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Discrete Particle Simulation and Visualized Research of the Gas-Solid Flow in an Internally Circulating Fluidized Bed Yin Bin,* M. C. Zhang, B. L. Dou, Y. B. Song, and J. Wu Department of Energy Engineering, Shanghai Jiaotong University, Minhang, Shanghai 200240, People’s Republic of China
In this paper, the DEM (discrete element method) is used to simulate the gas-solid flow of a two-dimensional internally circulating fluidized bed. Eulerian and Lagrangian methods are used to deal with gas-field and discrete particles, respectively. The gas flow is determined by the local average Navier-Stokes equation, the motion of individual particle is obtained by solving Newton’s second law of motion, and particle collisions are taken into account. The simulated results indicate that there is particle circulation from the high-velocity region to the low-velocity region in internally circulating fluidized beds with uneven air distributions; therefore, the mixing characteristics of the bed material are good. Through visualization experiments, we confirm that there is transverse movement of bubbles from the high-velocity side to the low-velocity side in internally circulating fluidized beds. Thus, particles in the trailing vortex of bubbles also move transversely. This is an important factor that results in particle circulation in internally circulating fluidized beds. Introduction At present, the discrete element method (DEM) is a powerful tool for simulating dense gas-solid flow. In the past, two-fluid models were frequently used to describe hydrodynamic phenomena in fluidized beds. Examples of such models include those of Gidaspow1 and Kuipers et al.2 In two-fluid models, the gas and solid phases are considered to be continuous and fully interpenetrating, and the two phases are described in terms of separate sets of conservation equations with appropriate interaction terms representing the coupling between the phases. However, in two-fluid models, the assumptions of local equilibrium and solid-phase continuum essentially impair the characteristics of gassolid flow, so such models cannot describe discrete particles. The discrete element method established by Cundall and Strack3 is referred to as a discrete particle method (DPM) when it is applied to granular systems. In the DPM, the motion of individual particle is obtained by solving Newton’s second law of motion. Many researchers incorporate DEM and computational fluid dynamics (CFD) simultaneously in considering the interactions of gas and solid.4 The Eulerian method is used for the gas field, the Lagrangian method is used for the discrete particles, and every particle is directly tracked.5 In recent years, with the rapid development of computer technology, the simulation of dense gas-solid flow with the DEM has attracted many researchers. From different considerations, two types of DEM can be distinguished: the soft-sphere model and the hardsphere model. In the soft-sphere model, the motion of particles is calculated by numerical integration of the Newtonian equations of motions, and interparticle forces are of key importance in comparison with the forces of drag and gravity. Some researchers, such as Tsuji et al.,6 have simulated gas-solid fluidized beds with this * Corresponding author. Tel.: 0086-21-54742847. Fax: 008621-54742996. E-mail:
[email protected].
model and obtained simulated results that are in good agreement with those of experiments.6 One deficiency of the soft-sphere model is that the spring coefficient of the particles is small, which artificially increases the time of particle collisions. If a large or real value springconstant value is used, then the time step must be made smaller than the interaction time of the particles. This results in greatly increased computation times, so that the simulations cannot even be completed with ordinary computers. The advantage of the soft-sphere model is that multiparticle collisions can be considered realisticically. As the basis of the collision model of Wang and Mason, which is used to compute the dynamics of inelastic collisions with friction, Hoomans et al.7 proposed the hard-sphere model. In this model, particle collisions are binary and quasi-rigid, and the shape of the particles is assumed to be retained after impact. Two major drawbacks of the hard-sphere model are that detailed information about interparticle forces is not considered completely and the assumption of binary collisions is unrealistic. In the DEM model, the gas flow is determined by the local average Navier-Stokes equation, and the motions of individual particles are obtained by solving Newton’s second law of motion. The coupling between gas and solid is achieved directly by applying the principle of Newton’s third law. Many researchers use the wellknown Ergun correlation for < 0.8, where is porosity, and the Wen-Yu correlation8 for > 0.8. The DEM has been used in the field of fluidization by some researchers for several years. However, up-todate reports on the gas-solid flow of internally circulating fluidized beds simulated by the DEM are still missing. The moving characteristics of particles in internally circulating fluidized beds are not completely understood by many researchers. The movement of particles in an internally circulating fluidized bed is first simulated with the DEM in this paper; therefore, our research is innovative work.
10.1021/ie020435q CCC: $25.00 © 2003 American Chemical Society Published on Web 11/22/2002
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Internally circulating fluidized beds are a new type of fluidized bed. The circulation of bed material is achieved by distributing air unevenly. Inclined distributors are generally employed in internally circulating fluidized beds, and the fluidized velocity is high on the low side of the distributors where many bubbles appear. These bubbles grow during upward movement, and particles in the trailing vortex also move upward following these bubbles. At the same time, particle exchange between the bubble phase and the emulsion phase occurs.9 In this paper, the DEM and computational fluid dynamics (CFD) have been combined for the simulation of gas-solid flow in an internally circulating fluidized bed with unevenly distributed air. Then, an imageprocessing approach is applied to examine the movement of the bubbles in the internally circulating fluidized bed. At the same time, the phenomenon of particle circulation is analyzed.
material. When the displacement reaches a maximum value, the particle cease forward motion and rebound along the original direction. If particle collisions are not complete elasticity, then some kinetic energy will be lost upon impact. The magnitude of the loss is related to the damping coefficient and the relative velocity of the particles. When noncentral collision occurs, a tangential torque appears because of the tangential force. This tangential torque enables the particles to rotate. The above physical processes can be described mathematically as follows10,11
F Bcnij ) (-kδn - ηv bij‚n bij)n bij bij‚B t ij)B t ij F Bctij ) (-kδt ) -ηv F Bctij ) -µf|F Bcnij|Bt ij
In the DEM model, the gas flow is described by the Navier-Stokes equation of two-phase coupling, the motion of individual particles is described by the random trajectory model, and the change in the momentum of particles due to collisions is considered. Particle Model. When no collisions occur, the forces acting on the particles are mainly drag and gravity. The motion of the particles is described by
m
dv b ) Fg + mg dt
(2)
and the drag coefficient is expressed as
C′d ) Cd-4.7
(3)
The drag coefficient for an isolated particle depends on the particle Reynolds number as given by Rowe and Henwood
{
24 [1 + 0.15(Rep)0.687] Rep < 1000 Re Cd ) p Rep g 1000 0.44
(4)
Fg|m b -b v |dp µg
I
d$ ) dt
∑T,
b v tij |v btij|
Bt ij ) b nij )
xj - xi
2 I ) mR2 5
i+
x(xj - xi)2 + (yj - yi)2
(5)
Through eq 1, the acceleration of the particles can be calculated by numerical integration, and then the velocities and positions of the particles can be obtained. In terms of the laws of classical physics, when two spherical particles move in opposite directions and impact each other, elastic deformation at the contact point occurs. The extent of deformation depends on the relative velocity of the particles and their stiffness. The particles are subjected to an elastic resistance after the collision in their original direction of motion. The resistance force is directly proportional to the displacement of the deformation and the stiffness of the particle
(8) (9) (10)
(11) yj - yi
x(xj - xi)2 + (yj - yi)2
(12)
where F Bcnij is the normal force; F Bctij is the tangential force; k is the stiffness of the individual particles; η is the damping coefficient; δn is the normal deformation displacement; δt is the tangential deformation displacement; b nij is the normal unit vector; B tij is the tangential unit vector; µf is the friction coefficient; b vij is the relative velocity of particles i and j; b vtij is the tangential relative velocity; $i ($j) is the angular velocity of particle i (j); T is the rotational torque; I is the moment of inertia; and (xi, yi) and (xj, yj) are the coordinates of particles i and j, respectively. Through eqs 6-8, the normal and tangential forces due to interparticle collisions can be obtained. In terms of Newton’s second law, the acceleration during the collision can be calculated, and then the velocity after the collision can be expressed as
a∆t b v)b v0 + b
where the particle Reynolds number is defined as
Rep )
(|F Bctij| g µf|F Bcnij|)
(1)
where the drag force Fg can be written as
1 b-b v |(u b-b v) Fg ) πR2C′dFg2|u 2
(|F Bctij| < µf|F Bcnij|) (7)
b v tij ) b v ij - (v bij‚n bij)n bij + R($i + $j) × b nij
Mathematical Model
(6)
(13)
where b v0 is the velocity before the collision, b a is the acceleration, and ∆t is the time step. Gas-Phase Flow. The calculation of the gas motion follows a generalization of the Navier-Stokes equations for a gas interacting with particles, as shown by the following equations for mass and momentum conservation in vector form
∂(Fg) + 3‚(Fgb u) ) 0 ∂t
(14)
∂(Fgu b) + 3‚(Fgu bb u) ) ∂t -3p - β(u b-b v ) + 3‚(τ) - Fgg (15) where is the porosity, τ is the viscous stress tensor,
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Table 1. Parameters Used for Present Simulation solid phase particle shape particle diameter spring coefficient coefficient of restitution time step
sphere 4 mm 6000 N/m 0.9 1.0 × 10-5 s
gas phase number of particles particle density damping coefficient friction coefficient
1000 2500 kg/m3 0.17 0.3
density viscosity number of cells cell width cell height
1.205 kg/m3 1.82 × 10-5 N s/m2 10 × 22 0.012 m 0.012 m (below), 0.02 m (above)
where n is the number of particles in the cell; W and H are the width and height of the cell, respectively; and R is the radius of the individual particles. The value of the porosity obtained from eq 17 is in the range (0, 1), which is unrealistic because the porosity in a fluidized bed should be higher than mf. Thus, the value of 2 must be corrected. In terms of Hoomans et al.’s model,7 the true porosity can be achieved by considering the threedimensional porosity. That is7
3 ) 1 -
and β is the interphase momentum transfer coefficient. In terms of an empirical correlation, β can be expressed as
{
Fg (1 - )2 µg 150 + 1.75(1 - ) |u b-b v | < 0.8 2 d d p p
3 (1 - ) C Fg|u b-b v |-2.65 4 d dp
g 0.8 (16)
Computation Procedure. (i) First, solve eqs 14 and 15 are for a given time step with the SIMPLEC method to obtain the fluid velocities. (ii) Using eq 1, calculate the position and velocity of each particle at that time step with the four-step Runge-Kutta method. (iii) Determine whether collisions occur. If there are collisions between particles, calculate the particles’ resulting velocities using eqs 6-8 and 13. (iv) Count the porosity of every cell, calculate the average velocities of the particles in every cell , and then calculate the reflected term of the particles to the fluid. (v) Correct the NavierStokes equation of two-phase coupling, and then obtain new fluid velocities. (vi) Repeat steps ii-v. Computation Strategy. When searching for collision partners, a nearest-neighbor list is employed to save CPU time. At the central particle, a square, whose dimension is 8 times the particle’s diameter is established. All particles in the square region are considered for possible collision with the particle inthe center. If the collision rate is determined directly by counting collision events and the absolute time scale,12 the time required to search for collision partners will be very long. For each cell of the computation domain, the porosity can be calculated from the area occupied by the particles in the cell. The two-dimensional porosity is defined as
2 ) 1 -
(1 - 2)3/2
xπx3
(18)
After such a treatment, 3 ranges from 0.394 to 1. This value of porosity is in agreement with that for true fluidized systems.
Figure 1. Scale of the bed.
β)
x2
nπR2 WH
(17)
Simulation of a Two-Dimensional Internally Circulating Fluidized Bed Object of Simulation. For the simulation, an internally circulating fluidized bed, which includes an inclined distributor, is employed. The bed’s width is 0.096 m, the height is 0.432 m, and the tilt angle of the distributor is 20°. At the high and low sidea of the distributor are arranged two inlets so that the fluidized velocity of the low side is high and that of the high side is low. The scale of the bed is shown in Figure 1. Conditions of Simulation. The values of simulation parameters are important to the simulated results. In terms of previous reports,4,6,7 some parameters were selected by our repeating the tests. The parameter values used are reported in detail Table 1. Results of Simulation. The simulated particle flow is shown in Figure 2. At the initial time, a bubble appears on the low side of the distributor. As the bubble moves upward, it grows continuously. Particles in the trailing vortex of the bubble also move upward following this bubble. New particles are entrained from the emulsion phase when particles in the trailing vortex drop back down. After the bubble breaks at the top of the bed, the particles in the trailing vortex are dispersed in all directions. From Figure 2, an important phenomenon can be observed: there is rightward component of the velocity of many particles. In other words, as particles descend, some particles simultaneously move rightward. The low-side velocity of the distributor is high, and the high-side velocity of the distributor is low, which generally results in critical fluidization or small bubbling fluidization. Thus, upward particle flow occurs only on low side of the distributor. In the upper region of the bed, the particle concentration on the left side is higher than that on the right side; therefore, particles will diffuse from left to right in the upper region. In addition, because an inclined distributor is employed, particles in the lower region of the right side of the bed will slide to the left along the distributor as a result of
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Figure 2. Particle flow in the bed.
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Figure 3. Vectors of particles.
gravity. When these particles move to the low side of the distributor, they become entrained upward by bubbles. Therefore, internal circulation of particles occurs. For conventional bubbling fluidized beds, bubbles appear uniformly because horizontal distributors are employed and the gas is distributed evenly. Therefore, local particle circulation results. In contrast, internally circulating fluidized beds can achieve macroscopic particles circulation, so the mixing characteristic of the particles in the horizontal direction is better. The vectors indicating the particles’ motion are shown in Figure 3. At the initial time, the particles on left side move upward. When these particles reach the top of the
bed, they gradually move to the right. At the same time, particles on right side of the bed move downward. When these particles reach the bottom of the bed, they slowly move to the left. Therefore, the particle circulation is clockwise. This particle circulation in internally circulating fluidized beds is an advantage in many respects, including mixing and diffusion of particles, combustion, and chemical reactions in beds. Consequently, internally circulating fluidized beds are adapted for the incineration solid waste materials for power. The simulated results obviously indicate that the particle circulation in internally circulating fluidized
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Figure 4. Experimental system.
beds is significant. In the following section, this particle circulation is explained through a visualization experiment. Visualization Experiment The results of our experiments are introduced in this section. At the base of a cold experimental device that is a two-dimensional internally circulating fluidized bed, we record snapshots of the moving progress of a bubble with a CCD (charge-coupled device) camera, thus obtaining a photographic sequence of the bubble’s motion, and then analyze this sequence with image-processing software. Finally, we obtain the moving characteristics and velocities of the bubbles and investigate the particle circulation in the bed in terms of these results. Experimental System. The experimental system is shown in Figure 4. The system consists of a twodimensional internally circulating fluidized bed, a CCD camera, an image-collecting card, a computer, and a light source. The width, height, and depth of the bed are 400, 2500, and 50 mm, respectively. Two independent air chambers are designed in the bed where the side of low velocity is a moving bed and the side of high velocity is a fluidized bed. An inclined distributor with a tilt angle of 20° is employed. The main body of the bed is constructed of organic glass for visual observation. The type of the card used to capture photographs is VideoKING KCE-9971, and the time interval between two consecutive frames is 0.04 s. During the experiment, digital photographs are recorded with the CCD camera and then transferred to the computer through the image-collecting card. Therefore, these digital photographs could be conveniently processed using image-processing software. Image Processing. Because the brightness of the system was not uniform, the digital images obtained were blurry. The visualized effect of the images could be improved through image enhancement. The contrast of the images was adjusted with a linear transform based on the gray level of the images. The mathematical description of this transformation is as follows
g(x,y) ) [(n - m)/(b - a)][f(x,y) - a] + m a e f(x,y) e b f(x,y) < a m f(x,y) > b n (19)
{
where f(x,y) is the gray value of the original images and g(x,y) is the gray value after the transformation. [a, b] represents the gray level range of the original images, and [m, n] gives the gray level range of the images after the transformation.
Figure 5. Images after binary operation.
The next step is binary operation of the images. The purpose of the binary operation is to investigate the change in the bubbles’ scale along the bed’s height. The threshold method is commonly used for binary operations; this method is especially adapted to images whose object and background are different. First, select an appropriate threshold, and then compare the threshold with each gray value in the image. If the gray value is greater than the threshold, it is assigned the maximum gray value. Similarly, if the gray value is lower than the threshold, it is assigned the minimum gray value. A new binary image is formed through this operation. The mathematic description is as follows
g(x,y) )
{
1 f(x,y) g T 0 f(x,y) < T
(20)
where f(x,y) is the gray value of the original images, T is the threshold, and g(x,y) is the gray value after the binary operation. Images in the sequence after the binary operation are shown in Figure 5. After the binary images are formed, the scale of the bubbles can be obtained by calculating the area of the white region in the images. The detailed process is described as follows: In terms of calibration, the true area of shooting region is width × height ) 41 cm × 30 cm. The true area represented by each pixel is pa ) 0.1165 cm × 0.1041 cm when the distinguishability of the CCD is considered. Because the images are binary images and the pixel values of these binary images are 1 or 0, the true scale of the bubble can be obtained simply by counting the number of pixels with a value of 1 and multiplying that number by pa. The changing tendency of the bubble is shown in Figure 6. We can observe that the bubble grows along the bed’s height.13-15 Pathways and Velocities of the Bubbles. Every image in the above sequence of images is successively recorded at a time interval of 0.04 s. The scale of these images is equal, and the number of pixels is also same. Therefore, as long as the pixel value of the bubble’s center is determined and compared to the calibration value, the bubble’s position in the bed can be obtained. If the central pixel’s coordinates of two continuous
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Figure 6. Change in the bubble’s scale along the bed’s height.
Figure 8. Change in the bubble’s velocity.
From Figure 7, we can obviously see that the bubbles not only move upward, but also move transversely in internally circulating fluidized beds. This is an important phenomenon. Particles in the trailing vortex will also move transversely following the bubbles. This is the main factor that results in particle circulation in internal circulating fluidized beds. Furthermore, comparing the plots in Figure 7, a conclusion can be drawn that the transverse movement of the bubbles becomes more apparent as the fluidized velocity in the low side of the distributor is increased. In terms of the above-proposed method, the average velocities of the bubbles in every time interval can be obtained. The results are shown in Figure 8. The conditions are that the fluidized velocity in the low side of the distributor is 0.83 m/s, and the fluidized velocity in the high side of the distributor is 0.52 m/s (this value is near the critical fluidized velocity). From Figure 8, we can see that the vertical component of the bubble’s velocity increases along the bed’s height. When the bubble is close to the bed’s interface, the vertical component decreases. The horizontal component of the bubble’s velocity fluctuates significantly and is smaller than the vertical component of the bubble’s velocity. Figure 7. Moving pathways of bubbles.
Conclusions
bubbles are (xi,yi) and (xj,yj), then xj - xi and yj - yi represent the displacements in the horizontal and vertical directions, respectively. By dividing these two values by 0.04, the average velocity in this time interval can be obtained.16,17 The moving pathway of the bubbles is shown in Figure 7, where the horizontal axis represents the bed’s width and the vertical axis represents the bed’s height. The upper plot is obtained from the sequence of images in Figure 5.
The incorporation of the DEM and CFD is a powerful approach for researching dense gas-solid flow. Using this method to simulate the movement of particles in an internally circulating fluidized bed, the results indicate that there is large-scale particle circulation in internally circulating fluidized beds. This particle circulation improves the mixing of particles in the transverse direction, and it will deeply affect particle dispersion, heat transfer, mass transfer, and chemical reaction in internally circulating fluidized beds.
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The visual observation of the movement of bubbles in an internally circulating fluidized bed with a CCD camera is a new type of method. The movement and change of bubbles can be tracked in very small time intervals, so the accuracy of the experiments is exceptional. The results obtained indicate that there is transverse movement of bubbles in internally circulating fluidized beds. Furthermore, this transverse movement becomes more obvious as the fluidized velocity in the low side of the distributor is increased. It is a main factor contributing to particle circulation in internally circulating fluidized beds. Acknowledgment This work was sponsored by ministry of science and technology of People’s Republic of China. Notation a ) particle acceleration, m s-2 C′d ) effective drag coefficient dp ) particle diameter, mm e ) restitution coefficient Fg ) drag force, N F Bcnij ) normal interaction force, N F Bctij ) tangential interaction force, N g ) gravitational acceleration, 9.8 m s-2 H ) height of the cell, m I ) moment of inertia, kgm2 K ) spring coefficient, Nm-1 m ) particle mass, kg n ) number of particles b nij ) normal unit vector p ) gas-phase pressure, kg m-1 s-2 R ) particle radius, m Rep ) particle Reynolds number Btij ) tangential unit vector b u ) gas-phase velocity, m s-1 b v ) relative velocity of collision particles, m s-1 b v ) tangential component of the relative velocity of collision particles, m s-1 W ) width of the cell, m Greek Letters β ) interphase momentum transfer coefficient, kg m-3 s-1 ) porosity δn ) normal deformation displacement, m δt ) tangential deformation displacement, m η ) damping coefficient µg ) gas viscosity, m s-1 s-1 Fg ) gas density, kg m-3 Fp ) particle density, kg m-3 τ ) gas-phase stress tensor, kg m-1 s-2 ∆t ) time step, s
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Received for review June 12, 2002 Revised manuscript received October 18, 2002 Accepted October 18, 2002 IE020435Q
