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Numerical study on microscopic mixing characteristics in fluidized beds via DEM
Fuel Processing Technology 88 (2007) 187 – 198 www.elsevier.com/locate/fuproc
Numerical study on microscopic mixing characteristics in f luidized beds via DEM Fengguo Tian ⁎, Mingchuan Zhang, Haojie Fan, Mingyan Gu, Lei Wang, Yongfeng Qi Institute of Thermal Energy Engineering, Shanghai Jiaotong University, Minhang, Shanghai 200240, People's Republic of China Received 26 September 2005; received in revised form 27 March 2006; accepted 7 April 2006
Abstract In this paper, discrete element method (DEM), combined with computational fluid dynamics (CFD), is used to investigate the micro-mixing process in fluidized beds (FBs) of uniform particles. With the aid of snapshots and adoption of Lacey and Ashton indexes, mixing evolvement for two cases, fluidized bed using horizontal distributor with even gas supply and fluidized bed using inclined distributor with uneven gas supply, is discussed in detail. Results indicate that the Ashton index appears to be more effective in assessing the mixing dynamics in this work. Further analyses illustrate that in the case of horizontal distributor incorporated with even gas supply, diffusive mixing pattern is predominant, since bubbles lateral motion is reduced in such a bed; whereas, there is a faster convective mixing process in a fluidized bed using inclined distributor with uneven gas feed, followed by shear mixing. Generally, localized air supply induces the density gradient of particle distribution in the bed, which is the basic agent of convective particle stream. The analyses are confirmed by the comparison of solid flux during the simulations of the two cases. In addition, the mixing mechanism and the mixing time scale agree well with published experimental results. © 2006 Elsevier B.V. All rights reserved. Keywords: Fluidized bed; Mixing; Simulation; Discrete element method (DEM)
1. Introduction For their good adaptability and prominent processing ability, fluidized beds (FBs) are popular in many industrial fields, including energy [1], environment, chemical engineering [2], etc. Solids mixing process plays an important role in the design and operation of fluidized bed, e.g., how the fresh materials blend with previous background particles, knowledge about the residence time distribution (RTD) of the feed, the spatial differences of heat transfer and chemical reactions in the bed. Various experimental methods have been explored in the gas– solid flow research area, which are crucial for comprehensive understanding about the fluidization hydrodynamics. Arena and Cammarota [3] and Martin et al. [4] implemented direct measurement of the solids concentration, employing quick closing valves with which the column of interest can be portioned in sections of suitable length. Mann and Crosby [5] developed a strain gauge to measure local particle concentrations ⁎ Corresponding author. Tel.: +86 21 3420 5696; fax: +86 21 3420 6115. E-mail address: [email protected] (F. Tian). 0378-3820/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fuproc.2006.04.006
and velocities in gas–solid flows, deduced from the impact frequency with surrounding particles and the deformation magnitude of the probe nose. By withdrawing particles from gas–solid two-phase flow with extraction probes, Bader [6] evaluated the local mass flow rate of the particles. Laser Doppler velocimetry (LDV) is based on the detection of optical interference phenomena, arising from two interacting laser beams [7]. Ishida et al. [8] developed reflective optical probe systems which enable measurement of velocities of bubbles and individual particles in bubbling fluidized bed. However, some of them are intrusive, e.g., quick closing valves, extraction probes and strain gauge, which alter the flow pattern in the bed. Others are confined to strict occasions, due to the property of gas–solid system or to the complicated procedure of the experiment. For example, LDV technique doesn't work in dense suspensions [9]. While, calibration for the nonlinear transform in optical experiments should be carefully treated. Although the introduction of radioactive tagging and X-ray methods offers the possibility of measuring the solids velocity directly [10], they have not found widespread application due to the considerable cost of the equipment. Moreover, most available information
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provides only the overall characterizations about the flow pattern and mixing phenomena. Systemic discussions engaged in the microscopic mixing phenomena of dense gas–solid flows, in temporal and spatial details, are believed unavailable. Owe to increasing computer power, computational fluid dynamics (CFD) has become a versatile tool in studying the gas–solid flow pattern [11]. Two-fluid model (TFM), also referred as Eulerian–Eulerian (EE) model, has been frequently used to describe fluid dynamic phenomena in fluidized beds [12]. TFM supposes that the gas and the solid phase are considered to be continuous and fully inter-penetrating. Both phases are described in terms of separate sets of conservation equations with appropriate interaction terms representing the coupling between the phases. Though a physical characteristic of the solid particles such as the shape and size is included in the continuum representation through the empirical relations for the interfacial friction, these models do not recognize the discrete character of the solid phase and are farfetched for dense gas–solid flow [13,14]. Also, in the two-fluid approach, assumptions need to be made concerning the solids rheology, where often Newtonian behavior is assumed in the absence of a more detailed knowledge [15]. Free of the continuum hypothesis for the solid phase, discrete element method emphasizes the subtle details of individual moving body in particulate assemblies. The origins of DEM technique actually lie in the field of Molecular Dynamics (MD) [16]. As a development, Monte Carlo (MC) techniques make use of the random nature of the molecular motion in gases or dilute gas–solid phase to approach the problem from a statistical point of view [17]. In the Monte Carlo method, it is assumed that the velocities of individual particles are independently distributed within a velocity distribution function without regard to history or to the behavior of neighboring particles. It is unlikely that this is true in dense concentrations where particles will experience many collisions with their nearest neighbors and it is therefore unlikely that there would be no correlation between their velocities or positions. Numerical simulation of dense-phase discrete particle systems was first reported by Cundall and Strack [18]. In their simulations, it was postulated that particulate flow occurs in the absence of an interstitial fluid. Attracted by the discrete treatment of the particle phase in DEM, Tsuji et al. [19] introduced such concept into the simulation of dense gas–solid flow in fluidized bed, combined with traditional CFD, also referred as Eulerian–Lagrangian (EL) model. The DEM implement is constrained by a large computational cost associated with detection of inter-particle contact. Whereas, such kind of numerical method enables us to examine data which are normally inaccessible, and provide substantial information in a particulate system. With regard to DEM investigations engaged in two-phase flow, Tsuji et al. [19], Hoomans et al. [13], Xu and Yu [20], and Ouyang and Li [21] reasonably predicted the typical behaviors in fluidized beds. Limtrakul et al. [22] paid attention to the time-averaged axial velocity distribution of the tracer particles. Rhodes [23] analyzed the effects of gas velocity, particle density, and particle diameter on mixing rate by means of Lacey mixing index. To date, the transient description of the mixing process in fluidized beds is still limited.
In this paper, a comparison is made between the Lacey and Ashton mixing indexes, with respect to their sensitivity in characterizing the particles mixing process. For its better effectiveness in evaluating the mixing development, the Ashton index is introduced to study the influences of gas supply pattern on the micro-mixing mechanism in a fluidized bed, incorporated with analysis of tracer particles' fields. The simulations are also compared with previous experimental results, which suggests that this work is supposed to give satisfactory predictions. The paper ends with a detailed discussion about the mixing dynamics in the fluidized column, which is thought as still absent in relevant literatures. 2. Simulation of mixing process in FBs 2.1. Mathematical model Discrete element method (DEM) is a relatively new but now widely available tool for researching into gas–particle systems. Particles are traced individually by direct solution of Newton's equations of motion in the simulation. When no collisions occur, the forces acting on the particles are mainly from the gas phase and gravity. The motion of the particles is described by m
dY v Y ¼ F g þ mg dt
ð1Þ
where Y v is the particle velocity; the particle–fluid interaction Y force F g , can be written as: Y F g ¼ Vp
b Y Y ð ug − v Þ−jp 1−e
ð2Þ
ug is the gas velocity. The where Vp is the volume of a particle; Y inter-phase momentum transfer coefficient, β depends on the void fraction. The well-known Ergun equation is used for the packed bed region, ε ≤ 0.8. When ε N 0.8, and the particle's motion is taken to be only weakly affected by other particles, and a modified Wen–Yu equation of the fluid resistance for a single particle is used for the dilute region. The expressions of β are as follows: 1−e ð1−eÞl b ¼ 2 150 þ 1:75qejY ug − Y vj ðe V 0:8Þ ð3aÞ e dp qð1−eÞjY ug − Y v j −2:65 3 b ¼ CD e 4 dp
ðe N 0:8Þ
ð3bÞ
where CD is the drag coefficient, and can be written as: CD ¼
24ð1 þ 0:15Re0:687 Þ=Re 0:43
ðRe b 1000Þ ðRe z 1000Þ
ð4Þ
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 collisions
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between particles and between a particle and a wall are simulated by Hooke's linear springs and dashpots [18]. Governing equations can be expressed as: Y vij ÞY nij F cnij ¼ ð−kdn − gY Y X dϖ Y ¼ I F ctijd rp dt Y Y Y F ctij ¼ ð−kdt − gY vijd Y t ij ; j F ctij jblf j F cnij j t ij ÞY Y Y Y Y Y j F ctij jzlf j F cnij j F ctij ¼ −lf j F cnij j t ij ;
ð5Þ ð6Þ
Table 2 Parameters for gas phase Density Viscosity Number of cells Cell width Cell height
1.205 kg/m3 1.82 × 10− 5 N s/m2 10 × 29 12 mm 12 mm (lower) 20 mm (upper)
ð7Þ
Y Y where F cnij is the normal force; F ctij is the tangential force; I is the moment of inertia. The motion of the gas phase is calculated from the volume averaged gas phase governing equations as put forward by Anderson [24]. Equations for gas mass and momentum conservation in vector form are given by: Aðqg eÞ þ jd ðqg eY ug Þ ¼ 0 At Aðqg eY ug Þ þjd ðqg eY ug Y ug −Y vÞ ug Þ ¼ −ejp−bðY At −jd ðesÞ þ qg eg
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ð8Þ ð9Þ
v Þ represents the inter-phase In Eq. (9), the term −bðY ug −Y momentum transfer between the gas phase and each individual particle. Here, β is same as defined in Eqs. (3a) and (3b). 2.2. Computational conditions
Current numerical experiments are dedicated to comprehending the mixing behavior of monosize particles in a fluidized setup, which is difficult to implement in a laboratory. In order to make the computation time bearable, a smaller spring is often assigned. Discussions [25] revealed that such assumption does not lead to a larger difference in a gas–solid pattern. In this work, spring coefficient is set as 2000 N/m. As for the damping coefficient, restitution coefficient and friction coefficient, they are often determined on the base of experiences or energy loss test with DEM simulation [20]. Interestingly, these parameters are quite close to each other correspondingly in reported works [13,19,20]. Computation conditions and physical parameters used are given in Tables 1 and 2. The initial field is setup by randomly allocating particles in the bed and then allowing the particles to fall freely under gravity in the absence of a gas flow. In the present paper, two-dimensional (2D) beds of 420 mm in width and 96 mm in height are used. There are two gas inlets at the bottom of the air distributor, with their size and position shown in Fig. 1. The air velocity mentioned later is the average value, with mass flow of each gas inlet responsible for its
corresponding half (left and right). The thickness of the beds is the same as the single particle diameter. Meanwhile, in order to study the influence of the air supply and distributor on the mixing character in the bed, simulations are conducted with horizontal and inclined distributor successively. For their agreeable ability in enhancing the diffusion pattern of bed materials, inclined air distributors are widely employed, including the municipal solid waste (MSW) incinerator [26], the cone-shaped gas distributor in fluidized bed gasifiers [27] and fluidized bed driers for agricultural purposes. To make the present work more practically meaningful, the angle of the inclined distributor is equal to 18°, comparable to that in industrial applications. Step grid arrangement at the bottom is adopted for the inclined distributor. 2.3. Simulation description For gas motion, the local averaged Navier–Stokes equations [24] are discretized in finite volume form on a staggered grid. The Crank–Nicholson scheme is used for the time discretization, the power-law scheme for the convective term and a central differences scheme for diffusion terms. And, the conventional SIMPLE algorithm [28] is adopted to solve the equations for the fluid phase. In the computation, the Newtonian equations of motion of individual particles are integrated with fourth-order Runge–Kutta algorithm. The inter-particle force models discussed in Section 2.1 are also applied to the collision between a particle and wall, with the corresponding wall properties used during the calculation. For the fluid velocity, the no-slip
Table 1 Particle parameters Particle diameter Particle density Number of particles Spring coefficient
4 mm 1500 kg/m3 1000 2000 N/m
Damping coefficient Restitution coefficient Friction coefficient Time step
0.17 0.9 0.3 1.0 × 10− 5 s
Fig. 1. Schematic of the object simulated (unit: mm).
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boundary condition applies to the left, bottom and right walls and zero normal gradient condition to the top exit. The computational flowchart of global algorithm description of mixing process is illustrated in Fig. 2. The solution for each time step, with the voidage and particle velocity field from the discrete particle scheme, evolves through a series of computational cycles consisting of (i) explicit calculations of fluid velocity components for all interior computational cells and (ii) implicit determination of pressure distributions using an iterative procedure. For each time step the velocity and position of each particle are refreshed using the equation of motion and the forces acting on the particle. Fig. 3 outlines the procedure in solving particle movement. In the implicit stage of gas motion solving, mass residuals are calculated according to the fluid continuity equation. If a residual deviates from a prescribed convergence criterion for all the interior computational fluid cells, a whole field pressure correction is applied. New densities are calculated and new fluid velocities obtained from the momentum equation and, using recomputed mass residuals, the convergence criterion is reche-
Fig. 3. Sub-flowchart of particle motion.
cked. This sequence continues until convergence is attained or an iteration limit is reached. In the discrete element model, the size of the time step is critical to the calculations since it determines the stability and accuracy. The oscillation period of the spring–mass system used to model contacting particles is determined as [19]: Dt ¼ 2p
Fig. 2. Main flowchart of DEM simulation.
pffiffiffiffiffiffiffiffiffi m=k
ð10Þ
It is found that the calculation became unstable when the time step is near the limit value given by this equation. In the current simulation, the time step is 1/100 of the characteristic time scales of the spring constant as recommended by Eq. (10) to meet the required accuracy. In this DEM program, Verlet (or neighbor) list is employed for inter-particle contact detection. The neighbor lists are updated periodically to keep up with the motion of the particles inside the simulation domain. The interval for such updating is controlled by the size of the neighborhood, accounting for the
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highest possible velocity of the particles in the system. During the implementation, let vmax,n denote the maximum moving velocity of all particles for the integration time step tn. Assume ti is the time step for the last update of the neighbor lists. The next moment for the neighbor lists to be updated again, tj, is determined as: j X
1 jvmax;n jDt z rnb 2 n¼i
ð11Þ
where rnb is the radius of the neighborhood. A typical 1 s realtime simulation conducted on the PC (CPU 2.0 GHz, RAM 526 M) takes about 4 h to complete. 2.4. Mixing process in the bed This paper is interested in the mixing pattern within the bubbling regime of a fluidized bed. According to experimental
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observation, the average gas velocities lie among 1.5 umf and 2.5 umf. Instantaneous snapshots of particle distribution in Fig. 4 visually reproduce the development of solids flow in the bed. In this case, horizontal distributor and even gas supply are employed, with the superficial gas velocity being 2 umf, with umf = 1.289 m/s according to Wen and Yu [29]. At the early stage, the bed surface expands from fixed bed and reaches a basic stable surface during the fluidization. Initiated at the orifice with small size, bubbles ascend through the bed, break up or coalesce with their neighbors turbulently, and finally burst at the bed surface, due to the pressure difference between inner space of the bubble and the freeboard. Bubbles eruption induces lateral dispersion of part of the wake's particles over a certain area, and the remainder of these is ejected into the freeboard. Because of bubbles' pronounced axial activities and the inherent direction of gravity, axial mixing potentials are predominant. Correspondingly, improving the lateral mixing
Fig. 4. Mixing process in the case of horizontal gas distributor with even air supply.
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performance in fluidized beds earns particular meaning, which is also the interest of the present research. With the goal mentioned above, the bed materials are renumbered into two equal parts in lateral direction, according to their position relative to the centerline in width at the beginning. The particles in the left half of the bed are tagged with a black color and the rest green. Obviously, the two groups of particles are completely segregated at the moment t = 0 s. As time proceeds, such a completely segregated state gradually gives way to one in which the particles of different colors are partially mixed. A dynamic equilibrium is established after about 5 s of mixing. For detailed interpretation, snapshots at t = 0.5, 1.0, 2.0 s indicate that material transfer between the two colored zones can be divided into three terms: (1) the shear effects on the interface of these zones; (2) dispersion in the upper freeboard; (3) convective mixing caused by bubble turbulence, which is thought to be important for mixing capability in the entire bed. In a bubbling fluidized bed, each bubble carries a wake of particles, which is ultimately deposited on the bed surface, and good mixing of solids occurs in the wake where movement is generated. This contributes partially to the lateral mixing of solids. Since fluidizing gas is evenly supplied, bubbles in the two sides share similar frequency and activity. Such symmetrical characteristic weakens the bubbles' lateral movement, resulting in poor convective effects. The discussion is also approved by information from the following snapshot (namely, t = 2.0 s in Fig. 4). It is clearly illuminated that after a mixing period of 2 s, the vicinity to the centerline enjoys a well mixed state of the two groups of particles, with high density of original host particles near the relevant wall. In other words, particles of the two particularly colored assemblies gradually penetrate into each other to complete the blending. It can be concluded that, in the case of horizontal distributor and even gas injection, diffusive effect is a controlling factor in particulate lateral mixing process. A time-series of particle field enriches our visual understanding about mixing phenomena, and deepens our insight into its mechanism. Meanwhile, the longer the mixing goes on, the more difficult the mixing extent to be assessed qualitatively, see snapshots of t = 4.0, 5.0, 6.0 s in Fig. 4. Therefore, an appropriate mathematical description is important for accurate and full comprehension of the mixing process. 3. Quantitative evaluation of the mixing degree
standard deviations of a specified property, which can be defined as [30]: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u1 X r¼t ðCi −CÞ2 ; N i¼1
C ¼ lim
N Yl
N 1X Ci N i¼1
! ð12Þ
where σ is standard deviation when the number of sampling cell, N, is infinite; Ci is the concentration of tracer particles in the sampling unit i; C the average value of Ci in N sampling cells. Since parameter C is an ideal limit (N is infinite), it is also referred to as the true value of the tracer particles' concentration. In practical measurement, infinite sampling units mean impossible. Consequently, for the finite sampling procedure, standard deviation is modified to: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N N u 1 X 1X S¼t C¯ ¼ ðCi − C¯ Þ2 ; Ci ð13Þ N −1 1 N i¼1 where S is the standard deviation when the number of sampling units is finite, C¯ is the average concentration of tracer particles in the sampling cells. Standard deviation, S, just represents the absolute degree of the departure from average concentration, C¯, which is inadequate in reflecting the mixing status. In order to describe the closeness of a mixture to a ‘completely random mixture’ reasonably, the well-known Lacey index [30] is widely used: M¼
r 20 −S 2 r 20 −r2r
ð14Þ
wherepσffiffiffiffiffiffiffiffiffiffiffiffiffi 0 is ffithe is the standard deviation before mixing, ¯ r0 ¼ Cð1− C¯Þ; σr is the standard deviation of tracer particleffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¯ ¯ n, CÞ=¯ concentration for a complete random mixture, rr ¼ Cð1− where ¯n is the average number of particles contained in a sampling cell. The value of M is 0 for a completely segregated mixture, and 1 for a completely random mixture. Nevertheless, the Lacey index was reported to be insensitive in a small scale mixing system [32]. Hence, a substitute was suggested by [32]: A¼
logr0 −log S logr0 −logrr
ð15Þ
Both the Lacey index and the Ashton index are calculated to elucidate the dynamic mixing process in a fluidized bed in this study.
3.1. Statistical mixing index 3.2. Application of mixing index in solids mixing in FBs According to Lacey [30], a completely random mixture is one in which the probability of finding a particle of a constituent in every point is identical. When the components of the mixture are completely segregated, it can be assumed that any sample withdrawn from the mixture will be entirely composed of a pure component. Various mixing indexes are applied to describing the effectiveness of different mixers in the process industry [31]. Most of these indexes are developed based on statistical analysis and especially on the definitions of
As we know, during the DEM simulation, an individual particle is tracked, so the location and velocity of every particle in the bed can be extracted at any time, which is the very information for calculating the mixing indexes mentioned above. Furthermore, continuous and non-intrusive data acquirement is another remarkable advantage over experimental methods. A sampling cell is defined by dividing the fluidized column into 4 × 15 subzones, and particles are counted based on their
F. Tian et al. / Fuel Processing Technology 88 (2007) 187–198
Fig. 5. Sampling grid used in mixing index calculation.
center positions, see Fig. 5. The width and height of the sampling unit are 24 mm and 28 mm, respectively. It should be noted here that if there is no particle in a sampling cell, Ci is not available and that if there are only a few particles in a sampling cell, the error in the standard deviation value is large. Accordingly, only the sampling cells where more than 20 particles are contained are chosen to calculate the mixing indexes. Fortunately, even with such limitation, up to 94% of particles are included throughout the simulation. Obviously, such sampling is considered to enable the overall picture of mixing in the bed to be revealed. By the use of DEM simulation, particles in the bed are almost equally separated into two assemblies and are labeled with individual color by the procedure mentioned in Section 2.4. The black colored particles are designated as tracer particles. Information about the mixing indexes was collected every 0.01 s. 3.3. Results and discussion
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from a completely segregated mixture to a completely random mixture in the bed as the time going on, with their values increasing from 0 to 1. On the other hand, careful comparison of these curves tells us that the two curves take on different developing characters, except for the appreciable coincidence during the earlier period of about 1.0 s. Obviously, the curve of the Lacey index develops faster and more smoothly than that of the Ashton index. The moment the homogenous distribution of the trace particles occurs is t = 4.0 s and t = 5.0 s more or less, for the curve of Lacey index and that of Ashton index respectively, which is explicated using dashed lines in Fig. 6. A review of the snapshots of t = 3.0–7.0 s in Fig. 4 reveals that t = 5.0 s is more convincing. Additionally, the smooth Lacey index curve makes an impression that the complete random mixture seems to be in a stagnant state. Actually, deviations from the completely random mixture, due to coalescence of neighboring tracer particles, are unavoidable in a practical mixing process. Sharp fluctuations in the upper Ashton index curve vividly embody such dynamic characteristics of the mixing balance. In other words, the Ashton index offers more precise prediction for the mixing evolvement, which agrees well with the analysis of [32]. Here, it needs to be pointed out that the fluctuation magnitude is considered to be within permission, with the summit value being 1.0838 and time-averaged value equal to 0.9908 between t = 5.0 s and 10.0 s. Since the Ashton index, A, appears to be more effective in describing the mixing behavior in the present work, the following discussions are mainly based on this parameter. 3.3.2. Effect of gas distribution Many efforts have been done to improve the comprehensive mass transfer in fluidized beds, among which inner structures and uneven air supply are commonly employed in industry reactors [33–35]. Results of these investigations enrich the understanding about the inner gas–solid flow pattern of the beds. Nevertheless, because of the complex interaction of the particles in the bed, few discussions, in the writers' knowledge,
3.3.1. Comparison between Lacey and Ashton indexes Fig. 6 exhibits the curves of Lacey index and Ashton index as functions of mixing time, sharing the same simulation conditions mentioned in Section 2.4. Generally, both curves reflect the process
Fig. 6. Comparison between Lacey index (M ) and Ashton index (A).
Fig. 7. Effect of gas distribution on mixing process, horizontal distributor & even gas supply and inclined distributor & uneven gas supply. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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were made about the mixing mechanism on individual particle level in detail, which is thought to be very important, especially for chemical engineering, following work attempts to elucidate the micro-process of the mixing in fluidized beds operated under uneven gas distribution. Curves of the Ashton index in two simulation cases are displayed in Fig. 7. The black colored curve represents mixing evolvement under condition of inclined distributor incorporated with uneven gas supply, the average air velocity in the left half being 2.5 umf and that in the right 1.5 umf (see Fig. 1). The red colored curve repeats the simulation in Section 2.4 for the sake of further comparison. It is evident that these two curves are quite different. It takes only about 1.5 s for the black one to reach A = 0.9, or a well mixing state. Whereas, for the other the durance is about 4 s, which is marked by a corresponding colored dashed line in Fig. 7. Namely, the case of inclined distributor associated with nonuniform air feed enjoys a much more faster mixing rate than that of horizontal distributor together with uniform air feed does. Deeper observations of the black colored curve manifest that the mixing development using inclined distributor experiences a period, 2.5 s approximately, of small scale vibration between t = 1.5 s and t = 4 s, which is
somewhat invisible for its counterpart having horizontal air distribution plate. To find out what is happening behind the unlikeness between the two conditions, attention falls back on the visual snapshots gathered during the mixing course. 3.3.3. Hydrodynamic analysis Snapshots along the mixing time in Fig. 8 directly illustrate the evolvement of solids mixing in the fluidized bed using inclined distributor. The superficial gas velocity in the left and right half equals to 2.5 umf and 1.5 umf, respectively. Fig. 8 discloses that a bubble causes a drift of particles to be drawn up as a spout below it. Particles appear to be pulled into the wake and drift, carried up the bed for a distance, and then shed. New particles are entrained from the emulsion phase when particles in the trailing vortex drop back down. With higher gas velocity, the left side is a bubble active zone, where there is an ascending solids flow consequently. Because of the gradient in particle concentration between the two sides, particles in the right side move to the left. The design of inclined distributor helps to enhance such a descending flow over the distributor as a result of gravity. New arrivals in the left half (the bubble active zone) become entrained upward again. Subsequently, there is a
Fig. 8. Mixing process in the case of inclined gas distributor with uneven air supply.
F. Tian et al. / Fuel Processing Technology 88 (2007) 187–198
primary descending stream of solids in the right side of the bed, where there are no bubbles. So that, an overall convective circulation is setup, particle velocity field in Fig. 9(b) makes these abstract descriptions more live. Solids mixing in a freely bubbling fluidized bed is caused not only by the vertical movement of bubbles and bursting of bubbles at the bed surface, but also by the lateral motion of bubbles as a result of the interaction and coalescence of neighboring bubbles. The lateral mixing of solids is augmented by the lateral motion of bubbles. As hinted by Fig. 9(a), due to the uniform activity of bubbles in width of the bed with even air supply, convective mixing is localized. By contrast, visual experiment [36] manifests that there is a transverse movement of bubbles in internally circulating fluidized beds, which is considered to be a main factor contributing to particle circulation in internally circulating fluidized beds. For the intention to validate previous discussions, solid flux at the lower part of the central line (see Fig. 9(c), (d)) is also monitored during the computation, having a positive value leftward. Fluctuations of solids flux in Fig. 10 reveal the turbulent bubbles movement, indirectly. Remarkably, there is a substantial difference between the time-averaged solid fluxes of the two cases discussed above. For inclined distributor with uneven gas supply, the time-averaged solid flux value is up to 123.804 kg/(m2 s), which implies that a regular particle circulation does exist in the bed. Whereas, in the case of even gas supply, the value is 1.308. Apparently, the relatively small value indicates that lateral convection between the subzones, left and right half of the bed, seems feeble under such a condition. Up to now, it is evident that: (i) the large scale solids circulation in the fluidized bed using inclined distributor incorporated with uneven gas supply leads to a faster convective mixing process, (ii) bubbles transverse movement is restricted because of the uniform distribution of the air for the case of horizontal
Fig. 9. Vector field of particle velocity at t = 1.0 s. (a) Horizontal distributor; (b) inclined distributor; (c), (d) are the schematics during the solid flux calculation for horizontal and inclined distributor, respectively (unit: mm).
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Fig. 10. Solid flux curves as a function of mixing time, horizontal distributor & even gas supply and inclined distributor & uneven gas supply.
air distribution plate, and the lateral convective mass transfer is reduced notably. 3.3.4. Discussion The advent of bubbles symbolizes the essential transition from packed beds to fluidized beds. Bubble hydrodynamics plays a fundamental role in fluidization engineering. To much extent, bubbles' parameters (such as size, velocity, frequency, etc) and their behaviors (such as generation, enlargement, breakage, coalescence, eruption, vertical and lateral movement, distribution, etc.) determine the particle mixing pattern, and, sequentially, the performance of mass transfer, heat exchange and chemical reactions in the beds [37,38]. On a macroscopical view, localized air supply induces the density gradient of particle distribution in the bed, which is the basic agent of convective particle stream. In a homogenous system, diffusive effect is a natural matter. Based on an overall review of the above paragraphs, the mixing process in fluidized beds mainly involves the following three categories. (1) Convective mixing: under the effect of external forces, particle materials become flexible and move in batches from their original locations to others (see snapshots at t = 0.2–1.0 s in Fig. 8), resulting in rapid mixing within the whole system. As far as the mixing phenomena in fluidized beds are concerned, such forces mainly include drag force from the gas phase, interactions among particles and sole weight of particles. (2) Diffusive mixing: particles of segregated components creep in the vicinity of their interfaces, and penetrate into each other slowly in a diffusive manner (see snapshots at t = 0.5–2.0 s in Fig. 4). During such a kind of mixing process, mixtures in local area, where diffusion occurs, achieve a relatively uniform density distribution of different components, which is more clearly illustrated in a snapshot at t = 2.0 s in Fig. 4.
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Table 3 Comparison between literature [39] and present simulation Parameter
Literature [39]
Present simulation
80 mm (depth) × 600 mm (width) × 1200 mm (height), 2D
4 mm (depth) × 96 mm (width) × 420 mm (height), 2D 1500 kg/m3, black, 4.0 mm in diameter 1500 kg/m3, green, 4.0 mm in diameter 190 mm
Sketch
many nearly unmixed clusters composed of identical component particles, which are highlighted with dotted circles in snapshots at t = 1.3 s, 1.5 s in Fig. 8. Such a local segregated state leads to a high standard deviation value, and, consequently, the vibrating mixing index after primary mixing is finished, at t = 1.5 s approximately. Obviously, according to the mechanism of shear mixing (relative slipping movement among local particles clusters), the rest of the task is left to it. The other way around, the slow diffusive mixing pattern in case of even air supply provides a gradual change in the distribution of trace particles, and, then, a smoother mixing index curve before t = 4.0 s in Fig. 7. 4. Comparison with reported experiment
Bed configuration
Bed materials Tracer particles Static bed Surface height Air velocity
Polypropylene, white, 3.0 mm in diameter Polypropylene, dyed black, 3.0 mm in diameter 300 mm 5.4 umf in both sides of the bed 1.8 umf at the center of the bed
2.5 umf in the lower part of the distributor 1.5 umf in the higher part of the distributor
(3) Shear mixing: relative slipping movement among various particles clusters is the major mechanism for shear mixing, which contributes to local mixing performance. Strictly speaking, all these three mixing patterns are concurrent in practical applications, with some prominent, which is also confirmed by Figs. 7 and 8. The fluctuation of the Ashton mixing index curve between t = 1.5 s and t = 4.0 s more or less, under the condition of uneven air distribution, actually reflects the shear mixing process. Although uneven air distribution brings about faster convective particle flow, the mixing quality is rude, leaving
Due to the instantaneously changing gas–solid flow structure, continuous sampling at second-level is still quite a challenge for the time being. 2D visualization experiments appear to be more feasible to study the micro-mixing process in fluidized beds, through image analysis technology. Tian [39] investigated the diffusive character of tracer particles in an internally circulating fluidized bed with CCD snapshots. Table 3 details the major parameters used in both literature [39] and this work. Data in Table 3 suggest that the two researches share nearly identical principles, including the 2D bed structure, the application of inclined air distributor, uneven gas feed, D-group particles, etc. From this point of view, present simulation results are comparable with those of a published experiment [39]. Ref. [39] found that there were two internal tracer particles stream over the ‘/\’ type air distributor, clockwise in the left half of the bed and anticlockwise right. Fig. 11 lists the continuous snapshots in the left half, which is thought to be similar to the case of inclined distributor in this paper. Fig. 11 fully embodies the pronounced convective mixing pattern under uneven gas feed, as predicted in this DEM simulation. Moreover, different mixing patterns, analyzed in Section 3.3.4, can also found their proofs among the snapshots in Fig. 11. For example, convective mixing t = 0.4–1.6 s, and shear mixing for t = 2.0–3.2 s in Fig. 11, with
Fig. 11. The diffusive process of tracer particles in internally circulating fluidized bed [39].
F. Tian et al. / Fuel Processing Technology 88 (2007) 187–198
their counterpart of t = 0.4–1.5 s, and t = 1.5–3.0 s more or less in Fig. 8, respectively. The degree of the mixing process can also be clearly recognized along the time-series snapshots in Fig. 11. In terms of the quantitative description, the mixing is relatively developed at a time around t = 3.0 s, which is 2.0 s in the present calculation. Parameters in Table 3 give a reason for the difference between experimental and predicted mixing time scales. On one hand, approximately, the simulation and a reported experiment have the same lateral velocity gradient, with 10.4 umf /m ((2.5–1.5) / (96 × 10− 3)) for the simulation and 12 umf /m ((5.4–1.8) / (600 × 10− 3 / 2)) for the experiment. On the other hand, the static bed surface in the present simulation, 190 mm, is lower than that of the experimental bed, 300 mm. As a result, with the nearly equal velocity gradient, the tracer particles in the simulated fluidized bed will experience a smaller scale circulation, achieving homogeneous distribution about 1 s earlier than those in the experimental setup. Comparisons demonstrate that this work reasonably reflects the complicated mixing phenomena in a fluidized bed, both qualitatively and quantitatively. The mixing time scale, evaluated with the Ashton mixing index, deserves its practical meaning. 5. Conclusions This paper makes full use of abundant particulate information in DEM simulation, and introduces Lacey and Ashton indexes to quantitatively evaluate the mixing characteristics of fluidized beds with uniform particles. Compared with the Lacey index, the Ashton index predicts the mixing mode more precisely. Ashton index curves illustrate that: in fluidized bed using inclined distributor together with uneven gas feed, rapid convective mixing process occurs at early stage, followed by shear mixing; for the case of horizontal distributor incorporated with even gas supply, the relatively smooth curve implies that diffusive mixing is the controlling factor in such a bed. Hydrodynamic information from the snapshots argues that different behaviors of bubbles movement cause the unlikeness between the two cases (uneven and even gas injection). Solid flux suggests there is a large scale regular circulating stream between the bubbles active and inactive subzones under the condition of nonuniform gas supply; however, in the case of uniform gas supply, particles lateral activities are reduced, since bubbles horizontal motion is confined to a local area. The mixing patterns and the mixing time scale predicted by this work are in good agreement with reported experiment. Nomenclature A Ashton index, dimensionless M Lacey index, dimensionless C true value of the tracer particles' concentration, dimensionless n¯ average number of particles contained in sampling units average concentration of tracer particles in sampling C¯ cells, dimensionless Y nij normal unit vector Ci concentration of tracer particles in the sampling cell i,
N CD p dp rp Y F cnij S Y F ctij t Y Fg Y t ij g umf I Y ug k Y v m Y vij
197
dimensionless number of sampling cell drag coefficient, dimensionless gas pressure, Pa particle diameter, m particle radius, m is the normal force, N standard deviation when N is finite tangential force, N time, s particle–fluid interaction force, N tangential unit vector 2 gravitational acceleration, m s− 1 minimum fluidization velocity, m s− moment of inertia; kg m2 1 gas velocity, m s− 1 particle stiffness, N m− particle velocity, m s−1 particle mass, kg relative velocity between colliding particles, m
Greek letter β inter-phase momentum transfer coefficient, kg m− 3 s− 1 ρg gas density, kg m− 3 δn normal deformation displacement, m σ standard deviation when N is infinite δt tangential deformation displacement, m σ0 standard deviation of the completely segregated states, dimensionless ε voidage, dimensionless σr standard deviation of fully randomized states, dimensionless η damping coefficient, dimensionless τ viscous stress tensor, kg m− 1 s− 2 μf friction coefficient, dimensionless Y ϖ angular velocity of particle, s− 1 References [1] K. Laursen, J.R. Grace, Fuel Processing Technology 76 (2002) 77–89. [2] Rahmat Sotudeh-Gharebagh, Navid Mostoufi, Fuel Processing Technology 85 (2003) 189–200. [3] U. Arena, A. Cammarota, L. Pistone, Proc. 1st Int. Conf. Circulating Fluidized Bed Technology, Pergamon, 1986, pp. 119–126. [4] M.P. Martin, P. Turlier, J.R. Bernard, G. Wild, Powder Technology 70 (1992) 249–258. [5] U. Mann, E.J. Crosby, Industrial & Engineering Chemistry Process Design and Development 16 (1977) 9. [6] R. Bader, J. Findlay, T. Knowiton, Proc. 2nd Int. Conf. Circulating Fluidized Bed Technology, Pergamon, Oxford, 1988, pp. 123–138. [7] M. Yinneskis, Powder Technology 49 (1987) 26–269. [8] M. Ishida, T. Shirai, A. Nishiwaki, Powder Technology 27 (1980) 1–6. [9] J.J. Nieuwlan, R. Mejer, J.A.M. Kuipers, W.P.M. van Swaajj, Powder Technology 87 (1996) 27–139. [10] Hisashi Umekawa, Mamoru Ozawa, Nobuyuki Takenaka, et al., Nuclear Instruments & Methods in Physics Research. Section A, Accelerators, Spectrometers, Detectors and Associated Equipment 424 (1999) 77–83. [11] A. Samuelsberg, B.H. Hjertager, International Journal of Multiphase Flow 22 (1996) 575–591.
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[12] E. Helland, R. Occelli, L. Tadrist, International Journal of Multiphase Flow 28 (2002) 199–223. [13] B.P.B. Hoomans, J.A.M. Kuipers, W.J. Briels, W.P.M. van Swaaij, Chemical Engineering Science 51 (1996) 99–118. [14] Matteo Chiesa, Vidar Mathiesen, Jens A. Melheim, et al., Computers and Chemical Engineering 29 (2005) 291–304. [15] J.A.M. Kuipers, K.J. van Duin, F.P.H. V.A.N. Beckum, W.P.M. van Swaaij, Chemical Engineering Science 47 (1992) 1913–1924. [16] R. Smith, K. Beardmore, A. Gras-Mati, Vacuum 46 (1995) 1195–1199. [17] M.A. Hopkins, H. Shen, Journal of Fluid Mechanics 244 (1992) 477–491. [18] P.A. Cundall, O.D.L. Strack, Geotechnique 29 (1979) 47–65. [19] Y. Tsuji, T. Kawaguchi, T. Tanaka, Powder Technology 77 (1993) 79–87. [20] B.H. Xu, A.B Yu, Chemical Engineering Science 52 (1997) 2785–2809. [21] Jie Ouyang, Jinghai Li, Chemical Engineering Science 54 (1999) 2077–2083. [22] Sunun Limtrakul, Ativuth Chalermwattanatai, Kosol Unggurawirote, et al., Chemical Engineering Science 58 (2003) 915–921. [23] M.J. Rhodes, X.S. Wang, M. Nguyen, et al., Chemical Engineering Science 56 (2001) 2859–2866. [24] T.B. Anderson, R. Jackson, I and EC Fundamentals 6 (1967) 527–539. [25] Yasunobu Kaneko, Takeo Shiojima, Masayuki Horio, Chemical Engineering Science 54 (1999) 5809–5821. [26] Thomas Malkow, Waste Management 24 (2004) 53–79. [27] Wen-Ching Yang, Dale L. Keairns, Powder Technology 111 (2000) 168–174.
[28] S.V. Pantakar, Numerical Heat Transfer and Fluid Flow, Hemisphere, New York, 1980. [29] C.Y. Wen, Y.H. Yu, Chemical Engineering Progress Symposium Series 62 (5) (1966) 100–105. [30] P.M.C. Lacey, Developments in the theory of particle mixing, Journal of Applied Chemistry 4 (1954) 257. [31] M. Poux, P. Fayolle, J. Bertrand, D. Bridoux, J. Bousquet, Powder Technology 68 (1991) 213–234. [32] M.D. Ashton, F.H.H. Valentin, Transactions of the Institution of Chemical Engineers 44 (1966) T66. [33] J.M.D. Merry, et al., Transactions of the Institution of Chemical Engineers 51 (4) (1973) 361–368. [34] T. Ohshita, T. Higo, S. Kosugi, et al., Heat-Transfer Japanese Research 23 (4) (1994) 349–363. [35] Chun-Hua Luo, Kenichi Aoki, Shigeyuki Uemiya, et al., Fuel Processing Technology 55 (1998) 193–218. [36] Y. Bin, M.C. Zhang, B.L. Dou, et al., Industrial & Engineering Chemistry Research 42 (2003) 214–221. [37] P.N. Rowe, B.A. Partridge, A.G. Cheney, G.A. Henwood, E. Lyall, Transactions of the Institution of Chemical Engineers 43 (1965) T271. [38] H.V. Nguyen, A.B. Whitehead, O.E. Potter, AIChE Journal 23 (1977) 913. [39] W.D. Tian, X.L. Wei, M.D. Sun, et al., Mechanics and Engineering 23 (3) (2001) 35–38 (in Chinese).
Numerical study on microscopic mixing characteristics in f luidized beds via DEM Fengguo Tian ⁎, Mingchuan Zhang, Haojie Fan, Mingyan Gu, Lei Wang, Yongfeng Qi Institute of Thermal Energy Engineering, Shanghai Jiaotong University, Minhang, Shanghai 200240, People's Republic of China Received 26 September 2005; received in revised form 27 March 2006; accepted 7 April 2006
Abstract In this paper, discrete element method (DEM), combined with computational fluid dynamics (CFD), is used to investigate the micro-mixing process in fluidized beds (FBs) of uniform particles. With the aid of snapshots and adoption of Lacey and Ashton indexes, mixing evolvement for two cases, fluidized bed using horizontal distributor with even gas supply and fluidized bed using inclined distributor with uneven gas supply, is discussed in detail. Results indicate that the Ashton index appears to be more effective in assessing the mixing dynamics in this work. Further analyses illustrate that in the case of horizontal distributor incorporated with even gas supply, diffusive mixing pattern is predominant, since bubbles lateral motion is reduced in such a bed; whereas, there is a faster convective mixing process in a fluidized bed using inclined distributor with uneven gas feed, followed by shear mixing. Generally, localized air supply induces the density gradient of particle distribution in the bed, which is the basic agent of convective particle stream. The analyses are confirmed by the comparison of solid flux during the simulations of the two cases. In addition, the mixing mechanism and the mixing time scale agree well with published experimental results. © 2006 Elsevier B.V. All rights reserved. Keywords: Fluidized bed; Mixing; Simulation; Discrete element method (DEM)
1. Introduction For their good adaptability and prominent processing ability, fluidized beds (FBs) are popular in many industrial fields, including energy [1], environment, chemical engineering [2], etc. Solids mixing process plays an important role in the design and operation of fluidized bed, e.g., how the fresh materials blend with previous background particles, knowledge about the residence time distribution (RTD) of the feed, the spatial differences of heat transfer and chemical reactions in the bed. Various experimental methods have been explored in the gas– solid flow research area, which are crucial for comprehensive understanding about the fluidization hydrodynamics. Arena and Cammarota [3] and Martin et al. [4] implemented direct measurement of the solids concentration, employing quick closing valves with which the column of interest can be portioned in sections of suitable length. Mann and Crosby [5] developed a strain gauge to measure local particle concentrations ⁎ Corresponding author. Tel.: +86 21 3420 5696; fax: +86 21 3420 6115. E-mail address: [email protected] (F. Tian). 0378-3820/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fuproc.2006.04.006
and velocities in gas–solid flows, deduced from the impact frequency with surrounding particles and the deformation magnitude of the probe nose. By withdrawing particles from gas–solid two-phase flow with extraction probes, Bader [6] evaluated the local mass flow rate of the particles. Laser Doppler velocimetry (LDV) is based on the detection of optical interference phenomena, arising from two interacting laser beams [7]. Ishida et al. [8] developed reflective optical probe systems which enable measurement of velocities of bubbles and individual particles in bubbling fluidized bed. However, some of them are intrusive, e.g., quick closing valves, extraction probes and strain gauge, which alter the flow pattern in the bed. Others are confined to strict occasions, due to the property of gas–solid system or to the complicated procedure of the experiment. For example, LDV technique doesn't work in dense suspensions [9]. While, calibration for the nonlinear transform in optical experiments should be carefully treated. Although the introduction of radioactive tagging and X-ray methods offers the possibility of measuring the solids velocity directly [10], they have not found widespread application due to the considerable cost of the equipment. Moreover, most available information
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provides only the overall characterizations about the flow pattern and mixing phenomena. Systemic discussions engaged in the microscopic mixing phenomena of dense gas–solid flows, in temporal and spatial details, are believed unavailable. Owe to increasing computer power, computational fluid dynamics (CFD) has become a versatile tool in studying the gas–solid flow pattern [11]. Two-fluid model (TFM), also referred as Eulerian–Eulerian (EE) model, has been frequently used to describe fluid dynamic phenomena in fluidized beds [12]. TFM supposes that the gas and the solid phase are considered to be continuous and fully inter-penetrating. Both phases are described in terms of separate sets of conservation equations with appropriate interaction terms representing the coupling between the phases. Though a physical characteristic of the solid particles such as the shape and size is included in the continuum representation through the empirical relations for the interfacial friction, these models do not recognize the discrete character of the solid phase and are farfetched for dense gas–solid flow [13,14]. Also, in the two-fluid approach, assumptions need to be made concerning the solids rheology, where often Newtonian behavior is assumed in the absence of a more detailed knowledge [15]. Free of the continuum hypothesis for the solid phase, discrete element method emphasizes the subtle details of individual moving body in particulate assemblies. The origins of DEM technique actually lie in the field of Molecular Dynamics (MD) [16]. As a development, Monte Carlo (MC) techniques make use of the random nature of the molecular motion in gases or dilute gas–solid phase to approach the problem from a statistical point of view [17]. In the Monte Carlo method, it is assumed that the velocities of individual particles are independently distributed within a velocity distribution function without regard to history or to the behavior of neighboring particles. It is unlikely that this is true in dense concentrations where particles will experience many collisions with their nearest neighbors and it is therefore unlikely that there would be no correlation between their velocities or positions. Numerical simulation of dense-phase discrete particle systems was first reported by Cundall and Strack [18]. In their simulations, it was postulated that particulate flow occurs in the absence of an interstitial fluid. Attracted by the discrete treatment of the particle phase in DEM, Tsuji et al. [19] introduced such concept into the simulation of dense gas–solid flow in fluidized bed, combined with traditional CFD, also referred as Eulerian–Lagrangian (EL) model. The DEM implement is constrained by a large computational cost associated with detection of inter-particle contact. Whereas, such kind of numerical method enables us to examine data which are normally inaccessible, and provide substantial information in a particulate system. With regard to DEM investigations engaged in two-phase flow, Tsuji et al. [19], Hoomans et al. [13], Xu and Yu [20], and Ouyang and Li [21] reasonably predicted the typical behaviors in fluidized beds. Limtrakul et al. [22] paid attention to the time-averaged axial velocity distribution of the tracer particles. Rhodes [23] analyzed the effects of gas velocity, particle density, and particle diameter on mixing rate by means of Lacey mixing index. To date, the transient description of the mixing process in fluidized beds is still limited.
In this paper, a comparison is made between the Lacey and Ashton mixing indexes, with respect to their sensitivity in characterizing the particles mixing process. For its better effectiveness in evaluating the mixing development, the Ashton index is introduced to study the influences of gas supply pattern on the micro-mixing mechanism in a fluidized bed, incorporated with analysis of tracer particles' fields. The simulations are also compared with previous experimental results, which suggests that this work is supposed to give satisfactory predictions. The paper ends with a detailed discussion about the mixing dynamics in the fluidized column, which is thought as still absent in relevant literatures. 2. Simulation of mixing process in FBs 2.1. Mathematical model Discrete element method (DEM) is a relatively new but now widely available tool for researching into gas–particle systems. Particles are traced individually by direct solution of Newton's equations of motion in the simulation. When no collisions occur, the forces acting on the particles are mainly from the gas phase and gravity. The motion of the particles is described by m
dY v Y ¼ F g þ mg dt
ð1Þ
where Y v is the particle velocity; the particle–fluid interaction Y force F g , can be written as: Y F g ¼ Vp
b Y Y ð ug − v Þ−jp 1−e
ð2Þ
ug is the gas velocity. The where Vp is the volume of a particle; Y inter-phase momentum transfer coefficient, β depends on the void fraction. The well-known Ergun equation is used for the packed bed region, ε ≤ 0.8. When ε N 0.8, and the particle's motion is taken to be only weakly affected by other particles, and a modified Wen–Yu equation of the fluid resistance for a single particle is used for the dilute region. The expressions of β are as follows: 1−e ð1−eÞl b ¼ 2 150 þ 1:75qejY ug − Y vj ðe V 0:8Þ ð3aÞ e dp qð1−eÞjY ug − Y v j −2:65 3 b ¼ CD e 4 dp
ðe N 0:8Þ
ð3bÞ
where CD is the drag coefficient, and can be written as: CD ¼
24ð1 þ 0:15Re0:687 Þ=Re 0:43
ðRe b 1000Þ ðRe z 1000Þ
ð4Þ
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 collisions
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between particles and between a particle and a wall are simulated by Hooke's linear springs and dashpots [18]. Governing equations can be expressed as: Y vij ÞY nij F cnij ¼ ð−kdn − gY Y X dϖ Y ¼ I F ctijd rp dt Y Y Y F ctij ¼ ð−kdt − gY vijd Y t ij ; j F ctij jblf j F cnij j t ij ÞY Y Y Y Y Y j F ctij jzlf j F cnij j F ctij ¼ −lf j F cnij j t ij ;
ð5Þ ð6Þ
Table 2 Parameters for gas phase Density Viscosity Number of cells Cell width Cell height
1.205 kg/m3 1.82 × 10− 5 N s/m2 10 × 29 12 mm 12 mm (lower) 20 mm (upper)
ð7Þ
Y Y where F cnij is the normal force; F ctij is the tangential force; I is the moment of inertia. The motion of the gas phase is calculated from the volume averaged gas phase governing equations as put forward by Anderson [24]. Equations for gas mass and momentum conservation in vector form are given by: Aðqg eÞ þ jd ðqg eY ug Þ ¼ 0 At Aðqg eY ug Þ þjd ðqg eY ug Y ug −Y vÞ ug Þ ¼ −ejp−bðY At −jd ðesÞ þ qg eg
189
ð8Þ ð9Þ
v Þ represents the inter-phase In Eq. (9), the term −bðY ug −Y momentum transfer between the gas phase and each individual particle. Here, β is same as defined in Eqs. (3a) and (3b). 2.2. Computational conditions
Current numerical experiments are dedicated to comprehending the mixing behavior of monosize particles in a fluidized setup, which is difficult to implement in a laboratory. In order to make the computation time bearable, a smaller spring is often assigned. Discussions [25] revealed that such assumption does not lead to a larger difference in a gas–solid pattern. In this work, spring coefficient is set as 2000 N/m. As for the damping coefficient, restitution coefficient and friction coefficient, they are often determined on the base of experiences or energy loss test with DEM simulation [20]. Interestingly, these parameters are quite close to each other correspondingly in reported works [13,19,20]. Computation conditions and physical parameters used are given in Tables 1 and 2. The initial field is setup by randomly allocating particles in the bed and then allowing the particles to fall freely under gravity in the absence of a gas flow. In the present paper, two-dimensional (2D) beds of 420 mm in width and 96 mm in height are used. There are two gas inlets at the bottom of the air distributor, with their size and position shown in Fig. 1. The air velocity mentioned later is the average value, with mass flow of each gas inlet responsible for its
corresponding half (left and right). The thickness of the beds is the same as the single particle diameter. Meanwhile, in order to study the influence of the air supply and distributor on the mixing character in the bed, simulations are conducted with horizontal and inclined distributor successively. For their agreeable ability in enhancing the diffusion pattern of bed materials, inclined air distributors are widely employed, including the municipal solid waste (MSW) incinerator [26], the cone-shaped gas distributor in fluidized bed gasifiers [27] and fluidized bed driers for agricultural purposes. To make the present work more practically meaningful, the angle of the inclined distributor is equal to 18°, comparable to that in industrial applications. Step grid arrangement at the bottom is adopted for the inclined distributor. 2.3. Simulation description For gas motion, the local averaged Navier–Stokes equations [24] are discretized in finite volume form on a staggered grid. The Crank–Nicholson scheme is used for the time discretization, the power-law scheme for the convective term and a central differences scheme for diffusion terms. And, the conventional SIMPLE algorithm [28] is adopted to solve the equations for the fluid phase. In the computation, the Newtonian equations of motion of individual particles are integrated with fourth-order Runge–Kutta algorithm. The inter-particle force models discussed in Section 2.1 are also applied to the collision between a particle and wall, with the corresponding wall properties used during the calculation. For the fluid velocity, the no-slip
Table 1 Particle parameters Particle diameter Particle density Number of particles Spring coefficient
4 mm 1500 kg/m3 1000 2000 N/m
Damping coefficient Restitution coefficient Friction coefficient Time step
0.17 0.9 0.3 1.0 × 10− 5 s
Fig. 1. Schematic of the object simulated (unit: mm).
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boundary condition applies to the left, bottom and right walls and zero normal gradient condition to the top exit. The computational flowchart of global algorithm description of mixing process is illustrated in Fig. 2. The solution for each time step, with the voidage and particle velocity field from the discrete particle scheme, evolves through a series of computational cycles consisting of (i) explicit calculations of fluid velocity components for all interior computational cells and (ii) implicit determination of pressure distributions using an iterative procedure. For each time step the velocity and position of each particle are refreshed using the equation of motion and the forces acting on the particle. Fig. 3 outlines the procedure in solving particle movement. In the implicit stage of gas motion solving, mass residuals are calculated according to the fluid continuity equation. If a residual deviates from a prescribed convergence criterion for all the interior computational fluid cells, a whole field pressure correction is applied. New densities are calculated and new fluid velocities obtained from the momentum equation and, using recomputed mass residuals, the convergence criterion is reche-
Fig. 3. Sub-flowchart of particle motion.
cked. This sequence continues until convergence is attained or an iteration limit is reached. In the discrete element model, the size of the time step is critical to the calculations since it determines the stability and accuracy. The oscillation period of the spring–mass system used to model contacting particles is determined as [19]: Dt ¼ 2p
Fig. 2. Main flowchart of DEM simulation.
pffiffiffiffiffiffiffiffiffi m=k
ð10Þ
It is found that the calculation became unstable when the time step is near the limit value given by this equation. In the current simulation, the time step is 1/100 of the characteristic time scales of the spring constant as recommended by Eq. (10) to meet the required accuracy. In this DEM program, Verlet (or neighbor) list is employed for inter-particle contact detection. The neighbor lists are updated periodically to keep up with the motion of the particles inside the simulation domain. The interval for such updating is controlled by the size of the neighborhood, accounting for the
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highest possible velocity of the particles in the system. During the implementation, let vmax,n denote the maximum moving velocity of all particles for the integration time step tn. Assume ti is the time step for the last update of the neighbor lists. The next moment for the neighbor lists to be updated again, tj, is determined as: j X
1 jvmax;n jDt z rnb 2 n¼i
ð11Þ
where rnb is the radius of the neighborhood. A typical 1 s realtime simulation conducted on the PC (CPU 2.0 GHz, RAM 526 M) takes about 4 h to complete. 2.4. Mixing process in the bed This paper is interested in the mixing pattern within the bubbling regime of a fluidized bed. According to experimental
191
observation, the average gas velocities lie among 1.5 umf and 2.5 umf. Instantaneous snapshots of particle distribution in Fig. 4 visually reproduce the development of solids flow in the bed. In this case, horizontal distributor and even gas supply are employed, with the superficial gas velocity being 2 umf, with umf = 1.289 m/s according to Wen and Yu [29]. At the early stage, the bed surface expands from fixed bed and reaches a basic stable surface during the fluidization. Initiated at the orifice with small size, bubbles ascend through the bed, break up or coalesce with their neighbors turbulently, and finally burst at the bed surface, due to the pressure difference between inner space of the bubble and the freeboard. Bubbles eruption induces lateral dispersion of part of the wake's particles over a certain area, and the remainder of these is ejected into the freeboard. Because of bubbles' pronounced axial activities and the inherent direction of gravity, axial mixing potentials are predominant. Correspondingly, improving the lateral mixing
Fig. 4. Mixing process in the case of horizontal gas distributor with even air supply.
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performance in fluidized beds earns particular meaning, which is also the interest of the present research. With the goal mentioned above, the bed materials are renumbered into two equal parts in lateral direction, according to their position relative to the centerline in width at the beginning. The particles in the left half of the bed are tagged with a black color and the rest green. Obviously, the two groups of particles are completely segregated at the moment t = 0 s. As time proceeds, such a completely segregated state gradually gives way to one in which the particles of different colors are partially mixed. A dynamic equilibrium is established after about 5 s of mixing. For detailed interpretation, snapshots at t = 0.5, 1.0, 2.0 s indicate that material transfer between the two colored zones can be divided into three terms: (1) the shear effects on the interface of these zones; (2) dispersion in the upper freeboard; (3) convective mixing caused by bubble turbulence, which is thought to be important for mixing capability in the entire bed. In a bubbling fluidized bed, each bubble carries a wake of particles, which is ultimately deposited on the bed surface, and good mixing of solids occurs in the wake where movement is generated. This contributes partially to the lateral mixing of solids. Since fluidizing gas is evenly supplied, bubbles in the two sides share similar frequency and activity. Such symmetrical characteristic weakens the bubbles' lateral movement, resulting in poor convective effects. The discussion is also approved by information from the following snapshot (namely, t = 2.0 s in Fig. 4). It is clearly illuminated that after a mixing period of 2 s, the vicinity to the centerline enjoys a well mixed state of the two groups of particles, with high density of original host particles near the relevant wall. In other words, particles of the two particularly colored assemblies gradually penetrate into each other to complete the blending. It can be concluded that, in the case of horizontal distributor and even gas injection, diffusive effect is a controlling factor in particulate lateral mixing process. A time-series of particle field enriches our visual understanding about mixing phenomena, and deepens our insight into its mechanism. Meanwhile, the longer the mixing goes on, the more difficult the mixing extent to be assessed qualitatively, see snapshots of t = 4.0, 5.0, 6.0 s in Fig. 4. Therefore, an appropriate mathematical description is important for accurate and full comprehension of the mixing process. 3. Quantitative evaluation of the mixing degree
standard deviations of a specified property, which can be defined as [30]: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u1 X r¼t ðCi −CÞ2 ; N i¼1
C ¼ lim
N Yl
N 1X Ci N i¼1
! ð12Þ
where σ is standard deviation when the number of sampling cell, N, is infinite; Ci is the concentration of tracer particles in the sampling unit i; C the average value of Ci in N sampling cells. Since parameter C is an ideal limit (N is infinite), it is also referred to as the true value of the tracer particles' concentration. In practical measurement, infinite sampling units mean impossible. Consequently, for the finite sampling procedure, standard deviation is modified to: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N N u 1 X 1X S¼t C¯ ¼ ðCi − C¯ Þ2 ; Ci ð13Þ N −1 1 N i¼1 where S is the standard deviation when the number of sampling units is finite, C¯ is the average concentration of tracer particles in the sampling cells. Standard deviation, S, just represents the absolute degree of the departure from average concentration, C¯, which is inadequate in reflecting the mixing status. In order to describe the closeness of a mixture to a ‘completely random mixture’ reasonably, the well-known Lacey index [30] is widely used: M¼
r 20 −S 2 r 20 −r2r
ð14Þ
wherepσffiffiffiffiffiffiffiffiffiffiffiffiffi 0 is ffithe is the standard deviation before mixing, ¯ r0 ¼ Cð1− C¯Þ; σr is the standard deviation of tracer particleffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¯ ¯ n, CÞ=¯ concentration for a complete random mixture, rr ¼ Cð1− where ¯n is the average number of particles contained in a sampling cell. The value of M is 0 for a completely segregated mixture, and 1 for a completely random mixture. Nevertheless, the Lacey index was reported to be insensitive in a small scale mixing system [32]. Hence, a substitute was suggested by [32]: A¼
logr0 −log S logr0 −logrr
ð15Þ
Both the Lacey index and the Ashton index are calculated to elucidate the dynamic mixing process in a fluidized bed in this study.
3.1. Statistical mixing index 3.2. Application of mixing index in solids mixing in FBs According to Lacey [30], a completely random mixture is one in which the probability of finding a particle of a constituent in every point is identical. When the components of the mixture are completely segregated, it can be assumed that any sample withdrawn from the mixture will be entirely composed of a pure component. Various mixing indexes are applied to describing the effectiveness of different mixers in the process industry [31]. Most of these indexes are developed based on statistical analysis and especially on the definitions of
As we know, during the DEM simulation, an individual particle is tracked, so the location and velocity of every particle in the bed can be extracted at any time, which is the very information for calculating the mixing indexes mentioned above. Furthermore, continuous and non-intrusive data acquirement is another remarkable advantage over experimental methods. A sampling cell is defined by dividing the fluidized column into 4 × 15 subzones, and particles are counted based on their
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Fig. 5. Sampling grid used in mixing index calculation.
center positions, see Fig. 5. The width and height of the sampling unit are 24 mm and 28 mm, respectively. It should be noted here that if there is no particle in a sampling cell, Ci is not available and that if there are only a few particles in a sampling cell, the error in the standard deviation value is large. Accordingly, only the sampling cells where more than 20 particles are contained are chosen to calculate the mixing indexes. Fortunately, even with such limitation, up to 94% of particles are included throughout the simulation. Obviously, such sampling is considered to enable the overall picture of mixing in the bed to be revealed. By the use of DEM simulation, particles in the bed are almost equally separated into two assemblies and are labeled with individual color by the procedure mentioned in Section 2.4. The black colored particles are designated as tracer particles. Information about the mixing indexes was collected every 0.01 s. 3.3. Results and discussion
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from a completely segregated mixture to a completely random mixture in the bed as the time going on, with their values increasing from 0 to 1. On the other hand, careful comparison of these curves tells us that the two curves take on different developing characters, except for the appreciable coincidence during the earlier period of about 1.0 s. Obviously, the curve of the Lacey index develops faster and more smoothly than that of the Ashton index. The moment the homogenous distribution of the trace particles occurs is t = 4.0 s and t = 5.0 s more or less, for the curve of Lacey index and that of Ashton index respectively, which is explicated using dashed lines in Fig. 6. A review of the snapshots of t = 3.0–7.0 s in Fig. 4 reveals that t = 5.0 s is more convincing. Additionally, the smooth Lacey index curve makes an impression that the complete random mixture seems to be in a stagnant state. Actually, deviations from the completely random mixture, due to coalescence of neighboring tracer particles, are unavoidable in a practical mixing process. Sharp fluctuations in the upper Ashton index curve vividly embody such dynamic characteristics of the mixing balance. In other words, the Ashton index offers more precise prediction for the mixing evolvement, which agrees well with the analysis of [32]. Here, it needs to be pointed out that the fluctuation magnitude is considered to be within permission, with the summit value being 1.0838 and time-averaged value equal to 0.9908 between t = 5.0 s and 10.0 s. Since the Ashton index, A, appears to be more effective in describing the mixing behavior in the present work, the following discussions are mainly based on this parameter. 3.3.2. Effect of gas distribution Many efforts have been done to improve the comprehensive mass transfer in fluidized beds, among which inner structures and uneven air supply are commonly employed in industry reactors [33–35]. Results of these investigations enrich the understanding about the inner gas–solid flow pattern of the beds. Nevertheless, because of the complex interaction of the particles in the bed, few discussions, in the writers' knowledge,
3.3.1. Comparison between Lacey and Ashton indexes Fig. 6 exhibits the curves of Lacey index and Ashton index as functions of mixing time, sharing the same simulation conditions mentioned in Section 2.4. Generally, both curves reflect the process
Fig. 6. Comparison between Lacey index (M ) and Ashton index (A).
Fig. 7. Effect of gas distribution on mixing process, horizontal distributor & even gas supply and inclined distributor & uneven gas supply. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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were made about the mixing mechanism on individual particle level in detail, which is thought to be very important, especially for chemical engineering, following work attempts to elucidate the micro-process of the mixing in fluidized beds operated under uneven gas distribution. Curves of the Ashton index in two simulation cases are displayed in Fig. 7. The black colored curve represents mixing evolvement under condition of inclined distributor incorporated with uneven gas supply, the average air velocity in the left half being 2.5 umf and that in the right 1.5 umf (see Fig. 1). The red colored curve repeats the simulation in Section 2.4 for the sake of further comparison. It is evident that these two curves are quite different. It takes only about 1.5 s for the black one to reach A = 0.9, or a well mixing state. Whereas, for the other the durance is about 4 s, which is marked by a corresponding colored dashed line in Fig. 7. Namely, the case of inclined distributor associated with nonuniform air feed enjoys a much more faster mixing rate than that of horizontal distributor together with uniform air feed does. Deeper observations of the black colored curve manifest that the mixing development using inclined distributor experiences a period, 2.5 s approximately, of small scale vibration between t = 1.5 s and t = 4 s, which is
somewhat invisible for its counterpart having horizontal air distribution plate. To find out what is happening behind the unlikeness between the two conditions, attention falls back on the visual snapshots gathered during the mixing course. 3.3.3. Hydrodynamic analysis Snapshots along the mixing time in Fig. 8 directly illustrate the evolvement of solids mixing in the fluidized bed using inclined distributor. The superficial gas velocity in the left and right half equals to 2.5 umf and 1.5 umf, respectively. Fig. 8 discloses that a bubble causes a drift of particles to be drawn up as a spout below it. Particles appear to be pulled into the wake and drift, carried up the bed for a distance, and then shed. New particles are entrained from the emulsion phase when particles in the trailing vortex drop back down. With higher gas velocity, the left side is a bubble active zone, where there is an ascending solids flow consequently. Because of the gradient in particle concentration between the two sides, particles in the right side move to the left. The design of inclined distributor helps to enhance such a descending flow over the distributor as a result of gravity. New arrivals in the left half (the bubble active zone) become entrained upward again. Subsequently, there is a
Fig. 8. Mixing process in the case of inclined gas distributor with uneven air supply.
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primary descending stream of solids in the right side of the bed, where there are no bubbles. So that, an overall convective circulation is setup, particle velocity field in Fig. 9(b) makes these abstract descriptions more live. Solids mixing in a freely bubbling fluidized bed is caused not only by the vertical movement of bubbles and bursting of bubbles at the bed surface, but also by the lateral motion of bubbles as a result of the interaction and coalescence of neighboring bubbles. The lateral mixing of solids is augmented by the lateral motion of bubbles. As hinted by Fig. 9(a), due to the uniform activity of bubbles in width of the bed with even air supply, convective mixing is localized. By contrast, visual experiment [36] manifests that there is a transverse movement of bubbles in internally circulating fluidized beds, which is considered to be a main factor contributing to particle circulation in internally circulating fluidized beds. For the intention to validate previous discussions, solid flux at the lower part of the central line (see Fig. 9(c), (d)) is also monitored during the computation, having a positive value leftward. Fluctuations of solids flux in Fig. 10 reveal the turbulent bubbles movement, indirectly. Remarkably, there is a substantial difference between the time-averaged solid fluxes of the two cases discussed above. For inclined distributor with uneven gas supply, the time-averaged solid flux value is up to 123.804 kg/(m2 s), which implies that a regular particle circulation does exist in the bed. Whereas, in the case of even gas supply, the value is 1.308. Apparently, the relatively small value indicates that lateral convection between the subzones, left and right half of the bed, seems feeble under such a condition. Up to now, it is evident that: (i) the large scale solids circulation in the fluidized bed using inclined distributor incorporated with uneven gas supply leads to a faster convective mixing process, (ii) bubbles transverse movement is restricted because of the uniform distribution of the air for the case of horizontal
Fig. 9. Vector field of particle velocity at t = 1.0 s. (a) Horizontal distributor; (b) inclined distributor; (c), (d) are the schematics during the solid flux calculation for horizontal and inclined distributor, respectively (unit: mm).
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Fig. 10. Solid flux curves as a function of mixing time, horizontal distributor & even gas supply and inclined distributor & uneven gas supply.
air distribution plate, and the lateral convective mass transfer is reduced notably. 3.3.4. Discussion The advent of bubbles symbolizes the essential transition from packed beds to fluidized beds. Bubble hydrodynamics plays a fundamental role in fluidization engineering. To much extent, bubbles' parameters (such as size, velocity, frequency, etc) and their behaviors (such as generation, enlargement, breakage, coalescence, eruption, vertical and lateral movement, distribution, etc.) determine the particle mixing pattern, and, sequentially, the performance of mass transfer, heat exchange and chemical reactions in the beds [37,38]. On a macroscopical view, localized air supply induces the density gradient of particle distribution in the bed, which is the basic agent of convective particle stream. In a homogenous system, diffusive effect is a natural matter. Based on an overall review of the above paragraphs, the mixing process in fluidized beds mainly involves the following three categories. (1) Convective mixing: under the effect of external forces, particle materials become flexible and move in batches from their original locations to others (see snapshots at t = 0.2–1.0 s in Fig. 8), resulting in rapid mixing within the whole system. As far as the mixing phenomena in fluidized beds are concerned, such forces mainly include drag force from the gas phase, interactions among particles and sole weight of particles. (2) Diffusive mixing: particles of segregated components creep in the vicinity of their interfaces, and penetrate into each other slowly in a diffusive manner (see snapshots at t = 0.5–2.0 s in Fig. 4). During such a kind of mixing process, mixtures in local area, where diffusion occurs, achieve a relatively uniform density distribution of different components, which is more clearly illustrated in a snapshot at t = 2.0 s in Fig. 4.
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Table 3 Comparison between literature [39] and present simulation Parameter
Literature [39]
Present simulation
80 mm (depth) × 600 mm (width) × 1200 mm (height), 2D
4 mm (depth) × 96 mm (width) × 420 mm (height), 2D 1500 kg/m3, black, 4.0 mm in diameter 1500 kg/m3, green, 4.0 mm in diameter 190 mm
Sketch
many nearly unmixed clusters composed of identical component particles, which are highlighted with dotted circles in snapshots at t = 1.3 s, 1.5 s in Fig. 8. Such a local segregated state leads to a high standard deviation value, and, consequently, the vibrating mixing index after primary mixing is finished, at t = 1.5 s approximately. Obviously, according to the mechanism of shear mixing (relative slipping movement among local particles clusters), the rest of the task is left to it. The other way around, the slow diffusive mixing pattern in case of even air supply provides a gradual change in the distribution of trace particles, and, then, a smoother mixing index curve before t = 4.0 s in Fig. 7. 4. Comparison with reported experiment
Bed configuration
Bed materials Tracer particles Static bed Surface height Air velocity
Polypropylene, white, 3.0 mm in diameter Polypropylene, dyed black, 3.0 mm in diameter 300 mm 5.4 umf in both sides of the bed 1.8 umf at the center of the bed
2.5 umf in the lower part of the distributor 1.5 umf in the higher part of the distributor
(3) Shear mixing: relative slipping movement among various particles clusters is the major mechanism for shear mixing, which contributes to local mixing performance. Strictly speaking, all these three mixing patterns are concurrent in practical applications, with some prominent, which is also confirmed by Figs. 7 and 8. The fluctuation of the Ashton mixing index curve between t = 1.5 s and t = 4.0 s more or less, under the condition of uneven air distribution, actually reflects the shear mixing process. Although uneven air distribution brings about faster convective particle flow, the mixing quality is rude, leaving
Due to the instantaneously changing gas–solid flow structure, continuous sampling at second-level is still quite a challenge for the time being. 2D visualization experiments appear to be more feasible to study the micro-mixing process in fluidized beds, through image analysis technology. Tian [39] investigated the diffusive character of tracer particles in an internally circulating fluidized bed with CCD snapshots. Table 3 details the major parameters used in both literature [39] and this work. Data in Table 3 suggest that the two researches share nearly identical principles, including the 2D bed structure, the application of inclined air distributor, uneven gas feed, D-group particles, etc. From this point of view, present simulation results are comparable with those of a published experiment [39]. Ref. [39] found that there were two internal tracer particles stream over the ‘/\’ type air distributor, clockwise in the left half of the bed and anticlockwise right. Fig. 11 lists the continuous snapshots in the left half, which is thought to be similar to the case of inclined distributor in this paper. Fig. 11 fully embodies the pronounced convective mixing pattern under uneven gas feed, as predicted in this DEM simulation. Moreover, different mixing patterns, analyzed in Section 3.3.4, can also found their proofs among the snapshots in Fig. 11. For example, convective mixing t = 0.4–1.6 s, and shear mixing for t = 2.0–3.2 s in Fig. 11, with
Fig. 11. The diffusive process of tracer particles in internally circulating fluidized bed [39].
F. Tian et al. / Fuel Processing Technology 88 (2007) 187–198
their counterpart of t = 0.4–1.5 s, and t = 1.5–3.0 s more or less in Fig. 8, respectively. The degree of the mixing process can also be clearly recognized along the time-series snapshots in Fig. 11. In terms of the quantitative description, the mixing is relatively developed at a time around t = 3.0 s, which is 2.0 s in the present calculation. Parameters in Table 3 give a reason for the difference between experimental and predicted mixing time scales. On one hand, approximately, the simulation and a reported experiment have the same lateral velocity gradient, with 10.4 umf /m ((2.5–1.5) / (96 × 10− 3)) for the simulation and 12 umf /m ((5.4–1.8) / (600 × 10− 3 / 2)) for the experiment. On the other hand, the static bed surface in the present simulation, 190 mm, is lower than that of the experimental bed, 300 mm. As a result, with the nearly equal velocity gradient, the tracer particles in the simulated fluidized bed will experience a smaller scale circulation, achieving homogeneous distribution about 1 s earlier than those in the experimental setup. Comparisons demonstrate that this work reasonably reflects the complicated mixing phenomena in a fluidized bed, both qualitatively and quantitatively. The mixing time scale, evaluated with the Ashton mixing index, deserves its practical meaning. 5. Conclusions This paper makes full use of abundant particulate information in DEM simulation, and introduces Lacey and Ashton indexes to quantitatively evaluate the mixing characteristics of fluidized beds with uniform particles. Compared with the Lacey index, the Ashton index predicts the mixing mode more precisely. Ashton index curves illustrate that: in fluidized bed using inclined distributor together with uneven gas feed, rapid convective mixing process occurs at early stage, followed by shear mixing; for the case of horizontal distributor incorporated with even gas supply, the relatively smooth curve implies that diffusive mixing is the controlling factor in such a bed. Hydrodynamic information from the snapshots argues that different behaviors of bubbles movement cause the unlikeness between the two cases (uneven and even gas injection). Solid flux suggests there is a large scale regular circulating stream between the bubbles active and inactive subzones under the condition of nonuniform gas supply; however, in the case of uniform gas supply, particles lateral activities are reduced, since bubbles horizontal motion is confined to a local area. The mixing patterns and the mixing time scale predicted by this work are in good agreement with reported experiment. Nomenclature A Ashton index, dimensionless M Lacey index, dimensionless C true value of the tracer particles' concentration, dimensionless n¯ average number of particles contained in sampling units average concentration of tracer particles in sampling C¯ cells, dimensionless Y nij normal unit vector Ci concentration of tracer particles in the sampling cell i,
N CD p dp rp Y F cnij S Y F ctij t Y Fg Y t ij g umf I Y ug k Y v m Y vij
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dimensionless number of sampling cell drag coefficient, dimensionless gas pressure, Pa particle diameter, m particle radius, m is the normal force, N standard deviation when N is finite tangential force, N time, s particle–fluid interaction force, N tangential unit vector 2 gravitational acceleration, m s− 1 minimum fluidization velocity, m s− moment of inertia; kg m2 1 gas velocity, m s− 1 particle stiffness, N m− particle velocity, m s−1 particle mass, kg relative velocity between colliding particles, m
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