Advanced Powder Technology 20 (2009) 227–233

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Advanced Powder Technology

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Original Research Paper

Particle dynamics and granular temperatures in dense fluidized beds as revealedby diffusing wave spectroscopy

V. Zivkovic a, M.J. Biggs b,*, D.H. Glass a, L. Xie a

a Institute for Materials and Processes, The University of Edinburgh, Sanderson Building, King’s Buildings, Mayfield Road, Edinburgh EH9 3JL, UKb School of Chemical Engineering, The University of Adelaide, Engineering North Building, Adelaide 5005, Australia

a r t i c l e i n f o

Article history:Received 16 March 2009Accepted 17 March 2009

Keywords:Gas-fluidized bedVibro-fluidized bedLiquid-fluidized bedSlurryGranular temperatureHeat transferSegregationErosionAttritionAggregationGranulationAgglomerationMultiple light scattering

0921-8831/$ - see front matter � 2009 The Society ofdoi:10.1016/j.apt.2009.03.003

* Corresponding author. Tel.: +61 8 8303 5447; faxE-mail address: mark.biggs@adelaide.edu.au (M.J.

a b s t r a c t

Diffusing wave spectroscopy (DWS) can be used to elucidate the fundamentals of particle motion indynamic dense granular media and the associated mean of the square of the particle velocity fluctuationsabout the mean flow velocity, which is related directly to the so-called ‘granular temperature’ that under-pins many theories for dynamic granular processes. In this paper, we provide an overview of DWS and, bydrawing on our previous and new work, demonstrate its application to various fluidized bed (FB)configurations.

� 2009 The Society of Powder Technology Japan. Published by Elsevier BV and The Society of PowderTechnology Japan. All rights reserved.

1. Introduction

Flowing dense granular media are ubiquitous – just a few exam-ples include fluidized beds, screw conveyors, hopper flows, granu-lators and soils during seismic events. In these and many othercontexts, the motion of the individual particles relative to their lo-cal mean motion is important. For example, this relative motionplays a major role in momentum transfer, heat transfer, segrega-tion, erosion, attrition and aggregation; see for example [1–6]. Itis for this reason that theories for these and other particulate pro-cesses include this phenomenon via Ogawa’s granular temperature[7,8], which is related directly to the mean of the square of the par-ticle velocity fluctuations about the mean flow velocity, hdv2i. Thedependence of various theories on the Ogawa granular tempera-ture means their full validation requires its experimental valida-tion and measurement. We have, therefore, applied diffusingwave spectroscopy (DWS), a multiple light scattering technique,to the elucidation of the particle dynamics for various dense gran-

Powder Technology Japan. Publish

: +61 8 8303 4373.Biggs).

ular systems and, where relevant, measure the granular tempera-ture. In this paper, we provide an overview of DWS and somekey results obtained for gas, liquid, dry and slurry vibro-fluidizedbeds (FBs) using the method.

2. Apparatus and Method

2.1. Overview of diffusing wave spectroscopy

Diffusing wave spectroscopy (DWS), which is described in detailby Weitz and Pine [9], is one of a number of multiple light scatter-ing techniques that has been used extensively to study the dynam-ics of turbid colloids [10,11] and, to a much lesser extent, densegranular systems [12–19]. In this technique, following irradiationof the medium of interest with monochromatic laser light, theintensity fluctuations of the backscattered or transmitted light,which arise from the motion of the scatterers (the particles here),are measured through time, Fig. 1a. As Fig. 1b illustrates, this inten-sity fluctuation data can be characterised by an intensity autocor-relation function, g2(t), which can be transformed into anormalized electric-field autocorrelation function, g1(t), via theSiegert relationship [9]

ed by Elsevier BV and The Society of Powder Technology Japan. All rights reserved.

10-12 10-10 10-8 10-6 10-4 10-2 100

10-17

10-16

10-15

<Δr2 >

(m2 )

t2 (s2)

Siegert relationship,

equation (1).

Ballistic regionδv = 3.17 mm/s

10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 1011.0

1.1

1.2

1.3

1.4

1.5

g 2

t (s)

z = 0 z = L

Incident light of intensity, I0

I0(t)

Transmitted light

of intensity IT(t)

IT(t)

Backscattered light

of intensity IB(t)

IB(t)

10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 1010.0

0.2

0.4

0.6

0.8

1.0

g 1

t (s)

2 2

(0) ( )( )

(0)

I I tg t

I= 1 2

(0) ( )( )

(0)

E E tg t

E=

Model linking g1(t ) to motion of scatters such as equation (2).

(a)

(b)

(c)

10-12 10-10 10-8 10-6 10-4 10-2 100

10-17

10-16

10-15

<Δr2 >

(m2 )

t2 (s2)

Siegert relationship,

equation (1).

Ballistic regionδv = 3.17 mm/s

10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 1011.0

1.1

1.2

1.3

1.4

1.5

g 2

t (s)

z = 0 z = L

Incident light of intensity, I0

I0(t)Incident light of intensity, I0

I0(t)

Transmitted light

of intensity IT(t)

IT(t)

Backscattered light

of intensity IB(t)

IB(t)

10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 1010.0

0.2

0.4

0.6

0.8

1.0

g 1

t (s)

2 2

(0) ( )( )

(0)

I I tg t

I= 1 2

(0) ( )( )

(0)

E E tg t

E=

Model linking g1(t ) to motion of scatters such as equation (2).

(a)

(b)

(c)

< v2>1/2 = 3.17 mm/s

Fig. 1. Schematic illustrating how granular temperature is obtained by diffusing wave spectroscopy (DWS). See text for full explanation of symbols.

228 V. Zivkovic et al. / Advanced Powder Technology 20 (2009) 227–233

g2ðtÞ ¼ 1þ bjg1ðtÞj2 ð1Þ

where b is a constant. By assuming the photons undergo an unbi-ased random walk (i.e. diffusive) process as they pass through themedium, mathematical models relevant to the experimental condi-tions can be derived that relate the electric-field autocorrelationfunction to the mean square displacement of the scatterers, hDr2(t)i,Fig. 1c. For example, assuming a weakly light-absorbing medium ofthickness, L, is uniformly illuminated by light of wavelength, k = 2p/k0,it can be shown that when using the transmitted light [9]

g1ðtÞ ¼Ll� þ 4

3

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

0hDr2iq

1þ 49 k2

0hDr2i� �

sinh Ll�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

0hDr2iq� �

þ 43

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

0hDr2iq

cosh Ll�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

0hDr2iq� �

ð2Þ

provided l*/L << 1, where l* is the distance the photons must travelin order for their direction to be completely randomized. Variousother models also exist, including for when backscattered ratherthan transmitted light is sampled and when the medium is stronglylight-absorbing [9], both of which we have used in our work.

The particle velocity fluctuations about the mean flow velocity(i.e. a multiple of the Ogawa granular temperature) can be derivedstraightforwardly from the ‘‘ballistic” region of the mean squaredisplacement [12], Fig. 1c, where

hDr2i ¼ hdv2it2 ð3Þ

2.2. Overview of DWS apparatus

Typical DWS apparatus for use in the transmission mode isillustrated in Fig. 2. A beam of monochromatic light of �2 mmdiameter from a laser (e.g. Ar ion laser of power �1 W operatingin a single longitudinal mode at a wavelength of 514.5 nm) is direc-ted normally at the transparent wall of the vessel containing thegranular medium. The light passes through the medium as perFig. 1a before exiting the back of the bed as a spot of �20 mmdiameter for beds of 10–20 mm thickness. The transmitted lightis collected using a single mode optical fibre (e.g. OZ Optics Ltd.,Ottawa, Canada) and then passed to two photomultiplier tubes(PMT) that transform the light into electronic signals. The intensityoutputs I(t) are then fed to a multi-tau digital correlator (e.g. Flex05, Correlator.com, US) which evaluates the intensity autocorrela-

laser

Transparent walls

PMT

transmittedlight

optical fibre

correlator

laserbeam

PMT

beam splitter

y

Forcing such as vibrations or fluid flow

Scattering per Figure 1(a)

laser

Transparent walls

PMT

transmittedlight

optical fibre

correlator

laserbeam

PMT

beam splitter

y

Forcing such as vibrations or fluid flow

Scattering per Figure 1(a)

Fig. 2. Schematic illustrating typical DWS apparatus.

10

12

14

Linear fits

V. Zivkovic et al. / Advanced Powder Technology 20 (2009) 227–233 229

tion function, g2(t), in real time to be stored on a PC for laterprocessing.

3. Results and discussion

3.1. Dry vibro-fluidized bed

Vibrated dry-granular systems have received much attentionover many years because they are a simple example of a dissipativenon-equilibrium system that demonstrates rich and complexbehaviour. We have used DWS to determine the granular temper-ature and dynamics of mono-disperse glass particles(dp = 0.95 ± 0.05 mm; q = 2500 kg/m3) in a non-evacuated bed196 mm wide, 14.5 mm thick and 75 mm deep, subject to verticalsinusoidal vibrations [17].

Fig. 1 shows the intensity autocorrelation function and associ-ated electric-field autocorrelation function and mean square dis-placement for the vibro-FB when driven at a frequency andacceleration of f = 96.535 Hz and C = 3g, respectively. The meansquare displacement is characterised by three well defined regions.The first, at short times, is associated with the ballistic motionof the particles where hdv2i1/2 = 3.17 mm/s as per Eq. (3). Beyondthe ballistic region, the mean square displacement remains un-changed up to t � 0.3 s, indicating that the particles are in effectmoving about a point in space during this period. The third regionof the mean square displacement, at long times, appears to be asub-diffusive where hDr2i � ta, with a < 1. Inadequate statistics atlong times means it is not possible to say if a diffusive region –i.e. where a = 1 – exists.

Both correlation functions are characterised by a double decay.Using results obtained from other multiple light scattering tech-

λc

λc

Time

λc

λc

λc

λc

λc

λc

(a) (b) (c)

TimeTime

Fig. 3. Schematic showing the three regimes experienced by the particles in thevibro-FB: (a) ballistic dynamics at short times in which the distance travelledbetween collisions is kc; (b) ‘rattling’ around in cages defined by neighbouringparticles at intermediate times; and (c) particles moving between cages in acooperative manner at long times. (after Zivkovic et al. [17]).

niques for non-granular systems [20,21] we have hypothesisedthat this double decay is indicative of caged motion at intermedi-ate times followed by cage breaking due to collective re-arrange-ment of the particles at longer times as illustrated in Fig. 3 [17].If the particle dynamics depicted in Fig. 3 do prevail, the Ogawagranular temperature may not be a suitable basis for describingprocesses such as mixing and segregation in such systems. Instead,one may need to look to alternative definitions of the granulartemperature [22] such as, for example, that of Ciamarra et al. [23].

The granular temperature associated with the ballistic regionwas determined for selected frequency/acceleration (f and C,respectively) combinations to allow systematic study of the effectof peak vibrational velocity, vp, and amplitude, A, over wide ranges.As Fig. 4 shows, the particle velocity fluctuations were found toscale directly with the peak vibration velocity independent of theacceleration and, indeed, the other two vibrational parameters, inline with the kinetic theory of Kumaran [24]. The change in slopebelow vp � 18 mm/s is indicative of a jamming transition wherethe particle dynamics slow down dramatically. This transitionand slowdown is not predicted by current kinetic theories.

3.2. Submerged vibro-fluidized bed

Although the granular temperature of dry vibrated beds hasbeen intensively investigated, especially computationally and

0 20 40 60 80 100 120 1400

2

4

6

8

< δv2 >

1/2 (

mm

/s)

vp (mm/s)

Fig. 4. Variation of the particle velocity fluctuations, hdv2i1/2, with peak vibrationvelocity, vp, of the base of the vibro-FB at C = 1.55 g (circle), 2.16 g (square), 3.00 g(diamond) and 4.17 g (triangle). (after Zivkovic et al. [17]).

50 75 100 125 150 175 2000.0

0.5

1.0

1.5

2.0

<δ v

2 >1/2 [

mm

/s]

f [Hz]

Fig. 6. Variation of the particle velocity fluctuations, hdv2i1/2, with the frequency, f,at a centreline point 50 mm above the base for various accelerations C = 0.75 g(squares), 1.25 g (triangles), 2 g (diamonds), 3 g (hexagons), and 4 g (invertedtriangles). Lines are guide for the eye only. The peak around f = 175 Hz is due to theresonant frequency of the setup, which was determined using transmissibilitymeasurements [25].

230 V. Zivkovic et al. / Advanced Powder Technology 20 (2009) 227–233

theoretically, submerged vibro-fluidized beds have received virtu-ally no attention. We have, therefore, recently applied DWS tostudy the dynamics of mono-disperse glass particles(dp = 0.475 mm; q = 2650 kg/m3) in a bed 200 mm wide, 15 mmthick and 95 mm deep filled with water up to 120 mm and subjectto vertical sinusoidal vibrations [19].

Our work shows there are both strong similarities and differ-ences between the dry and submerged vibrated beds. For example,similar to the dry vibro-FB (see previous sub-section), a double de-cay in the autocorrelation functions, which is indicative of cageddynamics as depicted in Fig. 3, was also observed in the submergedvibro-FB for a range of conditions. Also as seen for the dry vibro-FB[17,25], decaying oscillations were observed in the electric-fieldautocorrelation function whose period matches that of the vibra-tions (not shown here). The relative heights of these oscillations,of which the height of the first peak (i.e. at tf = 1) is indicative, re-veals the degree of fluidization [25]. Fig. 5, which shows g1(tf = 1)for various heights above the bed base as a function of the vibra-tion acceleration for the submerged FB, indicates that, inline withthe dry-FB [17,25], the degree of fluidization, increases with dis-tance above the bed base and the vibrational acceleration, C, oncea critical (height dependent) value is exceeded. In contrast withobservations for dry vibro-FBs, on the other hand, Fig. 5 shows thatfor the points near the free surface, the critical acceleration Cm issmaller than the gravitational acceleration which is considered asa threshold for fluidization in a dry system [25]. Perhaps one ofthe most striking differences between dry and submerged FBs isour observation that the granular temperature in the latter scaleswith the vibrational acceleration rather than the peak vibrationalvelocity, Fig. 6, as observed for the former (see Fig. 4). In particular,we have observed that the granular temperature scales with thesquare of the forcing acceleration, a result that is not predictedby any theory as far as we are aware.

3.3. Gas fluidized bed

Menon and Durian [13], who used DWS to determine the gran-ular temperature of Geldart Group A and Group B particles in a gasFB, only observed particle velocity fluctuations when bubbling waspresent, prompting them to claim that gas FBs are essentially solid-like until the onset of bubbling. More recently, on the other hand,Valaverde et al. [26] observed particle diffusion in a non-bubblinggas FB of 8.53 lm flow-conditioned toner particles, and Spinewineet al. [27] were able to detect particle velocity fluctuations in a

0.00.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

g 1(t

f = 1

)

Γ (g)

Fig. 5. Variation of the height of the first echo in the electric-field autocorrelationfunction, g1(tf = 1), with the forcing acceleration, C, at centreline points 40 mm(squares), 50 mm (circles), 60 mm (triangles) and 70 mm (diamonds) above thebase of the vibro-FB.

water FB of 6.1 mm particles using a video-based method. One pos-sible reason why Menon and Durian did not observe velocity fluc-tuation prior to onset of bubbling was the small interval betweenthe minimum fluidization velocity, Umf, and minimum bubblingvelocity, Umb, for the particles they considered. In an effort to re-solve this controversy, we applied DWS to a deep air FB of semi-transparent hollow glass particles (dp = 60 lm, q = 200 kg/m3) forwhich Umb >> Umf [15,16].

A typical variation of the relative bed expansion, (h�h0)/h0, andpressure drop, DP, across the air FB with superficial velocity, Us, isshown in Fig. 7. Bubbling was first visually observed at a relativebed expansion of �100% and superficial velocity of �7.5 mm/s.Defining the minimum fluidization velocity, Umf, is not straightfor-ward for the system considered here. If onset of fluidization is con-sidered to occur where bed height first changes, then Umf would be1.4–1.5 mm/s. If, however, it is considered to occur where the pres-sure drop across the bed ceases to change, then Umf would beapproximately 4.5 mm/s. This and visual observations describedelsewhere [16] suggest that substantial fluidization of the bed oc-curred nearer the lower of the two possible minimum fluidizationvelocities, giving Umf � 1.45 mm/s (i.e. Umb � 5Umf).

0 2 4 6 8 1 0 1 20

1

2

Umb

Us ( mm/s )

(h− h

0)/ h0

Umf

0

50

100

150

200

ΔP

( N/m

2)

Fig. 7. Variation of relative bed height (squares), (h�h0)/h0, and pressure drop(cross), DP, with superficial velocity, Us, at a point 140 mm above the distributorand 45 mm from the edge of the gas-FB. (after Biggs et al. [16]).

0

2

4

6

8

10

0.1 0.2 0.3 0.4 0.5 0.6

Us

(mm

/s)

<φ>

V. Zivkovic et al. / Advanced Powder Technology 20 (2009) 227–233 231

Fig. 8 shows the variation of the particle velocity fluctuationswith superficial velocity at a point well above the distributor.The particle velocity fluctuations, which were first detected atUs � 1.67 mm/s, rise with superficial velocity in a sigmoidal fashionto a plateau in the region of the minimum bubbling velocity – it isclear from this figure that bubbling is not necessary for particlevelocity fluctuations and air FBs are not solid-like prior to the onsetof bubbling. The initial rise reflects the increasing energy that isavailable to the particles from the fluidizing gas as the interstitialgas velocity rises. At the onset of bubbling, however, further in-creases in gas flow rate pass through the bed as bubbles. The small-ness of the number, size and speed of the bubbles in the region ofthe minimum bubbling velocity means they are expected to havelittle effect on the particles, which leads to the observed plateau.Although we have not probed well beyond the minimum bubblingvelocity (the increasingly large bubbles disrupt data gathering), itis expected that once the number of bubbles and their size andspeed increase sufficiently, the particle velocity fluctuations willonce again begin to increase with superficial velocity.

The variation of the particle velocity fluctuations with superfi-cial velocity seen in Fig. 8 convolutes two important effects – thelocal strain rate and the solid fraction, u. Batchelor [28] proposedthat the mean particle velocity averaged over a cross-section of auniform FB, v, and the particle velocity fluctuations are relatedby a function of the solid fraction only

0 2 4 6 8 100

5

10

15

<δv2 >1/

2(m

m/s

)

Us ( mm/s )

Fig. 8. Variation of particle velocity fluctuations, hdv2i1/2, with superficial velocity,Us, at a point 140 mm above the distributor and 45 mm from the edge of the gas-FB.(after Biggs et al. [16]).

0.30 0.35 0.40 0.45 0.50 0.55 0.600.0

0.5

1.0

1.5

2.0

h (φ

)

φ

Fig. 9. h(u) in Eq. (5) obtained using data in Figs. 7 and 8.

hdv2i ¼ Hð/Þv2 ð4Þ

If we assume that the mean particle velocity, v, is proportional tothe mean interstitial velocity, Ui = Us/(1�u), Eq. (3) can be re-writ-ten as [16]

hdv2i ¼ hð/ÞU2i ð5Þ

Fig. 10. Liquid FB expansion data (points) and Richardson–Zaki fit (line).

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.550

1

2

3

4

5

6

7

8

<δv

2>1/

2(m

m/s

)<

δv2>1/

2(m

m/s

)

<φ >

0 1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

8

9

Us (mm/s)

(a)

(b)

Fig. 11. The variation of height averaged values of the particle velocity fluctuations,hdv2i1/2, with superficial velocity, Us, (a) and with mean particle concentration, hui,(b) of the liquid-FB. Error bars are standard deviations of the measurements. (afterZivkovic et al. [18]).

0.302

0.304

0.306

0.308

0.310

0.312φ

0.17

0.18

0.19

0.20

0.21

φ

0 50 100 150 2006.50

6.75

7.00

7.25

7.50

7.75

8.00

y (mm)

y (mm)y (mm)

y (mm)

0 50 100 150 200

0 20 40 60 80 100 120 140

0 20 40 60 80 100 120 140

3.30

3.35

3.40

3.45

3.50

3.55

<δv

2>1/

2(m

m/s

)

<δv

2>1/

2(m

m/s

)

(a)

(c) (d)

(b)

Fig. 12. Particle concentration, u, as a function of height above the distributor of the liquid-FB, y, for two typical mean solid volume fraction (a) hui = 0.306 and (b) hui = 0.183.The particle velocity fluctuations, hdv2i1/2, vertical profiles for the same concentration (c) hui = 0.306 and (d) hui = 0.183. The right borders of graphs represent the meanfluidized bed heights. Error bars are standard deviations of the measurements. (after Zivkovic et al. [18]).

232 V. Zivkovic et al. / Advanced Powder Technology 20 (2009) 227–233

Fig. 9 shows the function h(u) obtained by combining the data inFigs. 7 and 8. Whilst this derived function does suffer somewhatfrom the fluctuations in the original experimental data, it behavesvery much as described by Batchelor [28] – it increases with solidsloading to a maximum at u � 0.35, reflecting the growing impor-tance of particle-particle collisions, and then tails off to zero as uapproaches the close-packed limit where particles are unable tomove freely.

3.4. Liquid fluidized bed

Unlike gas FBs, liquid FBs tend to expand more uniformly andhomogenously without bubbling. This enables study of the varia-tion of the particle velocity fluctuations over wider ranges of thesolids fraction, u. We have, therefore, applied DWS to a deep FB(200 mm wide by 20 mm thick) of glass particles (dp = 165 lm,q = 2500 kg/m3) fluidized by water at 20 �C [18]. The distributor,which consists of a stainless steel mesh of 40 lm aperture and a5 cm deep packed bed of 1.5 mm stainless steel beads, was de-signed to provide highly uniform and homogeneous fluidizationthat follows the Richardson–Zaki correlation [29], Fig. 10.

Fig. 11a shows variation of height averaged values of the parti-cle velocity fluctuations with superficial velocity. The particlevelocity fluctuations are the same order of magnitude as the driv-ing velocity as observed in the other FB systems, and increase withsuperficial velocity up to a maximum at Us = 7.5 mm/s. In order toexplain the observed maximum, we re-plot the data to obtain thevariation of hdv2i1/2 with the mean solid fraction, hui, as shown inFig. 11b. This exhibits a maximum at hui = 0.175, although there is

only one point below this maxima solid fraction. This is in line withthe simulations of Gevrin et al. [30], who observed a maximum inthe granular temperature at a solid fraction close to u = 0.2. A sim-ilar trend was observed when the local solid fraction was plottedagainst the local velocity fluctuation data [18], indicating that theparticle velocity fluctuation may be described solely in terms ofthe solids volume fraction, u, at least for liquid FBs.

In addition to the granular temperature measurements, the so-lid concentration was measured using the light transmission mea-surement technique [31–33]. It was observed that the local solidconcentration and granular temperature varied little with heightabove the distributor for mean solid volume fractions greater thana critical value of hui = 0.238, Fig. 12a and b. However, for a meansolid volume fraction below this critical solid volume fraction,strong variation of both u and hdv2i1/2 with height was observedas can be seen in Fig. 12c and d. The vertical profiles of the granulartemperature are a consequence of observed solid concentrationprofile with corresponding maximum of a local granular tempera-ture for a local solid fraction of u � 0.18 as predicted by heightaveraged data, Fig. 11. The concentration stratification is in linewith other experimental observations [33,34].

4. Conclusions

We have applied the multiple light scattering technique of dif-fusing wave spectroscopy (DWS) extensively to a variety of denseFB systems [15–19]. By way of example of the results that can beobtained using DWS, we have shown how it has provided a basisfor us to hypothesise that particles in a vibro-FB (both dry and sub-

V. Zivkovic et al. / Advanced Powder Technology 20 (2009) 227–233 233

merged) rattle around in cages formed by their neighbours untiltheir collective re-arrangement occurs at long times. Given thatthis hypothesis would render the Ogawa granular temperature[7,8] irrelevant to mixing and segregation in such systems, furtherwork using other multiple light scattering techniques more suitedto probing slower dynamics (e.g. time resolved spectroscopy [34]and speckle visibility spectroscopy [35]) and other techniques iswarranted.

We have also provided examples of how the granulartemperature varies with the level of forcing for various FBs. Thisdata reveals that the granular temperature scales with, and is ofthe same order as, the forcing velocity (i.e. superficial velocity orpeak vibration velocity) for all except the submerged vibro-FBwhere it scales with the forcing acceleration. The former appearto be in accord with theory, whilst no theory as yet (as far as weare aware) is able to predict the latter, indicating there is still moreto be done theoretically for dynamic granular systems.

We have also shown how this data may be deconvoluted to re-veal how the solids fraction affects the granular temperature in gasand liquid FBs. We demonstrated, for example, that vertical pro-files of the granular temperature are a consequence of observedconcentration stratification in a liquid FB.

Acknowledgement

We thank to the EPSRC (EP/C546849) for support of thisresearch.

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