Hydrodynamic Modeling of Gas−Particle Flows in D−D Calciners

Ind. Eng. Chem. Res. 2005, 44, 3033-3041

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Hydrodynamic Modeling of Gas-Particle Flows in D-D Calciners Zheng Jianxiang,† Lu Huilin,*,† Sun Xiaoquan,† He Yurong,† Ding Jianmin,‡ and Wang Shuyan† School of Energy Science and Engineering, Harbin Institute of Technology, Harbin, 150001 China, and Agere Systems Inc., 555 Union Boulevard, Allentown, Pennsylvania 18109

A numerical parametric study was performed on the influence of various physical aspects over the hydrodynamics of gas-solid two-phase flow in the D-D calciner. An Eulerian continuum formulation, incorporated with the kinetic theory of granular media to represent the transport properties of the solid phase, was applied for both phases. The distributions of particle concentration and velocities of gas and particle phases were predicted. The numerical results show that the velocity of the primary air jet and throat diameter affect the hydrodynamics of gas-solid flow in the D-D calciner. Wavelet analysis was used to predict the nonlinear flow behavior of gas-solid mixture in the D-D calciner. 1. Introduction Cement producers and equipment manufacturers have focused on the effort of improving the energy efficiency of industrial cement calciners since the calciners are heavy fuel consumers.1-3 In China, thousands of small calciners dominate the cement industry, and their role has been increasing. Most plants are very small, and the unit capacity of a calciner is small. Hence, diversified production technologies are used in the cement industry to improve the thermal efficiency and increase the capacity of calciners. Furthermore, fluctuations of the degree of calcination of the raw mix may cause instability in the whole cement process as well as increased carbon monoxide production, which is a significant pollution factor that under certain circumstances may lead to explosive gas mixtures.4 Hence, hydrodynamic behavior of gas and solid flow in the calciner of cement production is of great importance because the flow structure of gas and particles phases affects the operation control and the energy requirements of the calciners. Computational fluid dynamics (CFD) has become an indispensable tool for gathering information to be used for design and optimization. Thus, computational twophase flow has emerged as an important research area with unique characteristics and issues. There are many ways to formulate a two-fluid model of gas and solids flow. The general idea is to first formulate the integral balances for mass, momentum, and energy for a fixed control volume containing both phases. This balance must be satisfied at any time and at any point in space. Studies that examined some of the numerical aspects of multiphase flow include those by Soo,5 Gidaspow,6 Crowe et al.,7 and Jackson.8 Generally speaking, there are two basic Eulerian-Eulerian modeling approaches to dense gas-solid flow, the model with constant solidphase viscosity and the two-fluid model using the kinetic theory of granular flows for the solid phase. Both * To whom correspondence should be addressed. E-mail: [email protected]. † Harbin Institute of Technology. ‡ Agere Systems Inc.

approaches are based on Eulerian descriptions of the gas and particles phases. With suitable viscosity inputs, the constant solid viscosity model is able to predict the large-scale flow patterns and mixing on the order of equipment scale of gas-solid flow systems.9-11 In most recent continuum models, constitutive relations according to the kinetic theory of granular flow are incorporated to study the complex hydrodynamics of gas-solid flow. This theory is basically an extension of the classical kinetic theory of dense gases12 to particulate flows, which takes nonideal particle-particle collisions and gas-particle drag into account. The kinetic theory of granular flow has been widely used to model gassolid flow in fluidized beds and risers (e.g., Sinclair and Jackson;13 Ding and Gidaspow;14 Pita and Sundaresan;15 Bolio and Sinclair;16 Nieuwland et al.;17 Benyahia et al.;18 Neri and Gidaspow;19 Huanpeng et al.20). In this work, a numerical parametric study is performed on the influence of various physical aspects over the hydrodynamics of gas-solid two-phase flow in the D-D calciner using a transient two-fluid model incorporated with the kinetic theory approach considering the effect of the interaction of particle collisions. The flow structure of gas and particles in the D-D calciner has been predicted. Frequency analysis is performed to analyze the dynamics of gas-solids flow by means of the fast Fourier transform (FFT) method. The effects of the primary air jet velocity and throat diameter on the hydrodynamics of gas and particles are discussed in the D-D calciner. 2. Mathematical Model and Solution Method 2.1. Two-Fluid Model of Gas-Solid Flow. Equations used are the conservation of mass and momentum for the solid and for the gas phase. A summary of the governing equations is given in Table 1.6,21 In this model, solid viscosity and pressure are derived from the kinetic theory of granular flow by considering the random fluctuation of particle velocity and its variations due to particle-particle collisions. Such a random kinetic energy, or granular temperature θ, can be predicted by solving, in addition to the mass and momentum equations, a fluctuating kinetic energy equation for particles. The viscosity and pressure of the

10.1021/ie040168j CCC: $30.25 © 2005 American Chemical Society Published on Web 03/25/2005

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Table 1. Mathematical Model of Gas-Solid Flow A. Conservation Laws (1) Continuity Equations (a) fluid phase (b) particulate phase

(a) fluid phase (b) particulate phase

∂ (gFg) + ∇‚(gFgvg) ) 0 ∂t ∂ (sFs) + ∇‚(sFsvs) ) 0 ∂t (2) Momentum Equations ∂ (gFgvg) + ∇‚(gFgvgvg) ) -g∇P + ∇‚τg + gFgg - βgs(vg - vs) ∂t ∂ (sFsvs) + ∇‚(sFsvsvs) ) -s∇P + ∇‚τs - ∇Ps + sFsg - βgs(vs - vg) ∂t (3) Equation of Conservation of Solids Fluctuating Energy 3 ∂ ( F θ) + ∇‚(sFsθ)vs ) (-∇psIh + τs):∇vs + ∇‚(ks∇θ) - γs + φs + Dgs 2 ∂t s s

[

]

(1) (2)

(3) (4)

(5)

B. Constitutive Equations (a) fluid phase stress (b) gas shear viscosity (subgrid scale model, SGS)

(c) particulate phase stress (d) dissipation fluctuating energy (e) radial distribution function at contact (f) solid pressure (g) shear viscosity of solids (h) bulk solids viscosity (i) rate of energy dissipation per unit volume (j) exchange of fluctuating energy between gas and particles (k) fluid-particulate interphase drag coefficients21

2 τg ) µg[∇vg + (∇vg)T] - µg(∇‚vg)I 3 µg ) µg,l + Fg(Ct∆)2(τg‚τg)

(7)

∆ ) (∆x∆y)1/2

(8)

2 τs ) ξs∇‚vsI + µs[(∇vs + (∇vs)T) - (∇‚vs)I] 3 4 θ γs ) 3(1 - e2)s2Fsgoθ - ∇‚vs d π s 1/3 -1 go ) 1 s,max ps ) sFsθ[1 + 2gos(1 + e)]

(9)

[ ( )]

(x

(6)

)

(10) (11) (12)

10Fsdxπθ 4 θ 4 µs ) s2Fsdgo(1 + e) 1 + gos(1 + e) + 5 π 96(1 + e)sgo 5 θ 4 ξs ) s2Fsdgo(1 + e) 3 π dFs 18µg 2 Dgs ) |vg - vs|2 2 4xπθ d Fs φs ) -3βθ

( )

x x

[

]

2

βgs ) φβgs|Ergun + (1 - φ)βgs|Wen&Yu g g 0.8: g < 0.8:

(14) (15) (16) (17)

Fgs|vg - vs| 3 g-2.65 βgs|Wen&Yu ) CD 4 d (1 - g)sµg Fgs|vg - vs| βgs|Ergun ) 150 + 1.75 gd ( d)2

(18) (19)

g

(

)

150[1.75(0.2 - s)] φ ) arctan + 0.5 π 24 CD ) (1 + 0.15Re0.687) Re < 1000 Re CD ) 0.44 Re g 1000

(20) (21) (22)

Fgg|vg - vs|d Re ) µg

solid can then be computed as a function of granular temperature at any time and position. A more complete discussion of the implemented kinetic theory model is presented in Gidaspow’s book.6 The turbulent flow of gas phase is modeled using a simple subgrid scale (SGS) model shown in eqs 7 and 8.22 Deardorff22 suggested Ct be in the range of 0.1 to 0.2. The value of Ct ) 0.1 was used in this study. 2.2. Boundary Conditions. At the inlet, all velocities of gas and particles phases were specified. The inlet concentration of particles was calculated from the inlet mass flux of solids. At the outlet, the pressure was specified (atmospheric). Initially, the velocities of gas and particles were set to zero. At the wall, the gas tangential and normal velocities were set to zero (no slip condition). The normal velocity of particles was also

(13)

(23)

set to zero. The following boundary equations apply for the tangential velocity and granular temperature of particles at the wall:13

vt,w ) -

6µss,max

∂vs,w πφFssgox3θ ∂n

ksθ ∂θw x3πφFssvsgoθ3/2 + θw ) ew ∂n 6s,maxew

(24)

(25)

where ew is the restitution coefficient at the wall. 2.3. Simulation Code and Computation Procedures. The simulations were carried out with the CFD code that was previously used to model gas-solid flow

Ind. Eng. Chem. Res., Vol. 44, No. 9, 2005 3035 Table 2. Parameters Used for Simulations symbol

description

computer runs

Fs Fg d ujet e φ µg,l s,max h D1 D D2 D3

particle density gas density average diameter of particles inlet velocity of primary air restitution coefficient of particles specularity coefficient laminar gas viscosity maximum concentration of solids height of calciner inlet diameter of calciner diameter of cylinder throat diameter of calciner outlet diameter of calciner

1600 kg/m3 1.2 kg/m3 30.0 µm 25.0 m/s 0.99 0.5 1.5 × 10-5 Pa‚s 0.64 18.3 m 1.54 m 4.18 m 2.8 m 3.18 m

distributions of flow variables are computed covering a period of 60 s corresponding to 1-2 weeks of computational time on a PC (40 GB hard disk, 128 MB Ram, and 600 MHz CPU). 3. Simulation Results and Discussions

Figure 1. Illustration of a D-D calciner geometry.

in the riser.21 The model partial equations with their corresponding boundary equations were differenced over staggered meshes of computational cells and solved by the ICE (implicit continuum Eulerian) technique. In this method, the continuity equations are differenced explicitly, while the momentum equations of both phases are differenced explicitly except the implicit gas pressure gradient terms and the partial implicit coupling between two phases. This software allows free implementation of extra equations, boundary conditions, and differencing schemes. This modeling work is concerned with the D-D calciner typically used in the cement industry. Figure 1 schematically shows the D-D calciner system of which the capacity was about 2000 tuns per day. The diameters of cylinder and throat are 4.18 and 2.8 m, respectively. The total height is 18.3 m. The primary air and the mixture of pulverized coal and calcined raw (limestone) particles enter the D-D calciner through the bottom. The inlet diameter of the primary air jet is 1.54 m. The diameter and density of pulverized coal and limestone particles are different and change with combustion in the calciner. We assumed that the average diameter and density of the mixture of pulverized coal and limestone particles are 30 µm and 1600 kg/m3 in the simulations. The properties used in the simulations are listed in Table 2. Labous et al.23 reported the experiments of collisional properties of 25.4 mm nylon spheres using high-speed video analysis. They found that the restitution coefficient of particles was e ) 0.97 ( 0.03. By measuring granular temperature and particle velocity distributions by means of a CCD camera in the circulating fluidized bed, Tartan and Gidaspow24 show that the restitution coefficient of 530 mm glass beads was to be 0.99. In this study, the values of restitution coefficient between particle-particle and particle-wall of 0.99 and 0.90 were used. For simplicity, a two-dimensional computational domain is assumed in the following simulations. The stretched grid is composed of 95 × 320 nodes. In this simulation, a constant time step of 1.0 × 10-5 was used. Time-averaged

Several cases have been modeled to investigate the effect of operating conditions on the gas-solid flow pattern in the D-D calciner. Figure 2 illustrates the instantaneous particle concentration at the time of t ) 10, 15, 20, 30, and 40 s at the primary air jet velocity and inlet mass flux of particles of 25 m/s and 60 kg/(m2 s), respectively. At the base of the column the injections of the primary air jet push the solids from the bottom into the calciner in the form of a central jet. These give rise to a relative high concentration at the center and a low concentration at the walls. This results from a nonuniform distribution of particle concentration in the calciner. The local instantaneous concentrations of particles in the calciner are never stable, and they are always oscillations. These elementary structures are in evolution during their passing through the calciner. It can be seen that at the base of the calciner two zones of low particle concentrations are formed between the primary air jet and walls. Their formations are strongly influenced by primary air arrangement. At the throat, the particles are passed through with a high velocity, and the secondary air-solid jet is formed at the exit of the throat. Two zones of low particle concentrations are formed between the secondary air-solid jet and walls. Their formations are strongly influenced by the throat diameter. Hence, the flow structure of gas and particles

Figure 2. Instantaneous particle concentration distributions at the primary air jet velocity of 25 m/s.

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Figure 5. Particle concentration as a function of time at the primary air jet velocity of 25 m/s.

Figure 3. Instantaneous gas velocity distributions at the primary air jet velocity of 25 m/s.

Figure 4. Instantaneous particle velocity distributions at the primary air jet velocity of 25 m/s.

in the lower and upper portions of the D-D calciner is dominated by the primary air jet and throat arrangement. Figure 3 shows the instantaneous gas velocity as a function of time at the primary air jet velocity and inlet mass flux of particles of 25 m/s and 60 kg/(m2 s), respectively. The gas velocity is high in the center because of the effect of the primary air jet at the lower portion of the calciner. Two vortexes are formed between the primary air jet and walls at the inlet. These are induced to form gas circulations. It can be seen that gas velocity is positive in the center, which means that gas flows up, and negative at the walls, which indicates that gas flows down. At the upper portion, the gas velocity in the center is high because of the effect of the throat by reducing cross-sectional area. The cross-sectional area of the contractor is gradually reduced, and gas is accelerated as it passes through the throat. A secondary air jet is formed at the exit of the throat. This results from the high gas velocity in the center of the upper portion of calciner. Figure 4 shows the instantaneous particle velocity as a function of time at the primary air jet velocity and inlet solid mass flux of 25 m/s and 60 kg/(m2 s). It can be seen that the particle velocity is

high in the center because of the effect of the primary air jet. The particle velocity is positive in the center, which indicates that particles flow up with the gas phase, and negative at the walls, which means that particles flow down. At the lower portion, two vortexes of particles are formed between the primary air jet and walls, the same as the gas phase. The particle concentrations in these two vortexes are low. These induced particle recirculation in the lower portion. As the crosssectional area of the flow passage is gradually reduced at the throat, the particles are accelerated and the particle velocity is increased. This results from the positive velocity of particles in the center and negative at the walls in the upper portion. Hence, particle recirculations are also formed. The recirculation of particles will increase the residence time of particles in the D-D calciner. Figure 5 shows the instantaneous particle concentration as a function of time at the primary air jet velocity and inlet solid mass flux of 25 m/s and 60 kg/(m2 s). The initial particle concentration in the calciner is zero. We see that the calciner will take about 10 s to fill up and come to a stable operating condition. However, the local particle concentration oscillates and changes with time. Time-averaged distribution of variables is then computed considering the last 50 s of simulations. A grid refinement study has been performed to assess the accuracy and sensitivity of the predictions of gas and particles flow in the calciner. Three grid systems (grid-A with 55 × 240 nodes, grid-B with 95 × 320 nodes, and grid-C with 145 × 430 nodes) are generated. Figure 6 shows the distributions of time-averaged velocity of particles in the calciner for three grid systems at the primary air jet velocity and inlet solid mass flux of 25 m/s and 60 kg/(m2 s). As can be seen from the plot, the impact of grid resolution on the velocity of particles is not significant for grid-B and grid-C systems. The down-flow of particles appears near the walls. However, the profile of particle velocity indicates the value of particle velocity is small at the walls for grid-A. The reason is primarily due to the coarse grid used near the walls for grid-A. It should be emphasized that for the grid-B and grid-C the distribution of concentration of particles is not sensitive to the grid resolution used. Although the flow structure is better resolved with gridC, we utilize grid-B for the rest of the study because of the negligible impact on distributions between two grid systems and less computing time used in the simulations.

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Figure 6. Grid refinement study for particle flows in the D-D calciner.

Figure 7. Time-averaged particle concentrations at the primary air jet velocity of 25 m/s.

Figure 7 shows the time-averaged particle concentration in the different locations at the primary air jet velocity and inlet solid mass flux of 25 m/s and 60 kg/ (m2 s). In the lower portion of calciner, the timeaveraged concentration of particles is higher in the center. The concentration of particles decreases along the lateral direction, reaches a minimum, and then increases toward the walls. In the upper portion of calciner, the distributions of time-averaged particle concentration become uniform. The aerations of the primary air jet lead to a high particle concentration in the center and low concentration between the center and walls in the lower portion of calciner. The more uniform distribution of particle concentration in the upper portion of calciner is contributed by the effect of the secondary air-solid jet at the exit of the throat. The time-averaged gas velocity distribution is shown in Figure 8 at the primary air jet velocity and inlet solid mass flux of 25 m/s and 60 kg/(m2 s). The simulated results showed that gas flows upward with a high velocity at the center because of the effect of primary air jet. At the walls, gas velocity is negative where gas circulation is formed. The distribution of time-averaged particle velocity is shown in Figure 9 at the primary air jet velocity and inlet solid mass flux of 25 m/s and 60 kg/(m2 s). In the center part of the calciner, particles flow upward because of the effect of primary air jet. At the walls, the particle velocity is negative, which means that particles flow down. The particle recirculations are formed in the lower and upper portions of the calciner. Comparing Figure 8 with Figure 9, it can be seen that

Figure 8. Time-averaged axial gas velocity distributions at the primary air jet velocity of 25 m/s.

Figure 9. Time-averaged axial particle velocity at the primary air jet velocity of 25 m/s.

Figure 10. Time-averaged gas velocity at the primary air jet velocity of 25 m/s.

the slip velocity, ug - us, was very small since a small diameter of particles and high primary air jet velocity were used in this study. The computed time-averaged gas velocity obtained using a laminar gas viscosity replacing the effective gas viscosity, which was determined from the SGS turbulence model, was shown Figure 10 at the primary air jet velocity and inlet solid mass flux of 25 m/s and 60 kg/(m2 s), respectively. It can be seen that the predicted gas velocity using the laminar gas viscosity is larger than that using the effective gas viscosity in the center region. The trends predicted by the SGS turbulence model and a laminar gas viscosity are the same. The

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Figure 11. Distribution of time-averaged concentrations of particles for difference models of particle viscosity.

Figure 13. Time-averaged axial particle velocity as a function of primary air jet velocity.

Figure 12. Time-averaged particle concentrations as a function of primary air jet velocity.

Figure 14. Profile of particle concentration as a function of throat diameter.

SGS model includes several empirical constants. Because of a lack of experimental data, these constants are usually taken from single gas-phase flow. Its validity in gas-solid turbulent flow is questionable. The hydrodynamics of the gas phase in the D-D calciner was clearly observed in present 2-D simulation. It is important to mention that the turbulent flow of the gas phase is always three-dimensions in the calciners. Therefore, three-dimensional simulations are required to obtain more accurate simulation results. The particle viscosity in the momentum equation of solids can be predicted by either the kinetic theory of granular flow or empirical equations. Sun and Gidaspow10 and Huilin and Gidaspow11 used the empirical viscosity correlation to model the flow of gas and particles phases in the circulating fluidized bed. Figure 11 shows the computed time-averaged particle concentration distributions using the different equations of particle viscosity at the primary air jet velocity and inlet solid mass flux of 25 m/s and 60 kg/(m2 s), respectively. The profile of the particle concentration indicates that the value of the particle concentration between the center and walls is higher without considering the viscosity of particles than that with considering the particle viscosity predicted by the equations of Sun and Gidaspow,10 those of Huilin and Gidaspow,11 and the kinetic theory of granular flow. It can be seen that the computed particle concentrations using the particle viscosity equation of Sun and Gidaspow10 and Huilin and Gidaspow11 and the kinetic theory of granular flow are not significantly different. This indicates that the

model with constant solid-phase viscosity inputs should be considered in the simulations of the D-D calciners. 3.1. Effect of Primary Air Jet Velocity. The influence of primary air jet velocity on the timeaveraged particle concentration is shown Figure 12 at the inlet solid mass flux of 60 kg/(m2 s). It can be seen that the time-averaged particle concentration is high at the center, reaches a minimum, and then increases at the walls. The particle concentration decreases with the increase of primary air jet velocity in the calciner since the inlet particle mass flux stays constant. Figure 13 shows the distribution of time-averaged vertical velocity of particles as a function of primary air jet velocity at the inlet solid mass flux of 60 kg/(m2 s). At the center of the calciner, the particle vertical velocity increases with the increase of the primary air jet velocity. However, the vertical velocity of particles near the walls decreases with the decrease of primary air jet velocity. We see that the primary air jet velocity affects gas-solid flow in the D-D calciner. 3.2. Effect of Throat Diameter. Figure 14 shows the time-averaged particle concentration distribution as a function of the throat diameter at the primary air jet velocity and inlet solid mass flux of 25 m/s and 60 kg/ (m2 s). As the throat diameter increases, the location of minimum particle concentration is drafted toward the walls. The value of the minimum particle concentration is increased, and a more uniform distribution of particle concentration is formed in the calciner. Figure 15 shows the time-averaged vertical particle velocity as a function of the throat diameter at the primary air jet velocity

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the fine scale variability (the detail coefficients or wavelet coefficients) and the coarser scale smoothness (the smooth coefficients or mother-function coefficients). The complete wavelet transform is a process from the finest to the coarsest wavelet level (scale). This describes a scale-by-scale extraction of the variability information at each scale.25 A wavelet H(t) (with real values in our case) transforms a time function (signal)(t) as

∫

WT(a,b) ) |a|-0.5 x(t)H

Figure 15. Distribution of vertical particle velocity a function of throat diameter.

Figure 16. Power spectrum density of instantaneous particle concentration.

and inlet solid mass flux of 25 m/s and 60 kg/(m2 s). The time-averaged vertical particle velocity decreases with the increase of throat diameter. From these figures, we see that the throat diameter affects the distributions of velocity and concentration of gas and particle phases in the D-D calciner. 3.3. Analysis of Power Spectrum Density of Instantaneous Particle Concentration. Power spectra density (PSD) of instantaneous particle concentrations is illustrated in Figure 16 at the primary air jet velocity and inlet solid mass flux of 25 m/s and 60 kg/ (m2 s) by means of the fast Fourier transform (FFT) method. These show distribution of energy with frequency in the calciner. It can be seen that the PSD of the local particle concentration fluctuation exhibits a broad-band character with many spikes over a wide frequency range. Compared with the profile of power spectrum density at the center, a high exchange of momentum and energy between gas and particle phases occurs at the locations of x ) 1.46 m and x ) 1.9 m. Since the hydrodynamics of gas-solid flow in the calciner are a complicated nonlinear dynamical system, a detailed understanding of the behavior of gas and particles flow is important. To reveal the nonlinear dynamical characteristics of gas-solid flow in the D-D calciner, the wavelet transform (WT) and wavelet multiresolution analysis can be used. Wavelet transform is a transformation of information from a fine scale to a coarser scale by extracting information that describes

(t -a b) dt

(26)

where the two-parameter set of functions, Ha,b(t), is obtained from a single one, H(t), called the basic (mother) wavelet, through dilations by the factor a and translations by the factor b. The factor |a|-0.5 is used for normalization purposes whereby all the wavelets in the generated family have the same energy. Equation 26 implies that the WT can be thought of as a convolution of a function (signal)(t) with an analysis window H(t) (mother wavelet) shifted in time by b and dilated by a scale parameter a. Each wavelet is located at a different position along the time axis; also each wavelet is localized in the sense that it decreases rapidly to zero when sufficiently far from its center. The scale parameter a, which can take any value on the positive real axis, is usually chosen such that it is inversely proportional to frequency. Accordingly, large values of a correspond to wide (low frequency) wavelets, while small values of a correspond to short (high frequency) wavelets. Thus by changing the scale parameter a while keeping b fixed, one allows the wavelet Ha,b(t) to cover the desired frequency range around the time t ) b. By changing the translation parameter b as well, one can move the time localization center b to the desired position so that at, for example, b ) b1, each of the wavelets in the generated family Ha,b1(t) is centered around the desired time t ) b1. Accordingly, any particular local event of the function (signal) can be identified from the scale and position of wavelets into which its composed, and the appropriate selection of the two parameters a and b make the WT extract the localized conditions, that is, individual (local in time) frequency events of the time varying function (signal)(t). Figure 17 shows the distributions of the original signal (s), detailed coefficients (di, i ) 1, 7), and scale signal (a7) at the primary air jet velocity and inlet solid mass flux of 25 m/s and 60 kg/(m2 s). According to wavelet transform, the original signals can be decomposed into different scale signals and detail signals (wavelet transform), and the original signal may be reconstructed from the information contained in the last scale. First, the original signal (s) is resolved into scale 1 signal (a1) and scale 1 detail signal (d1). Scale 1 detail signal captures the information with high frequency (5.0 Hz). The scale 1 signals, that is, the remainder of the original signals through wavelet filter, can be further decomposed into scale 2 signals (a2) and scale 2 detail signals (d2). Through a family of wavelet filters, a series of detail signals is obtained with different frequency band. Scale 7 signal is the remainder signal by wavelet transform. Hence, the original signal can be reconstructed from the scale 7 signal and other detail signals. From the profile of the detailed coefficients in Figure 17, it can be seen that the alternative larger positive and negative peaks are in the range of 0.3125-0.625

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Figure 17. Wavelet coefficients analysis of fluctuating particle concentrations.

Hz. This gives a low oscillation of particle concentration in the D-D calciner. 4. Conclusion The hydrodynamic behavior of a gas-solid flow in the D-D calciner was modeled using a two-fluid model incorporated with the kinetic theory of granular flow to represent the transport properties of the solid phase. Simulations illustrate the ability of the model to represent gas-solid flow in the D-D calciner. The up-flow in the center and down-flow at the walls of gas and particle phases were predicted. The flow structure of gas-solid flow is strongly influenced by the primary air jet velocity and arrangement of throat in the D-D calciner. Wavelet analysis provides an effective tool for analyzing the simulated instantaneous particle concentrations in a D-D calciner. The simulated fluctuating particle concentrations can be decomposed into their approximations and details at different resolutions. Wavelet analysis gives the scale 7 detail signals, which reflect the hydrodynamic behavior of gas-solid flow in the D-D calciner.

Acknowledgment This work was supported by the National Science Foundation in China through Grant No. 50376013 and Project No. MD2002.31 supported by the multidiscipline scientific research foundation of Harbin Institute of Technology. Notation CD ) drag coefficient Ct ) Smagorinsky constant d ) particle diameter D ) diameter of calciner e ) restitution coefficient g ) gravity go ) radial distribution function I ) unit tensor ks ) conductivity of fluctuating energy n ) normal direction P ) fluid pressure Ps ) particle pressure q ) fluctuating energy flux Re ) Reynolds number t ) time

Ind. Eng. Chem. Res., Vol. 44, No. 9, 2005 3041 ujet ) gas velocity of primary air jet vg ) gas velocity vs ) particle velocity x ) transverse distance from axis ∆x ) grid size in x direction y ) vertical distance ∆y ) grid size in y direction Greek Letters τg ) gas stress tensor τs ) particle stress tensor θ ) granular temperature µg ) gas viscosity µg,l ) laminar gas viscosity µs ) shear viscosity φ ) specularity coefficient g ) porosity s ) particle concentration s,max ) maximum concentration of solids γ ) collisional energy dissipation Fs ) particle density Fg ) gas density β ) drag coefficient

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Received for review May 26, 2004 Revised manuscript received December 7, 2004 Accepted December 21, 2004 IE040168J