Powder Technology 254 (2014) 22–29

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Experimental characterizing the residence time distribution of largespherical objects immersed in a fluidized bed

Cai Rongrong, Zhang Yanguo⁎, Li Qinghai, Meng AihongKey Laboratory for Thermal Science, Power Engineering of Ministry of Education, Beijing Key Laboratory for CO2 Utilization Reduction Technology, Department of Thermal Engineering, TsinghuaUniversity, Beijing 100084, China

⁎ Corresponding author.E-mail address: zhangyg@tsinghua.edu.cn (Y. Zhang).

0032-5910/$ – see front matter © 2014 Published by Elsehttp://dx.doi.org/10.1016/j.powtec.2013.12.050

a b s t r a c t

a r t i c l e i n f o

Article history:Received 8 September 2013Received in revised form 29 November 2013Accepted 31 December 2013Available online 8 January 2014

Keywords:Residence time distributionLarge spherical objectFluidized bedInclined air distributorDimensionless

In many applications of fluidized bed, large objects are coexisting with small emulsion solids. The object motionpatterns and residence time distribution (RTD) in the bed have a paramount effect in the performance of the re-actor. In this paper, a series of experiments were conducted to study the influence of the superficial gas velocityand the constitutive properties of large spherical objects on their RTD in a fluidized bed with inclined air distrib-utor using Electrical Capacitance Tomography (ECT) tracing technique. The objects were larger and denser thanthe fine particles of the dense phase, with the ratio of object diameter to bed characteristic length approximately0.14 and the ratio of object density to static bed density ranging from 1.5 to 5.0. Experimental results show thatthe RTD curves of large objects have a relatively large discrepancy from the ideal normal distribution. As anobject's size and density increases, themean residence time (MRT) of the object decreases initially and increasessubsequently. With the increasing of the superficial gas velocity, theMRT decreases with a trend of falling downfast at first then getting slow. Based on amechanicalmodel of the object on an inclined air distributor, the behav-iors of the object were properly explained. Finally, an empirical correlation was derived for the MRT using someof experimental data and dimensionless analysis. The validations of the correlation by other independent exper-imental data show that the predicted values of MRT are well in accordance with the experimental values, andmost of the relative errors are within ±30%.

© 2014 Published by Elsevier B.V.

1. Introduction

Fluidized beds have been applied in many processes involvinggasification, granulation, separation and combustion of a wide rangeof particulate materials including biomass, solid waste, sewage sludgeand catalyst.Mixtures of dissimilar particles often exist in these process-es, but the problems due to the presence of more than one particulatespecies that differ in their constitutive properties e.g. size, density orshape in the same unit remain less addressed.

Many attempts have been made to better understand the hydrody-namics of gas–solid flow in a multi-component fluidized system. Mostof these studies focused on the overall mixing and segregation phenom-ena with analyses of the mixing and segregation progress, patterns andmechanisms under various conditions, [1–8] accepting that they areessentially driven by the action of bubbles. Other studies have focusedon the motion behavior of single object with unusual constitutiveproperties (e.g. special shapes, large sizes or low densities) in a multi-component system, [9–15] since its motion patterns and ability tomove throughout the bed are key factors influencing the performanceof the bed. Sanderson et al. [12] studied the motion of a single large

vier B.V.

object in a 2-D bed, with characterization of cycles sinking down fromthe surface of the bed and rising back again. Similarly, Soria-Verdugoet al. [13] experimentally studied the effect of gas velocity, bed height,density and object shape on the motion of an object submerged in thebed with a rotating distributor. Takuya et al. [15] simulated the motionof a large sphere in a bubblingfluidized bed using DEM-CFDmesoscopicmodel combined with volume penalization method. However, theirstudies mainly concentrated on the low density objects and verticalmotion in the dilute zone of a fluidized bed, rather than on the lateralmotion behavior of large heavy objects next to the air distributor.

Several researchers [16,17] have noticed that large objectswith largedensity may stop fluidizing in the bed and be captured in a stagnantzone over the distributor, therefore, investigation of their motionbehavior within the dense zone in order to discharge them effectivelyis of significance for normal operation of fluidized bed. Uneven air distri-bution, such as inclined air distributor, induces an internal particlecirculation inside the bed which can drive the large objects to thedischarging hole [18,19]. Therefore, the residence time of objects inthe uneven air distribution bed becomes one of themost important pa-rameters. Some researchers[20–22] have studied the effect of operatingparameters on the RTD of large objects, however, the investigationswere all qualitative with few consistent conclusions; in addition, therehave been few attempts on developing the mechanical model to depictthe large object motion and explain the experimental results.

Notation

CD drag coefficientCV coefficient of variationdp bed material diameter, mmdo object diameter, mmf friction coefficientfs static friction coefficientFd drag force, NFf friction force, NFG gravitational force, NFN normal force, NFP bed pressure difference force, Ng acceleration of gravity, m/sk coefficientL bed length, mN repeated test timess3 skewnesst residence time, stm mean residence time, stmin the minimum residence time, sVo object volume, m3

v velocity of gas–solid flow, m/svcr critical fluidization velocity, m/svo object velocity, m/svt turning gas velocity, m/s

Greek lettersα, β, γ exponentθ inclined angle of air distributorρ bed density during fluidization, kg/m3

ρb initial bulk density of bed materials, kg/m3

ρp skeletal density of bed material, kg/m3

ρo object density, kg/m3

ρt turning density, kg/m3

ε bed voidageμ dynamic viscosity of gas, N · s/m2

σ2 variance

Subscripts⊥ direction normal to the air distributor:∥ direction parallel to the air distributor:

Fig. 1. Schematic of experimental setup.

Fig. 2. Structure of air distributor and air caps.

23R. Cai et al. / Powder Technology 254 (2014) 22–29

The main objective of the present work is to examine the RTD of alarge spherical object in a fluidized bed with an inclined air distributorand oriented caps. The emphasis is laid on the influence of the superfi-cial gas velocity and the object properties (density, size) on their resi-dence time. A dimensionless empirical correlation of MRT is proposedbased on some of the experimental data. Along with the experiment, amechanical model is established, trying to identify the fundamentalprinciples controlling the RTD of the object in a multi-component fluid-ized system.

2. Experimental

2.1. Setup

The experiments were conducted in a rectangular-shaped fluidizedbed setup with a length (L) of 0.28 m, a width (W) of 0.24 m, and aheight (H) of approximately 1.4 m, as shown in Fig. 1. Fluidizing air en-ters the bed through a 5° inclined air distributor from a fan after passingthrough a rotameter and a valve. The distributor wasmade of plexiglass

and comprises of 194 equally spaced air caps, each of them consisting ofsix horizontal openings of 0.002 m in diameter, ensure the air flows inthe downward direction parallel to the distributor, as detailed in Fig. 2.

2.2. Materials

Polypropylene particles with relative permittivity as low as 1.5were used as the bed materials. They had an average diameter (dp)

24 R. Cai et al. / Powder Technology 254 (2014) 22–29

of 0.83 mm and a skeletal density (ρp) of 760 kg/m3, belonging toGroup B in terms of the Geldart classification and with a measuredcritical fluidization velocity (vcr) of 0.27 m/s. The static bed height(Hs) was 0.12 m from the low side of the air distributor and thebulk density (ρb) was measured to be 470 kg/m3.

The large spherical objects were made by filling molds with varioushigh permittivity materials. Their shapes and sizes (do) were deter-mined by the mold outside and their densities (ρo) could be adjustedby mixing materials in different proportions inside. Their constitutiveproperties andmanufacturingmaterials are given in Table 1. The objectswere larger and denser than the fine particles of the dense phase, withthe ratio of object diameter to bed characteristic length approximately0.14 and the ratio of object density to static bed density ranging from1.5 to 5.0.

2.3. Experimental procedure and RTD determination

During experiment, an object fell vertically from the upper part ofthe fluidized bed, due to the large density of the object, it always quicklysank to the bottom zone of the bed, and moved down along the topsurface of the air distributor until reaching the lower side because ofits own gravity and internal solids circulation induced by uneven airdistribution, [19] as labeled sequentially No. 1, No. 2, No. 3 in Fig. 1.

The residence time of an object was measured using ECT tracingtechnique [23]. The ECT system consisted of an electrical capacitancesensor unit, a data acquisition module (PTL300E-SP-G ECT) and a per-sonal computer equipped with custom communication software. Twosets of capacitance sensors each including 8 measurement electrodesand several corresponding axial/radial guard electrodes were wrappedaround the circumference of the bed at different heights. Details oftheir sizes can be seen in Fig. 3. All the electrodes were connected tothe PC through the data acquisition module with a capturing rate of50 frames per second. The upper set of capacitance sensor recordedthe initial falling time (t1) of an object, while the lower set traced itsmo-tion trajectory in the dense zone of bed, until it reached the low side ofthe air distributor and recorded the time (t2), thus the residence timecould be calculated as t = t2 − t1.

Due to the structure heterogeneity, regimemultiplicity and behaviornonlinearity of the fluidization systems, the residence time of an objectin each experimental test is different even at the same operating condi-tion, thus repeated tests are required to study its residence time distri-bution (RTD). According to the Law of Large Numbers (LLN) theorem,the mean and the distribution of the residence time obtained from alarge number of tests should be close to the theoretical value, andthey tend to become closer as more tests are performed. Therefore, tobetter describe the RTD of an object in the bed and retain the inherentdynamic nature of flow structure, Eq. (1) was used to determine theaccepted minimum number of tests for each experimental condition,

N ¼ minf tð ÞN− f tð Þ N−10ð Þ��� ���

f tð Þ N−10ð Þb10%

8<:

9=; ð1Þ

Table 1Object properties.

Diameterdo (cm)

Densityρo (g/cm3)

Materials

Mold Fillings Overall relativepermittivity

3,4,5 0.56 PS Acid PMMA powder, PS foam 153,4,5 0.90 Water, ethanol 203,4,5 1.63 Acid PMMA powder,

PbO powder16

3,4,5 2.35 Acid PMMA powder,PbO powder

16

4 3.00 PbO powder 15

where f(t)N is the probability density distribution of the residence timeof N tests. Eq. (1) implies that the RTD curves from N tests and (N − 1)tests have a similarity over 90%.

Previous experimental studies indicate that themean residence timeof an object is dependent on the fluidization velocity v, object density ρo,object size D, bed material characteristic and air distribution type[21,22]. However, the structure and sizes of their experimental appara-tus, as well as the bedmaterials used were different from each other, sotheir conclusions are poor in comparability. In this work, dimensionlessanalysis was adopted for experimental condition design, using three di-mensionless variables (v/vcr, ρ/ρb, do/L) with critical fluidization velocityvcr, static bed density ρb and bed characteristic length L as independentvariables, as shown in Table 2.

2.4. Characterization of the RTD

The RTD is a probability density function that may be characterizedusing statistical moments. The moments are a mathematical expecta-tion over the interval [0, ∞] of the general type:

Ex t−t0ð Þkh i

¼Z ∞

0t−t0ð ÞkE tð Þdt: ð2Þ

Eq. (2) is the kth moment of a distribution of the residence time (t).The mean (tm), variance (σ2) and skewness (s3) of the distribution arethen defined as

tm ¼

Z ∞

0tE tð ÞdtZ ∞

0E tð Þdt

¼Z ∞

0tE tð Þdt ð3Þ

σ2 ¼Z ∞

0t−tmð Þ2E tð Þdt ð4Þ

s3 ¼Z ∞

0t−tmð Þ3E tð Þdt: ð5Þ

Combinations of the moments are commonly used to describe theprobability distribution. The coefficient of variation, defined as

CV ¼ σtm

ð6Þ

is the ratio of the standard deviation to the mean which expresses thedispersing extent of a distribution. Buffman and Mason [24] advocatedCV as the most appropriate single measure of the extent of spreadingof an experimental RTD. Thus RTD measurements were characterizedby their mean, variance, skewness and coefficient of variation deter-mined from the moments.

3. Mechanical model

The motion of an object in the bed includes two processes, as illus-trated 1–2 and 2–3 in Fig. 1. Process (i) is dominated by gravitationalforce, since the investigated objects are far denser than the fluidizingbed density, thus the time cost in process (i) is small enough to beneglected. Therefore, attention is only paid on discussion of the forcecondition of an object in process (ii).

Consider a spherical object sliding on the surface of the air distributorand assume the surrounding gas–solid flow incompressible and pseudo-homogeneous, with its density calculated as ρ = ρp(1 − ε) + ρgε. Then2-dimensional mechanical model of an object moving in process (ii) canbe expressed as:

FG cosθ−FP⊥−Fd⊥−FN ¼ 0 ð7Þ

Fig. 3. Schematic diagram of ECT sensor.

25R. Cai et al. / Powder Technology 254 (2014) 22–29

π6do

3ρog cosθ−π6do

3ρg cosθ−12CDπ

do2

� �2ρ v⊥−vo⊥ð Þ2−FN ¼ 0 ð8Þ

for the direction normal to the air distributor;

ma ¼ FG sinθ‐Fd∥ þ FP∥−F f ð9Þ

Table 2Experimental results for the moments of the residence time distributions.

No. ρoρb

vvcr

doL N tm/s σ/s s/s Cv

1 1.5 3.06 0.143 50 26.42 10.12 1.02 0.382 3.67 0.107 50 103.65 52.35 0.65 0.513 3.67 0.143 50 69.96 36.38 0.50 0.524 3.67 0.179 50 31.25 10.25 1.23 0.335 4.13 0.143 50 83.57 35.62 1.35 0.436 1.9 2.59 0.143 50 333.70 126.36 0.59 0.387 2.76 0.143 50 251.76 153.47 0.57 0.618 3.06 0.143 70 44.70 20.54 1.04 0.519 3.37 0.143 60 20.14 12.23 1.66 0.6110 3.67 0.143 60 14.18 4.68 1.05 0.3311 4.13 0.143 70 9.74 5.25 1.32 0.5412 3.67 0.107 50 36.38 21.46 0.58 0.5913 3.67 0.179 80 10.53 5.41 1.18 0.5114 3.5 2.76 0.143 50 327.58 132.49 0.89 0.4015 2.91 0.143 50 150.31 45.88 0.93 0.3116 3.06 0.143 50 99.59 46.54 1.23 0.4717 3.37 0.143 60 69.51 41.69 1.61 0.6018 3.67 0.143 50 31.78 9.48 1.20 0.3019 4.13 0.143 50 24.73 13.29 1.40 0.5420 3.67 0.107 50 21.25 12.13 1.62 0.5721 3.67 0.179 50 39.46 14.42 0.85 0.3722 5.0 2.91 0.143 50 306.56 113.43 0.83 0.3723 3.06 0.143 50 141.29 53.68 0.67 0.3824 3.37 0.143 50 75.83 29.26 0.55 0.3925 3.67 0.143 50 59.95 19.29 0.52 0.3226 4.13 0.143 55 36.65 16.77 0.54 0.4627 4.59 0.143 50 28.15 14.36 0.51 0.5128 3.67 0.107 50 42.91 14.19 0.36 0.3329 3.67 0.179 50 120.44 68.68 0.50 0.57

ρoVodvo∥dt

¼ π6do

3ρog sinθ−12CDπ

do2

� �2ρ v∥−vo∥ð Þ2−Vo

∂P∂x− f FN ð10Þ

for the direction parallel to the air distributor.Where FG is due to gravityand is determined by the object size and density; FP is the sum of staticpressure distribution on the whole object from surrounding gas–solidflow; Fd is the sum of the drag forces acting on the object from the sur-rounding gas and fine particles; FN is the normal force of air distributor(actually the air caps) on the object; Ff is the friction force between theair distributor, bed wall and the object; θ is the inclined angle of the airdistributor. Several other forces also act on the object, such as Bassetforce, the additionalmass force and collision force. Since the investigatedobjects are larger and denser than the surrounding particles, in addition,their motion velocity is slow, thus these forces are small enough to beneglected here. The schematic of the mechanical model can be seen inFig. 4.

4. Results and discussion

Table 2 summarizes the detailed experimental results for tm (togetherwith σ, s and CV) of the objects in the fluidized bed with various fluidiza-tion velocities. All the operating variables were made dimensionless by

Fig. 4. Mechanical model of a large object on air distributor.

0 20 40 60 80 100 120

0.02

0.04

0.06

0.08

0.1

0.12

t/s

pdf/s

−1

v/vcr

=3.06

v/vcr

=3.37

v/vcr

=3.67

v/vcr

=4.13

0 50 100 150 200

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

t/s

pdf/s

−1

v/vcr

=3.06

v/vcr

=3.37

v/vcr

=3.67

v/vcr

=4.13

0 50 100 150 200

2

4

6

8

10

12

14

16

18

x 10−3

t/s

pdf/s

−1

v/vcr

=3.06

v/vcr

=3.37

v/vcr

=3.67

v/vcr

=4.13

gdf

ρo/ρ

b=1.9 ρ

o/ρ

b=3.5 ρ

o/ρ

b=5.0

Fig. 5. Residence time distribution of objects (do/L = 0.143).

26R.Caietal./Pow

derTechnology

254(2014)

22–29

Fig. 6. A comparison between RTD and normal distribution.

Fig. 8. Influence of object size on MRT (v/vcr = 3.67).

27R. Cai et al. / Powder Technology 254 (2014) 22–29

dividing by vcr, ρb or L as independent variables. In the following sections,the influence of each variable on the RTD will be discussed accordingly.

4.1. Residence time distribution of objects

Examples of the measured residence time distribution for someobjects in the bed are presented in Fig. 5. Each curve was filtered to re-move random noise and normalized to give ∫ 0

∞E(t)dt = 1. The charac-teristics, such as the maximum peak height, the spread and the generalshape of the RTD curves of the objects under different experimentalconditions, differ significantly fromeach other. Even for the same exper-imental condition, the residence time can differ by as much as an orderofmagnitude, which reveals that the gas–solidflow in the dense zone ofthe fluidized bed is dominated by both deterministic and randommechanisms.

Previous studies on the RTD of the particles in a fluidized bed rarelyconcentrated on single large heavy object, but on the overall homoge-neous or heterogeneous particles [25–27]. Results concluded that theRTD of homogeneous particles is approximately subjected to a normaldistribution, however, for mixed heterogeneous particles such as mu-nicipal solid waste, there exist a relatively large discrepancy. A compar-ison between experimental results and theoretical calculation for RTD ofthe objects is shown in Fig. 6. It is noted that, there exists a relativelylarge discrepancy between the experimental value and the theoreticalcurve. The Shapiro–Wilk normality test reveals that RTD curves are

Fig. 7. Influence of object density on MRT (do/L = 0.143).

approximately but not exactly normal distribution at detection level of0.5. In addition, the object tends to have a broad distribution and along tail when at a low fluidization velocity.

The detailed σ, s and CV of objects under various operating condi-tions are given in Table 2. More valuable information can be obtainedas follows. (1) Most of RTD curves have skewness to the right with along tail, and it's longer for lighter objects. (2) With the increasing ofthe superficial gas velocity, σ of the RTD decreases remarkably, but theimpact on CV is less notable. (3) As object density or size increases, thechanging tendency of σ is uncertain, while CV varies or keeps in arange from 0.3 to 0.6. However, it must be noted that the measurederror of RTD's σ, s and CV is quite high, meanwhile, the measured preci-sion of MRT can doubtlessly reach an expected level because of theirstatistical averaged characteristics. Therefore, more attention is paid todiscussions on MRT rather than σ, s and CV in the following section.

4.2. Influence of object density on the MRT

The impact of the object density on theMRT is shown in Fig. 7.Whenthe gas velocity is lowwith v/vcr at 3.06,MRT increaseswith the increas-ing of object density. Once the v/vcr increases to 3.67 ormore, the impactbecomes complicated. MRT decreases at first and then increase again asobject density increases. This conclusion is different from those obtain-ed from the light objects (ρ/ρb ≤ 1),whereMRT is negatively correlatedwith the object density [21]. It can be explained that light objects inves-tigated in literature [21] can be involved in the diffusion and convectionprocess of the fluidized bed, thus the lower density it was, the easier itgot involved in and the longer MRT it was.

Fig. 9. Influence of superficial gas velocity on MRT (do/L = 0.143).

28 R. Cai et al. / Powder Technology 254 (2014) 22–29

When the density increases to a certain value, the object will sink tothe bottom of the bed, and then its gravitational force and the frictionforce from the air distributor become dominated factors influencing itsmotion behavior. Substitution of Eq. (9) for FN derived from Eq. (7)gives the motion acceleration of the object as a ¼ g sinθ−fg cosθþf FP⊥þ f Fd⊥−Fd∥þFP∥

ρoVo. Thus, when the fluidization velocity keeps constant

(Fp and Fd is constant), increasing object density decreases the motionacceleration and results in a long residence time. Therefore, if the inves-tigated density span is broad enough, there exists a turning density (ρt),where the residence time is the smallest. It's verified in the experimentthat when the object density is even small (ρ/ρb ≤ 1.5), it may neversettle on the low side of the air distributor, but keep moving all overthe air distributor.

4.2. Influence of object size on the MRT

Object size exerts complicated impact on theMRT, as shown in Fig. 8.When the object density is smallwithρo/ρb at 1.5,MRT decreases sharp-ly with the increasing of the object size, which is consistent with litera-ture [21].When ρo/ρb increases to 1.9, the decreasing tendency becomesnot apparent. However, with further increase of ρo/ρb, MRT increases asthe object size increases.

It can be explained that the motion behavior of the objects with vari-ous sizes is different, suspending in the bed, sliding, rolling or transientstagnating on the air distributor. As observed in the experiments, theobject with small density and size performs more actively in the bed. Itparticipates in the fluidization process sometime, visits the whole regionof the air distributor and has more trajectories repeated. Therefore whenρo/ρb is as small as 1.9, the smaller object moves more actively in the bedand has a longer MRT.While, object with high density and large size onlysits on the surface of the air distributor and slide/roll discontinuously tothe low side. Combination of Eq. (8) and Eq. (10) gives the motion

acceleration of the object, a ¼ g sinθ− f g cosθ− ρg cosθþ3CD4do

ρ v⊥−vo⊥ð Þ2ρo

� �−

3CD4do

ρb v∥−vo∥ð Þ2−∂P∂x

ρo. So the denser objects have an opposite tendency, and

the larger object has a longer MRT.

4.3. Influence of superficial gas velocity on the MRT

Plot of MRT against the object density with the superficial gas veloc-ity as a parameter is shown in Fig. 9. With the decrease of v/vcr, MRTincreases gradually at first; when v/vcr decreases to a certain value,MRT increases sharply. As can be seen from Eq. (7), when gas velocityincreases, the drag force acted on the object in direction normal to the

Fig. 10. Comparison of the experimental values with the calculated MRT from Eq. (13).

air distributor (Fd⊥) increases as well, thus the friction force betweenthe object and the air distributor, namely (Ff) decreases. On the otherhand, high gas velocity strengthens the internal circulation of bedmate-rials in bed which will more quickly drive the object to the low side ofthe air distributor. Thus, MRT decreases with the increasing of the fluid-ization velocity. In addition, if thefluidization velocity is too low, the bedpressure difference (Fp) and drag force (Fd) are very small but thefriction force (Ff) becomes large.When the force acted on the object sat-isfies the formula as FG sin θ b Fd∥ − FP∥ + f(FG cos θ − FP⊥ − Fd⊥), theobject may stagnate on the bottom of bed, and never reaches the lowside of the air distributor.

It's also noted from Fig. 9 that there exists a turning gas velocity (vt)for each object, lower thanwhichMRT increases sharply. When the gasvelocity is small, the objectmoves discontinuously in the bed due to thehigh static friction with the air distributor, therefore the residence timeis long. As the gas velocity increases, the object moves more continu-ously. When the gas velocity increases to the turning gas velocity, theobject can move uninterrupted to the low side of the air distributor,thus the effect of the gas velocity on the MRT becomes less significant.According to the above analysis, equation as FG sin θ + FP∥ − Fd∥ =fs(FG cos θ − FP⊥ − Fd⊥) holds at the turning gas velocity, and it isimpacted by object properties, the static friction coefficient (fs) andthe inclined angle of the air distributor.

Further observation of the plots in Fig. 9 can also find that with theincrease of the gas velocity, the MRT decreases and finally reduces inasymptote to a constant value. In addition, the value increases withthe object density because denser objects require more inertial force.

4.4. Dimensionless correlation of the MRT

As mentioned in Sections 4.2 and 4.3, there are two cases of motionbehavior of an object in the fluidized bed. (i) Objects with small densityand size participate in the diffusion and convection process, performingactively in the bed. (ii) Objects with large density and size slide, roll andsometimes bounce up on the surface of the air distributor until reachingthe low side. Since case (i) has been studied in other literatures, hereweonly concentrate on case (ii). According to Eq. (8), if the force acting onan object satisfies the formula (11), it belongs to case (ii). Therefore, amathematical model as Eq. (12) is used for correlating MRT with thedata of Exp. nos. 6–9, 14–29 which corresponding to case (ii). Where,μ is the dynamic viscosity of the gas. The correlation result is given inEq. (13), with the correlation coefficient (R2) of 0.9007 and the averagestandard error (S) of 0.175 and Fisher's statistic (F) of 23.56. It is shownthat the empirical correlation is statistically significant andfitswell withthe experimental data.

ρog cosθ N ρg cosθþ 3CD

4doρ v⊥−vo⊥ð Þ2 ð11Þ

tm−tmin

d2pρb=μ¼ k v�

vcr

� αρo=ρbð Þβ D=Lð Þγ ð12Þ

tm ¼ 4:23� 107 v�vcr

� −11:04ρo=ρbð Þ1:05 D=Lð Þ0:89 þ 20:07 ð13Þ

In order to verify the stability and predictive ability of the empiricalcorrelation, some other independent experimental data whose operat-ing conditions corresponding to case (ii) were adopted to test themodel. Fig. 10 presents the comparison of experimental values withthe calculated ones obtained from Eq. (13). Reasonable agreementbetween them is obtainedwithmost of the relative errorswithin±30%.

29R. Cai et al. / Powder Technology 254 (2014) 22–29

5. Conclusion

The influence of the superficial gas velocity and the density, size oflarge spherical objects on their RTDs in thefluidized bedwas investigatedexperimentally. The conclusions are the followings:

(1) The residence timedistribution (RTD)of anobject of repeated testshas a relatively large discrepancy from the ideal normal distribu-tion, for it has a long tail. The coefficients of variation of the RTDsunder different experimental conditions range from 0.3 to 0.6.

(2) As object's size and density increase, the MRT decreases ini-tially and increases subsequently with a turning point. Lightand small objects may be involved in the diffusion and con-vection process, while, dense and large objects slide, roll andsometimes bounce up on the surface of the air distributoruntil reaching the low side. The turning condition can becalculated as ρog cosθ N ρg cosθþ 3CD

4doρ v⊥−vo⊥ð Þ2.

(3) With the increase of the superficial gas velocity, the MRT de-creases with a trend of reducing fast at first then getting slow.There exists a turning gas velocity (vt) for each object, and theMRT increases sharply when the velocity is lower than it.

(4) The predicted values obtained from the dimensionless empiricalcorrelation of MRT are well in accordance with the other inde-pendent experimental values, and most of the relative errorswere within ±30%.

Acknowledgments

Financial support from theNational Basic Research Programof China(973 Program, No. 2011CB201502) and the National Natural ScienceFoundation of China (No. 21376134) are gratefully acknowledged.

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