gas–solid flow and heat transfer in fluidized beds with tubes effects of material properties and...

8/16/2019 Gas–Solid Flow and Heat Transfer in Fluidized Beds With Tubes Effects of Material Properties and Tube Array Settings

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Gas–solid ow and heat transfer in uidized beds with tubes: Effects of

material properties and tube array settings

Q.F. Hou ⁎, Z.Y. Zhou, A.B. Yu

Laboratory for Simulation and Modelling of Particulate Systems, School of Materials Science and Engineering, The University of New South Wales, Sydney, NSW 2052, Australia

Laboratory for Simulation and Modelling of Particulate Systems, Department of Chemical Engineering, Monash University, Clayton, VIC 3800, Australia

a b s t r a c ta r t i c l e i n f o

Available online xxxx

Keywords:

Computational uid dynamics

Discrete element method

Tube array

Fluidization

Heat transfer

Theeffects of material properties andtube array settingson gas–solid ow andheat transfer characteristics inu-

idized beds with tubes are investigated by the combined approach of computationaluid dynamics and discreteelement method, incorporated with heat transfer models. First, the effect of material properties is illustrated by

considering cohesive and non-cohesive powders with different particle sizes. The contributions of different heat

transfer mechanisms are discussed at two tube temperatures. Signicant differences of gas–solid ow between

cohesive and non-cohesive powders are observed. The results reveal that conductive heat transfer between a

uidized bed and a tube is dominant for small cohesive particles while convective heat transfer is dominant

for large non-cohesive particles. Then, the uniformity of particle velocity and temperature elds is analyzed. It

is shown that material properties and gas velocity affect the uniformity of particle velocity and temperature in

a complicated manner. Finally, the effect of tube array settings is examined in terms of two geometrical param-

eters for both in-line and staggered settings. Complicated gas–solid ow and heat transfer characteristics are

observed. An effort is made to link macroscopic observations to microscopic information such as local porosity

and contact number between uidized particles and tubes. The ndings should be helpful for the optimization

of operation and design of uidized systems with tubes.

© 2015 Elsevier B.V. All rights reserved.

1. Introduction

Fluidized bed reactors are widely used in industries mainly due to

their high heat and mass transfer capability. Immersed surfaces such

as vertical or horizontal tubes,ns, and water walls are usually adopted

to control ow and heat transfer. Heat transfer performance is affected

by many factors such as material properties of gas and solid phases,

geometrical settings and operating conditions. In the past, many macro-

scopic studies have been carried out in this eld, leading to the formu-

lation of various correlations to determine the heat transfer coef cient

(HTC) of uidized beds as, for example, summarized by Kunii and

Levenspiel [1] and Molerus and Wirth [2]. These correlations have

shown their value in solving some practical problems. However, the

predictions by some correlations show signicant differences partly

due to negligence of certain parameters and unknown experimental

set-up and conditions [3]. To produce equations that can be generally

applied to different systems, microscopic understanding of ow

and heat transfer mechanisms at a particle scale is helpful. Such under-

standing can be obtained through experimental and/or numerical

approaches. In recent years, experimental examination of the heat

transfer at a particle scale has been attempted by various investigators,

often doneby measuring the temperature evolution of a tracing particle

[4–6]. The resulting information is useful for fundamental understand-

ing and model validation. To generate a more comprehensive picture

of heat transfer, in recent years numerical studies have been carried

out for various uidized systems based on the combined computational

uid dynamics (CFD) and discrete element method (DEM) approach

[7–21]. The models developed vary in some details and have different

advantages and limitations. However, they all demonstrate that the

combined CFD–DEM approach, incorporated with heat transfer models,

is an effective technique for investigating heat transfer in uidized

systems at a particle scale.

Fluidization andrelated heat transfer behaviors vary with thetype of

powders as classied by Geldart [22]. However, most of the previous

investigations are focused on large particles. Only a few investigators

studied heat transfer characteristics of ne particles theoretically or

experimentally, focusing on macroscopic HTC of packed beds rather

than uidized beds [23–29]. Di Natale et al. [24] found that HTC

between a uidized bed of ne particles and an immersed spherical or

cylindrical surface increases with the increase in particle Archimedes

number (which is a function of particle density and size, and uid

density and viscosity). Recently, heat transfer between a tube/probe

and a uidized bed has been investigated by the combined CFD–DEM

Powder Technology xxx (2015) xxx–xxx

⁎ Corresp onding author at: Laboratory for Simulation and Modelling of Particulate

Systems, Department of Chemical Engineering, Monash University, Clayton, VIC 3800,

Australia. Tel: + 61 3 99050845.

E-mail address: [email protected] (Q.F. Hou).

PTEC-10888; No of Pages 13

http://dx.doi.org/10.1016/j.powtec.2015.03.028

0032-5910/© 2015 Elsevier B.V. All rights reserved.

Contents lists available at ScienceDirect

Powder Technology

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / p o w t e c

Pleasecitethis article as: Q.F.Hou,et al., Gas–solidow and heat transferin uidized beds with tubes:Effects of material properties andtube arraysettings, Powder Technol. (2015), http://dx.doi.org/10.1016/j.powtec.2015.03.028

http://dx.doi.org/10.1016/j.powtec.2015.03.028http://dx.doi.org/10.1016/j.powtec.2015.03.028http://dx.doi.org/10.1016/j.powtec.2015.03.028mailto:[email protected]://dx.doi.org/10.1016/j.powtec.2015.03.028http://www.sciencedirect.com/science/journal/00325910http://www.elsevier.com/locate/powtechttp://dx.doi.org/10.1016/j.powtec.2015.03.028http://dx.doi.org/10.1016/j.powtec.2015.03.028http://www.elsevier.com/locate/powtechttp://www.sciencedirect.com/science/journal/00325910http://dx.doi.org/10.1016/j.powtec.2015.03.028http://localhost/var/www/apps/conversion/tmp/scratch_6/Journal%20logomailto:[email protected]://dx.doi.org/10.1016/j.powtec.2015.03.028


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approach [11,12,14]. The contributions of different heat transfer mech-

anisms are discussed [11], and the effects of some material properties

such as particle size and particle thermal conductivity are examined

[12,14]. In uidized beds, uniform particle velocity and temperature

distributions are often desired for heat transfer and chemical reactions.

If theuniformity is not good enough, hot spot,as pointed out by Kaneko

et al. [30], could be formed. Somehow, this importantissue has not been

addressed in detail in the previous studies.

A tube array rather than a single tube is often used in

uidizationsystems. One major concern is the setting of a tube array, related to

heat transfer and tube erosion [31–33]. Previous studies of such systems

have been mainly conducted by using two-uid models [34–36] or

by experimental approaches [37–39]. Recent studies on the setting

of a tube array are carried out by means of the CFD–DEM approach

[40,41]. Some interesting ndings are presented, but controversies can

also be identied. For example, the signicant effect of tube pitch on

the erosion has been demonstrated [35]. While no signicant difference

in terms of bubbling behaviors or heat transfer between different tube

settings (in-line and staggered) is observed [34,41], quite different

factors underlying heat transfer such as particle impacts and bubble be-

haviors are predicted by the CFD–DEM approach [40]. A possible reason

could be that particle scale interactions are not suf ciently considered.

Thedifferent observationsindicate that there is a need for further inves-

tigation of the effect of tube array settings on gas–solid ow behavior. In

particular, the effect on heat transfer and the underlying mechanisms

should be properly understood.

In this work, in connection with our previous efforts [14,15], two

signicant concerns relevant to gas–solid ows and heat transfer charac-

teristics are addressed by using the combined CFD–DEM approach. Firstly,

the effect of material properties for different types of particles including

non-cohesive and cohesive particles is investigated for a uidized bed

with a horizontal tube. The uniformity of velocity and temperature elds

is quantied. Secondly, the effect of tube array settings is investigated for

a uidized bed with multiple horizontal tubes. The complicated variation

of heat ux between the uidized bed and tubes is discussed in terms of

microscopic information such as local porosity and contact number be-

tween particles and tubes. Thendings should be useful for better under-

standing and prediction of heat transfer in gas uidization.

2. Model description

2.1. Governing equations for solid phase

Here, gasuidizationis considered to be composed of a discrete solid

phase and a continuum gasphase. The solidphase is described by DEM,

originally proposed by Cundall and Strack [42]. At any given time t , the

equations governing the translational and rotational motionsof particle

i can be written as:

mid v i=dt ¼ ∑ j f e;ij þ f d;ij þ f v;ij

þ f pf ;i þ mig ; ð1Þ

and

I idωi=dt ¼ ∑ j T t ;ij þ T r ;ij

; ð2Þ

where the equation for the van der Waals force is written as:

f v;ij ¼ −H a6 64R

3i R

3 j h þ Ri þ R j

h2 þ 2Rih þ 2R jh

2h2 þ 2Rih þ 2R jh þ 4RiR j

2⋅n: ð3Þ

The forces involved are: particle–uid interaction force f pf ,i, the grav-

itational force mig and the forces between particles (and between parti-

cles and walls) which include the elastic force f e,ij, the viscous damping

force f d,ij and the cohesive force f v,ij. Note that the cohesive force f v,ij,

considered here is the van der Waals force given by Eq. (3), which de-

pends on the Hamaker constant H a and the separation h of the interacting

surfaces along the line joining the centers of particles i and j. Ri and R j are

the radii of particles i and j respectively. A minimum separation hmin is

used in the calculation of f v,ij to represent the physical repulsive nature

and avoid the singular attractive force when h = 0. This treatment has

been proved to be valid for particles down to 1 μ m [43–45]. The torque

acting on particle i due to particle j includes two components: T t ,ij

which is generated by the tangential force and causes particle i to rotate,and T r ,ij which, commonly known as the rolling friction torque, is gener-

ated by asymmetric normal contact forces and slows down the relative

rotation between contacting particles [46,47]. If particle i undergoesmul-

tiple interactions, the individual interaction forces and torques are

summed up for all particles interacting with particle i. The equations

used to calculate the particle–particle interaction forces and torques,

and particle–uid interactionforces havebeen well established as, for ex-

ample, reviewed by Zhu et al. [48]. The equations used for the present

work are the same as those used in our previous studies [20,49].

The heat transfer between particle i and its surroundings have three

modes: convection with uid, conduction with other particles, tubes or

walls, and radiation with its local environment. According to the energy

balance, the governing equation for particle i can be written as [10]:

mic p;idT i=dt ¼ ∑ jQ ·

i; j þ Q ·

i; f þ Q ·

i;rad þ Q ·

i;wall þ Q ·

i;tube; ð4Þ

where Q ·

i; j is the conductive heat exchange rate between particles i

and j; Q ·

i; f is the convective heat exchange rate between particle i and

its local surrounding uid; Q ·

i;rad is the radiative heat exchange rate

between particle i and its local surrounding environment; Q ·

i;tube is the

conductive heat exchange rate between particle i and tubes; and

Q ·

i;wall is the conductive heat exchange rate between particle i and

wall. Mathematically, Eq. (4) is the same as the so-called lumped-

capacity formulation, where the thermal resistance within a particle is

neglected [50]. This condition is valid when the Biot number, dened

as h ⋅ (V i/ Ai)/ k pi, is less than 0.1, where h is the heat transfer coef cient;V i is the particle volume; Ai is theparticle surface area; and k pi is thepar-ticle thermal conductivity. However, as noted by Zhou et al. [10], Eq. (4)

is established on the basis of energy balance at the particle scale. So, the

values of parameters (e.g. m i, c pi, T i, and k pi) involved should be the

representative properties of the particle at this scale, which may need

further studies in the future. So is the case for the equations used to

calculate heat exchange rates involved.

The equations to calculate heat exchange rates in Eq. (4) are listed in

Table 1, andthe treatments for heat transfer between a tube and a uid-

ized bed have been discussed and used in the previous studies [10,

13–15]. Four conductive heat transfer mechanisms are considered

in the present work, including the conduction through particle–uid–

particle path: (1) between non-contacted particles, or (2) between

contacted particles; and the conduction through particle–particle

path: (3) between particles in enduring contact, or (4) betweenparticles in collisional contact. Note that Eq. (c) in Table 1 isfor the con-

ductive heat transfer mechanisms (1) and (2) between particles i and j;

and Eqs. (d) and (e) are for the conductive heat transfer mechanisms

(3) and (4), respectively.

The treatments of a tube are outlined below. A tube is treated as

walls because its size is much larger than a particle or a computational

cell used in CFD; otherwise, it can be treated as a particle. The conduc-

tion between a tube and a particle is considered in a similar manner to

that between particles. The equation used for evaluating the local

convection heat transfer between the tube and uid is the same as

those between a wall and uid. For the present study, the domain size

for the radiative heat transfer between particlesis the same as a compu-

tational cell 2d p. The denition of bed temperature (T bed) and tube

environmental temperature (T e) isthesame as T local,i. The local porosity

2 Q.F. Hou et al. / Powder Technology xxx (2015) xxx– xxx

Please citethisarticle as: Q.F.Hou,et al., Gas–solidow andheat transfer inuidized beds withtubes: Effects of material propertiesand tube arraysettings, Powder Technol. (2015), http://dx.doi.org/10.1016/j.powtec.2015.03.028

http://dx.doi.org/10.1016/j.powtec.2015.03.028http://dx.doi.org/10.1016/j.powtec.2015.03.028


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and T e in the vicinity of the tube are obtained for an annular region

around the tube with a thickness of 5d p in its radial direction. All these

treatments have been used in our previous study and proved to work

satisfactorily [14].

2.2. Governing equations for uid phase

Theuid phase, air to be specic for this study, is treated as a contin-

uum phase and modeled in a way similar to the one widely used in the

conventional two-uid model [51]. In this connection, there are three

sets of governing equations, developed by Anderson and Jackson [51].

Different governing equations may lead to different results, depending

on the systems considered. According to Zhou et al. [20], Set II and in

particular Set I can be used generally, and Set III can only be used condi-

tionally. In this work, Set I is used. Thus, the conservations of mass and

momentum in terms of the local averaged variables over a computa-

tional cell are given by:

∂ ρ f ε f .

∂t þ ∇⋅ ρ f ε f u

¼ 0; ð5Þ

and

∂ ρ f ε f u .

∂t þ ∇⋅ ρ f ε f uu

¼ −∇ p−F f p þ ∇⋅τ þ ρ f ε f g : ð6Þ

The corresponding energy equation for heat transfer can be written

as:

∂ ρ f ε f c pf T .

∂t þ∇⋅ ρ f ε f uc pf T

¼ ∇⋅ ke∇T ð Þ þ Q ·;

ð7Þ

where u, ρ f , p andF fparetheuid velocity, density,pressureand volumetric

uid–particle interaction force, re spectively; τ ¼ μ f ∇uð Þ þ ∇uð Þ−1

h i−

23 μ f ∇⋅uð ÞδkÞ and ε f ¼ 1−∑

kvi¼1V i=ΔV

are the uid viscous stress

tensor and porosity respectively, with V i representing the volume

of particle i (or part of the volume if the particle is not fully in a

CFD cell), and kV the number of particles in the computational

cell of volume ΔV . Note that ε i is the local porosity for particle i to

calculate particle–uiddragforce andε f is determined overa compu-

tational cell for uid phase. Theoretically, the two porosities are not

necessarily the same. For convenience, in the present work, ε i = ε f . keis the effective uid thermal conductivity, dened by (k f + c pf μ t / σ T ),

and σ T the turbulence Prandtl number, which is set to 1.0 for this

work. μ e (= μ f + C μ ρ f ε f k2/ ε ) is the uid effective viscosity and μ t is

the turbulent viscosity, which are determined by a widely used stan-

dard k–ε turbulence model [52]. The effect of gas turbulence on solid

phase is not considered, and this treatment has been tested as

acceptable for the current systems using a similar approach byKuang and Yu [53]. The volumetric particle–uid interaction force

F fp in Eq. (6) can be determined as F f p ¼ ∑kvi¼1 f d;i þ f pg ;i

.ΔV . The

volumetric heat exchange rate Q

in Eq. (7) can be determined as

Q ·

¼ ∑kvi¼1 Q ·

f ;i þ Q ·

f ;wall þ Q ·

f ;tube þ Q ·

f ;rad

=ΔV

, where Q

·

f ;i is the con-

vective heat exchange rate between uid and particle i ; Q ·

f ;tube is

the convective heat exchange rate between uid and tubes; Q ·

f ;wall

is the convective heat exchange rate between uid and a wall; and

Q ·

f ;rad is the radiative heat exchange rate between uid and its envi-

ronment. In the present work, because of the low emissivity of

uid, the radiative heat transfer between uid and its environment

is ignored for simplicity.

2.3. CFD–DEM coupling scheme

The methods of numerical solutions to problems requiring CFD–

DEM coupling have been well established [19,20,54]. Heat transfer

modelshave also been incorporated into this approach as demonstrated

Table 1

Equations to calculate heat exchange rates.

Heat exchange rates Equation

Convective Q ·

i; f ¼ 2:0 þ aRebi Pr

1=3

k f AiΔT =d pi(a)

Q ·

f ;wall ¼ 0:037Re0:8 Pr

1=3k f AwΔT =L

(b)

ConductiveQ ·

i; j ¼ T j−T i

∫r sf

r sij2π ⋅r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2i −r 2

q −r R2i þ H

r ij

⋅

1=k pi þ 1=k pj

þ2 R2i þ H

−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiR2i −r 2

q k f

−1dr

(c)

Q ·

i; j ¼ 4r c T j−T iÞ= 1=k pi þ 1=k pj (d)Q ·

i; j ¼ c T j−T i

π r 2c t −1=2c =

ρ pic pik

−1= 2

þ ρ pj c pjk pj

−1= 2 (e)

Radiative Q ·

i;rad ¼ σ eAi T 4local;i−T 4i

; Q

·

f ;rad ¼ σ e f A f T 4local;i−T 4 f

where T local;i ¼ ε f T f ;Ω þ 1−ε f

∑kΩ j¼1T j j≠ið Þ=kΩ (f)

Table 2

Physical and geometrical parameters used in the simulations.a

Variables Values

Bed width × height, d p 100 × 1280

Tube position, d p Z = 50

Total CFD cells, – 50 × 640

Cell size (Δ x × Δ z ), d p 2 × 2

Number of particles (N ), – 30,000

Tube diameter, d p 40

Particle diameter d p, mm 0.1

Particle density ρ, kg/m

3

1440Thermal conductivity of particles k p, W/(m K) 1.1

Thermal conductivity of tube k p, W/(m K) 380

Specic heat of particles c p, J/(kg K) 840.0

Specic heat of tube c p, J/(kg K) 24.4

Temperature of hot tube T s, °C 200

Particle–particle/wall sliding friction μ s, – 0.3

Particle–particle/wall rolling friction μ r , – 0.01

Restitution coef cient, – 0.8

Particle Young's modulus E , kg/(m s2) 1 × 107

Particle Poisson ratio ν , – 0.3

Hamaker constant H a, J 2.10 × 10−21

Fluid density ρ f , kg/m3 PM /(RT f )

Fluid molecular viscosity μ f , Pa s 1.511 × 10−6T f

3/2/(T f + 120.0)

Fluid thermal conductivity k f , W/(m K) 2.873 × 10−3 + 7.760 × 10−5 × T f

Fluid specic heat c pf , J/(kg K) 1002.737 + 1.232 × 10−2 × T f

a These are forthe basecase. Some parameters may vary in different cases, as specied

in the text or gure caption.

Table 3

Properties of three powders.

Parameter d p, mm H a, J umf , m/s

Powder

A 0.1 2.10 × 10−21 0.0072

A0 0.1 0 0.0072

B 0.5 2.10 × 10−21 0.15

3Q.F. Hou et al. / Powder Technology xxx (2015) xxx– xxx




4/13

elsewhere [10,14,15]. The present work further extends this approach

to consider a tube array in a uidized bed. The coupling scheme used

here is the sameas before, which is briey described as follows for com-

pleteness. At each time step, DEM will produce information such as the

positions, velocities, and temperature of individual particles, which will

be used for the evaluation of porosity, particle–uid interaction force,

andheat exchange rate in a computational cell. CFDwill then usethis in-formation to determine the uid ow and temperature eld, which in

turn can beusedto nd particle–uidinteractionforce andheat transfer

between uid and particles or tubes. Incorporation of the resulting

forces and heat exchange rates into DEM will produce information

about the motion and temperature of individual particles for the next

time step.

3. Simulation conditions

Table 2 lists the physical and geometrical parameters for the study of

the effect of material properties, unless otherwise specied. Three

powders of different types, namely A, A0 and B, are given in Table 3.

The number of particles is constant for all the cases in this work. It

should be noted that the van der Waals force is the only considered co-

hesive force. In principle, the Hamaker constant depends on many var-

iables related to physical and chemical properties, such as the surface

roughness or asperity, medium chemistry. In the present study the

Hamaker constant of 2.10 × 10−21 is adopted, which has been used in

our previous study to reasonably reproduce the behaviors of cohesiveparticles [49]. Walls are assumed to have the same material properties

as the particles for convenience. Spherical particles at a temperature of

25 °C are used as the initial solid phase uidized in a container with a

thickness of four particle diameter (d p). The periodic boundary condi-

tion is applied to the front and rear directions to eliminate the effect

of walls. To remove the effect of the side walls, the selected bed widths

should be suf cient as the side wall can only affect the ow up to 10d peven in a rather dense particulate ow (see, for example, [55]). For

the above geometry, two-dimensional CFD and three-dimensional

DEM are used as done by Feng et al. [54]. This treatment should be rea-

sonable, given that the bed width (100d p) is much larger than its thick-

ness (4d p) and the tubes are set horizontally. For the CFD computation

Fig. 1. Selection of unit cells for different settings: (a) square and (b) triangular.

t =0s 2s 6s 2s 6s 2s 6s

(a) (b) (c)

Fig. 2. Gas–solid ow pattern in uidizedbeds fordifferentpowders when u f / umf = 5 andtube temperature T s = 200 °C: (a) PowderA, (b)PowderA0,and (c) Powder B.All particlesare

shown, colored by their coordination number (CN). (For interpretation of the references to color in this gure legend, the reader is referred to the web version of this article.)





5/13

two major treatments are adopted for the transfer of information be-

tween 2D CFD and 3D DEM. The rst treatment is used to obtain the

local porosities with only one control volume assumed in the thickness

direction. The particles in a given CFD cell are determined only by the

coordinates in the x and z directions. The second treatment is used for

the transfer of momentum and energy sources assuming that the

sources in the bed thickness direction are negligible. The non-slip

boundary condition is applied to the walls, and zero diffusion ux

condition to the outlet for ow and heat transfer.

To investigate the effectof tubearray settings, particle diameter d p is

set to 0.6 mm with a numerically determined minimum uidization ve-

locity(umf ) of0.36 m/s.The bed of105 particleshas a width of 160d p and

a height of 1,000d p. Tubediameteris 20d p. Asthemain aim ofthis partis

to examine the effect of tube array settings, all the cases are carried out

at a low tube temperature (T s) of 200 °C where radiative heat transfer is

negligible [14]. The inlet gas velocity is set to 3umf . Two types of tube

array settings are considered as shown in Fig. 1. One setting is square

(in-line) and the other is triangular (staggered). Two unit cells are cho-

sen accordingly for analysis as shown in Fig. 1. It should be noted thatthis treatment is reasonable, as shown in the present study that the in-

teraction between neighboring tubes is insignicant if pitch length is

larger than 2.5D (here, D is tube diameter). The relative tube position

is quantied by two parameters: one is angle α between the line joining

the centers of the tubes and the horizontal line in the cross-section of

the tubes, and the other is pitch length L. Angles of 30, 45 and 60° and

center-to-center pitch lengths in the range of 1.5–4D are adopted for

triangular settings.

A simulation is started with the generation of particles without any

overlap in the bed, followed by a gravitational settling process where

all the applicable interparticle forces are considered. This particle set-

tling process continues until the rotational and translational velocities

of particles decrease to zero or a negligible value (b 10−5 m/s for trans-

lational velocity). Then, the bed is used as a base for the simulation.

There are different designs for the introduction of gas into uidized

beds. In thepresent study, air with a pre-set temperature (25°C) is uni-

formly injected at the bottom touidize the bed at a given velocity. The

container walls are assumed to be adiabatic for simplicity. The time step

in eachcaseis constant, which isso chosen to ensurethe accuracy of the

numerical simulation [56].

4. Results and discussion

A tubeinuidized beds usually actsas a heat sinkor source. The tube

temperature is higher than discrete solid and continuous gas phases in

the present study. The tube heats the particle bed by exchanging heat

rstly with its surroundings. Then, heat is transferred to the bed awayfrom the tube by different heat transfer mechanisms. Uniform distribu-

tion of temperatureswithinbeds is desired in many applications such as

pharmaceutical and chemical processes. To achieve this, the effects of

material properties such as particle size and the Hamaker constant,

and thesetting of a tubearrayare important.Theseeffectsare examined

in this section.

(a)2.0 2.5 3.0 3.5 4.0

0.00.51.01.52.02.5

0.0

0.1

0.2

0.0

0.1

0.2

Powder B Convection Conduction

Time (s)

Convection

ConductionPowder A0

H e a t

e x c h a n g e r a t e ( W )

Convection

ConductionPowder A

(b)

2.0 2.5 3.0 3.5 4.0

0

20

40

60

80

100

0

20

40

60

80

100

Powder A

Powder A0

Powder B

Time (s)

P e r c e n t a g e o f c o n v e c t i v e

h e a t e x c h a n g e r a t e ( % )

P e r c e n t a g e o f c o n d u c t i v e

h e a t e x c h a n g e r a t e ( % )

Powder A

Powder A0

Powder B

Fig. 3. Heat exchange rates between the uidized bed and the tube (a), and their percentage contributions to the total heat exchange for different powders (b).

3.0 3.2 3.4 3.6 3.8 4.0

0

50

100

0.6

0.8

1.0

P o r o

s i t y ( - )

C o n t a c t n u m b e r ( - )

Time (s)

Powder A

Powder A0

Powder B

Fig. 4. Evolution of the local porosity and the contact number between particles and the

tube for different powders.

2.0 2.5 3.0 3.5 4.0

0.000

0.005

0.010

0.000

0.005

0.010

0.000.01

0.02

0.03

0.0

0.1

0.2

Time (s)

(d)

(c)

(b)

H e a t e x c h a n g e r a t e ( W )

(a)

Fig. 5. Evolutionof heat exchange rates by different conductive heat transfer mechanisms:

(a) particle–uid–tube under non-contact condition, (b) particle–uid–tube under

contact condition, (c) particle–tube with collisional contacts, and (d) particle–tube with

static (enduring) contacts.





6/13

4.1. Effects of material properties and tube temperature on gas–solid ow

and heat transfer

The horizontal tube has a signicant effect on gas–solid ow charac-

teristic as shown in Fig. 2 for Powders A, A0 and B. Deuidized solid cap

in the downstream and gas lm in the upstream is observed as general

transient features of uidization with a tube. Some differences between

these powders are observed. Theuidization of Powder A0 (Fig. 2(b)) is

similar to that of Powder A (Fig. 2(a)). It should be noted that

the Hamaker constant used has been selected largely to match the

common observations [49]. The conditions used can provide a reason-

able comparability for different powders in the uidized bed ow

regime at the same ratio of gas velocity to minimum uidization veloc-

ity (u f / umf = 5). For Powder B, a vigorously bubbling uidized bed is

observed, which is more expanded than the beds for Powders A and

A0. One possible reason for this observation is that the particle–uid

interaction force varies with particle size, and hence, different parti-

cle–particle contact conditions are generated [49]. It can also be

reected in particle coordination number (CN) and different sizes of

bubbles or large voids within the bubbling uidized beds at the same

ratio of u f / umf . These differences can generate different heat transfer

characteristics as observed in the cases of uidized beds without tubes

[15].

The tube exchanges heat with its surroundings mainly through

conduction with particles and convection to the gas

ow at low temper-atures [14,57]. Heat exchange rates through thetwo modes areuctuat-

ing temporally for different powders, as shown in Fig. 3(a). For Powders

A and A0, conductive heat transfer between the uidized bed and

the tube is dominant. But for Powder B, convective heat transfer is

dominant. These canbe clearly observed in terms of thepercentage con-

tribution of heat exchange rates, as given in Fig. 3(b). Forsmall particles,

the van der Waals force at the given Hamaker constant affects heat

transfer slightly. Only minor differences can be observed between Pow-

ders A and A0 in terms of the contributions of different heat transfer

modes. This is clearly shown in Table 5 in Section 4.2 from the time-

averaged percentage contributions of different heat transfer modes. It

should be noted that if the Hamaker constant is large enough, different

heat transfer characteristics can be observed as demonstrated in our

previous study [15].

These observations are related to the local porosity and the contacts

between particles and the tube. As shown in Fig. 4, the local porosity

aroundthe tube fordifferent powders has only minor differences except

that large uctuations are observed for Powder B. This also applies to

the contact number between particles and the tube. Smaller particles

Table 4

Time-averaged percentage contributions of different heat transfer mechanisms.

Conditions 5 umf 10 umf

Mechanisms A A0 B A A0 B

Conduction 94.4% 94.2% 20.1% 83.4% 83.2% 8.7%

Convection 5.6% 5.8% 79.9% 16.6% 16.8% 91.3%

Particle–uid–tube non-contact 91.7% 91.7% 91.7% 93.6% 93.7% 91.8%

Particle–uid–tube contact 5.7% 5.7% 5.7% 5.0% 4.9% 7.7%

Particle–t ub e c ollision al c ont ac t 0.5% 0.5% 0.5% 0 .3 % 0.3% 0.3%

Particle–tube static contact 2.1% 2.1% 2.1% 1.1% 1.1% 0.2%

t =0.02s t =1s t =2s t =3s t =4s t =5s t =6s

Fig. 6. Solid ow pattern for Powder A when T s = 600 °C. All particles are shown, colored by their temperatures. (For interpretation of the references to color in this gure legend, the

reader is referred to the web version of this article.)

(a)3.0 3.2 3.4 3.6 3.8 4.0

0.0

0.2

0.4

H e a t e x c h a n g e r a t e ( W )

Time (s)

Conduction

Convection

Radiation

(b)3.0 3.2 3.4 3.6 3.8 4.0

0

20

40

60

80

100

P e r c e n t a g e ( % )

Time (s)

Conduction

Convection

Radiation

Fig. 7. Evolution of heat exchange rates and their percentage contributions at T s = 600 °C.





7/13

can enhance conductive heat transfer between the tube and the uid-

ized bed due to their relatively larger total contact area at a given poros-

ity [15]. The differences in ow behavior and contact conditions will

result in different heat transfer behaviors, as discussed in the following

sub-sections.

Conductive heat transfer between the tube and the uidized bed is

dominant forthe small particles. Thecontributions of fourdifferent con-

ductive heat transfer mechanisms to the total conductive heat transfer

are examined here. The evolution of different conductive heat exchange

rates is shown in Fig. 5. It can be observed that the heat transfer through

the tube–uid–particle path under non-contact condition is the largest

one. The second largest is the heat transfer through the path of tube–

uid–particle under contact condition. These results are consistent

with those reported in the literature, where gas lm around the tube

plays an important role in heat transfer between an immersed surface

and a uidized bed [1]. Due to the short contact time and small solid

contact area, the conductive heat exchange rates through the particle–

tube path are small. The contact time could vary with the velocities of

colliding particles and the gas–solid ow. For an approximation of im-

pact contact duration t c , Eq. (8) can be used, where m is the equivalent

mass, E is the equivalent elastic modulus, R is the equivalent radius, and

v is the normal impact velocity [58].

t c ¼ 2:94 5m

4E

2=5

Rvð Þ−1=5

ð8Þ

The contributions of different mechanisms vary for different pow-

ders as listed in Table 4. The contribution of each conductive heat trans-fer mechanism to the total conductive heat transfer is also quantied

(here ∑14 xi = 100 % where xi is the contribution of conductive heattransfer by mechanism i). For Powders A and A0, the conductive heat

transfer is dominant with its percentages around 90%. For Powder B,

the convective heat transfer is dominant with its percentage around

80%. With the increase of gas velocity, the contribution of convective

heat transfer increases and that of conductive heat transfer reduces.

One reason for this observation is the enhanced dilute gas–solid ow

due to the increase of gas velocity. Although different powders have

different dominant heat transfer modes, the results indicate that heat

transfer through the particle–uid–tube path is dominant for conduc-

tion in all the cases studied. The contribution of heat exchange rate

through the particle–uid–tube path increases with the increase of

gas velocity while that through the particle–tube path decreases.

Heat transfer between a tube and a uidized bed by radiation be-

comes important at high temperatures [14,57]. The proposed model

can account for this factor. As an example, this work also investigates

the heat transfer at a high tube temperature (T s) of 600 °C for different

powders. As shown in Fig. 6, particles near the tube are rst heated, and

then these ‘hot’ particles move into other parts of the bed, exchanging

heat with other ‘cold’ particles. In this process, particle temperature

Table 5

Time-averaged percentage contributions of different heat transfer modes at T s = 600 °C.


Mechanisms A A0 B A A0 B

Conduction 41.9% 44.0% 10.5% 32.0% 32.7% 5.0%

Convection 3.8% 3.5% 46.0% 5.7% 5.5% 51.7%

Radiation 54.3% 52.5% 43.5% 62.3% 61.8% 43.3%

t =0.1s 5s 6s

(a)

DoV

(b)

1s

Fig. 8. Deviation of velocity (DoV) of all particles in uidized beds during heating process by a tube at T s = 600 °C and u f / umf = 10: (a) Powder A and (b) Powder B.





8/13

distribution is not uniform, and areas with relatively high or low tem-

peratures are observed.

Theevolution of heat exchange rates at a high temperature and their

contributions are demonstrated in Fig. 7 for T s = 600 °C.At this temper-ature, the conductive heat exchange rate is the largest, the convective

one is the smallest, and the radiative one has an intermediate value

varying steadily. The conductive heat exchange rate uctuates with

values smaller than that of radiative heat transfer at some instants.

Their contributions to the total heat exchange rate by all heat transfer

modes are analyzed as shown in Fig. 7(b). The percentage contributions

uctuateand the radiative heat transfer has the largest value. Their con-

tributions are compared quantitatively in termsof time-averaged values

in Table 5 for different powders and gas velocities at T s = 600 °C. It in-

dicates that the radiative heat transfer is an important mode in all the

cases studied. For small particles, the contribution of radiation is larger

than that of conduction which is dominant at low temperatures. How-

ever, for large particles, the convective heat transfer is still dominant

under the current conditions while radiation plays an important role.

Withthe increase of gas velocity, the bed expands higher. The contribu-

tion of radiative heat transfer increases while that of conductive heat

transfer decreases for small particles (Powders A and A0). For largeparticles (Powder B), minor variations of the contribution of radiative

and convective heat transfer are observed.

4.2. Effects of material properties and gas velocity on particle velocity and

temperature elds

The predictions for different powders are usually compared at the

same ratio of inlet gas velocity to their minimum uidization velocities.

Even under this condition, a large difference in particle velocity can be

observed for different powders. The particle temperature also varies

and shows non-uniformity. To quantify the variation of particle velocity

and temperature, two dimensionless parameters are dened. One

parameter is the deviation of velocity (DoV) from the mean velocity

t =0.1s 1s 5s 6s

(a)

DoT

(b)

Fig. 9. Deviation of temperature (DoT) of all particles in uidized beds during heating process by a tube at T s = 600 °C and u f / umf = 10: (a) Powder A and (b) Powder B.

(a)0 1 2 3 4

0.0

0.1

0.2

0.4

0.6

0.8

1.0

D o V ( - )

D o T ( - )

Time (s) (b)0 1 2 3 4

0.0

0.1

0.2

0.4

0.6

0.8

1.0

D o V ( - )

D o T ( - )

Time (s)

2 .

8 2

Fig. 10. Evolution of bed averaged DoV and DoT of individual particles: (a) Powder A and (b) Powder B.





9/13

for individual particles, dened as |1− ui/ b ui N | where ui is the magni-tude of particle velocity and b N is the average over all particles. The

other parameter is the deviation of temperature (DoT) from the mean

temperature, dened as |1 − T i/ b T i N |.Heat transfer characteristics and the uniformity of particle tempera-

ture areclosely related to themotionof particles. Thedeviation from the

mean velocity of individual particles is shown in Fig. 8 for Powders A

and B. The particles in the vicinity of the tube have large DoVs because

of the disturbance induced by the tube. The deviation diminishes in

other areas as a result of the momentum exchange between particles.

Large DoVs of small particles are often observed in themiddle of the u-

idized bed; conversely, DoVs are large near the walls for large particles.

The deviation from the mean temperature of individual particles is

shown in Fig. 9 for Powders A and B, under the conditions correspond-

ing to those of Fig. 8. Compared to DoV, DoT is rather small as a result

of high heat transfer capability of uidized beds. However, a spatial

distribution of DoT can still be observed. A large DoT occurs in the

vicinity of the tube due to heat exchange between the bed and the

tube. This indicates the possibility of observing hot spots with non-

uniform particle temperatures in the bed.

To quantify the uniformity of particle velocities and temperatures at

a bed scale, the bed averaged DoV and DoT of individual particles

are shown in Fig. 10 for Powders A and B. It can be seen that the bed

averaged DoV decreases rather quickly to a small value while the bed

averaged DoT takes a longertime. Thevaluesvary for different powders.

To show the effect of material properties on DoV and DoT, the time-

averaged values are listed in Table 6. The time-average is carried out

withina xed time frame (0–6 s)in which period DoV and DoT decreaseto rather small values. It is found that a time-average with a longer time

will give a smaller value because thenegligible small values after 6 s are

included. Here, macroscopically steady states are assumed when the

small values are achieved. It should be noted that even in the macro-

scopically steady state, the results may uctuate slightly, if different

instants are used.

The effects of gas velocity, tubetemperature and material properties

on DoV and DoT are also examined, as discussed below. It can be seen

from Table 6 that gas velocity has a negligible effect on DoV for different

powders while DoT decreases with the increase of gas velocity. Tube

temperature affects DoV insigni

cantly, and DoT becomes smaller at ahigher tube temperature. Material properties affect DoT and DoV in a

more complicated manner. Larger particles have larger DoV and smaller

DoT. The cohesive force has little effect on DoV and DoT under the

current conditions. It can be seen from these results that DoT can be

reduced if a large gas velocity or a large particle size is used, giving a

more uniform temperature eld.

4.3. Effect of tube array settings on gas–solid ow and heat transfer

Fig. 11(a) shows the snapshots of gas–solid ow pattern in uidized

bedswitha tube array. The mainfeatures of gas–solidowina uidized

bed with multiple tubes can be observed: solid cap in the downstream

of the tubesand air lm upstream,consistent with theresults of thepre-

vious studies [14,59]. Theparticlesare colored by their identity number,

indicating the mixingof particles. Forthe powders studied, goodmixing

is achieved within a short period, which is one of the advantages of u-

idized beds. Fig. 11(b) shows the gas–solid ow pattern with different

tube array settings determined by angle α. The solid cap and air lm

for different tubes change with α as a result of the variation of the

wake of the gas–solid ow downstream. The differences are reected

in heat exchange rates related to local porosity and contact number, as

demonstrated in the following discussion. The results in Fig. 11 verify

thecapability of the current method to study gas–solid owin uidized

beds with a tube array and provide a sound basis for heat transfer

analysis.

Particles and gas both can exchange heat with the tubes. Heat

transfer between each tube and the uidized bed is considered in the

simulations. For simplicity, Fig. 12 gives the evolution of the total and

conductive heat exchange rates of two representative tubes (#1 and#2 as indicated in theinset in Fig. 12(a)); both tubes show largeuctu-

ations in the heat exchange rates. The differences observed are mainly

Table 6

Bed averaged DoV and DoT under different conditions.


Index A A0 B A A0 B

T s = 200 °C DoV 0.129 0.126 0.232 0.125 0.121 0.239

T s = 600 °C DoV 0.129 0.125 0.235 0.127 0.127 0.245

T s = 200 °C DoT 0.033 0.031 0.016 0.019 0.022 8.49E−3T s = 600 °C DoT 0.006 0.006 2.86E−4 1.74E−3 2 .0 1E−3 1 .1 0E−4

(a)

(b)

t =0.4 st =0 s t =0.8 s t =1.6 s t =6.0 s

Fig. 11. Gas–solid ow patterns in uidized beds with a tube array: (a) for different times when α = 45° and (b) for different settings when t = 6.0 s. The settings include a square one

(α = 0) and three triangular ones (α = 30°, 45°, and 60°) from the left to right. All particles in each case are shown.





10/13

due to the change of gas–solid ow around the tubes in the uidizedbed. The results indicate that conductive heat transfer between the

tube and the uidized bed has a large contribution to the total heat

transfer in the present system, different from our previous study

where different settings for the ratio of bed width to tube diameter

and tube temperature are used [14]. It should be noted that the relative

importance of different heat transfer modes may vary with operating

conditions and geometrical settings. Some of them have been investi-

gated by the combined CFD–DEM approach previously [14,15]. Because

of contradictory views about the effect of tube array settings in the

literature, there is a need for a more systematic study of the effect of

geometrical settings on heat transfer. This work addresses this needby a detailed analysis of the heat transfer between the uidized bed

and the tube (#2) located in the center of the selected unit cell, as

given below.

The contact number and local porosity around a given tube can be

obtained for understanding the variation of heat exchange rates. The

conductive heat exchange rate is closely related to the contact number

between uidized particles and a tube. As shown in Fig. 13(a) for the

two representative tubes, the contact numbers vary temporally with

different trends. Local porosity obtained for an annular region of 5d pthickness in the radial direction of a given tube [14] varies vigorously

(a)2.0 2.5 3.0 3.5 4.0

0.0

0.5

1.0

1.5

2.0#1

T o t a l h e

a t e x c h a n g e r a t e ( W )

Time (s)

Tube #1

Tube #2 #2

(b)2.0 2.5 3.0 3.5 4.0

0.0

0.2

0.4

0.6

0.8

1.0

C o n d u c t i v e h e a t e x c h a n g e r a t e ( W )

Time (s)

Tube #1

Tube #2

(c)2.0 2.5 3.0 3.5 4.00

20

40

60

80

100

P e r c e n t a g e ( % )

Time (s)

Tube #1

Tube #2

Fig. 12. Evolutionof heatexchangerates fortwo representative tubes in a uidizedbed: (a) total heat exchangerate,(b) conductiveheat exchange rate, and(c) percentage of conductive

heat exchange rate to the total. The positions of two representative tubes (#1 and #2) are illustrated in the inset in (a). The square setting for this case has a pitch length of 1.5D.

(a)2.0 2.5 3.0 3.5 4.0

0

25

50

75

100

C o n t a c t n u m b e r ( - )

Time (s)

Tube #1

Tube #2

(b)

2.0 2.5 3.0 3.5 4.00.4

0.5

0.6

0.7

0.8

0.9

1.0

L o c a l p o r o s i t y ( - )

Time (s)

Tube #1

Tube #2

Fig. 13. Evolution of: (a) the contact numbers between two representative tubes and uidized particles and (b) local porosity. The square setting for this case has a pitch length of 1.5D.





11/13

as shown in Fig. 13(b). The time-averaged values of the contact number

and local porosity can be obtained accordingly.

The setting of a tube array is an important factor affecting the capa-

bility of heat transfer between the tube array and uidized beds. Here,

the effects of two geometrical parameters are examined: pitch length

(L) and angle (α), as illustrated in Fig. 1(b).

Heat exchange rates vary with the increase of pitch in a complicatedmanner (Fig. 14). For the square setting (in-line, α = 0), the total and

convective heat exchange rates decrease gradually. The conductive

heat exchange rate varies complicatedly. It decreases rst, and then in-

creases slightly from a pitch length of 2D, and nally decreases slightly

from a pitch length of 3D. These observations are consistent with the

change of contact number and local porosity shown in Fig. 15. For the

triangular (staggered) settings, complex variations are also observed.

Some turning points are observed at different pitch lengths at different

angles. Withthe increase of pitch length, theuctuation becomes weak-

er. It is observed that the uctuation becomes insignicant when the

horizontal or vertical pitch is larger than 2.5D, which is slightly different

from a previous study [31]. The reason could be that ifthe pitch length is

larger than 2.5D, the effect of neighboring tubes becomes insignicant,

and hence the heat transfer between the uidized bed and the tube

considered is mainly determined by the surrounding gas–solid ow

and takes place without much variation.

Heat exchange rates vary with angle α in a complicated manner at

different pitch lengths as shown in Fig. 14. Generally, total heatexchange rate decreases with the increase of α. The convective heat

exchange rate shows a similar trend to that of the total heat exchange

rate. The conductive heat exchange rate varies complicatedly for

different pitch lengths, as discussed below. It can be seen that the con-

ductive heat exchange rate plays a role as important as the convective

heat exchange rate for the system considered.

Heat exchange rates vary differently with α in a certain range of

pitch lengths. For convenience, the pitch length is classied into three

categories according to the variation of conductive heat exchange

rate: small (1.5 and 2D), intermediate (2.5 and 3D), and large (3.5 and

4D). For the small pitch range, the conductive heat exchange rate

Fig. 14. Heat exchange rates as a function of pitch length and angle: (a) convection, (b) conduction, and (c) total.

Fig. 15. Contact number (a) and local porosity (b) as a function of pitch length and angle.





12/13

increases rst, and then decreases with the increase of α. The turning

points vary with the pitch length. Theconvective heat exchange rate re-

duces consistently with the increase of α. As the sum of conductiveand

convective heat exchange rates, the total heat exchange rate decreases

consistently for a pitch length of 1.5D or increases a little rst and

then decreases for a pitch length of 2D with the increase of α. For the in-

termediate range, the convective heat exchange rate decreases consis-

tently while the conductive heat exchange rate decreases rst and

then increases slightly with the increase of α. The total heat exchangerate decreases rst and then becomes constant for a pitch length of

2.5D, or decreases consistently for a pitch length of 3D. In the large

pitch range, variations in all heat exchange rates are insignicant.

The change of heat exchange rates can be understood from the

microscopic information such as contact number and local porosity.

Contact numbers under various conditions are shown in Fig. 15(a),

indicating consistent variations with the change of conductive heat

exchange rate shown in Fig. 14(b). Conductive heat transfer occurs

when particles are in direct contact with or close enough to a given

tube according to the heat transfer mechanisms discussed. Hence, as

expected, a connection between conductive heat exchange rate and

contact number is observed, although other factors such as colliding ve-

locity and contact time could affect the conductive heat transfer. Based

on the predictions, contact number can be considered as a main index

determining the conductive heat exchange rate for the system investi-

gated. The time-averaged local porosity (Fig. 15(b)) is closely related

to the convective heat transfer. However, only small change is observed

at a given inlet gas velocity, which is consistent with the variation of

corresponding convective heat exchange rate. Such information may

be helpful for the design and optimization of tube arrays for heat

transfer in uidized beds.

5. Conclusions

The combined CFD–DEM approach with heat transfer models incor-

porated is used to investigate the effects of material properties and

geometrical settings on gas–solid ow and heat transfer characteristics

in uidized beds with tubes. The following conclusions can be drawn

from the present study:

• The convective heat transfer is dominant for large, non-cohesive

particles while the conductive heat transfer is dominant for small, co-

hesive particles at low temperatures, when the radiative heat transfer

is negligible. Heat exchange ratethrough the particle–uid–tube path

under non-contact condition is dominant of the conductive heat

transfer mode. Radiative heat transfer becomes important at high

temperatures (higher than ~600 °C). Radiative heat transfer can be

dominant for small particles under certain conditions.

• The effect of material properties on the uniformity of particle velocities

and temperatures is signicant. Large particles have a low uniformity

of particle velocities but high uniformity of particle temperatures. For

a given Hamaker constant, the cohesive force affects both the velocity

and temperature elds insignicantly. Material properties and operat-ing conditions should be selected carefully by considering the unifor-

mity of particle velocities and temperature elds.

• The effect of gas velocity on the uniformity of particle velocities and

temperatures is complicated. On one hand, gas velocity has little effect

on the uniformity of particle velocities. On the other hand, a large gas

velocity can improve the uniformity of particle temperatures. A high

tube temperature can also result in a high uniformity of particle

temperatures.

• The effect of tube array setting is complicatedas reectedin thechang-

es in the gas–solid ow characteristics and heat exchange rates. It can

be related to microscopic information such as the local porosity, and

contact number between particles and a given tube. Conductive heat

exchange rate is closely related to the contact number while convective

rate is closely related to the local porosity.

It is considered that the results areusefulfor better understanding of

the coupled ow and heat transfer in a uidized bed. More importantly,

they demonstrate the capability of particle scale study in this direction,

although further developments are needed in order to produce results

that candirectly help the design and optimization of industrialuidized

processes of different types. Generally uidized bed reactors might be

classied into continuous or batch operation according to the methods

to load solids. The continuous operation possesses many advantages

such as continuous operation. The batch operation might be selectedto match theupstream anddownstream processesor to meet special re-

quirements of nal products. In the present study, the settings are for

batch operation with a constant number of particles in the system.

The transient heat transfer and mixing from the startup to the steady

state are investigated. The ndings from the batch operation might be

applied to continuous operation, subjected to further verication and

study.

Acknowledgments

The authors are grateful to the Australian Research Council

(FF0883231) for the nancial support. This work was supported by an

award under the Merit Allocation Scheme on the NCI National Facility

at the ANU.

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Pleasecitethis article as: Q F Hou et al Gas–solidow and heat transferin uidized beds with tubes:Effects of material properties andtube array
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