Forces acting on a single introduced particle in a solid–liquid fluidised bed

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Forces acting on a single introduced particle in a solid-liquid fluidised BEd Zhengbiao Peng, Swapnil V. Ghatage, Elham Doroodchi, Jyeshtharaj B. Joshi, Geoffrey M. Evans, Behdad Moghtaderi

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Received date: 9 February 2014 Revised date: 17 April 2014 Accepted date: 27 April 2014 Cite this article as: Zhengbiao Peng, Swapnil V. Ghatage, Elham Doroodchi, Jyeshtharaj B. Joshi, Geoffrey M. Evans, Behdad Moghtaderi, Forces acting on a single introduced particle in a solid-liquid fluidised BEd, Chemical Engineering Science, http://dx.doi.org/10.1016/j.ces.2014.04.040 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

1

Forces Acting on a Single Introduced Particle in a Solid-liquid Fluidised Bed

*

Zhengbiao Peng1, Swapnil V. Ghatage2, 3, Elham Doroodchi1 , Jyeshtharaj B. Joshi2,4, Geoffrey M. Evans1 and Behdad Moghtaderi1

1. Discipline of Chemical Engineering, University of Newcastle, Callaghan, NSW 2308, Australia.

2. Department of Chemical Engineering, Institute of Chemical Technology, Matunga, Mumbai 400 019, India.

3. Department of Chemical Engineering, Indian Institute of Technology, Gandhinagar, Gujarat 382424, India.

4. Homi Bhabha National Institute, Anushaktinagar, Mumbai 400 094, India.

*Corresponding author at: Priority Research Centre for Advanced Particle Processing & Transport, The University of Newcastle, Australia. Tel.: +61 2 4033 9066; Fax: +61 2 4033 9095; E-mail address: [email protected]

2

Abstract In a liquid fluidised bed system, the motion of each phase is governed by fluid-particle and particleparticle interactions. The particle-particle collisions can significantly affect the motion of individual particles and hence the solid-liquid two phase flow characteristics. In the current work, computational fluid dynamics-discrete element method (CFD-DEM) simulations of a dense foreign particle introduced in a monodispersed solid-liquid fluidised bed (SLFB) have been carried out. The fluidisation hydrodynamics of SLFB, settling behaviour of the foreign particle, fluid-particle interactions, and particle-particle collision behaviour have been investigated. Experiments including particle classification velocity measurements and fluid turbulence characterisation by particle image velocimetry (PIV) were conducted for the validation of prediction results. Compared to those predicted by empirical correlations, the particle classification velocity predicted by CFD-DEM provided the best agreement with the experimental data (less than 10% deviation). The particle collision frequency increased monotonically with the solid fraction. The dimensionless collision frequency obtained by CFD-DEM excellently fit the data line predicted by the kinetic theory for granular flow (KTGF). The particle collision frequency increased with the particle size ratio (dP2/dP1) and became independent of the foreign particle size for high solid fractions when the fluidised particle size was kept constant. The magnitude of collision force was 10 – 50 times greater than that of gravitational force and maximally 9 times greater than that of drag force. A correlation describing the collision force as a function of bed voidage was developed for Stp > 65 and dP2/dP1  2. A maximum deviation of less than 20% was obtained when the correlation was used for the prediction of particle collision force. Keywords: Fluidisation; Collision frequency and collision force; Discrete element method; Fluidparticle interactions; Classification velocity

3

1. Introduction Fluidised beds are widely used in chemical, petrochemical and process industries. They are preferred over other reactors for carrying out various gas-solid, solid-liquid and gas-solid-liquid processes due to the enhanced contact between the fluid and solid particles. However, the efficient operation of a fluidised bed requires the accurate choices of design and operating conditions, and the accuracy of prediction of fluidisation behaviour. With increasing computational power and development in computational methods, the use of Computational fluid dynamics-Discrete element method (CFDDEM) in the Eulerian-Langrangian framework for detailed simulations of a fluidised bed is increasing extensively. It provides important insight into the fluid flow, particle flow as well as the effects of fluid drag and particle-particle collisions on the transport phenomena. A particle in the solid-liquid fluidised bed (SLFB), mainly experiences two types of forces, i.e. drag applied by the fluid along the fluid streamlines, and collisions with other fluidised particles. The quantification of the relative contribution of drag and collision is very complex but essential. In literature, many researchers have studied the effect of turbulence generated by fluid on the motion of the particle (see e.g., Brucato et al., 1998; Doroodchi et al., 2008; Ghatage et al., 2013; Gidaspow, 1994; Joshi, 1983). Attempts have also been directed to quantify the effect of fluid drag and propose correlations to predict the motion of individual particles based on theoretical and empirical approaches (see e.g., Di Felice et al., 1991; Grbavcic et al., 2009; Joshi, 1983; van der Wielen et al., 1996). However, very few studies have been reported discussing the collision effect on the motion of the settling particle in the SLFB (Grbavcic et al., 2009; van der Wielen et al., 1996). It was thought desirable to quantify the particle-particle collision effect and the relative importance of drag and collision forces both experimentally and through modelling. In this work, a dense foreign particle has been introduced into the SLFB. CFD-DEM simulations have been carried out on the settling of the foreign particle in the SLFB. The fluid-particle interactions were

4 solved by a fully two-way coupling algorithm (Kafui et al., 2002; Tsuji et al., 1993; Xu and Yu, 1997), i.e., effects of solid volume fraction on fluid flow and interphase momentum exchange have been rigorously implemented in the governing equations of the solid-liquid two-phase flow. The particleparticle interactions were solved based on the contact mechanics of rigid bodies (Peng et al., 2013; Tsuji et al., 1993). The motion of the foreign particle was monitored continuously whilst it moved through the bed and the detailed information on particle dynamics was recorded. The fluidisation hydrodynamics of SLFB, settling behaviour of the foreign particle, fluid-particle interactions and particle-particle collision behaviour have been investigated. The model predictions were compared with experimental measurements. A number of CFD-DEM studies of gas fluidised beds were reported in the literature (see e.g., Kafui et al., 2002; Peng et al., 2013; Tsuji et al., 1993; Wu et al., 2006; Xu and Yu, 1997; Ye et al., 2004), however, similar studies of SLFBs are very scarce. Notably, there have been, to our knowledge, no published numerical studies focusing on the settling of a foreign particle in the SLFB. Therefore, an attempt has been made below to bring out the knowledge gaps and the significant attempts in similar studies in the published literature. In the past few decades the focus of the research in such systems is on the drag applied on the particles (Ghatage et al., 2013). The readers may like to read published literature for greater insight into the turbulence effects on the particle motion (Brucato et al., 1998; Doroodchi et al., 2008; Ghatage et al., 2013; Gidaspow, 1994; Joshi, 1983). In the view of present study, the literature review has been presented focusing on particle-particle collisions within SLFBs and numerical approaches for the simulation of SLFBs.

5

2. Literature Review 2.1. Particle-particle collisions in SLFB Researchers have investigated the settling of a foreign particle in SLFB (see e.g., Grbavcic et al., 2009; Grbavcic and Vukovic, 1991; Joshi, 1983; van der Wielen et al., 1996). These authors have studied the effect of fluid drag on the settling of the foreign particle in the presence of fluidised particles. Many have proposed correlations to predict the particle classification velocity. For detailed discussion on the effect of fluid drag on the settling of a foreign particle in the SLFB and other turbulence devices, readers are referred to Ghatage et al. (2013). Most of these studies enlightened on the effect of fluid drag on the motion of the foreign particle. However, the importance of particle-particle collisions was often overlooked. Very few attempts have been reported in the literature wherein the estimation of the collisional force was discussed. Some researchers have tried to quantify this effect in terms of collisional frequency, number of collisions and collision intensity. Pozo et al. (1993) developed an experimental technique to measure the particle collision frequency based on the perturbations in the diffusion current of tethered electrodes. A tethered particle near its anchorage point was used and gave a good estimate of the collision frequency of a free floating particle. The collision frequency between an electrode and the bed particles was obtained from the measured number of collisions Nc between the nc tracer particles and the electrode over time t: f coll

( N c / t )(nt / nc )

(1)

where nt is the total number of bed particles which was obtained from the particle mass in the bed. Their experimental results showed that the collision frequency monotonically decreased as the superficial liquid velocity was increased. Gidaspow (1994) has proposed a correlation for the prediction of collision frequency in a binary mixture based on the analogy with the kinetic theory of granular flow (KTGF) as, N 12

2 4n1 n 2 d 12 g 0 S

(2)

6 where the granular temperature, 4 was defined as, 

1 2 up 3

(3)

and g0 was defined as a function of solid fraction as, g 0 D S

ª § Ds «1  ¨¨ ¬« © D s0

·º ¸» ¸ ¹¼»

1

(4)

Gjaltema et al. (1997) assumed homogeneous isotropic turbulence and the particle size being larger than the turbulent microscale in three-phase inverse fluidisation. They proposed a correlation for computing the collision frequency in a binary mixture as, 2 N12 Sn1n2 d12 Ur

(5)

where Ur is the relative velocity between the fluid and particle. For cases of solid-liquid particulate flows where the collisional effects can be considered dominant, the particle collision behaviour was also described by the classic kinetic theory for granular flow (KTGF) (Ding and Gidaspow, 1990; Koch, 1990; Simonin, 1991). A prediction of a dimensionless collision frequency can be obtained from the KTGF, * f coll

24

2 DS g0 3S

(6)

where g0 is the pair correlation function defined as,

g0

§ D ¨1  S ¨ D Sm ©

· ¸ ¸ ¹

2.5D Sm

(7)

where Sm is the solid fraction in packed beds of particles. The results predicted by the above two correlations (Gidaspow, 1994; Gjaltema et al., 1997) and those by KTGF consistently showed that the collision frequency continuously decreased as the bed voidage increased. Zenit et al. (1997) developed a measuring technique based on the use of a dynamic pressure

7 transducer, to measure the collisional particle pressure (characterising particle-particle collisions) in a two-phase fluidised bed. The value of the collisional particle pressure was shown to reach a maximum at a solid fraction between 0.37 and 0.43. The increasing number of collisions and the decreasing collision intensity was thought to be responsible for the maximum value of the collisional particle pressure as the solid hold up was increased. It is clearly understandable that the number of collisions increases with the solid fraction, up to a given value where the motion of particles becomes hampered by the high packing density (i.e., in the packed bed). However, Buffière and Moletta (2000) carried out experiments using a hydrophone to estimate the frequency of collisions, which was found to increase with an increase in solid fraction, attain a peak around S=0.35 and further decrease as solid fraction approached the packed bed state. Buffière and Moletta (2000) attributed this phenomenon to the significant effect of particle-particle interactions on the structure of flow field at high values of solid fraction. However, uncertainties about the validity of the above explanation were addressed by the authors, who also stated that more theoretical work was needed to understand the underlying mechanisms at a high solid amount. Aguilar-Corona et al. (2011) conducted experiments to measure the particle collision frequency by particle tracking in an index-matched array. Collision detection was based on the acceleration threshold of the instantaneous speed of coloured tracers. The measured particle collision frequency also showed a peak value at the solid fraction of 0.3, which apparently deviated from the prediction results by KTGF. Aguilar-Corona et al. (2011) attributed this behaviour of their experimental results to the detection limits of collisions from the particle acceleration signal process. Specifically, the particle Stokes number was estimated based on the particle root-mean-square velocity (i.e.,

qp2 ) to evaluate the

effect of collisions that occurred with rebound. The particle Stokes number was defined as, St p

Up

qp2 d p

9P f

(8)

8 For solid fractions higher than 0.3, Stp in their study was smaller than 10, which is considered as the critical value below which a particle collision will not lead to a rebound (ec = 0). Therefore, for high solid phase fractions (S > 0.3), particle collisions can no longer be accurately detected by particle acceleration signals, due to the fact that the total particle collision frequency increases but collisions with rebound (i.e. acceleration signal) are less frequent. They also claimed that their approach based on the acceleration threshold cannot be applied for high solid fractions (S > 0.3). van der Wielen et al. (1996) proposed a correlation for estimating the collision force between a foreign particle and fluidised particles in SLFB based on the force balance. This collision force was defined as, FP P

FG  FB  FF-P

(9)

where, the fluid-particle interaction force was described as, FF P

CD

S 4

2 d P2

2 U LU S2

2

(10)

The buoyancy force was defined in literature based on the fluid density (Clift et al., 1987; Epstein, 1984; Fan et al., 1987; Joshi et al., 2001) or mixture density (Foscolo and Gibilaro, 1984; Foscolo et al., 1983; van der Wielen et al., 1996). So, the particle-particle collision (or friction) force was given as, FP  P

VP2 U P 2 g

1

2 U M 0.75C D U LU S2  U P2 d P2 gU P2

(11)

Based on the experimentally noted classification velocity, they plotted the ratio of FP-P/FG (from Eq. (11)) against the product of liquid hold-up (L) and particle classification velocity (V2) and observed a linear dependency. The particle classification velocity refers to the constant settling velocity of the foreign particle through a liquid fluidised bed of different particles. The slope was defined as the interaction coefficient () and its dependency on the ratio of dense particle diameter to fluidising particle diameter was defined as,

9



§d · 10.5¨¨ P1 ¸¸ © d P2 ¹

0.95

(12)

Grbavcic et al. (2009) performed experiments to study the single particle settling velocities in a fluidised bed. Due to collisions between the foreign particle and fluidised particles, an additional collision force was taken into account in their theoretical analysis. As a settling particle must displace N fluidised particles (i.e. non-elastic collisions), they assumed that the collision force is proportional to the total drag forces acting on the N displaced particles moving through the fluidised suspension. The number of displaced particles N was given as, § 1  D L ¨¨ d P2 © d P1

N

· ¸¸ ¹

3

(13)

Therefore, the collision force was expressed as, FP  P

ª1 § S º · k 1  f N « U f ¨ d p2 ¸U S2f » ¹ ¬2 © 4 ¼

(14)

where f is the Ruzicka (2006) transition function. k is the collision coefficient and was defined as,

k

2.2D

 5.07 L

§ d P2 · ¨¨ ¸¸ © d P1 ¹

1.48

§ UM ¨¨ © U eff

· ¸¸ ¹

2.47

(15)

where M is the mixture density and eff is the effective or relevant density and can be obtained as defined by Ruzicka (2006). Grbavcic et al. (2009) claimed that the collision force can be one order of magnitude higher than the drag force between these particles and significantly affects the classification velocity of the foreign particle. They also pointed out that for high bed voidages the collision forces between particles of fluidised bed and the transiting particle are relatively small but their influence increases as the bed voidage decreases. 2.2. Numerical approaches for simulation of SLFB The motion of both the fluid and the solid particles in SLFBs has been simulated by the direct numerical simulation (DNS) (see e.g. Hu et al., 2001; Reddy et al., 2010; Reddy and Joshi, 2008, 2009;

10 Reddy et al., 2013; Sangani and Didwania, 1993; Sangani and Prosperetti, 1993). The hydrodynamic forces acting on the particles were directly computed from the surrounding fluid flow, and the motion of the fluid flow and particles were fully coupled. The DNS of the exact particle motion may be the only theoretical tool capable of studying the nonlinear and geometrically complicated phenomena of particle-particle interactions (Hu et al., 2001). However, due to the prohibitive CPU time, the number of particles involved in the simulation has been largely limited. Brady and Bossis (1988) have developed numerical techniques (Stokesian dynamics) for simulating the motion of a relatively large number of particles in Stokes flows. Their model was appropriate for colloidal suspensions in the limit of zero particle Reynolds number. Hu and co-workers (Hu, 1996; Hu et al., 1993; Hu et al., 1992) developed the arbitrary Lagrangian-Eulerian (ALE) particle mover for simulations of fluid-solid systems at finite Reynolds numbers. In this scheme both the fluid and solid equations of motion were incorporated into a single coupled variational equation. At each time step a new mesh was generated when the old one became too distorted, and the flow field was projected onto the new mesh. Glowinshi et al. (1999) developed a different approach based on the concept of fictitious domain using a fixed grid, independent of remeshing. Applying this concept to particulate flows, the particle domain was treated as a fluid with additional constraints to impose proper rigid body motion. Later, Pan et al. (2002) applied the fictitious domain method in a 3D simulation with 1204 particles. Peskin (1972) proposed an immersed boundary method (IBM) that allows the use of rectangular grids for arbitrarily complex geometries. Recently, Guo et al. (2013) incorporated IBM into their existing CFD-DEM code to simulate slightly compressible gas-solid flows with complex and/or moving boundaries. It was demonstrated that the capacity of conventional CFD-DEM has been enhanced with the incorporation of IBM. However, as a whole, all of the above methods that are capable to reveal more details of the fluid flow around a solid particle are very computationally demanding, which renders them impractical to simulate some real problems.

11 The most common and efficient approach for simulating particulate flows is to use the continuum theory (i.e. the Eulerian approach) that views the solid and the liquid as interpenetrating mixtures (see e.g. Drew and Passman, 1999; Fan and Zhu, 1998; Gidaspow, 1994; Lu and Gidaspow, 2003; Reddy and Joshi, 2009). However, this approach leads to unknown terms that represent the interactions between the phases. As a result, the nature of detailed interactions between fluid and particles and those between particles and particles cannot be understood from the application of mixture theories alone. A second approach for the simulation of SLFBs is Lagrangian numerical simulation (LNS), which provides a direct description of the particulate flow by tracking the motion of individual particles, thus allowing for special focus and treatments on particle-particle and particle-fluid interactions. The fluid flow satisfies the continuum equations that are solved on a fixed field in the conventional Eulerian way. Andrews and O’Rourke (1996) introduced a multiphase particle-in-cell (MP-PIC) method which used the mapping of Langrangian particles to the computational grid. On grid points, continuum approach was applied for particles considering the particle phase as the continuum and mapped back to individual particles. Such approach can be used to simulate dense, as well as dilute granular systems. However, it was assumed that particle-particle collisions are impulsive events that do not depend on the local flow field of the continuous phase. This assumption is only valid in cases where the inertia of the continuous phase is negligible compared to that of the dispersed phase (e.g. in gas-fluidised beds). Seibert and Burns (1998) used a discrete particle simulation technique to study structural phenomena in liquid fluidised beds. Their model, however, was developed in a simplified form. The net force acting on a particle that was used to predict particle motion was given by the difference between the equilibrium value of hydrodynamic forces and the calculated value at the current conditions. Moreover, the particle-particle interaction forces were not considered in their model. Zhang et al. (1999) have shown that the generally used collision models need to be modified when the continuous phase is a liquid, in order to account for the drainage of the fluid between colliding particles and the acceleration of the fluid surrounding the particles. The adaptations of discrete particle model (DPM) involved two

12 additional forces, i.e. the virtual mass force and the pressure gradient force. They found that when these two forces were incorporated, the strong influence of the liquid surrounding the particles on the particle trajectories before and after a collision can be captured. Malone et al. (2007) developed the fully coupled CFD-DEM model and applied the model in the simulations of hydrodynamics and heat transfer of liquid-fluidised beds. Their simulation results indicated that modifications to account for the liquid effect have a significant improvement in terms of the micro-scale particle mixing behaviour. The CFD-DEM model has since been applied to study the liquid-solid interactions in the context of the analysis of gas bubble formation in a gas-liquid-solid system (Chen and Fan, 2004; Li et al., 1999, 2000; van Sint Annaland et al., 2005). Later, Di Renzo and Di Maio (2007) applied the CFD-DEM model to investigate the characteristics of particulate and aggregative fluidisation as well as the transition hydrodynamic stability of gas and liquid fluidised beds. Given the above similar studies in the literature and the existing knowledge gaps, the objective of present work is to provide insight into particle-particle and fluid-particle interactions as well as their relative importance in the settling process of a foreign particle within the SLFB using a CFD-DEM model of high fidelity. In the next section, details pertinent to the CFD-DEM model are presented, followed by the description on the experimental set-up.

3. Computational 3.1. Model formulation: Computational fluid dynamics – discrete element method 3.1.1. Governing equations In a dense fluid-solid flow, a single particle is interacting with neighbouring particles, surrounding fluid and computational domain boundaries. The equations for the translational and rotational motion of a particle are,

13 dv i dt

mi

f c, i  f f ,i  mi g

(16)

Tc,i  Tr ,i

(17)

and,

Ii

d i dt

A soft-sphere model, namely the linear spring-dashpot model (Tsuji et al., 1993; Xu and Yu, 1997), was employed to calculate the collision contact force fc,i and the contact torque Tc,i caused by the tangential component of the contact force. Tr,i is the rolling resistance torque caused by particle rolling motion (Ai et al., 2011; Zhou et al., 1999) and is calculated by, N pc

Tr ,i



¦P

rol

f cn ,ij

j 0

 ij |  ij |

(18)

ri

where μrol is the rolling friction coefficient depending on the particle material. Npc is the total number of the particles that are contacting with particle i at the time instant. fcn, ij is the normal contact force; ij is the relative angular velocity between particle i and j as,

 ij

 i ri   j r j

(19)

ri  r j

The fluid-particle interaction force (ff, i) results from the distortion of fluid streamlines passing around the particle and the subsequent variation of local fluid stress tensor at the particle surface. According to Anderson and Jackson (1967), the total fluid force acting on particle i was expressed as, f f,i

Vi ’p  Vi ’ x  f  D L f d,i

(20)

where f is the fluid viscous stress tensor and expressed as,

f

>

@

§

2 3

·

P L ’u L  ’u L 1  ¨ O  P L ¸’ x u L I ©

¹

(21)

14 It should be noted that the fluid force acting on a particle were calculated as point forces at the centroid of a particle to consider the distribution of particles in a cell (van der Wielen et al., 1996; Wu et al., 2006; Ye et al., 2004). To this end, the Eulerian variables of fluid flow field need to be converted to point values at the particle centre (see Section 3.1.2). Compared to the Eulerian-Eulerian approach, no closure is required for the solid phase stress tensor in the DEM method, as the motion of individual particles is solved directly. However, empirical correlations have to be used for the fluid drag force, since the hydrodynamics of fluid phase is resolved on a length scale larger than the particle size. In this study, the Gidaspow drag law (Gidaspow, 1994) was employed for the calculation of fluid drag force fd,i,

f d,i

Vi

1  D L

E u L  v i

(22)

where  is the interphase momentum exchanging coefficient and calculated by the Gidaspow draw law through the following expressions,

E

­1501  D L 2 P L 1.751  D L U L u L  v i  ° dp d p2D L ° ® U g D L 1  D L u L  v i  2.65 ° DL °0.75C d dp ¯

D L d 0.8 (23)

D L ! 0.8

where Cd is the drag coefficient and givens as,

Cd

­0.44 ° ® 24 0.687 ° Re 1  0.15 Re p p ¯





Rep d 1000 Rep ! 1000

(24)

Rep is the particle Reynolds number based on the superficial slip velocity between the particle and the fluid and calculated by,

Re p

U g d pD L u L  v i PL

(25)

15 The local averaged continuity and momentum equations of the continuum can be written as (Anderson and Jackson, 1967), w D L U L  ’ x D L U L u L 0 wt

(26)

wD L U L u L  ’ x D L U L u L u L ’p  ’ x  f  f S-L  D L U L g wt

(27)

where fS-L is the local mean particle-fluid interaction force, which is calculated based on the Newton’s third law by,

f S-L

§ ¨ ¨ ©

· f f' ,i ¸ / 'Vc ¸ 1 ¹

Np

¦ i

(28)

where Np is the total number of particles with a partial or entire body locating in the cell and Vc is the cell volume. f f' ,i is the fluid force acting on the partial particle body (i.e. when the particle is sitting over the cell boundaries) or the entire particle body (i.e. when the particle is enclosed by the cell boundaries). In the conventional approach, the entire body of a particle is treated to be in that cell if the particle centroid is inside the cell, thus we have f f' ,i f f ,i . However, in this study the scenarios where a particle is sitting over the cell boundaries were considered and the particle segment volume cut by cell boundaries was calculated accurately by a quasi-analytical approach, as to be detailed in Section 3.1.3. Accordingly, f f' ,i is calculated as, f f' ,i

G i f f ,i

(29)

where i is the fractional volume of particle i in the cell (Kuang et al., 2008). 3.1.2. Mapping scheme from Eulerian variables to point values The point values of Eulerian variables were calculated based on their spatial gradient distribution. The point values of these variables at the particle centroid position, i.e.

p,

are thus calculated by:

16 Ip Ic  dr x ’GR

(30)

where dr = xp – xc. ’ GR is the gradient of an Eulerian variable that is also stored at the cell centre and computed using the divergence theorem, i.e., Nf

’ GR

¦I

f ,i A f ,i

i 1

(31)

'Vc

where Nf is the number of cell faces; Af, i is the area normal of cell face i. at the cell centre and

f is

c is

the variable value stored

the face value interpolated by adjacent cells following a certain numerical

scheme, e.g. central-differenced, second-order upwind and third-order Monotone Upstream-centred Scheme for Conservation Laws (MUSCL). 3.1.3. Calculation of cell void fraction Conventionally the cell void fraction was calculated by affiliating each particle with a cell as the particles were tracked transiently (Tsuji et al., 1993). This affiliation is determined based on the position of the particle centroid, that is, if a particle centroid is inside a cell, the entire body of the particle is treated to be in that cell and hence the cell void fraction is calculated based on the total volume of the particle rather than the actual partial volume of the particle within the cell. This treatment may lead to errors of up to 50% in the calculation of segment volume of one particle in a computational cell when the particle centroid is near the cell boundaries. Such large errors in particle segment volume may result in the incorrect prediction of local void fraction and consequently lead to strange behaviour in the model outputs if the particle size is comparable with the computational cell size (Peng et al., 2014). In this study a quasi-analytical method based on the particle meshing technique was applied to calculate the cell void fraction. In the particle meshing method (PMM), the particle is meshed into small particle grids. Then each particle grid is affiliated with a cell based on the position of their centroids. The volume of the solid

17 phase in a computational cell is calculated by counting the number of particle grids and adding up their volumes. The method to implement PMM in the CFD-DEM simulation is presented below. Assume a particle is evenly meshed into Npm particle grids with each particle grid having a volume of Vpm = Vp/Npm. The equivalent particle grid size is,

d pm

3

6Vpm

S

3

6 Vp S N pm

3

1 dp N pm

(32)

Hence, for the same cell size (i.e., Sc), the effective ratio of Sc/dpm is much greater than that in the conventional approach (i.e. Sc/dp), reducing the error associated with the calculation of particle segment volume (Peng et al., 2014). In order to calculate the particle segment volume in computational cells, the centroid position and the volume of each particle grid need to be known. To reduce the computational time, a template particle was introduced. The template particle has a radius of r0 and the centroid position defined as x0 = (0, 0, 0). The origin of local Cartesian coordinate is set at the centroid of the template particle. The template particle is meshed into a number of small particle grids. The data of each grid (i.e., centroid xeo and volume Veo) are calculated and stored in arrays of X and V before running the CFD-DEM simulation. In each time step, the data stored in X and V are mapped into the data of real particles that are moving around in the system. To this end, the coordinate transformation and magnification are conducted to convert the data of template particle grids into the local values of a real particle by, xe  xp

Ve

9 x e0  x 0

9 3V e0

(33) (34)

where xe and Ve are the local centroid position and volume of particle grids of real particles, respectively.  is the magnification factor defined as  = rp/r0 where rp is the real particle size. A

18 meshed template particle (with 1214 particle grids) and the acquisition of centroid position and particle grid volume are illustrated in Fig. 1.

Fig. 1. Particle meshing and acquisition of centroid position and volume of particle grids in particle meshing method: (a) template particle meshed by 1214 grids; (b) real particle and its particle grid.

After meshing, the spherical surface of a particle is meshed into a number of grids with planar surfaces. The sum of the volume of particle grids is thus slightly smaller than the volume of that particle. For this reason, the volume of each particle grid is normalised by a scaling factor of = Vsum/Vp. 3.1.4. Robust implementation of particle-fluid interactions The dynamic effects of the presence of discrete particles in a fluid flow are commonly taken into account by the terms involving cell void fraction and particle-fluid interaction forces. These two terms are often treated explicitly in the numerical solution (see e.g., Kafui et al., 2002; Tsuji et al., 1993; Xu and Yu, 1997). However, the explicit treatment of particle-fluid forces may induce convergence difficulty in the solution of a dense particulate flow (Hu and Joseph, 1992). In the present study, we calculated the particle-fluid forces semi-implicitly by treating the particle velocity (vi) implicitly, as detailed below. Since the non-linear momentum exchanging coefficient  is a function of uL and vi, we first explicitly linearise  using uL and vi at the previous time step according to the drag laws of Gidaspow (1994). Other terms such as pressure gradient and interphase slip velocities are evaluated implicitly. The linearisation of  leads to a system of linear equations. If only the fluid drag force and pressure gradient force are considered as fluid forces, the particle velocity can thus be calculated by,

19



mi v in1  v in



§ VEn ¨¨  Vi ’p n1  Vi ’ x  f n1  i in1 u Ln1  v in1 1  DL ©



·¸¸'t ¹

s

(35)

Based on the value of v in 1 and abiding by the Newton’s third law, the fluid-particle interaction forces are obtained. The interphase momentum exchange terms are subsequently calculated by summing up the fluid-particle interaction forces on each particle in the cell, weighted by the particle fractional volume (see Sections 2.2 and 2.3). Through the synchronized calculations of particle velocity and liquid flow velocity, the two-way coupling gets numerically enhanced and hence significantly increase the solution convergence time (Hu and Joseph, 1992; Wu et al., 2009). 3.2. Computational geometry and numerical methodologies and strategies The computational geometry for the fluid flow was kept consistent with the experimental set-up except the height of the vertical cylinder, which was reduced to 1.0 m to minimise the computational time. A non-uniform but symmetric mesh was generated on the bottom inlet face with dense meshes close to the boundary. The face mesh was then coopered along the axial direction of the computational domain. Consequently, a total number of 5600 hexahedral meshes were generated. The computational geometry and mesh for the calculation of fluid flow are depicted in Fig. 2.

Fig. 2 Computational geometry: (a) domain and (b) mesh.

The Semi-Implicit Method for Pressure-Linked Equation (SIMPLE) algorithm was used to solve the pressure-velocity coupling equations of the fluid flow, namely the continuity and momentum conservation equations (i.e., Eqs. (26) and (27)). The quadratic upwind interpolation of convective kinematics (QUICK) scheme was employed for the spatial discretisation of the convection term. The diffusion term was discretised by a central-differenced scheme that always is of second-order accuracy.

20 The Green-Gauss Node Based method was employed to calculate the variable gradients for constructing values of a scalar at cell faces and also for computing secondary diffusion terms and velocity derivatives. In each time step, the globally scaled residual of 10-5 has been set as the convergence criteria for solving fluid phase equations. The fluidised particles with the same total mass (i.e. 0.5255 kg) in experiments were randomly distributed in the domain. A simulation on the sedimentation process of the fluidised particles was conducted separately and the final steady packed bed was used as the initial condition of the solid phase in each CFD-DEM simulation. To obtain the steady state of the fluidised bed, 20 s (with water) or 30 s (with NaI) of simulation was completed before dropping the dense foreign particle from the top of the computational domain. A further 15 s of simulation was conducted to allow the foreign particle to settle through the fluidised bed. However, to save computational resources, the simulation will automatically abort if the foreign particle reaches the bottom. For the sake of accuracy, a value of 104 for the spring stiffness has been chosen for all computational cases in this study, corresponding to a solid time step of 5×10-5 s and a maximum particle normal overlap of 0.1074% rp (averaging over 10 s simulation; dP1 = 6 mm, dP2 = 5 mm, UL = 0.06 m/s). The simulation conditions and parameters used in the simulations are listed in Table 1.

Table 1. Simulation conditions and parameters.

Multi-thread parallel computation (OpenMP) based on the shared memory was conducted to improve the computational efficiency. For a typical parallel simulation with 5600 meshes and 16668 particles using two computational cores (clock speed 2.66 GHz, Smart Cache 12 M and QPI speed 6.4GT/s), it expends approximately 66 hours to complete the simulation of 23 s solid-liquid flow.

21

4. Experimental 4.1. Measurement of classification velocity The schematic of the experimental set-up of the SLFB is shown in Fig. 3. It consisted of a glass circular column (1), with an inner diameter of 50 mm and a height of 1500 mm. The circular test section of the fluidised bed was encased in a square column (2) which was filled with the same liquid to ensure proper photographic images. The distributor was a perforated plate (3) containing 128 holes of 2 mm diameter on a triangular pitch of 3.1 mm. A calming section (4) packed with either 6 or 8 mm glass beads of 0.2 m height was provided to homogenise the liquid flow before it reached the liquid distributor. The required liquid flow rate through the column was maintained using a rotameter (5) and globe valve at the bed inlet. A 2 HP centrifugal pump (6) was used for pumping the liquid from storage tank (7). Borosilicate glass beads with mean diameter of 3±0.03 and 5±0.03 mm have been used as fluidised particles. The glass beads were purchased from Sigmund Lindner (Germany) under the trade name Silibeads type P having refractive index of 1.472. Proper arrangements (8) were made for the insertion of the foreign particle from top of the fluidised bed. A heat exchanger (9) was provided in circulation loop to maintain the temperature of fluid.

Fig. 3 Schematic of the experimental set-up of the SLFB.

Precision-diameter dense (steel) particles with mean diameters (±2 μm) of 4 mm and 6 mm were used for the classification velocity measurements. The settling velocities of single dense particle were measured over a vertical distance of approximately 100 mm to 200 mm from bottom using a high speed video camera (model: Photron Fastcam Super-10K) and a strong LED lighting behind the bed. The classification velocity was calculated by tracking the centre-motion of the dense particle from images

22 (512×420 pixels) captured at 500 frames per second giving a time resolution of 0.002 sec. For 5 mm particles, it was observed that the dense particle can be easily seen even at high solid concentrations. However, with 3 mm fluidised particles, the refractive index of the particles needed to be matched to that of the liquid in order to visualise the foreign particle clearly. Aqueous sodium iodide (NaI) solution was used to match the refractive index with borosilicate spheres. Lab reagent grade sodium iodide powder was purchased from Sigma Aldrich. To prepare the solution, sodium iodide powder was weighed in proportion to required amount of water. Powder was gradually added to water and dissolved with rod stirrer. The best RI match was obtained for RI = 1.472 corresponding to 58.5% w/w sodium iodide in water. To maintain the clarity of solution, sodium thiosulphate is added to the solution. Addition of approximately 0.1 g of sodium thiosulphate is sufficient to clarify 1 litre of sodium iodide solution. Due care was taken to ensure that the particles had reached terminal velocity prior to entering the test section and not influenced by the mean flow. The fractional liquid hold-up was varied from 0.42 to 0.80. Each settling velocity reported is the average of five measurements, where the reproducibility was observed to be within r5 per cent. 4.2. Turbulence characterisation using particle image velocimetry (PIV) The PIV measurements were undertaken using the experimental set-up shown in Fig. 4. The RI matched sodium iodide solution was filled in the fluidised bed and the surrounding square column, with the fluidised bed particles being 3 mm or 5 mm diameter high-precision borosilicate glass beads. Dantec PIV system was used consisting of Litron LDY 300 laser capable of generating 30 mJ/pulse energy at 1000 Hz. Phantom v640 camera capturing image pairs with resolution of 1600 x 1600 pixels at 900 Hz and BNC 575 synchroniser were employed. The high speed system provided time resolved data, giving a closer look at the dynamics liquid velocity within the fluidised bed. The field of view was 50 × 50 mm covering the complete column diameter. The refractive index matched sodium iodide solution was seeded with fluorescent seeding particles.

23

Fig. 4 Experimental set-up for PIV measurements.

Post-processing of the captured raw PIV images was undertaken to determine the velocity vectors. The raw PIV images were processed using the image processing routines programmed in MATLAB R2011a. Out-of-plane motion of the seeding particles and strong local velocity gradients caused some spurious velocity vectors. Median filtering, with a threshold value 1.5 times the median of surrounding vectors, was applied to filter the high spurious vectors. A signal-to-noise ratio of 4 was applied to filter the low spurious vectors. Parameters like time difference between laser pulses, light sheet thickness and seeding density were optimised so that spurious vectors remained below 2 percent.

5. Results and discussion 5.1. Fluidisation of solid-liquid systems The hydrodynamics of solid-liquid fluidised bed was first investigated to ensure the established model is capable to correctly capture the complex interactions in the multi-particle systems, namely particleparticle and particle-fluid interactions. Fig. 5 shows the total pressure drop of the liquid fluidised bed (in water) as a function of superficial liquid velocity ranging between 0.03 m/s and 0.24 m/s. As typically exhibited in fluidised beds, the pressure drop keeps increasing until the onset of fluidisation, after which the pressure drop fluctuates around a constant value. As for each liquid superficial velocity, the simulation was run for a sufficient length of time (20 s) to obtain the steady state of fluidisation, the simulation process is equivalent to the de-fluidisation process and the hysteresis effect was not reflected in Fig. 5.

24 Fig. 5 Pressure drop versus liquid superficial velocity (dP1 = 5 mm, liquid: water).

As commonly applied in the literature to determine the minimum fluidisation velocity, two fitting lines (y = 51750x – 676.42 and y = 1520) were drawn for the packed bed regime and the fluidisation regime, respectively. The intersection point (i.e. at x = 0.0424) is the minimum fluidisation velocity Umf. The theoretical minimum fluidisation velocity can be calculated by equating the expression for the apparent weight of the particles with the revised Ergun equation (Doroodchi et al., 2012):

1  D L UP  UL g

2 § · ¨18 ˜ PLU mf ˜ 1  D L  0.33 ˜ ULU mf ˜ 1  D L ¸ 2 4.8 4.8 ¸ ¨ dP dP DL DL ¹ ©

(36)

where L is the packed bed voidage, P and L are the particle density and the fluid density, respectively;

L is the fluid viscosity. The predicted Umf by Eq. (36) is 0.0423 m/s, which agrees well with the value predicted by the CFD-DEM model (i.e. Umf = 0.0424 m/s). The total pressure drop calculated based on the apparent weight of the particles is 1450 Pa, which is 4.6% lower than the predicted value, i.e. 1520 Pa. The results verify the capability of the model to correctly capture the fluidisation hydrodynamics of SLFB. Fig. 6 illustrates the transient snapshot of the foreign particle in the solid-liquid fluidised bed at the end of the calculations for superficial liquid velocities around the minimum fluidisation velocity of the fluidised particles (Umf = 0.042 m/s). The foreign particle (dP2 = 6 mm) is coloured in red, and the fluidised particles (dP1 = 5 mm) are coloured in white and displayed with 20% transparency. The starting point of the timer is when the foreign particle was released from the top of the column (height: 1 m). It can be seen the foreign particle rests at the top of the solid-liquid fluidised bed at UL = 0.03 and 0.04 m/s (Figs. 6(e) and 6(d)), i.e. below the minimum fluidisation velocity. There is a less than 1 mm penetration depth due to the inertia of the foreign particle when it reaches the bed surface. The penetration depth decreases as the superficial liquid velocity increases due to the increasing upwards

25 drag force and the consequent smaller inertia of the foreign particle. At UL = 0.05 m/s (Fig. 6(c)), that is just above the minimum fluidisation velocity, the foreign particle continues to fall at the end of the simulation (t = 15 s). At higher liquid superficial velocities (UL = 0.06 and 0.07 m/s, i.e. Figs. 6(b) and 6(a)), the foreign particle settles to the bottom very quickly, i.e. 4.54 s and 2.79 s for UL = 0.06 m/s and UL = 0.07 m/s, respectively.

Fig. 6 Final snapshot of particle settling in the liquid fluidised bed at superficial liquid velocities around the minimum fluidisation velocity (dP1 = 5 mm, dP2 = 6 mm, liquid: water, Umf = 0.042 m/s): (a) UL = 0.07 m/s; (b) UL = 0.06 m/s; (c) UL = 0.05 m/s; (d) UL = 0.04 m/s; (e) UL = 0.03 m/s.

Fig. 7 shows the evolution of vertical position of the foreign particle for the superficial liquid velocities around the minimum fluidisation velocity (Umf = 0.042 m/s). It can be seen that at superficial liquid velocities of UL = 0.03 and 0.04 m/s, the foreign particle stops falling shortly after reaching the surface of the solid-liquid bed (H0 = 0.2 m). At UL = 0.05 m/s the bed expands and the foreign particle starts falling through the bed of particles. Due to the significant hampered effect of particle-particle interactions, it takes a long period of time (> 30 s) for the foreign particle to settle to the bottom. As UL increases further, the bed expands more and leads to a higher bed voidage. Subsequently, the foreign particle settles to the bottom very quickly, although the upwards drag force increases as UL increases. It implies that in the cases shown in Fig. 6, the particle collisional effects are dominant over the fluid viscous effects. The above results also explain the liquid-like properties of fluidised beds and reveal that the solid hold-up or bed voidage plays a key role in the settling process of the foreign particle.

Fig. 7 Evolution of vertical position (height) of the foreign particle (dP1 = 5 mm, dP2 = 6 mm, liquid: water).

26

5.2. Classification velocity of the foreign particle The data related to the foreign particle were monitored through the entire settling process. Fig. 8 shows the evolution of transient data of the foreign particle, including particle vertical position (xp, z), particle vertical velocity (vp, z), magnitude of particle angular velocity (|p|), liquid volume fraction (L, @p) and liquid vertical velocity (vL,

z, @p)

at the particle position, and particle Reynolds number (Rep). The

particle experienced three sequent processes: acceleration after being released from the top, steady falling after reaching the terminal velocity, and settling through the liquid fluidised bed. After being released from the top, the particle started falling under the effect of gravity. The velocity of the particle continuously increased until it reached the terminal value when major forces, i.e., buoyancy force, drag force and gravity, balance. The fluid velocity surrounding the particle was also observed to be varying slightly due to the motion of the particle. After the particle reached the surface of the liquid fluidised bed, there was a significant drop of particle vertical velocity due to collisions with fluidised particles. However, the particle re-reached the steady state very quickly due to the collisions with fluidised particles (or greater viscosity of the pseudo-fluid) (Gibilaro et al., 2007). Thereafter, the foreign particle was settling through the fluidised bed at the constant settling velocity, i.e., particle classification velocity (V2). The local liquid void fraction was fluctuating around 0.55 in the range of 0.46 – 0.8, which implies the liquid fluidised bed at this superficial liquid velocity (i.e. UL = 0.1 m/s) has a heterogeneous structure (as detailed below).

Fig. 8 Settling of the foreign particle through the liquid fluidised bed column (dP1 = 5 mm, dP2 = 6 mm, UL = 0.1 m/s, liquid: water).

27 Several approaches can be applied in the simulations to calculate the classification velocity of the foreign particle, e.g. averaging the particle vertical velocity over a time period or dividing distance by the total settling time. In experiments, the particle classification velocity was measured through image processing: the time that is taken for the particle to pass through a region with the height of 0.1 m – 0.2 m. To keep consistent with experimental measurements and also ensure that the simulation data were sampled in the steady state, the particle classification velocity was calculated in the same way as that in experiments. Specifically, the particle vertical position (i.e. xp, z) was plotted against time and the slope of the fitting line (i.e. dxp,z/dt) was considered as the classification velocity of the foreign particle. Fig. 9 illustrates the above methodology for the calculation of particle classification velocity for superficial liquid velocities from 0.06 m/s to 0.24 m/s. Table 2 lists the equation and the R-squared value of the fitting lines. It can be seen the majority of R-squared values of the fitting lines are around 0.99 with a minimum value of 0.954. It indicates that the above approach can be applied to calculate the particle classification velocity in SLFB.

Fig. 9 Particle classification velocity based on the evolution of particle vertical position (dP1 = 5 mm, dP2 = 6 mm, liquid: water).

Table 2. Linear fitting to the data of particle vertical position vs. time (dP1 = 5 mm, dP2 = 6 mm, liquid: water).

The particle classification velocities are plotted in Fig. 10 as a function of superficial liquid velocity. Consistent with the findings in the literature (see e.g. Grbavcic et al., 2009; van der Wielen et al., 1996), the plot reveals that the particle classification velocity increases with the superficial liquid velocity. An exponential function was applied to fit the scattered simulation data. It shows that the particle classification velocity increases faster in the early stage (UL < 0.09 m/s). At the incipient

28 fluidisation (for superficial liquid velocities just above the minimum fluidisation velocity) the packed bed slightly expands. The space between fluidised particles may be even smaller than the foreign particle size. In such cases, particle-particle collisions are the dominant mechanism controlling the settling process of the foreign particle. As the superficial liquid velocity increases, the fluidised bed expands more, and additional space is available for the particle to pass through. Therefore, the particle classification velocity increases sharply. At this moment, the particle is suffering from fluid drag hindrance rather than colliding and displacing fluidised particles, i.e. particle collisional hindrance. As the superficial liquid velocity increases further, the particle can easily settle through the space between fluidised particles and thereby the particle classification velocity increases smoothly. The experimental data of particle classification velocities for superficial liquid velocities of 0.06 m/s – 0.22 m/s are also shown in Fig. 10. It can be seen that a good agreement between simulation and experiment is obtained as the experimental data scatter closely around the exponential fitting line. However, it is worth noting that both predicted results and experimental data of particle classification velocities vary non-monotonically and the data are more scattered around the fitting line for high superficial liquid velocities, i.e. the maximum fitting error is 8.5% for UL  0.08 m/s and 20.6% for UL  0.09 m/s. This is associated with the transition of solid-liquid fluidisation regime, i.e. from homogeneous to heterogeneous, which is to be discussed in detail in the next section. Specifically, at low superficial velocities, a homogeneous liquid fluidised bed was obtained. In such cases, the measured classification velocity was time independent, as the distribution of fluidised particles in the bed and the fluid flow were almost the same over time. In other words, the particle settling history through the region of h = 0.1 m – 0.2 m of the liquid fluidised bed remained the same for measurements at different times. However, at high superficial liquid velocities, the local solid concentration and fluid flow structure in the sampled region (i.e. 0.1 m – 0.2 m) varied significantly with time, resulting in the fluctuating value of particle classification velocity. Therefore the particle

29 classification velocities appear more scattered for high superficial liquid velocities and the deviation between simulation and experiment becomes larger.

Fig. 10 Classification velocity as a function of superficial liquid velocity (dP1 = 5 mm, dP2 = 6 mm, liquid: water).

Fig. 11 plots the particle classification velocities predicted by CFD-DEM simulations and various empirical correlations against the experimental data. It can be seen that the data predicted by CFDDEM in this work provide the best agreement with the experimental data with all deviation points less than 10%. The correlations proposed by Joshi (1983) and Grbavcic et al. (2009) provide a good agreement with experimental data for dilute systems. In dense systems (e.g. at the smallest superficial liquid velocity of UL = 0.06 m/s) a remarkable deviation is observed (far beyond 20%). The rest correlations developed in literature include those by K&B (1966), K&L (1969), Di Filece et al. (1991) and van der Wielen et al. (1996) provide very poor predictions due to different operating conditions (e.g. size and density of particles) used in the experiments for the development of these empirical correlations.

Fig. 11 Predicted particle classification velocity versus experimental measured data (dP1 = 5 mm, dP2 = 6 mm, liquid: water).

30 5.3. Transition of solid-liquid fluidisation regime The fluid velocity (uL) of fluid flow in a small window (x

[-0.025, 0.025], y = 0, z

[0.1, 0.2]) of the

middle xz cross-section (y = 0) was averaged over a time period of t = 10 – 30 s (in NaI). The fluctuating fluid velocity (u’L) of fluid flow was calculated by,

u L'

1 Nc

§ 1 ¨ ¨ t  t / 't bs f 1 © es

Nc

¦ c

¦ u

·

t es

t t bs

L

c, t  u L (c) ¸¸

(37)

¹

where Nc is the total cell number of the window. u L (c ) is the average fluid velocity in computational cell c over a time period and uL(c, t) is the fluid velocity in cell c at time t. t is the time at each time step, thus is discrete. tbs is the beginning time to process data and tes is the ending time of data processing. In this study, tbs and tes are set as 10 s and 30 s, respectively. tf is the fluid time step. The fluctuating fluid velocity is shown in Fig. 12 as a function of average bed voidage. The first 10 s of simulation time was set for the solid-liquid fluidised bed to reach the steady state. It can be seen that for low superficial liquid velocities, i.e. low bed voidages, the values of fluctuating fluid velocity are much smaller compared to the values at high bed voidages. It implies the turbulence in the solid-liquid fluid bed is nearly homogeneous and stationary in dense fluidised beds. Remarkably, a sharp increase of u’L is observed for both dp1 = 3 mm and dp1 = 5 mm when the bed voidage increases from 0.52 to 0.62. It would be reasonable to expect that the solid-liquid flow structure transits from homogeneous to heterogeneous after the bed voidage becomes higher than a critical value, which is sitting between 0.52 – 0.62. The available experimental data of the fluctuating fluid velocity measured by PIV is also included in Fig. 12. It can be seen that a good agreement is obtained between simulation and experiment with a maximum deviation around 15%. The deviation may be attributed to the ideal (theoretical) conditions used in the simulations (e.g., uniform fluid flow profile applied at the inlet of the column, ideal particle size and shape, and ideal fluid properties), which are different to some extent from practical or actual cases in experiments. Moreover, the limited operation error and inherent flaws

31 of methodologies for data acquisition and processing in experiments might have also contributed to the deviation.

Fig. 12 Averaged value of fluctuating liquid velocity as a function of bed voidage (liquid: NaI).

To explore the relationship between bed voidage and the solid-liquid fluidisation regime, the superficial velocity used in simulations was increased gradually from 0.06 m/s with a step of 0.01 m/s, providing a wide range of bed voidage from 0.46 to 0.8. Fig. 13 shows the flooded contour plots of local liquid volume fraction of the middle xz cross-section (y = 0) for superficial liquid velocities of 0.06 – 0.11 m/s at t = 20 s. At the lowest superficial liquid velocity (i.e. UL = 0.06 m/s, Fig. 13(a)), the particles are fluidised and evenly distributed in the column, thus a typical homogeneous fluidisation is obtained. As the superficial liquid velocity increases (i.e. UL = 0.07 - 0.08 m/s, Fig. 13(b)-13(c)), some tiny local voids appear in the region close to the wall. As the superficial liquid velocity continuously increases, the voids grow bigger and expand to the central region. For superficial liquid velocities greater than 0.09 m/s (Fig. 13(e)-13(f)), the voids spread over the entire fluidised bed, leading to a heterogeneous solid distribution in the column and hence the heterogeneous fluidisation regime.

Fig. 13 Contour plots of liquid volume fraction (dP1 = 5 mm, dP2 = 6 mm, t = 20 s, liquid: water): (a) UL=0.06 m/s, L=0.46; (b) UL=0.07 m/s, L=0.48; (c) UL=0.08 m/s, L=0.51; (d): UL=0.09 m/s, L=0.54; (e) UL=0.10 m/s, L=0.56; (f): UL=0.11 m/s, L=0.58.

The contour line of vorticity (component at y direction) of the middle xz cross-section (y = 0) is calculated and plotted in Fig. 14. A small window (z

[0.1, 0.2]) of the xz cross-section is zoomed in

32 with the flooded contours of fluid flow velocity magnitude and flow vector for superficial velocities of 0.06 m/s and 0.11 m/s, as shown in Figs. 14(A) and 14(B). For superficial liquid velocities lower than 0.09 m/s (Figs. 14(a) – 14(c)), the vorticity magnitude is small and the liquid flows upwards in order (Fig. 14(A)). No large gradients of vorticity are observed. The liquid flow is thus homogeneous. As the superficial liquid velocity increases beyond 0.09 m/s, the vortices distribute asymmetrically in the domain and more small vortices appear in the flow field (Figs. 14(d) – 14(f)). The magnitude of vorticity varies over a wide range, which indicates the existence of heterogeneity. The typical heterogeneous fluid flow field with large vortices can be observed for UL = 0.11 m/s (Fig. 14(f) and Fig. 14(B)). The results indicate that the structure of the liquid fluidised bed transits from homogeneous to heterogeneous at UL = 0.09 m/s, which corresponds to an average bed voidage of 0.54.

Fig. 14 Contour plots of y-component vorticity, fluid velocity magnitude and fluid flow vectors (dP1 = 5 mm, dP2 = 6 mm, t = 20 s, liquid: water): (a) UL=0.06 m/s, L=0.46; (b) UL=0.07 m/s, L=0.48; (c) UL=0.08 m/s, L=0.51; (d): UL=0.09 m/s, L=0.54; (e) UL=0.10 m/s, L=0.56; (f): UL=0.11 m/s, L=0.58.

5.4. Particle-particle collision 5.4.1. Particle-particle interactions and collision events Fig. 15 shows snapshots of the motion of the foreign particle in the LSFB for average bed voidages of 0.45, 0.65 and 0.79. The dense foreign particle (dP2 = 6 mm) is coloured in red and the fluidised particles (dP1 = 3 mm) are coloured in blue. The entire domain is clipped along the y direction, thus a part of the SLFB is shown in Fig. 15 in order to have a close look at the interactions between the foreign particle and surrounding fluidised particles. At the low bed voidage of L = 0.45 (Fig. 15(a)) the liquid fluidised bed is in the homogeneous fluidisation regime with fluidised particles nearly evenly

33 distributed in the bed. The foreign particle collides with particles below and pushes the particles aside. A void space is subsequently generated following the motion of the foreign particle, as labelled with ellipses in Fig. 15(a). The trajectory of the foreign particle is almost linear, which implies that the collision intensity is too small to overcome the vertical inertia of the foreign particle and change its moving direction. As the bed voidage increases further to 0.65 (Fig. 15(b)), the bed becomes diluter and the solid-liquid flow turns into the heterogeneous regime. The particle number density around the foreign particle is smaller compared to those in Fig. 15(a). The fluidised particles in the vicinity of the foreign particle are observed flowing downwards along with the foreign particle, and the particles in other regions mostly are flowing upwards. As the bed voidage increases to 0.79 (Fig. 15(c)), the fluidised bed is very dilute, and the foreign particle moves downwards much faster and more freely (i.e. less collisions). In turn, the motion of fluidised particles is not influenced by the motion of the foreign particle. However, the trajectory of the foreign particle alters a lot at L = 0.79. For instance, the transverse motion of the foreign particle becomes more intensely, e.g. more fluidised particles are observed in the region of y  0.1. This is due to the increasing relative velocity between the particles, which results in the increase in the collision intensity and leads to the change in the transient moving direction of the foreign particle.

Fig. 15 Snapshot of the foreign particle motion through the SLFB (dP1 = 3 mm, dP2 = 6 mm, fluid: water): (a) L = 0.45, y  0; (b) L = 0.65, y  0; (c) L = 0.65, y  0.1.

It can be seen that a particle often collides with multiple particles at the same time, especially in dense systems. This poses a huge challenge to experimental measurements of collision events. However, in the simulations, the number of contacting particles and any new contact of the foreign particle can be detected and recorded to describe the collision behaviour. Fig. 16 shows the average and the maximum

34 number of particles that are contacting with the foreign particle in the entire settling process. It is shown that the maximum number of contacting particles reaches 8 at L = 0.45 and keeps constant at 2 for L  0.65. The average number of contacting particles decreases from 3.54 to 0.06 as the bed voidage increases from 0.45 to 0.79. The results imply that the particle-particle collision happens very rarely in dilute systems.

Fig. 16 Average and maximum number of fluidised particles that are contacting with the foreign particle during the entire settling process (dP1 = 3 mm, dP2 = 6 mm, liquid: water).

Fig. 17 shows the evolution of the cumulative number of contacts of the foreign particle and the corresponding instant particle velocity. An increase in the cumulative number of contacts denotes the occurrence of a collision event between the foreign particle and fluidised particles. It can be seen that the particle collision immediately incurs the pulsation of particle velocity. The z-component of the particle velocity (i.e. vpz) responds instantly to every collision event. However, the x- and y-components of particle velocity (i.e. vpx and vpy) may not sense some particle collision events, e.g. vpy in the time periods I (t = 1.004 s – 1.008 s) and IV (t = 1.068 s – 1.076 s); vpx in the time period III (t = 1.048 s – 1.056 s). It means that the energy dissipated due to particle collisions is mainly reflected by the dampened particle vertical velocity. In other words, the pulsation of particle vertical velocity can be used as a good indication of particle collision events, as already employed by Aguilar-Corona et al. (2011) in their experiments. However, it should be noted that the collisions often occur in a group, i.e. more than a single collision. This causes inconsistency between the pulsation of particle velocity and the number of collision events if this approach is used in experiments to describe the particle collision frequency. Moreover, it is worth noting that the velocity of the foreign particle varies or decreases linearly before and after the collision sections. It implies that the collisional effects play the dominant

35 role in the particle settling process in the context of this study (with Stp > 65). A good knowledge of particle collision events is the key to well understand the particle settling behaviour.

Fig. 17 Particle collisions and consequent variation of particle velocity (dP1 = 5 mm, dP2 = 5 mm, UL = 0.20 m/s, liquid: NaI).

5.4.2. Collision frequency In the simulation, the transient velocity of particles was monitored during the motion of the foreign particle in the SLFB. With the data of particle instantaneous velocities and particle size, the dimensionless collision frequency was calculated as (Aguilar-Corona et al., 2011): * f coll

f colld P q

(38)

2 p

where dp is the particle diameter and qp2 is the “small scale” fluctuating kinetic energy of the solid phase in the fluidised bed. The fluctuating kinetic energy can be calculated as:

qp2

3 '2 up 2

(39)

where up'2 is the particle velocity variance. Fig. 18 shows the variation of particle fluctuating kinetic energy (estimated using Eq. (39)) over a wide range of solid fraction (i.e., 0.2 – 0.6). It can be seen that the particle fluctuating kinetic energy decreases with solid fraction, which implies the effects of turbulence on the particle motion become significant and the effects of particle collision events are diminishing due to the decreasing solid fraction.

36

Fig. 18 Fluctuating kinetic energy of the particle phase as a function of solid fraction S (dP1 = 5 mm, dP2 = 6 mm, liquid: water).

The dimensionless particle collision frequency predicted by CFD-DEM simulations for solid fractions ranging between 0.2 and 0.8 is shown in Fig.19. The figure shows that the collision frequency increases monotonically with the solid fraction. The dimensionless collision frequency was also calculated by the classic KTGF (Simonin, 1991), as shown in Fig. 19. The maximum solid fraction, i.e. Sm used in Eq. (7) for the calculation, is 0.575, which was measured in experiments. It can be seen that the dimensionless collision frequency obtained by CFD-DEM excellently fit the data line predicted by KTGF.

Fig. 19 Dimensionless collision frequency as a function of the solid fraction (dP1 = 5 mm, dP2 = 6 mm, liquid: water).

The variation of collision frequency is shown in Fig. 20 as the foreign particle size (dP2) and fluidised particle size (dP1) are altered and the particle size ratio (dP2/dP1) varies between 0.6 and 2.0. The data on the effect of fluid medium, namely water and NaI, has also been included. It can be seen that as the particle size ratio increases, the collision frequency increases for a fixed solid fraction. This can be easily understood as the foreign particle needs to displace N fluidised particles when it settles through the liquid fluidised bed (Gidaspow, 1994). Since non-elastic collisions occur between the particles, N becomes greater as dP2/dP1 increases, indicating the occurrence of more collisions between the foreign particle and fluidised particles. The collision frequency is observed decreasing steeply at high solid phase fractions (S > 0.4), e.g. fcoll = 1649/s at S = 0.54 and fcoll = 101/s at S = 0.42. At low solid

37 fractions the collision frequency decreases gently as solid fraction decreases. It can be attributed to the two counteracting effects at low solid phase fractions. On the one hand, as the solid fraction decreases, the average distance between neighbouring particles increases, which leads to a drop in the particle collision frequency. On the other hand, as the liquid velocity increases, the bed hydrodynamics becomes more turbulent and the rate at which the particles circulated in the bed increases, which in turn increases the collision frequency (Pozo et al., 1993). However, the former effect is more dominant than the latter effect and as a result, the particle collision frequency decreases as the solid phase fraction decreases, but not as steeply as those at high solid phase fractions. When the size of fluidised particles is the same, the particle collision frequency is independent of the foreign particle size, remaining almost constant at high solid phase fractions (S > 0.54). As discussed above, for S > 0.54 the homogeneous fluidisation regime is obtained (Section 5.3). The fluidised particles are evenly distributed in the domain. Therefore, the chance for the foreign particle to come across fluidised particles in a space of certain volume should be equal, giving rise to a constant collision frequency. It can also be seen that the effect of fluid medium on the particle collision frequency is negligible with the trend lines (fcoll versus S) overlapping for a fixed particle size ratio (e.g., dP2/dP1 = 2.0 or dP2/dP1 = 1.2). This result may be explained by the high particle Stokes number (Stp > 65) investigated in this study. Under the high Stokes number the viscous effects of interstitial fluid on the settling of the foreign particle are negligible and the collisional effects can be considered dominant. In such an inertia flow regime, the motion of the foreign particle is not influenced by the fluid medium when the bed voidage of the liquid fluidised bed keeps constant. Therefore, the change in the fluid medium does not lead to any significant difference in the particle collisional behaviour.

Fig. 20 Collision frequency as a function of the solid fraction, particle sizes and fluid medium.

38 5.4.3. Collision force acting on the foreign particle 5.4.3.1. Significance of collision force The collision force acting on the foreign particle normalised by the weight force of foreign particle (mg) was monitored in the simulation at a frequency of 1000 Hz. Fig. 21 shows the evolution of the normalised collision force for various bed voidages ranging between 0.46 and 0.79. Unlike other forces (e.g. drag and gravitational forces), the evolutional line of collision force is not continuous with zero values lasting over a period of time. This is because the foreign particle does not collide with any particles in some time periods, especially in dilute systems. The non-zero valued data dots denote the contacting status of the foreign particle with fluidised particles. It can be seen the non-zero valued data dots are denser at lower bed voidages, corresponding to more frequent particle collisional events. However, the magnitude of the normalised collision force increases with the bed voidage. It is worth noting that the magnitude of the normalised collision force is much greater than the gravitational force with the ratio |Fc|/mg ranging between 10 and 50. The results suggest that the collision force is significant in SLFBs and must be accounted for while conducting analysis based on the force balance. Also, a simple correlation should be available to estimate the collision force as a function of bed voidage.

Fig. 21 Evolution of collision force acting on the foreign particle (dP1 = 5 mm, dP2 = 6 mm, liquid: water): (a) L = 0.46; (b) L = 0.51; (c) L = 0.60; (d) L = 0.67; (e) L = 0.71; (f) L = 0.79.

Fig. 22 shows the magnitude of fluid drag force and pressure gradient force as a function of bed voidage (0.46  L  0.8). The magnitude of the forces was calculated as the average value of the total forces acting on the foreign particle since it falls into the SLFB. The drag force increases at a decreasing rate with the bed voidage due to the increased upwards fluid velocity. The magnitude of

39 drag force increases from 0.1 mg to 0.5 mg as the bed voidage increases from 0.46 to 0.8. The magnitude of pressure gradient force does not change significantly, decreasing from 0.11 mg to 0.07 mg as the bed voidage increases from 0.46 to 0.8. In the homogeneous fluidisation regime (i.e. L < 0.54), the pressure gradient force is merely due to the static pressure gradient (equal to the buoyancy force of the particle, i.e. f/p = 998.2/7800 = 0.128). The result implies that as the bed voidage increases, the solid distribution and the fluid flow in the bed are no longer homogeneous. The turbulence caused by the increased superficial liquid velocity becomes more significant and leads to the non-uniformly distributed local pressure gradients, which renders the decreasing pressure gradient force.

Fig. 22 Magnitude of fluid drag force and pressure gradient force as a function of bed voidage (dP1 = 5 mm, dP2 = 6 mm, liquid: water).

As the collision force acting on the foreign particle is not continuous, the magnitude of particle collision force was calculated by averaging the total collision force accumulated during the entire process of particle settling through the SLFB. The comparison of collision force and fluid drag force is shown in Fig. 23 as a function of bed voidage. It can be seen the ratio of collision force to fluid drag force decreases exponentially with the bed voidage. The collision force is the dominant force with the maximum magnitude of 9 times greater than that of the drag force for the bed voidages (0.45 < L < 0.8) investigated in this study. These estimates clearly provide insights about the significance of collision force which has been overlooked to date in the literature. It is also noted that the force ratio decreases very gently at high bed voidages (L > 0.6) due to the two counteracting effect as mentioned above, i.e. increasing average distance between neighbouring particles and increasing particle circulation rate. As the increased superficial liquid velocity also leads to the increase of fluid drag force, the evolution line

40 of the force ratio (|Fc|/|Fd|) appears even more flatly than that of the particle collision frequency, as shown in Fig. 20.

Fig. 23 Normalised collision force by fluid drag force as a function of bed voidage (dP1 = 5 mm, dP2 = 6 mm, liquid: water).

5.4.3.2. Relationship between collision force and bed voidage The collision force acting on the foreign particle is determined by two parts: how frequent the particle collides with fluidised particles (i.e. collision frequency) and how strong the collisions are (i.e. collision intensity). As discussed earlier, the particle collision frequency is related to local solid concentration or bed voidage. The collision intensity is mainly determined by the relative velocity between the foreign particle and fluid (Pozo et al., 1993; van der Wielen et al., 1996) and particle material properties. The solid-liquid relative velocity is the difference between particle classification velocity and the interstitial fluid velocity; both of them are a function of bed voidage. Considering the particle properties are known and identical and the collisional effects are dominant for Stp > 65 in this study, the collision force is thus described as a sole function of the average bed voidage. Fig. 24 shows the variation of the normalised collision force (by the weight force of foreign particle) as a function of bed voidage for a 6 mm dense foreign particle settling in the fluidised bed of 5 mm glass beads in water. It can be seen that the normalised collision force continuously decreases with an increase in bed voidage. As discussed above, this is because as the bed expands, the collision frequency decreases, which is more dominant over the increased collision intensity as the bed voidage increases. Consequently, a drop in the net collision force is obtained. The quadratic fitting method was applied to fit the scattered data to relate the normalised collision force to the bed voidage as:

41 Fc m p g

1.08  0.26D L  1.34D L2

St p ! 65,

d p2 d p1 d 2



(40)

Fig. 24 Relationship between the normalised collision force and bed voidage (dP1 = 5 mm, dP2 = 6 mm, liquid: water).

For two extreme cases: (i) Empty bed, where L = 1.0, |Fc|/mpg = 0, |Fc| = 0; (ii) Packed bed, where, L = 0.425, |Fc|/mpg = 0.948, |Fc| =0.948mpg. In this case, |Fc| balances the gravity together with fluid forces. As the particles are nearly stationary, the supporting forces from the fluidised particles around the foreign particle are considered as the total collision force acting on the foreign particle. It was thought desirable to compare the estimated normalised collision force in the present study with those predicted using various correlations from literature. Fig. 25 depicts the comparison which clearly shows that the magnitude of the normalised collision force predicted in present study (by Eq. (40)) matches that predicted by the correlation of van der Wielen et al. (1996). However, the predictions of Grbavcic et al. (2009) are much less as compared to others. Moreover, the present study clearly shows the normalised collision force decreases monotonically as the bed voidage increases. Considering the strong decline in the number of collisions as the bed voidage increases (as shown in Fig. 19), the effect of the increase in collision intensity would be drown out. The overall result would be expected to be a decrease in the normalised collision force. However, the predictions using the correlations by van der Wielen et al. (1996) and Grbavcic et al. (2009) appear non-monotonic and somehow reveal a peak value of the normalised collision force. The inconsistency between the predictions by correlations and

42 that in the present study by CFD-DEM simulations may be attributed to the assumptions and operating conditions used in the development of these correlations. Grbavcic et al. (2009) developed their correlation based on the assumption that the collision force is proportional to the total drag force acting on the N fluidised particles that have been displaced by the foreign particle during the settling process. This is valid only in the homogeneous fluidisation regime where the foreign particle must squeeze away a certain number of fluidised particles to fall down. In the heterogeneous regime, however, the liquid fluidised bed is dilute and the particle collision number significantly decreases (see Fig. 21). The foreign particle settles mainly through the space between the fluidised particles. In such cases, the assumption does no longer hold and the correlation may provide unreasonable outputs. Moreover, the collision coefficient (i.e. k in Eqs. (14) and (15)) depends on the particle size ratio (dP2/dP1) and the properties of the fluidise bed, which may also contribute to the deviation and inconsistency. The correlation by van der Wielen et al. (1996) was developed based on the force balance on the foreign particle, which seems more plausible. However, in a real liquid fluidised bed, especially in the heterogeneous regime, the forces acting on the foreign particle may be more than those considered in the correlation. Moreover, the heterogeneous structure of fluid flow and the non-uniform distribution of solid particles may result in the inapplicability of the formulations used for the calculation of the forces. For example, the pressure gradient force is equal to the buoyancy force in the homogeneous fluidisation regime, but is not the case in the heterogeneous fluidisation regime, as shown in Fig. 22.

Fig. 25 Comparison of the normalised collision force with different correlations from literature (dP1 = 5 mm, dP2 = 6 mm, liquid: water).

Fig. 26 shows the comparison between the prediction results of |Fc|/mg by the correlation (Eq. 40) and those obtained in the CFD-DEM simulations. It can be seen that the majority of the deviation fall into

43 the range of 10% and all the data points are within 20%, which verifies the validity of the correlation in the prediction of collision force as a sole function of the average bed voidage for these specific cases. The randomness of the sampled data and the hydrodynamical stability state of the fluidised bed are considered the major factors that are responsible for the deviations.

Fig. 26 Validation of Eq. (40) in the prediction of collision force based on the bed voidage.

Eq. (40) correlates the magnitude of the collision force only with the average bed voidage and should be used with due care in other situations. As discussed above, in all cases investigated in this study, the particle collisional effects on the settling of the foreign particle are dominant with the high particle Stokes numbers (Stp > 65). The interstitial fluid viscous effects are deemed negligible. However, in cases where the fluid viscous effects are dominant with a small Stokes number (Stp < 5) (Joseph et al., 2001; Rao et al., 2011), the fluid drag force is playing a significant role in determining the particle collision behaviour. As such, the applicability of this correlation needs to be verified in these cases. Moreover, the ratio of the foreign particle size and the fluidised particle size, i.e. dP2/dP1 is between 0.6 and 2. For dP2/dP1 > 2.0, the correlation may not be valid. In the work by Grbavcic et al. (2009), the particle collision force tended to be zero for particle size ratios (dP2/dP1) greater than 10. In such cases the correlation developed in this study would be invalid. Furthermore, Eq. (40) was developed based on the data of bed voidages up to 0.8. However, as indicated earlier, the correlation by itself is valid for cases with the bed voidages up to the ideally maximum bed voidage, i.e., L = 1.0 in empty beds.

6. Conclusions The settling behaviour of a dense foreign particle in a monodispersed solid-liquid fluid bed (SLFB) has been investigated by a fully coupled CFD-DEM model. Specifically, the particle motion was solved

44 and tracked by DEM and the fluid flow was solved by CFD; the fluid-particle interactions were solved by a full two-way coupling algorithm. The fluidisation hydrodynamics of SLFB, settling behaviour of the foreign particle, fluid-particle interactions and particle-particle collision behaviour have been investigated. Experiments including particle classification velocity measurements and fluid turbulence characterisation by PIV were carried out to verify the validity of prediction results. The CFD-DEM model established in this study was capable to correctly capture the solid-liquid fluidisation hydrodynamics. Compared to those predicted by various empirical correlations, the particle classification velocity predicted by CFD-DEM provided the best agreement with the experimental data with the maximum deviation of less than 10%. When the bed voidage was below 0.54, the solid-liquid flow exhibited typical homogeneous flow characteristics. The particles were evenly distributed in the bed and the fluid flew upwards in order with small vorticity gradients. The solid-liquid fluidisation transited to the heterogeneous regime for bed voidages (L) over 0.54 with local voids and large vorticity gradients spreading over the entire bed. The heterogeneous regime of solid-liquid fluidisation regime caused the large fluctuation of particle classification velocity. The particle collision immediately incurred the pulsation of particle velocity. The vertical component (z-direction) of particle velocity was very sensitive to every collision event, whist the horizontal components (x- and y- directions) of the particle velocity did not respond to some particle collision events. The collisions often occurred in a group, i.e. more than a single collision, in dense systems. The collision frequency increased monotonically with the solid fraction. The dimensionless collision frequency obtained by CFD-DEM excellently fit the data line predicted by KTGF. As the particle size ratio (dP2/dP1) increased, the collision frequency increased for a fixed solid fraction. The particle collision frequency increased steeply at high solid phase fractions (S > 0.4); at low solid fractions the particle collision frequency decreased gently as the solid fraction decreased. When the size of fluidised particles was the same, the particle collision frequency was independent of the foreign particle size at

45 high solid phase fractions (S > 0.54). The effect of fluid medium on the particle collision frequency was negligible for high Stokes numbers (Stp > 65). The magnitude of collision force acting on the foreign particle was much greater than the gravitational force with the ratio of |Fc|/mg ranging between 10 and 50. The collision force decreased as the bed voidage increased. The collision force was the dominant force when the bed voidage was less than 0.8, with the maximum magnitude being 9 times greater than that of the drag force. The collision force was described as a sole function of the average bed voidage when the particle collisional effects were dominant. The correlation proposed to relate the collision force solely to the bed voidage provided a maximum error of 20% in the prediction of collision force for Stp > 65 and dP2/dP1  2. Limitations of the correlation have also been addressed.

Acknowledgements Thanks to Richard Dear for his assistance with the usage of high performance cluster (HPC) facilities at the University of Newcastle. The authors wish to acknowledge the financial support of the University of Newcastle and the Australian Research Council for the work presented in this paper.

Nomenclature CD

drag coefficient, -

dP1

diameter of particles comprising fluidised bed, m

dP2

diameter of the foreign particle, m

d12

distance between the centre of the particles, m

dpm

equivalent particle grid size, m

D

column diameter, m

46 ec

normal restitution coefficient, -

f

Ruzicka (2006) transition function, -

fc

collision contact forces, N

ff

total fluid forces, N

fd

fluid drag force, N

fS-L

forces acting on fluid by the solid particles, N

fcoll

collision frequency, s-1

f*coll

dimensionless collisional frequency

FB

buoyancy force, N

FD

drag force, N

FG

gravitational force, N

g

gravitational acceleration, m/s2

g0

function of solid hold-up, -

I

moment of inertia, kg/m2

I

unit tensor, -

k

collision coefficient, -

n1

number of particles of phase 1 per unit volume, -

n2

number of particles of phase 2 per unit volume, -

N

number of displaced particles, -

N12

collision frequency of binary mixture, Hz

47 Npm qp2

number of particle grids, “small scale” fluctuating kinetic energy of the solid phase, m2/s2

rp

particle radius, m

Re

liquid Reynolds number, -

ReP

particle Reynolds number, -

Tc

torque due to the tangential component of contact force, N·m

Tr

torque due to the rolling resistance, N·m

ur

relative particle velocity during a collision, m/s

u’

bulk turbulence velocity, m/s

u p2

mean square velocity of the particles, m/s

u p'2

particle velocity variance, m/s

Vp

volume of single particle, m3

Vpm

volume of particle grid, m3

V2

particle classification velocity, m/s

Vsum

total particle grid volume, m3

uL

liquid velocity vector, m/s

up

mean velocity of the particles, m/s

UL

superficial liquid velocity, m/s

Umf

minimum fluidisation velocity, m/s

48 Ur

relative velocity between fluid and particle, m/s

US2

settling velocity of dense particle, m/s

US

terminal velocity of particle, m/s

v

particle velocity vector, m/s



particle angular velocity, rad/s

x

particle position, m

Greek letters 

fractional phase hold-up, -

S0

solid hold-up in the fixed bed, -



particle fractional volume in a cell, -



momentum exchanging coefficient, -



scaling factor, -

ts

solid time step, s



liquid bulk viscosity, kg/(m·s)

μ

viscosity of fluid, kg/(m·s)

M

mixture density, kg/m3

eff

effective or relevant density, kg/m3

f

viscous stress tensor, -



interaction coefficient, -

4

granular temperature, m2/s2

49 Subscripts 1

fluidised particles

2

dense foreign particle

f

fluid

L

liquid

i

index

P

particle

r

relative

S

solid

x, y, z direction component

infinite medium

50

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53 Table 1. Simulation conditions and parameters.

Table 2. Linear fitting to the data of particle vertical position vs. time (dP1 = 5 mm, dP2 = 6 mm, liquid: water). Fluid phase Liquid density, kg/m3

998.2 (water), 1800 (NaI)

Liquid viscosity, Pa·s

0.001 (water), 0.0018 (NaI)

Fluid time step, s

1×10-4

Solid phase Foreign particle diameter, mm

6, 5, 4, 3

Fluidized particle diameter, mm

3, 5

Foreign particle density, kg/m3

7800

Fluidized particle density, kg/m3

2300

Fluidized particle mass, kg

0.05255

Normal spring-stiffness, N/m

104

Restitution coefficient, -

0.9

Sliding friction coefficient, -

0.3

Rolling friction coefficient -

0.0175

Solid time step, s

5×10-5

54 UL (m/s)

Eq. of fitting line

R2

UL (m/s)

Eq. of fitting line

R2

0.06

y=-0.097x+2.269

0.99

0.15

y=-0.268x+6.024

0.9779

0.07

y=-0.147x+3.324

0.9988

0.16

y=-0.276x+6.335

0.9790

0.08

y=-0.191x+4.290

0.9966

0.17

y=-0.264x+6.335

0.954

0.09

y=-0.229x+5.103

0.9996

0.18

y=-0.304x+6.898

0.9982

0.10

y=-0.214x+4.779

0.9997

0.19

y=-0.359x+8.158

0.9988

0.11

y=-0.216x+4.824

0.9937

0.20

y=-0.339x+7.730

0.9956

0.12

y=-0.250x+5.566

0.9907

0.21

y=-0.372x+8.535

0.9877

0.13

y=-0.252x+5.642

0.9983

0.22

y=-0.371x+8.490

0.9544

0.14

y=-0.239x+5.356

0.9803

0.24

y=-0.422x+9.612

0.9936

Highlights

Particle collision frequency increased with solid fraction and particle size ratio x The collision force decreased as the bed voidage increased x Magnitude of collision force was 10-50 times greater than that of weight force x Magnitude of collision force was maximally 9 times greater than that of drag force x A correlation describing collision force as a function of bed voidage was developed

55 Fig. 1. Particle meshing and acquisition of centroid position and volume of particle grids in particle meshing method: (a) template particle meshed by 1214 grids; (b) real particle and its particle grid. Fig. 2 Computational geometry: (a) domain and (b) mesh. Fig. 3 Schematic of the experimental set-up of the SLFB. Fig. 4 Experimental set-up for PIV measurements. Fig. 5 Pressure drop versus liquid superficial velocity (dP1 = 5 mm, liquid: water). Fig. 6 Final snapshot of particle settling in the liquid fluidised bed at superficial liquid velocities around the minimum fluidisation velocity (dP1 = 5 mm, dP2 = 6 mm, liquid: water, Umf = 0.042 m/s): (a) UL = 0.07 m/s; (b) UL = 0.06 m/s; (c) UL = 0.05 m/s; (d) UL = 0.04 m/s; (e) UL = 0.03 m/s. Fig. 7 Evolution of vertical position (height) of the foreign particle (dP1 = 5 mm, dP2 = 6 mm, liquid: water). Fig. 8 Settling of the foreign particle through the liquid fluidised bed column (dP1 = 5 mm, dP2 = 6 mm, UL = 0.1 m/s, liquid: water). Fig. 9 Particle classification velocity based on the evolution of particle vertical position (dP1 = 5 mm, dP2 = 6 mm, liquid: water). Fig. 10 Classification velocity as a function of superficial liquid velocity (dP1 = 5 mm, dP2 = 6 mm, liquid: water). Fig. 11 Predicted particle classification velocity versus experimental measured data (dP1 = 5 mm, dP2 = 6 mm, liquid: water). Fig. 12 Averaged value of fluctuating liquid velocity as a function of bed voidage (liquid: NaI).

56 Fig. 13 Contour plots of liquid volume fraction (dP1 = 5 mm, dP2 = 6 mm, t = 20 s, liquid: water): (a) UL=0.06 m/s, L=0.46; (b) UL=0.07 m/s, L=0.48; (c) UL=0.08 m/s, L=0.51; (d): UL=0.09 m/s, L=0.54; (e) UL=0.10 m/s, L=0.56; (f): UL=0.11 m/s, L=0.58. Fig. 14 Contour plots of y-component vorticity, fluid velocity magnitude and fluid flow vectors (dP1 = 5 mm, dP2 = 6 mm, t = 20 s, liquid: water): (a) UL=0.06 m/s, L=0.46; (b) UL=0.07 m/s, L=0.48; (c) UL=0.08 m/s, L=0.51; (d): UL=0.09 m/s, L=0.54; (e) UL=0.10 m/s, L=0.56; (f): UL=0.11 m/s, L=0.58. Fig. 15 Snapshot of the foreign particle motion through the SLFB (dP1 = 3 mm, dP2 = 6 mm, fluid: water): (a) L = 0.45, y  0; (b) L = 0.65, y  0; (c) L = 0.65, y  0.1. Fig. 16 Average and maximum number of fluidised particles that are contacting with the foreign particle during the entire settling process (dP1 = 3 mm, dP2 = 6 mm, liquid: water). Fig. 17 Particle collisions and consequent variation of particle velocity (dP1 = 5 mm, dP2 = 5 mm, UL = 0.20 m/s, liquid: NaI). Fig. 18 Fluctuating kinetic energy of the particle phase as a function of solid fraction S (dP1 = 5 mm, dP2 = 6 mm, liquid: water). Fig. 19 Dimensionless collision frequency as a function of the solid fraction (dP1 = 5 mm, dP2 = 6 mm, liquid: water). Fig. 20 Collision frequency as a function of the solid fraction, particle sizes and fluid medium. Fig. 21 Evolution of collision force acting on the foreign particle (dP1 = 5 mm, dP2 = 6 mm, liquid: water): (a) L = 0.46; (b) L = 0.51; (c) L = 0.60; (d) L = 0.67; (e) L = 0.71; (f) L = 0.79. Fig. 22 Magnitude of fluid drag force and pressure gradient force as a function of bed voidage (dP1 = 5 mm, dP2 = 6 mm, liquid: water).

57 Fig. 23 Normalised collision force by fluid drag force as a function of bed voidage (dP1 = 5 mm, dP2 = 6 mm, liquid: water). Fig. 24 Relationship between the normalised collision force and bed voidage (dP1 = 5 mm, dP2 = 6 mm, liquid: water). Fig. 25 Comparison of the normalised collision force with different correlations from literature (dP1 = 5 mm, dP2 = 6 mm, liquid: water). Fig. 26 Validation of Eq. (40) in the prediction of collision force based on the bed voidage.

Figure 1

z

a

y

x

r

0

x

x

0

b z xe

rp xp

Ve

y x

O

e0

Ve0

Figure 2

0.05 m

a

b 1.0 m

z

y x

Figure 3

Foreign particle insertion 8 arrangement Outlet

50

Heat 9 Exchanger

50

Glass column 1500

1

Outer square column 2

Distributor 200

3

Calming section 4 Vent Inlet

Rotameter 5

Water 7 PUMP 6

Figure 4

Figure 5

1800 1700

y = 1520

1600

∆p (Pa)

1500 1400 1300 y = 51750x - 676.42 1200 1100 x = 0.042443 1000 900 800 0.02

0.06

0.1

0.14

UL (m/s)

0.18

0.22

Figure 6

Figure 7

1 0.9

UL (m/s): 0.03 0.04 0.05 0.06 0.07

0.8

xp, z (m)

0.7 0.6 0.5 0.4

Packed bed height

0.3 0.2 0.1 0 20

25

30

t (s)

35

Figure 8

Acceleration

Steady

Settling through LFB

6 7000 Rep, -

Transient information on the classifying particle (-)

0 (0.3) 5

vL, z, @p, m/s

1 (-0.1) 4

εL, @p, -

50 (0.4) 3 |ωp|, rad/s

2 0 (0)

vp, z, m/s 1 1 (-1)

xp, z, m

00 0

0.5

1

Time (s)

1.5

2

Figure 9

vL (m/s):

0.24

0.22 0.2

xp, z (m)

0.18

0.16 0.14 0.12

0.1 0.08 0.06 21.2

21.7

22.2

t (s)

22.7

0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.24 Linear (0.06) Linear (0.07) Linear (0.08) Linear (0.09) Linear (0.1) Linear (0.11) Linear (0.12) Linear (0.13) Linear (0.14) Linear (0.15) Linear (0.16) Linear (0.17) Linear (0.18) Linear (0.19) Linear (0.2) Linear (0.21) Linear (0.22) Linear (0.24)

Figure 10

0.45 0.40 0.35

V2 (m/s)

0.30 0.25 0.20 0.15 Exponential fit of sim. data

0.10

Simulation Experiment

0.05 0.00 0.05

0.10

0.15

UL (m/s)

0.20

0.25

Figure 11

Predicted particle classifiication velocity (m/s)

0.8

0.7

0.6

0.5

0.4

20%

10% 0.3 DEM-CFD K&B (1966) K&L (1969) Joshi (1983) Di Felice (1991) van der Wielen (1996) Grbavcic (2009)

0.2

0.1

0 0

0.1

0.2

0.3

0.4

0.5

0.6

Experimental particle classifiication velocity (m/s)

0.7

0.8

Figure 12

0.08

u'L (m/s)

0.06

0.04

Simulation, d_p1=3mm

0.02

Experiment (PIV), d_p1=3mm Simulation, d_p1=5mm

0 0.4

0.5

0.6

0.7

αL (-)

0.8

0.9

Figure 13

a

0.4

b

0.4

c

0.4

d

0.4

e

0.4

f

0.4

αL (-):

0.2

0.2

0.2

0.2

0.2

0.2

0

0

0

0

0

0

0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6

Figure 14

a

0.4

b

0.4

c

0.4

d

0.4

e

0.4

f

0.4 -1

Vorticity (s ):

0.2

0.2

0.2

0.2

0.2

A

0

A

0.2

0.18

0.2

B

0

0

0

B

0

90 80 70 60 50 40 30 20 10 0 -10 -20 -30 -40 -50 -60 -70 -80

0

0.2

0.18

|uL| (m/s): 0.16

0.16

0.14

0.14

0.12

0.12

0.1

0.1

0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Figure 15

22.05 s

22.09 s

22.13s

(a) αL = 0.45, 0 ≤ y 21.67 s

21.71 s

21.75s

(b) αL = 0.65, 0 ≤ y 21.86 s

21.88 s

21.90s

(c) αL = 0.79, 0.1 ≤ y

Figure 16

10

9 8

N_a, con N_max, con 1

Na, con (-)

6 5 4

0.1

3 2 0.01

1 0.4

0.5

0.6

αL (-)

0.7

0.8

Nmax, con (-)

7

Figure 17

Particle velocity (m/s)

2.3 0.2

vpx

1.6 0.2 (-0.5)

vpy

0.2 (-0.5) 0.9

Collisioon number (-)

vpz

510 (-0.5) 0.2

I

IV

III

II

Nc

V

-0.5 430 1

1.02

1.04

1.06

t (s)

1.08

1.1

Particle fluctuating kinetic energy, m2/s2

Figure 18

0.01

0.001 0.15

0.25

0.35

α S, -

0.45

0.55

Figure 19

300

250

f*col (-)

200 KTGF (Simonin, 1991)

150

DEM-CFD simulation

100

50

0 0

0.1

0.2

0.3

αS (-)

0.4

0.5

0.6

Collision frequency, fcoll (s-1)

Figure 20

dP2, mm dP1, mm dP2/dP1, 6in3W 6 3 2.0 6in4W 6 4 1.5 6in5W 6 5 1.2 5in5W 5 5 1.0 4in5W 4 5 0.8 3in5W 3 5 0.6 6in3N 6 3 2.0 6in5N 6 5 1.2

3000

Liquid water water water water water water NaI NaI

300

30 0.1

0.2

0.3

0.4

αS (-)

0.5

0.6

Figure 21

50

50

a

b

40

|Fc|/mg (N)

|Fc|/mg (N)

40 30 20 10

30 20 10

0

0 21

21.05

21.1

21.15

21.2

21

21.05

t (s)

21.2

50

c

d

40

|Fc|/mg (N)

40

|Fc|/mg (N)

21.15

t (s)

50

30 20 10

30 20 10

0

0 21

21.05

21.1

21.15

21.2

21

21.05

t (s)

21.1

21.15

21.2

t (s)

50

50

e

40

f

40

30

|Fc|/mg (N)

|Fc|/mg (N)

21.1

20 10 0

30 20 10 0

21

21.05

21.1

t (s)

21.15

21.2

21

21.05

21.1

t (s)

21.15

21.2

Figure 22

0.5 Drag force Pressure gradient force

0.45 0.4

|F|/mg (-)

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0.4

0.5

0.6

0.7

αL (-)

0.8

Figure 23

10

9 Exponential fitting Prediction data

8

|Fc|/|Fd| (-)

7 6 5 4 3 2 1 0 0.4

0.5

0.6

αL (-)

0.7

0.8

Figure 24

1 Eq. (40) Prediction data

0.9

|Fc|/mg (-)

0.8 0.7 0.6 0.5

0.4 0.4

0.5

0.6

0.7

αL (-)

0.8

0.9

Figure 25

1.0 Present study van der Wielen (1996) Grbavcic (2009)

0.9 0.8

|Fc|/mg (-)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.40

0.50

0.60

0.70

αL(-)

0.80

0.90

Figure 26

1

10%

0.9

|Fc|/mg in Simulation (-)

0.8 0.7 0.6 0.5

20% dP2, mm dP1, mm Liquid

0.4

6 5 6mm 5 5 5mm 4 5 4mm 3 5 3mm 6 4 Series5 6 3 6in3mm 6 5 6in5mminNaI 6 3 6in3_NaI

0.3 0.2 0.1

water water water water water water NaI NaI

0 0

0.1

0.2

0.3

0.4

0.5

0.6

|Fc|/mg by Eq. (40) (-)

0.7

0.8

0.9

1

22.05s 





22.09s



 22.13s







(a) L = 0.45, 0  y 21.67s 





21.71s



 21.75s







(b) L = 0.65, 0  y 21.86s 





21.88s



21.90s





(c) L = 0.79, 0.1  y