16 shows the variation of the degree of inertial migration with particle21 of 33 concentration for different dp and Re0. It is observed that Pf equals one for all the Re0 at a low particle concentrationFigure 15. Snapshots (φ 5%), of while the particleP remains positions around with 0.5 different ~ 0.6 at particleφ = 50% concentrations, for all the Re (a) ,ϕ implying = 1%, (b) aϕ huge ≤ f 0 impact= 10%, of the(c) ϕ particle = 40%. The concentration particle diameter on theis dp lateral = 6 and migration.all the snapshots A full are migrationcaptured at isthe always steady state. accessible when the particle suspension is not dense, but it is completely suppressed with a very large particle To quantify the influence of the particle concentration, the degree of inertial migration can be concentration for the range of Re0 in the present study. However, when φ is between 5% and 50%, definedFigure as [94], 14. Snapshots of the migration process of particle suspensions with different channel Reynolds the variation of Pf strongly depends on Re0. With a fixed φ, a larger Re0 leads to a larger Pf, i.e., a higher numbers, (a) Re0 = 4, (b) Re0 = 40, and (c) Re0 = 100. The particle diameter is dp = 6 and the concentration degree of lateral migration. 1 is ϕ = 10%. =≥ (43) PPDFyHf  (), 2 which estimates the total probability distribution function (PDF) of the particles in the upper and lower quarter of the channel. Since the lateral equilibrium position is around 0.6 according to the Segré and Silberberg effect, the degree of the migration must be Pf = 1 if all the particles are laterally focused. On the other hand, if the particle suspension remains randomly distributed in the channel, Pf is expected to be 0.5. Figure 16 shows the variation of the degree of inertial migration with particle concentration for different dp and Re0. It is observed that Pf equals one for all the Re0 at a low particle concentration (ϕ ≤ 5%), while Pf remains around 0.5 ~ 0.6 at ϕ = 50% for all the Re0, implying a huge impact of the particle concentration on the lateral migration. A full migration is always accessible when the particle suspension is not dense, but it is completely suppressed with a very large particle concentrationFigure 15. forSnapshots the range of theof Re particle0 in the positions present with study. different However, particle when concentrations, ϕ is between (a) φ 5%= 1%, and (b )50%,φ = the 10%, (c) φ = 40%. The particle diameter is d = 6 and all the snapshots are captured at the steady state. variation of Pf strongly depends on Re0. Withp a fixed ϕ, a larger Re0 leads to a larger Pf, i.e., a higher degree of lateral migration.

FigureFigure 16. The degreedegree of of inertial inertial migration migrationPf asPf aas function a function of particle of particle concentration concentration for diﬀ forerent different particle particlesizes dp sizesand channeldp and channel Reynolds Reynolds numbers numbersRe0. The Re data0. The was data originally was originally reported reported in [37]. in [37].

TheThe above above discussion discussion reveals reveals that that the the degree degree of of the the migration migration is is determined determined by by the the particle particle concentrationconcentration and and channel channel Reynolds Reynolds number. number. A A dimensionless dimensionless focusing focusing number number was was proposed proposed to to characterisecharacterise the the degree degree of of migration migration [37], [37], Rem = Re00 FFc = ,, (44) (44) c φφnn wherewhere mm == 0.360.36 and and nn == 2.332.33 are are obtained obtained from from the the fitting ﬁtting of of simulation data. Figure Figure 1717 showsshows the the degreedegree of of migration migration PPf asf as a afunction function of of the the focusing focusing number number Fc,F wherec, where an anempirical empirical fitting ﬁtting is given is given by −×0.0043 (logF )6.51 =− × c an exponential equation Pf 10.510 . It can be seen that the numerical results coalesce onto a single master curve and three distinct regimes can be identified based on the value of Fc: a laterally unfocused regime (Fc < 30), a partially migration regime (30 < Fc < 300) and a fully migration regime (Fc > 300). ChemEngineering 2020, 4, 55 22 of 33

6.51 by an exponential equation P = 1 0.5 10 0.0043 (log Fc) . It can be seen that the numerical results f − × − × coalesce onto a single master curve and three distinct regimes can be identiﬁed based on the value of Fc: a laterally unfocused regime (Fc < 30), a partially migration regime (30 < Fc < 300) and a fully migration regime (Fc > 300). ChemEngineering 2020, 4, 55 22 of 33

Figure 17.17. TheThe degreedegree of of inertial inertial migration migration as a as function a func oftion the of focusing the focusing number. number. The data The was data originally was originallyreported in reported [37]. in [37].

6.2. Agglomeration Agglomeration of of Adhesive Adhesive Particles Particles in in Channel Channel Flow Flow The inertial focusing introduced in in the prev previousious section has prompted a variety of novel applications in in the the microfabrication, microfabrication, such such as particle as particle filtration, filtration, segregation segregation and flow and cytometry flow cytometry in the microin the channels micro channels [107–109]. [107 An–109 inevitable]. An inevitable problem problem of particle of particle agglomeration agglomeration due to the due cohesive to the cohesive nature whennature the when size the reduces size reduces to the tomicron the micron scale scalemust must be taken be taken into into consideration. consideration. The The agglomeration agglomeration is ubiquitousis ubiquitous in in nature nature and industry,industry, and and sometimes sometimes will will cause cause unfavourable unfavourable effects effects in industrial in industrial facility. facility.For example, For example, the agglomeration the agglomeration of asphaltenes of asph or gasaltenes hydrates or gas results hydrates in the pipelineresults in blockage the pipeline in the blockageoil and gas in engineeringthe oil and gas [110 engineering,111]. Therefore, [110,111]. it is Th wortherefore, investigating it is worth the investigating fundamental the mechanism fundamental of mechanismmicro-particle of micro-part agglomeration.icle agglomeration. As displayed in Figure 18,18, a stream of dilute particle suspension flowing flowing through a 3D square channel is considered. Periodic Periodic boundary boundary conditions conditions are are imposed imposed at at the the inlet inlet and and outlet, outlet, while while other other fourfour faces faces of of the the channel channel are are set set as as ‘no-slip’ ‘no-slip’ walls. walls. The The flow flow is is driven driven by by a a co constantnstant pressure pressure gradient. gradient. The particles are neutrally buoyant and adhesive, with initial positions randomly distributed in the channel. The computational parametersparameters areare set set as as follow. follow. The The dimension dimension of of the the channel channel is Lis L H× H H× H= × × =201 201 ×61 61 ×61. 61. By By varying varying the the pressure pressure gradient, gradient, the the channel channel Reynolds Reynolds number number ranges ranges between between 5.4 and5.4 × × and108. 108. The particleThe particle size issizedp =is5 d andp = 5 theand particle the particle concentration concentration is fixed is atfixed 1%, correspondingat 1%, corresponding to a particle to a particlenumber number of 110. Theof 110. strength The strength of the particle of the adhesionparticle adhesion is represented is represented by the surface by the energy,surfacewhich energy, is whichchosen is as chosen 0.075 andas 0.075 0.0075. and The 0.0075. SRT modelThe SRT and model the JKR and contact the JKR theory contact are theory used inare the used LBM in andthe DEMLBM andfor this DEM problem, for this respectively.problem, respectively.

Figure 18. Schematic of agglomeration of adhesive particles in channel flow. ChemEngineering 2020, 4, 55 22 of 33

Figure 17. The degree of inertial migration as a function of the focusing number. The data was originally reported in [37].

6.2. Agglomeration of Adhesive Particles in Channel Flow The inertial focusing introduced in the previous section has prompted a variety of novel applications in the microfabrication, such as particle filtration, segregation and flow cytometry in the micro channels [107–109]. An inevitable problem of particle agglomeration due to the cohesive nature when the size reduces to the micron scale must be taken into consideration. The agglomeration is ubiquitous in nature and industry, and sometimes will cause unfavourable effects in industrial facility. For example, the agglomeration of asphaltenes or gas hydrates results in the pipeline blockage in the oil and gas engineering [110,111]. Therefore, it is worth investigating the fundamental mechanism of micro-particle agglomeration. As displayed in Figure 18, a stream of dilute particle suspension flowing through a 3D square channel is considered. Periodic boundary conditions are imposed at the inlet and outlet, while other four faces of the channel are set as ‘no-slip’ walls. The flow is driven by a constant pressure gradient. The particles are neutrally buoyant and adhesive, with initial positions randomly distributed in the channel. The computational parameters are set as follow. The dimension of the channel is L × H × H = 201 × 61 × 61. By varying the pressure gradient, the channel Reynolds number ranges between 5.4 and 108. The particle size is dp = 5 and the particle concentration is fixed at 1%, corresponding to a particle number of 110. The strength of the particle adhesion is represented by the surface energy, ChemEngineeringwhich is chosen2020, 4, as 55 0.075 and 0.0075. The SRT model and the JKR contact theory are used in the LBM23 of 33 and DEM for this problem, respectively.

ChemEngineering 2020Figure, 4Figure, 55 18. 18.Schematic Schematic of of agglomeration agglomeration of of adhe adhesivesive particles particles in channel in channel flow. ﬂow. 23 of 33

FiguresFigures 1919 andand 2020 displaydisplay thethe snapshotssnapshots ofof adhesiveadhesive particleparticle suspensionssuspensions withwith didifferentﬀerent surfacesurface energiesenergies γγ == 0.0750.075 andand γγ == 0.0075,0.0075, respectively.respectively. It can be foundfound thatthat largelarge sizedsized agglomeratesagglomerates areare formedformed withwith relativelyrelatively higherhigher surfacesurface energyenergy ( γ(γ= = 0.075).0.075). Some particles eveneven stickstick onon thethe wallwall duedue toto strongstrong adhesion.adhesion. However, However, when when the the surface surface energy energy is reduced is reduced by one by order one oforder magnitude of magnitude (γ = 0.0075), (γ = the0.0075), size ofthe the size agglomerates of the agglomerates becomes becomes smaller andsmaller the and particles the particles appear toappear focus to around focus around a certain a lateralcertain position lateral position with Re with= 54 andRe = 10854 and (see 108 Figure (see 20 Figurec,d), which 20c,d), resembles which resembles the Segr éthe-Silberberg Segré-Silberberg eﬀect as discussedeffect as discussed in the last in section. the last section.

FigureFigure 19.19. Snapshots ofof adhesive particleparticle suspensionssuspensions withwith γγ == 0.075 and didifferentﬀerent ReynoldsReynolds numbersnumbers ((aa)) ReRe == 5.4,5.4, ((bb)) ReRe == 21.6,21.6, ((cc)) ReRe == 5454 andand ((dd)) ReRe == 108.108. The left columnscolumns areare sideside viewsviews andand thethe rightright columnscolumns areare cross-sectionalcross-sectional view.view. ChemEngineering 2020, 4, 55 24 of 33

ChemEngineering 2020, 4, 55 24 of 33

FigureFigure 20. Snapshots 20. Snapshots of adhesiveof adhesive particle particle suspensions suspensions with γγ == 0.00750.0075 and and different different Reynolds Reynolds numbers numbers (a) Re(a=) Re5.4, = (5.4,b) Re(b)= Re21.6, = 21.6, (c) (Rec) Re= 54= 54 and and ( d(d))Re Re == 108. The The left left columns columns are are side side views views and andthe right the right columns are cross-sectional view. columns are cross-sectional view. Figure 21 shows the lateral PDF of the particle suspensions based on the following definition, Figure 21 shows the lateral PDF of the particle suspensions based on the following definition, Nll(,+Δ l ) = (45) fl()N(l, l + ∆l) , f (l) = N , (45) where N is the total number of particles in the channelN and N(l, l+Δl) is the number of particles in the wheresquareN is theannulus total between number l ofand particles l + Δl. For in comparison, the channel the and channelN(l, l +flow∆l) with is the non-adhesive number of particlesparticles is in the squarealso annulus modelled between and thel andresultsl + are∆l .included. For comparison, It can be theseen channel from Figure flow 21a with that non-adhesive a single peak particlesat the is lateral position around 0.6 is observed in the PDF for non-adhesive particles, which moves closer to also modelled and the results are included. It can be seen from Figure 21a that a single peak at the the wall as Re increases and is in consistent with the Segré and Silberberg effect. However, the PDFs lateralare position quite distinct around for 0.6 adhesive is observed particles. in the The PDF single for non-adhesive peak distribution particles, is not which clearly moves observed closer for to the wall asparticlesRe increases with γ and= 0.075 is in(see consistent Figure 21b). with Moreover, the Segr éthereand Silberbergappears to be eff ect.a peak However, close to thethe PDFswall are quiteposition, distinct forwhere adhesive a number particles. of particles The are single found peak to stick distribution to wall because is not clearlyof the strong observed adhesion, for particles as with γindicated= 0.075 (seeby Figure Figure 19. 21 Onb). the Moreover, other hand, there for appears much less to beadhesive a peak particles close to the(γ = wall 0.0075), position, the PDFs where a numberbecome of particles similar to are those found of tonon-adhesive stick to wall particles because (see of Figure the strong 21c). adhesion,The averag ase lateral indicated position by Figure as a 19. On thefunction other of hand, Re is forfurther much shown less adhesivein Figure 22. particles Note that (γ the= 0.0075), relatively the larger PDFs value become of average similar lateral to those of non-adhesiveposition for Re particles = 5.4 is caused (see Figure by the 21 statisticalc). The effect average [38], lateral because position the fluid as inertia a function is relatively of Re smallis further shown in Figure 22. Note that the relatively larger value of average lateral position for Re = 5.4 is caused by the statistical effect [38], because the fluid inertia is relatively small to drive the lateral migration and the particles almost remain randomly distributed in the cross-sectional area. Except for ChemEngineering 2020, 4, 55 25 of 33 ChemEngineering 2020,, 44,, 5555 25 of 33 to drive the lateral migration and the particles almost remain randomly distributed in the cross- theto drive data fortheRe lateral= 5.4, migration it is observed and thatthe theparticles average almost lateral remain position randomly increases distributed monotonically in the with cross- the sectional area. Except for the data for Re = 5.4, it is observed that the average lateral position increases increasesectional of area.Re forExcept both for non-adhesive the data for Re and = less5.4, it adhesive is observed particles that the (γ average= 0.0075), lateral which position demonstrates increases a monotonically with the increase of Re for both non-adhesive and less adhesive particles (γ = 0.0075), similarmonotonically dynamics with between the increase non-adhesive of Re for and both weak non-adhesive adhesive particles. and less However,adhesive particles for strongly (γ = adhesive 0.0075), which demonstrates a similar dynamics between non-adhesive and weak adhesive particles. particleswhich demonstrates (γ = 0.075), the a monotonicsimilar dynamics increase ofbetween the average non-adhesive lateral position and weak disappears adhesive because particles. of the However, for strongly adhesive particles (γ = 0.075), the monotonic increase of the average lateral agglomeration.However, for strongly adhesive particles (γ = 0.075), the monotonic increase of the average lateral position disappears because of the agglomeration.

Figure 21. LateralLateral probability probability distribution distribution functions functions for for different diﬀerent cases, cases, (a) (non-adhesivea) non-adhesive particles, particles, (b) (adhesiveb) adhesive particles particles with with γ = γ0.075,= 0.075, and and (c) adhesive (c) adhesive particles particles with with γ = γ0.0075.= 0.0075.

Figure 22. The average lateral position as a function of Reynolds number. Figure 22. The average lateral position as a function of Reynolds number.

The agglomerationagglomeration mechanism mechanism for for such such channel channel flow flow can be can attributed be attributed to the competition to the competition between thebetween interparticle the interparticle adhesive adhesive contact forcecontact and force the and hydrodynamic the hydrodynamic force. Theforce. adhesive The adhesive force sticksforce particlessticks particles together, together, which which imposes imposes a positive a positive effect oneffecteffect the onon agglomeration. thethe agglomeration.agglomeration. The hydrodynamic TheThe hydrodynamichydrodynamic force exertsforce exerts two opposite two opposite effects oneffects the particleon the suspensions.particle suspensions. First, it induces First, it the induces particle the lateral particle migration, lateral whichmigration, motivates which the motivates interparticle the interparticle collision and collision facilitates and the facilitates agglomeration. the agglomeration. Secondly, the Secondly, fluid inertia the alsofluid tears inertia particles also tears apart particles from each apart other, from resulting each other, in the resulting breakup in of the agglomerates. breakup of agglomerates. As a consequence, As a aconsequence, new dimensionless a new adhesiondimensionless number adhesionAd is proposed number based Ad isis on proposedproposed the balance basedbased of the onon critical thethe balancebalance pull-o ff ofofforce thethe andcritical the pull-off drag force force [11 and,12, 38the], drag force [11,12,38], γ Ad == γ , (46) Ad = µ f U , (46) μ U (46) ffU where µf is the fluid dynamic viscosity and U is the mean fluid velocity. Using the adhesion number, thewhere degree μff isis of thethe the fluidfluid agglomeration dynamicdynamic viscosityviscosity can be and characterisedand U isis thethe meanmean and represented fluidfluid velocity.velocity. by UsingUsing the agglomerate thethe adhesionadhesion ratio number,number, that is definedthe degree as theof the ratio agglomeration of the number can of be particles characterised involved and in represented agglomerate by to the the agglomerate total particle ratio number. that Figureis defined 23 shows as the theratio relationship of the number between of particles the agglomerate involved ratio in agglomerate and the adhesion to the number.total particle A monotonic number. Figure 23 shows the relationship between the agglomerate ratio and the adhesion number. A monotonic increase of the agglomerate ratio as a function of Ad isis observed.observed. AfterAfter Ad reachesreaches aa criticalcritical ChemEngineering 2020, 4, 55 26 of 33

ChemEngineering 2020, 4, 55 26 of 33 increase of the agglomerate ratio as a function of Ad is observed. After Ad reaches a critical value, value,the agglomerate the agglomerate ratio seems ratio toseems converge to converge to one. Therefore,to one. Therefore, two diﬀ erenttwo different regimes canregimes be identiﬁed, can be identified,i.e., a partial i.e., agglomeration a partial agglomeration regime where regime the hydrodynamic where the hydrodynamic force dominates force over dominates the adhesive over force, the adhesiveand a complete force, agglomerationand a complete regime agglomeration where all the regime particles where are involvedall the particles in agglomerates. are involved in agglomerates.

FigureFigure 23. The agglomerateagglomerate ratio ratio as as a functiona function of theof adhesionthe adhesion number. number. The vertical The vertical dashed dashed line denotes line denotesthe critical the valuecritical of value adhesion of adhesion number number in the present in the study.present The study. data The was data originally was originally reported reported in [38]. in [38]. 6.3. Sedimentation of Particle Suspensions in Cavity Flow 6.3. SedimentationSedimentation of Particle is extensively Suspensions used in Cavity in industries Flow to separate particles from fluid or other particlesSedimentation with different is extensively sizes or velocities, used in industries such as separating to separate particles particles and from cells fluid in or microfluids other particles [112], withdewatering different coal sizes slurries or velocities, [113], and post-treatingsuch as sepa wastewaterrating particles [114– 116and]. Furthermore,cells in microfluids sediments [112], are dewateringalso physical coal pollutants slurries [113], in the and river, post-treatin whichg could wastewater have great[114–116]. influence Furthermore, on the ecosystemsediments andare alsohuman physical health pollutants [117–120 ].in Therefore,the river, which it is of coul greatd have importance great influence to study on the the sedimentation, ecosystem and which human is healtha very [117–120]. complicated Therefore, process, it is due of togreat the import complexance fluid to study mechanics the sedimentation, involved with which the is sediment a very complicatedparticles. With process, LBM-DEM, due to the a complex numerical fluid investigation mechanics involved is then performed with the sediment to gain particles. some insights With LBM-DEM,to the sedimentation a numerical problem. investigation is then performed to gain some insights to the sedimentation problem.As shown in Figure 24, we consider a sedimentation system composed of a cuboid channel and a cavity,As whichshown is in placed Figure in 24, the we middle consider of the a sedimentation channel. The boundaries system composed in x and ofy directionsa cuboid channel are periodic, and aand cavity, all the which other is faces placed are setin asthe no-slip middle walls. of the The channel. particles The are boundaries initially generated in x and at randomy directions positions are periodic,in the main and channel all the other (no particles faces are in set the as cavity). no-slip The walls. particle The particles density is are larger initially than generated the fluid, soat thatrandom they positionscan sedimentate in the main into channel the cavity (no under particles the in gravity. the cavity). The sizeThe ofparticle the main density channel is larger is L thanW theH fluid,= 321 × × so 81that they49. The can cavity sedimentate has the into same the width cavity as theunder main the channel. gravity. TheThe length size of of the the main cavity channel is l = 40 is andL × W 80, × × ×while H = 321 the × height 81 × 49. is Theh = cavity16 and has 32, the which same gives width rise as to the four main configurations channel. The of length the cavity. of the Thecavity channel is l = 40Reynolds and 80, while number, the definedheight is as h Re= 16= andUxH 32,/ν varieswhich betweengives rise 28.44 to four and configurations 88.32. The particle of the cavity. diameter The is channeldp = 6 and Reynolds the density number, is in thedefined range as 1.1 Re ~ 2.0.= U ThexH/ν total varies number between of particles 28.44 and is fixed 88.32. at 110,The whichparticle is diameterequal to anis d overallp = 6 and concentration the density is less in the than range 1%. 1.1 The ~ SRT2.0. The model total and number the Hertz of particles model areis fixed used at in 110, the whichLBM andis equal DEM to for an thisoverall particular concentration problem, less respectively. than 1%. The SRT model and the Hertz model are used in the LBM and DEM for this particular problem, respectively. ChemEngineering 2020, 4, 55 27 of 33 ChemEngineering 2020, 4, 55 27 of 33

ChemEngineering 2020, 4, 55 27 of 33

Figure 24. Schematic of the sedimentation system. FigureFigure 24. 24.Schematic Schematic ofof thethe sedimentationsedimentation system. system. Figure 25 shows the snapshots of the particle suspensions at different time. The particles are FigureFigure 25 shows 25 shows the the snapshots snapshots of of the the particle particle suspensionssuspensions at at different different time. time. The The particles particles are are initiallyinitially located located at random at random positions positions in in the the main main channelchannel withwith no no particles particles trapped trapped inside inside the cavity.the cavity. initially located at random positions in the main channel with no particles trapped inside the cavity. Then Thenthe particlesthe particles settle settle to tothe the bottom bottom ofof thethe channelchannel andand flowflow over over the the cavity. cavity. During During the the Then the particles settle to the bottom of the channel and flow over the cavity. During the transportation transportationtransportation of the of thesuspensions, suspensions, some some particles particles fallfall into thethe cavity cavity and and stay stay stationary stationary inside, inside, while while ofsome the someother suspensions, other keep keep circulating some circulating particles around around fall an intoan orbit orbit the in cavityin the the centralcentral and stay vortexvortex stationary inside inside the inside, the cavity. cavity. while The The total some total number other number keep circulatingof particlesof particles around trapped trapped an orbitinside inside in thethe the central cavity cavity vortex isis evaluatedevaluated inside the afterafter cavity. aa Thesufficientlysufficiently total number long long time, of time, particles when when trappedthe the insidetransportationtransportation the cavity of isthe of evaluated theparticles particles reaches after reaches a su a affisteady steadyciently state. state. long From time, FigureFigure when 25 25 the the typical transportation typical trajectories trajectories of of the particle of particles particle reachessuspensionssuspensions a steady in the state.in thecavity Fromcavity are Figureare analysed, analysed, 25 the where where typical threthre trajectoriese distinct dynamic ofdynamic particle behavi behavi suspensionsoursours can can be in identified: thebe identified: cavity are resuspension, deposition and circulation. Resuspension means that the particles fall into the cavity analysed,resuspension, where deposition three distinct and circulation. dynamic behaviours Resuspensi canon bemeans identified: that the resuspension, particles fall depositioninto the cavity and and then come out due to the lift force from the fluid. Deposition happens mostly at the rear edge of circulation.and then come Resuspension out due to the means lift force that thefrom particles the fluid. fall Deposition into the cavity happens and mostly then come at the out rear due edge to the of lift forcethe fromcavity, the where fluid. the Deposition particles deposit happens to the mostly corner at of the the rear cavity edge and of stay the cavity,motionless. where Circulation the particles the cavity,indicates where that the particlesparticles are deposit captured to theby the corner central of vortexthe cavity inside and the staycavity motionless. and keeps movingCirculation deposit to the corner of the cavity and stay motionless. Circulation indicates that the particles are indicatesalong that an orbitthe particlesin a periodic are way. captured Both deposition by the ce andntral circulation vortex inside are treated the cavity as the entrapmentand keeps ofmoving a captured by the central vortex inside the cavity and keeps moving along an orbit in a periodic way. alongparticle. an orbit in a periodic way. Both deposition and circulation are treated as the entrapment of a Bothparticle. deposition and circulation are treated as the entrapment of a particle.

Figure 25. Snapshots of the particle suspensions for Re = 56.88, h = 32, l = 80 and ρp = 1.5. The time point for each subplot is t = 0, t = 50,000, t = 200,000 and t = 500,000 from top to bottom, respectively. The flow field only contains a slice at the mid-plane in y direction. Some particles may be sheltered and look smaller. Figure 25. SnapshotsSnapshots of of the the particle particle suspensions suspensions for for ReRe = 56.88,= 56.88, h =h 32,= 32,l = 80l = and80 andρp = 1.5.ρp = The1.5. time The point time pointfor each for subplot each subplot is t = is0, tt == 0,50,000,t = 50,000, t = 200,000t = 200,000 and t and= 500,000t = 500,000 from fromtop to top bottom, to bottom, respectively. respectively. The Theflow ﬂow field ﬁeld only only contains contains a slice a slice at the at the mid-plane mid-plane in iny ydirection.direction. Some Some particles particles may may be be sheltered sheltered and look smaller. ChemEngineering 2020, 4, 55 28 of 33

ChemEngineering 2020, 4, 55 28 of 33 Figure 26 shows the trap efficiency η, defined as the ratio of the trapped particle number to the total particleFigure number, 26 shows as a functionthe trap ofefficiency the particle η, defined density as for the di ratiofferent of cavitythe trapped configurations particle number and Reynolds to the totalnumbers. particle It is number, clear that as as a the function particle of density the particle increases, density the trap for e ffidifferentciency increases cavity configurations continuously fromand Reynolds0 to 1. For numbers. a fixed Reynolds It is clear number, that as the the largest particle cavity density (h = 32,increases,l = 80)has the the trap highest efficiency trap eincreasesfficiency, continuouslywhile the smallest from 0 cavity to 1. For (h = a fixed16, l = Reynolds40) has thenumber, lowest the trap largest efficiency. cavity ( Theh = 32, results l = 80) of has the the other highest two trapcavity efficiency, configurations while the are smallest in between, cavity which (h = indicates16, l = 40) that has increasing the lowest either trap efficiency. length or depthThe results leads of to thea higher other traptwo ecavityfficiency. configurations Furthermore, arefor in between, a fixed cavity which configuration, indicates that an increasing increase ineither the Reynoldslength or depthnumber leads results to a inhigher a decrease trap efficiency. in the trap Furthermore, efficiency. Withfor a afixed larger cavity Reynolds configuration, number, an the increase particle’s in thetranslational Reynolds velocitynumber results in x direction in a decrease becomes in the larger, trap so efficiency. that it has With less a time larger to Reynolds enter the cavitynumber, in the particle’svertical direction. translational Meanwhile, velocity a in larger x direction Reynolds becomes number larger, produces so that a largerit has liftless force. time to As enter a consequence, the cavity inthe the particles vertical are direction. less easily Meanwhile, trapped in thea larger cavity. Reynolds number produces a larger lift force. As a consequence, the particles are less easily trapped in the cavity.

Figure 26. 26. TheThe trap trap efficiency eﬃciency as a as function a function of partic of particlele density density for different for diﬀ erentReynolds Reynolds numbers: numbers: (a) Re =(a 28.44,) Re = (28.44,b) Re = (b 56.88,) Re = (56.88,c) Re = ( c85.32.) Re = 85.32.

7. Summary Summary An overview of the coupled discrete element method with the lattice Boltzmann method is presented together together with with its its applications applications in insolid–liquid solid–liquid flows. flows. The fundamentals The fundamentals of DEM of DEMand LBM, and includingLBM, including the contact the contact mechanics mechanics based based on Hert on Hertzz model model and and JKR JKR theory, theory, the the lattice lattice Boltzmann equation with both single-relaxation-time model model and and multi-relaxation-time multi-relaxation-time model model are are introduced. introduced. Several solid–fluid solid–fluid interaction models, i.e., the coupling between DEM and LBM, are discussed and compared inin detail.detail. The The validity validity and and accuracy accuracy of the of numericalthe numerical method method are validated are validated for three for classical three classicalsolid–liquid solid–liquid flow problems, flow problems, and the and results the result are ins goodare in quantitativegood quantitati agreementve agreement with thosewith those from fromexperiments experiments and other and numerical other numerical approaches. approaches Three case. Three studies, case including studies, theincluding inertial migrationthe inertial of migrationparticle suspensions, of particle thesuspensions, agglomeration the agglomeration of adhesive particles, of adhesive and the particles, sedimentation and the problem, sedimentation are also problem,performed are to also demonstrate performed the to capability demonstrate of the the coupled capability DEM-LBM of the coupled to solid–liquid DEM-LBM flow to problems solid–liquid and flowpotential problems engineering and potential applications. engineering applications.

Author Contributions: Author Contributions: Conceptualization,Conceptualization, C.-Y.W.; C.-Y.W.; formal formal analys analysis,is, W.L.; W.L.; funding funding acquisition, acquisition, C.-Y.W.; C.-Y.W.; methodology, W.L. and C.-Y.W.; project administration, C.-Y.W.; supervision, C.-Y.W.; validation, W.L.; methodology,writing—original W.L. draft, and W.L.; C.-Y.W.; writing—review project administration, and editing, C.-Y.W.; C.-Y.W. Allsupervision, authors have C.-Y.W.; read andvalidation, agreed toW.L.; the writing—originalpublished version draft, of the W.L.; manuscript. writing—review and editing, C.-Y.W. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Engineering and Physical Sciences Research Council (EPSRC), grant Funding:number No: This EP research/N033876 was/1. funded by the Engineering and Physical Sciences Research Council (EPSRC), grant numberAcknowledgments: No: EP/N033876/1.The authors acknowledge Duo Zhang and Nicolin Govender in the University of Surrey for their helpful suggestions and fruitful discussion in developing the numerical approach. Acknowledgments: The authors acknowledge Duo Zhang and Nicolin Govender in the University of Surrey for theirConﬂicts helpful of Interest:suggestionsThe and authors fruitful declare discus nosion conﬂict in developing of interest. the numerical approach.

Conflicts of Interest: The authors declare no conflict of interest.

ChemEngineering 2020, 4, 55 29 of 33

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