Orientation of cylindrical particles in gas–solid circulating fluidized bed

Particuology 10 (2012) 89–96

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Orientation of cylindrical particles in gas–solid circulating fluidized bed Jie Cai a,∗ , Qihe Li a , Zhulin Yuan b a b

School of Energy and Mechanical Engineering, Nanjing Normal University, Nanjing 210042, Jiangsu, China Thermal-Energy Institute, Southeast University, Nanjing 210096, Jiangsu, China

a r t i c l e

i n f o

Article history: Received 13 October 2010 Received in revised form 24 January 2011 Accepted 21 March 2011 Keywords: Euler–Lagrange model Rigid kinetics Impact kinetics Orientation of cylindrical particles Nutation angle

a b s t r a c t The orientation of cylindrical particles in a gas–solid circulating fluidized bed was investigated by establishing a three-dimensional Euler–Lagrange model on the basis of rigid kinetics, impact kinetics and gas–solid two-phase flow theory. The resulting simulation indicated that the model could well illustrate the orientation of cylindrical particles in a riser during fluidization. The influences of bed structure and operation parameters on orientation of cylindrical particles were then studied and compared with related experimental results. The simulation results showed that the majority of cylindrical particles move with small nutation angles in the riser, the orientation of cylindrical particles is affected more obviously by their positions than by their slenderness and local gas velocities. The simulation results well agree with experiments, thus validating the proposed model and computation. © 2011 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

1. Introduction Circulating fluidization of cylindrical particles has been extensively employed across a wide range of industries, including combustion of biomass stalks, adjustment of moisture content of cut-tobacco and drying of pills in a fluidized bed. The orientation of cylindrical particles is an important behavior in fluidization, which affects heat and mass transfer between particle and hot gas, control of dryer efficiency and the quality of the final product. The orientation of a cylindrical particle can be represented by the Euler angles, composed of precession angle ( ), nutation angle () and spin angle (˚). As shown in Fig. 1, there are two reference frames, that is, the body axes (,  and ) and the fixed reference frame (x, y and z). nutation angle is the angle between -axis of body axes and z-axis of the fixed reference frame. If the centroid of cylindrical particle is on the origin and the axis of cylindrical particle is along the -axis in body axes, nutation angle is just the angle between the axis of a cylindrical particle and the axis of the riser. Orientation is an important parameter of cylindrical particles while describing their fluidization behavior. Early theoretical challenge on orientation of fibers was based on a two-dimensional dilutephase flow field, for which a rotation equation of fibers based on slender-body theory (Batchelor, 1970) was established, and then a phenomenological term to describe the interactions between fibers was introduced (Givler, 1983; Givler, Crochet, & Pipes, 1983;

∗ Corresponding author. Tel.: +86 15850551890. E-mail address: [email protected] (J. Cai).

Folgar & Tucker, 1984; Jackson, Advani, & Tucker, 1986). Some valuable two-dimensional models were then formulated in succession and were soon used in moment estimation, orientation analysis of wedge-shaped suspension flow field, and dynamics of suspensions of fibers sedimenting for the limiting case of zero Reynolds number, etc. (Butler & Shaqfeh, 2002; Cloitre & Monqruel, 1999; Krushkal and Gallily, 1988; Lin & Zhang, 2002; Lin, Zhang, & Olson, 2007; Saintillan & Shelley, 2007; Saintillan, Shaqfeh, & Darve, 2006). The question in the mean probability function was solved with characteristics analysis, and drag reduction induced by rigid fibers in turbulent channel flow was studied with continuity equations (Paschkewitz, Dubief, Dimitropoulos, Shaqfeh, & Moin, 2004; Zhang, Lin, & Zhang, 2006). The finite difference scheme was introduced to study the orientation of fibers in spherical co-ordinates leading to the diffusivity transformation (Zhou & Lin, 2007). Experimental studies were carried out on the orientation of fiber suspensions at low Reynolds numbers, on both laminar and turbulent fiber suspension flows, on fiber suspension flowing through a T-shaped branching channel, and on elongational flows through direct observations and measurements involving different spatial orientations and lengths of the fibers (Bernstein & Shapiro, 1994; Nishimura, Yasuda, & Nakamura, 1997). In spite of the significant achievements on the study of the movement of cylindrical particles at low Reynolds numbers, no result on the fluidization behavior of cylindrical particles in a CFB has been presented. In this paper, a 3D Euler–Lagrange model was first established according to rigid kinetics, impact kinetics and gas–solid two-phase flow theory. Rigid kinetics is the dynamic foundation of studying movement of aircrafts, rockets, secondary planets, and other

1674-2001/$ – see front matter © 2011 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

doi:10.1016/j.partic.2011.03.012

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J. Cai et al. / Particuology 10 (2012) 89–96

Nomenclature Aaxis Aradial Hriser Vair Vcp Wriser air stalk

vair t ϕsp ωcp  

axial region distribution in the riser radial region distribution in the riser height of riser air velocity at the entrance initial velocity of cylindrical particles width of riser air density stalk density air kinematic viscosity time step equivalent diameter of cylindrical particles initial angular velocity of cylindrical particle nutation angle slenderness ratio of cylindrical particles

celestial bodies. It effectively describes the orientation behavior of a traveling object in 3D coordinates. The model fully considers the effect of inter-cylindrical particle impact by introducing a collision probability function and rigid impact kinetics. Compared to a sphere, the force and movement of a cylindrical particle in a flow field is more often than not affected by its orientation. This study focuses on both academic values and engineering guidance, on design and performance as well as optimization of structure parameters. 2. Experimental 2.1. Experimental equipment In order to validate the theoretical model, a visual fluidized bed was constructed for experimental study, as shown in Fig. 2, consisting of a CFB riser made of Plexiglass to facilitate highspeed photography and provided with CAD software to study the

Fig. 1. Nutation angle between body axes and fixed reference frame.

Fig. 2. Schematic diagram of visual experimental system. 1, Rotary blower; 2, valve; 3, manometer; 4, flow equalizer; 5, wind box; 6, riser; 7, high-speed camera; 8, news lamp; 9, A/D converter; 10, differential pressure transducer; 11, data acquisition computer; 12, flowmeter.

orientation of cylindrical particles in the two-phase flow field. The experimental riser includes 3 parts: two rectangular parallelepipeds with a quadrangular frustum in between. The upper and lower rectangular parallelepipeds are, respectively 320 and 260 mm in side length and 2000 and 500 mm in height, and the height of the quadrangular frustum is 500 mm. The cylindrical particles used in the experiments are homogeneous biomass matchsticks, with a density of 0.6 kg/m3 and a diameter of 4 mm. The entrance wind velocity range is 9–11 m/s. 2.2. Experimental results 2.2.1. Data treatment Compared to prior investigations of fibers in suspensions at relatively low velocities, in the case of fluidizing cylindrical particles, both the velocity and the amount of cylindrical particles are much higher, so the orientation of cylindrical particles could not be easily captured by using current methods. In this experiment, both the concentration and velocity distribution of cylindrical particles are symmetrical owing to the quadrate sections of the riser. If the axis of a cylindrical particle is parallel to the shooting wall, the nutation angle is just the angle between the axes of the cylindrical particle and the riser. A shot photo was imported into AutoCAD and all the lengths of cylindrical particles in the photo were measured. If the measured length of a cylindrical particle ranges from 90% to 100% of its actual length, the cylindrical particle would be considered to be approximatively parallel to the shooting wall. Then the nutation angles of the cylindrical particles could be measured. The number of cylindrical particles whose nutation angles belong to a specific nutation angle domain could be obtained by this way. A large number of experimental photos were taken in order to obtain more reliable statistical average of data. 2.2.2. Experimental results Fig. 3 shows the snapshots of fluidized cylindrical particles taken with a high-speed camera, and Fig. 4 shows the statistical results of the orientation of all fluidized cylindrical particles parallel to the shooting wall in the whole riser. As shown in Fig. 4, most of cylindrical particles move with small nutation angles between 0◦ and 30◦ , that is, the axes of cylindrical particles are closely upright during fluidization.

J. Cai et al. / Particuology 10 (2012) 89–96

91

Fig. 3. Experimental snapshots of fluidized cylindrical particles.

3. Mathematical models 3.1. Gas-phase model The density of fluid is considered constant, so the continuity equation can be written as:

∇ · uc = 0.

the particle-phase, and uc remains constant once the calculation of continuous phase has converged. The two-way coupling is currently a challenge for suspension of irregular particles. In this paper, though unidirectional coupling may result in more errors compared to two-way coupling, it still possesses important merits for qualitative experimental forecast and engineering guidance.

(1) 3.2. Particle-phase model

The Navier–Stokes equations are given as: ∂p du =− + ∇ · ( ∇ u), dt ∂x ∂p dv =−  + ∇ · ( ∇ v), dt ∂y

Particle-phase models include two parts: one is the model of force and movement of a cylindrical particle in a flow field and the other is the inter-cylindrical particle collision.





(2)

dw ∂p + ∇ · ( ∇ w), =− dt ∂z

where uc (u, v, w) is the velocity vector and p is the pressure of flow field, and is the dynamic viscosity of the fluid. The 3D Euler–Lagrange model is one-way coupled, that is, only the drag force that acts on the discrete phase is taken into account, but the reactions acting on the flow field are neglected. Iteration of the continuous phase model is carried out before calculation of

Fig. 4. Experimental orientation distribution of cylindrical particles in riser.

3.2.1. Mathematical model In order to study the force and movement of a cylindrical particle in a flow field, the cylindrical particle is divided into several equal parts along the axis, whose lengths are set similar to the diameter of the cylindrical particle, thus each part is approximately disposed as a spherical particle, and the force analysis of each part is carried out respectively. Lastly, based on rigid kinetics, the total force acting on the cylindrical particle can be calculated by vector addition of all components acting on each part. The force analysis of a cylindrical particle in a flow field, based on the slender body theory, is thus carried out. The force differences along the axis of a cylindrical particle result in the rotation of the cylindrical particle and the torque acting on a cylindrical particle is calculated in the light of the rotation theory around a fixed point. The force analysis of a cylindrical particle in a flow field is shown in Fig. 5. It can be seen that the grid of flow field must be dense enough. One needs to know the velocity of each part of a cylindrical particle when analyzing the force acting on a cylindrical particle in a

Fig. 5. Schematic diagram of force analysis of a cylindrical particle in a flow field.

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flow field. According to the theory of gas–solid two-phase flow, the fluid drag force acting on a cylindrical particle is given as:

⎧   CDi Ai  uc − ui  (uc − ui ) ⎪ ⎪ ⎪ Fi = , ⎪ ⎪ 2 ⎪ ⎪ ⎪ n ⎪  ⎨ F=

(3)

0

where Fi is the force acting on part i; CDi is the drag force coefficient of local flow field; Ai is the force area of part i;  is the density of local flow field; F is the resultant drag force acting on a cylindrical particle; L is the torque acting on the cylindrical particle in body axes; ui is the velocity of part i, which is given as: ui = u + (I · ␻) × ri ,

(4)

in which u is velocity of the centroid of the cylindrical particle; I is the cosine matrix of conversion of coordinates; ␻ is the angular velocity in body axes; ri is the position vector between part i and the centroid of a cylindrical particle in body axes. So, the translation of a cylindrical particle is given as: d(mu) = F + mg, dt

(5)

in which m is the mass of a cylindrical particle; dt is the time step; g is the acceleration of gravity. Based on rigid kinetics, the rotation of a cylindrical particle can be solved using Euler dynamical equation, which is expressed as (Jia, 1987):

⎧ ⎪ ⎨ J ω˙  = L + (J − J )ω ω ,

J ω˙  = L + (J − J )ω ω ,

(6)

J ω˙  = L + (J − J )ω ω ,

˙ on the axis , where ω˙  , ω˙  , ω˙  are respectively the projections of ␻ ,  of body axes; ω , ω , ω the projections of ␻ on the axis , ,  of body axes; L , L , L the projections of L on the axis , ,  of body axes; J , J , J are respectively the components of rotary inertia of cylindrical particle along the axis , ,  of body axes. Singular point is a nonexistent point in fact, and it will result in the divergence of calculation of Eq. (6). In order to avoid the occurrence of singular point, Euler parameters were adopted. The relations between Euler parameters and Euler angles are given as follows: + 2 − 2 − 2

2

+ 2



˙ 0

⎤

⎡



(7) ,



−1 − 2 − 3 0

⎢˙ ⎥ ⎢ 1 ⎥ 1 ⎢ 0 − 3 2 1 ⎢ ⎥= ⎢ ⎢ ˙ ⎥ 2 ⎢ ⎣ 3 0 − 1 2 ⎣ 2⎦ ˙ 3

−2 1 0 3

Pri =

.

⎤⎡

ω

⎤

⎥ ⎢ ω ⎥ ⎥⎢ ⎥ ⎥⎢ ⎥, ⎦ ⎣ ω ⎦ 0

(8)

N  j−1

(9)

 ,

1



Prij =

N  n j−1

N





li (Di + Dj ) cos i  Gij t,

(10)

where n is the real number of cylindrical particles; N is the number of sampled cylindrical particles; li is the length of cylindrical particle i; Di is the diameter of cylindrical particle i; i is the nutation angle of cylindrical particle i; Gij is the relative velocity between cylindrical particles i and j (Tsuji et al., 1998). A random number R is extracted from a generator which has a uniform distribution ranging from zero to unity. If 0 < Pri < 1 and R > j/(N − Prij ), the collision between cylindrical particles i and j is decided. The two closest discrete parts of cylindrical particle i and j are chosen as the collision point between the two cylindrical particles. The post-collision velocities and angular velocities of cylindrical particles i and j can be determined according to the rigid impact kinetics (Liu, 1994). First, the pre-collision velocities ui and uj of collision points of cylindrical particles i and j are given as:



,

,

where are the precession angle, nutation angle and spin angle after one time step. Accordingly, singular point is avoided during conversion of coordinates due to the introduction of Euler parameters. With Eqs. (3)–(9), the force and movement of a cylindrical particle in a flow field can be calculated. Since there are so many inter-cylindrical particle collisions during fluidization, it is impossible to absolutely determine all collision events between real cylindrical particles in a flow field because of the huge calculating work involved. The direct simulation Monte Carlo (DSMC) method, often used in gas–solid two-phase flow dealing with collision problems of particle clusters (Tsuji, Tanaka, & Yonemura, 1998), was introduced and then modified accordingly so that it can be used for the collision problems of cylindrical particles. The modified collision probability between cylindrical particles i and j is:

,

So the equations of rotation around a fixed point are replaced by Euler parameters as:

⎡

+ arctan

0 1 ⎪ ⎪     ⎪ ⎪ 3 2 ⎩  = arctan − arctan , ,

i=1

  ⎧  ⎪ ⎪ 0 = cos cos ⎪ ⎪ 2 ⎪ ⎪ ⎪   ⎪ ⎪ ⎪  ⎪  = sin cos ⎪ 1 ⎨ 2   ⎪  ⎪ ⎪ 2 = sin sin ⎪ ⎪ 2 ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ ⎪ 3 = cos  sin ⎩

⎧    = 2 × arccos( 20 + 23 ), ⎪ ⎪ ⎪ ⎪     ⎨ 3 2  = arctan

Fi ,

⎪ i=1 ⎪ ⎪ ⎪ n ⎪  ⎪ ⎪ ⎪ L= (I−1 · Fi ) × ri , ⎪ ⎩

⎪ ⎩

in which ˙ 0 , ˙ 1 , ˙ 2 and ˙ 3 are Euler parameter accelerations. Then the Euler angles can be calculated as follows:

ui = Vi0 + (Ii · ␻i0 ) × ri = Vi0 − ri × Ii · ␻i0 , uj = Vj0 + (Ij · ␻j0 ) × rj = Vj0 − rj × Ij · ␻j0 ,

(11)

where Vi0 , Vj0 are the pre-collision velocities of the centroids; Ii , Ij the cosine matrixes of conversion of coordinate; ␻i0 , ␻j0 the precollision angular velocities in body axes; ri , rj the relative position vectors between collision point and the centroid. V is the velocity of collision point of a cylindrical particle when it is in the biggest deformation period. The component of ui − V parallel to the line through collision point and the centroid of cylindrical particle i and the component of uj − V parallel to the line through the collision point and the centroid of cylindrical particle j are expressed as:



(ui − V) = ri0 [ri0 · (ui − V)] = Pi · (ui − V), (uj − V) = rj0 [rj0 · (uj − V)] = Pj · (uj − V),

(12)

in which ri0 , rj0 are the unit vectors of ri , rj ; Pi , Pj the dyadic tensors of Vi0 , Vj0 . The component of ui − V perpendicular to the line through the collision point and the centroid of cylindrical particle

J. Cai et al. / Particuology 10 (2012) 89–96

i and the component of uj − V perpendicular to the line through the collision point and the centroid of cylindrical particle j are expressed as:





(ui − V)⊥ = (ui − V) − (ui − V) = ( −Pi ) · (ui − V), (uj − V)⊥ = (uj − V) − (uj − V) = ( −Pj ) · (uj − V).

(13)

93

Table 1 Experimental conditions of simulation. air (kg/m3 ) air (m2 /s) e Wriser (m) Vcp (m/s)

1.205 1.502×10–5 0.8 0.5 0

stalk (kg/m3 ) t (s) Hriser (m) ωcp (rad/s)

3.0×102 1.0×10–3 4.363 0–2

Then the following equation was obtained after vector calculation:



(ui − V)⊥ = Vi⊥ + ␻i⊥ × ri = Vi⊥ − ri × J−1 · (ri × mi Vi⊥ ), i (uj − V)⊥ = Vj⊥ + ␻j⊥ × rj = Vj⊥ − rj × J−1 · (rj × mj Vj⊥ ), j

(14)

in which Vi⊥ , Vj⊥ are the velocity variations along the vertical directions of ri0 , rj0 ; Ji , Jj the rotary inertia in fixed reference frame. If we let

⎧   −1 A = ( −Ti ) · ( −Pi ) + Pi , ⎪ ⎪   ⎨ −1 B=(

−Tj )

·(

T = m r × J−1 × r , ⎪ ⎪ ⎩ Ti = mi ri × Ji−1 × ri , j

j j

j

−Pj ) + Pj ,

(15)

j

where is the unit tensor in fixed reference frame, then the impulses qi , qj are expressed as:

⎧ ⎨ qi = mi A · [ui − (mi A + mj B)−1 · (mi A · ui + mj B · uj )],

q = m B · [uj − (mi A + mj B)−1 · (mi A · ui + mj B · uj )],

⎩ qj + q j= 0. i j

(16)

Then post-collision velocities Vi , Vj and the post-collision angular velocities in body axes ␻i , ␻j are given as:

⎧ 1 ⎪ ⎪ ⎪ Vi = Vi0 − mi (1 + e)qi , ⎪ ⎪ ⎨ ␻ = ␻ − (1 + e)J −1 · I−1 · (r × q ), i i0 i i i i 1 ⎪ Vj = Vj0 − (1 + e)qj , ⎪ ⎪ mj ⎪ ⎪ ⎩  −1 −1 j

· Ij

(17)

· (rj × qj ),

in which, e is restitution coefficient; Ji , Jj are the rotary inertia in body axes:



−1

Ii · J i

−1 Ij · J j

 (◦ ) Aaxis (m) Aradial (m)  Vair (m/s)

0–15, 15–30, 30–45, 45–60, 60–75, 75–90 0–0.763, 0.763–1.263, 1.263–1.883, 1.883–2.503, 2.503–3.123, 3.123–3.743, 3.743–4.363 0–0.1, 0.1–0.2, 0.2–0.3, 0.3–0.4, 0.4–0.5 6, 8, 10, 12, 14 9.0, 9.5, 10.0, 10.5, 11.0

4. Simulation results and discussion 4.1. Simulation of cylindrical particle fluidization



␻j = ␻j0 − (1 + e)J

Table 2 Experimental parameter domain of variation.

· Ii −1 = J−1 , i

· Ij −1 = J−1 . j

(18)

4.1.1. Air velocity profile Fig. 6 shows the air velocity profile in the riser, indicating that air velocity decreases from the entrance to the exit, with an evident axial velocity gradient in the transitional section, i.e. the reducer pipe. There is no evident radial variation in axial air velocity except for the regions near walls, where axial air velocity declines quickly due to wall effect, implying that the movements of cylindrical particles in those regions are unstable due to many collisions between cylindrical particles and walls. 4.1.2. Fluidization of cylindrical particles in the riser Fig. 7 shows some animated photos of the fluidization of cylindrical particles in the riser at different time after the onset of fluidization. It is found that, because the air velocity is lower in upper region than in bottom region, bulk cylindrical particles quickly pass through the bottom region and then concentrate on the upper region, resulting a higher number fraction of cylindrical particles in upper region than that in bottom region. Horizontal movement of suspended cylindrical particles due to the instable collisions between cylindrical particles and wall (Lin, Wang, & Shi, 2003), lead to fewer cylindrical particles near wall.

3.2.2. Boundary treatment and model solution The position and orientation of a cylindrical particle after hitting the wall is also analyzed with rigid impact kinetics. In Eqs. (10)–(18), particle j can be presumed as the wall if its mass is infinite and its velocity is zero, and the end of particle i that is out of the boundary is considered as the collision point of particle i with the wall. With Eqs. (3)–(18), the features of kinematics, involving inter-particle collisions, are depicted well. The partial differential equations are solved with the 4th-order Runge–Kutta algorithm and calculation programs are implemented with c++. 3.3. Calculation conditions and parameters When a time step for particle settling finishes, the sampled cylindrical particles are uniformly distributed on the bottom of bed. The three Euler angles are all random numbers in the range of 0 to ␲. Other experiment conditions and experiment parameters are given in Tables 1 and 2, respectively.

Fig. 6. Simulation of air velocity profile in the riser.

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Fig. 7. Photos of fluidization of cylindrical particles in the riser (V = 9.0 m/s).

4.2. Assessment of simulated orientation of cylindrical particles in the riser

large, and can thus be used in the following numerical study on the orientation of cylindrical particles.

Fig. 8 shows both experimental and simulated results of the orientation of cylindrical particles in the riser. The error bar of simulated orientation shown in Fig. 8 is 10%, that is, about 10% difference may exist between simulated and experimental results, which may be attributed to the assumption of one-way-coupling for drag force adopted in the Euler–Lagrange model, as well as the uncertainties caused by inevitable experimental errors. It can be seen that most of cylindrical particles move with small nutation angles in the range of 0–30◦ in a riser. In fact, the movement feature of cylindrical particles conforms to the principle of minimum potential energy. The comparison between experimental and simulated results proved that the established 3D Euler–Lagrange model is reasonable by and

4.3. Axial position dependence of orientation of cylindrical particles Fig. 9 shows the orientation of cylindrical particles in different axial positions of the riser. The region near the entrance is the bottom of the riser, the transitional region is just the quadrate reducer pipe and the 3rd and 4th axial regions are above the transitional region, as shown in Table 2. In the region near the entrance, orientation fraction of cylindrical particles decreases first and then increases as the nutation angle increases with a minimum of fraction around 45–60◦ . In transitional region, there is a wavecurve distribution in orientation fraction of cylindrical particles vs.

J. Cai et al. / Particuology 10 (2012) 89–96

Fig. 8. Simulated and experimental results of orientation distribution of cylindrical particles.

95

Fig. 10. Radial position dependence of orientation of cylindrical particles.

4.5. Effect of air velocity on orientation of cylindrical particles 15–30◦

nutation angles, with a peak and a valley value at and 60–75◦ , respectively. Above the transitional region, the majority of cylindrical particles move with small nutation angle, preferably around 0–15◦ . Nutration angle of cylindrical particles becomes smaller with increasing the height in the riser, and approaches a steady value and fluctuates about that value. 4.4. Radial position dependence of orientation of cylindrical particles Fig. 10 shows the orientation of cylindrical particles in different radial positions of the riser. In centre zones (Regions 1–3), the orientation distribution of cylindrical particles vs. nutation angle is relatively uniform along radial direction. Due to relatively large velocity gradient in the regions near walls, as shown in Fig. 6, the number fractions of cylindrical particles moving with small nutation angles are much bigger in the regions near walls than in centre zones. In the region near the walls, the percentage of cylindrical particles moving with small nutation angles is much higher.

Fig. 9. Axial position dependence of orientation of cylindrical particles.

In Fig. 11, the curve for velocity of 9.0 m/s is chosen as the datum curve and the interval value of error bar is 15%. As shown in this figure, some data points of orientation fraction of cylindrical particles with the same nutation angle but at different air velocities are beyond this error interval of 15%, and there is no monotonic relation between orientation fraction of cylindrical particles and air velocity. The reason may be because that increasing air velocity does not necessarily mean the bigger velocity gradient in a flow field, and air velocity cannot evidently affects the dynamic equilibrium of fluidized cylindrical particles. 4.6. Effect of slenderness on orientation of cylindrical particles The slenderness of a cylindrical particle is the ratio of its length to diameter. Usually, if its slenderness is less than 4, a cylindrical particle is treated as a spherical particle (Shi, 2002). Fig. 12 shows the orientation distributions of cylindrical particles of different

Fig. 11. Simulated results of orientation distribution of cylindrical particles at different air velocities.

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Fig. 12. Simulated results of orientation distribution of cylindrical particles of different slenderness ratios.

slenderness ratios, Indicating that the orientation distribution of cylindrical particles is not evidently affected by the slenderness of cylindrical particles, which is consistent with the results presented in literature (Zhang & Lin, 2003). 5. Conclusions In order to study the fluidization behavior of cylindrical particles in gas–solid two-phase flows, a three-dimensional Euler–Lagrange model is established according to rigid kinetics, impact kinetics and gas–solid two-phase flow theory. Both experimental and simulated results showed that most of cylindrical particles move with small nutation angles in a riser during fluidization, indicating that the established 3D Euler–Lagrange model is applicable for study of fluidization behavior of cylindrical particles in a riser. It is found that the orientation of cylindrical particles is affected more obviously by their position in the riser than by their slenderness and air velocity. References Batchelor, G. K. (1970). Slender-body theory for particles of arbitrary cross-section in Stokes flow. Journal of Fluid Mechanics, 44(3), 419–440. Bernstein, O., & Shapiro, M. (1994). Direct determination of the orientation distribution function of cylindrical particles immersed in laminar and turbulent shear flows. Journal of Aerosol Science, 25(1), 113–136.

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