Three-Dimensional Modeling of GasâSolid Motion in a Slot

Article pubs.acs.org/IECR

Three-Dimensional Modeling of Gas−Solid Motion in a SlotRectangular Spouted Bed with the Parallel Framework of the Computational Fluid Dynamics−Discrete Element Method Coupling Approach Shiliang Yang, Kun Luo,* Mingming Fang, Ke Zhang, and Jianren Fan* State Key Laboratory of Clean Energy Utilization, Zhejiang University, Hangzhou 310027, P. R. China ABSTRACT: The gas−solid flow in a three-dimensional slot-rectangular spouted bed is numerically investigated under the parallel framework of the computational fluid dynamics−discrete element method coupling approach. The fluid phase is solved with the k−ε turbulence model at the computational grid level, while the solid phase is tracked individually with the discrete element method at the particle scale. Millions of particles are calculated using a total number of 112 CPUs in a cluster. The predicted pressure drop profiles and the minimum spouting velocities of the calculations under different bed heights agree well with the experimental data. The startup procedure and the solid flow pattern are identified. Then, wall effects on the distribution properties of the gas−solid hydrodynamics are explored. Moreover, the interaction between spout and annulus and the effects of the superficial velocity and bed height on the bed hydrodynamics are discussed. The results show that a larger pressure drop appears in the system with a higher bed height. A bubble with its shape influenced by the slot geometry is formed in the startup procedure. Moreover, the bed hydrodynamics show a strong 3-D behavior along the depth direction. Large vertical solid flux mainly concentrates in the central region of the spout. The spout−annulus interface is dramatically influenced by the slot shape and shows different evolutionary tendencies in the different directions, and this influence becomes weaker along the bed height. The larger the superficial velocity is, the higher the bed height and the larger the spouting height appear. Besides, solid flow patterns under different bed heights with the superficial velocity the same ratio to the corresponding minimum spouting velocity show a similar behavior. the flow regimes in the slot-rectangular spouted bed has been reported by Freitas et al.12 with pressure fluctuation analysis based on the experimental data obtained. Luo et al.13 conducted an experimental investigation of the flow characteristics in a slot-rectangular spouted bed with draft plates. Chen et al.11 undertook an experiment in a slot-rectangular spouted bed with slots of equaling area but different length-to-width ratios to explore the hydrodynamics of the system. The results indicated that the local flow structure of the slot-rectangular spouted bed becomes similar to that in the conventional spouted bed as the height increases. With the advancement of the computational technology in the last two decades, mathematical modeling of the two-phase dense flow in the spouted bed has been more and more popular. There exist two main approaches for modeling the two-phase dense flow, namely, the two-fluid model (TFM) and the discrete element method (DEM). In the TFM, both the fluid and solid phases are treated as continuums and solved at the scale of the computational grid. For the DEM approach, the fluid phase is modeled at the computational grid level, while the motion of the each particle is tracked individually at the particle scale.14−16 Plenty of research has been carried out to explore the hydrodynamics of the conventional spouted bed based on

1. INTRODUCTION Due to its high gas−solid contacting efficiency and easy operation, the conventional conical-based spouted bed consisting of a cylindrical column has been widely utilized in many industrial processes, such as the drying of granular material, the coating of tablets, the mixing of particles, and combustion of coal and wastage.1−6 However, a large-capacity spouted bed is not commonly seen in industrial utilization due to its scale-up difficulties, such as the instability to obtain an excellent spouting in the large-scale unit and the difficulty in predicting the performance of the system with a diameter larger than 0.3 m.7−9 To overcome the scale-up difficulties of the cylindrical conical-based spouted bed, the slot-rectangular spouted bed, originally proposed in the Soviet Union, has been paid much attention due to its significant threedimensional effects as the bed thickness increases and the possibility to achieve different flow regimes by changing the inlet dimensions.7,10,11 Extensive experiments had been conducted to evaluate the hydrodynamic parameters of the slot-rectangular spouted bed, such as the minimum spouting velocity, bed height, and the maximum pressure drop. Dogan et al.7 carried out experimental research on the bed hydrodynamics of a thin slot-rectangular column with width 150 mm and slot width ranging from 2 to 20 mm for four types of particles. They pointed out that the important hydrodynamics of the spouted bed, such as the maximum spouting bed height and pressure drop, are remarkably influenced by the slot width. The identification of © XXXX American Chemical Society

Received: June 8, 2013 Revised: August 11, 2013 Accepted: August 15, 2013

A

dx.doi.org/10.1021/ie401811y | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

Article

these two methods.17−22 However, only a few numerical studies are reported on the dense gas−solid flow in the slot-rectangular spouted bed, while nearly all of them were performed for the 2D case. Zhao et al.23 presented a numerical research on the particle dynamics in a 2-D spouted bed with a rectangular column and a tapered base using the discrete element method. The results demonstrated that the particles exhibit the greatest drag and acceleration magnitudes near the spout entrance and that the drag forces continuously reduce as particles progress upward in the spout. Zhao et al.24 carried out experimental and DEM researches on the flow patterns of solids in a twodimensional rectangular spouted bed with draft plates. They pointed out that the addition of draft plates results in the steeper particle streamline in the annulus but also a smaller velocity magnitude as well. Zhu et al.25 recently reported the mixing characteristics of the dry and wet particles in a rectangular spouted bed based on the DEM approach to investigate the effects of the superficial velocity and moisture content on the particle mixing process. However, due to the geometrical difference between the conventional spouted bed and slot-rectangular spouted bed, the hydrodynamics in the slot-rectangular spouted bed, which strongly depends on the slot shape and the bed depth,7 cannot be simply treated as a 2-D plane flow. Furthermore, numerical investigation of the hydrodynamics in a fully 3-D lab-scale slotrectangular spouted bed at the particle scale level is relatively rare due to the limitation of the needed computational resource to model the system with millions of particles when the depth direction is taken into account. This obviously affects the deep understanding of the latent solid transporting mechanism in the system. To overcome this gap, numerical evaluation of the hydrodynamics in the 3-D slot-rectangular spouted bed is performed on the basis of the parallel framework of computational fluid dynamics−discrete element method (CFD−DEM) coupling approach. The motion of the fluid phase is solved with the k−ε turbulence model, while the motion of the solid phase is solved by the discrete element method. The minimum spouting velocity, the start-up procedure, and the flow behaviors of the solid phase in both the macroscopic and microscopic views are explored. Subsequently, the distribution properties of the velocity, the concentration, and the flux of solid phase are presented. Then, the interaction between spout and annulus is evaluated. Finally, the effects of the superficial velocity and the bed height on the hydrodynamics of the system are discussed.

∂(εgρg u iu j) ∂ (εgρg u i) + ∂t ∂xj = −εg

⎤ ⎛ ∂u j ∂u i ⎞⎥ ⎜⎜ ⎟⎟ + ∂xj ⎠⎥⎦ ⎝ ∂xi

μt = cμρg k 2/εt

∂t

∂(εgρg u i) ∂xi

=0

(3)

where cμ is a constant, cμ = 0.09. k and εt are, respectively, the turbulent kinetic energy and turbulent dissipation rate, for which the governing equations can be illustrated as ∂(εgρg k u j) ∂ (εgρg k) + ∂t ∂xj =

μ ⎞ ∂k ⎤ ∂u ⎛ ∂u j ∂u i ⎞ ∂ ⎡⎢ ⎛ ⎟ εg ⎜μ + t ⎟ ⎥ + εgμt i ⎜⎜ + ∂xj ⎢⎣ ⎝ σk ⎠ ∂xj ⎥⎦ ∂xj ⎝ ∂xi ∂xj ⎟⎠ − εgρg εt

(4)

∂(εgρg εt u j) ∂ (εgρg εt) + ∂t ∂xj =

εgc1εt ∂u i ⎛ ∂u j μ ⎞ ∂ε ⎤ ∂u i ⎞ ∂ ⎡⎢ ⎛ ⎟ ⎜⎜ εg ⎜μ + t ⎟ t ⎥ + μt + k ∂xj ⎢⎣ ⎝ σk ⎠ ∂xj ⎥⎦ ∂xj ⎝ ∂xi ∂xj ⎟⎠ − εgc 2ρg

εt 2 k

(5)

In the above equations, σk and σε are turbulent Prandtl numbers for k and εt, σε = 1.3 and σk = 1.0. c1 and c2 are the model constants, c1 = 1.44, c2 = 1.92. The momentum sink item Sp in the momentum equation represents the momentum transfer between the fluid and particles, which is calculated as n

Sp =

∑ (fd,i + f p,i)/ΔV i=1

(6)

where n is the total number of particles in the current computational grid. ΔV is the volume of the current cell. fd,i and fp,i stand for the drag force and pressure gradient force exerted on particle i by the fluid phase, respectively, which will be discussed later. 2.2. Governing Equations of Solid Motion. The solid phase is tracked individually at the particle scale level. The forces exerted on a particle in the dense flow can be described as the drag force (fd), the pressure gradient force (fp), the gravitational force (mpg), the particle−particle or particle−wall interaction force (fc). The translational and rotational motions of a particle are governed by the Newton’s second law, which can be expressed as

2.1. Governing Equations for Fluid Phase. Gas motion is mathematically modeled using the Navier−Stokes equations at the computational grid scale level with the consideration of the existence of the solid phase. The governing equations can be formulated as +

(2)

where εg = 1 − ((∑i n= 1Vpi)/ΔV) is the voidage of the gas phase. ρg, u, p, and μ are the density, the velocity vector, the pressure, and the dynamic viscosity of the fluid phase, respectively. g is the gravitational acceleration. μt is the turbulent viscosity, which is modeled with the k−ε turbulence model in the current work as

2. COMPUTATIONAL MODELS

∂(εgρg )

⎡ ∂p ∂ ⎢ − Sp + ρg εg g + εg(μ + μt ) ∂xi ∂xj ⎢⎣

(1) B


Industrial & Engineering Chemistry Research mp

Ip

dvp dt

dωp dt

Article

= m p g + f p + fd + fc

(7)

= Tp

(8)

F3(εp) = 0.0673 + 0.212εp +

D fcnij = f Scnij + f cn ij = ( − k nδnij − ηn vtij·nij)nij

(16)

3. SIMULATION SETUPS 3.1. Simulation Conditions. The simulation is carried out in a 3-D slot-rectangular spouted bed, which has the same geometrical structures as the experimental work conducted by Chen.28 The sketch of the geometry is shown in Figure 1a. The

where mp, vp, Ip, and ωp are the mass, the translational velocity, the momentum of inertia, and the rotational velocity of the particle, respectively. Tp Stands for the torque exerted on the particle by the other colliding particles. For the calculation of the contacting force in the colliding particle−particle or particle−wall pair, the soft-sphere contacting model originally proposed by Cundall and Strack26 is applied in the current work to mimic the contacting procedure. The contacting force can be divided into the tangential and normal components, which can be formulated as

fcij = fcnij + fctij

0.0232 εg 5

(9) (10)

fctij = f Sctij + f ctDij = min(−k tδtijtij − ηt (vtij·tij)tij , μp |fcnij|tij) (11)

where the subscripts of n and t represent the variables in the normal and tangential directions of the contact, respectively. k and η are the spring constant and the viscous damping coefficient for the contact model, respectively. δij is the displacement of the colliding pair. μp is the friction coefficient, with which the Coulomb friction law is used for the calculation of the tangential contacting force when sliding occurs between the contacting pair. t and n stand for the unit vector in the tangential and vertical directions, respectively. The fluid−particle interaction force for each individual particle in the dense flow is estimated as fd =

Vpβ εp

(u − vp)

Figure 1. (a)Schematic diagram of the 3-D slot-rectangular spouted bed. (b) Grid representation of the calculation domain. (12)

where Vp is the volume of particles in the current cell. εp is the volume fraction of solid phase. β is the drag force coefficient, for which the correlation proposed by Koch and Hill27 is adopted in the current work. It can be expressed as βKoch&Hill =

18μεgεp ⎛ ⎞ 1 ⎜F (ε ) + F3(εp)Rep⎟ 0 p ⎠ 2 d p2 ⎝

vessel has a cross-section of 150 mm × 100 mm (width and depth), a height of 1000 mm, and an inclined angle of 60°. The bottom of the spouted bed has a cross-section of 50 mm × 100 mm (width and depth) together with a centrally located slot with dimensions of 4 mm × 30 mm (width and depth). The open-source MIFX-DEM program29,30 is used for the calculation. The grid representation of the calculation domain is shown in Figure 1b. Uniform grid with size of 5 mm is used in the vertical direction of the unit. In the other directions, a nonuniform grid with the grid size in the range of 3−7.5 mm is adopted, which is nearly 2.25−5.26 times the particle diameter. A total number of 49 × 192 × 20 grid cells is used in the calculation. For the solid phase, the diameter and density of the glass sphere used in the calculation are 1.33 mm and 2490 kg/m3, respectively. Millions of particles are randomly generated in the calculation domain and then fall freely to accumulate at the bed bottom. Finally, bed heights of 0.15, 0.2, and 0.25 m are formed with the total kinematic energy of the solid phase dissipated out, respectively. The corresponding particle numbers are, respectively, 938 488, 1 315 293, and 1 691 480. Detailed physical and numerical parameters used in the simulation are illustrated in Table 1.

(13)

where Rep is the particle Reynold’s number, which is estimated as Rep =

εgρg |u − vp|d p μ

(14)

The functions F0 and F3 in eq 13 are computed as follows: ⎧ εp 135 ⎪ 1 + 3 2 + 64 εp ln(εp) + 16.14εp if εp < 0.4 ⎪ ⎪ 1 + 0.681εp − 8.48εp2 + 8.16εp3 F0(εp) = ⎨ ⎪ 10εp ⎪ 3 if εp ≥ 0.4 ⎪ εg ⎩ (15) C



Article

Table 1. Physical and Numerical Parameters for Simulation Gas Phase temp, K superficial gas velocity, m/s pressure, atm

298 1.0−0.2

viscosity, Pa·s molecular weight, kg/mol

1.8 × 10−5 28.8

1 Particles

number diameter, m density, kg/m3 interparticle restitution coefficient

938 488, 1 315 293, 1 691 480 1.33 × 10−3 2490 0.97

interparticle friction coefficient spring constant, N/s particle−wall restitution coefficient particle−wall friction coefficient

0.3

thickness, cm cells in x-direction cells in z-direction

10 49 20

800 0.97 0.25

Geometry width, cm height, cm cells in y-direction bed height, cm

15 100 192 15, 20, 25

3.2. Initial and Boundary Conditions. Followed by the discretization of the governing equations for the fluid phase, proper initial and boundary conditions should be assigned for the enclosure of the algebraic equations. Fully developed gas is uniformly supplied to the slot, while the no-slip boundary condition is used for the walls. The pressure outlet is applied for the outlet of the geometry while the zero-gradient Neumann boundary condition is chosen for the walls and the inlet of the system. The calculation time steps for the fluid and solid phases are 1 × 10−4 and 2 × 10−6 s, respectively. A real time of 20 s is calculated for each individual simulation case with the calculation data of the initial 5 s discarded to avoid the influence of the start-up effect on the statistical analysis. Moreover, each simulation case is conducted on the basis of the parallel CFD−DEM coupling framework with a total number of 112 CPUs and 15 days’ calculation time in the cluster. Each computational node of the cluster consists of eight cores and has a shared memory of 12G RAM. The type of CPU is the IntelXeonE5335 at 2.0 GHz. A domain decomposition strategy is adopted for the parallel calculation, and the parallel communication between the calculation nodes is achieved via the OpenMPI library.

Figure 2. (a) Pressure drop curves under different bed heights with decreasing the superficial velocity. (b) Comparison of the predicted minimum spouting velocity with the experimental data under different bed heights.

The pressure drop is enlarged slowly with diminishing the superficial velocity initially, which is corresponding to the continuous spouting regime in the bed. Subsequently, the pressure drop increases rapidly with the spouting velocity decreasing into a transition region, which is connected to the spouting collapse. When some critical point is reached, the pressure drop suddenly increases until a maximum value is obtained, followed by a slowly decreasing process with the velocity reduction. Given a specific superficial velocity, the larger the initial bed height is, the higher the pressure drop appears. Moreover, the superficial velocity corresponding to the spouting collapse is larger with a higher initial packed bed height. The pressure drop profile with a bed height of 0.20 m is compared to the available data in the literature. As shown in Figure 2a, the predicted one agrees quite well with the experimental profile. On the other hand, Figure 2b presents the comparison of the minimum spouting velocities between the data obtained from the simulation and the experiment. It exhibits that the simulation precisely captures this important system parameter, indicating that the present work can be used to study the hydrodynamics in the slot-rectangular spouted bed. 4.2. Startup Effect. The solid motion in the startup procedure is important for the design and especially the safe and reliable operation of the bed. Figure 3 shows the time evolutionary snapshots of the solid phase in slice Z = 0.05 m of the bed. Particles above the inlet are flushed by the introduced gas, resulting in a void in the bed bottom. Then, the void enlarges and rises upward, followed by a gas jet channel connecting the void and the bed inlet. Moreover, the bed height enlarges with the gas continuously injected from the inlet. During the time interval of 0.2−0.3 s, the rising void is choked by the falling particles, and part of the particles is injected into the domain of the void, resulting in the

4. RESULTS AND DISCUSSION 4.1. Model Verification. The minimum spouting velocity of the spouted bed is a key parameter in the design and operation of the system. In the practical operation, it is obtained by reducing the superficial velocity until the pressure drop of the system is enlarged suddenly. The pressure drop profile is obtained with the superficial velocity decreasing stepwisely. As the calculation is time-consuming and the available computational resources are relatively limited, a compromise approach is adopted to decrease the superficial velocity. Under the steady spouting regime, the velocity diminishes with a fraction of about 10%. In the vicinity of the maximum value, nearly 5% reduction of the superficial velocity is applied. However, when the internal jet regime appears, the superficial velocity decreases with a percentage of 20%. Figure 2a illustrates the pressure drop profiles with decreasing the gas velocity under three different bed heights. D



Article

Figure 4. 3-D view of the spout evolution during the startup pprocedure, Hb = 0.2 m, Ug = 0.75 m/s. Figure 3. Snapshots of solid distribution and spout evolution in slice Z = 0.05 m of the system during the startup procedure, Hb = 0.2 m, Ug = 0.75 m/s: (a) t = 0.1 s, (b) t = 0.2 s, (c) t = 0.3 s, (d) t = 0.4 s, (e) t = 0.5 s, (f) t = 0.6 s, (g) t = 0.7 s, (h) t = 0.75 s, (i) t = 0.8 s, (j) t = 0.9 s, (k) t = 1.0 s, and (l) t = 1.1 s.

the spout. Meanwhile, the velocity magnitude of the solid phase in the spout is several times that in the annulus. Particles in the annulus are entrained into the spout mainly from the bed bottom and the spout−annulus interface along the axial direction. Followed the rising process in the spout region, particles are injected into the fountain region. Then, the solid vertical velocity diminishes along the axial direction due to the effect of gravity. 4.4. Distribution Property of Fluid Phase. Fluid phase mainly concentrates in the central part of the spout. The gas leakage from spout to annulus through the spout−annulus interface occurs along the axial direction, resulting in a reduction of the velocity magnitude along the spout and plenty of gas rising upward from the gap of the solid phase in the annulus. Figure 6 shows the gas vertical velocity along the axial direction in the slice X = 0.075 m. It is clearly observed that gas vertical velocity in the central line is the largest and the velocity near the wall is nearly zero. While a significant difference appears in the near central region of the bed, gas velocity decreases with increasing height due to gas leakage from spout to annulus. The influence of the bed thickness on the voidage distribution is shown in Figure 7. It can be clearly observed that voidage in the central line of slice Z = 0.05 m decreases initially and then increases, reflecting that solid concentration in the central region of spout goes large initially due to the falling of particles from annulus to spout. Moreover, voidage along the central line of Z = 0.035 m first increases and then decreases, followed by an increasing tendency. It means that the area of

concurrence of a void and a jet in the bed. After the bed surface is reached, the void breaks up with particles injected into the free domain above the bed surface. The system maintains an internal jet and the jet is enlarged until the bed height is reached, as illustrated in Figure 3e−g. Subsequently, the internal jet breaks through the whole bed. After a short adjustment, as can be seen in Figure 3i−l, a steady flow pattern is established in the bed. In order to capture the startup characteristics of the whole field, Figure 4 illustrates the snapshots of the gas motion at several time points corresponding to those in Figure 3. A bubble is generated in the vicinity of the inlet with larger size in the depth direction as compared with that in the width direction due to the influence of the slot shape. As its rising, the bubble grows larger and the influence of the slot shape diminishes especially for the bubble at t = 0.3 s. After the bed surface is reached, the bubble with the largest size erupts. Then, the gas jet enlarges and penetrates the whole bed. Meanwhile, the symmetrical distribution properties of the bubble and jet can be observed in both the X and Z directions. 4.3. Flow Behavior of Solid Phase. The detailed local solid motion is illustrated in Figure 5. Obviously, particles fall downward slowly in the annulus while they rise vigorously in E



Article

Figure 7. Voidage distribution along the axial direction in the lines Z = 0.05, 0.035, and 0.01 m of slice X = 0.075 m, Hb = 0.2 m, Ug = 0.75 m/ s.

understanding of the solid circulation mechanism in the system. Figure 8 shows the lateral distribution profiles of

Figure 5. Detailed illustration of the local solid motion in slice Z = 0.05 m, Hb = 0.2 m, Ug = 0.75 m/s: (a) spout region and (b) fountain region.

Figure 6. Gas vertical velocity along the axial direction in the lines Z = 0.05, 0.035, and 0.01 m of slice X = 0.075 m, Hb = 0.20 m, Ug = 0.75 m/s.

Figure 8. Lateral distribution of the vertical solid velocity in slice Z = 0.05 m, Hb = 0.20 m, Ug = 0.75 m/s: (a) in the conical region, (b) in the rectangular region with the spout, and (c) in the fountain region.

the spout expands with bed elevation. However, voidage in the vicinity of the wall exhibits a distinct behavior, with a constant value in the annulus region while increasing sharply when the bed surface is reached. 4.5. Distribution Properties of Solid Phase. 4.5.1. Velocity Distribution. Detailed information on the distribution characteristic of the solid phase is important for the

solid vertical velocity in slice Z = 0.05 m with Ug = 0.75 m/s. The maximum solid velocity exists in the central part of the bed at all the heights investigated, which is due to the large drag force exerted by the fluid phase. Along the horizontal direction, solid velocity changes sharply from the central region to the annulus. Smoother decreasing tendency from the central region to the periphery can be observed for the solid velocity in the F



Article

fountain. The wall effect on the solid motion can be referred from the profile comparison illustrated in Figure 9. In the

Figure 9. Solid vertical velocity along the axial direction in the lines Z = 0.05, 0.035, and 0.01 m of slice X = 0.075 m, Hb = 0.2 m, Ug = 0.75 m/s.

central line of the plane Z = 0.05 m, particles are accelerated by the injected gas after being entrained from the annulus to spout in the bed bottom, followed by a rapid acceleration procedure. After the maximum value is reached, the solid velocity decreases gradually along the axial direction. Significantly different behavior can be obtained for solid velocity in different lines along the depth direction. For the vertical velocity profile in the central line of slice Z = 0.035 m, particles are accelerated sharply until a maximum value is reached and then decrease gradually. After the bed surface is reached, the vertical velocity enlarges gradually again, followed by a decreasing procedure. In the central line of slice Z = 0.01 m, the vertical velocity is nearly zero in the annulus. Above the bed surface, the vertical velocity shows a negative value that corresponds to the falling of particles in this region. Hence, the three-dimensional flow behavior of the solid phase is strongly influenced by the bed thickness in slot-rectangular spouted bed. 4.5.2. Solid Flux Distribution. It has been obtained that the solid phase in the central region shows large velocity but small concentration while an opposite distribution characteristic exists in the annulus and the periphery of the fountain, which makes the investigation of solid flux important for the understanding of the carry ratio of the gas phase. The solid flux in a specific cell is estimated with the multiplication of the density, the concentration, and the velocity of the solid phase in the current cell. Then, the time statistical solid flux is obtained by averaging the instantaneous solid flux over time, Nt, which is calculated as ⟨ϕp(x)⟩ =

1 Nt

Figure 10. Vector plots of the solid flux in the system with Hb = 0.2 m, Ug = 0.75 m/s: (a) plot in Z = 0.05 m and (b) plot in X = 0.075 m.

Figure 11. Vertical solid flux along the axial direction in the lines Z = 0.05, 0.035, and 0.01 m of slice X = 0.075 m, Hb = 0.2 m, Ug = 0.75 m/ s.

tendency but an explicitly small scale of the solid flux can be observed along the central line of Z = 0.035 m like that of Z = 0.05 m. However, the downward solid flux is observed along the line in the slice near the wall. 4.6. Characteristic of Spout−Annulus Interaction. The spout−annulus interface is an important parameter concerning the hydrodynamics and the design of the system, such as the insertion of the draft tube. The interface divides the system into two parts with opposite solid moving behaviors. Meanwhile, the spout−annulus interface can be identified with the contour plot of voidage of fluid phase below the bed surface, as described in the literature.31 Figure 12 shows the 3-D view of the contour surfaces with a voidage of 0.8. It can be observed that the spout increases from the bottom to the bed surface with its shape strongly influenced by the slot geometry. Different variations can be captured for the spout boundary in the different directions. Along the width direction, the spout boundary is observed to behave in a diverging tendency continually along the bed height. However, spout boundary in the side view, as shown in Figure 12c, increases initially and then shrinks until the minimum, followed by the diverging tendency along the bed height. The reason is mainly due to the influence of slot

Nt

∑ (εp(x, t )ρp vp(x, t )) 0

(17)

Figure 10 shows the vector plots of solid flux in slices of Z = 0.05 m and X = 0.075 m in the spouted bed under Ug = 0.75 m/s. The systematic circulation pattern can be captured. Vertical solid flux exists in the central region of the bed while the downward movement appears in the periphery of the bed. On the other hand, the maximum of the vertical solid flux appears in the spout exit. The magnitude of maximum upward solid flux is nearly 1100 kg/(m2·s), while the downward one is only 100 kg/(m2·s). The wall effect on the distribution of solid flux is clearly illustrated in Figure 11. Along the central line of Z = 0.05 m, the vertical solid flux first increases sharply with the maximum value obtained at nearly the bed height and then decreases gradually in the fountain region. The similar variation G



Article

Figure 12. 3-D view of the spout−annulus interfaces in the system with Hb = 0.2 m, Ug = 0.75 m/s: (a) global view, (b) front view, and (c) side view. Figure 14. Flow patterns of solid phase in slices Z = 0.05 m and X = 0.075 m with bed height Hb = 0.2 m under different superficial velocities: (a) Ug = 0.75 m/s, (b) Ug = 0.85 m/s, and (c) Ug = 0.95 m/ s.

shape in the slot-rectangular spouted bed, which is also an explicit difference compared with the traditional conical spouted bed. The variations of the spout diameters under three superficial velocities in slices Z = 0.05 m and X = 0.075 m along the axial direction are shown in Figure 13. It can be observed that the

increasing the superficial velocity, demonstrating that more particles with larger velocity are injected into the fountain region under a higher superficial velocity. 4.8. Effect of Bed Height. The effect of bed height on solid motion in the spouted bed is shown in Figure 15, in which the solid distribution patterns under three different bed heights with the superficial velocity being the same ratio to the minimum spouting velocity are compared in slices Z = 0.05 m and X = 0.075 m. The similar fluidizing behavior of the solid phase can be observed in all the three systems investigated. For each case, the flow behavior changes from the steady spouting

Figure 13. Spout diameters along the axial direction under three superficial velocities in slices Z = 0.05 m and X = 0.075 m of the spouted bed with Hb = 0.20 m: (a) slice Z = 0.05 m and (b) in slice X = 0.075 m.

spout boundary enlarges at nearly all the bed heights with increasing the superficial velocity in the rectangular region of the bed. However, this increasing tendency in the conical region is not explicit. 4.7. Effect of Spouting Velocity. The effect of superficial velocity on the solid motion in the slot-rectangular spouted bed is illustrated in Figure 14. With the increase of the superficial velocity, the bed height enlarges due to the increase of voidage under a large superficial velocity. Moreover, the spouting height and the number of particles in the fountain enlarge upon

Figure 15. Flow patterns of the solid phase in slices Z = 0.05 m and X = 0.075 m of the systems under three bed heights. H



Article

regime to the internal jet and then the packed bed with diminishing the superficial velocity. In the situation of Ug/Ums > 1.0, as shown in Figure 15a−c, steady spouting is captured with three regions appearing in the bed. The spout−annulus interface in slice Z = 0.05 m shows an increasing tendency with the enlargement of bed height. The obvious influence of the bed height on solid motion can be captured from the solid concentration in the spout outlet and the spout−annulus interface. It demonstrates that the solid concentration in the upper part of the spout becomes larger with increasing bed height, as shown in Figures 15a−c, resulting from more particles falling into the spout from the spout−annulus interface in the higher bed height. In the situation of Ug/Ums < 1.0, as shown in Figure 15d−f, the steady spouting phenomenon disappears. Gas jet cannot penetrate the whole bed, leading to the appearance of the internal jet flow regime. In this condition, the jet is formed above the inlet of the bed, and particles near this region move in an internal circulating pattern while particles in other regions move with the status as that in a packed bed. When the superficial velocity decreases continuously, as shown in Figure 15g−i, the gas jet disappears. Particles in the system are in a motionless status, and the introduced gas escapes from the gap of the packed bed.

(5) Increasing the superficial velocity results in the enlargements of bed height and the maximum spouting height. The flow pattern of the solid phase under different bed height shows a similar distribution behavior with the superficial velocity being the same ratio to the Ums. Moreover, the flow regime of the system changes from the steady spouting regime to the internal jet and then the packed bed with decreasing the superficial velocity.

■

AUTHOR INFORMATION

Corresponding Author

*Fax: +86-0571-87991863. E-mail: [email protected] (J.R.F.); [email protected] (K.L.). Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS Financial support from the National Natural Science Foundations of China (Grant Nos. 51176170, 50976098) and the Zhejiang Provincial Natural Science Foundation for Distinguished Young Scholars (Grant No. LR12E06001) is sincerely acknowledged.

■

5. CONCLUSIONS The parallel framework of the CFD−DEM coupling approach has been used to model the gas−solid flow in a 3-D slotrectangular spouted bed. The motion of the solid phase is tracked with the DEM approach, and gas phase is solved with the k−ε turbulence model. The simulated results show excellent agreement with the data in the literature. On the basis of the simulation results, several conclusions can be drawn: (1) The pressure drop enlarges gradually with decreasing gas velocity. When some critical point is reached, a sudden increase of the pressure drop can be obtained until a maximum value is reached, followed by a gradual decrease procedure with reducing the spouting velocity. The variation tendency of pressure drop with superficial velocity reflects the alternation of the flow regime. The higher the bed height is, the larger the pressure drop appears under the same superficial velocity. (2) In the startup procedure, a bubble is generated above the inlet with its shape strongly influenced by the slot geometry. In the rising procedure, the bubble grows larger, followed by a gas channel connecting the bubble and the bed inlet. After the breakup of the bubble and a short adjustment, a steady flow pattern can be constructed. This phenomenon is useful for the design and safe operation of the system. (3) Gas velocity exhibits a large value in the central region and decreases along the axial direction. However, the vertical velocity of the solid phase increases sharply and then decreases in the central part along the axial direction. A large vertical solid flux mainly concentrates in the central part of the spout. The strong 3-D effect of the bed thickness on the distribution properties of gas−solid phases can be captured. (4) The spout−annulus interface shows a strong dependence on the slot shape in the lower part of the bed with a larger size in the depth direction. Along the axial direction, the influence of the slot reduces or even disappears at the higher level of the bed. The higher the spouting velocity is, the larger the size of the spout boundary that can be obtained.

NOMENCLATURE dp = particle diameter, m fc = contact force, N fcnij = normal contact force, N fctij = tangential contact force, N fd = drag force, N fp = far field pressure force, N g = gravitational acceleration, m/s2 Hb = bed height, m Ip = particle moment of inertia, kg·m2 k = turbulent kinetic energy, m2/s2 kn = spring coefficient in normal direction, N/m kt = spring coefficient in tangential direction, N/m mp = particle mass, kg N = number of time interval for statistic n = normal unit vector p = pressure, Pa r = position, m R = particle radius, m Rep = particle Reynolds number Sp = particle drag sink term, N/m3 Tp = particle torque, N·m t = tangential unit vector u = velocity vector, m/s Ug = superficial gas velocity, m/s Ums = minimum spouting velocity, m/s Vp = particle velocity, m/s ΔV = volume of current computational cell, m3 Vp = particle volume, m3

Greek symbols

β = interphase momentum transfer coefficient, kg/(m3·s) δnij = normal displacements between particle I and particle j, m δti = tangential displacements between particle I and particle j, m εg = void fraction εt = turbulent dissipation rate, m2/s3 εp = solid concentration μ = gas dynamic viscosity, kg/(m·s)

I



Article

μp = friction coefficient between particles or particle-wall μt = gas turbulence viscosity, kg/(m·s) ρg = gas density, kg/m3 ρp = particle density, kg/m3 ω = particle angular velocity, 1/s ηn = damping coefficients in normal direction, kg/s ηt = damping coefficients in tangential direction, kg/s

(17) Da Rosa, C. A.; Freire, J. T. Fluid Dynamics Analysis of a DraftTube Continuous Spouted Bed with Particles Bottom Feed using CFD. Ind. Eng. Chem. Res. 2009, 48, 7813−7820. (18) Béttega, R.; Da Rosa, C. A.; Corrêa, R. G.; Freire, J. T. Fluid Dynamic Study of a Semicylindrical Spouted Bed: Evaluation of the Shear Stress Effects in the Flat Wall Region Using Computational Fluid Dynamics. Ind. Eng. Chem. Res. 2009, 48, 11181−11188. (19) Cunha, F. G.; Santos, K. G.; Ataìde, C. H.; Epstein, N.; Barrozo, M. A. S. Annatto Powder Production in a Spouted Bed: An Experimental and CFD Study. Ind. Eng. Chem. Res. 2008, 48, 976−982. (20) Yang, S.; Luo, K.; Fang, M.; Fan, J. Discrete Element Simulation of the Hydrodynamics in a 3D Spouted Bed: Influence of Tube Configuration. Powder Technol. 2013, 243, 85−95. (21) Szafran, R. G.; Kmiec, A. CFD Modeling of Heat and Mass Transfer in a Spouted Bed Dryer. Ind. Eng. Chem. Res. 2004, 43, 1113− 1124. (22) Luo, K.; Yang, S.; Zhang, K.; Fang, M.; Fan, J. Particle Dispersion and Circulation Patterns in a 3D Spouted Bed with or without Draft Tube. Ind. Eng. Chem. Res. 2013, 52, 9620−9631. (23) Zhao, X. L.; Li, S. Q.; Liu, G. Q.; Yao, Q.; Marshall, J. S. DEM Simulation of the Particle Dynamics in Two-Dimensional Spouted Beds. Powder Technol. 2008, 184, 205−213. (24) Zhao, X. L.; Li, S. Q.; Liu, G. Q.; Song, Q.; Yao, Q. Flow Patterns of Solids in a Two-Dimensional Spouted Bed with Draft Plates: PIV Measurement and DEM Simulations. Powder Technol. 2008, 183, 79−87. (25) Zhu, R. R.; Zhu, W. B.; Xing, L. C.; Sun, Q. Q. DEM Simulation on Particle Mixing in Dry and Wet Particles Spouted Bed. Powder Technol. 2011, 210, 73−81. (26) Cundall, P. A.; Strack, O. D. L. A Discrete Numerical Model for Granular Assemblies. Geotechnique 1979, 9, 47−65. (27) Koch, D. L.; Hill, R. J. Inertial Effects in Suspension and PorousMedia Flows. Annu. Rev. Fluid Mech. 2001, 33, 619−647. (28) Chen, Z. Hydrodynamics, Stability and Scale-up of SlotRectangular Spouted Beds; University of British Columbia, Vancouver, 2008. (29) Garg, R.; Galvin, J.; Li, T.; Pannala, S. Open-Source MFIX-DEM Software for Gas−Solids Flows: Part IVerification Studies. Powder Technol. 2012, 220, 122−137. (30) Li, T.; Garg, R.; Galvin, J.; Pannala, S. Open-Source MFIX-DEM Software for Gas−Solids Flows: Part IIValidation Studies. Powder Technol. 2012, 220, 138−150. (31) Link, J. M.; Deen, N. G.; Kuipers, J.; Fan, X.; Ingram, A.; Parker, D. J.; Wood, J.; Seville, J. PEPT and Discrete Particle Simulation Study of Spout−Fluid Bed Regimes. AIChE J. 2008, 54, 1189−1202.

Subscripts

c = contact force d = drag force g = gas phase p = particle phase t = turbulence

■

REFERENCES

(1) Olazar, M.; Arandes, J. M.; Zabala, G.; Aguayo, A. T.; Bilbao, J. Design and Operation of a Catalytic Polymerization Reactor in a Dilute Spouted Bed Regime. Ind. Eng. Chem. Res. 1997, 36, 1637− 1643. (2) Olazar, M.; San José, M. J.; Alvarez, S.; Morales, A.; Bilbao, J. Measurement of Particle Velocities in Conical Spouted Beds Using an Optical Fiber Probe. Ind. Eng. Chem. Res. 1998, 37, 4520−4527. (3) Vuthaluru, H. B.; Zhang, D. Effect of Coal Blending on Particle Agglomeration and Defluidisation during Spouted-Bed Combustion of Low-Rank Coals. Fuel Process. Technol. 2001, 70, 41−51. (4) San José, M. J.; Alvarez, S.; de Salazar, A. O.; Olazar, M.; Bilbao, J. Spout Geometry in Shallow Spouted Beds with Solids of Different Density and Different Sphericity. Ind. Eng. Chem. Res. 2005, 44, 8393− 8400. (5) Calo, J. M.; Madhavan, L.; Kirchner, J.; Bain, E. J. Arsenic Removal via ZVI in a Hybrid Spouted Vessel/Fixed Bed Filter System. Chem. Eng. J. 2012, 189−190, 237−243. (6) Piccinini, N.; Gracet, J. R.; Mathur, K. B. Vapour Phase Chemical Reaction in Spouted Beds: Verification of Theory. Chem. Eng. Sci. 1979, 34, 1257−1263. (7) Dogan, O. M.; Freitas, L. A. P.; Lim, C. J.; Grace, J. R.; Luo, B. Hydrodynamics and Stability of Slot-Rectangular Spouted Beds. Part I: Thin Bed. Chem. Eng. Commun. 2000, 181, 225−242. (8) Konduri, R. K.; Altwicker, E. R.; Morgan, M. H., III Design and Scale-up of a Spouted-Bed Combustor. Chem. Eng. Sci. 1999, 54, 185− 204. (9) Du, W.; Xu, J.; Ji, Y.; Wei, W.; Bao, X. Scale-up Relationships of Spouted Beds by Solid Stress Analyses. Powder Technol. 2009, 192, 273−278. (10) Freitas, L.; Dogan, O. M.; Lim, C. J.; Grace, J. R.; Luo, B. Hydrodynamics and Stability of Slot-Rectangular Spouted Beds Part II: Increasing Bed Thickness. Chem. Eng. Commun. 2000, 181, 243− 258. (11) Chen, Z.; Lim, C. J.; Grace, J. R. Hydrodynamics of SlotRectangular Spouted Beds: Effect of Slot Configuration on the Local Flow Structure. Can. J. Chem. Eng. 2008, 86, 598−604. (12) Freitas, L. A. P.; Dogan, O. M.; Lim, C. J.; Grace, J. R.; Bai, D. Identification of Flow Regimes in Slot-Rectangular Spouted Beds Using Pressure Fluctuations. Can. J. Chem. Eng. 2004, 82, 60−73. (13) Luo, B.; Lim, C. J.; Freitas, L. A. P.; Grace, J. R. Flow Characteristics in Slot-Rectangular Spouted Beds with Draft Plates. Can. J. Chem. Eng. 2004, 82, 83−88. (14) Luo, K.; Fang, M.; Yang, S.; Zhang, K.; Fan, J. LES−DEM Investigation of an Internally Circulating Fluidized Bed: Effects of Gas and Solid Properties. Chem. Eng. J. 2013, 228, 583−595. (15) Fang, M.; Luo, K.; Yang, S.; Zhang, K.; Fan, J. Computational Fluid Dynamics−Discrete Element Method Investigation of Solid Mixing Characteristics in an Internally Circulating Fluidized Bed. Ind. Eng. Chem. Res. 2013, 52, 7556−7568. (16) Fang, M.; Luo, K.; Yang, S.; Zhang, K.; Fan, J. LES−DEM Investigation of Gas−Solid Flow Dynamics in an Internally Circulating Fluidized Bed. Chem. Eng. Sci. 2013, 101, 213−227. J


Three-Dimensional Modeling of GasâSolid Motion in a Slot

Recommend Documents

Three-Dimensional Modeling of GasâSolid Motion in a Slot