Powder Technology 210 (2011) 87–102

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Predicting agglomerate fragmentation and agglomerate material survival influidized beds

Sarah Weber a, Sébastien Josset a, Cedric Briens a,⁎, Franco Berruti a, Murray Gray b

a Department of Chemical and Biochemical Engineering, The University of Western Ontario, London, Ontario, Canada, N6A 5B9b Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, Canada, T6G 2G6

⁎ Corresponding author.E-mail address: cbriens@uwo.ca (C. Briens).

0032-5910/$ – see front matter © 2011 Elsevier B.V. Aldoi:10.1016/j.powtec.2011.02.010

a b s t r a c t

a r t i c l e i n f o

Article history:Received 5 April 2010Received in revised form 11 December 2010Accepted 11 February 2011Available online 18 February 2011

Keywords:Agglomerate stabilityAgglomerate destructionAgglomerate growthFragmentationErosion

In fluid coking, agglomeration is undesirable because it causes heat and mass transfer limitations whichincrease reactor fouling and increase coke formation. Fluid coking efficiency is reduced because valuableproduct yields decrease. Agglomeration is desirable in some industries where it is used to reduce dustproblems and stabilize mixtures of various particulate components. This technology is used for thecommercial production of pharmaceuticals, detergents, specialty chemicals, fertilizers and food. In both cases,it is important to know whether agglomerates and granules that form during fluidization fragment or remainintact and whether they increase or decrease in size. The objective of this study was to determine methods topredict agglomerate behavior during fluidization. Breakage behavior was predicted using the Stokesdeformation number, fluidized bed shear rate, and agglomerate strength. After the properties of the breakageproduct were estimated, the growth/reduction behavior was estimated using two approaches: polynomialapproximation and non-dimensionalization. The predictions were compared with experimental dataobtained from previous studies conducted in the absence of reaction. The experimental results werepredicted fairly well and the correlations can be used as a predictive tool for agglomerate behavior.

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© 2011 Elsevier B.V. All rights reserved.

1. Introduction

For many granulation processes, it is important to know whathappens to agglomerates and granules in the granulation equipment.Determining the amount of agglomerate material and the size ofagglomerate fragments that survive at certain fluidized bed condi-tions is important information for process design whether agglom-erates are desired or undesired. This work describes the developmentof a predictive tool to determine the average agglomerate fragmentvolume after breakage, as well as the amount of agglomerate materialpresent at a given time within the fluidized bed. The behaviorpredicted by the method presented in this work applies to a specificcase in the fluid coking process. It applies to the behavior of agglom-erates after they are formed in and pass through the fluid coker heavyfeed spray zone.

Several investigations have studied specific aspects of thegranulation and agglomeration processes focusing mostly on granulecoalescence and growth. Iveson [1] investigated the effect of dis-tributed impact separation forces and bond strengthening duringgranule coalescence. This work showed that coalescence will onlyoccur if a minimum bond strength is attained during impact that canresist breakage during subsequent impacts within the granulationequipment. Lian et al. [2] investigated coalescence using computer

simulations. Their study found that after coalescence, the structure ofthe agglomerate was disordered. The structure was dependent on theimpact velocity and the interstitial fluid viscosity [2]. Schaafsma et al.[3] investigated the growth of an agglomerate from one droplet ofbinder and developed a model to describe the general agglomerategrowth as a function of time. Agglomerate growth was found to occurin two stages. The first stage was initially fast and occurred as theliquid wet the powder. The second stage of growth occurred at amuch slower rate as liquid migrated from inside the granule structureto the granule surface [3]. Some published research has focused onagglomerate destruction. Stein et al. [4] investigated porous glassparticle attrition in a fluidized bed. They found that attrition was themajor process and that breakage was not occurring to a great extent.They also found that the attrition rate was proportional to theexcess gas velocity when a multi-orifice distributor was used. It wasobserved that the attrition rate for this type of distributor wasapproximately two orders of magnitude greater than that of a porousplate distributor [4]. This is not directly applicable to the currentstudy because the particles undergoing attrition in the work by Steinet al. [4] comprised the fluidized bed and the material strength ofthese particles was higher than the agglomerates considered in thiswork.

Work has also been done to model the behavior of agglomerates inparticulate operations. This has been done using different methods.One method that has been used is population balances. Populationbalances can be used to describe agglomerate growth, attrition, and

Nomenclature

a Agglomerate radius (m)A⁎ Agglomerate attrition rate (m3/s)A0e Parameter for Darton's bubble correlationacr Critical agglomerate radius (m)b•vð Þbreak Birth rate distribution due to breakage (no.s−1 m−4)

b•vð Þcoal Birth rate distribution due to coalescence (no.s−1 m−4)

b•vð Þnuc Birth rate distribution due to nucleation (no.s−1 m−4)

CVmass Coefficient of variation for massdb Bubble diameter (m)db,i Bubble diameter at stage i (m)db0e Bubble diameter when it detaches from the

distributor (m)dbo Initial bubble diameter (m)Dcolumn Column diameter (m)dpsm Sauter mean diameter of solid particles in

agglomerate (m)d•vð Þbreak Death rate distribution due to breakage (no.s−1 m−4)

d•vð Þcoal Death rate distribution due to coalescence (no.s−1 m−4)

fbtrack,i Bubble frequency at bubble track entering stage i (no/s)Fpendular Pendular force exerted by liquid bridge (N)g Gravitational acceleration constantG⁎ Agglomerate growth rate (m3/s)h0e Height at which bubble detaches from distributor (m)kHB Apparent viscosity for the Herschel–Bulkley modelKJMP Coefficient in Eqs. (22)–(24)L/S Ratio of the mass of liquid to the mass of solid in an

agglomerate (kg/kg)m Mass of agglomerate material after fluidization (g)mo Mass of agglomerate after agglomerate formation (g)nex Number density of outlet (no.m−4)nHB Flow index for Eq. 12nin Number density of inlet (no.m−4)Nor,i Number of bubble tracks at stage iNor_initial Number of bubbling sites at the distributorQ•ex Flow rate of agglomerates out of the system (m3/s)

Q•in Flow rate of agglomerates into the system (m3/s)

rb Bubble radius (m)rp Particle radius (m)S Saturation (−)s Vertical bubble spacing (m)si Vertical bubble spacing at stage i (m)St⁎def Critical Stokes deformation number (−)Stdef Stokes deformation number (−)t Time (s)U Superficial gas velocity (m/s)Ub Bubble velocity (m/s)Ub,i Bubble velocity entering stage i (m/s)Umf Minimum fluidization velocity (m/s)Vfrag Volume of agglomerate fragment (m3)Vgr Granulator volume (m3)z Fluidized bed vertical coordinate (m)γ• Fluidized bed shear rate (1/s)γ Liquid surface tension (N/m)δ Dimensionless bubble spacingεL Volume fraction liquid in the agglomerate (−)εS Volume fraction of solid in the agglomerate (−)η Coefficient in Eqs. (22)–(24)θ Contact angle of liquid with the solids (°)θb Constant in Eq. (6)μ Viscosity (Pa s)ρagglomerate Agglomerate density (kg/m3)ρL Liquid density, (kg/m3)

ρp Particle density (kg/m3)τgranule Agglomerate yield stress (Pa)τy Granule yield strength (Pa)φ Sphericity (−)φb Constant in Eq. (6)ψ Coefficient in Eqs. (22)–(24)ω Coefficient in Eq. (22)Ω Coefficient in Eq. (24)

88 S. Weber et al. / Powder Technology 210 (2011) 87–102

fragmentation. The general population balance equation is shownin Eq. (1) [5].

∂Vgrn v; tð Þ∂t = Q

•

in nin vð Þ−Q•

ex nex vð Þ−Vgr∂ G�−A�ð Þn v; tð Þ

∂v+ Vgr b

•vð Þnuc + b

•vð Þcoal + b

•vð Þbreak−d

•vð Þcoal−d

•vð Þbreak

� � ð1Þ

Eq. (1) is the population balance on a volume basis. The first twoterms on the right hand side of the equation are the inlet and outletflowrates of particles into the volume, G⁎ and A⁎ are the volumetric growthand attrition rates respectively, and the last term consists of the birthand death rates of agglomerates of a certain size due to nucleation,coalescence, and breakage. Several researchers have used this approachto determine granule growth, reduction, and change in the number ofgranules over time. Saleh et al. [6] used population balances to predictthe time evolution of the particle size distribution. Biggs et al. [7]applied population balances to describe the processes occurring in highshear granulators. Other investigators have studied how to best applypopulation balances. Kumar et al. [8] examined different calculationmethods to improve accuracy without increasing computationalrequirements. One challenge in using population balance equations isthat the knowledge of the kinetic expressions for the processesoccurring during granulation is required. These can be very difficult todetermine for granulation systems because they require precisemeasurements and very good knowledge of what is occurring withina very complicated particulate environment. Another challenge forusing population balances is that the solution to the equations can bedifficult. This is especially true for coalescence problems [5]. Iveson [9]identified some problems with one-dimensional population balances.Most population balances assume uniformity throughout the equip-ment. One-dimensional balances also assume that granule size is theonly factor that affects granule growth. One-dimensional popula-tion balances must be expanded to include effects such as granuleporosity andbinder content and also consider segregation andmixing ingranulation equipment to get accurate results [9]. This makes thepopulation balances more complex and difficult to solve.

Granulation processes have also been modeled using othercomputer based simulations. Kafui and Thornton [10] used a 3Ddiscrete element method (DEM) approach to describe fluidized bedspray granulation. Their simulations were able to predict realisticgranule size trends for early granulation. This approach improves onthe population balance method because it does not rely on theassumption that the system is spatially uniform and does not rely onempirically fitted parameters that are limited only to the systems towhich they were fitted [10]. It does, however, require extensivecalculations and computer processing capacity. Thornton et al. [11]numerically simulated the fracture and fragmentation of agglomer-ates during impact outside the granulation equipment. While theirwork was only directly applicable to two-dimensional monodisperseagglomerates, the results were thought to be directionally correct formore realistic agglomerate structures [11].

The purpose of this work was to determine an empirical procedurethat could be used as a predictive tool for determining the amount ofagglomerate material in a fluidized bed at any specific time.

Table 1Particle sizes and liquid viscosities used in previous experimental studies [12–14].

Particle type Sauter mean diameter (μm) Liquid viscosity (cP)

Glass beads 171 1Silica sand 165 1

6.2Fluid coke 140 6.4

89S. Weber et al. / Powder Technology 210 (2011) 87–102

2. Development

From previous studies [12–14], agglomerates have been observedto undergo three processes in different combinations: fragmentation,growth, and erosion/abrasion. The overall procedure for determiningthe amount of agglomerate material in a fluidized bed is illustratedin Fig. 1. The experimental results presented in Refs. [12], [13] and [14]were used to develop the correlations presented in this study. InRefs. [12–14], the fluidized bed diameter was 0.10 m, the staticfluidized bed height was 0.15 m, and the U/Umf values tested rangedfrom 6.5 to 30. The particle Sauter mean diameter and liquid viscosityof the binders used in the experiments are shown in Table 1.

The agglomerates were made artificially outside of the fluidizedbed to control initial agglomerate parameters; therefore, no liquidbinder was sprayed into the fluidized bed.

The procedure requires information provided by the user about thefluidized bed conditions and information about the agglomeratesformed. In fluid coking, agglomerates are formed by nozzles thatintroduce liquid feed horizontally into the fluidized bed. Informationabout the properties of the agglomerates formed by the spray nozzlesis a required input for this procedure. In previous experimental work[12–14], initial agglomerate properties were controlled and thisinformationwas used for the development of the prediction procedure.In the future, information about the initial properties of agglomeratesproduced by commercial nozzles, such as liquid content and agglom-erate size, can be used in the prediction.

2.1. Determining fragmentation

The first question that must be addressed is whether the agglom-eratewill fragmentor remain intact. For thepurposeof this procedure, itis assumed that once an agglomerate is exposed to the fluidized bedenvironment, fragmentation, if it occurs, will happen relatively quickly.This was observed experimentally (b30 s of fluidization). From aprevious work [12,13], agglomerate fragmentation was observed whena “critical superficial gas” velocity was exceeded. This critical velocitywasdependent on the viscosity of the liquid binder, the liquid content ofthe agglomerate, and the liquid binder surface tension.

Some work has been done to investigate the breakage of agglom-erates in the granulation equipment. One study by Tardos et al. [15]

Fig. 1. Schematic overview for the prediction of agglomerate material survival.

investigated the critical parameters in binder granulation of finepowders. The authors looked at several agglomerate processes,including an assessment of granule breakage. These authors defineda breakage condition based on the Stokes deformation number. TheStokes deformation number is a dimensionless number representingthe ratio of the externally applied kinetic energy to the energyrequired for deformation. Eq. (2) is the mathematical definition of theStokes deformation number, Stdef [15].

Stdef =ρagglomerate aγ•

� �22τgranule

ð2Þ

In Eq. (2), γ• is the shear rate in the granulation equipment, a isthe radius of the agglomerate or granule, ρagglomerate is the agglom-erate density and τgranule is the granule yield stress. Tardos et al. [15]found that when granules were exposed to a constant shear field,several outcomes could occur: no agglomerates could break, someagglomerates could break, or all agglomerates could break, dependingon the Stokes deformation number. When the Stokes deformationnumber was greater than a critical value, breakage of all agglomerateswas observed [15]. This agrees with the observations of Benali et al.[16] who studied granule growth kinetics in high shear mixers. Theyfound that below an optimal impeller speed, granules will uncontrol-lably grow. Above the optimal impeller speed, the granules breakextensively, illustrating the importance of the shear produced bygranulating equipment. The shear generated by the granulationequipment exerts force on the agglomerates and granules through ashear field. The amount of force exerted by the shear field depends onthe size of the agglomerates in the field. The relationship in Eq. (2) isthe combination of these factors that results in the dimensionlessStokes number. In the work by Tardos et al. [15], the critical Stokesnumber, Stdef⁎, where complete breakage was observed, was found tobe 0.2 for wet agglomerates. The critical Stokes deformation numberdoes not change with agglomerate properties because it is a func-tion of agglomerate strength. Agglomerate strength is dependent onthe saturation of the agglomerate and the properties of the liquidbinder. Because the Stokes deformation number is a function of theagglomerate radius, a, a critical agglomerate radius, acr, can be deter-mined by substituting and rearranging Eq. (2). The relationship isshown in Eq. (3).

acr =

2τySt�def

ρagglomerate

" #1=2

γ•ð3Þ

To solve Eq. (3), the shear rate produced by the fluidized bed andthe agglomerate strength is required. For a fluidized bed, the shearrate can be found using the relations in Eqs. (4) and (5) [15].

γ•=

18Ub

dbδ2 ð4Þ

δ =srb

ð5Þ

Just as the shear generated by the impeller greatly affectedagglomerate growth and breakage behavior in high shear mixers [16],

Fig. 2. Definition of the dimensionless bubble spacing (adapted from Ennis et al. [16]).

Fig. 3. Calculated shear rates throughout the fluidized bed. Glass beads (dpsm=171 μm,ρp=2500 kg/m3) were used in the calculation.

90 S. Weber et al. / Powder Technology 210 (2011) 87–102

the shear in a fluidized bed generated by bubbles affects agglomerategrowth and breakage behavior. Eq. (4) relates the velocity andproximity of bubbles to the shear rate. Eqs. (4) and (5)were determinedusing two-phase inviscid flow theory to estimate velocity gradients formean collisional velocities [17]. Eq. (4) is a simplification derived byconsidering equal sized bubbles and finding an average value ofcollisional velocities [17]. Eq. (4) was also reported by Ennis et al. [17],however, those authors used the constant 12 instead of 18. For theprocedure described here, the relationship used by Tardos et al. [15] hasbeen used.

The shear rate produced by a fluidized bed is a function of thebubble velocity, the bubble diameter, and the bubble spacing. Thebubble velocity and bubble diameter were calculated using standardcorrelations. The bubble velocity was found using a correlation fromHilligardt and Werther [18]:

Ub = φb U−Umfð Þ + 0:711θb gdbð Þ0:5: ð6Þ

The correction factors θb andφb in Eq. (6) are dependent on the ratioof thefluidized bed height tofluidized bed diameter, the diameter of thefluidized bed, and the Geldart classification of the particles [18]. Thebubble diameter was calculated using a correlation from Darton [19]:

db = 0:54g−0:2 U−Umfð Þ0:4 z−h0e + 4 A0eð Þ0:5h i0:8

: ð7Þ

In Eq. (7), h0e is the height at which the bubble detaches from thedistributor. The experimental results were obtained using porousstyle distributors, therefore, h0e=0. A0e is a parameter of Darton'scorrelation and defined as:

A0e =

db0e

1:64

� �2:5g0:5

U−Umfð Þ : ð8Þ

Eqs. (6) and (7) were used to calculate bubble velocity and bubblediameter because they are well established equations that are gen-eralized enough to take into account various fluidized bed configura-tions and fluidized bed particle properties.

To calculate the bubble spacing, s, the number of bubbling siteswas required. Because porous distributors were used to obtain theexperimental data, the number of bubbling sites was not fixed as it isfor a perforated distributor. To determine the number of bubblingsites, the assumption was made that the bubbles formed initiallyat the distributor will just touch in the horizontal direction. Withthis criterion, a relationship from Kunii and Levenspiel [20] was usedto calculate the initial number of bubbling sites at the distributor(Eq. 9).

Nor initial =

πD2column

42ffiffiffi

3p

d2bo

ð9Þ

As the bubbles rise in a fluidized bed, the number of bubble trackschanges due to bubble coalescence. A coalescence stage was definedas a volume in the fluidized bed where the bubble volume leaving thestage was double the bubble volume entering the stage. The numberof bubbles leaving the coalescence stage was assumed to be half thenumber of bubbles entering the coalescence stage from the previousstage (i.e. each coalescence stage represented the merging of twobubbles into one). This information was used to calculate the bubblefrequency of each bubble track entering each coalescence stage i(Eq. 10).

fbtrack;i =3D2

column U−Umfð Þ2d3

b;iNor;ið10Þ

Ennis et al. [17] defined the dimensionless bubble separation, s, asshown in Fig. 2. The bubble spacing was measured between thecenters of the bubbles as shown in Fig. 2. The bubble spacing at thebottom of each coalescence stage i is defined by Eq. (11).

si =Ub;i

fbtrack;ið11Þ

It should be noted that using the vertical center-to-center spacingof the bubbles does not always guarantee that there will be no physicaloverlap in the bubbles.

Considering Eqs. (6)–(11), it can be seen that the parameters inEq. (4) change with the fluidized bed height, causing the shear rate tochange as well. The shear rate was calculated using Eq. (4) for differentheights in a fluidized bed of glass beads for several different superficialgas velocities. The results are shown in Fig. 3.

Fig. 3 shows that the initial shear rate increased with increasingsuperficial gas velocity. It can also be seen that the shear rate increasedwith vertical position for all superficial gas velocities until a maximumvalue was reached, after which the shear rate decreased slightly. Theincrease in the shear rate occurred because the increase in the bubble

Table 2Characteristic shear rates for a fluidized bed of glass beads (ρp=2500 kg/m3,dpsm=171 μm).

Superficial gasvelocity (m/s)

Maximum shearrate (1/s)

Vertical position ofmaximum (m)

Bed height(m)

0.10 153.81 0.24933 0.249330.15 271.14 0.19533 0.285330.20 381.04 0.19533 0.318670.25 490.4 0.13933 0.349330.30 597.46 0.13933 0.3780.35 693.95 0.13933 0.405330.40 785.32 0.13933 0.4320.45 871.97 0.13933 0.45733

91S. Weber et al. / Powder Technology 210 (2011) 87–102

velocity with height was greater than the decrease in bubble spacing.After coalescence stopped, the constraint that the bubble diameter nolonger increases with vertical position was imposed on the system. Thenumber of bubble tracks was also set to 1 when coalescence stopped.Evenwith these conditions, the shear rate gradually decreased from thismaximum value, especially at the highest superficial gas velocities,because the bubble velocity calculation has a correction factor thatchanges with vertical position (φb). This caused the bubble velocity tocontinue to increase after the bubble diameter stopped increasing. Themaximum values of the fluidized bed shear rate were found andpresented in Table 2 for several superficial gas velocities in a fluidizedbed of glass beads (ρp=2500 kg/m3, dpsm=171 μm). The method ofcalculation of the shear rates is dependent only on bubble properties.The only effect of the distributor on the shear rate is its effect on theinitial properties of the bubbles formed at the distributor.

Eq. (3) uses the fluidized bed shear rate to determine the criticalagglomerate radius that can survive at that fluidization condition. Forthe purpose of this prediction procedure, a characteristic shear rate foreach superficial gas velocity was required. This would give one criticalagglomerate radius. The maximum fluidized bed shear rate producedat each superficial gas velocity was selected as the input for Eq. (3) toobtain the minimum critical agglomerate radius for that condition.During fluidization, the agglomerates and fragments travel through-out the fluidized bed and will likely experience the maximum shearconditions.

The characteristic granule stress, τgranule, in Eq. (3) can be evaluatedaccording to the Herschel–Bulkley model shown in Eq. (12) [15].

τgranule = τy + kHBγ• nHB ð12Þ

In Eq. (12), kHB is the apparent viscosity, nHB is the flow index, andτy is the agglomerate yield strength. When considering unsolidifiedagglomerates, which was the case for the experimental results [12–14],the system is complex and exhibits both a yield strength and a non-Newtonian behavior [15]. A first approximation of Eq. (12) can bemadeby assuming the apparent viscosity, kHB, is negligible compared to theyield strength. This assumption can bemadeby considering the agglom-erates as highly concentrated slurries of binder and particles [15]. Thisassumption equates the characteristic granule stress to the granule yieldstrength, τy.

Tardos et al. [15] experimentally measured the agglomerate yieldstrength,τy, using a compression testingmethod. Thiswas an acceptabletest because the agglomerates in that study were made from carbowax(PVP). The viscosity of the binder was greater than the viscosity of thebinder used in a previous experimental work [12–14]. The agglomeratesin those studies were much weaker and could not be subjected to thesame mechanical testing methods as easily as the agglomerates madewith more viscous binders. To obtain the granule yield strength fromexperimental results, the number of agglomerate fragments wasestimated by counting. As a first approximation, the agglomerate wasassumed to fragment into equal sized pieces. This information thenyields the average size of the agglomerate fragments produced by

fragmentation by taking the initial agglomerate volume before fluid-ization and dividing it by the number of fragments that were counted.The critical agglomerate radius was determined by assuming thatthe agglomerate fragments were spherical. During experiments,the agglomerates did not always fragment into the same number ofpieces or into equal sized pieces. An average value of agglomerate yieldstrength was used in subsequent calculations.

Eq. (3) was rearranged and used to solve for the agglomeratestrength for a few different experimental cases. This provided someinformation to determine the relationship for the agglomerate yieldstrength, τy. The forces generated by liquid bridges are determined bythe saturation state of the agglomerate or granule. Agglomeratesaturation is defined as the pore volume occupied by liquid divided bythe total void volume of the agglomerate, and can be expressed as apercentage [21]. It is defined mathematically in Eq. (13):

S =εL

1−εSð Þ =L = Sð Þρagglomerateρp

ρL ρp + L = Sð Þρp−ρagglomerate

� �0@

1A: ð13Þ

When Sb25%, the agglomerate is in the pendular state of saturation.Because the saturation is only based on the ratio of the liquid volumeand the total volume, this calculation is not affected directly by particleshape. Particle shape may change the packing structure of theagglomerate and affect the void space which would then affect thesaturation. In the pendular state of saturation, liquid exists as discretebridges. When SN80%, the agglomerate is in the capillary state ofsaturation and the forces holding the agglomerate together aregenerated by capillary suction. The state between these two extremesis the funicular state and the forces holding the agglomerate structuretogether are a combination of these two forces [21].

In previous experiments [12,13], the highest liquid saturationinvestigated was 32%. Although this is technically in the funicularstate, it was assumed that it was close enough to the pendular state tobe described only by liquid bridge forces to simplify the problem.Some discrepancies in the results may be attributed to the effect ofcapillary suction beginning to influence the results. In the pendularstate of saturation, the force exerted by a pendular liquid bridge isdescribed by Eq. (14) [21].

Fpendular =2πrpγ

1 +tanθ2

ð14Þ

In the current work, the particles were wetted well by therespective liquids and the Sauter mean diameter of the particles wasused to estimate the true particle radius. The force exerted by thependular bridges becomes Eq. (15).

Fpendular = πdpsmγ ð15Þ

The force described by Eq. (15) is for a single liquid bridge. Eq. (15)was also reported by Seville et al. [22] in their review of inter-particleforces in fluidization. In that review, Eq. (15) was the same as themaximum static liquid bridge force. When the saturation of theagglomerate increases, the liquid begins to fill the voids. It also meansthat there is more liquid available to form bridges, so more bridges areformed. The effect of saturation was required to describe the overallagglomerate strength. The empirical correlation for the agglomeratestrength was based on Eq. (15) and added the agglomerate saturationand liquid viscosity as parameters. The coefficients for the correlationwere found by comparing the resulting agglomerate strengthwith theagglomerate strength found from the experimental results. Theexperimental agglomerate strength was found by using the averagenumber of agglomerate fragments at a certain superficial gas velocityand finding the critical agglomerate radius. These values were used in

92 S. Weber et al. / Powder Technology 210 (2011) 87–102

Eq. (3) and it was solved for agglomerate strength. The empiricalcorrelation for agglomerate strength is shown in Eq. (16).

τy = 8:5 × 1012μ0:9 S� 100ð Þ0:35 πdpsmγ� �2 ð16Þ

Eq. (16) is only valid when the maximum shear rate produced inthe fluidized bed is used. At this time, the effect of the particle shapewas not considered in the equation to find the agglomerate strength.Iveson and Page [23] found that the flow stresses for non-sphericalparticles were higher than those of spherical particles, however, thetrends were identical for both particle shapes. Both also exhibited thetransition from brittle to plastic failure as was found by uniaxialcompression tests [23]. Iveson and Page [24] investigated the strengthdevelopment between two liquid bound particles during compressionat low strain rates. They used a cold welding analogy to describe thestrengthening behavior. This analogy was not used in the currentwork because the saturation considered here was much lower (32%maximum versus 70%) and because the method of applying the loadin fragmentation is different from compression which was tested inthe study by Iveson and Page [24]. The predicted values determinedby Eq. (16) were compared with the experimental values (Fig. 4).

Fig. 4 shows that the predicted and measured values for the ag-glomerate strength are relatively similar. There were some discre-pancies which were not surprising because the agglomerate strengthdetermined from experimental results was very sensitive to thecounted number of fragments recovered from the fluidized bed.The fragmentation behavior was very changeable. The number offragments and the size of fragments changed and it was also verysensitive to how the agglomerate material was prepared. In Fig. 4,Eq. (16) has the most difficulty in predicting the silica sandagglomerates made with a higher viscosity liquid. Iveson et al. [25]studied the dynamic strength of partially saturated powder assem-blies. That study found that the viscous forces have little effect onthe dynamic strength at low strain rates, however, they becomedominant at higher strain rates [25]. Seville et al. [22] also reported adynamic liquid bridge force equation, however, the particle–particlerelative velocity is required to find this value. The liquid bridge force iscomprised of both static and dynamic forces. At low strain rates, thestatic forces dominate the dynamic forces. At high strain rates and

Fig. 4. Comparison of the predicted and measured values of the agglomerate strength.Error bars are the standard deviation of 5 measurements.

when viscous liquids are used, the dynamic forces dominate. This mayexplain the difficulties accurately predicting the agglomerate yieldstrength for the more viscous liquid in the silica sand system. In orderto observe breakage in this system, a high superficial gas velocity wasrequired. This would increase the strain rate experienced by theagglomerate and the viscous forces would become dominant andchange the behavior of the agglomerate yield strength. This was notobserved to a great extent in the fluid coke system. This wassurprising because the viscosity of that liquid is just as high as inthe silica sand case. Breakage in the coke system happened at lowersuperficial gas velocities [14]. The strain rates were not as high andthis may have reduced the influence of the dynamic bridge forces.Because this procedure focused on agglomerate systems with awater binder, Eq. (16) was used as it appears and viscous effects werenot included. To further extend this work to systems with differentbinders, viscous forces would have to be considered. The purpose ofdetermining the fragmentation condition is to get an idea about howmany fragments are produced by fluidization to determine theirapproximate volume for application in the erosion and growthpredictions that will be discussed in the next section.

To determine how well fragmentation was estimated, thepredicted number of fragments was plotted versus the actual numberof fragments for some of the experimental conditions. The conditionsselected for the comparison had the most data points and the mostsuperficial gas velocities tested. This was done to see how well thispart of the correlation did predicting the observed results. Using thecorrelation in Eq. (16) to estimate the agglomerate material yieldstrength, the average agglomerate radius was found using Eq. (3). Theradius was used to find the average agglomerate fragment sphericalvolume and the total initial agglomerate volume was divided by thisvalue to obtain an average number of fragments. The results areshown in Fig. 5.

Generally, the procedure did a relatively good job predicting thenumber of fragments when the liquid content was high at allsuperficial gas velocities. As can be seen from the wide error bars,there was significant variability in the number of fragments producedby fluidization. The predicted number of fragments is close to theaverage number of fragments experimentally observed. There was alot more discrepancy between the predicted and experimental valueswhen the liquid content was low. The predicted number of fragmentsis generally higher than the actual number of fragments, especially athigh velocities. This occurred because some agglomerate fragmentsbecame so small due to secondary erosion that they were no longerrecovered from the fluidized bed. Higher superficial gas velocitiesincrease the erosion rate experienced by the fragments in thefluidized bed. The observations were true in both the glass beads–water and fluid coke–biodiesel agglomerate systems.

The calculation procedure does a reasonable job predicting thenumber of agglomerate fragments. This information was then appliedto the fluidized bed erosion and growth processes experienced by theagglomerates.

2.2. Determining fluidized bed growth and erosion behavior

For the purpose of this prediction procedure, it is assumed that theagglomerates will fragment to an average stable agglomerate sizedepending on the fluidized bed conditions. This is implied by Eq. (3).Although agglomerate yield strength is not affected by agglomeratesize, fragmentation is strongly affected by agglomerate size because ofthe interaction of the granule within the shear field. Once fragmenta-tion has stopped or when it does not happen, agglomerates can eithergrow or erode. For the conditions considered here, it is assumed thatagglomerate growth occurs only by layering, not coalescence, andagglomerate reduction occurs due to erosion only.

The purpose of this section is to predict the amount of agglomeratematerial recovered from the fluidized bed with fluidization time. In

93S. Weber et al. / Powder Technology 210 (2011) 87–102

order to do this, the effects of agglomerate saturation, agglomerate size,superficial gas velocity, fluidization time and particle sphericity wereconsidered. Previouswork has shown that liquid viscosity, liquid surfacetension, and the contact angle of the liquid with the solids affectagglomeratematerial survival [12–14]. Thesewerenot considered at thistime because more information and data are required. When experi-ments concerning the effects of these parameters were performed, thevariable under investigation did not always change in isolation. Forexample, when selecting a liquid to use to investigate liquid viscosity inboth silica sand and coke systems that wets the particles well, the liquidsurface tension is difficult to match as well. More experimental resultswhere the liquid parameters change in isolation are required. Furtherinvestigation is required to meaningfully describe both systems.

A few different approaches were used to determine empiricalcorrelations to predict the relative amount of agglomerate material

Fig. 5. Comparison of the predicted and measured values of the number of aggloDagglomerate=0.0169 m, t=30 s, (B) glass beads and water, Dagglomerate=0.0120 m, t=30biodiesel, Dagglomerate=0.0169 m, t=60 s. Error bars are the standard deviation of 3–5 meas

recovered from the fluidized bed (m/mo). One method used a poly-nomial equation as a local approximation of the true nonlinear input/output relationship, which is the response surface method. For thiswork, a second degree polynomial was used for the approximationand the coefficients were found using a least squares solution inMatlab. The second approach that was considered removed the unitsin a mathematical function by non-dimensionalization.

2.2.1. Polynomial equation to approximate the true relationshipSeveral different parameters were considered in the correlation

to determine the evolution of the m/mo versus time graph. Theyincluded the agglomerate saturation, agglomerate size, the ratio ofthe superficial gas velocity to the minimum fluidization velocity,particle sphericity, and the fluidization time. Agglomerate fragmentvolumes predicted from the previous procedure were used as the

merate fragments produced during fragmentation. (A) Glass beads and water,s, (C) glass beads and water, Dagglomerate=0.0067 m, t=30 s, and (D) fluid coke andurements. The legend values are the agglomerate liquid contents on a mass basis (wt.%).

Fig. 6. Comparison of the predicted andmeasured values form/mo using Eq. (17) for glassbeads. (A) Data used to generate Eq. (16) and (B) data not used to generate Eq. (17).

94 S. Weber et al. / Powder Technology 210 (2011) 87–102

input for the agglomerate size. It was very difficult to find a cor-relation that would predict the results well for both glass beads(spherical particles) and silica sand (non-spherical particles). In ourexperiments, the sphericity of the fluidized bed particles was seen tobe one of the most influential parameters on the erosion rates in thefluidized bed. For this reason, the data were divided based on particlesphericity. When the particle sphericity was 1, Eq. (17) was the cor-relation that best determined the relative agglomerate mass recoveredfrom the fluidized bed.

mmo

= −8:50 × 10−4 S×100ð Þ2−4:77 × 10−3 UUmf

� �2

+ 5:12 × 10−6t2−3767 S × 100ð Þ Vfrag

� �+ 3:80

× 10−3 S × 100ð Þ UUmf

� �+ 1:83

× 10−4 S × 100ð Þ tð Þ−5163VfragUUmf

� �+ 542:98Vfrag tð Þ−9:72

× 10−4 UUmf

� �t + 2:14 × 10−2 S × 100ð Þ + 1:20 × 105Vfrag

+ 6:18 × 10−2 UUmf

� �−4:74 × 10−4t + 0:47

ð17Þ

When the fluidized bed particle sphericity was 0.7, then thecorrelation that was found to best determine the relative massrecovered from the fluidized bed is shown in Eq. (18).

mmo

= −1:01 × 10−2 S×100ð Þ2−0:115UUmf

� �2−3:40 × 10−5t2−2:11

× 104 S × 100ð Þ Vfrag

� �+ 0:242 S × 100ð Þ U

Umf

� �+ 1:24

× 10−3 S × 100ð Þ tð Þ + 2:55 × 105VfragUUmf

� �−9:69

× 103Vfrag tð Þ−1:51 × 10−2 UUmf

� �t−0:877 S × 100ð Þ

−1:24 × 106Vfrag + 0:514UUmf

� �+ 7:93 × 10−2t

ð18Þ

Figs. 6 and 7 illustrate howwell Eqs. (17) and (18) respectively didin predicting themeasured results. Figs. 6(A) and 7(A) show howwellthe data used to generate the correlations was predicted. Some datafrom each of the conditions was not used to generate the correlationsto see howwell the correlations predicted these results. Figs. 6(B) and7(B) show how well the data not used to generate the correlationswas predicted.

The polynomial approximations did a relatively good job inpredicting many of the values for m/mo for both the glass beads andthe silica sand systems. Fig. 6 does show that there were some troublespredicting results at the extreme values of m/mo, especially erosion, inthe glass beads system. This was not observed to the same degree inFig. 7 for the silica sand system. The difficulties predicting the extremeconditions were not surprising. At extremely high and extremely lowvalues of m/mo, the agglomerates and fragments are undergoingextreme changes in their structures and it cannot be assumed that theinitial values for agglomerate volume and agglomerate saturation arestill good predictors. Another problem that may have occurred duringerosionwas recovering all of thematerial successfully from thefluidizedbed. When agglomerate destruction is so extensive, this can becomedifficult. The coefficient of variation for the agglomerate mass wascalculated using Eq. (19). The results that were less than −50% andgreater than +50% of the initial mass were removed and thecorrelations for glass beads and silica sand were re-determined(Eqs. 20 and 21 respectively). The results are shown in Figs. 8 and 9.

CVmass =m−mo

mo

� �× 100 ð19Þ

mmo

= −7:36 × 10−4 Sñ100ð Þ2−5:46 × 10−3 UUmf

� �2

+ 1:58 × 10−5t2−3:30 × 103 S × 100ð Þ Vfrag

� �+ 1:26 × 10−3 S × 100ð Þ U

Umf

� �+ 1:15

× 10−4 S × 100ð Þ tð Þ−1:84 × 104VfragUUmf

� �

+ 6:14 × 102Vfrag tð Þ−3:12 × 10−4 UUmf

� �t + 3:68

× 10−2 S × 100ð Þ + 1:99 × 105Vfrag + 0:111UUmf

� �−5:07

× 10−3t + 0:190

ð20Þ

Fig. 7. Comparison of the predicted andmeasured values form/mo using Eq. (18) for silicasand. (A) Data used to generate Eq. (18) and (B) data not used to generate Eq. (18).

Fig. 8. Comparison of the predicted and measured values for m/mo using Eq. (20) forglass beads with extreme high and low values removed. (A) Data used to generateEq. (20) and (B) data not used to generate Eq. (20).

95S. Weber et al. / Powder Technology 210 (2011) 87–102

mmo

= 2:11 × 10−3 Sñ100ð Þ2−0:178UUmf

� �2+ 8:52

× 10−5t2−9:44 × 104 S × 100ð Þ Vfrag

� �

+ 0:157 S × 100ð Þ UUmf

� �+ 1:09 × 10−2 S × 100ð Þ tð Þ

+ 5:02 × 105VfragUUmf

� �−1:43 × 105Vfrag tð Þ + 8:27

× 10−2tUUmf

� �−0:455 S × 100ð Þ−0:177t

ð21Þ

The residuals for the correlation were plotted versus the predictedvalues for both glass beads and silica sand results. The residuals were

also compared with normal distributions. The results are shown inFigs. 10–13.

Figs. 8 and 9 show that the removal of the extreme data doesimprove the agreement between the correlation and the observeddata. There are still some points that are not predicted very well. Thismay have occurred because of some variability in the experimentaldata. Fig. 10 shows that the residuals for the glass beads data arerandomly distributed and Fig. 11 shows that the majority of theresiduals for the glass beads data do have a normal distribution. Thereare some deviations from this behavior at the tails of Fig. 11. This wascaused by discrepancies between the experimental data and thepredicted values. Experimental error contributed to this effect. Itagrees with the findings of Fig. 8 which shows that some data pointsare not predicted very well. The deviation from the normal

Fig. 9. Comparison of the predicted and measured values for m/mo using Eq. (21) forsilica sand with extreme high and low values removed. (A) Data used to generateEq. (21) and (B) data not used to generate Eq. (21).

Fig. 10. Comparison of the residuals with the predicted values for the polynomialapproximation for glass beads data.

Fig. 11. Comparison of the residuals for the polynomial fit with normal distributions forglass bead data.

96 S. Weber et al. / Powder Technology 210 (2011) 87–102

distribution means that the correlation does not account for all of thebehaviors, although the majority of the data is explained well by thepolynomial approximation. For the silica sand data, Fig. 12 shows thatthe residuals were not as randomly distributed. The residualsalso deviated more from the normal distribution than the glassbeads results. This indicates that the polynomial approximation is notaccounting for all of the effects that are occurring in the fluidized bedsystem. Not as much data was available for the silica sand system asfor the glass beads system. This was not ideal because the polynomialapproximation required the estimation of many coefficients. Usingmore data to generate the polynomial approximation is expected tohelp make this correlation more accurate and account for more of theeffects occurring in the fluidized bed system. This will make theresiduals more normally distributed.

Eqs. (20) and (21) predict the behavior of glass beads and silica sandagglomerates respectively well. A drawback of using a polynomialapproximation to predict agglomerate material survival in the fluidizedbed is its dependency on the equipment and experimental conditionsused to derive the correlations.While it is easy to use, it is not as generaland cannot be applied to many processes and systems. For this reason,analternativemethodof describing the survival of agglomeratematerialin a fluidized bed was used.

2.2.2. Non-dimensionalizationThe second approach used to try to predict the evolution of the m/

mo versus time graph was to use non-dimensionalization to combinethe factors of interest so that both sides of the equation have the sameunits. In this case, the equation should be dimensionless because therelative mass recovered from the fluidized bed is dimensionless.Eq. (22) is the result of non-dimensionalization. From previousexperiments [12–14], it was seen that superficial gas velocity,fluidization time, liquid content, and agglomerate size affected theamount of agglomerate material recovered from the fluidized bed.These variables were selected to be included in the correlation.

Fig. 12. Comparison of the residuals with the predicted values for the polynomialapproximation for silica sand data.

Table 3Coefficients obtained from JMP non-linear least squares analysis. The ranges are the 95%confidence intervals of the parameter estimates.

Coefficients Glass beads Silica sand

KJMP 0.617±0.137 0.525η 0.469±0.040 0.542ψ 0.181±0.043 −0.090ω −0.199±0.112 0.795

97S. Weber et al. / Powder Technology 210 (2011) 87–102

From the experiments, it was observed that increasing liquid contentand agglomerate size increased the amount of agglomerate materialrecovered from the fluidized bed. These variables were put in thenumerator of the equation. Also from the experiments, increasingfluidization time and superficial gas velocity were seen to decreasethe amount of agglomerate material recovered from the fluidizedbed and were put in the denominator of the equation. The two termswith superficial gas velocity were included to see which velocityrelationship to minimum fluidization velocity would give the bestpredictions.

mmo

= KJMP

S×100ð ÞηVψ = 3frag

tψ U−Umfð Þψ UUmf

� �ω ð22Þ

In Eq. (22), KJMP, η, ψ, and ω are fitting parameters that can beadjusted. The excess gas velocity (U−Umf) was used because theunits of this term were more conducive to make the equation non-dimensional. Because the ratio of the superficial gas velocity to theminimum fluidization velocity was dimensionless, it was included inthe equation to see if the resulting fit would be better. The problem

Fig. 13. Comparison of the residuals for the polynomial fit with normal distributions forsilica sand data.

was set up in JMP statistical software. The data that was supplied tothe software was divided based on the particle type as was done forthe polynomial approximations. Using non-linear least squares fitting,the values for the parameters in Eq. (22) that gave the best fit werefound. The results are given in Table 3.

Fig. 14. Comparison of the predicted versus actual values of m/mo for glass beads usingEq. (22). (A) Data used to generate the equation and (B) data not used to generate theequation.

Table 4Coefficients obtained from JMP non-linear least squares analysis with extreme dataremoved for use with Eq. 23. The ranges are the 95% confidence intervals of theparameter estimates.

Coefficients Glass beads Silica sand

KJMP 0.777±0.094 0.217±0.061η 0.288±0.024 0.375±0.036ψ 0.069±0.018 −0.098±0.047

98 S. Weber et al. / Powder Technology 210 (2011) 87–102

The software was unable to calculate the confidence intervals forthe parameters in the silica sand system so they were not reported.The predicted values were calculated for both the data used for the fitand data not used in the fit and were compared with experimentalresults as shown in Figs. 14 and 15.

Once again, Fig. 14 shows that there are discrepancies betweenthe predicted and actual values of the amount of agglomerate mate-rial recovered from the fluidized bed when a lot of erosion is occur-ring. A double tail was observed in the erosion regime. This was aninteresting result, however, the prediction of the experimental valueswas not as good as was found using a polynomial to approximate thetrue function. It was also found that excluding the ratio of the super-ficial gas velocity to the minimum fluidization velocity did not greatlyaffect the predicted values of the amount of agglomerate material

Fig. 15. Comparison of the predicted versus actual values of m/mo for silica sand usingEq. (22). (A) Data used to generate the equation and (B) data not used to generate theequation.

recovered. To simplify the equation as much as possible, the velocityratio was removed from the equation. As was done for the polynomialapproximation, the extreme values were eliminated from the fittingdata and the JMP software was used to fit Eq. (23) using a least-squares method. Table 4 presents the coefficients for Eq. (23) for both

Fig. 16. Comparison of the predicted versus actual values of m/mo for glass beads usingEq. (23). (A) Data used to generate the equation and (B) data not used to generateEq. (23).

Fig. 17. Comparison of the predicted versus actual values of m/mo for silica sand usingEq. (23). (A) Data used to generate the equation and (B) data not used to generateEq. (23).

Fig. 18. Comparison of the residuals with the predicted values for the non-dimensionalapproximation for glass beads data.

Fig. 19. Comparison of the residuals for the non-dimensional fit with normaldistributions for glass beads data.

99S. Weber et al. / Powder Technology 210 (2011) 87–102

glass beads and silica sand, and Figs. 16 and 17 show the results forglass beads and silica sand respectively.

mmo

= KJMP

Sñ100ð ÞηVψ = 3frag

tψ U−Umfð Þψ ð23Þ

As was done for the polynomial approximation, the residuals forthe non-dimensional correlation were plotted versus the predictedvalues for both glass beads and silica sand results. The residuals werealso compared with normal distributions. The results are shown inFigs. 18–21.

As was observed for the polynomial approximation, most data fellvery close to the line when the extreme values were eliminated. Therewere still, however, some points that deviated from the line, but thismay have been caused by experimental variation. The correlationpredicts the experimental data not used in the fit well. As was seen forthe polynomial approximation, the residuals for the glass beadsresults nearly had a normal distribution. One difference was that theonly deviation from the normal distribution was observed at theextreme left side of the graph, unlike the polynomial approximationresiduals which deviated at both the left and right hand sides of thegraph. Another difference was that the residuals for the silica sand datawere relatively normally distributed, unlike the silica sand residuals forthe polynomial approximation. Although there was less data available,fewer parameters were estimated in the non-dimensional approach.The non-dimensional correlation accounted for the processes occurringduring fluidization well.

While Eq. (23) predicts m/mo for glass beads and silica sand usingdifferent coefficients, the final option that was considered was tocombine both systems together and to use the sphericity in Eq. (23).Experimental observations have shown that when the sphericity ofthe particles decreased, the amount of agglomerate material

Fig. 20. Comparison of the residuals with the predicted values for the non-dimensionalapproximation for silica sand data.

Table 5Coefficients obtained from JMP non-linear least squaresanalysis with extreme data removed for use with Eq. 24.The ranges are the 95% confidence intervals of theparameter estimates.

Coefficients

K 0.751±0.090η 0.295±0.023ψ 0.067±0.018Ω 0.320±0.219

100 S. Weber et al. / Powder Technology 210 (2011) 87–102

recovered from the fluidized bed decreased. Eq. (24) is thedimensionless expression that accounts for this behavior.

mmo

= KJMP

S×100ð ÞηVψ3fragϕ

Ω

tψ U−Umfð Þψ ð24Þ

Once again, JMP software was used to fit the equation. Table 5gives the coefficients that were found to give the best fit using theleast squaresmethod and Fig. 22 shows how Eq. (24) fit the data (bothincluded and not included in the fit).

Fig. 22 shows that including the sphericity in the non-dimensionalequation predicts the results for both glass beads and silica sandmaterial well. There are still some conditions that do not fall on theline, however, the number and amount of these deviations haveremained consistent for all methods of applying the non-dimensionalequation. Table 5 shows that the confidence intervals for theestimates of the parameters in Eq. (24) are very narrow in mostcases. The widest confidence interval was observed in the sphericityexponent. This only affects the fit for the silica sand results becausethe sphericity of the glass beads was 1.

Fig. 21. Comparison of the residuals for the non-dimensional fit with normaldistributions for silica sand data.

3. Summary

Two approaches were considered to empirically predict theamount of agglomerate material recovered from a fluidized bed in

Fig. 22. Comparison of the predicted versus actual values of m/mo for both glass beadsand silica sand using Eq. (24). (A) Data used to generate Eq. (24) and (B) data not usedto generate Eq. (24).

Table 6Residual sum of squares for the different methods considered in this study. The resultspertain to the case where the extreme large and small values were eliminated.

Residual sum of squares R2

Polynomial approximationGlass beads

11.04 0.664

Polynomial approximationSilica sand

0.74 0.831

Non-dimensional approachGlass beads

13.38 0.593

Non-dimensional approachSilica sand

0.10 0.977

Non-dimensional approachGlass beads and silica sand

14.05 0.625

101S. Weber et al. / Powder Technology 210 (2011) 87–102

this study. The residual sum of squares (SSE) and the R2 values werecalculated for both the polynomial approximation and the non-dimensional approach. The results are shown in Table 6.

When comparing the polynomial and non-dimensional approachesin the glass beads system, it can be seen that the non-dimensionalapproach does not predict the experimental results quite as well asthe polynomial approach. While this makes the polynomial approxi-mation approach seembetter, itmust be noted that 14 coefficientsmustbe estimated for the correlation. In comparison, the non-dimensionalapproach requires only 3 parameters. To compare how well eachcorrelation fitted the experimental data while considering the numberof estimated coefficients required, an F-test was completed to a sig-nificance level of 0.05 [26]. From the F-test, the polynomial approxima-tion was found to be the better correlation for the glass beads resultsand the non-dimensional approach was the better correlation for thesilica sand results. For the glass beads system, a lot of data was availableso the estimation of the parameters for the polynomial approxima-tion accounted for most of the behaviors occurring in the fluidized bed.For the silica sand system, less data was available and the non-dimensional approach required the estimation of fewer parameters,allowing better prediction of the majority of behaviors occurring in thefluidized bed.

From Table 6, it can be seen that the R2 value is approximately0.60 in the glass beads system. Some reasons why the correlationshad some difficulties predicting the experimental values are partiallycaused by the inputs into the system. Several parameters wereestimated, including the agglomerate fragment volume. Experimentalresults showed that the agglomerates did not consistently fragmentinto the same number of pieces nor was the agglomerate fragmentsize equal. Obtaining more information about the fragmentationproduct may help increase the accuracy in predicting the massrecovered from the fluidized bed. Part of the error in the predictionmay also have been caused by the estimate of the fluidized bed shearrate and the interaction of the shear with the agglomerates. This canbe improved by using more complicated fluidized bed hydrodynamicmodels to obtain more accurate values for some fluidization param-eters such as the one proposed by Di Renzo and Di Maio [27]. Theirwork only reported how the fluidized bed height and bed voidagechanged with time, but this could lead to more accurate determina-tion of the shear fields in a fluidized bed by extending the model.Another issue that may have decreased the ability of the procedure topredict the experimental values is that the fragment volume andsaturation are changing with time. During extreme growth, a largemass of dry particles were added to the agglomerate structure fromthe fluidized bed which decreased the saturation. In the opposite case,erosion is greatly decreasing the initial agglomerate and fragmentdiameter. To account for these processes, the change in these prop-erties with time is required.

The purpose of the procedure was to propose a relatively easymethod to obtain some idea about the fluidized bed agglomeratebehavior. The empirical prediction method outlined here does allow

the prediction of a fluidized bed agglomerate behavior in some cases.Adding more complexities to account for the changing agglomerateproperties with time will help make it more applicable to other situa-tions and agglomerating systems.

4. Conclusions

Several different approaches were taken to try to describe the datathat was obtained during fluidized bed experiments at ambienttemperature. Dividing the data by the sphericity helped find a set oftwo polynomials that described the actual relationship governing theresults in the glass beads–water and silica sand–water systems. Thedrawback to this approach is that the polynomial correlationsestimated in this study are system and equipment specific and maynot be generally applied to other systems. Further research resultswould allow the incorporation of binder viscosity, surface tension, andliquid contact angle into the correlations. This would make it mucheasier to extend the correlations to predict the behavior of agglom-erates in other solid–liquid systems.

A non-dimensional approachwas considered next. The data used forthe fitting was first divided based on sphericity. Good predictionof results was observed in most cases, however, the correlation had adifficulty in predicting the results when extreme erosion or growthprocesses were occurring. Removing these extremes from the dataallowed the non-dimensional correlations to better predict the experi-mental results. After confirming that the non-dimensional analysiswasworkingwell, thedatawas recombined and sphericitywas added tothe correlation. This general non-dimensional equation did a good jobpredicting the experimental results for both glass beads and silica sandagglomerates.

F-tests indicated that the polynomial approximation was thebetter correlation for glass beads results while the non-dimensionalapproach was the better correlation for the silica sand results. Whilethe polynomial approximations predicted the results well and wereeasy to use, the correlation derived from dimensional analysis alsogenerally predicted the results well and has the potential to be moreuniversal. The non-dimensional approach should be tested usingfluidized bed results obtained using different distributors to see if theresults are still predicted well. Further extension into other granula-tion equipment should also be considered. Particle shape effects in thedetermination of the granule strength were not considered at thistime. This would be a good extension of this work, however, it wouldbe more meaningful when used in conjunction with better measure-ment and understanding of the fragmentation product.

Acknowledgements

The authors acknowledge the support of the Natural Sciences andEngineering Research Council of Canada and Syncrude Canada Ltd. fortheir funding and support for this study.

References

[1] S.M. Iveson, Granule coalescence modeling: including the effects of bondstrengthening and distributed impact separation forces, Chemical EngineeringScience 56 (2001) 2215–2220.

[2] G. Lian, C. Thornton, M.J. Adams, Discrete particle simulation of agglomerateimpact coalescence, Chemical Engineering Science 53 (1998) 3381–3391.

[3] S.H. Schaafsma, P. Vonk, P. Segers, N.W.F. Kossen, Description of agglomerategrowth, Powder Technology 97 (1998) 183–190.

[4] M. Stein, J.P.K. Seville, D.J. Parker, Attrition of porous glass particles in a fluidizedbed, Powder Technology 100 (1998) 242–250.

[5] J. Litster, B. Ennis, The Science and Engineering of Granulation Processes, KluwerAcademic Publishers, Boston, USA, 2004.

[6] K. Saleh, D. Steinmetz, M. Hemati, Experimental study and modeling of fluidizedbed coating and agglomeration, Powder Technology 130 (2003) 116–123.

[7] C.A. Biggs, C. Sanders, A.C. Scott, A.W. Willemse, A.C. Hoffman, T. Instone, A.D.Salman, M.J. Hounslow, Coupling granule properties and granulation rates inhigh-shear granulation, Powder Technology 130 (2003) 162–168.

102 S. Weber et al. / Powder Technology 210 (2011) 87–102

[8] J. Kumar, G. Warnecke, M. Peglow, S. Heinrich, Comparison of numerical methodsfor solving population balance equations incorporating aggregation and breakage,Powder Technology 189 (2009) 218–229.

[9] S.M. Iveson, Limitations of one-dimensional population balance models of wetgranulation processes, Powder Technology 124 (2002) 219–229.

[10] D.K. Kafui, C. Thornton, Fully-3D DEM simulation of fluidized bed spraygranulation using an exploratory surface energy-based spray zone concept,Powder Technology 184 (2008) 177–188.

[11] C. Thornton, K.K. Yin, M.J. Adams, Numerical simulation of the impact fracture andfragmentation of agglomerates, Journal of Physics D: Applied Physics 29 (1996)424–435.

[12] S. Weber, C. Briens, F. Berruti, E. Chan, M. Gray, Agglomerate stability in fluidizedbeds of glass beads and silica sand, Powder Technology 165 (2006) 115–127.

[13] S. Weber, C. Briens, F. Berruti, E. Chan, M. Gray, Effect of agglomerate properties onagglomerate stability in fluidized beds, Chemical Engineering Science 63 (2008)4245–4256.

[14] S. Weber, Agglomerate Stability in Fluidized Beds. PhD Thesis. The University ofWestern Ontario (2009).

[15] G.I. Tardos, M.I. Khan, P.R. Mort, Critical parameters and limiting conditions inbinder granulation of fine powder, Powder Technology 94 (1997) 245–258.

[16] M. Benali, V. Gerbaud, M. Hemati, Effect of operating conditions and physico-chemical properties on the wet granulation kinetics in high shear mixer, PowderTechnology 190 (2009) 160–169.

[17] B.J. Ennis, G. Tardos, R. Pfeffer, A microlevel-based characterization of granulationphenomena, Powder Technology 65 (1991) 257–272.

[18] K. Hilligardt, J. Werther, Local bubble gas hold-up and expansion of gas/solidfluidized beds, German Chemical Engineering 9 (1986) 215–221.

[19] R.C. Darton, Bubble growth theory of fluidized bed reactors, Transaction of theInstitution of Chemical Engineers 57 (1979) 134–138.

[20] D. Kunii, O. Levenspiel, Fluidization Engineering, 2nd EditionButterworth-Heinemann, Boston, USA, 1991.

[21] P.J. Sherrington, R. Oliver, “Granulation”. Monographs in Powder Science andTechnology. Ed. A.S. Goldberg. Philadelphia, U.S.A.:Heyden & Son, Ltd., 1981. 7–59,153–165.

[22] J.P.K. Seville, C.D. Willet, P.C. Knight, Interparticle forces in fluidization: a review,Powder Technology 113 (2000) 261–268.

[23] S.M. Iveson, N.W. Page, Dynamic strength of liquid-bound granular materials: theeffect of particle size and shape, Powder Technology 152 (2005) 79–89.

[24] S.M. Iveson, N.W. Page, Tensile bond strength development between liquid-boundpellets during compression, Powder Technology 117 (2001) 113–122.

[25] S.M. Iveson, J.A. Beathe, N.W. Page, The dynamic strength of partially saturated powdercompacts: the effect of liquid properties, Powder Technology 127 (2002) 149–161.

[26] J. Neter, M. Kutner, W. Wasserman, C. Nachtsheim, Applied Linear StatisticalModels, 4th EditionIrwin, Chicago, USA, 1996.

[27] A. Di Renzo, F.P. Di Maio, Homogenous and bubbling fluidization regimes in DEM-CFD simulations: hydrodynamic stability of gas and liquid fluidized beds,Chemical Engineering Science 62 (2007) 116–130.