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Short communication

Particle dispersion in a partially ﬁlled rotating cylindrical tank Yunfu Chen a,c , Cheng Yu a , Meifang Liu b , Sufen Chen b , Yongping Chen a,∗ a

Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, School of Energy and Environment, Southeast University, Nanjing, Jiangsu 210096, PR China b Research Center of Laser Fusion, China Academy of Engineering Physics, Mianyang, Sichuan 621900, PR China c College of Engineering, Nanjing Agricultural University, Nanjing, Jiangsu 210031, PR China

a r t i c l e

i n f o

a b s t r a c t

Article history:

Particles-liquid ﬂow involving a free surface in partially ﬁlled rotating cylindrical tank is

Received 17 September 2015

common in chemical engineering, which attracts numerical studies on it in an effort to

Received in revised form 12 March

provide useful information for the design and optimization of operational conditions in the

2016

real application. However, the dynamics of particles dispersion in partially ﬁlled rotating

Accepted 20 March 2016

cylindrical tank under small particles-liquid density ratio has not been fully understood by

Available online 12 April 2016

performing a reasonable 3D numerical simulation. A three-dimensional unsteady model of a solid–liquid dispersed ﬂow involving a free surface has been developed and numerically

Keywords:

analyzed to investigate the particle dispersion in a partially ﬁlled rotating cylindrical tank.

Solid–liquid ﬂow

In the model, the discrete element method (DEM) and the volume of ﬂuid (VOF) method have

Rotating cylindrical tank

been applied. The ﬂow ﬁeld and the particle dispersion process in a rotational cylindrical

Particle dispersion

tank have been presented. The effects of the tank rotation speed and the liquid height on

Free surface

the dispersion behaviors of particles are examined and analyzed. The results indicate that vortexes have formed in the liquid phase that entrain the particles and induce their gradual dispersion throughout the entire region of the liquid phase. The average mass concentration of particles is high in the rotating-out side of the liquid phase and in the liquid–wall contact region, while it is low in the rotating-in side of the liquid phase and in the vortex generation region. In addition, as the rotation speed increases, the entrainment capability of the liquid phase has been enhanced, which improves the dispersion performance of the particles. The selection of an appropriate liquid height is beneﬁcial for the enhancement of particle dispersion in a partially ﬁlled rotating cylindrical tank. © 2016 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

1.

Introduction

Solid–liquid ﬂows involving a free surface are common in chemical engineering, including the extraction, segregation, and wet ball milling (Liu et al., 2014; Chou et al., 2010; Sakai et al., 2012). For examples, Liu et al. (2014) disperses capsules encapsulating inner water by a partially ﬁlled rotating

∗

cylindrical tank, so as to facilitate the extraction of inner phase out of the capsules by the continuous phase in the cylindrical tank. Chou et al. (2010) utilized a rotating drum partially ﬁlled with liquid to orderly segregate mixed particles, which indicated that the viscosity and the ﬁlling ratio of the liquid have a signiﬁcant effect on the transverse motion mode of the particles. In addition, Sakai et al. (2012) investigate the mixing

Corresponding author. Tel.: +86 25 8379 2483. E-mail address: [email protected] (Y. Chen). http://dx.doi.org/10.1016/j.cherd.2016.03.023 0263-8762/© 2016 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

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performance of particles in a partially ﬁlled rotating cylindrical tank, which can be used to reﬂect whether and how a homogeneous wet ball milling can be achieved. Note that, these previous works demonstrated that the dynamics of solid–liquid ﬂow with free surfaces in a rotating cylinder involves the coupled effects of liquid–solid interaction, liquid–gas free surface, and solid–solid collision, which determines the performance of the corresponding chemical and industrial process. In this context, it is crucial to understand the characteristics of solid–liquid ﬂow involving free surfaces in a cylindrical tank, which is useful both for the design and optimization of operational conditions in the industrial processes. To understand the suspension quality of particles in the liquid-phase during the mixing process, several theoretical attempts have been focused on the investigation of the dynamic behaviors of particles in liquid–solid mixture ﬂows (Kasat et al., 2008; Tamburini et al., 2011; Klenov and Noskov, 2011; Tamburini et al., 2012; Kalaga et al., 2012). In these attempts, the Eulerian–Eulerian methods (such as the twoﬂuid model, the homogeneous model, and the small slip model) are typically employed to solve solid–liquid ﬂows in which the particles are treated as continua. One major disadvantage of these methods is that empirical relationships are required to describe the constitutive behavior of the particles, which makes it impossible to accurately evaluate the solid stress. Thus, it is difﬁcult for these approaches to accurately simulate the solid–liquid ﬂows involving a high volume fraction of solid phase in which the contact force acting on a particle becomes important (Sun et al., 2014). In contrast to the Eulerian–Eulerian methods, the Lagrangian methods are capable of tracing discrete particles individually in Lagrangian coordinates. As a typical Lagrangian method, the discrete element method (DEM) (Cundall and Strack, 1979; Fan et al., 2016; Liu et al., 2015; Paulick et al., 2015; Maione et al., 2015) has proven to be a versatile tool to obtain the trajectory of particles with high accuracy via Newton’s second law of motion. In DEM, the particles are represented by soft spheres, and the contact forces are derived from their overlap between particles (or particle–wall) during a ﬁnite-time collision (Cundall and Strack, 1979; Stevens and Hrenya, 2005; Tsuji et al., 1993; Zhang et al., 2009; Zhang and Vu-Quoc, 2002; Weerasekara et al., 2016; Cleary, 2015). DEM has been systematically applied and examined in various powder systems, such as ball mills and pneumatic conveying (Sakai et al., 2005, 2006; Mishra, 2003; Sakai and Koshizuka, 2009; Chen et al., 2015; Zhou et al., 2015; Tsuji et al., 1992; Fraige and Langston, 2006; Kuang and Yu, 2011). Therefore, in the current study, the DEM is utilized to simulate the motion trajectory of particles in a liquid phase and in a particle–particle (and particle–wall) collision. In addition to the appropriate treatment of particle behaviors, the free surface hydrodynamics should be efﬁciently resolved in the exploration of solid–liquid ﬂows involving free surfaces in a rotational cylindrical tank. Fortunately, hybrid numerical methods have been proposed to investigate the particulate ﬂows involving a free surface. Sakai et al. (2012) and Sun et al. (2014) utilized the DEM with the moving particle semi-implicit (MPS) method to model the solid–solid and liquid–solid interactions involved in the solid–liquid ﬂows in a two-dimensional rotational cylindrical tank. Special attention was focused on the effect of the spring constant and hydrodynamic forces (such as lubrication and virtual mass forces) on the macroscopic behavior of the particles. Subsequently, based on the DEM-SPH (smoothed particle hydrodynamics) method, Sun et al. (2013, 2014) performed three-dimensional

simulations to investigate the effects of the friction coefﬁcient, cutoff radius and rotational velocity on the dynamic behavior of particle beds. Note that the above investigation focused on the dense granular ﬂows in which the share of the particle bed is very large in the liquid continuous phase, and the particle density is larger than the liquid phase. In this case, the particles accumulate, and it is difﬁcult to follow the motion of the liquid continuous phase. In contrast to dense granular ﬂows, the liquid–solid dispersed ﬂow possesses the feature that the share of particles in the liquid phase is low, and the density of particles is slightly different from the liquid. In such a rotating cylinder, the particles are capable of dispersing in the liquid continuous phase and could follow the rotational ﬂow of the matrix ﬂuid. The interface hydrodynamic behaviors of gas and liquid have more inﬂuence on the particle behaviors. To date, limited attention has been focused on the liquid–solid dispersed ﬂow involving a free surface in a rotating cylinder. In addition, the liquid ﬂow behaviors and particle dispersion characteristics are less understood, especially the effects of the gas–liquid interface height and rotational speed of the cylinder on the dispersion characteristics of particles. In summary, several attempts have been made to study the solid–liquid ﬂow involving a free surface in a rotating cylindrical tank using numerical algorithms based on hybrid methods. Note that, the available hybrid numerical methods suppose the continuous ﬂuid as the micelles of discrete liquid and gas particles (Ge and Li, 2001; Monaghan, 2005), which is deviated from the actual condition under the low Reynolds number. Furthermore, the computational accuracy by these methods is too dependent on the size of the discrete liquid and gas particles. In addition, these numerical studies are mainly concentrated on the particles behaviors under large density ratio of particles to the continuous liquid (Liu and Liu, 2010). However, an insight into the dispersion mechanisms for multiparticles in a partially ﬁlled rotating cylindrical tank has not been gained under small particle–liquid density ratio. In addition, the effects of several inﬂuencing factors, e.g., rotation speed and liquid height, are still not fully understood. Therefore, a three-dimensional theoretical model for solid–liquid ﬂows involving a free surface in a partially ﬁlled rotating cylindrical tank is developed based on the volume of ﬂuid (VOF) method and the DEM in a Euler–Lagrange framework in an effort to investigate the particle dispersion behaviors in the liquid phase. In the proposed model, the liquid–solid ﬂows with two-way coupling and gas–liquid interactions, as well as particle–particle and particle–wall collisions, are considered. The effects of the rotating speed of the cylindrical tank and the liquid height on the dispersion performance of the particles in the liquid phase are examined and analyzed. In addition, a visualization experiment is conducted to provide a comparison and validation for the simulation results.

2. Particle dispersion in the partially ﬁlled rotating cylindrical tank To study the dispersion characteristics of the particles in a liquid continuous phase, a three-dimensional solid–liquid ﬂow involving a free surface in a rotating cylindrical tank is investigated. As shown in Fig. 1(a), driven by an external power device, the cylindrical tank rotates along the y-axis at a certain rotational speed, and the height and inner diameter of the cylindrical tank are h and D, respectively. The cylindrical tank is ﬁlled with a certain amount of liquid phase with height

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2.1.1.

Liquid and gas ﬂows

The cylindrical tank is partially ﬁlled with liquid, and its upper part is ﬁlled with gas. The liquid is in rotational motion when the tank begins to rotate. The liquid ﬂow is subjected to several action forces induced by the wall shear, solid–liquid interactions and gas–liquid interactions. The gas ﬂow is subjected to wall shear and gas–liquid interactions. For a locally averaged description of the continuous liquid motion (Anderson and Jackson, 1967), the three-dimensional unsteady mass and momentum conservation equations can be written as follows:

Fig. 1 – Particle dispersion in a partially ﬁlled rotating cylindrical tank: (a) schematic of rotating cylindrical tank and (b) initial position of particles.

H, and the remainder is ﬁlled with gas. A certain amount of particles is mixed into the liquid. For the solid–liquid ﬂow, a slight difference between the solid density and liquid density is considered in which the particle density is assumed as p = 1100 kg/m3 and the liquid density as l = 1000 kg/m3 . In the current work, the height and inner diameter of the cylindrical tank are assumed as h = 50 mm and D = 195 mm, respectively. As shown in Fig. 1(b), the rotating-in side is the region in liquid phase near the circumferential inner wall of container rotating into the liquid phase. Correspondingly, the rotating-out side is the region in liquid phase near the circumferential inner wall of container rotating out of the liquid phase.

∂(εl l ) + ∇ · (εl l ul ) = 0 ∂t

(1)

∂(l εl ul ) + ∇ · (l εl ul ul ) = −εl ∇p + ∇ · (εl ) + l εl g − f pf ∂t

(2)

where t is the time, εl is the liquid volume fraction, l is the liquid density, g is the gravity acceleration, ul is the liquid velocity vector, p is the pressure, is the stress tensor for the liquid, and fpf represents the momentum exchange with particles. For the gas motion, the above mass and momentum conservation equations could also be utilized, but εl , l and ul in Eqs. (1) and (2) are substituted for εg (gas volume fraction), g (gas density) and ug (gas velocity vector), respectively, and the volume fraction εg = 1, while the momentum exchange with particles fpf = 0. The VOF method is utilized to represent the free interface between gas and liquid. The motion of ﬂuid treated as incompressible viscous ﬂuid is calculated based on the timeaveraged Navier–Stokes equations, considering the volumetric forces at the interface resulting from the surface tension. The portion of the two ﬂuids in a computational cell is represented by the volume fraction. The volume fraction is governed by the transport equation, ∂˛i + ∇(˛i · v) = 0 ∂t

(3)

where ˛i is the volume fraction in each cell, and the summation of the each phase’s volume fraction ˛i is unity, ˛g + ˛l = 1. For ˛g = 1.0, the cell represents the gas region, and for 0 < ˛g < 1, the cell represents the interface region. The parameter v is the velocity vector governed by the mass and momentum equations for incompressible Newtonian ﬂuid. ∇ ·v=0

2.1.

(4)

Mathematical model

This study develops a three-dimensional unsteady model of solid–liquid ﬂow involving free surfaces in a rotating device based on the VOF method and the DEM. For such a complex solid–liquid–gas multiphase system, several dynamic processes are involved, including the rotational liquid and gas ﬂows, particle motion followed by liquid ﬂow, liquid–solid interaction, liquid–gas interaction, as well as solid–solid and solid–wall collisions. The Navier–Stokes equation is applied to describe the rotational liquid and gas ﬂows in which the gas–liquid interface is captured by VOF methods. The drag force and the pressure gradient force are utilized to model the liquid–solid interaction with two-way coupling. DEM is applied to capture the particle dispersion motion in solid–liquid ﬂow in which the solid–solid and solid–wall interactions are described by a soft sphere approach.

∂(v) 2 + ∇(vv) = −∇p + ∇ (∇v + ∇vT ) − ∇ · vI + g + f s ∂t 3 (5) where fs represents the volumetric forces at the interface resulting from the surface tension, is the density, and is the coefﬁcient of dynamic viscosity. The properties in each cell are determined as follows: = ˛g g + (1 − ˛g )l

(6)

= ˛g g + (1 − ˛g )l

(7)

v=

˛g g vg + ˛l l vl

(8)

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The volume force at the free surfaces is given by the CSF model (Brackbill et al., 1992) as follows: f s = ∇˛i

i

1 |n|

(9)

n |n|

j

· ∇ |n| − (∇ · n)

slider

spring spring

where is the surface force coefﬁcient, and k is the free surface curvature, which can be given as follows: k=

i

Normal direction

j

dash-pot dash-pot Tangential direction

(10) Fig. 2 – Model of contact force.

Here, n is the normal vector, n = ∇ ˛i .

2.1.2.

The gravity forces can be expressed as follows:

Particle motion

In the cylindrical tank, the particles are carried by the liquid motion. In the particle motion, the particles collide with other particles and with the solid wall. In this study, the DEM is utilized to trace the particle motion with inter-particle and particle–wall collisions. In DEM, the particle trajectories can be obtained via the numerical integration of the particle motion equation (Cundall and Strack, 1979). Considering the effect of the liquid–solid interaction force at the particle positions, the governing equation for the particles is given as follows: dup mp = F c + F lp + F g dt

(11)

where up is the particle velocity, mp is particle mass, Flp is the liquid–solid interaction force (see Section 2.1.3), Fc is the contact force (including inter-particle and particle–wall collisions, see Section 2.1.4), and Fg is the gravitational force.

2.1.3.

1 C dp |ul − up |(ul − up ) 8 d l

(12)

⎧

4 1−ε ⎪ 150 + 1.75 ε ≤ 0.8 ⎪ ⎨ 3

ε Rep

0.687 ⎪ ⎪ ⎩ 24 + 3.6 · Rep ε−3.65 ε > 0.8

(13)

Rep

where ε is the void fraction of particles, and Rep is the particle Reynolds number, which is deﬁned as follows: Rep =

l dd |ul − up | l

(14)

The pressure gradient force is given by the following equation: 1 Fp = − d3p ∇ps 6

(16)

The added mass force accounts for the resistance of the ﬂuid mass that is moving at the same acceleration as the particle. For a spherical particle, the volume of the added mass is equal to one-half of the particle volume (Auton et al., 1988; Milne-Thompson, 1968): Ff =

d 1 d3 (u − up ) 12 p l dt l

(17)

Based on Newton’s third law of motion, the forces acting on particles yield a reaction force on the liquid. Therefore, the momentum transfer from particles to liquid is considered by the volumetric liquid–solid interaction force term Fpf , which can be determined as follows:

kc

where dp is the particle diameter, and Cd is the drag coefﬁcient of the sphere particle, which is given by the following equation (Gidaspow, 1994):

Cd =

1 3 d g 6 p d

Liquid–solid interaction

In this study, the drag force and the pressure gradient force are considered to model the liquid–solid interaction. A drag model to deﬁne the constitutive rule of interphase momentum transfer is employed based on a combination of the equations of Ergun (1952) and Wen and Yu (1966). The drag force acting on a suspended particle is proportional to the relative velocity between the phases: Fd =

Fg =

(15)

where ps is the gradient of the static pressure in the continuous liquid phase.

F pf =

F i=1 lp

V

(18)

where V is the volume of a computational cell, and kc is the number of particles located in this cell. In the present study, Flpi is the ﬂuid force acting on particle i, including the drag force Fd , the added mass force Ff and the pressure gradient force Fp .

2.1.4.

Solid–solid interaction

To consider the solid–solid interaction, the DEM based on a soft sphere approach is used to describe the particle collision. In the DEM, the particle–particle or particle–wall contact force is determined from the overlap accumulated during the collision using a viscoelastic model consisting of a spring, dashpot and friction slider, as shown in Fig. 2. The contact force of a solid–solid collision can be decomposed into the normal contact force Fnij and the tangential contact force Ftij . In the simulation, the normal contact force is determined by the overlap and relative velocity: F nij = (−Kn ın − n G · ne )ne

(19)

The tangential contact force is constructed from trial values that test whether the points in contact are slipping. F tij = −Kt ıt − t Gct

(20)

If the following relationship is satisﬁed: |F tij | > f |F nij |

(21)

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where jn is the total number of particles, representing the simultaneous collisions with the particle i.

where f is the coefﬁcient of friction, then sliding occurs. The tangential contact force is given by the following equation: F tij = −f |F nij |T = −f |F nij |

Gct |Gct |

2.1.5. (22)

where Kn , Kt , n and t are normal spring stiffness, tangential spring stiffness, normal damping coefﬁcient and tangential damping coefﬁcient, respectively. The parameters ın , ıt , G, Gct are normal displacement, tangential displacement, relative velocity, and slip velocity, Gct = G − (G · ne )ne . The viscous damping coefﬁcient is derived from the following equation:

= 2 ln e

Km∗

(23)

2

(ln e) + 2

where e is the restitution coefﬁcient, and m* is the reduced mass. For the pairing of particles i and j, m* = mi mj /(mi + mj ). For the ith particle, m* = mi when collision occurs between this particle and a rigid wall. A comprehensive review of the available DEM contact models can be found in the literature (Thornton et al., 2011, 2013). In the particle motion, the particles possibly undergo multiple, simultaneous collisions. The total contact force acting on a particle i is given by the following equation:

F ci =

jn

(F nij + F tij )

Boundary conditions

To investigate the solid–liquid–gas ﬂow inside the rotational cylindrical tank, boundary conditions are required to close the mathematical formulation of solid–liquid ﬂows involving a free surface in a rotating cylindrical tank, including the initial boundary condition, the ﬂow boundary conditions. At the initial time, the liquid is at the bottom region, gas is located in the upper part, and both the gas and liquid are in the static state. In addition, the particles (numbered at 581) are dispersed in the liquid phase with given positions, which are mostly located in the lower region of the liquid phase. Driven by an external power device, the cylindrical tank revolves along the y-axis at ˝ rpm. No-slip boundary condition is assumed on the inner wall of the cylindrical tank: 0 ≤ y ≤ h and

x2 + z2 = D/2 : ωr = 0,

ω =

2 · ˝, 60

ωy = 0 (25)

The effects of wall adhesion at the ﬂuid interface in contact with rigid boundaries in equilibrium are estimated within the framework of the CSF model in terms of the equilibrium contact angle between the ﬂuid and wall, (Brackbill et al., 1992). At the wall, the surface normal at the live cell next to the wall

(24)

j=1

Start

Initial values of fluid field and particles

Determine the volume fraction Using VOF method to compute volume fraction i

Does the particle contact other particles

No

Yes Solve Eqs.(1) and (2) a-a Using DEM model to compute contact force Fci

Calculate inter particle contact force

Does the particle contact walls Yes

Calculate solid-liquid interaction terms Flp

Calculate contact force

Update particle states based on the sum of all forces acting on individual particle

a-a No

Time=total time Yes stop Fig. 3 – Flow chart of numerical calculation.

No

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Table 1 – The parameters used in the numerical simulation. Parameter D h g g l l p dp f e Kn Kt

Value Inner diameter of cylindrical tank (mm) Height of cylindrical tank (mm) Gas density (kg/m3 ) Gas viscosity (Pa s × 10−3 ) Liquid density (kg/m3 ) Liquid viscosity (Pa s) Particle density (kg/m3 ) Particle diameter (mm) Friction coefﬁcient Restitution coefﬁcient Normal spring stiffness (N/m) Tangential spring stiffness (N/m)

195 50 1.169 0.0185 1000 0.001 1100 1.5 0.3 0.9 1000 600

is deﬁned in terms of the contact angle, , between the tangent to the interface and the wall as follows: n = nw cos + tw sin

(26)

where nw and tw are the unit vectors normal and tangential to the wall, respectively. The combination of the contact angle with the normally calculated surface normal one cell away from the wall determines the local curvature of the surface. All of the results in this study were obtained with the contact angle of 45◦ . In the present numerical simulation, the corresponding parameters are listed in Table 1.

2.2.

Numerical solution

The numerical simulation of a solid–liquid ﬂow involving a free surface is conducted in a rotating cylindrical tank. Based on the VOF method and the DEM, a three dimensional Euler–Lagrange model is applied. In the simulation, the ﬁnite volume method is used to transform a set of governing equations for the continuous phase, including the gas phase and liquid phase, into a form that can be solved numerically. To enhance the computational accuracy, the governing equations are discretized using the second-order upwind scheme. The interpolation of the pressure values at the cell faces is performed using the PRESTO scheme, which improves the convergence rate and the stability of the computation. Pressure–velocity coupling is accomplished using the SIMPLE algorithm. The piecewise linear interface calculation (PLIC) interface reconstruction technique is used to track the geometry of the interface in all of the cases simulated. The convergence criterion is set at 10−5 for all of the equations. The explicit time integration method is applied to solve the particle motion equations in the Lagrange frame. The trajectory tracking of each particle is performed in every time step. Compared with the mean free time of particles, the time step for numerical integration is set sufﬁciently small. The solution algorithm for the solid–liquid ﬂow in a rotating cylindrical tank includes the following steps. First, at the initial time step, the positions and velocities of particles are assumed to be known. Fluid properties are updated based on the current (initialized) solution. Next, the governing equations of continuity and the momentum equation for ﬂuid are solved to obtain the pressure and velocity ﬁelds. The momentum transfer from particles to liquid is considered using the volumetric liquid–solid interaction force term. In the VOF

Fig. 4 – Grid geometry.

method, the volume fraction in each computational cell of gas–liquid interface is based on the known ﬂow ﬁeld. Surface tension is added to the momentum equations using a source term in the momentum equations. Then, for the solid phase, the contact force between particles (or between particles and walls) is evaluated using the DEM model. The forces with liquid acting on particles are obtained following the description in Section 2.1.3. The sum of all of the forces acting on

Fig. 5 – Experimental setup of particle distribution in the cylindrical tank.

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7

Fig. 6 – Comparison of particle distribution between simulation results and experimental data. individual particles is accumulated to update the particle states, such as velocities and positions. In the simulation, the computation of a solid–liquid ﬂow involving a free surface in a rotating cylindrical tank is shown in Fig. 3. In the current work, as shown in Fig. 4, the computational domain is discretized by hexahedral grids. To ensure that the numerical results are independent of the grid size, a grid independence test is conducted using different mesh sizes. The test indicates that the numerical results based on the ﬁnal grid system presented in this study are independent of the mesh size. Based on the grid independence test, a grid number of 222750 are utilized in current study, taking both consideration of simulation precision and computing cost.

2.3.

Experimental validation

To verify the present model, the experiment of a solid–liquid ﬂow involving a free surface in a rotating cylindrical tank is conducted. As shown in Fig. 5, the experimental device consists of a cylindrical tank, motor, decelerator, high-speed video camera, light source and minor subsidiary equipment. A cylindrical tank with the same dimensions as that in the simulated system is ﬁxed at the connection plate by bolts. To allow visualization from the side, the cylindrical tank is constructed of polymethyl methacrylate. Polyethylene (PE) with an average diameter and density of approximately dp = 2 mm and p = 948 kg/m3 , respectively, is used as the particles. The liquid is a mixture of water and alcohol with an average density of 930 kg/m3 . In the experiment, the particles are placed in the cylindrical tank, which is partially ﬁlled with liquid. The height of the liquid phase is H = 86 mm. The rotation of the cylindrical tank is controlled by a motor assisted by a speed frequency controller. The experiment is performed at room temperature. A high-speed video camera is used to record snapshots of the particles at different times. Photographs are taken to obtain the particle distribution.

To validate the present simulation method, a comparison of particle distribution in the liquid phase between the simulation results obtained by the present model and the corresponding experimental data are plotted in Fig. 6. As shown, the simulation results of the particle distribution in a rotating cylindrical tank agree well with the experimental data, which validates the reasonability of the present model.

3.

Results and discussion

3.1.

Particle dispersion behavior

The gas and liquid ﬂow ﬁelds in a rotating cylindrical tank provide an understanding of the particle dynamics of solid–liquid ﬂow in a cylindrical tank. Fig. 7(a) presents the ﬂow ﬁeld inside a rotating cylindrical tank. As shown, due to the action of shear force arising from the inner wall of the rotating cylindrical tank, the vortex formations are observed in the gas and liquid regions. To analyze the particle motion behaviors in a rotating cylindrical tank, Fig. 7(b) illustrates the evolution of the particle distribution during the dispersion of particles involving the free surfaces. As shown, the particles are initially stacked at the bottom of the cylindrical tank. When the cylindrical tank begins to revolve, the particles initially move near the wall of the cylindrical tank along the rotational direction due to the drag force of liquid acting on the particles and due to the wall frictional force. As a result, the particles are rolled up along the wall of the cylindrical tank (t = 1.0 s, 2.0 s), and they follow the rotational liquid ﬂow. When the liquid ﬂows close to the upper gas–liquid interface at the rotating-out side (t = 3.0 s), the ﬂow reverses because it is blocked at the interface. As a result, the particles tend to turn round and stack where the particle–particle interactions and the particle–wall interactions frequently occur. Subsequently, following the liquid ﬂow, the particles are re-dispersed into the half of the liquid region at the rotating-out side (t = 5.0 s). Over

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Fig. 7 – Particle dispersion process in a cylindrical tank (H = 84 mm, ˝ = 20 rpm): (a) velocity ﬁeld (y = 25 mm) and (b) snapshot of instantaneous particle spatial distribution. time, the particles gradually disperse into the entire region of the liquid phase. In a rotating cylindrical tank, the particle motion trajectories are determined by the combined effect of liquid–particle interactions, inter-particle collisions and particle–wall collisions. To provide a detailed analysis of particle motion in a rotational ﬂow, Fig. 8 shows the particle motion trajectories of three typical particles in a liquid phase for the duration of 20 s. As shown, during the motion process, the particles (numbers 1–3) are primarily located within the middle region of the

liquid phase and the rotating-out side, and some particles (i.e., number 2) could occasionally move to the rotating-in side. It can also be observed that although every particle has its own trajectory, including both axial and radial motions, they generally obey the liquid ﬂow direction in the rotating cylindrical tank (Fig. 7(a)). Accordingly, we can conclude that the particles follow the vortex of the liquid phase and gradually disperse into the entire region of the liquid phase, and the ﬂow ﬁeld of the liquid phase determines the motion trajectories of the particles.

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Fig. 8 – Motion trajectories of typical particles (H = 84 mm, ˝ = 20 rpm).

3.2.

Effect of the tank rotation speed

As mentioned above, the gas and liquid ﬂow ﬁelds in a rotating cylindrical tank play an important role in the particle motions and dispersion characteristics. The vortex in the

liquid phase can entrain the particles, which is beneﬁcial for the dispersion of particles. To provide further insight into the particle dispersion behavior, Fig. 9 presents the effect of rotation speed on the average mass concentration at the quasi steady state and spatial particle distribution. Note that the

Fig. 9 – Effect of rotation speed on solid–liquid ﬂow (H = 84 mm).

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average mass concentration is deﬁned as the statistical average of the mass concentration over the time and space for particles. As shown in the ﬁgure, the average mass concentration of particles is high in the rotating-out side of the liquid phase and in the liquid–wall contact region, while it is low in the rotating-in side of the liquid phase and in the vortex generation region. This implies that the particles are mainly located in the rotating-out side and near the wall region, and they seldom pass through the region of the rotating-in side. This phenomenon can also be qualitatively reﬂected in the motion trajectories of typical particles, as shown in Fig. 8. For a very slow rotation speed (˝ = 5 rpm), the entrainment capability of the rotational ﬂow is so low that nearly all of the particles accumulate at the rotating-out side of the liquid phase, and a peak value of mass concentration appears in the region of the rotating-out side. As the rotation speed increases, the entrainment capability is improved, reducing the aggregation level of the particles. In this case, the particles begin to distribute in the middle region of the liquid phase, and some particles may move to the rotating-in side. When the rotation speed is fast (e.g., ˝ = 60 rpm), the distribution of particles is relatively uniform in the liquid phase, indicating that the dispersion performance of the particles has improved. To further quantify particle dispersion performance in liquid phase inside cylindrical tank under different rotation speed of the cylindrical tank, the relative standard deviation (RSD) of the particle concentration in sampling cell is introduced here (Jovanovic´ et al., 2014), which is deﬁned as

4

3 RSD

10

2

1 0

15

30 -1 / r· min

45

60

Fig. 10 – particle concentration distribution RSD vs. rotational speed (H = 84 mm).

RSD = , cav

=

M (c i=1 i

− cav )

M−1

2

(27)

where M is the number of samples, ci is the concentration in sample cell i, and cav is the average concentration of all cells. It can be seen from the deﬁnition of RSD that small RSD means good dispersion of the particles in the computational domain. Accordingly, the particle concentration RSD under different rotation speed of current cylindrical tank is summarized and depicted in Fig. 10. As shown, the RSD of particle concentration in continuous liquid phase decreases gradually with increasing rotation speed, implying that dispersion performance of the particles has improved.

Fig. 11 – Effect of liquid height on solid–liquid ﬂow (˝ = 40 rpm).

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3.3.

Effect of liquid height

The liquid height in a rotating cylindrical tank affects the gas and liquid velocity distribution, hence determining the particle motions. Fig. 11 shows the effect of liquid height on the dispersion performance of particles in the liquid phase with the same number of particles. As shown, the liquid height plays a signiﬁcant role in the velocity distribution of the liquid phase and the dispersion behaviors of particles in the liquid phase. When the liquid height is low (H = 44 mm), the velocity distribution of the liquid phase is irregular, and several vortexes appear in the liquid space. The aggregation of particles is observed at the rotating-out side of the liquid phase, and virtually no particles go through the rotating-in side of the liquid phase. In this case, the mass concentration is very high in the region of the rotating-out side, while it is nearly zero in the rotating-in side of the liquid phase. As the liquid height increases (e.g., H = 84 mm), the velocity distribution of the liquid phase tends to be regular, and only one vortex is generated in the middle region of the liquid phase. The particles could penetrate and disperse into the entire region of the liquid phase, and no accumulation of particles is detected in the liquid phase. This induces the relatively uniform mass concentration in the entire region of the liquid phase. Thus, an increase in liquid height causes the main particle distribution region to transfer from the rotating-out side to the entire region of the liquid phase. Once the liquid height is very high, the velocity distribution of the liquid phase is more regular, and only one vortex is generated in the middle region of the liquid phase. In this case, it is observed that the particles tend to aggregate all around the liquid phase, i.e., the particles frequently penetrate the wall–liquid contact region and the gas–liquid interface region. As a result, the mass concentration is high in the wall–liquid contact region and in the gas–liquid interface region, and it is low in the middle region of the liquid phase. Therefore, we can infer that the selection of an appropriate liquid height is beneﬁcial for the enhancement of particle dispersion in a partially ﬁlled rotating cylindrical tank.

4.

Conclusions

A three-dimensional Euler–Lagrange model for a solid–liquid ﬂow involving a free surface in a rotating cylindrical tank is developed to investigate the particle dispersion behaviors in the liquid phase in a rotating cylindrical tank. In this model, the DEM is applied to consider the particle collisions, and the VOF method is applied to represent the solid–liquid interface. The motion behaviors and concentration distribution of the particles in the liquid phase are explored, and the velocity ﬁeld of gas and liquid in a rotating cylindrical tank is presented. The effects of the rotation speed of the cylindrical tank and the liquid height on the dispersion characteristics of particles involving a free surface in a rotating cylindrical tank are examined and analyzed. In addition, a visualization experiment is conducted to verify the present theoretical model. The conclusions can be summarized as follows:

(1) In a rotating cylindrical tank, due to the action of shear force arising from the inner wall, the vortex formations are observed in the gas and liquid regions. The particles follow the vortex of the liquid phase and gradually disperse into

11

the whole region of the liquid phase. The ﬂow ﬁeld of liquid phase determines the motion trajectories of particles. (2) For solid–liquid ﬂow involving a free surface in a partially ﬁlled rotating cylindrical tank, the average mass concentration of particles is high in the rotating-out side of liquid phase and the liquid–wall contact region, while it is low in the rotating-in side of liquid phase and vortex generation region. (3) The rotation speed of tank plays a signiﬁcant role in the particle dispersion behaviors in the liquid phase in a rotating cylindrical tank. For small rotation speed, the particles accumulate at the rotating-out side of liquid phase. As the rotation speed rises, the entrainment capability of liquid phase is enhanced, so the particles mainly distribute in the middle region of liquid phase and some particles could move to the rotating-in side, which ultimately improves the dispersion performance of the particles. (4) The liquid height is also an important parameter affecting the solid–liquid ﬂow in a rotating cylindrical tank. When the liquid height is low, the aggregation of particles is observed at the rotating-out side of the liquid phase. As the liquid height increases, the particles could penetrate and disperse into the entire region of the liquid phase, which induces the relatively uniform mass concentration in the entire region of the liquid phase. Once the liquid height is very high, the mass concentration is high in the wall–liquid contact region and at the gas–liquid interface region, and it is low in the middle region of the liquid phase. The selection of an appropriate liquid height is beneﬁcial for the enhancement of particle dispersion in a partially ﬁlled rotating cylindrical tank.

Acknowledgements The authors gratefully acknowledge the supports provided by NSAF (No. U1530260) and China Academy of Engineering Physics (No. 2014B0302052).

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