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Collision of single particle in rotating flow field
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Collision of single particle in rotating flow field Xin Ma a,∗ , Yixuan Peng a , Qi Yu a , Ke Lai a , Jie Li b , Di Wu c a
The Ministry of Education Key Laboratory for Oil and Gas Equipment, School of Mechanical Engineering, Southwest Petroleum University, Chengdu, Sichuan 610500, PR China b School of Automotive Engineering, Qiannan Polytechnic for Nationalities, Duyun, Guizhou 558022, PR China c Liaohe Oil Field Drilling Technology Research Institute, Panjin, Liaoning 124010, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history:
The studies on motion and collision of particle in rotating flow field are of great importance
Received 13 July 2019
for chemical engineering. As the lack of work concerning the single particle, the motion
Received in revised form 20
trajectory and collision characteristics of single particle have not been understood. In this
December 2019
article, a coupled discrete element method (DEM) and the computational fluid dynamics
Accepted 24 December 2019
(CFD) method model of a single particle in rotating flow field has been developed and ver-
Available online xxx
ified by experimental data to investigate the collision characteristics of the particle. The
Keywords:
by experiment. In The numerical simulation, the particle motion process has been simu-
Rotating flow field
lated. The effects of particle density and liquid viscosity on collision and particle collision
effects of rotation speed and particle diameter on particle motion have been investigated
Single particle
force have been investigated. Furthermore, the effects of wall and rotating fluid on particle
Collision
rotation have been discussed and analyzed. The results indicate that the rotation speed of
Numerical simulation
flow field and single particle diameter play significant roles in the particle motion behaviors.
Particle motion
In the process of particle motion, the particle has collided with the wall at the bottom of the Y-axis of the trajectory. The selection of an appropriate particle material density and Liquid medium is beneficial for the decreasing of the number of collisions in a fully filled rotating flow field. The plastic deformation of wall will not be caused by collision force. In addition, the particle rotation is significant affected by the particle-wall contact, while it is hardly affected by rotating fluid. They thereby can provide a technical basis for prediction and control of particle motion and maintenance of equipment in centrifugal separation and helical transportation. © 2019 Published by Elsevier B.V. on behalf of Institution of Chemical Engineers.
1.
Introduction
The lack of work concerned these aspects are partly due to difficulties in making quantitative experiment of particles in rotating flow field. It
Particles-liquid flow in rotating flow field is common in chemical engineering. The liquid flow induces the particles motion by acting forces
is also difficult to accurately simulate the particle motion and collision in this system.
on particles, as well as particle-particle and particle-wall collisions occur in the process of particles motion. In this context, an insight into the motion and collision characteristics of particle makes an effort
The discrete element method (DEM) (Cundall and Strack, 1979) has proven to be a versatile tool to simulate the motion and collisions of particles with high accuracy. The particles are represented by soft spheres and the contact forces are derived from their overlap between
to provide useful information for the design operation, maintenance and optimization of operational conditions in the real application,
others (or walls) during a finite-time collision. Zhang et al. (2015) inves-
including the centrifugal separation (Kou and Dejam, 2019; Dejam, 2019a,b; Dejam et al., 2015a,b), helical transportation and rotary agi-
tigated the effects of particles collisions on particle distribution and particle resuspension rate.) Li et al. (2016) studied the contact forces
tation (Zverev and Ushakov, 1968; Kasat et al., 2008; Mang et al., 1998).
between the particles and the wall with the increase of flattening and rotation speed by DEM. With the development of advanced computational techniques, the information of particle-fluid, particle-particle
∗
and particle-wall interaction forces can be generated by use coupling
Corresponding author. of Computational Fluid Dynamics (CFD) and Discrete Element Method E-mail address: [email protected] (X. Ma). https://doi.org/10.1016/j.cherd.2019.12.024 0263-8762/© 2019 Published by Elsevier B.V. on behalf of Institution of Chemical Engineers.
Please cite this article in press as: Ma, X., et al., https://doi.org/10.1016/j.cherd.2019.12.024
Collision of single particle in rotating flow field. Chem. Eng. Res. Des. (2020),
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(DEM) approach (Tsuji et al., 1993). Recently, CFD-DEM coupling method has been proven to be effective in modeling particle-particle, particlewall and particle-fluid interactions with high precision (Wang et al., 2009). Ren et al. (2012) investigated the migration of particles under a cylindrical jet bed based on the DEM-CFD approach. Liu et al. (2016) studied particle-particle and particle-wall collision frequencies considering the different shape based on the DEM–CFD approach. However, the particle motion and collision in rotating flow field have not been conducted based on DEM-CFD method. The effects of influencing factors on particle collisions have not been sufficiently considered. In previous studies, several attempts have been made to study the motion of particles in a rotating flow field using experimental and numerical method. Chou et al. (Chou et al., 2011, 2014; Chou et al., 2019) experimental studied the solid particle group motion in rotating
The cylindrical tank, driven by the external device, rotates along the z-axis at a certain rotating speed, of which the inner diameter D is 120 mm and the height H is 20 mm shown in Figure (b). The single particle placed in the cylindrical tank is carried by the rotating liquid flow. Moreover, particle density and liquid density is considered that the liquid density is set as l = 1000 kg/m3 and particle material density as p = 2230 kg/m3 . It is noted that the densities of liquid and particle are just equal to their physical values, where the liquid is water, and the particle is made by high borosilicate glass. The rotating-in side is the region in liquid phase of left side of y-axis. Correspondingly, the rotating out side is the region in liquid phase of right side of y-axis.
flow field considering several influencing factors. Sakai et al. (2012) and Sun et al. (2014) studied the influence of spring coefficient on particles behavior and the influence of hydrodynamic force on the macroscopic particle movement behavior and compared it with the simulated data. Sun et al. (2013) verified the accuracy of DEM-SPH model through experiments, and considered that DEM-SPH model could accurately solve the motion characteristics of particle groups in the rotating flow field by comparing the macroscopic behaviors of particle group movement in experiments and simulations. Chen et al., 2016a developed a threedimensional theoretical model based on the volume of fluid (VOF) method and the DEM validated by visualization experiment to investigate dispersion characteristics of particles where the trajectory of single particle was obtained. Note that, these previous works investigated the performance of the macroscopic behavior of solid particle group in a rotating flow field, which is crucial to understand the motion of particles in rotating flow field. However, few investigations concerning the macroscopic motion of single particle, as well as the present works have some limitations to acquire motion trajectory in experiment and directly investigate the characteristics of single particle. In summary, several attempts have been made to study the particles in rotating flow field which are mainly focused on particle group macroscopic motion. Although some previous researchers focused on liquid forces acting on single particle (Hsu et al., 2005; Brown and Lawler, 2003; Maxey and Riley, 1983; Fukada et al., 2014), enough information cannot be obtained from these previous investigations, particularly for the information of single particle macroscopic motion and collision characteristics which are seemingly little studied respects. Considering the study on single particle is to lay the foundation for our future work for investigating the particle swarm motion and collision characteristics. In contrast to the previous works, the collaboration of quantitative experiment and numerical simulation of single particle in rotating flow field in this paper possesses the feature that can acquire the motion trajectory regularity with high precision and have a convenience to investigate the characteristics of single particle. In addition, comparing with particle group, the motion of single particle has little effects on liquid flow. However, the particle-particle collisions have not been considered in this study which cannot adequate represent the general problem of sparse solid–liquid flow. As shown in Fig. 1, the three dimensional numerical model of single particle in rotating flow field based on the DEM and CFD method is
2.1.
Mathematical model
In this study, a three-dimensional CFD-DEM numerical model of single particle in rotating flow field has been developed using EDEM and ANSYS Fluent software packages. In this model, several dynamic processes are involved, including rotational liquid flow, particle motion, particle-liquid interaction, particle–wall collisions and particle rotation. The Navier–Stokes equation is applied to describe the rotational liquid. The interaction forces including particle-liquid interaction forces and particle–wall collisions forces are considered to couple the particle with rotating flow field. In addition, rotational moment balance equation is applied to describe the rotation of particle.
2.1.1.
Liquid flow
The liquid is in rotational motion which is subjected to several action forces induce by the wall shear when flow field begins to rotate. The fluid phase is governed by the Navier–Stokes equation for an incompressible fluid, so that three-dimensional mass and momentum equations shown as (Anderson and Jackson, 1967a):
→ ∂l + ∇ • l u 1 ∂t
→
∂ l u 1 ∂t
=0
→ →
+ ∇ • l u 1 u 1
(1)
= l g − ∇p + ∇ − fpf
(2)
Where t is the time, l is the liquid density, g is the gravity →
acceleration, u 1 is the liquid velocity vector, p is the pressure, is the stress tensor for the liquid, and fpf represents the forces of the particle acting on liquid. The coupling scheme can be given as (Xu and Yu, 1997):
developed firstly. Then, the visualization experiment is conducted to investigate the effects of rotating speed and particle size on motion
Fpf
trajectory of particle and validate for the simulation results. In the proposed numerical model, the single particle collisions characteristics including collision forces and effects of particle density and liquid
fpf =
viscosity on the particle-wall collisions are investigated. In addition, the effects of particle-wall collisions and fluid on particle rotation are discussed.
Where Vc is the volume of a computational cell, and Fpf is the fluid forces acting on particle. The velocity of the liquid at the edge of the flow field can be equal to the wall velocity (Chen et al., 2016b), which can be given as:
2. The numerical simulation of single particle in rotating flow field
Vc
vr = 0 To study the collision characteristics of single particle in fully filled rotating flow field, the single particle motion and collision in rotating flow field fully filled with liquid is investigated. Please cite this article in press as: Ma, X., et al., https://doi.org/10.1016/j.cherd.2019.12.024
v =
2 ωr 60
(3)
(4)
(5)
Collision of single particle in rotating flow field. Chem. Eng. Res. Des. (2020),
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Fig. 1 – Single particle in rotating flow field: (a) Sketch of collision of single particle in rotating flow field and (b) Sketch of cylindrical tank. Where vr is the radial velocity, v is the tangential velocity, ω is the rotation speed, and r is the radius of the experimental section.
particle. Moment balance equation of rotating particle can be written as (Thornton, 2015):
2.1.2.
Particle motion
In the rotating flow field, the motion of a single particle follows the Newton’s second law. The governing equation for the particle is given as follows (Cundall and Strack, 1980):
dup F = dt →
mp
(6)
→
Where mp is particle mass, u p is particle velocity and F is the vector sum of the forces, including particle gravity force, particle-liquid interaction forces and particle-wall contact forces, shown in Appendixes A and B.
2.1.3.
→ dωP = Fti rp = T dt →
IP
Particle rotation moment balance equation
The rotation of particle has great effects on the motion trajectory in the fluid, especially for high rotational inertia or heavy Please cite this article in press as: Ma, X., et al., https://doi.org/10.1016/j.cherd.2019.12.024
Ip =
(7)
p d5p 60
(8) →
Where Ip is moment of inertia, T is torque, rp is particle radius, Fti is tangential contact force, and ωp is particle angular velocity.
2.2.
Simulation conditions
The CFD-DEM three-dimensional simulation of a single particle in rotating flow field is carried out by coupling ANSYSFluent with EDEM software packages. At initial time, the liquid fully filled in the cylindrical tank is in the static state, of which the viscosity is set as 0.001 Pa s. In addition, the single particle is located in the bottom region of the liquid, of which the velocity up is set as 0 m/s, the density p as 2230 kg/m3 and angular velocity ωp as 0 rad/s. As for the three-dimensional computational model, the direction of gravity forces is set in the negative direction of Y-
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Fig. 2 – Grid diagram.
Table 1 – The parameters used in the numerical simulation. Parameter D
Value
l l
Inner diameter of cylindrical tank (mm) Height of cylindrical tank (mm) Cylindrical tank rotation speed (r/min) Liquid density (kg/m3 ) Liquid viscosity (Pa s)
p
particle density (kg/m3 )
dp f e
particle diameter(mm) The coefficient of friction Coefficient of restitution
h ω
120 20 60 1000 0.001, 0.002, 0.003, 0.004, 0.005, 0.006 900, 1200, 2000, 2230, 2550, 3000 1.5 0.3 0.9
axis. Without fluid inlet and outlet, the inner diameter D and height H are set as 120 mm and 20 mm respectively. The fluid computational domain contains 108394 hexahedral structuring grid cells that the simulation precision and computing cost have been considered, shown in Fig. 2. The interface (wall) of the geometric model captured is recognized as solid boundary, which rotates along the Z-axis at ω rpm. The simulation conditions parameters are set as follows (Table 1).
2.3.
Numerical solution
The coupling between rotating flow field and the single particle allows the particle-liquid interaction forces to act on
particle, particle to collide with wall and particle motion to influence the fluid flow. The coupling implemented includes the follow steps. The CFD calculation for one Fluent time-step are firstly performed with initial single particle information such as position and velocity given by DEM and subsequently the particle-liquid interaction forces acting on single particle are evaluated based on the updated fluid flow field. The resulting forces are then incorporated into DEM as well as the single particle is advanced at a smaller EDEM time-step until synchronized with Fluent time. The particle motion and collision forces have been calculated in EDEM, as well as yield the updated single particle information. These information are passed to Fluent and update momentum exchange term for CFD calculation at next time-step. In this case, the flow field is recalculated every 100 time steps of the EDEM. The EDEM time step tp is set as 5 × 10−6 s which is determined from Rayleigh time step. Correspondingly, the fluent time step tl is 5 × 10-4 s for the total simulation time of 25 s. At each time-step, the convergence criterion for each governing equation is controlled below 10−5 for convergence. In addition, pressure–velocity coupling is accomplished by using the SIMPLE algorithm. To enhance the stability of calculation, the governing equations are discretized using the second-order upwind scheme. Without fluid inlet and outlet, the linear rotation wall is applied to the bounding wall which rotates counterclockwise around the Z axis. Furthermore, RNG k- model is adopted to simulate flow field since the shear flow motion existing in rotating flow field (Yakhot and Orszag, 1986). Non-Equilibrium Wall Function is applied to calculate the flow field near the wall, considering the effects of Pressure
Fig. 3 – Experimental device. Please cite this article in press as: Ma, X., et al., https://doi.org/10.1016/j.cherd.2019.12.024
Collision of single particle in rotating flow field. Chem. Eng. Res. Des. (2020),
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Fig. 4 – Comparison between experimental data and the simulated data: (a) experimental particle motion, (b) simulated particle motion, (c) experimental data, (d) simulated data, (e) Comparison of X-coordinate value of single particle between experimental and simulated data and (f) Comparison of Y-coordinate value of single particle between experimental and simulated data. gradient. The contact of the single particle and wall is modeled by the Hertz-Mindlin Non-slip contact model based on a soft sphere approach in DEM (Hertz, 1949).
3.
Experimental validation
To verify the accuracy of the coupling model, a visualization experiment is conducted. As shown in Fig. 3, the experiment Please cite this article in press as: Ma, X., et al., https://doi.org/10.1016/j.cherd.2019.12.024
device consists of frequency converter, motor, rotating flow field, coordinate frame and high-speed video camera, with the center of these in the same axis. The high borosilicate glass is used as the spherical single particle with an average diameter and density of approximately 1.5 mm and 2230 kg/m3 , respectively. The liquid is deionized water with density of 1000 kg/m3 fully filled the cylindrical tank. The rotating flow field with the same dimensions as that in the numerical sim-
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Fig. 5 – Motion trajectory of a single particle in a rotating flow field. ulation rotates at a certain rotation speed that is controlled by a motor assisted by a frequency converter and fixed at the connection plate by bolts. The cylindrical tank is constructed of polymethyl methacrylate to acquire visualization of single particle. The experiment is performed at room temperature. The visualization collected by the high-speed video camera on the obverse side of the rotating flow field is outputted at 25 frames per second through the software that the particle coordinate position and motion trajectory data are obtained. In addition, the coordinate frame is used to calculate the particle position of each frame with X and Y coordinates. In this study, a comparison of motion trajectory of particle with a diameter of 1.5 mm and rotation speed of 60 r/min in the rotating flow field between the simulated data obtained by numerical simulation and the experimental data are plotted (Fig. 4). As shown, the simulation data of the particle motion in a rotating flow field agree well with the experimental data, which validate the reasonability of the coupling model.
4.
Results and discussion
4.1.
The effects of rotation speed and particle diameter
In this section, we focus on some macroscopic aspects of the particle motion has been attained. Several groups of particle motion trajectory images in the rotating flow field from static state to stabilized circumferential motion are clearly provided by visualization experiment. As shown in Fig. 5, composite images of each frame are presented. It can be seen in Please cite this article in press as: Ma, X., et al., https://doi.org/10.1016/j.cherd.2019.12.024
Fig. 5(a), (b) and (c), for a low rotation speed (ω = 20 r/min), the single particle makes small scale motion near the wall. Due to the increases of rotation speed, the single particle entrained by rotating liquid moves to the intersection of X-axis and wall on the rotating-out side, where the radial component of resultant forces acting on particle point toward the center and increase accordingly. These factors induce the particle moving away from the wall gradually and the trajectory range of particle expanded. It can be seen in Fig. 5(d), (e) and (f), the particle with diameter of dp = 1.5 mm, dp = 4 mm and dp = 4.8 mm respectively are investigated at a certain rotating speed ( = 60 r/min). As shown, trajectory range of particle has been narrowed with the increasing of the particle diameter. For a particle of large diameter (dp = 4.8 mm), due to an increase in gravity force and drag force, the particle velocity has decreased that single particle makes small scale motion near the wall. As shown, we can conclude that the rotation speed and particle diameter play a significant effects on the particle motion in rotating flow field, which provides a basis for prediction and control of particle motion in centrifugal separation. However, only the obverse view was analyzed because there was a limitation in measuring the side view in the experiment.
4.2.
Particle motion and collision
To analyze the collision during particle motion, the variation of particle velocity and particle trajectory with time within 25 s in a rotating flow field (ω = 60 r/min, dp =1.5 mm) are inves-
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Fig. 6 – Particle motion process and collision between particle and the wall.
tigated. It should be noted that we focused on the plane consisted by X and Y-axis for clearly observing the particlewall collisions. Fig. 6(a) illustrates the evolution of the particle motion trajectory in a rotating flow field. As shown, the particle is static which located in the lower region of the liquid (t = 0 s). When the rotating flow field begins to revolve, the particle moves near the wall along rotating direction initially and makes a small scale motion (t = 5 s). Due to the combined actions of liquid and wall on particle, the motion trajectory of particle increases and eventually makes a stable motion (t = 20 s∼t = 25 s). As the particle downwards, the particle velocity gradually increases and reaches the maximum value at the bottom of the Y-axis direction of the particle trajectory. Subsequently, particle collides with wall surface at the bottom of the particle trajectory with the particle velocity decreasing rapidly Please cite this article in press as: Ma, X., et al., https://doi.org/10.1016/j.cherd.2019.12.024
(Fig. 6(b)). When the particle moves upward along the wall, the velocity gradually decreases and reaches the minimum value at the top of Y-axis direction of the particle trajectory. As a result, we can clearly confirm that collisions frequently occurred in the bottom of the Y-axis of the trajectory, where the particle velocity is the maximum and the particle velocity is the minimum at the top of the trajectory.
4.3. The effects of particle material density and liquid viscosity The particle material density and liquid viscosity affects the force exerted on particle in the rotating flow field, hence determining the particle motion and collision. To understand the effects of particle material density and liquid viscosity on
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Fig. 7 – The effects of particle material density and liquid viscosity on number of collision.
Fig. 8 – The variation of collision force.
Fig. 9 – The effects of material density and liquid viscosity on Mas stress. collision, the effects of particle material density and liquid viscosity on collision are investigated. Fig. 7(a) shows the effect of particle material density (900 kg/m3 , 1200 kg/m3 , 2000 kg/m3 , 2230 kg/m3 , 2550 kg/m3 and 3000 kg/m3 ) on the number of collisions between single particle and wall surface in fully filled rotating flow field within 25 s (ω = 60 r/min, dp = 1.5 mm). As
Please cite this article in press as: Ma, X., et al., https://doi.org/10.1016/j.cherd.2019.12.024
shown, with the increasing of particle material density which is less than 2230 kg/m3 , the particle quality and resistance on particle motion increase, which decrease the range of single particle motion accordingly. Moreover, when the particle moves upward along the wall after the particle-wall collisions, it is easier to contact closely with wall instead of forming
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in each condition that we can infer the plastic deformation of the wall will not be affected by collision. However, more case studies such as different rotating speed and particle shape are required to better elucidate effects of particle collisions on wall surface.
4.5.
In the rotating flow field, the rotation of particle is affected by the contact between particle and wall and field velocity gradient which cause the different force on both sides of particle. To understand the variation of particle rotation, the angular velocity that characterizes the particle rotation is investigated. As shown in Fig. 10, there follows a peak of angular velocity as the particle moves upward along the wall when the rotation of flow field begins. Soon afterwards, the particle angular velocity fluctuated with circumferential motion of particle. To provide insight into the effects of wall and fluid on the rotation of particle, the relationship between the angular velocity and the particle Y coordinate value with time (t = 23 s–t = 25 s) is investigated. Based on the particle trajectory analysis (Fig. 6), after the collision between particle and the wall at the bottom of the Y-axis of the particle motion trajectory (the particle Y coordinate value is the minimum), the angular velocity of the particle decreases sharply. Subsequently, the angular velocity increases rapidly with particle moving upward along the wall. When the particle deviates from the wall surface, the angular velocity of particle is nearly balanced until the next collision between particle and wall. As a result, the particle rotation has been hindered by collision between particle and wall, and accelerated by particle moving upward along the wall, while it is hardly affected by rotating fluid in rotating flow field (Fig. 11).
Fig. 10 – Angular velocity vs. Time.
Fig. 11 – Angular velocity vs. Time and Particle Y coordinate value. multiple collisions. These factors induce the number of collisions decreases. When particle material density is greater than 2230 kg/m3 and less than 3000 kg/m3 , the number of collisions varies in a small range. Fig. 7(b) shows the effect of liquid viscosity (0.001, 0.002, 0.003, 0.004, 0.005, and 0.006) on the number of collisions within 25 s (ω = 60 r/min, dp = 1.5 mm). As shown, the number of collisions increases with the increasing of liquid viscosity. Therefore, we can infer that the selection of an appropriate particle material density and Liquid medium is beneficial for the decreasing of the number of collisions in a fully filled rotating flow field. As for the problem that the damage and instability of rotating flow field resulted by frequent collisions and strong collision forces, this study could provide useful information for maintenance in real application.
4.4.
Collision force
The plastic deformation of the wall surface may be affected by the collision force. To analyze the effects of collision force on wall surface, the tangential force and normal force decomposed by collision force are investigated. As shown in Fig. 8(a), the normal force fluctuates in the case of collision between a single particle and the wall, with a maximum value of 2.77 × 10−2 N. As shown in Fig. 8(b), the tangential force fluctuates as well in this case, with a maximum value of 1.27 × 10−2 N. As shown in Fig. 9(a) and FIg. 9(b) respectively, the maximum stress affected by particle material density and liquid viscosity is calculated. As a result, the value of maximum stress is far less than the yield limit of the wall ( p = 355 Mpa) Please cite this article in press as: Ma, X., et al., https://doi.org/10.1016/j.cherd.2019.12.024
Particle angular velocity
5.
Summary and conclusions
(1) The rotation speed of flow field and single particle diameter play significant roles in the particle motion behaviors. As the rotating speed increases, the particle motion has been enhanced that the trajectory range of single particle has been expanded. As the particle diameter increases, the particle motion has been weakened that the trajectory range of single particle has been narrowed. In addition, under the condition of ω = 60 r/min and dp = 1.5 mm, the particle collides with the wall at the bottom of the Y-axis of the trajectory, where the particle velocity is the maximum and the particle velocity is the minimum at the top of the trajectory. (2) Particle material density and liquid viscosity are important parameters affecting the collisions between particle and wall. With the increasing of particle material density which is less than 2230 kg/m3 , the number of collisions decreases. When particle material density is greater than 2230 kg/m3 and less than 3000 kg/m3 , the number of collisions varies in a small range, as well as the number of collisions increases rapidly with the increasing of liquid viscosity. (3) The normal collision force and the tangential collision force fluctuate in the case of collision between the single particle and wall. With the variation of particle material density and liquid viscosity, the plastic deformation of the wall will not be caused by collision force.
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(4) The particle-wall contact plays a significant role in the particle rotation which has been hindered by collision between particle and wall, and accelerated by particle moving upward along the wall. While the particle rotation is hardly affected by rotating fluid in rotating flow field. (5) It should be noted that this study has some limitations in representing the actual condition of engineering accurately because that we examine only one particle in rotating flow field. Therefore, a more accurate and advanced experiment should be developed, as well as the particle-particle collisions should be considered in further investigations to give a better understanding the characteristics of particle motion and collision.
accounted by the added mass force. For a spherical particle, the volume of the added mass is equal to one-half of the particle volume that the Additional mass force is calculated as (Auton, 1988; Milnethomson and Rott, 1968): Fvm =
dup 1 du Vp ( l − ) 2 l dt dt
(A7)
Where l is the liquid density. In the process of liquid rotating, there is a velocity gradient as liquid flow move around the particle. This velocity gradient produces a force perpendicular to the direction of the single particle motion, namely the Saffman force (Saffman, 1965): Fs = 1.61( l )
0.5 2 dp (ul
0.5 dul dy
− up )
Conflict of interests
(A8)
du
The authors declare that they have no conflict of interests.
Appendix A. Particle-liquid interaction In this study particle-liquid interaction forces have been considered to investigate the single particle in rotating flow field. Gravity force is calculated by: Fg =
Fm = l ul
1 3 d p g 6 p
1 CD p d2p (ul − up ) (ul − up ) 8
=
(A2)
Where ul is fluid velocity, up is particle velocity and CD is the drag coefficient of the sphere particle which is given by the following equation:
CD =
⎧ ⎪ ⎨ ⎪ ⎩
24
24/Rep 1 + 0.125Re0.687 p 0.44
Rep =
=
ul dS =
1 ωp d3p 3
/Rep (0.5 < Rep ≤ 1000)
(A3)
(Rep ≥ 1000)
where is velocity circulation on the surface of a spherical particle andωp is particle angular velocity. In this study, the liquid viscosity influences on particle collision has been considered. As the particles accelerate or decelerate in viscous fluid, there is instantaneous flow resistance acting on the whole process of particle acceleration or deceleration, namely Basset force, which have a great effect on particle motion. The Basset force is calculated by (Zapryanov and Tabakova, 1999):
3 d p ω2 rp 6 p
Fba
KB 2 √ d l = 4 p
t
dul d
du
− dp d √ t−
(A11)
−∞
l dp (ul − up )
(A10)
S
(Rep ≤ 0.5)
(A4)
Where l is the liquid density and is the liquid viscosity. Centrifugal force acting on single particle is calculated by: Fcent =
(A9)
(A1)
Where p is the particle material density and dp is particle diameter. Based on the drag force formula proposed by Di Felice (1994), the drag force can be written as: → Fd
Where dyl is the velocity gradient of the liquid. The single particle rotates with the liquid effects and particle-wall collision in rotating flow field. Therefore, the single particle affected by Magnus force, which is perpendicular to the plane consists of the particle rotation angular velocity vector and particle motion velocity vector. Magnus force is calculated by (Tsuji et al., 1985):
(A5)
Where ω is rotation speed of rotating flow and rp is the distance from particle to center of rotation. The pressure gradient force generally includes buoyancy force and acceleration pressure gradient in fluid, which is given by (Anderson and Jackson, 1967b):
Where is integral variable, and KB is parameter which can be given as: KB = 2.88 +
3.12 (Ac + 1)
3
(A12)
Where Ac is determined by the ratio of the hydrodynamic force to the force producing the particle acceleration, which can be given as:
Ac =
ul − up ap dp
(A13)
Where ap is particle acceleration. Fp = −Vp ∇p
(A6)
Where ∇pis dynamic pressure difference and Vp is particle volume. In the rotating flow field, the resistance of the liquid mass that is moving at the same acceleration as the particle is Please cite this article in press as: Ma, X., et al., https://doi.org/10.1016/j.cherd.2019.12.024
Appendix B. Particle-wall interaction To investigate the particle-wall collision force, the DEM based on a soft sphere approach is used to describe the collision between particle and inner wall of tank. The collision force
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of a particle-wall can be decomposed into the normal contact force Fct and the tangential contact force Fcn , which can be written as follows: Fct = −kt dt − t [(r − r n) n + rωs n]
(B1)
Fcn = −kn dn − n (vr × n) n
(B2)
Where kt , kn , dt , and dn are tangential elasticity coefficient, normal elasticity coefficient, tangential relative velocity and normal relative velocity, respectively. The parameters nt , nn , ωs , and vr are tangential damping coefficient, normal damping coefficient, particle relative angular velocity and particle relative linear velocity, respectively. Based on the Hertz theory, the maximum stress is concentrated in the radial direction of the contact surface, as well as the collision between single particle and wall surface is random. In this theory, it is assumed that the collision contact radius between particle and wall surface is the same as that of particle. The maximum stress on particle collision can be written as (Hertz, 1881): 1 =
4F1 d2p
(B3)
Where 1 is maximum stress, dp is the particle diameter and F1 is the collision force.
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Contents lists available at ScienceDirect
Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd
Collision of single particle in rotating flow field Xin Ma a,∗ , Yixuan Peng a , Qi Yu a , Ke Lai a , Jie Li b , Di Wu c a
The Ministry of Education Key Laboratory for Oil and Gas Equipment, School of Mechanical Engineering, Southwest Petroleum University, Chengdu, Sichuan 610500, PR China b School of Automotive Engineering, Qiannan Polytechnic for Nationalities, Duyun, Guizhou 558022, PR China c Liaohe Oil Field Drilling Technology Research Institute, Panjin, Liaoning 124010, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history:
The studies on motion and collision of particle in rotating flow field are of great importance
Received 13 July 2019
for chemical engineering. As the lack of work concerning the single particle, the motion
Received in revised form 20
trajectory and collision characteristics of single particle have not been understood. In this
December 2019
article, a coupled discrete element method (DEM) and the computational fluid dynamics
Accepted 24 December 2019
(CFD) method model of a single particle in rotating flow field has been developed and ver-
Available online xxx
ified by experimental data to investigate the collision characteristics of the particle. The
Keywords:
by experiment. In The numerical simulation, the particle motion process has been simu-
Rotating flow field
lated. The effects of particle density and liquid viscosity on collision and particle collision
effects of rotation speed and particle diameter on particle motion have been investigated
Single particle
force have been investigated. Furthermore, the effects of wall and rotating fluid on particle
Collision
rotation have been discussed and analyzed. The results indicate that the rotation speed of
Numerical simulation
flow field and single particle diameter play significant roles in the particle motion behaviors.
Particle motion
In the process of particle motion, the particle has collided with the wall at the bottom of the Y-axis of the trajectory. The selection of an appropriate particle material density and Liquid medium is beneficial for the decreasing of the number of collisions in a fully filled rotating flow field. The plastic deformation of wall will not be caused by collision force. In addition, the particle rotation is significant affected by the particle-wall contact, while it is hardly affected by rotating fluid. They thereby can provide a technical basis for prediction and control of particle motion and maintenance of equipment in centrifugal separation and helical transportation. © 2019 Published by Elsevier B.V. on behalf of Institution of Chemical Engineers.
1.
Introduction
The lack of work concerned these aspects are partly due to difficulties in making quantitative experiment of particles in rotating flow field. It
Particles-liquid flow in rotating flow field is common in chemical engineering. The liquid flow induces the particles motion by acting forces
is also difficult to accurately simulate the particle motion and collision in this system.
on particles, as well as particle-particle and particle-wall collisions occur in the process of particles motion. In this context, an insight into the motion and collision characteristics of particle makes an effort
The discrete element method (DEM) (Cundall and Strack, 1979) has proven to be a versatile tool to simulate the motion and collisions of particles with high accuracy. The particles are represented by soft spheres and the contact forces are derived from their overlap between
to provide useful information for the design operation, maintenance and optimization of operational conditions in the real application,
others (or walls) during a finite-time collision. Zhang et al. (2015) inves-
including the centrifugal separation (Kou and Dejam, 2019; Dejam, 2019a,b; Dejam et al., 2015a,b), helical transportation and rotary agi-
tigated the effects of particles collisions on particle distribution and particle resuspension rate.) Li et al. (2016) studied the contact forces
tation (Zverev and Ushakov, 1968; Kasat et al., 2008; Mang et al., 1998).
between the particles and the wall with the increase of flattening and rotation speed by DEM. With the development of advanced computational techniques, the information of particle-fluid, particle-particle
∗
and particle-wall interaction forces can be generated by use coupling
Corresponding author. of Computational Fluid Dynamics (CFD) and Discrete Element Method E-mail address: [email protected] (X. Ma). https://doi.org/10.1016/j.cherd.2019.12.024 0263-8762/© 2019 Published by Elsevier B.V. on behalf of Institution of Chemical Engineers.
Please cite this article in press as: Ma, X., et al., https://doi.org/10.1016/j.cherd.2019.12.024
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(DEM) approach (Tsuji et al., 1993). Recently, CFD-DEM coupling method has been proven to be effective in modeling particle-particle, particlewall and particle-fluid interactions with high precision (Wang et al., 2009). Ren et al. (2012) investigated the migration of particles under a cylindrical jet bed based on the DEM-CFD approach. Liu et al. (2016) studied particle-particle and particle-wall collision frequencies considering the different shape based on the DEM–CFD approach. However, the particle motion and collision in rotating flow field have not been conducted based on DEM-CFD method. The effects of influencing factors on particle collisions have not been sufficiently considered. In previous studies, several attempts have been made to study the motion of particles in a rotating flow field using experimental and numerical method. Chou et al. (Chou et al., 2011, 2014; Chou et al., 2019) experimental studied the solid particle group motion in rotating
The cylindrical tank, driven by the external device, rotates along the z-axis at a certain rotating speed, of which the inner diameter D is 120 mm and the height H is 20 mm shown in Figure (b). The single particle placed in the cylindrical tank is carried by the rotating liquid flow. Moreover, particle density and liquid density is considered that the liquid density is set as l = 1000 kg/m3 and particle material density as p = 2230 kg/m3 . It is noted that the densities of liquid and particle are just equal to their physical values, where the liquid is water, and the particle is made by high borosilicate glass. The rotating-in side is the region in liquid phase of left side of y-axis. Correspondingly, the rotating out side is the region in liquid phase of right side of y-axis.
flow field considering several influencing factors. Sakai et al. (2012) and Sun et al. (2014) studied the influence of spring coefficient on particles behavior and the influence of hydrodynamic force on the macroscopic particle movement behavior and compared it with the simulated data. Sun et al. (2013) verified the accuracy of DEM-SPH model through experiments, and considered that DEM-SPH model could accurately solve the motion characteristics of particle groups in the rotating flow field by comparing the macroscopic behaviors of particle group movement in experiments and simulations. Chen et al., 2016a developed a threedimensional theoretical model based on the volume of fluid (VOF) method and the DEM validated by visualization experiment to investigate dispersion characteristics of particles where the trajectory of single particle was obtained. Note that, these previous works investigated the performance of the macroscopic behavior of solid particle group in a rotating flow field, which is crucial to understand the motion of particles in rotating flow field. However, few investigations concerning the macroscopic motion of single particle, as well as the present works have some limitations to acquire motion trajectory in experiment and directly investigate the characteristics of single particle. In summary, several attempts have been made to study the particles in rotating flow field which are mainly focused on particle group macroscopic motion. Although some previous researchers focused on liquid forces acting on single particle (Hsu et al., 2005; Brown and Lawler, 2003; Maxey and Riley, 1983; Fukada et al., 2014), enough information cannot be obtained from these previous investigations, particularly for the information of single particle macroscopic motion and collision characteristics which are seemingly little studied respects. Considering the study on single particle is to lay the foundation for our future work for investigating the particle swarm motion and collision characteristics. In contrast to the previous works, the collaboration of quantitative experiment and numerical simulation of single particle in rotating flow field in this paper possesses the feature that can acquire the motion trajectory regularity with high precision and have a convenience to investigate the characteristics of single particle. In addition, comparing with particle group, the motion of single particle has little effects on liquid flow. However, the particle-particle collisions have not been considered in this study which cannot adequate represent the general problem of sparse solid–liquid flow. As shown in Fig. 1, the three dimensional numerical model of single particle in rotating flow field based on the DEM and CFD method is
2.1.
Mathematical model
In this study, a three-dimensional CFD-DEM numerical model of single particle in rotating flow field has been developed using EDEM and ANSYS Fluent software packages. In this model, several dynamic processes are involved, including rotational liquid flow, particle motion, particle-liquid interaction, particle–wall collisions and particle rotation. The Navier–Stokes equation is applied to describe the rotational liquid. The interaction forces including particle-liquid interaction forces and particle–wall collisions forces are considered to couple the particle with rotating flow field. In addition, rotational moment balance equation is applied to describe the rotation of particle.
2.1.1.
Liquid flow
The liquid is in rotational motion which is subjected to several action forces induce by the wall shear when flow field begins to rotate. The fluid phase is governed by the Navier–Stokes equation for an incompressible fluid, so that three-dimensional mass and momentum equations shown as (Anderson and Jackson, 1967a):
→ ∂l + ∇ • l u 1 ∂t
→
∂ l u 1 ∂t
=0
→ →
+ ∇ • l u 1 u 1
(1)
= l g − ∇p + ∇ − fpf
(2)
Where t is the time, l is the liquid density, g is the gravity →
acceleration, u 1 is the liquid velocity vector, p is the pressure, is the stress tensor for the liquid, and fpf represents the forces of the particle acting on liquid. The coupling scheme can be given as (Xu and Yu, 1997):
developed firstly. Then, the visualization experiment is conducted to investigate the effects of rotating speed and particle size on motion
Fpf
trajectory of particle and validate for the simulation results. In the proposed numerical model, the single particle collisions characteristics including collision forces and effects of particle density and liquid
fpf =
viscosity on the particle-wall collisions are investigated. In addition, the effects of particle-wall collisions and fluid on particle rotation are discussed.
Where Vc is the volume of a computational cell, and Fpf is the fluid forces acting on particle. The velocity of the liquid at the edge of the flow field can be equal to the wall velocity (Chen et al., 2016b), which can be given as:
2. The numerical simulation of single particle in rotating flow field
Vc
vr = 0 To study the collision characteristics of single particle in fully filled rotating flow field, the single particle motion and collision in rotating flow field fully filled with liquid is investigated. Please cite this article in press as: Ma, X., et al., https://doi.org/10.1016/j.cherd.2019.12.024
v =
2 ωr 60
(3)
(4)
(5)
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Fig. 1 – Single particle in rotating flow field: (a) Sketch of collision of single particle in rotating flow field and (b) Sketch of cylindrical tank. Where vr is the radial velocity, v is the tangential velocity, ω is the rotation speed, and r is the radius of the experimental section.
particle. Moment balance equation of rotating particle can be written as (Thornton, 2015):
2.1.2.
Particle motion
In the rotating flow field, the motion of a single particle follows the Newton’s second law. The governing equation for the particle is given as follows (Cundall and Strack, 1980):
dup F = dt →
mp
(6)
→
Where mp is particle mass, u p is particle velocity and F is the vector sum of the forces, including particle gravity force, particle-liquid interaction forces and particle-wall contact forces, shown in Appendixes A and B.
2.1.3.
→ dωP = Fti rp = T dt →
IP
Particle rotation moment balance equation
The rotation of particle has great effects on the motion trajectory in the fluid, especially for high rotational inertia or heavy Please cite this article in press as: Ma, X., et al., https://doi.org/10.1016/j.cherd.2019.12.024
Ip =
(7)
p d5p 60
(8) →
Where Ip is moment of inertia, T is torque, rp is particle radius, Fti is tangential contact force, and ωp is particle angular velocity.
2.2.
Simulation conditions
The CFD-DEM three-dimensional simulation of a single particle in rotating flow field is carried out by coupling ANSYSFluent with EDEM software packages. At initial time, the liquid fully filled in the cylindrical tank is in the static state, of which the viscosity is set as 0.001 Pa s. In addition, the single particle is located in the bottom region of the liquid, of which the velocity up is set as 0 m/s, the density p as 2230 kg/m3 and angular velocity ωp as 0 rad/s. As for the three-dimensional computational model, the direction of gravity forces is set in the negative direction of Y-
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Fig. 2 – Grid diagram.
Table 1 – The parameters used in the numerical simulation. Parameter D
Value
l l
Inner diameter of cylindrical tank (mm) Height of cylindrical tank (mm) Cylindrical tank rotation speed (r/min) Liquid density (kg/m3 ) Liquid viscosity (Pa s)
p
particle density (kg/m3 )
dp f e
particle diameter(mm) The coefficient of friction Coefficient of restitution
h ω
120 20 60 1000 0.001, 0.002, 0.003, 0.004, 0.005, 0.006 900, 1200, 2000, 2230, 2550, 3000 1.5 0.3 0.9
axis. Without fluid inlet and outlet, the inner diameter D and height H are set as 120 mm and 20 mm respectively. The fluid computational domain contains 108394 hexahedral structuring grid cells that the simulation precision and computing cost have been considered, shown in Fig. 2. The interface (wall) of the geometric model captured is recognized as solid boundary, which rotates along the Z-axis at ω rpm. The simulation conditions parameters are set as follows (Table 1).
2.3.
Numerical solution
The coupling between rotating flow field and the single particle allows the particle-liquid interaction forces to act on
particle, particle to collide with wall and particle motion to influence the fluid flow. The coupling implemented includes the follow steps. The CFD calculation for one Fluent time-step are firstly performed with initial single particle information such as position and velocity given by DEM and subsequently the particle-liquid interaction forces acting on single particle are evaluated based on the updated fluid flow field. The resulting forces are then incorporated into DEM as well as the single particle is advanced at a smaller EDEM time-step until synchronized with Fluent time. The particle motion and collision forces have been calculated in EDEM, as well as yield the updated single particle information. These information are passed to Fluent and update momentum exchange term for CFD calculation at next time-step. In this case, the flow field is recalculated every 100 time steps of the EDEM. The EDEM time step tp is set as 5 × 10−6 s which is determined from Rayleigh time step. Correspondingly, the fluent time step tl is 5 × 10-4 s for the total simulation time of 25 s. At each time-step, the convergence criterion for each governing equation is controlled below 10−5 for convergence. In addition, pressure–velocity coupling is accomplished by using the SIMPLE algorithm. To enhance the stability of calculation, the governing equations are discretized using the second-order upwind scheme. Without fluid inlet and outlet, the linear rotation wall is applied to the bounding wall which rotates counterclockwise around the Z axis. Furthermore, RNG k- model is adopted to simulate flow field since the shear flow motion existing in rotating flow field (Yakhot and Orszag, 1986). Non-Equilibrium Wall Function is applied to calculate the flow field near the wall, considering the effects of Pressure
Fig. 3 – Experimental device. Please cite this article in press as: Ma, X., et al., https://doi.org/10.1016/j.cherd.2019.12.024
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Fig. 4 – Comparison between experimental data and the simulated data: (a) experimental particle motion, (b) simulated particle motion, (c) experimental data, (d) simulated data, (e) Comparison of X-coordinate value of single particle between experimental and simulated data and (f) Comparison of Y-coordinate value of single particle between experimental and simulated data. gradient. The contact of the single particle and wall is modeled by the Hertz-Mindlin Non-slip contact model based on a soft sphere approach in DEM (Hertz, 1949).
3.
Experimental validation
To verify the accuracy of the coupling model, a visualization experiment is conducted. As shown in Fig. 3, the experiment Please cite this article in press as: Ma, X., et al., https://doi.org/10.1016/j.cherd.2019.12.024
device consists of frequency converter, motor, rotating flow field, coordinate frame and high-speed video camera, with the center of these in the same axis. The high borosilicate glass is used as the spherical single particle with an average diameter and density of approximately 1.5 mm and 2230 kg/m3 , respectively. The liquid is deionized water with density of 1000 kg/m3 fully filled the cylindrical tank. The rotating flow field with the same dimensions as that in the numerical sim-
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Fig. 5 – Motion trajectory of a single particle in a rotating flow field. ulation rotates at a certain rotation speed that is controlled by a motor assisted by a frequency converter and fixed at the connection plate by bolts. The cylindrical tank is constructed of polymethyl methacrylate to acquire visualization of single particle. The experiment is performed at room temperature. The visualization collected by the high-speed video camera on the obverse side of the rotating flow field is outputted at 25 frames per second through the software that the particle coordinate position and motion trajectory data are obtained. In addition, the coordinate frame is used to calculate the particle position of each frame with X and Y coordinates. In this study, a comparison of motion trajectory of particle with a diameter of 1.5 mm and rotation speed of 60 r/min in the rotating flow field between the simulated data obtained by numerical simulation and the experimental data are plotted (Fig. 4). As shown, the simulation data of the particle motion in a rotating flow field agree well with the experimental data, which validate the reasonability of the coupling model.
4.
Results and discussion
4.1.
The effects of rotation speed and particle diameter
In this section, we focus on some macroscopic aspects of the particle motion has been attained. Several groups of particle motion trajectory images in the rotating flow field from static state to stabilized circumferential motion are clearly provided by visualization experiment. As shown in Fig. 5, composite images of each frame are presented. It can be seen in Please cite this article in press as: Ma, X., et al., https://doi.org/10.1016/j.cherd.2019.12.024
Fig. 5(a), (b) and (c), for a low rotation speed (ω = 20 r/min), the single particle makes small scale motion near the wall. Due to the increases of rotation speed, the single particle entrained by rotating liquid moves to the intersection of X-axis and wall on the rotating-out side, where the radial component of resultant forces acting on particle point toward the center and increase accordingly. These factors induce the particle moving away from the wall gradually and the trajectory range of particle expanded. It can be seen in Fig. 5(d), (e) and (f), the particle with diameter of dp = 1.5 mm, dp = 4 mm and dp = 4.8 mm respectively are investigated at a certain rotating speed ( = 60 r/min). As shown, trajectory range of particle has been narrowed with the increasing of the particle diameter. For a particle of large diameter (dp = 4.8 mm), due to an increase in gravity force and drag force, the particle velocity has decreased that single particle makes small scale motion near the wall. As shown, we can conclude that the rotation speed and particle diameter play a significant effects on the particle motion in rotating flow field, which provides a basis for prediction and control of particle motion in centrifugal separation. However, only the obverse view was analyzed because there was a limitation in measuring the side view in the experiment.
4.2.
Particle motion and collision
To analyze the collision during particle motion, the variation of particle velocity and particle trajectory with time within 25 s in a rotating flow field (ω = 60 r/min, dp =1.5 mm) are inves-
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Fig. 6 – Particle motion process and collision between particle and the wall.
tigated. It should be noted that we focused on the plane consisted by X and Y-axis for clearly observing the particlewall collisions. Fig. 6(a) illustrates the evolution of the particle motion trajectory in a rotating flow field. As shown, the particle is static which located in the lower region of the liquid (t = 0 s). When the rotating flow field begins to revolve, the particle moves near the wall along rotating direction initially and makes a small scale motion (t = 5 s). Due to the combined actions of liquid and wall on particle, the motion trajectory of particle increases and eventually makes a stable motion (t = 20 s∼t = 25 s). As the particle downwards, the particle velocity gradually increases and reaches the maximum value at the bottom of the Y-axis direction of the particle trajectory. Subsequently, particle collides with wall surface at the bottom of the particle trajectory with the particle velocity decreasing rapidly Please cite this article in press as: Ma, X., et al., https://doi.org/10.1016/j.cherd.2019.12.024
(Fig. 6(b)). When the particle moves upward along the wall, the velocity gradually decreases and reaches the minimum value at the top of Y-axis direction of the particle trajectory. As a result, we can clearly confirm that collisions frequently occurred in the bottom of the Y-axis of the trajectory, where the particle velocity is the maximum and the particle velocity is the minimum at the top of the trajectory.
4.3. The effects of particle material density and liquid viscosity The particle material density and liquid viscosity affects the force exerted on particle in the rotating flow field, hence determining the particle motion and collision. To understand the effects of particle material density and liquid viscosity on
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Fig. 7 – The effects of particle material density and liquid viscosity on number of collision.
Fig. 8 – The variation of collision force.
Fig. 9 – The effects of material density and liquid viscosity on Mas stress. collision, the effects of particle material density and liquid viscosity on collision are investigated. Fig. 7(a) shows the effect of particle material density (900 kg/m3 , 1200 kg/m3 , 2000 kg/m3 , 2230 kg/m3 , 2550 kg/m3 and 3000 kg/m3 ) on the number of collisions between single particle and wall surface in fully filled rotating flow field within 25 s (ω = 60 r/min, dp = 1.5 mm). As
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shown, with the increasing of particle material density which is less than 2230 kg/m3 , the particle quality and resistance on particle motion increase, which decrease the range of single particle motion accordingly. Moreover, when the particle moves upward along the wall after the particle-wall collisions, it is easier to contact closely with wall instead of forming
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in each condition that we can infer the plastic deformation of the wall will not be affected by collision. However, more case studies such as different rotating speed and particle shape are required to better elucidate effects of particle collisions on wall surface.
4.5.
In the rotating flow field, the rotation of particle is affected by the contact between particle and wall and field velocity gradient which cause the different force on both sides of particle. To understand the variation of particle rotation, the angular velocity that characterizes the particle rotation is investigated. As shown in Fig. 10, there follows a peak of angular velocity as the particle moves upward along the wall when the rotation of flow field begins. Soon afterwards, the particle angular velocity fluctuated with circumferential motion of particle. To provide insight into the effects of wall and fluid on the rotation of particle, the relationship between the angular velocity and the particle Y coordinate value with time (t = 23 s–t = 25 s) is investigated. Based on the particle trajectory analysis (Fig. 6), after the collision between particle and the wall at the bottom of the Y-axis of the particle motion trajectory (the particle Y coordinate value is the minimum), the angular velocity of the particle decreases sharply. Subsequently, the angular velocity increases rapidly with particle moving upward along the wall. When the particle deviates from the wall surface, the angular velocity of particle is nearly balanced until the next collision between particle and wall. As a result, the particle rotation has been hindered by collision between particle and wall, and accelerated by particle moving upward along the wall, while it is hardly affected by rotating fluid in rotating flow field (Fig. 11).
Fig. 10 – Angular velocity vs. Time.
Fig. 11 – Angular velocity vs. Time and Particle Y coordinate value. multiple collisions. These factors induce the number of collisions decreases. When particle material density is greater than 2230 kg/m3 and less than 3000 kg/m3 , the number of collisions varies in a small range. Fig. 7(b) shows the effect of liquid viscosity (0.001, 0.002, 0.003, 0.004, 0.005, and 0.006) on the number of collisions within 25 s (ω = 60 r/min, dp = 1.5 mm). As shown, the number of collisions increases with the increasing of liquid viscosity. Therefore, we can infer that the selection of an appropriate particle material density and Liquid medium is beneficial for the decreasing of the number of collisions in a fully filled rotating flow field. As for the problem that the damage and instability of rotating flow field resulted by frequent collisions and strong collision forces, this study could provide useful information for maintenance in real application.
4.4.
Collision force
The plastic deformation of the wall surface may be affected by the collision force. To analyze the effects of collision force on wall surface, the tangential force and normal force decomposed by collision force are investigated. As shown in Fig. 8(a), the normal force fluctuates in the case of collision between a single particle and the wall, with a maximum value of 2.77 × 10−2 N. As shown in Fig. 8(b), the tangential force fluctuates as well in this case, with a maximum value of 1.27 × 10−2 N. As shown in Fig. 9(a) and FIg. 9(b) respectively, the maximum stress affected by particle material density and liquid viscosity is calculated. As a result, the value of maximum stress is far less than the yield limit of the wall ( p = 355 Mpa) Please cite this article in press as: Ma, X., et al., https://doi.org/10.1016/j.cherd.2019.12.024
Particle angular velocity
5.
Summary and conclusions
(1) The rotation speed of flow field and single particle diameter play significant roles in the particle motion behaviors. As the rotating speed increases, the particle motion has been enhanced that the trajectory range of single particle has been expanded. As the particle diameter increases, the particle motion has been weakened that the trajectory range of single particle has been narrowed. In addition, under the condition of ω = 60 r/min and dp = 1.5 mm, the particle collides with the wall at the bottom of the Y-axis of the trajectory, where the particle velocity is the maximum and the particle velocity is the minimum at the top of the trajectory. (2) Particle material density and liquid viscosity are important parameters affecting the collisions between particle and wall. With the increasing of particle material density which is less than 2230 kg/m3 , the number of collisions decreases. When particle material density is greater than 2230 kg/m3 and less than 3000 kg/m3 , the number of collisions varies in a small range, as well as the number of collisions increases rapidly with the increasing of liquid viscosity. (3) The normal collision force and the tangential collision force fluctuate in the case of collision between the single particle and wall. With the variation of particle material density and liquid viscosity, the plastic deformation of the wall will not be caused by collision force.
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(4) The particle-wall contact plays a significant role in the particle rotation which has been hindered by collision between particle and wall, and accelerated by particle moving upward along the wall. While the particle rotation is hardly affected by rotating fluid in rotating flow field. (5) It should be noted that this study has some limitations in representing the actual condition of engineering accurately because that we examine only one particle in rotating flow field. Therefore, a more accurate and advanced experiment should be developed, as well as the particle-particle collisions should be considered in further investigations to give a better understanding the characteristics of particle motion and collision.
accounted by the added mass force. For a spherical particle, the volume of the added mass is equal to one-half of the particle volume that the Additional mass force is calculated as (Auton, 1988; Milnethomson and Rott, 1968): Fvm =
dup 1 du Vp ( l − ) 2 l dt dt
(A7)
Where l is the liquid density. In the process of liquid rotating, there is a velocity gradient as liquid flow move around the particle. This velocity gradient produces a force perpendicular to the direction of the single particle motion, namely the Saffman force (Saffman, 1965): Fs = 1.61( l )
0.5 2 dp (ul
0.5 dul dy
− up )
Conflict of interests
(A8)
du
The authors declare that they have no conflict of interests.
Appendix A. Particle-liquid interaction In this study particle-liquid interaction forces have been considered to investigate the single particle in rotating flow field. Gravity force is calculated by: Fg =
Fm = l ul
1 3 d p g 6 p
1 CD p d2p (ul − up ) (ul − up ) 8
=
(A2)
Where ul is fluid velocity, up is particle velocity and CD is the drag coefficient of the sphere particle which is given by the following equation:
CD =
⎧ ⎪ ⎨ ⎪ ⎩
24
24/Rep 1 + 0.125Re0.687 p 0.44
Rep =
=
ul dS =
1 ωp d3p 3
/Rep (0.5 < Rep ≤ 1000)
(A3)
(Rep ≥ 1000)
where is velocity circulation on the surface of a spherical particle andωp is particle angular velocity. In this study, the liquid viscosity influences on particle collision has been considered. As the particles accelerate or decelerate in viscous fluid, there is instantaneous flow resistance acting on the whole process of particle acceleration or deceleration, namely Basset force, which have a great effect on particle motion. The Basset force is calculated by (Zapryanov and Tabakova, 1999):
3 d p ω2 rp 6 p
Fba
KB 2 √ d l = 4 p
t
dul d
du
− dp d √ t−
(A11)
−∞
l dp (ul − up )
(A10)
S
(Rep ≤ 0.5)
(A4)
Where l is the liquid density and is the liquid viscosity. Centrifugal force acting on single particle is calculated by: Fcent =
(A9)
(A1)
Where p is the particle material density and dp is particle diameter. Based on the drag force formula proposed by Di Felice (1994), the drag force can be written as: → Fd
Where dyl is the velocity gradient of the liquid. The single particle rotates with the liquid effects and particle-wall collision in rotating flow field. Therefore, the single particle affected by Magnus force, which is perpendicular to the plane consists of the particle rotation angular velocity vector and particle motion velocity vector. Magnus force is calculated by (Tsuji et al., 1985):
(A5)
Where ω is rotation speed of rotating flow and rp is the distance from particle to center of rotation. The pressure gradient force generally includes buoyancy force and acceleration pressure gradient in fluid, which is given by (Anderson and Jackson, 1967b):
Where is integral variable, and KB is parameter which can be given as: KB = 2.88 +
3.12 (Ac + 1)
3
(A12)
Where Ac is determined by the ratio of the hydrodynamic force to the force producing the particle acceleration, which can be given as:
Ac =
ul − up ap dp
(A13)
Where ap is particle acceleration. Fp = −Vp ∇p
(A6)
Where ∇pis dynamic pressure difference and Vp is particle volume. In the rotating flow field, the resistance of the liquid mass that is moving at the same acceleration as the particle is Please cite this article in press as: Ma, X., et al., https://doi.org/10.1016/j.cherd.2019.12.024
Appendix B. Particle-wall interaction To investigate the particle-wall collision force, the DEM based on a soft sphere approach is used to describe the collision between particle and inner wall of tank. The collision force
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of a particle-wall can be decomposed into the normal contact force Fct and the tangential contact force Fcn , which can be written as follows: Fct = −kt dt − t [(r − r n) n + rωs n]
(B1)
Fcn = −kn dn − n (vr × n) n
(B2)
Where kt , kn , dt , and dn are tangential elasticity coefficient, normal elasticity coefficient, tangential relative velocity and normal relative velocity, respectively. The parameters nt , nn , ωs , and vr are tangential damping coefficient, normal damping coefficient, particle relative angular velocity and particle relative linear velocity, respectively. Based on the Hertz theory, the maximum stress is concentrated in the radial direction of the contact surface, as well as the collision between single particle and wall surface is random. In this theory, it is assumed that the collision contact radius between particle and wall surface is the same as that of particle. The maximum stress on particle collision can be written as (Hertz, 1881): 1 =
4F1 d2p
(B3)
Where 1 is maximum stress, dp is the particle diameter and F1 is the collision force.
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Collision of single particle in rotating flow field. Chem. Eng. Res. Des. (2020),