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    Chemical Engineering Science 62 (2007) 3397–3409www.elsevier.com/locate/ces

    Computation of gas and solid dispersion coefficients in turbulent risers andbubbling beds

    Veeraya Jiradilok a, Dimitri Gidaspowa,∗, Ronald W. Breaultba Illinois Institute of Technology, Chicago, IL, USAbUS Department of Energy, Morgantown, WV, USA

    Received 1 July 2006; received in revised form 30 January 2007; accepted 30 January 2007

    Available online 24 March 2007

    Abstract

    A literature review shows that dispersion coefficients in fluidized beds differ by more than five orders of magnitude. To understand the

    phenomena, two types of hydrodynamics models that compute turbulent and bubbling behavior were used to estimate radial and axial gas

    and solid dispersion coefficients. The autocorrelation technique was used to compute the dispersion coefficients from the respective computed

    turbulent gas and particle velocities.

    The computations show that the gas and the solid dispersion coefficients are close to each other in agreement with measurements. The

    simulations show that the radial dispersion coefficients in the riser are two to three orders of magnitude lower than the axial dispersion

    coefficients, but less than an order of magnitude lower for the bubbling bed at atmospheric pressure. The dispersion coefficients for the bubbling

    bed at 25 atm are much higher than at atmospheric pressure due to the high bed expansion with smaller bubbles.

    The computed dispersion coefficients are in reasonable agreement with the experimental measurements reported over the last half century.

     2007 Elsevier Ltd. All rights reserved.

    Keywords:  Fluidization; Gas-particle flow; Computational fluid dynamics; Reynolds stresses

    1. Introduction

    Traditional design of gasifiers for the FutureGen project and

    other reactors requires the knowledge of dispersion coefficients,

    as demonstrated by Breault (2006). However, they are known to

    vary by 5 orders of magnitudes (Gidaspow et al., 2004; Breault,

    2006).

    From experimental investigations over the last half-century,

    the dispersion coefficients are known to be large for large diam-

    eter bubbling beds and small at low gas velocities. Surprisingly,

    they differ by two to three orders of magnitudes at the same

    gas velocity. Hence, a better understanding of the phenomena

    causing such large differences is needed. This study presents a

    ∗  Corresponding author. Department of Chemical Engineering, Illinois In-stitute of Technology, 10 west 33rd street, Chicago, IL 60616, USA.

    Tel.: +1312 5673045; fax: +1312 5678874.

     E-mail address:  [email protected] (D. Gidaspow).

    0009-2509/$- see front matter   2007 Elsevier Ltd. All rights reserved.

    doi:10.1016/j.ces.2007.01.084

    computational method of determining the gas and solid ax-

    ial and radial dispersion coefficients in bubbling beds and

    risers.

    The physical definition of dispersion coefficients is based

    on the kinetic theory of gases (Bird et al., 2002; Chapman

    and Cowling, 1970) and granular flow (Gidaspow, 1994). For

    diffusion of gases or particles the diffusivity, D, is defined as

    the mean free path, L, times the average velocity, C , as shown

    below:D = L × C, (1)where the peculiar velocity, C  is given by

    C = c − v, (2)where c is instantaneous velocity and  v is the velocity averaged

    over velocity space, as shown below:

    v = 1n

       cf (c)dc   with  n =

       f(c) dc. (3)

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    3398   V. Jiradilok et al. / Chemical Engineering Science 62 (2007) 3397– 3409

    Tartan and Gidaspow (2004).   measured the instantaneous

    particle velocities, c, in a riser and computed the hydrodynamic

    velocities, v, as a function of time. From these two types of 

    velocities the particle stresses,  CC, and the Reynolds stresses,

    vv, were computed, where v=v−v and v is the time averagedvelocity. In the center of the riser the velocity,   v, could be

    obtained from Poiseuille flow.The mean free path was obtained from the average velocity

    and collision time, :

    L = C × , (4)

    since C  and√ C2 differ by only 10%.

    The dispersion matrix can be defined as

    DP  = CC × . (5)

    Similarly for turbulent oscillations, the dispersion matrix is de-

    fined as

    DT  = vv × . (6)

    This definition is identical to that used in single phase tur-

    bulent flow (Hinze, 1959). For gas–particle flow we have two

    types of dispersion coefficients, that for the gas and for the par-

    ticles. In this paper we computed only the turbulent dispersion

    coefficients.

    Hence the dispersion coefficients in the  x  and  y  directions

    are computed from the normal Reynolds stresses in the  x  and

     y directions. For the riser, the normal Reynolds stresses in the

    direction of flow are shown here to be about two orders of 

    magnitude larger than the radial Reynolds stresses due to thefact that the radial velocities are small compared to the axial

    velocities. This provides an explanation for the large anisotropy

    of the dispersion coefficients in the riser and not in bubbling

    beds, where the velocities in radial and axial directions are

    similar.

    2. Hydrodynamics models

    The physical principles used are the conservation laws of 

    mass, momentum and energy for each phase, the fluid phase and

    the particulate phases. This approach is similar to that of  Soo

    (1967) for multiphase flow and of  Jackson (1985) for fluidiza-

    tion. A Newtonian or power law constitutive equation for the

    surface stress of phase “k ” will depend at least on its symmet-

    rical gradient of velocity. The kinetic theory of granular flow

    provides a physical motivation for such an approach (Gidaspow,

    1994). Hence, the general balance laws of mass and momen-

    tum for each phase, with phase change, are given by Eqs. (7)

    and (8) and the constitutive equation for the stress is given by

    Eq. (9).

    Continuity equation for phase k :

    (kk)

     + ∇ (kkvk)

     = ˙mk . (7)

    Table 1

    Hydrodynamic viscosity model

    Continuity equations

    (gg)

    t + ∇ · (ggvg) = 0

    (ss)

    t + ∇ · (ssvs) = 0

    Momentum equations

    (ggvg)

    t + ∇ · (ggvgvg) = −∇ PI  + ∇ · g − B

    vg − vs

    + gg(ssvs)

    t + ∇ · (ssvsvs ) = − ∇ P sI  + ∇ · s + B(vg − vs)

    + sg

    s −

    N k=g,s

    kg

    g

    Constitutive equations

    (1) Definitions

    g + s = 1

    (2) Gas pressure

    P g = g R̃T g

    (3) Stress tensor (i = gas or solid)

    i = 2iDi + (i −  23i) tr(Di )I 

    with

    Di =  12 [∇ vi + (∇ vi)T ](4) Empirical particulate phase viscosity and stress model

    ∇ P s = G(g)∇ s

    G(g) = 10−8.686g+8.577 dyne/cm2

    s = 0.1651/3s   g0  poise(5) Fluid-particulate interphase drag coefficients for < 0.8 (based on the

    Ergun equation)

     = 1502sg

    2gd 2p

    + 1.75gsgd p

    |g − s |

    for > 0.8 (based on the empirical correlation)

     = 34Cd 

    gs |g − s |d p

    −2.65g

    where

    Cd  =   24Rep

    [1 + 0.15Re0.697p   ]   for  Rep < 1000

    Cd  = 0.44 for  Rep > 1000

    Rep =ggd p|vg − vs |

    g

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    3400   V. Jiradilok et al. / Chemical Engineering Science 62 (2007) 3397– 3409

    These equations are similar to Bowen’s (1976) balance laws

    for multicomponent mixtures. The principle difference is the

    appearance of the volume fraction of phase “k ” denoted by k.

    In the case of phases not all the space is occupied at the same

    time by all phases, as it is by components due to the negligi-

    ble size of molecules. As in the case of the mixture equations

    for components, the mixture equations for phases show that thesum of the phase change productions in Eq. (7) is zero and the

    sum of the drag forces in Eq. (8) is zero. In convective form,

    the phase change momentum in Eq. (8) is zero, insuring in-

    variance under a change of frame of reference for translation.

    Eq. (9) is the usual Newtonian expression for the stress which

    arises from the assumption that the stress is a function of its

    own symmetrical gradient of velocity. For the fluid  P k  is the

    fluid pressure. When this form is substituted into the momen-

    tum equation, the result is not the usual momentum balance

    presented by Gidaspow (1986) and widely used in gas–liquid

    two-phase flow. It is a slightly modified version of the momen-

    tum balance called model B by Bouillard et al. (1989).

    In this study we are applying this model to single size

    particle–gas system with no reaction or phase change. The

    particle viscosity and the solid stress are input into the vis-

    cosity model given in Table 1. In the kinetic theory version,

    Table 2, the particle viscosity and solid stress are automatically

    System Geometry and System Properties

    Riser diameter 0.186 m.

    Riser inlet diameter 0.093 m.

    Riser height 8 m.

    Particle size 54 µm

    Particle density 1398 kg/m3

    Restitution coefficient, e 0.9

    Wall restitution coefficient, ew   0.6

    Specularity coefficient,    0.6

    Grid size, (∆ x  × ∆ y)   0.465 cm × 2.68 cm

    Grid number 42 (radial) × 300 (axial)

    Time step 5 × 10-5

    Outlet

    Y

    Inlet

    X

    Fig. 1. System geometry for simulations based on Wei et al. (1998a) experiments.

    computed. The restitution coefficient is a fitting parameter. In

    the simulation presented here the drag was also modified for

    flow of FCC particles as described in Jiradilok et al. (2006).

    The numerical scheme used in this study is the implicit con-

    tinuous Eulerian (ICE) approach. The model uses donor cell

    differencing. The conservation of momentum and energy equa-

    tions are in mixed implicit form. It means that the momentumequations are fully explicit. The continuity equations excluding

    mass generation are in implicit form.

    3. Simulations

    3.1. Flow of FCC particles in the riser 

    3.1.1. System properties

    Jiradilok et al. (2006)  have already shown that the kinetic

    theory model with the EMMS approach, as shown in  Table 2

    (Yang et al., 2004), is capable of computing turbulent fluidiza-

    tion of FCC particles in agreement with experimental data. The

    computed granular temperatures, particles viscosities, solid

    pressures and oscillation frequencies agreed with experiments.

    The simulations were carried out for the riser section of a

    circulating fluidized bed. A two-dimensional Cartesian coordi-

    nate system was used. Initially the riser column was empty and

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    0

    1

    2

    3

    4

    5

    6

    7

    8

       H  e   i  g   h   t ,  m .

    800

    700

    600

    500

    400

    300

    200

    100

    5 10 15

    0.4

    0.35

    0.3

    0.25

    0.2

    0.15

    0.1

    0.05

    0

    0.0001

    Laminar Granular Temperature m2 /s20.001 0.01 0.11   1

    Fig. 2. (a) Snapshot of solid volume fraction, (b) axial laminar granular temperature profile for  W s = 132kg/m2 s and  U g = 4.57 m/s.

    the velocities of both phases were assumed to be zero. At the

    outlet, atmospheric pressure was prescribed. Particularly im-

    portant is also the specification of appropriate boundary condi-

    tions at the wall. For the gas a non-slip boundary condition was

    used. For the granular temperature wall boundary condition the

    Johnson and Jackson (1987) boundary condition was used. It

    was obtained by equating the granular flux to collisional dissi-

    pation with a correction for slip.

    Fig. 1 gives the system geometry and the system properties

    for describing the experiments of  Wei et al. (1998a). The main

    fluidizing gas is air. The solid phase consisted of FCC particles.

    A restitution coefficient of 0.9 was used (Jiradilok et al., 2006).

    3.1.2. Flow structureWe have shown that the standard kinetic theory-based CFD

    model with a modified drag as suggested by Li group (Yang

    et al., 2004) is capable of correctly describing the coexistence

    of the dense and dilute regimes for flow of FCC particles in a

    riser in the turbulent regime (Jiradilok et al., 2006).

    Fig. 2(a) displays the snap shot at 7.2 s for the solid flux of 

    132kg/m2 s and the gas velocity of 4.25 m/s. The top part of 

    the riser is dilute and the bottom part is dense. The structure

    at the bottom part is core-annular. There is a low concentration

    of solid at the center and a high solid volume fraction near the

    wall, which approximately agrees with the experimental data.

    Using the kinetic theory model, the laminar granular tem-

    perature was computed. The axial profile of laminar granular

    Table 3

    Equations for obtaining the averaged velocity and stresses

    The mean velocity

    particle

    vi(r) =   1m

    mk=1

    vik (r, t)

    The normal Reynolds

    stress

    vivi =

      1

    m

    mk=1

    (vik (r, t) − v̄i (r))(vik (r, t) − v̄i (r))

    i represents x  or y directions, m is the total number of data over a given time

    period.

    temperature is shown as Fig. 2(b). The laminar granular tem-

    perature increases with increasing bed height due to the oscil-

    lation of individual particles.

    3.1.3. Reynolds stressesReynolds stresses are produced due to the random velocity

    fluctuations of the hydrodynamic velocity. This is the principal

    characteristic of turbulent flow. The Reynolds stresses are used

    to estimate the turbulent part of granular temperature and the

    dispersion coefficients. The Reynolds stress can be calculated

    as a function of hydrodynamic velocity and mean velocity, as

    shown in Table 3.

    The time-average values of normal Reynolds stresses per

    unit bulk density in the axial direction of gas and solid phases

    with various heights are shown as Fig. 3. The values vary as a

    function of the radial position for both phases. The oscillations

    on the top, 2 m, and bottom, 6m, parts occur due to the effects

    of the inlet and outlet.

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    3402   V. Jiradilok et al. / Chemical Engineering Science 62 (2007) 3397– 3409

    1

    2

    3

    4

    5

    6

    7

    8

    9

    10

      v                  ′  y

      v                  ′

      y  m

       2   /  s   2

    0

    1

    2

    3

    4

    5

    6

    7

    8

    9

    10

      v                  ′

      y  v

                      ′  y  m

       2   /  s   2

    0

    1

    2

    3

    4

    5

    6

    7

    8

    9

    10

      v                  ′  y

      v                  ′

      y  m

       2   /  s   2

    0

    -1

    Gas phase Solid phase

    -0.8

    r/R

    -0.6 -0.4 -0.2 0

    -1 -0.8 -0.6 -0.4 -0.2 0

    -1 -0.8 -0.6 -0.4 -0.2 0

    Fig. 3. Axial normal Reynolds stress per bulk density of gas and solid phases

    at (a) 2m, (b) 4m and (c) 6m.

    Due to the turbulences in the direction of flow, the normalReynolds stresses per unit bulk density in the axial direction

    are higher than those in the radial direction for both phases,

    as shown in Fig. 4. We see that the Reynolds stresses per unit

    bulk density for the gas and the solid are close to each other.

    3.2. Bubbling commercial size fluidized beds

    3.2.1. System properties

    To show that the model predicts the effects of pressure on

    dispersion coefficients, two bubbling fluidizations were simu-

    lated. Simulations of bubbling commercial size fluidized beds

    from a two-dimensional rectangular bed were performed using

    0

    0.05

    0.1

    0.2

    0.25

    0.15

      v                  ′  x

      v                  ′

      x  m

       2   /  s   2

    0

    0.05

    0.1

    0.2

    0.25

    0.15

      v                  ′

      x  v

                      ′  x  m

       2   /  s   2

    0

    0.05

    0.1

    0.2

    0.25

    0.15

      v                  ′

      x  v

                      ′  x  m

       2   /  s   2

    -1 -0.8 -0.6 -0.4 -0.2

    r/R

    Gas phase Solid phase

    0

    -1 -0.8 -0.6 -0.4 -0.2 0

    -1 -0.8 -0.6 -0.4 -0.2 0

    Fig. 4. Radial normal Reynolds stress per bulk density of gas and solid phases

    at (a) 2m, (b) 4m and (c) 6m.

    the viscosity model, as given in Table 1. The system geome-

    try and the system properties are summarized in  Fig. 5. For

    boundary conditions, the non-slip boundary condition was used

    for the gas phase at the wall. For the solid phase, the free-slip

    boundary condition was chosen. Initially, the bedwas filled with

    particles at a solid volume fraction of 0.58. In this study we

    simulated bubbling beds under two different pressures, a high

    pressure, 25 atm, and a low pressure, atmospheric pressure.

    3.2.2. Flow structure

    Instantaneous snapshots of the solid volume fraction profiles

    of the bubbling bed are shown in Fig. 6. Figs. 6(a) and (b) were

    obtained with atmospheric pressure and 25 atm, respectively.

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    Geometry of the reactors

    •   Height 12.19 m.

    •   Reactor Diameter (OD) 1.22 m.

    •   Reactor Diameter (ID) 1.14 m.

    •   Angle 75

    •   Number of inlet cell

    •   Position of outlet right hand side•   Outer diameter of the outlet 0.16 m.

    There is one cell of outlet on right hand side.

    Solid properties

    •   Density 2500 kg/m3

    •   Diameter 500 µm

    Fluid properties

    •   Air

    Grid size

    •   Radial ∆ x = 4.064 cm

    •   Axial ∆ y = 16.475 cm

    Number of Grid

    •   Radial × Axial 30 × 74 cells

    (Including boundary wall cells)Operating Condition

    •   Gas velocity 0.45 m/s

    •   Initial solid volume fraction 0.58

    •   Initial bed height 4 m.

    •   Temperature 300 K

    •   Pressure

    Case 1 Atmospheric pressure, 1 atm

    Case 2 25 atm

    4

    Fig. 5. System geometry and operating conditions for bubbling bed simulations.

    Fig. 6. Instantaneous plots of solid volume fraction field in bubbling bed

    simulations: (a) 1 atm, (b) 25 atm.

    The bed expansion ratio is calculated based on the fluidized

    bed height and the initial bed height, 

    =H f /H 0. Using David-

    son’s bubble-growth model (Darton et al., 1977), the equivalent

    Table 4A comparison of expanded height ratio and equivalent bubble diameter at

    two pressures

    P (atm) Expanded height ratio (H/Ho) Equivalent bubble diameter (m)

    1 1.21 0.70 ± 0.1425 2.40 0.58 ± 0.18

    bubble diameter can be estimated as a function of the distance

    above the distributor in a bubbling fluidized bed, as follows:

    DB = 0.54(U 0 − U mf )0.4(h + 4 A0)0.8/g0.2, (10)

    where DB is the equivalent bubble diameter, U 0 is the superfi-

    cial gas velocity,  U mf  is the minimum fluidization velocity, h

    is the initial bed height, 4√ A0 is 0.03 m for a porous-plate gas

    distributor.

    In the high-pressure system there is a high expansion and

    small bubbles, as shown in  Table 4. It is well known in the

    fluidization community (Sobreiro and Monteiro, 1982; Rowe

    et al., 1983; Piepers et al., 1984; Gidaspow, 1994) that under

    high pressure, Geldart’s Type B powders undergo considerable

    expansion before bubbling.

    At atmospheric pressure, the equivalent bubble diameter ob-

    tained from the above equation is 0.58 m. This shows a reason-

    able agreement with bubble size in Table 4.

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    0

    1

    2

    3

    4

    5

    20 25 30 35 40 45 50

       A  x   i  a   l   S  o   l   i   d   V  e

       l  o  c   i   t  y ,  m   /  s

    -0.8

    -0.6

    -0.4

    -0.2

    0

    0.2

    0.4

    0.6

    0.8

    20 25 30 35 40 45 50

       R  a   d   i  a   l   V  e   l  o  c   i   t  y ,  m   /  s

    Time, sec

    Time, sec

    Fig. 7. Typical time series of axial (a) and radial (b) hydrodynamic velocities

    (v)  for particles in the center region at a bed height of 4.

    3.2.3. Fluctuations

    Fig. 7 shows typical time series of axial and radial hydro-

    dynamic velocities for particles at 25 atm. The hydrodynamic

    velocities are obtained directly from the code. Their frequency

    distributions of hydrodynamic velocities are shown in Fig. 8(a).

    The main frequency for the axial direction is 0.233Hz, and

    the main frequency for the radial direction is 0.3Hz. Fig. 8(b)

    shows the power spectrum density of bed void at 25 atm. The

    dominant frequency   (f )  of porosity oscillations in the bub-bling bed can be estimated by an analytical solution, as follows

    (Gidaspow et al., 2001; Jung et al., 2005):

    f  =   12

     gH 

    1/2 (3s/g + 2)ss0

    1/2, (11)

    where s0  and  H 0  are some initial solid volume fraction and

    initial bed height.

    The time averaged solid volume fraction at the center of 

    column is approximately 0.80. The initial bed height is 4m. The

    calculated main frequency for porosity oscillations obtained

    from the above equation is 0.24Hz. This shows a reasonably

    good agreement with the main frequency in Fig. 8.

    0

    10

    20

    30

    40

    50

    60

    70

    80

    90

    100

    0 1 2 3 4 5

       P  o  w  e  r   S  p  e  c   t  r  a   l   M

      a  g  n   i   t  u   d  e

    Axial

    Radial

    0.000

    0.005

    0.010

    0.015

    0.020

    0.025

    0.030

    0 1 2 3 4 5

       P  o  w  e  r   S  p  e  c   t  r  a   l   M  a  g  n   i   t  u   d  e

    Frequency (Hz)

    Frequency (Hz)

    Fig. 8. Frequency and power spectral magnitude of (a) hydrodynamic ve-

    locities and (b) bed void. Main frequency: axial velocity = 0.233Hz; radialvelocity = 0.3 Hz; bed void = 0.3Hz.

    4. Dispersion coefficients

    There are two kinds of mixing in fluidization: that due to in-

    dividual particle oscillations and that due to cluster oscillations.

    An order of magnitude estimate of the dispersion coefficient

    due to individual particles oscillations can be obtained from

    the laminar granular temperature (Jiradilok et al., 2006). Tur-

    bulent dispersion coefficients can be obtained as a function of 

    normal Reynolds stress corresponding the Lagrangian integral

    time scale as described below.

    4.1. Turbulent dispersion coefficients calculation

    The dispersion coefficients in the radial and axial directions

    are expressed as in Hinze (1959), as follows:

    DT (a) = v(a)2T L, (12)

    where   v(a)2 is the mean square fluctuating velocity corre-sponding to normal Reynolds stress and  T L is the Lagrangian

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    -0.6

    -0.4

    -0.2

    0

    0.2

    0.4

    0.6

    0.8

    1

    0 0.1 0.2 0.3 0.4 0.5 0.6   A  x   i  a   l  c  o  r  r  e   l  a   t   i  o  n  c  o  e   f   f   i  c   i  e  n   t

     R L ( y,t ′)

    0

    -0.6

    -0.4

    0

    0.2

    0.4

    0.6

    0.8

    1

    0 0.1 0.2 0.3 0.4 0.5 0.6   R  a   d   i  a   l  c  o  r  e   l  a   t   i  o  n   C  o  e   f   f   i  c   i  e  n   t

    -0.2

     R L ( x,t ′)  0

    Time, sec

    0.20.40.6

    Time, sec

    0.6   0.4 0.2

    Fig. 9. Typical autocorrelation functions of solid phase (a) axial; (b) radial

    for  W s = 98.8 kg/m2

    s and  U g = 3.25 m/s.

    integral time scale of the particle and gas motion, defined by

    T L =  ∞

    0RL(a, t ) dt  =

      ∞0

    v(t)v(t  + t )v2

    dt , (13)

    where v  here is Lagrangian velocity fluctuations and the auto-correlation function, given by

    RL(a,t ) =  v

    (t)v(t  + t )v2

    . (14)

    Eulerian turbulence characteristics can be obtained from

    Lagrangian turbulence characteristic (Hinze, 1959). The rela-

    tionship between the Eulerian and the Lagrangian turbulence

    characteristics has been given by Hay and Pasquil as

    T L = T E , (15)where  is the coefficient, T E is the Eulerian integral time scale

    of the particle and gas motion.

    In order to estimate the order of magnitude of the disper-

    sion coefficient, the Eulerian integral time scale approximately

    equals Lagrangian integral time scale (Hinze, 1959):

    T L ≈ T E . (16)

    Eq. (12) is a special case of the second-order tensor disper-

    sion matrix, Eq. (6).

    Fig. 9 shows typical autocorrelation functions of solid phase

    in radial and axial directions for  W s = 98.8 kg/m2 s and  U g =3.25 m/s. The autocorrelation function decays with time from

    the maximum value of one, and goes to zero. For the radial

    autocorrelation, the profile dips below zero, then oscillates to astationary value of zero due to the wall limitation of  x  direction.

    For the direction of flow, the autocorrelation coefficient simply

    decayed exponentially, corresponding to Roy et al. (2005) in

    a liquid–solid system and Godfroy et al. (1999) in a gas–solid

    riser.

    4.2. Characteristic lengths

    Fig. 10 shows the snapshots of solid volume fractions of the

    computed clusters at 6.5, 7.5 and 8.5 s. The length and width

    of clusters can be approximated from characteristic lengths es-

    timated from the relation between the dispersion coefficientsand the oscillating velocity as

    Dispersion coefficients  (D)

    = characteristic length × oscillating velocity. (17)The oscillating velocities are obtained from the square root

    of normal Reynolds stress. Fig. 11 shows the radial distribution

    of characteristic lengths in the axial and the radial directions.

    The length and width of clusters depend on the position cor-

    responding to Fig. 10. The lengths and widths of clusters are

    approximately 10–100 and 0.5–4 cm, respectively.

    4.3. Turbulent dispersion coefficients

    The computed radial and axial particle and gas dispersion

    coefficients varied as a function of positions. The values of 

    dispersion coefficients are reported by averaging over the cross

    section of a riser or bubbling beds. The standard deviations are

    based on these lateral position variations.

    For flow of FCC particles in the riser, the experimental solid

    flux of 98.8 kg/m2 s and the gas velocity of 3.25m/s  (Wei

    et al., 1998a) was used for the computation of the gas and solid

    dispersion coefficients. The computation of gas dispersion co-

    efficients was studied based on Jiradilok et al. (2006) study. A

    comparison of radial and axial solid and gas dispersion coeffi-cients at various heights is summarized in Table 5. The solid

    and gas dispersions are of same order of magnitudes, because

    the Reynolds stresses per unit bulk density do not differ from

    each other. Substantially both dispersion coefficients vary from

    top to bottom of the riser, as expected. The simulations show

    that the radial dispersion coefficients in the riser are two to

    three orders of magnitude lower than the axial dispersion co-

    efficients.

    For bubbling commercial size fluidized bed simulations, the

    effects of pressure on dispersion coefficients were studied. A

    comparison of radial and axial solid and gas dispersion coef-

    ficients at 25 atm and at an atmospheric pressure is given in

    Table 6. They were calculated at 4 m above the distributor. The

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    3406   V. Jiradilok et al. / Chemical Engineering Science 62 (2007) 3397– 3409

    Fig. 10. Snapshots of solid volume fraction at 6.5, 7.5 and 8 s for  W s = 98.8 kg/m2

    s and  U g = 3.25 m/s.

    dispersion coefficients for the bubbling bed at 25atm are much

    higher than at atmospheric pressure due to the high bed expan-

    sion with smaller bubbles.

    Fig. 12 shows that solid dispersion coefficients increase with

    the bed diameter because the bubble diameter increases with

    the bed diameter. The computed axial solid dispersion coeffi-

    cients of bubbling bed at atmospheric pressure agree with the

    measured data. The differences between the simulations and

    the experiments are in part due to different definitions of the

    dispersion coefficients.

    Figs. 13 and 14 show the comparisons of computed axial

    and radial gas dispersion coefficients with the literature survey

    by Breault (2006). The computed dispersion coefficients are in

    the range of the literature data.

    The axial solid dispersion coefficients for 850 m cork parti-

    cle were generated at the National Energy Technology Labora-

    tory (NETL) and measured using the autocorrelation method.

    This is the same method used in this study. The data are re-

    ported in Fig. 15. A numerical simulation of the NETL riser is

    in progress.

    Comparisons between computed solid dispersion coefficients

    and the literature survey for both directions, axial and radial,

    are shown in Figs. 15 and 16, respectively. The computations

    show that the gas and the solid dispersion coefficients are close

    to each other in agreement with measurements. The simulations

    show that the radial dispersion coefficients in the riser are two

    to three orders of magnitude lower than the axial dispersion

    coefficients, but less than an order of magnitude lower for the

    bubbling bed at atmospheric pressure.

    5. Conclusions

    •  We have shown how to compute radial and axial particle andgas dispersion coefficients in the turbulent regime of a riserwith flow of FCC particles and in bubbling commercial size

    fluidized beds at low and high pressures.

    •  The dispersion coefficients were computed from the turbulentvelocity oscillations of the gas and the particles obtained

    by direct numerical solutions of the coupled Navier–Stokes’

    equations for gas–particle flow in the two fluid model.

    •  The computed dispersion coefficients are in reasonable agree-ment with the experimental measurements reported over the

    last half century. The CFD computations suggest that the re-

    ported differences in the dispersion coefficients may be due

    to geometrical effects of the risers and the bubbling beds,

    since the geometry strongly affects the local gas and particle

    velocities from which the dispersion coefficients are derived.

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    V. Jiradilok et al. / Chemical Engineering Science 62 (2007) 3397– 3409   3407

    •  The computed dispersion coefficients and the normal stressesallow the computation of characteristic lengths of clusters.

    The lengths and widths agree with snapshots of volume frac-

    tion of solid.

    0

    20

    40

    60

    80

    100

    0 0.2 0.4 0.6 0.8

       A  x   i  a   l   C   h  a  r  e  c   t  e  r   i  s   t   i  c   L  e  n  g   t   h   (  c  m   )

    0

    1

    2

    3

    4

    0 0.2 0.4 0.6 0.8

    r/R

       R  a   d   i  a   l   C   h  a  r   t  e  r   i  s   t   i  c   L  e  n  g   t   h   (  c  m   )

    200 cm 400 cm 600 cm

    r/R

    1

    1

    Fig. 11. Radial distributions of characteristic lengths (a) axial; (b) radial for

    W s = 98.8 kg/m2 s and  U g = 3.25 m/s.

    0.0001

    0.001

    0.01

    0.1

    1

    0.01 0.1

    Bed Diameter (m)

       S  o   l   i   d  s   D   i  s  p  e  r  s   i  o  n   C

      o  e   f   f   i  c   i  e  n   t  s   (  m   2   /  s   )

    10

    May (1959)

    Thiel and Potter (1978)Avidan et al (1985)Morooka et al (1972)Du et al (2002)

    Jung et al (2005)Lewis et al (1962)de Groot (1967)Liu and Gidaspow (1981)

    Lee and Kim (1990)Mostoufi et al (2001)Jung et al (2005)

    This study (Computation)

    Axial,

    Bubbling, 1 atm

    Computation

    A

    B

    1

    Fig. 12. Effect of the bed diameter on experimental and computed solid

    dispersion coefficients for bubbling and turbulent fluidized beds for Geldart

    A and B particles (Avidan and Yerushalmi, 1985; Du et al., 2002; de Groot,

    1967; Jung et al.,   2005; Lee and Kim, 1990; Lewis et al., 1962; Liu andGidaspow, 1981; May, 1959; Morooka et al., 1972; Mostoufi and Chaouki,

    2001; Thiel and Potter, 1978).

    1

    10

    0 1 2 3 4 5 6 7 8 9 10

       A  x   i  a   l   G

      a  s   D   i  s  p  e  r  s   i  o  n   C  o  e   f   f   i  c   i  e  n   t   (  m   2   /  s   )

    Dry, 1989

    Kim, 1998 & 1999

    Wei, 2001

    CFD-Riser-FCC particles

    Li , 1989

    4 m.

    2 m.6 m.

    CFD,1 atm(Bubbling)

    Gas Velocity (m/s)

    0.1

    0.01

    0.001

    0.0001

    Fig. 13. Effect of the gas velocity on experimental and computed axial gas

    dispersion coefficients (Dry and White, 1989; Kim and Namkung, 1998,

    1999; Li and Weinstein, 1989; Wei et al., 2001).

    Table 5

    A comparison of computed radial and axial solid and gas dispersion coefficients at various heights for FCC particles in a riser

    Height (m)   s   Solid dispersion coefficient (m2 /s) Gas dispersion coefficient (m2 /s)

    Axial Radial Axial Radial

    2 0.25 0.347 ± 0.120 0.002 ± 0.001 0.614 ± 0.323 0.004 ± 0.0044 0.14 1.221 ± 0.289 0.001 ± 0.001 2.032 ± 0.927 0.002 ± 0.0026 0.04 1.331 ± 0.757 0.009 ± 0.005 1.134 ± 0.821 0.004 ± 0.003

    Table 6

    A comparison of computed radial and axial solid and gas dispersion coefficients at two pressures for the bubbling beds

    P (atm)   s   Solids dispersion coefficient (m2 /s) Gas dispersion coefficient (m2 /s)

    Axial Radial Axial Radial

    1 0.41 0.069 ± 0.042 0.012 ± 0.005 0.06 ± 0.027 0.019 ± 0.01125 0.20 0.791 ± 0.373 0.030 ± 0.009 0.891 ± 0.464 0.040 ± 0.018

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    3408   V. Jiradilok et al. / Chemical Engineering Science 62 (2007) 3397– 3409

    1

    10

    0 1 2 3 4 5 6 7 8 9 10

       R  a   d   i  a   l   G  a  s   D   i  s  p  e  r  s   i  o

      n   C  o  e   f   f   i  c   i  e  n   t   (  m   2   /  s   )

    Adanez, 1997Leckner, 2002

    Werther, 1992

    Rhodes,1993

    Wei, 2001Leckner,(hot) 2000

    Leckner,(cold) 2000

    4 m.2 m.6 m.

    CFD, 1atm

    (Bubbling)

    CFD-riser(FCC particles)

    0.1

    0.01

    0.001

    0.0001

    Gas Velocity (m/s)

    Fig. 14. Effect of the gas velocity on experimental and computed radial

    gas dispersion coefficients (Adanez et al., 1997; Leckner et al., 2000, 2002;

    Rhodes et al., 1993; Wei et al., 2001; Werther et al., 1992).

    0.001

    0.01

    0.1

    1

    10

    100

    0.01 0.1

       A  x   i  a   l   S  o   l   i   d  s   D   i  s  p  e  r  s   i  o  n   (  m   2   /  s  e  c   )

    Duet al. (2002)Thiel and Potter (1978)Aviden and Yerushalmi (1985)Weiet al. (1998)Weiet al. (1995)Gidaspow et al. (2004), IITRiserNETL unit,850µm Cork particlesJiradiloket al. (2006), FCC particles2 m.4 m.6 m.This study (Bubbling Bed, 500 µm)

    Experiment

    Computation

    Bubble

    Computation, 1 atm

    Single particleoscillation

    FCCparticles-Cluster

    Computation

    NETL ,

    Cork particles

    1 10

    Gas Velocity (m/sec)

    Fig. 15. Effect of the gas velocity on experimental and computed radial solid

    dispersion coefficients (Du et al., 2002; Thiel and Potter, 1978; Avidan and

    Yerushalmi, 1985; Wei et al., 1995, 1998a, b; Gidaspow et al., 2004; Jiradilok 

    et al., 2006).

    1

       R  a   d   i  a   l   S  o   l   i   d  s   D   i  s  p  e  r  s   i  o

      n   (  m   2   /  s  e  c   )

    Du et al. (2002)Koenigdorff and Werther (1995)Wei et al. (1998)We i et al. (1995)Jiradilok et al. (2006), FCCparticles

    2 m.4 m.6 m.

    This study (Bubbling Bed, 500 µm)

    Experiment

    Computation

    BubbleComputation, 1 atm

    FCC particles-clusterComputation

    0.1

    0.01

    0.001

    0.1 1

    Gas Velocity (m/sec)

    10

    Fig. 16. Effect of the gas velocity on experimental and computed radial solid

    dispersion coefficients (Du et al., 2002; Koenigsdorff and Werther, 1995; Wei

    et al., 1995, 1998a, b; Jiradilok et al., 2006).

    •  The dispersion coefficients for the bubbling bed at 25 atm aremuch higher than at atmospheric pressure due to the high bed

    expansion with smaller bubbles.

    •  The computations show that the gas and the solid dispersioncoefficients are close to each other in agreement with mea-

    surements. The simulations show that the radial dispersion

    coefficients in the riser are two to three orders of magnitudelower than the axial dispersion coefficients, but less than an

    order of magnitude lower for the bubbling bed at atmospheric

    pressure.

    Notation

    a x  or y directions

    c   instantaneous velocity

    Cd    drag coefficient

    d k   characteristic particulate phase diameter

    DP    particle dispersion coefficients

    DT    turbulent dispersion coefficients, due to cluster os-

    cillationse   coefficient of restitution

    g   gravity

    g0   radial distribution function at contact

     L   mean free path

    P   continuous phase pressure

    P k   dispersed (particulate) phase pressure

    vi   hydrodynamic velocity in i direction

    vi   mean particle velocity in i direction

    vivj    Reynolds stress (i = j  normal Reynolds stress; i =

    j  shear Reynolds stress)

    Greek letters

    B   interphase momentum transfer coefficient

      energy dissipation due to inelastic particle collision

    k   volume fraction of phase k 

      granular temperature

      granular conductivity

      bulk viscosity of phase k 

    k   shear viscosity of phase k 

    k   density of phase k 

    k   stress of phase k 

      specularity coefficient

    Acknowledgment

    This study was supported by the US Department of Energy

    Grant (DE-FG26-06NT42736).

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