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these multiphase processes are widely applied detailed under-

standing of the relative motion between the particles and the

fluid and their corresponding influence on mass, heat and

momentum transfer [for instance, Joshi et al., 1980, 1981, 2003;

Thakre and Joshi, 1999; Murthy et al., 2007] has been limited for

two reasons. First, non-intrusive experimental observations are

very difficult to undertake, especially of the liquid motion in the

interstices between the particles, droplet or bubbles. Second, the

computational requirements for direct numerical modelling of such a complex flow have been prohibitively large. Recent

advances in both flow visualization techniques and computing

capability have reduced these limitations to such an extent that

detailed investigation of multiphase systems that more closely

reflect actual industrial practices is now possible.

In this study, particle image velocimetry (PIV) has been

utilized to quantify both the particle slip and interstitial liquid

velocity within a solid–liquid fluidized bed in the Reynolds

number range of 51–759. The experimental measurements are

complemented by in-house direct numerical simulations (DNS) to

quantify the influence of the presence of neighbouring particles

on the wake structure and settling velocity of freely falling

particles. The DNS is carried out using the in-house code reported

in Jin et al. (2009), whereby the hydrodynamic force between the

particle and the fluid is resolved without the need for assuming a

given drag law relationship.

1.1. Previous work

Numerous experimental and theoretical studies (e.g.

Richardson and Zaki, 1954; Hanratty and Bandukwala, 1957;

Garside and A1-Dibouni, 1977; Joshi, 1983; Pandit and Joshi,

1998; Gevrin et al., 2008; Reddy and Joshi., 2009; Zivkovic et al.,

2009) have investigated bed expansion characteristics of

solid–liquid fluidized beds; and whilst these studies are useful

they give no real insight into the flow behaviour in the interstices

between the particles. There have been a number of studies that

have attempted to address this issue. Handely et al. (1966)

undertook photographic measurements in beds of 3 mm glassbeads at ReN of 45, 87 and 182 in a 75 mm diameter glass column

and found that the root mean square fluctuating velocity in the

axial direction is about 2.5 times higher than that in the radial

direction. Chen and Kadambi (1990) studied solid–liquid slurry

flows in a horizontal pipe using Laser Doppler Velocimetry (LDV)

and refractive index matching of the solid and liquid phases. They

used silica-gel particles having an average size of 40 mm and 50%

W/W sodium iodide aqueous solution. They were able to measure

the average axial liquid velocity for silica-gel concentrations

between 5 and 50% W/W. Neither Handely et al. (1966) nor

Chen and Kadambi (1990) reported turbulence quantities.

Chen and Fan (1992) applied PIV to three phase gas–liquid–

solid fluidized beds to obtain 49 velocity vectors that provided an

excellent basis for quantifying the mean flow. However, thenumber was well below the $100,000 required for reliable

estimation of turbulence intensity, turbulent kinetic energy,

turbulent energy dissipation rate and structure functions. Haam

et al. (2000) combined PIV measurements with refractive index

matching to obtain velocity information inside solid–liquid flui-

dized beds at solids concentrations much higher than was pre-

viously possible. They measured the axial and radial velocity

components in a fluidized bed of glass beads for ReN values of

1040 and 1550. They found that the axial turbulent intensity of

the fluid increased by up to 70% due to the presence of the

glass beads.

Dijkhuizen et al. (2007) coupled their PIV measurements with

simultaneous measurement of the instantaneous velocity and

granular temperature fields. The PIV algorithm was specifically

optimized for dense granular systems and applied to a fluidized

bed at incipient fluidization conditions into which both single-

and multi-bubbles were injected. They observed that the highest

granular temperature was in the vicinity of the bubble(s), and for

the case of 1.5 mm glass particle bed the granular temperature

(Gidaspow, 1994) was in the range of 0.022–0.069 m2 sÀ2.

Shi (2007) applied PIV to investigate the particle motion and

cluster properties in a gas–solid two-phase flow in a circulating

fluidized bed riser. Visual images and micro-structure of variousclusters were captured. After the boundary of clusters was

determined by the gray level threshold method, clusters were

classified by the distance between particles and the shape and

position of clusters. In addition, the process of cluster formation

and breakup was described, and the sizes of clusters were also

obtained. With the Minimum Quadric Difference cross-correlation

algorithm suitable for high-density particles, the axial velocities

of the particles were obtained in the dilute phase section. Analysis

of the magnitude and distribution of particle axial velocity in the

radial direction showed at most radial cross-sections a parabolic

profile in the upward direction. The magnitude of axial velocity in

the core region was found to be higher than that in the near wall

region of the riser.

Muller et al. (2009) employed simultaneous PIV and Planar

Laser-Induced Fluorescence (PLIF) measuring techniques to

investigate the eruption of both a single and continuous stream

of bubbles in the freeboard region of a fluidized bed. The observed

bubble eruption patterns were in general agreement with the

bubble models published in the literature. Based on the calculated

vorticity of the gas in the freeboard it was found that the bubble

induced turbulence decays rapidly. Stereoscopic PIV measure-

ments of the out-of-plane component of the liquid velocity were

found to be not negligible.

Kashyap et al. (2011) used PIV to obtain laminar and turbulent

properties near the wall in the developing region of circulation of

Geldart B type particles in the riser part of circulating gas–solid

fluidized bed. Instantaneous velocities for the solid phase were

measured simultaneously in the axial and radial directions using

a CCD camera and a coloured rotating transparency. A novelmethod was used to determine axial and radial solid phase

dispersion coefficients using the autocorrelation technique.

The measured laminar and Reynolds stresses, laminar and turbu-

lent granular temperatures, laminar and turbulent dispersion

coefficients and energy spectra all exhibited anisotropy. The total

granular temperatures were in reasonable agreement with the

literature values. However, the axial and radial solid dispersion

coefficients measured near the wall were slightly lower than the

radially averaged values in the literature.

Hernandez-Jimenez et al. (2011) investigated both experimen-

tally and computationally the hydrodynamics of a rectangular,

bubbling air-fluidized bed of 5 mm thickness. The authors used

PIV and Digital Image Analysis (DIA) to obtain quantitative

information on bubble hydrodynamics, dense-phase probability,and time-averaged vertical and horizontal components of the

dense-phase velocity as a function of gas flow rate through the

bed. Simulations were also conducted using an Euler–Euler two-

fluid approach based on two different closure models for the gas–

particle interaction, namely the drag models of (1) Gidaspow

(1994) and (2) Syamlal and O’Brien (1989). Good agreement

between experimental observation and simulation results was

obtained for dense-phase probability, gas bubble diameter rise

velocity. The vertical component of the simulated dense-phase

velocity, however, was found to be nearly an order of magnitude

larger than that obtained from the PIV experiments.

Van Buijtenen et al. (2011) applied PIV, Positron Emission

Particle Tracking (PEPT), and Electrical Capacitance Tomography

(ECT) measuring techniques to quantify the flow in both quasi-2D

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and 3D spouted fluidized beds. In the pseudo-2D bed the

measurements were able to highlight the appearance of dead

zones in the annulus near the bottom of the bed in the spout-

fluidization regime. Measurements were also made in the jet-in-

fluidized-bed regime beds with spout heights of 0, 20, and 40 mm.

The results from both experimental systems were compared with

full 3D discrete particle modelling (DPM) simulations; with

generally good agreement being achieved except for slight over-

prediction of the velocities in the annular region for the quasi-2Dcase where wall effects in the experimental system are possibly

higher than assumed in the modelling.

In addition to the experimental investigations there have been

many studies involving simulations of multi-particle systems. For

a recent review of the topic, with particular focus on DNS, see

Reddy et al. (2010b). Some more recent studies include:

Deen et al. (2009) have reported DNS results for multi-fluid

flows in which simultaneously moving deformable (drops or

bubbles) and non-deformable (particles) elements are present,

possibly with the additional presence of free surfaces. They utilized

a front tracking (FT) model to circumvent the explicit computation

of the interface curvature. They also used an immersed boundary

(IB) model to incorporate both particle–fluid and particle–particle

interaction via a direct forcing method and a hard sphere Discrete

Particle approach. The capabilities of the hybrid FT-IB model are

demonstrated by the authors with a number of examples in which

complex topological changes in the interface are encountered.

Uhlmann and Pinelli (2009) conducted a DNS study of dilute

turbulent particulate flow in a vertical plane channel, considering

up to 8192 finite-size rigid particles with numerically resolved

phase interfaces. The particle diameter corresponded to approxi-

mately nine wall units, with a terminal Reynolds number of 136.

The bulk fluid upflow was maintained at a Re¼2700 to maintain

solids suspension. Two different density ratios were simulated,

which varied by a factor of 4.5. The corresponding Stokes

numbers for the two particles were O(10) in the near-wall region

and O(1) in the outer flow. The DNS simulations indicated that the

mean flow velocity profile tended towards a concave shape, and

anisotropy for both the turbulence intensity and normal stresses.Large-scale elongated streak-like structures were predicted by the

DNS, with stream-wise dimensions of the order of eight channel

half-widths and cross-stream dimensions of the order of one

channel half-width. There was no evidence for the formation of

particle clusters, which suggested that the large structures were

due to an intrinsic instability of the flow that was triggered by the

presence of the particles.

Xu and Subramaniam (2010) have reported DNS results for

turbulent flow past uniform and clustered configurations of fixed

particle assemblies at the same solid volume fraction. Their

approach was based on a discrete-time, direct-forcing immersed

boundary method that imposes no-slip and no-penetration

boundary conditions on each particle surface. Results are reported

for mean flow Reynolds number of 50 where the ratio of theparticle diameter to the Kolmogorov scale was 5.5. The DNS

confirmed experimental observations that the fluid-phase turbu-

lent kinetic energy was enhanced by clustered configurations

leading to increased levels of anisotropic turbulence.

Tenneti et al. (2011) have reported particle-resolved DNS

results of interphase momentum transfer in flow past fixed

random assemblies of mono-disperse spheres with finite fluid

inertia using a continuum Navier–Stokes solver. This solver is

based on a new formulation called the Particle-resolved Unconta-

minated-fluid Reconcilable Immersed Boundary Method (PURe-

IBM). The principal advantage of this formulation is that the fluid

stress at the particle surface is calculated directly from the flow

solution (velocity and pressure fields), which when integrated over

the surfaces of all particles yields the average fluid–particle force.

The PUReIBM is a consistent numerical method to study gas–solid

flow because it results in a force density on particle surfaces which

is reconcilable with the averaged two-fluid theory. The numerical

convergence and the accuracy of PUReIBM approach were estab-

lished through a comprehensive suite of validation tests. The

normalized average fluid–particle force, F , was obtained as a

function of solid volume and mean flow Reynolds number for

random assemblies of mono-disperse spheres. From the simula-

tions, a simple correlation for F , and hence drag coefficient, wasdeveloped for the particle Re range 100–300. Given that the

simulations were based on a fixed particle assembly, any effects

of mobility of the particles was not included. However, approach is

a good approximation for high Stokes number particles, which are

encountered in most gas–solid flows.

1.2. Research priorities

The great majority of industrial systems are operated under

turbulent conditions, wherein a compendium of eddies (turbulent

structures) of different length and time scales govern the momen-

tum, heat and mass transfer and mixing behaviour. For this reason

it is desirable to more fully understand the formation and

dynamics of these turbulent structures. All the above-mentioned

studies on PIV measurement and DNS computations have provided

an excellent foundation, and with recent advances in experimental

and mathematical techniques there is an opportunity to expand

that knowledge for multi-particle systems. The following list of 13

research priorities is aimed at coupling both experimental and

computational investigations. The priorities include:

1. DNS simulation for creeping flow (ReNo1) around a single

particle in both an infinite liquid and a solid–liquid fluidized

bed. Estimation of skin, form and total friction at all locations

on the surface of the particle. Estimation of total drag

coefficient at different solid volume fractions in the fluidized

bed to quantify the hindrance effect.

2. DNS simulation of flow around a particle in the ReN range of

1–103 to obtain 3D component instantaneous velocities attemporal resolutions up to the Kolmogorov scale. Simulation

of boundary layer separation and also the size, shape and

stability of wakes behind a particle with respect to ReN.

Estimation of skin and form drag on the entire surface of the

particle.

3. DNS simulation of flow around a single particle in the ReNrange of 103–106, capturing the sudden drop in drag coeffi-

cient, C D, at ReN of around 2 Â 105 due to the transition from

laminar to turbulent boundary layer separation.

4. DNS simulation of settling of a particle in a Hele-Shaw cell to

include the wall effect.

5. DNS simulation of flow around two neighbouring particles.

6. DNS simulation of a solid–liquid fluidized bed in the ReN

range of 1–106 to obtain 3D component instantaneous velo-cities at temporal resolutions up to a small scale permissible

by the computational power. Estimation of skin, form and

total friction at all locations on particles within the fluidized

bed. Simulation of boundary layer separation in multi-

particle systems and also understanding of size, shape and

stability of wakes with respect to ReN and solids volume

fraction.

7. DNS simulation of solid–liquid fluidized bed where the momen-

tum transfer is accompanied by mass/heat transfer at the wall.

Quantification of heat and mass transfer coefficients as a function

of ReN and ALutilizing u0u0, u0c 0 and u0T 0 simulated results.

8. Quantification of homogeneity, isotropy and energy spectrum

in solid–liquid dispersions as a function of ReN, AL and

particle diameter and shape.

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9. Development of scaling laws for the inertial, dissipation and

large scale sub-range in solid–liquid dispersions.

10. DNS simulation for estimation of interface forces, such as

drag, lift, virtual mass, and Basset; and subsequent applica-

tion in the closure formulation for RANS and LES simulations

of solid–liquid dispersions.

11. Measurement of instantaneous velocity within the fluidized

bed using techniques such as high speed PIV, at sampling

rates of up to 10 kHz, in combination with refractive indexmatching of the liquid and solid phases. Application of

mathematical techniques, such as discrete wavelet trans-

forms (DWT), continuous wavelet transforms (CWT) and

proper orthogonal decomposition (POD), to identify and

characterize the flow structures in terms of shape, size,

velocity and energy distributions.

12. To understand the mechanism of vorticity generation and

hence the origin of turbulence.

13. Relate momentum, heat and mass transfer to turbulent

structure dynamics.

Priorities 1 and 2 for the authors have already been reported in

Reddy et al. (2010a, 2010b). This study is focused on addressing

priorities 3, 6 and 11. In terms of DNS (Priorities 3 and 6),simulating up to 245 particles (as performed previously) with

Reynolds number range extended to 200. In terms of experimen-

tal development (Priority 11), standardization of PIV measure-

ment in terms of matching of refractive indices of the solid

particles and liquid; achieving sampling frequencies of 2 Hz over

a wide range of Reynolds numbers; and data processing in terms

of mean and turbulence characteristics which can be directly

compared to DNS results.

2. Experimental

2.1. Solid–liquid fluidized bed apparatus

The schematic of experimental apparatus is shown in Fig. 1. It

consisted of an acrylic circular column (1), with an inner diameter

of 50 mm and a height of 300 mm. A 0.5 HP centrifugal pump

(6) was used for pumping the refractive index-matched liquid.

A calming section packed (4) with 3–4 mm glass beads of 0.3 m

height was provided to homogenize the liquid flow before it

reached the liquid distributor. The distributor was a perforated

plate (3) containing 128 holes of 2 mm diameter on a triangular

pitch of 3.1 mm. The circular test section of the fluidized bed was

encased with a square tank (2) which was filled with a refractive

index-matched liquid to allow for Particle Image Velocimetry

(PIV) measurements within the bed.

2.2. Solid–liquid refractive index matching

PIV measurement within opaque fluidized beds is only possible

at solids concentrations of only a few percent by volume. However,

if the refractive index of the particle is matched1 to that of the fluid

then it is possible to take reliable measurements at solids con-

centrations greater than 50% by volume. In this study, different

mixtures of organic liquid and inorganic salts were screened for

matching the refractive index of 1.47 of high precision borosilicate

glass beads (d p¼3 mm and r p¼2230 kg mÀ3). As indicated in

Table 1, from the screening process it was found that Solution 12

(68% turpentine and 32% tetra-hydronaphthalene) provided the

best match for the RI of the borosilicate glass beads as well as being

compatible with the acrylic column. Paraffin oil, with very similar

RI to that of the solution and glass beads, was also added in small

quantities to change the viscosity of the solution and still maintain

the required RI.

2.3. PIV measurements

A schematic of the TSI PIV set-up is shown in Fig. 2. The laser

source was provided by a pulsed Nd:YAG laser that had a pulse

duration of 6 ns and was synchronized with the camera. The time

difference between the two laser pulses was optimized based on

Nyquist criteria. The optics included a combination of cylindrical

and spherical lenses that produced a thin laser sheet of 0.1 mmthickness. Images were captured at a frequency of 2 Hz using a

high-resolution 4 M CCD camera (2048Â 2048 pixels) placed

perpendicular to the laser sheet. The refractive index matched

liquid was seeded with silver-coated hollow glass particles with a

mean diameter equal to 20 mm. The interrogation area was set at

50 Â 100 mm (64 Â 64 pixels, with a 50% overlap) resulting in

approximately 1500 vectors per image.

Post-processing of the captured raw PIV images was under-

taken to determine the velocity vectors. Out-of-plane motion of

the seeding particles and strong local velocity gradients caused

some spurious velocity vectors. Median filtering, with a threshold

value 1.5 times the median of surrounding vectors, was applied to

filter the high spurious vectors. A signal-to-noise ratio of 4 was

applied to filter the low spurious vectors. Parameters like time

Outlet

50

100

300

InletRefractive index

matching liquid

PUMP

Acrylic column

Outer square column

Distributor

Gasket

VentCalming section

Flange

1

Heat

Exchanger

8

2

3

4

5

7

6

9

Fig. 1. Solid–liquid fluidized bed experimental setup (all dimensions are in mm)

[(1) acrylic column; (2) outer square column; (3) distributor; (4) calming section;

(5) flange; (6) pump; (7) refractive index matching liquid; (8) heat Exchanger;

(9) gasket].

1 An excellent review on the subject of RI matching has been written by

Wiederseiner et al. (2011).


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difference between laser pulses, light sheet thickness and seeding

density were optimized so that spurious vectors remained below

2%. The PIV measurements were performed in darkness to

minimize light contamination from external sources. Additional

information2 pertaining to the PIV measurements can be found in

Deshpande et al. (2009, 2010) and Sathe et al. (2010).

3. Direct numerical simulation

3.1. Simulation set-up

The physical properties of particles and liquid used to simulate

different particle Reynolds numbers within a fluidized bed are the

same as those used in Reddy et al. (2010b) and are listed in

Table 3. Direct numerical modelling was undertaken for 1, 9, 27,

100, 180 and 245 particles contained. For each system, corre-

sponding to a desired particle hold-up, the particles were initially

arranged in a regular lattice. At time, t ¼0, the particles were

allowed to settle through a rectangular domain with width and

depth of 10 and 20 particle diameters, respectively. A moving

reference frame method was employed, resulting in the rectan-gular domain to move downward with a velocity equal to the

average velocity of particle ensemble. At steady state, this situa-

tion is comparable to a fluidized bed where the heavier particles

are suspended by the upward velocity of the liquid. The compar-

ison between the particle and liquid velocity reference conditions

is shown in Fig. 3. For the fluidized bed (A), the liquid rises

upwards with particle settling velocity; whilst for the DNS

domain (B) the volume averaged liquid velocity is zero and the

particle is moving down with its settling velocity. The domain is

sliding downward with the same velocity as the particle. In both

cases, the relative velocity of particles with respect to the walls of

the domain is zero, and the relative velocity of the liquid with

respect to the domain walls is equal to settling velocity of the

particles, in the upward direction. The moving reference framehas two advantages for this application. First, very long settling

times can be simulated by using a domain with finite dimensions

making the simulations computationally affordable; and second,

it is a relatively straightforward to establish a fluidized bed with a

given particle hold-up. The initial conditions for the DNS are

summarized in Table 4. The side faces of the box have been

treated as wall boundary condition. The top face of the box was

treated as a pressure outlet, whilst the bottom face was treated as

the velocity inlet with zero velocity magnitude. The initial particle

positions are shown in Fig. 4.

Table 1

Screening of refractive index matching liquid for PIV experiments.

ID Liquid RI25 1C Remax Remarks

1 90% W/W glycerinþ10% W/W water 1.45 2 RI not exactly matched

2 Sodium idiode solution (55% W/W) 1.475 210 Highly corrosive

3 Paraffin oil light 1.465 23 For low ReN4 Potassium thiocyanate solution (42% W/W) 1.46 200 Harmful and corrosive

5 Ammonium thiocyanate solution (45% W/W) 1.47 200 Harmful and corrosive

6 P-cyamene 1.465 545 Specialty chemical7 Turpentine 1.44 850 RI not matched

8 Be nzene o r methyl benzoate with turpentine 1.47 900 Not compatib le with the acryl ic column (col umn was c racked)

9 30% W/W naphthale neþ70% W/W turpentine 1.46 5 750 Corrosive and not stable , turned in to pale yell ow

10 Turpentineþchloro-naphthalene 1.465 850 Chlorine compounds are not compatible with the acrylic column

11 Turpentineþbenzyl alcohol 1.47 850 Not compatible with the acrylic column (column was cracked)

12 68% Turpentineþ3 2% Tet ra hy dro n ap ht halene 1 .4 67 7 60 RI has b een matched and com patible wit h t he acry lic column

Table 2

PIV experimental conditions.

PIV exp

no.

Paraffin

light oil %

W/Wa

mL

(Pa s)

rL

(kg mÀ3)

RI (–) V S N(m sÀ1)

eL range

(–)

ReN(–)

1 85 0.01010 855 1.462 0.199 0.52–0.71 51

2 63 0.00420 870 1.471 0.266 0.46–0.75 164

3 48 0.00280 880 1.468 0.298 0.56–0.72 276

4 34 0.00193 890 1.469 0.309 0.53–0.73 437

5 18 0.00165 900 1.466 0.317 0.51–0.73 525

6 6.5 0.00142 910 1.470 0.325 0.46–0.66 625

7 0.0 0.00120 915 1.470 0.333 0.49–0.75 759

a Added to liquid 12 listed in Table 1.

Synchronizer

Laser

Solid-liquidfluidized bed

PumpRefractive

indexmatching

liquid

Camera

Fig. 2. Schematic diagram of PIV setup.

2 For a more general discussion on the topic see Tokuhiro et al. (1998), Deen

et al. (1999), Lindken and Merzkirch (1999, 2002) and Broder and Sommerfeld

(2003).


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3.2. Fictitious domain formulation

The DNS simulations were carried out using the fictitious

domain method described previously (see Diaz-Goano et al.,

2003; Veeramani et al., 2007). Briefly, the fluid and particles

occupy domains O1 and O2, respectively, and the relevant

equations of continuity and motion are applied to each domain

as appropriate. In the fictitious domain approach the equations

for the fluid are extended into the particle domain, such that

O¼(O1UO2), resulting in savings in computational time as the

liquid domain no longer needs to be re-meshed each time there is

particle motion. The relevant equations3 are

Du1

Dt

¼ Àr p1 þ 1Re r 2u1 þ

r2,iÀr1

r1G À F ð Þ, r Uu1 ¼ 0 inO ð1Þ

and

DU 1Dt

¼1

V i

Z O2,i

FdO ð2Þ

where

G ¼1Fr e g inO2,i, i ¼ 1,:::,n

0 inO1

(ð3Þ

and

F ¼1Fr e g þ

r1r2,iÀr1

^F inO2,i, i ¼ 1,:::,n

0 inO1

(ð4Þ

and

^F ¼À Du1

Dt þ 1Re r 2u1Àr p1 inO2,i, i ¼ 1,:::,n

0 inO1

(ð5Þ

where F is the interaction force. For collision of a particle with the

plane wall, the interaction force can be replaced by the lubrica-

tion force given by tenCate et al. (2002):

^ F W

i ¼À6pr iU ?m1

r i^h

À r ih

if ^hoh

0 otherwise

(ð6Þ

An operator splitting procedure is applied to the discretization

process, with a rigid body constraint being applied to the particles

in the last time step. Finally, the solver is parallelized using the

PETSc libraries ( Jin et al., 2008, 2009) on a shared memory system.

4. Results and discussion

4.1. PIV measurements

Seven experiments were performed using the refractive index

matched solid–liquid fluidized bed. A summary of the experi-

mental conditions is given in Table 2. Two sample raw images at

ReN of 625 and superficial liquid velocities of 0.046 and 0.116 m/s

are shown in Fig. 5. The post-processing of instantaneous velocityflow field at ReN of 625 and superficial liquid velocity of

0.046 m sÀ1 is shown in Fig. 6. The radial variation of the average

axial velocity, vL, at superficial liquid velocities of 0.046 and

0.116 m sÀ1is shown in Fig. 7. It can be observed that these

velocity profiles are almost flat in the radial direction.

It can be observed that at ReN of 625 and at superficial liquid

velocity, V L, of 0.047 m sÀ1 the axial turbulent intensity was on

average 69% and the radial turbulent intensity was around 42%

(Fig. 8A). An increase in the V L from 0.047 to 0.116 m sÀ1, results in

a decrease in the solid volume fraction of the bed from 0.54 to 0.355.

Therefore the turbulent intensity (ratio of root mean square velocity

to the average velocity) also decreases to about 47% in the axial

direction and around 28% in radial direction (Fig. 8B).

It has been observed that at ReN¼437 and at V L

of 0.047 m sÀ1

the axial turbulent intensity is 59% and radial turbulent intensity is

around 36%. In order to examine the effect of Reynolds number on

the turbulent intensity, the Reynolds number has been varied over

the range 51–759. The variation of average axial and radial

turbulent intensities with respect to Reynolds number is shown in

Fig. 9. At the lowest Reynolds number of 51 it was observed that the

axial turbulent intensity is only 7% whilst the radial turbulent

intensity is around 2%. When the Reynolds number is increased

from 51 to 164 it was observed that the axial and radial turbulent

intensities increase to 19% and 8%, respectively. Further increase in

Reynolds number from 164 to 759 led to an axial turbulent intensity

of 75% and a corresponding radial turbulent intensity of 47%. In the

turbulent regime, at ReN4500 (Clift et al., 1978; Joshi, 1983), it was

found that the axial turbulent intensity is 1.65 times that of the

radial turbulent intensity. This value is lower than the value of 2.5 reported by Handely et al. (1966). The image capture and

processing technique employed here, utilizing a laser light sheet

to clearly illuminate the particle boundary, is arguably more precise

than the photographic measurement of Handely et al. (1966). Based

on the limited number of experimental measurements, the follow-

ing simple relationships are proposed for the relationships between

the turbulence intensity in the axial, v0, and radial, u0, directions and

the solid phase holdup, AS , and slip velocity, V S :

v0

V S ¼ 1:29AS ð7Þ

u0

V S ¼ 0:78AS ð8Þ

Table 3

DNS physical properties.

ReN (–) 1 5 10 30 65 100 200

dP (mm) 15 15 15 15 15 15 15

rp (kg mÀ3) 1120 1120 1120 1120 1120 1120 1120

rL (kg mÀ3) 900 900 900 900 900 900 900

mL (Pa s) 0.5700 0.2210 0.1430 0.0680 0.0390 0.0282 0.0166

VS

VL=0

H

Liquid stationary,

Control volume

moving down

with velocity VS

VS=

VL

H

VL

Control volume

stationary,

Liquid moving

up with velocity

VL

Fig. 3. Schematic of fluidized bed and moving reference frame DNS simulation

framework.

3

See Veeramani et al. (2007) for definition of terms.


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The radial variation of turbulent kinetic energy at different

Reynolds numbers is shown in Fig. 10. It can be seen that

as Reynolds number increases from 51 to 759, the turbulent

kinetic energy also increases from 2 Â 10À5 to 4 Â 10À3 m2 sÀ2.

Moreover, for each Reynolds number there is practically no

variation in the turbulent kinetic energy profile along the radial

direction, which is similar to that for the turbulent inten-

sity profiles. Both of these observations are further validations of

the homogenous nature of turbulence in the solid–liquid

fluidized bed.

Fig. 4. Initial particle positions for (A) 1, (B) 9, (C) 27, (D) 100, (E) 245 particles [see Fig. 14A for orientation for 180 particles].

Fig. 5. Raw images at ReN¼ 625 for V L¼(A) 0.046 and (B) 0.116 m sÀ1.

Table 4

DNS initial conditions.

Number of particles 1 9 27 100 180 245

dP (mm) 15 15 15 15 15 15

r p (kg mÀ3) 1120 1120 1120 1120 1120 1120

Ln (–) —— 1.2 1.2 1.2 1.2 1.2

Comp. domain dimension (mm) Â (mm) Â (mm) 8 Â 16 Â 8 8 Â 16 Â 8 8 Â 16 Â 8 8 Â 16 Â 8 8 Â 16 Â 8 8 Â 16 Â 8

Number of nodes (–) 1.2Â 106 1.9 Â 106 2.6 Â 106 4.1 Â 106 5.2 Â 106 6.5 Â 106


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4.2. Direct numerical simulations

4.2.1. Wake structure

Fig. 11 shows the wake structure for a particle both in isolation

and with other eight particles present. It can be observed that at

the wake structure of a single particle at ReN¼200 in (A) an

infinite medium; and (B) surrounded by eight other particles at

centre-to-centre distance of 1.2 d p. It can be seen that whilst the

separation angle was similar for both particles the wake length

was reduced by almost 33% due to the presence of other particles.

For a single particle in an infinite medium the axisymmetric

toroidal vortex of the wake has been observed up to ReN¼210.

This same behaviour is exhibited for the isolated particle shown

in Fig. 11A, which is not surprising given that ReN¼200. For the

case of the nine particles this axisymmetry in the wake is broken

as shown in Fig. 11B. The reason for this early instability in the

wake is attributed to the asymmetric velocity gradients aroundthe particle, resulting in centrifugal acceleration in the core of the

toroidal vortex and an increase in the azimuthal velocity. The

axisymmetric toroidal vortex eventually breaks into two counter

rotating vortices. Additional DNS simulations were undertaken to

investigate the influence of particle spacing on the length of the

wake, and particularly to determine the minimum spacing

required so that the wake is no longer influenced by the

surrounding particles. The results (for ReN¼200) are shown in

Fig. 12 for dimensionless wake length, W *, as a function of

dimensionless particle spacing between particles, L*. It can be

seen for very close particle spacing then W *o1; and it steadily

increases to unity at L* equal to 6. Therefore, it can be concluded

that the effect of the surrounding particles can be neglected when

the centre-to-centre particle spacing is more than 6d p.

Axial velocity

(m s1)

Fig. 6. Instantaneous axial velocity flow field at ReN¼625 and V L¼ 0.047 m sÀ1.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.2 0.4 0.6 0.8 1

1

2

Fig. 7. Average axial velocity vs radial position at ReN¼625 and V L¼ (1) 0.047 and

(2) 0.116 m sÀ1.

0

10

20

30

40

50

60

70

80

0 0.2 0.4 0.6 0.8 1

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1

Fig. 8. Turbulent intensities vs axial and radial position at ReN¼625 and V L¼

(1) 0.047 and (2) 0.116 m sÀ1.


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4.2.2. Normalized settling velocity

The simulated normalized settling velocities (shown as solidlines) for 1, 2, 9, 27, 100, 180 and 245 particles are plotted as a

function particle Reynolds number in Fig. 13. It can be seen that

the time averaged settling velocity of the particle in the presence

of other particles decreases with an increase in the number of

particles surrounding it. This is in line with the hindrance effect

observed in particles settling in swarm. In the presence of eight

other particles, at ReN¼1, the normalized average settling velo-

city, V S , is 5.5% less than that for a single particle, V S N, settling in

an infinite liquid. For a system of 27 particles the average settling

velocity was found to be 0.606, which is 39.4% lower than V S N;

whilst for 100 particles V S is only 27% of V S N. The DNS results

show clearly that the particle settling velocity decreases with an

increasing number of particles, or solids volume fraction, AS ,

within the computational domain. Given that a decrease in

particle velocity is a consequence of an increase in the drag

coefficient, the simulation result is consistent with the findings of

Joshi (1983) and Pandit and Joshi (1998) who used an energybalance approach to derive the following relationship for the

particle drag coefficient, C D, as a function of AL:

C DC D1

¼ AÀ4:8L ð9Þ

In creeping flow the drag coefficient in an assemblage of

particle is increased due to (i) increased true fluid velocity within

the interstices between the particles, (ii) increased velocity

gradients resulting from more zero slip boundary condition

surfaces; and (iii) increased length of the fluid flow through the

assemblage of particles. In turbulent flow the same conditions

prevail, with enhanced momentum transfer, and hence higher

drag as the solids concentration is increased.

T U R B U L E N T I N T E N S I T Y ,

u i R

M S

/ ‹ u 2 › x 1 0 0 , ( - )

REYNOLDS NUMBER, Re∞ (-)

0

10

20

30

40

50

60

70

80

0 200 400 600 800

Fig. 9. Average (’) axial and radial (m) turbulent intensities vs ReN atV L¼ 0.047 m sÀ1.

NORMALIZED RADIAL DISTANCE, r/R, (-)

T U R B U L E N T K I

N E T I C

E N E R G Y , k , ( m 2 / s 2 )

0.00001

0.0001

0.001

0.01

0 0.2 0.4 0.6 0.8 1

Fig. 10. Turbulent kinetic energy radial profile vs ReN at V L¼ 0.047 m sÀ1

[ReN¼51(D), 164(J), 276(m), 437(’), 795()].

Fig. 11. Wake of the sphere at ReN¼200 [(A) settling in the infinite medium;(B) Settling in the presence of eight other surrounding particles].

D I M E N S I O N L E

S S W A K E L E N G T H ( - )

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 2 3 4 5 6

Fig. 12. Wake length vs particle spacing for ReN¼200.


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Also shown in Fig. 13, is the corresponding normalized settling

velocity predictions using the correlation of Richardson–Zaki

(1954) where the liquid void fraction, AL, used for the R–Z

calculation was that obtained from the DNS for the different

number of particles. It can be seen that the DNS simulations are in

good agreement (maximum deviation of 24%) with the R– Z

correlation. Interestingly, the R–Z correlation is based on a very

large number of experimental studies, typically for systems with a

homogeneous dispersion of a very large number of particles

(at least 2000). Whilst the current DNS simulations are for arelatively small number particles. It can be seen from Fig. 13 that

the agreement is increasing with increasing number of particles

where the influence of the wall on the wake of an individual

particle in the bulk assemblage is less. Ideally, more particles

should be included in the simulations. For this study, however, a

64 IBM Power4 processors having 256 GB RAM was required to

satisfactorily resolve the computational domain of 6.5 million

nodes for the 245 particles at ReN¼200. Further increase in either

the numbers of particles or Reynolds numbers will require much

bigger domains. To do this, the code will need to be fully

parallelized using domain decomposition technique in order to

perform simulations within a reasonable time frame.

4.2.3. Particle–particle and particle–wall collisions

The time sequence at 0, 3, 6, 9 and 12 s for the sedimentation

of 180 particles at ReN¼200 is shown in Fig. 14. The images show

that at different time intervals different particles come in contact

with each other, which is qualitatively in agreement with experi-

mental observations. For the simulations in this study a simple

lubrication model was used to account for both the particle–

particle and particle–wall collisions. The lubrication model is

really only valid for low Reynolds numbers, and at best an

approximation at higher Reynolds numbers. Moreover, the model

does not account for the experimentally observed clusters, espe-

cially at higher solids concentrations, where two or more particles

remain in contact with each other for some time. The

incorporation of a collision model that captures all of the

dynamics is an area of on-going DNS research.

4.2.4. Energy dissipation rate

The computed energy dissipation rate, e, was calculated from

the three components of instantaneous velocity and nine velocity

gradients at 12 points in the computational domain using the

expression:

e¼ 2@u1

@ x1

2

þ2@u2

@ x2

2

þ 2@u3

@ x3

2

þ@u1

@ x2

2

þ@u1

@ x3

2

þ@u2

@ x1

2

þ@u2

@ x3

2

þ@u3

@ x1

2

þ@u3

@ x2

2

þ2@u1

@ x2

@u2

@ x1

þ2

@u2

@ x3

@u3

@ x2

þ2

@u3

@ x1

@u01

@ x3

ð10Þ

The computed averaged energy dissipation rate for 245 parti-

cles at ReN¼51, with up to 3500 time steps was found to be

0.303 m2 sÀ3. This value compares favourably with the experi-

mental value of 0.36 m2 sÀ3 obtained by a volume-averaged

energy balance over the refractive index matched fluidized bed

used in this study4 at the same ReN¼51.

The local energy dissipation rate computed from the PIV mea-

surements at the point equivalent of 0.6, 8.7, 0.6 in the computa-tional domain is shown in Fig. 15 as a function of normalized time,

t *. Two peeks can be clearly seen at t *¼34.8 and 43.5 during the

measurement time. The first peak of approximately 4.0 m2 sÀ3

corresponded to the approach of a particle to the measurement

position, whilst the second peak of 5.5 m2 sÀ3 corresponded to the

departure of the particle. The period between these two peaks

corresponded to the time when the particle occupied the sampling

volume and hence a no liquid velocity vectors were measured and a

zero local energy dissipation rate was recorded. The peaks in the

local energy dissipation rate at the surface of the particle represent

an increase of between 10 and 18 times than that of the average

energy dissipation rate of 0.36 m2 s-3.

The flow patterns (including the energy dissipation rate) in the

vicinity of the particle interface and in the bulk have differentroles to play in terms of industrial operations. The flow pattern

near the particle surface governs the heat and mass transfer

characteristics at the particle–fluid interface, whereas the flow

pattern in the bulk governs the solid and liquid phase dispersions.

Typically, high values of particle–fluid heat and mass transfer

coefficients are desired along with low levels of particle and fluid

phase dispersion so that plug flow can be achieved. These two

characteristics are achieved by high and low levels of energy

dissipation rate, respectively. This behaviour is reflected in Fig. 15

as a particle transits through the sampling location, and for this

reason it is perhaps not surprising that fluidized beds are so

widely used for heat and mass transfer applications. However,

there are some limitations. First, the low dissipation rate in the

bulk results in low value of mass and heat transfer coefficients atthe container wall. Second, there may be some practical cases

where complete bulk mixing is desired in the bulk. For achieving

these process objectives an optimum selection of design (includ-

ing column diameter and height) and the operating parameters

(such as particle size and density, superficial liquid velocity, liquid

viscosity and density) is required.

5. Conclusions

Flow visualization experiments were performed using particle

image velocimetry and refractive index matching of the solid and

Fig. 13. Average velocity of the particles: 1, single particle; 2, nine particles; 3, 27

particles; 4, 100 particles; 5, 180 particles; 6, 245 particles, DNS (solid line),

————, Richardson and Zaki (1954) For 1oRe1o200 n ¼ 4:45þ18 d p=DÀ ÁÀ Á

ReÀ0:11

For 200oRe1o500 n ¼ ð4:45ÞReÀ0:11 .

4

PIV experiment 1 as listed in Table 2.


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liquid phases to understand the characteristics of turbulence in a

fluidized bed. For the Reynolds number in the range of 51–759

the experimental measurements revealed that the turbulence

intensity was constant in both the radial and axial directions,

thus establishing the homogenous nature system. Complemen-

tary DNS simulations provided increased spatial resolution of the

velocity field that could be obtained by the PIV measurements.

The computational domain, moving in the downward directionwith velocity equal to the averaged velocity of particles in the

swarm, comprised 1, 9, 27, 100, 180 and 245 particles. For each

case, the wake of individual particle was observed to attenuate

with increase in the volume fraction of particles. The averaged

particle slip velocity decreased with increase in the number of

surrounding particles, and compared reasonably with the

Richardson and Zaki (1954) correlation. Finally, the local energy

dissipation rate was computed from the DNS simulation, and forReN¼51, the energy dissipation rate near the surface of the

particle was found to be approximately 18 times the volume

averaged energy dissipation in the fluidized bed. Such a variation

in energy dissipation rate distribution would need to be taken

into consideration when designing fluidized beds given that heat

and mass transfer will be controlled by what is occurring at the

particle surface whilst mixing of the liquid and solid phases will

be controlled by the dissipation rate in the bulk.

Nomenclature

c 0 fluctuating concentration (kmol mÀ3)C DN drag coefficient of a single particle in infinite med-

ium (–)

C D drag coefficient of a single particle in the presence of

neighbouring particles (–)dP diameter of the particle (m)

e g unit vector in the direction of gravity

Fr Froude number (V2S1= gd p) (–)

F dimensionless redefinition of interaction force in the

most convenient form (–)

G representation of gravitational force in Eq. (6)^h distance between particle and wall or particle and

particle (m)

h grid size (m)L* dimensionless particle spacing between particles (–)

pL extension of the ^ pL to the entire domain O (–)ReN Reynolds number based on the particle (d p V sNrL / mL ) (–)

r i dimensionless radius of ith particle (–)

t time (s)

t * normalized time (–)U ? velocity component perpendicular to the wall or

particle (m sÀ1)

ui Velocity of fluid in ith direction (m sÀ1)

uL dimensionless fluid velocity (–)

uL extension of the uL to the entire domain O

u0 radial fluctuation velocity (m sÀ1)

v0 axial fluctuation velocity (m sÀ1)

V i volume of the ith particleVL superficial liquid velocity for fluidization (or) hin-

dered settling velocity of the particle in sedimenta-

tion (m sÀ1)

VS1 terminal settling velocity (m sÀ1)

Vs interstitial velocity (m sÀ1)

W wake length (m)

(W n¼W /d p) before the dimension (–)

Greek symbols

AL volume fraction of fluid

AS volume fraction of solid

Fig. 14. Sedimentation of 180 particles at different time intervals for ReN¼200 [(A) t ¼0; (B) t ¼3; (C) t ¼ 6; (D) t ¼9; (E) t ¼12].

Fig. 15. Local energy dissipation rate vs time for ReN¼51 [measurement taken at

the point (0.6, 8.7, 0.6)].


http://-/?-

http://-/?-

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r p density of solid particle (kg mÀ3)rS ,i density of ith solid particle (kg mÀ3)

rL density of fluid (kg mÀ3)

OL fluid domain

O entire computational domainOS solid domain

mL molecular viscosity of fluid (kg mÀ1 sÀ1)

e energy dissipation rate (m2/sÀ3)

Subscripts

i particle speciesL liquid phase

S solid phase

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