16
Direct numerical simulation of flow and heat transfer in dense fluid–particle systems Niels G. Deen n , Sebastian H.L. Kriebitzsch, Martin A. van der Hoef, J.A.M. Kuipers Multiphase Reactors Group, Department of Chemical Engineering and Chemistry, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands HIGHLIGHTS c Novel immersed boundary techni- que for fluid flow and heat transfer in dense suspensions. c Unique fluid–solid coupling through implicit incorporation of boundary conditions. c The technique does not require cali- bration of an effective particle dia- meter. c Fully resolved simulations of a sta- tionary random array of particles are presented. c Fully resolved simulations of a liquid fluidized bed are presented. GRAPHICAL ABSTRACT article info Article history: Received 8 February 2012 Received in revised form 23 May 2012 Accepted 26 June 2012 Available online 5 July 2012 Keywords: Fluidization Fluid mechanics Heat transfer Multiphase flow Immersed boundary method Multi-scale modeling abstract In this paper a novel simulation technique is presented to perform Direct Numerical Simulation (DNS) of fluid flow and heat transfer in dense fluid–particle systems. The unique feature of our fluid–solid coupling technique is the direct (i.e., implicit) incorporation of the boundary condition (with a second- order method) at the surface of the particles at the level of the discrete momentum and thermal energy equations of the fluid. Contrary to lattice Boltzmann or other commonly used immersed boundary implementations, our method does not require using any effective diameter. A fixed (Eulerian) grid is utilized to solve the Navier–Stokes equations for the entire computational domain. Dissipative particle–particle and/or particle-wall collisions are accounted via a hard sphere discrete particle approach using a three-parameter particle–particle interaction model accounting for normal and tangential restitution and tangential friction. Following the detailed verification of the method several dense multi-particle systems are studied in detail involving stationary arrays of particles and fluidized particles. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction Dense gas–particle flows are frequently encountered in a variety of industrial processes. More specifically these flows are encountered in gas–fluidized beds, which are frequently applied in the chemical, petrochemical, metallurgical, environmental and energy industries in large scale operations involving physical (coating, drying and granulation) and chemical (synthesis of fuels and base chemicals) transformations. The accurate prediction of dense gas–particle flows, which are inherently unsteady and heterogeneous, has proven notoriously difficult, which can be attributed to the wide variation of length scales existing in these flows. For the accurate prediction of these flows in engineering scale equipment, which in practice can only be achieved with con- tinuum models, it is essential that both the fluid–particle as well Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/ces Chemical Engineering Science 0009-2509/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ces.2012.06.055 n Corresponding author. Tel.: þ31 40 247 3681; fax: þ31 40 247 5833. E-mail address: [email protected] (N.G. Deen). Chemical Engineering Science 81 (2012) 329–344

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Page 1: Chemical Engineering Science - ssu.ac.irssu.ac.ir/.../Articles/EN/1-s2.0-S0009250912004204-main.pdf · of granular flow (model 2), the discrete particle model (model 3), the immersed

Chemical Engineering Science 81 (2012) 329–344

Contents lists available at SciVerse ScienceDirect

Chemical Engineering Science

0009-25

http://d

n Corr

E-m

journal homepage: www.elsevier.com/locate/ces

Direct numerical simulation of flow and heat transfer in densefluid–particle systems

Niels G. Deen n, Sebastian H.L. Kriebitzsch, Martin A. van der Hoef, J.A.M. Kuipers

Multiphase Reactors Group, Department of Chemical Engineering and Chemistry, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands

H I G H L I G H T S

G R A P H I C A L A

c Novel immersed boundary techni-que for fluid flow and heat transferin dense suspensions.

c Unique fluid–solid coupling throughimplicit incorporation of boundaryconditions.

c The technique does not require cali-bration of an effective particle dia-meter.

c Fully resolved simulations of a sta-tionary random array of particlesare presented.

c Fully resolved simulations of aliquid fluidized bed are presented.

09/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.ces.2012.06.055

esponding author. Tel.: þ31 40 247 3681; fax

ail address: [email protected] (N.G. Deen).

B S T R A C T

a r t i c l e i n f o

Article history:

Received 8 February 2012

Received in revised form

23 May 2012

Accepted 26 June 2012Available online 5 July 2012

Keywords:

Fluidization

Fluid mechanics

Heat transfer

Multiphase flow

Immersed boundary method

Multi-scale modeling

a b s t r a c t

In this paper a novel simulation technique is presented to perform Direct Numerical Simulation (DNS)

of fluid flow and heat transfer in dense fluid–particle systems. The unique feature of our fluid–solid

coupling technique is the direct (i.e., implicit) incorporation of the boundary condition (with a second-

order method) at the surface of the particles at the level of the discrete momentum and thermal energy

equations of the fluid. Contrary to lattice Boltzmann or other commonly used immersed boundary

implementations, our method does not require using any effective diameter.

A fixed (Eulerian) grid is utilized to solve the Navier–Stokes equations for the entire computational

domain. Dissipative particle–particle and/or particle-wall collisions are accounted via a hard sphere

discrete particle approach using a three-parameter particle–particle interaction model accounting for

normal and tangential restitution and tangential friction. Following the detailed verification of the

method several dense multi-particle systems are studied in detail involving stationary arrays of

particles and fluidized particles.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Dense gas–particle flows are frequently encountered in avariety of industrial processes. More specifically these flows areencountered in gas–fluidized beds, which are frequently appliedin the chemical, petrochemical, metallurgical, environmental and

ll rights reserved.

: þ31 40 247 5833.

energy industries in large scale operations involving physical(coating, drying and granulation) and chemical (synthesis of fuelsand base chemicals) transformations. The accurate prediction ofdense gas–particle flows, which are inherently unsteady andheterogeneous, has proven notoriously difficult, which can beattributed to the wide variation of length scales existing inthese flows.

For the accurate prediction of these flows in engineering scaleequipment, which in practice can only be achieved with con-tinuum models, it is essential that both the fluid–particle as well

Page 2: Chemical Engineering Science - ssu.ac.irssu.ac.ir/.../Articles/EN/1-s2.0-S0009250912004204-main.pdf · of granular flow (model 2), the discrete particle model (model 3), the immersed

N.G. Deen et al. / Chemical Engineering Science 81 (2012) 329–344330

as the particle–particle interactions are accurately accounted for.This requirement has led to adoption of a multi-scale modelingapproach for these flows (Van der Hoef et al., 2004, 2005) and isschematically presented in Fig. 1. At the most fundamental levelDirect Numerical Simulation (DNS) can be used, which offers thepossibility to directly compute the fluid–particle interaction andthe associated drag closure laws which are required at the morecoarse-grained levels of modeling. Due to CPU and memoryconstraints, typically O(103) particles can be treated simulta-neously in this type of simulation.

At the intermediate level the Discrete Particle (DP) model(Tsuji et al., 1993; Hoomans et al., 1996; Link et al., 2005; Ye et al.,2005) is used in which individual particles are tracked in thecomputational domain taking into account two-way coupling anddissipative collisions between the particles and/or confiningwalls. In the DP model the flow field computation is based onthe volume-averaged Navier–Stokes equations where the controlvolumes typically contain a large number of particles and as aconsequence the details of the flow in the vicinity of the particlesis lost, necessitating the incorporation of a fluid–particle dragclosure. For the description of the particle–particle interaction(i.e., the non-ideal collisions) two types of models are widelyused: the hard-particle model and the soft-particle model.

Fig. 1. Multi-scale approach for dense fluid–particle flows comprising the discrete

bubble model (model 1), the two-fluid model based on the kinetic theory

of granular flow (model 2), the discrete particle model (model 3), the immersed

boundary model (model 4) and the lattice Boltzmann model (model 5). (repro-

duced from Van der Hoef et al., 2008).

Table 1Overview of Direct Numerical Simulation (DNS) techniques for fluid–particle flows.

Method Advantages

Overset grid

method

Accurate incorporation of the fluid–solid interaction Can handle (ver

domains due to possibility to use a coarse background grid

Arbitrary

Lagrangian–

Eulerian

method

Implicit incorporation of the fluid–solid interaction Accurate represe

of the no-slip condition due to velocity nodes residing on particle

and possibility for efficient local grid refinement

Immersed

boundary

method

Very flexible with respect to incorporation of the degree of rigidity

particles Relatively easy to implement

Lattice

Boltzmann

method

Produces accurate results over a (very) wide range of particle pack

fractions

Typically the hard-particle model accounts for binary particle–particle interactions whereas the soft-particle model can handleencounters between more than two particles and as a conse-quence it is better suited to handle quasi static systems. The DPmodel offers the advantage that rather complicated particle–particle interaction models, including normal and tangentialrestitution and tangential friction, can be incorporated to assesstheir impact on the flow structure. The disadvantage of this typeof model is given by the fact that typically only O(106) particlescan be treated simultaneously, which is mainly due to CPUconstraints.

Finally for the simulation of engineering scale gas–fluidizedbeds a two-fluid or continuum model (Ding et al., 1990; Kuiperset al., 1992; Goldschmidt et al., 2001; Patil et al., 2005a, 2005b)based on the Kinetic Theory of Granular Flow (KTGF) is used.In this type of model both the continuous and the particulatephase are described as continuous media with mutual interaction.In the continuum models for dense particulate flows usually onlythe phase coupling due to drag is considered whereas for thedissipative particle–particle interactions only normal restitutionis considered.

The main emphasis in this paper is on the most fundamentallevel of modeling, namely the Direct Numerical Simulation offluid–particle flows, a field that has advanced considerably in thepast decade due to the advances in numerical simulation techni-ques and computer hardware. Most of the previous studies havefocused on isothermal systems; in the present study we extendthe DNS approach to non-isothermal particle-laden flows.At present several powerful DNS techniques exist, each with theirown particular advantages and disadvantages, which are sum-marized in Table 1, and briefly discussed below.

In overset grid methods (Chesshire and Henshaw, 1990;Henshaw and Schwendeman, 2003) for each (moving) particle abody conforming (spherical) grid is used as well as a fixedEulerian grid (termed the background grid), where interpolationbetween the grids is used to exchange the information. Theadvantage of this method is its capability to accurately computethe fluid–solid interaction terms. In addition rather large domainscan be treated since the fixed background grid can be relativelycoarse. The disadvantage of this method is given by the fact thatits implementation is rather complex whereas problems arise formultiple particles that closely approach each other.

The Arbitrary Lagrangian–Eulerian (ALE) technique (Hu et al.,2001) employs a moving unstructured finite element mesh withlocal mesh refinement around the particles. The ALE methodembeds fully coupled particle and fluid motion and usually acollision model to avoid very small elements between particles orbetween particles and confining walls. The advantage of this

Disadvantages

y) large Implementation is rather complex due to required interpolation

procedures Problems with multiple particle systems with closely

approaching particles

ntation

surface

Expensive remeshing operation Incorporation of a collision model is

necessary to avoid very small computational elements and hence

excessive small time steps

of the Explicit treatment of fluid–solid coupling leading to stiffness problems

for rigid bodies Appropriate values of fluid–solid interaction

parameters (spring stiffnes) need to be determined for each particular

class of problems

ing Calibration required

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N.G. Deen et al. / Chemical Engineering Science 81 (2012) 329–344 331

method is the implicit fluid–solid coupling. Moreover the velocitynodes reside on the particle surface, which is beneficial for theaccurate incorporation of the no-slip condition. Finally, efficientrefinement of the grid is possible. The ALE method also possessesa number of disadvantages including the expensive remeshingoperation and the necessity to include a collision model to avoidvery small computational elements, and hence avoid a (very)small permissible time step.

The Immersed Boundary (IB) method (Peskin, 1977; Saiki andBirlingen, 1996; , 2002; Mittal and Iaccarino, 2005) makes use of afixed Eulerian grid to solve for the flow field of the continuousphase and Lagrangian markers associated with the motion of theimmersed body that can be of flexible or rigid nature. In thesimplest version of the IB method two sets of Lagrangian markersare used, one set of markers tracks the motion of the immersedbody whereas the second set of markers tracks the motion of thecontinuous phase in the vicinity of the body. To enforce the no-slip condition at the surface of the body, the difference betweenthe (local) position vectors of two markers belonging to these twodifferent sets provides a measure to compute the local restoringforce. Following the computation of this local restoring force, theEulerian force density is computed by distributing to the Euleriangrid. The IB method has been widely used to study fluid–structureinteraction and was pioneered by Peskin (1977) to cardiac flowproblems. Subsequently the range of applications of this powerfulmethod has expanded considerably. For excellent reviews theinterested reader is referred to Peskin (2002) and Mittal andIaccarino (2005). The advantages of the IB method are itsflexibility with respect to the degree of rigidity (from elastic torigid) of the bodies. Moreover, this method is relatively easy toimplement. Disadvantages include the explicit treatment of thefluid–solid interaction, which leads to stiffness problems for rigidparticles. In addition appropriate values for the fluid–solid inter-action parameters (such as the spring stiffness) need to bedetermined for each particular class of problems.

A different class of IB technique is the Direct Forcing Method(DFM), first introduced by Mohd-Yusof (1997). Contrary to thevast literature on immersed boundary techniques to enforce theboundary conditions for the momentum equations, considerablyfewer studies have focused on the application of these techniquesto enforce the boundary conditions for the thermal energyequations. In the DFM the force densities are computed directlyfrom the discretized Navier–Stokes equations, taking into accountthe no-slip conditions at the particle surface, at Lagrangian points(or surface elements) that are distributed over the particle sur-face. Many different versions of the DFM have been presentedsince its introduction. For instance, Kim et al. (2001) introduced amass source/sink in the continuity equation which leads to adecreased magnitude of the error in the solution. However, theorder of their spatial discretization error remained second-order.An elegant method which combines the strong points of themethod of Peskin (1977) with the advantages of the DFM formoving rigid particles has been reported by Uhlmann (2005).

The Lattice Boltzmann Method (LBM) can be viewed as aspecial, particle-based discretization method to solve the Boltz-mann equation. This method is particularly attractive if multiplemoving objects or dispersed elements (particles, droplets orbubbles) have to be considered and avoids, contrary to theclassical finite difference and finite element methods, thedynamic remeshing, which becomes prohibitively expensivewhen dealing with a large number of moving objects. Ladd(1994a, 1994b) has used the LBM successfully to compute theeffective gas–particle drag in monodisperse particulate suspen-sions. Van der Hoef et al. (2005) have extended the study tobinary and polydisperse particle mixtures and reported that thecurrently used drag laws, which are based on modifications of the

drag laws for monodisperse systems, lead to a large underpredic-tion and overprediction of, respectively, the large and smallparticles in a dense assembly. The LB method offers the advantagethat accurate results are obtained over a (very) wide range ofpacking fractions. However, this method requires calibration,which constitutes a (minor) disadvantage.

Feng and Michaelides (2005) used a combined ImmersedBoundary-Lattice Boltzmann Method (IB-LBM) to simulate parti-culate flows. They used a direct forcing method to enforce theno-slip condition whereas the LB scheme was adopted for theflow computation. In the model of Feng and Michaelides, non-ideal particle–particle and/or particle-wall collisions were alsoaccounted for using a soft sphere encounter algorithm.

In addition to the abovementioned methods other techniqueshave been successfully applied to DNS of particulate flow.A particularly interesting development has been reported byZhang and Prosperetti (2005) who coupled analytical solutionsto the Stokes equations for the flow field in the vicinity of the(spherical) particles with finite difference solutions to the fullNavier–Stokes equations. They showed that accurate solutionscould be obtained with relatively few grid points distributed overthe radius of the particles, which makes the method particularlyattractive for systems involving ‘‘many’’ particles.

Glowinski et al. (2001) introduced the Distributed LagrangianMultiplier/Fictitious Domain (DLM/FD) method. In this methodthe rigid body motion is enforced throughout the domain occu-pied by the immersed solid objects via the introduction ofLagrangian multipliers in the combined momentum equations.As the coupling of the phases is implicit an iterative procedure isrequired to advance one time step.

The application of non-body fitted methods such as DFM orDLM/FD in the simulations of heat transfer has received much lessattention, however. Kim and Choi (2004) extended their DFMmethod to non-isothermal flows by introduction of a heat source/sink in the thermal energy equation. They present results fordifferent of forced and mixed convections problems for the flowaround cylinders in 2D where they have applied an constanttemperature or constant heat flux boundary condition on thesurface of the particles. Pacheco et al. (2005) used the DFMmethod of Kim and Choi (2004) in a non-staggered finite-volumemethod, which is used to discretize the momentum equationsthat are given in general non-orthogonal coordinates. Also theyintroduce a different interpolation method. They show results forisothermal flow around a cylinder in an unbounded fluid, naturalconvection in a 2D inclined cavity and a heated (constanttemperature) cylinder in a square enclosure.

In a series of papers Feng and Michaelides (2008, 2009)described the application of their direct forcing method to non-isothermal particulate flows. They used a finite-difference schemeon a staggered grid to discretize the equations of motion and thethermal energy equation. They assumed a uniform temperaturewithin a particle. Hence, they restrict the applicability of theirmodel to problems with a low Biot number. They studied theeffect of free convection on the sedimentation of single andmultiple particles with constant temperature as well as particlesthat have freely developing temperature due to the interactionwith the surrounding fluid.

Yu et al. (2006) extended the DLM/FD method to particulateflow with heat transfer and their method allows for a freelyvarying, non-uniform temperature across the particle. Theystudied the motion of a catalyst particle in a box with homo-genous heat production in the particle and the influence of thePeclet number on the effective thermal conductivity in shearednon-colloidal suspensions and nanofluids. Dan and Wachs (2010)extended this method to 3D problems and introduced somechanges in the algorithm to increase the robustness of the method.

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N.G. Deen et al. / Chemical Engineering Science 81 (2012) 329–344332

They validated their improved method against non-isothermal 2Dand isothermal 3D flow problems known from literature and thenstudied the influence of free convection on the sedimentation of aparticle in a semi-infinite channel.

Our model combines a second-order technique for the fluid–solid coupling and a hard sphere Discrete Particle (DP) modelto account for collisional interaction between the particles.The unique feature of our fluid–solid coupling technique is thedirect (i.e., implicit) incorporation of the boundary condition(with a second-order method) at the surface of the particles atthe level of the discrete momentum and thermal energy equa-tions of the fluid.

The organisation of this paper is as follows: first the descrip-tion of the model is given. The subsequent sections are devoted tothe numerical solution method and the verification of the methodfor several (3D) test cases for which analytical or numericalsolutions exist. Then, the results for gas–solid and liquid–solidsystems are presented and discussed and finally conclusions arepresented.

2. Model description

Our model consists of two main parts: one part deals with thesolution of the fluid phase equations, accounting for the presenceof the solid particles, whereas the other part deals with thesolution of the solid phase equations accounting for all forcesacting on them and the possible non-ideal collisions between theparticles themselves and/or confining walls. First the mainassumptions are given:

a)

The fluid phase is incompressible and Newtonian. b) All physical properties of both phases are constant. c) The solid phase consists of spherical particles with uniform

temperature.

2.1. Fluid phase equations

The transport phenomena in the fluid phase are governed bythe conservation equations for mass, momentum and thermalenergy, respectively, given by:

ðr � uÞ ¼ 0 ð1Þ

@rf u

@tþðr � rf uuÞ ¼�rpþmfr

2uþrf g ð2Þ

rf Cp,f

@Tf

@tþðr � uTf Þ

� �¼ lfr

2Tf ð3Þ

In these equations rf, mf, Cp,f and lf represent, respectively, thedensity, viscosity, heat capacity and the thermal conductivity ofthe fluid.

2.2. Solid phase equations

The translational and rotational motion of the suspended solidparticles is governed by the Newtonian equations of motion,respectively, given by:

mpdwp

dt¼mpgþFf-s ð4Þ

and

Ipdop

dt¼ Tf-s ð5Þ

where mp and Ip represent, respectively, the mass and themoment of inertia of the particle. The final terms on the right-hand sides in Eqs. (4) and (5) account for the fluid–particleinteraction (respectively, drag and torque) and are given by:

Ff-s ¼�

ZZSp

ðtf � nþpnÞdS ð6Þ

and:

Tf-s ¼�

ZZSp

ðr�rpÞ � ðtf � nþpnÞdS¼�

ZZSp

ðr�rpÞ � ðtf � nÞdS ð7Þ

where tf represents the viscous stress tensor given by:

tf ¼�mf ½ðruÞþðruÞT � ð8Þ

It should be noted that the buoyancy term is not included in anexplicit manner in the equation of motion for the particles since itis included in the calculation of the pressure force acting on theparticle surface. The particle temperature is governed by (assuminguniform particle temperature):

mpCp,sdTp

dt¼Ff-s ð9Þ

where the term on the right hand side represents the heat transferrate from the fluid to the particle given by:

Ff-s ¼�

ZZSp

ðlfrTf � nÞdS ð10Þ

For the evaluation of the total fluid–particle force Ff-s, thetorque Tf-s and heat transfer rate Ff-s the velocity gradients andthe temperature gradients need to be known. The evaluation ofthese quantities is detailed in Section 3.3.

3. Numerical solution method

3.1. Fluid phase equations

The fluid phase equations are solved in three dimensions (3D)on a Cartesian grid with uniform grid spacings in all directions.First we perform the time discretization of the momentumequation to obtain:

rf unþ1¼ rf un

�Dtrpnþ1�Dt 1:5Cn

m�0:5Cn�1

m

h i

þDt

2mfr

2unþ1þmfr

2unh i

þrf gDt ð11Þ

where n indicates the time index and Cm the net convectivemomentum flux given by:

Cm ¼ rf ðr � uuÞ ð12Þ

For the temporal discretization of the convective momentumtransport the Adams–Bashforth scheme is used whereas for theviscous term the Crank–Nicholson scheme is used. The solution ofEq. (11) is achieved via a two-step projection method where as afirst step the tentative velocity field unn is computed from:

rf unn¼ rf un�Dt 1:5C

n

m�0:5Cn�1

m

h iþDt

2mfr

2unnþmfr

2unh i

þrf gDt

ð13Þ

The Laplace operator is approximated with standard centralsecond-order finite difference representations whereas the con-vection terms are computed with a second-order flux limitedBarton-scheme (Centrella and Wilson, 1984). The enforcement ofthe no-slip condition at the surface of the immersed boundary ishandled at the level of the discretized momentum equations andwill be detailed in Section 3.3. We use a robust and very efficientIncomplete Cholesky Conjugate Gradient (ICCG) algorithm tosolve the resulting sparse matrix equation for each velocity

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N.G. Deen et al. / Chemical Engineering Science 81 (2012) 329–344 333

component. The velocity field at the new time level nþ1 is relatedto the tentative velocity field as follows:

unþ1¼ unn

�Dt

r rpnþ1 ð14Þ

Since unþ1 needs to satisfy the incompressibility constraint,upon taking the divergence of Eq. (14) the pressure Poissonequation is obtained:

1

rr2pnþ1 ¼

1

DtðrUunn

Þ ð15Þ

which is again solved using the ICCG algorithm to obtain thepressure at the new time level. Finally the velocity field at thenew time level is obtained from the second (correction) stepEq. (14). This correction step has a negligible effect on the correctenforcement of the no-slip condition at the fluid–particle bound-aries. It should be stressed here that the pressure and velocity areobtained for the entire domain (i.e., also for the grid pointsinterior to the particles).

Subsequently the thermal energy equation is solved using thesame temporal discretization as used for the solution of themomentum equations:

rf Cp,f Tnþ1f ¼ rf Cp,f Tn

f �Dt 1:5Cn

h�0:5Cn�1

h

h iþDt

2lfr

2Tnþ1þlfr

2Tnh i

ð16Þ

where the net convective heat flux Ch is given by:

Ch ¼ rf Cp,f ðr � uTÞ ð17Þ

For the spatial discretization of the convection and diffusionterms in Eq. (16) exactly the same schemes are used as for thesolution of the momentum equations.

Fig. 2. Incorporation of the boundary condition for a general fluid quantity f.

The five indicated cells are used for the discrete representation (in 2D) of the PDE

governing fluid quantity f (fluid velocity component or fluid temperature).

The ‘‘solid’’ node corresponding with f0 resides inside the particle whereas the

‘‘fluid’’ nodes corresponding to f1 and f2 reside in the fluid. The value of f0 is

expressed in terms of a second order polynomial according to f¼ ax2þbxþc

where the values of coefficients a, b and c are obtained from the known values of fat the particle boundary (x¼xs: f¼fp) and the two nodes in the fluid (x¼1:

f¼f1 and x¼2: f¼f2) leading to Eq. (23). Note that x¼0 corresponds to the

location of the ‘‘solids’’ node.

3.2. Solid phase equations

The source terms appearing in the Newtonian equations ofmotion are treated as known (explicit) terms and therefore theintegration of these equations can be conducted in principle withany integration technique for ordinary differential equations. Forthe simulations reported in this paper we have used a second-order trapezoidal rule producing translational and rotationalvelocities and the particle temperature at the new time levelcomputed, respectively as follows:

wnþ1p ¼wn

pþgDtþDt

2mpF

n

f-sþFnþ1

f-s

h ið18Þ

onþ1p ¼on

pþDt

2IpT

n

f-sþTnþ1

f-s

h ið19Þ

Tnþ1p ¼ Tn

pþDt

2mpCp,sFn

f-sþFnþ1f-s

h ið20Þ

This implies that the total fluid–particle interaction force, thetorque and the heat transfer rate needs to be evaluated for eachparticle at the beginning and the end of each time step. Once theparticle velocities and temperature are obtained an event drivenhard sphere collision model is invoked to account for dissipativeparticle–particle collisions. In this model it is assumed that theinteraction forces are impulsive and therefore all other finiteforces are negligible during collision. The closure for this collisionmodel involves three independent micro-mechanical parameters:the coefficients for normal and tangential restitution and thetangential friction coefficient, which in principle can be obtainedfrom impact experiments. Unresolved hydrodynamic interactionof the particles is not included in the current model.

3.3. Fluid–solid coupling

The fluid–solid coupling constitutes the key element of ourmethod and will be explained in more detail in this section.The discretization of the momentum equations Eq. (13) and thethermal energy equation Eq. (16) at the appropriate nodes leadsto algebraic equations of the following generic form:

acfcþXnb

anbfnb ¼ bc ð21Þ

where f corresponds to one of the fluid phase velocity compo-nents or the fluid phase temperature. This equation is obtainedfor all ‘‘fluid’’ nodes ‘‘c’’ exterior to the particles (see Fig. 2) andrelates quantity fc to six (3D) or four (2D) neighboring nodesindicated with ‘‘nb’’.

Discretization of Eqs. (13) and (16) leads to the followingexpression of the neighboring coefficients:

aj ¼�Dt

2Dx2j

Gf ac ¼ 1�Xnb

anb ð22Þ

where j indicates the coordinate direction in which the neighbor-ing cell is located, and Gf is the transport coefficient, which isequal to, respectively, mf/rf and lf/(rfCp,f) for the momentum andenergy equations. Note that all explicit terms are collected in thesource term bc.

For moving particles the detection of these nodes needs to becarried out during each time step taking into account the locationof the node in the staggered computational grid (velocity nodesare located at the faces of the cells, the node for scalar quantitiessuch as the temperature is located at the centre of the cells). Fromthe perspective of a particular ‘‘fluid’’ node the six surroundingnodes ‘‘nb’’ are examined to test whether one of these nodes ‘‘nb’’represents a ‘‘solid’’ node (i.e., a node located inside a particle). Inthat case a boundary condition has to be applied where the valueof fnb¼f0 at that particular ‘‘solid’’ node is expressed as a (1D)linear combination of the f values of the relevant ‘‘fluid’’ nodesf1 and f2:

f0 ¼�2xs

1�xsf1þ

xs

2�xsf2þ

2

ð1�xsÞð2�xsÞfp ð23Þ

where fp represents the desired value of f at the boundary of theparticle and xs a dimensionless distance that is known from theintersection of the surface of the particle and the grid line. Eq. (23)

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N.G. Deen et al. / Chemical Engineering Science 81 (2012) 329–344334

is obtained by fitting a second-order polynomial f¼ ax2þbxþc

to the values of f known at the particle boundary (x¼xs) and thetwo fluid nodes corresponding to f1 and f2 (see Fig. 2). For aspherical particle this intersection point can be calculated analy-tically whereas for a particle with arbitrary shape this needs to bedone numerically, for instance with the (bracketed) NewtonRaphson method to assure fast convergence. Eq. (23) is used toeliminate f0 from Eq. (21), which leads to an algebraic equationsimilar to Eq. (21), but with modified coefficients a. When weconsider that through its definition a2¼a0, the modified coeffi-cients are given by:

a2 ¼ a2þa0xs

2�xs¼ a2ð1þ

xs

2�xsÞ

a1 ¼ ac ¼ ac�a02xs

1�xs

bc ¼ bc�a02

ð1�xsÞð2�xsÞfp ð24Þ

This procedure needs to be carried out for all ‘‘solid’’ nodes toproperly account for the local boundary condition to be met at thesurface of the particle. In case f refers to the fluid velocity the no-slip condition at the surface of the particle needs to account fortranslation and rotation of the particle:

ujparticle-surface ¼wpþop � ðr�rpÞ ð25Þ

where r is the position vector of the intersection point of theparticle surface and the grid line and rp the position vector of thecentre of mass of the particle. It is possible that for fluid node ‘‘c’’boundary conditions need to be applied for multiple neighboringnodes (temporarily) coinciding with different (moving) particlesand therefore it is essential to keep track of the ‘‘connectivity’’ ofnodes to particles (see Fig. 3). This is achieved by storing for therelevant node (one of the three velocity nodes or the temperaturenode) the number of the particle coinciding with thatparticular node.

As mentioned earlier, for the integration of the solid phaseequations the total fluid–particle interaction force Ff-s, thetorque Tf-s and the heat transfer rate Ff-s need to be evaluated.These quantities are obtained by integrating the local forceexerted by the fluid on the particle and the local fluid-to-particleheat flux over the external surface of each particle and require theevaluation of, respectively, the velocity gradient and temperaturegradient at the particle surface. For a general quantity f the(local) dimensionless gradient at the particle surface is given by

Fig. 3. Incorporation of the boundary condition for a general fluid quantity f.

The five indicated nodes are used for the discrete representation (in 2D) of the PDE

governing fluid quantity f. For the nodes residing inside the (moving) particles

a and b the connectivity of these nodes to particle a or particle b needs to be

established during each time step. Due to the utilization of a staggered grid this

connectivity needs to be determined for each velocity component and tempera-

ture separately. For the situation shown the boundary condition (see Fig. 2) needs

to be applied for three neighbor nodes which reside inside the particles a (1 node)

or b (2 nodes).

the following expression:

dfdx

����x ¼ xs

¼ð2�xsÞ

ð1�xsÞf1�ð1�xsÞ

ð2�xsÞf2�

ð3�2xsÞ

ð1�xsÞð2�xsÞfp ð26Þ

For the evaluation of Ff-s the contribution due to the pressureneeds to be evaluated as well. Note that the viscous and pressurecontribution of the drag are obtained separately unlike theimmersed boundary method proposed by Uhlmann (2005). Wehave implemented two different methods that give identicalresults for Ff-s. In the first method the pressure at the surfaceof the particle is obtained by (linear) extrapolation of the pressureat the fluid nodes in the vicinity of the particle surface. In thesecond method the surface integral of the pressure Eq. (6) isconverted to a volume integral:

Ff-s ¼�

ZZSp

pndS¼�

ZZZVp

rpdV ð27Þ

The volume integral is evaluated by summing the values of thepressure gradient calculated at all nodes interior to the particleunder consideration. The components of the pressure gradient arecalculated at the velocity nodes in the staggered grid usingstandard (second-order) central finite difference approximations.In the limiting case of no motion (fluid–particle system at rest)the pressure gradient in Eq. (27) follows from hydrostaticsrp¼ rf g (see Eq. (2)) and the total fluid–particle interactionforce is given by the well-known buoyancy force:

Ff-s ¼�

ZZZVp

rpdV ¼�

ZZZVp

rf gdV ¼�rf Vpg ð28Þ

4. Verification

4.1. Heat transfer in laminar channel flow

In our first test the correctness of the implementation of thefluid flow equations was tested for heat transfer in fully devel-oped laminar flow in a square channel with diameter d. For thisparticular case the analytical expression for the Nusselt numberfor thermally developed conditions is given by the followingexpression:

Nu¼awd

lf¼ 2:98 ð29Þ

where lf the thermal conductivity of the fluid. The average heattransfer coefficient aw is defined as the heat flux averaged overthe wetted perimeter of the duct divided by the driving force forthe heat transfer process. Here we define the driving force as thedifference of the constant wall temperature and the flow-averageor mixing-cup temperature of the fluid. In Table 2 the data used

Table 2Data used for the simulations of thermally fully developed convective heat

transfer for laminar flow in a square duct.

Parameter Value Unit

Time step 10�4 s

Duct diameter 0.02 m

Duct length 0.2 m

Fluid density 1000 kg/m3

Fluid viscosity 1.0 kg/(m s)

Fluid heat capacity 1.0 J/(kg K)

Fluid thermal conductivity 1.0 W/(m K)

Fluid velocity 1.0 m/s

Fluid inlet temperature 293 K

Wall temperature 393 K

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Table 3Data used for the simulations of convective heat transfer to a stationary sphere.

Parameter Value Unit

Domain size 0.16 � 0.16 � 0.16 m

Time step 1.0 �10�4–2.5 �10�4 s

Sphere diameter 0.020 m

Fluid density 1000 kg/m3

Fluid viscosity 0.10 kg/(m s)

Fluid heat capacity 10 J/(kg K)

Fluid thermal conductivity 1.0 W/(m K)

Fluid inlet temperature 293 K

Sphere temperature 393 K

Table 4Comparison between heat transfer coefficients obtained from the simulations and

the Ranz–Marshall equation for various particle Reynolds numbers Rep.

Rep Simulated a(W/(m2 K))

Ranz–Marshalla (W/(m2 K))

Relativeerror (%)

20 225.08 234.16 �3.9

40 283.58 289.74 �2.1

60 328.01 332.38 �1.3

80 365.60 368.30 �0.7

100 399.18 400.00 �0.2

200 537.25 524.50 2.4

400 722.80 700.00 3.3

Fig. 4. Schematic representation of configuration used for simulations of a single

sphere sedimenting in an enclosure. Free-slip and adiabatic boundary conditions

were used for, respectively, the computation of the velocity and temperature

distributions.

0 1 2 3 4 50

2

4

6

8Exp. Rep=1.5Exp. Rep=4.1Exp. Rep=11.6Exp. Rep=31.9Sim. Rep=1.5Sim. Rep=4.1Sim. Rep=11.6Sim. Rep=31.9

h/d p

(-)

t (s)

0 1 2 3 4 5-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

Exp. Rep=1.5Exp. Rep=4.1Exp. Rep=11.6Exp. Rep=31.9Sim. Rep=1.5Sim. Rep=4.1Sim. Rep=11.6Sim. Rep=31.9

v p (m

/s)

t (s)

Fig. 5. Particle trajectories (top) and particle velocities (bottom) for single

sedimenting spheres in an enclosure (also see Fig. 4).

N.G. Deen et al. / Chemical Engineering Science 81 (2012) 329–344 335

for the simulations are given. Using grids of 10�10�100,20�20�200 or 40�40�400 cells in, respectively, the x-,y- and z-directions the computed Nusselt number amounted to,respectively, 3.037, 3.005 and 2.996 indicating excellent agree-ment with the analytical result given by Eq. (29).

More detailed results on the spatial convergence of ournumerical scheme can be found in Appendix A.

4.2. Convective heat transfer to a stationary sphere

In our second test we consider convective heat transfer to asingle stationary sphere in an enclosure. The data used for thenumerical simulations are given in Table 3. Laterally the spherewas positioned at the centre of the domain whereas in the flowdirection the centre of the sphere was positioned at 2dp from theinlet. For these simulations a 1603 grid was used with uniformgrid spacing in all directions. For a single sphere the expressionfor the particle Nusselt number Nup is given by the well-knownempirical Ranz–Marshall equation:

Nup ¼apdp

lf¼ 2:0þ0:6ðRepÞ

1=2ðPrÞ1=3

ð30Þ

where Rep is the particle Reynolds number and Pr the Prandtlnumber, respectively, given by:

Rep ¼rf u0dp

mf

and Pr¼mf Cp,f

lfð31Þ

In order to change the Reynolds number, the flow velocity u0

was varied from 0.1 m/s (Re¼20) to 2 m/s (Re¼400). The com-parison between the computed Nusselt numbers and the resultsobtained from the Ranz–Marshall equation is quite good (seeTable 4) despite the fact that the ratio of the particle radius andthe grid size amounts (only) to 10. If we use the flat plateapproximation to estimate the thermal boundary layer thicknessdt as:

dt

5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirf Cp,f u0x=lf

q ð32Þ

we find that there are 3 to 4 grid points in the thermal boundarylayer for x¼Rp at the highest Reynolds number, hence the goodagreement between the simulation results and values computedfrom Eq. (30). It is stressed that Eq. (30) is an empirical correla-tion, which might explain why we underpredict the value

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N.G. Deen et al. / Chemical Engineering Science 81 (2012) 329–344336

computed from Eq. (30) by about �4% at Rep¼20, whereas weoverpredict it by about 3% at Rep¼400. However, it is anticipatedthat local grid refinement is necessary for (substantially) higherparticle Reynolds numbers (i.e., exceeding 103) because thethickness of the thermal boundary layer decreases.

4.3. Single sedimenting spheres

As a next case we consider the sedimentation of single spheresin an enclosure and compare our simulation results with theexperimental measurements by Ten Cate et al. (2002). They alsoreported results of simulations using LBM combined with anadaptive forcing scheme. We consider a single spherical particlewith a diameter of 15 mm settling in a box of dimensions0.16�0.16�0.16 m3. The sphere was released from its restposition at a height h¼0.12 m from the bottom, as depicted inFig. 4. We used free-slip and adiabatic boundary conditions for,respectively, the computation of the velocity and temperaturedistributions. Due to the large ratio of the lateral domain size andthe particle diameter, the results are not influenced by theboundaries. Four cases were considered in which the density ofthe fluid ranged from 960 to 970 kg/m3 and the dynamic viscosityfrom 0.058 to 0.353 kg/(ms). The density of the particle was set to1120 kg/m3.

Fig. 5 shows the computed particle trajectories and settlingvelocities together with the experimental data reported byTen Cate et al. (2002). The terminal Reynolds number rangedfrom 1.5 to 31.9. From this figure it can be seen that theagreement between the simulations and the experiments is verygood. Note that in our method we do not make use of an effectivediameter, in contrast to the numerical simulations conducted byTen Cate et al. (2002). We will discuss this in more detail in thenext section.

4.4. Flow past periodic static arrays of spheres

For periodic arrays of spheres also a large body of data(obtained by combined analytical and computational approaches)is available and therefore this system provides a useful test casefor our model. We will focus here on the dimensionless drag forceF acting on a single sphere in periodic arrays, defined by:

F ¼Ff-s

3pmU0dpð33Þ

where U0 represents the superficial velocity. Zick and Homsy(1982) used a boundary integral method to obtain the drag actingon particles in periodic arrays and tabulated the dimensionlessdrag force Eq. (33) as a function of the particle volume fraction.See Table 5 for the data used in the simulations.

In Fig. 6 the particle configuration and the computed fluidtemperature distribution in a plane cutting through a layer ofspheres is shown together with the velocity map for the centralplane in the vicinity of the central particle. For the case shown in

Table 5Data used for the simulation of creeping flow through a simple cubic packing

at maximum packing fraction.

Parameter Value Unit

Computational grid 160�160�160¼4,096,000 (�)

Grid size 0.001 m

Time step 10�4 s

Particle diameter 0.032 m

Fluid density 100 kg/m3

Fluid viscosity 0.05 kg/(m s)

Fluid velocity 0.005 m/s

Fig. 6 (corresponding to the maximum packing fraction for theSimple Cubic (SC) system of p/6) the computed value of F

amounts 42.67, which agrees very well with the value of 42.10(error 1.35%) reported by Zick and Homsy (1982) and 42.6 (error0.2%) reported by Sørensen and Stewart (1974). For the evaluationof the pressure contribution to the drag in this case the methodbased on the evaluation of the volume integral Eq. (27) was used.A further comparison between the computed dimensionless dragforce and data from Zick and Homsy (1982) can be found in Fig. 7and Table 6.

We stress that in order to obtain these results, we have notmade use of any effective diameter. For the lattice Boltzmannmethod and also the Uhlmann implementation of the immersedboundary method, there is a small but significant discrepancybetween the simulation results and the exact result for the dragforce for periodic arrays at low Reynolds number and volumefraction. In the aforementioned methods, this is remedied byintroducing the concept of an effective diameter, which is thediameter such that a perfect match is obtained between the

Fig. 6. Particle configuration and fluid temperature map (top) and velocity map

(bottom) for creeping flow through a simple cubic packing at maximum packing

fraction of p/6.

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Table 7Data used for the simulations of the stationary random particle configuration.

Parameter Value Unit

Computational grid 100�100�400¼4,000,000 (�)

Grid size 0.0005 m

Time step 10�4 s

Particle radius 0.003 m

Initial sphere positions random configuration (1326 spheres) (�)

Fluid density 1.0 kg/m3

Fluid viscosity 2.0 �10�5 kg/(m s)

Fluid thermal conductivity 0.02 W/(m K)

Fluid heat capacity 1000 J/(kg K)

Superficial fluid velocity 0.10–0.40 m/s

Fluid inlet temperature 293 K

Particle temperature 393 K

N.G. Deen et al. / Chemical Engineering Science 81 (2012) 329–344 337

simulation data and the theoretical result in the limit of very lowsolids volume fractions. The diameter that follows from thiscalibration is then regarded as the ‘‘true’’ diameter of theparticles, and used for all simulations.

In Fig. 7 we show the dimensionless drag force on a particle forlow Reynolds number flow in a periodic simple cubic array ofparticles at different solids volume fractions and resolutions ofthe particle. These results were obtained using a modifiedUhlmann implementation of the immersed boundary method(Kriebitzsch, 2011). In the top graph no effective diameter wasused and one finds a strong increase in the error with increasingsolids volume fraction. As shown in the middle graph, this strongincrease is not observed when the effective diameter is employed.Although the origin of such an effective diameter can be under-stood, it remains a rather ad-hoc approach, also since thecalibration may modify other computational errors regardlesstheir origin. Moreover, for heat transfer, in principle a separatecalibration procedure would be required, which could potentiallylead to two effective diameters. The fact that our result agreeswith the theoretical results of Zick and Homsy (1982) without theuse of any effective diameter indicates that no calibration proce-dure is required, which we view as a significant advantage of ourmethod over other methods.

For the calculation at maximum packing fraction the ratio ofparticle radius and cell size amounted 16, but for much lowerresolutions the results are still reasonably accurate: calculationswhere this ratio was reduced to 8 and 4 produced, respectively, adimensionless drag F of 40.53 (�3.7%) and 38.91 (�7.6%). Similarresults were obtained for other test cases indicating that ourmethod produces already quite accurate results at relatively lowresolutions. This advantageous property is especially important incase simulations with many particles are of interest.

5. Results for dense random particle systems

In the previous section the model has been verified for anumber of cases in which the force exerted on the spheres or theNusselt number is known from well established sources and it wasfound that the agreement was very good for all cases examined.Subsequently the model will be applied to systems containing arelatively large number of bodies where the physics is substantiallymore complex. To show that the method is well suited for gas–solid and liquid–solid flow problems, in Section 5.1 the flow andheat transfer of a gas through a stationary array of particles isexamined whereas in Section 5.2 the liquid–fluidization and heattransfer of spheres in a thin pseudo two-dimensional fluidized bedis considered.

Table 6Parameters used for immersed boundary simulations of Stokes flows in simple

cubic packed arrays of spheres. The average non-dimensional drag force on the

spheres is plotted as a function of the solid volume fraction in Fig. 7. DF/F is the

deviation with the analytical results of Zick and Homsy (1982).

1�e(�)

dp (m) dp/Dx(�)

F (�)Zick and

Homsy (1982)

F (�)Sørensen and

Stewart (1974)

F (�)thiswork

DF/F(%)

0.5236 0.0320 32.0 42.14 42.6 42.67 1.3

0.450 0.0304 30.4 28.1 27.32 �2.8

0.343 0.0278 27.8 15.4 15.21 �1.2

0.216 0.0238 23.8 7.442 7.432 �0.1

0.125 0.0199 19.9 4.292 4.261 �0.7

0.064 0.0159 15.9 2.81 2.773 �1.3

0.027 0.0119 11.9 2.008 1.969 �1.9

5.1. Flow through a stationary random array of particles

In this section we will consider the flow gas through astationary random array of particles (see Table 7 for the dataused in the simulation). The spheres were distributed in a randomfashion over the computational domain to produce a predefinedvoidage of 0.70. A hard-sphere Monte-Carlo method has beenused to create the random configurations. For the simulation aprescribed uniform velocity and a prescribed pressure wereimposed at, respectively, the bottom and top boundaries. Forthe other boundaries the no-slip condition was imposed. Thetemperature of the particles Ts was kept constant in thesecalculations whereas the fluid entered with a uniform tempera-ture. At the solid walls adiabatic boundary conditions wereprescribed and a zero temperature gradient was set at the outflowboundary. The superficial velocity ranged from 0.1 m/s to 0.4 m/scorresponding to particle Reynolds numbers ranging from 36 to144. In Fig. 8 the particle configuration of 1326 spheres is shown(voidage e equals 0.70) together with the computed velocitydistributions at Rep¼144 for the central plane near the bottomof the array of particles. Inside the particles (not shown) thecomputed velocity field is very small due to the particularenforcement of the no-slip boundary condition.

For practical heat transfer calculations the evolution of thecup-mixing or flow-averaged fluid temperature /TfS is of con-siderable interest. This quantity is defined by:

/TfS¼

RRSuzðx,y,zÞTf ðx,y,zÞdxdyRR

Suzðx,y,zÞdxdyð34Þ

where the integration is performed over a surface Sf (i.e., that partof the surface occupied by the fluid) perpendicular to the mainflow direction (i.e., the Z-direction). In Fig. 9 the cup-mixingtemperature is shown as a function of the axial co-ordinate for therange of used superficial velocities (or particle Reynolds numbers,see Table 7). In the initial part of the system the fluid heats upquickly (due to the large driving force for fluid–particle heattransfer) and after Z¼0.10 m essentially thermal saturation hasoccurred.

In case the axial heat transport is dominated by fluid convec-tion (which is the case here) the differential equation describingthe evolution of /TfS is given by the following expression(assuming adiabatic boundary conditions at the walls confiningthe array in the lateral directions):

U0rf Cp,f

d/TfSdz

¼ apapðTs�/TfSÞ ð35Þ

where ap is the average (in plane Sf) fluid–particle heat transfercoefficient (based on the local driving force (Ts�/TfS)) and ap thespecific fluid–particle heat transfer surface (ap¼6(1�e)/dp).

Page 10: Chemical Engineering Science - ssu.ac.irssu.ac.ir/.../Articles/EN/1-s2.0-S0009250912004204-main.pdf · of granular flow (model 2), the discrete particle model (model 3), the immersed

Fig. 8. Particle configuration (left) and computed velocity map at Rep¼144 for

the central plane in the bottom section of the fixed bed (right).

0.00 0.05 0.10 0.15 0.20290300310320330340350360370380390400

Rep=36

Rep=72

Rep=108

Rep=144

T(k)

z(m)

Fig. 9. Cup-mixing temperature as a function of the axial co-ordinate for several

particle Reynolds numbers Rep.

Table 8Comparison between heat transfer coefficients obtained from the DNS simulations

and the Gunn correlation for various particle Reynolds numbers Rep for voidage

e¼0.70 and Pr¼0.80.

Rep Simulated ap

(W/(m2 K))Gunncorrelationap (W/(m2 K))

Relativedeviation(%)

Correlationcoefficientfitted ap

36 24.2 35.1 �31.1 0.999

72 37.4 44.2 �15.4 0.998

108 49.2 51.5 �4.50 0.998

144 62.1 58.0 þ7.10 0.999Fig. 7. Comparison of the average non-dimensional drag force on spheres in Stokes

flows for the case of a simple cubic packed array of spheres as a function of the solid

volume fraction. The top and middle graph show were obtained with an IB method

following the lines of Uhlmann. (Kriebitzsch, 2011). In the top graph diameter d0 of

the location of the marker points was equal to the particle diameter dp, whereas

an effective diameter deff obtained from calibration at very low solids volume fraction

was used to obtain the results. The bottom graph shows the forced that was obtained

with method described in this work, which does not use an effective diameter. The

solid line is the exact value of F given by Zick and Homsy (1982).

N.G. Deen et al. / Chemical Engineering Science 81 (2012) 329–344338

When we assume a constant heat transfer coefficient ap, Eq. (35)can be integrated with the inlet condition /TfS¼/Tf,0S at Z¼0leading to the following evolution equation for /TfS:

lnTs�/TfS

Ts�/Tf ,0S

� �¼�

apapz

U0rf Cp,fð36Þ

The DNS results were fitted to Eq. (36), where the averagefluid–particle heat transfer coefficient ap is the only unknownquantity. In Table 8 the fitted fluid–particle heat transfer coeffi-cients are compared with the results obtained from the empiricalcorrelation proposed by Gunn (1978) given by:

Nup ¼apdp

lf¼ ð7�10eþ5e2Þð1þ0:7ðRepÞ

0:2ðPrÞ1=3

Þ

þð1:33�2:40eþ1:20e2ÞðRepÞ0:7ðPrÞ1=3

ð37Þ

where Rep and Pr represent, respectively, the particle Reynoldsnumber (based on the superficial velocity U0) and the Prandtlnumber. For the fits the inlet zone (first particle layer) wasexcluded as well as the region where the fluid temperature wasvery close to the particle temperature (i.e., the region with nearlycomplete thermal saturation of the fluid). The agreement betweenthe fitted heat transfer coefficients and the results obtained fromthe empirical correlation of Gunn is quite reasonable, especially atthe higher Reynolds numbers. To our knowledge this is the firstnumerical validation of the widely-used correlation by Gunn,although only for a single Pr number and a single porosity.In future work we intend to perform a more elaborate comparisonwith Gunn’s correlation, which requires a large number ofsimulations in order to scan the parameter space (Re, Pr, e).

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N.G. Deen et al. / Chemical Engineering Science 81 (2012) 329–344 339

Of course, much more detailed information can be obtainedfrom the DNS results such as the (distribution of) heat transfercoefficients of individual particles. Here we define the local fluid–particle heat transfer coefficient ap by the following equation:

ap ¼Ff-s

4pR2pðTs�Tf Þ

ð38Þ

where Ff-s represents the heat flow rate of the particle to thesurrounding fluid with its local average temperature Tf. The localheat flow rate is obtained from:

Ff-s ¼�

ZZSp

ðlfrTf � nÞdS ð39Þ

The components of the temperature gradient at the particlesurface are obtained from Eq. (26). In this case the driving force isdefined as the difference between the (constant) surface tem-perature of the particle Ts and the local average fluid temperatureTf. The local average fluid temperature can be obtained from:

Tf ðrpÞ ¼

RRRVf

g ry�rp

�� ��� �TðryÞdVyRRR

Vfg ry�rp

�� ��� �dVy

ð40Þ

where the integration is performed over the local fluid volumesurrounding each particle. The function g(r) is a monotonicallydecreasing function of r for which we used g(r)¼exp[�(r/Rp)].

Note that a similar function has been used by Jackson (1997)to derive the volume-averaged conservation equations (i.e., theequations underlying the Two-Fluid Model) starting from thelocal instant formulation: the Navier–Stokes equations for thefluid and the Newtonian equations of motion for the particles.

For the practical usage of Eq. (40), a cubic box of size 2(1þb)Rp

is used with its center coinciding with the center of the (refer-ence) particle. Note that b¼0 corresponds to a cubic box in whichthe particle just fits. In Fig. 10 the average fluid–particle heattransfer coefficient computed from Eqs. (39) and (40) is shown asa function of the parameter b for several particle Reynoldsnumbers. From this figure it can be seen that (as expected) theeffect of b is relatively small and that for b44 essentially theaverage heat transfer coefficients converge to a constant values.Note that in their discrete particle simulations of a spout fluidizedbed, Link et al. (2005) also used a value of b¼4 to calculate (i) thelocal fluid properties at the position of the particle and (ii) thereaction force of the particle on the fluid. Using a well-defined testcase Link (2006) demonstrated that using b44, leads to an

2 4 6 8 10

30

40

50

60

70Re=36Re=72Re=108Re=144

β (-)

α p (W

/(m2 .

K))

Fig. 10. Average local fluid–particle heat transfer coefficient defined by Eqs. (38)

and (39) as a function of the size of the cubic averaging domain 2(1þb)Rp for

several particle Reynolds numbers Rep.

acceptable accuracy in the calculation of the local fluid properties(i.e., errors less than 0.5% in the resulting forces). This workconfirms his finding that b¼4 is a suitable value for mappingproperties between the particle and the fluid phase, and vice versa.

In addition one calculation was performed for a stationary arrayof 3000 particles (voidage equal to 0.6984) using a 150�150�400computational grid at Rep¼60. In Fig. 11 the distribution of the localheat transfer coefficient is given for Rep¼60 using a b value of 4. Forthis case the average fluid–particle heat transfer coefficientamounted 49.25 W/(m2K). From Fig. 11 we observe that a consider-able variation in the local fluid–particle heat transfer coefficientexists in the array. For these calculations again the inlet zone andthe zone with nearly complete thermal saturation of the fluid wereexcluded from the analysis.

5.2. Fluidization of spheres in a pseudo two-dimensional bed

In this section we will consider the liquid fluidization of 1296spheres (see Table 9 for the data used in the simulation). Initiallythe spheres were distributed over the bottom half of the compu-tational domain in a periodic array with uniform voidage of 0.65.Note that the thickness of the bed in the depth direction slightlyexceeds the diameter of the spheres. The height of the freeboardis the same as the initial bed height, allowing for expansion of thebed. For the simulation no-slip boundaries were taken at thewalls confining the bed in the lateral directions whereas aprescribed pressure boundary was taken at the top of the domain.Initially the particle bed was assumed to be at rest. The minimumfluidization velocity for the fluid–particle combination was esti-mated from the Ergun equation as 1.17 cm/s whereas the term-inal velocity of the particle amounted 39.6 cm/s. For thesimulation of the pseudo two-dimensional fluidized bed thesuperficial velocity was taken as 8 cm/s. The values of the normaland tangential coefficient of restitution were set to en¼0.9 andet¼0.3, whereas the friction coefficient was set to m¼0.1. Thesevalues represent typical values for glass beads.

The evolution of the bed structure is shown in Fig. 12 as afunction of time. From this figure it can be seen that initially thebed expands uniformly, however near the left and right walls ofthe bed (due to the no-slip condition) a disturbance in the particleconfiguration develops that eventually leads to the formation of asmall cavity, which propagates near the walls in the verticaldirection. During this process also in the central part of the beddisturbances in the particle configuration develop and severalcavities start to form and rise in the vertical direction, which soonleads to breakage of the symmetry and as a consequence a‘‘chaotic’’ movement and mixing of the particle bed commences.It should be noted here that the superficial velocity u0 is quitehigh leading to significant expansion of the particle bed. Once thebed expansion is complete, the total force exerted by the fluid onparticle bed in the vertical direction is always very close to thetotal weight of the particles (difference less than 1%) although atthe level of individual particles this force differs from the weightof an individual particle. Fig. 13 shows the velocity field of thefluidizing agent in the central plane for a small sub-domain ofthe bed at t¼2 s. From this figure it can clearly be seen that theflow field at the level of the particles is highly non-uniform.In principle this (type of) computation generates all the detailedinformation required to critically test assumptions and closuresadopted in models that do not resolve all the details of the flow ofthe fluidizing agent. The widely used discrete particle model is awell-known example since typically the size of the computationalcells in this type of model is much larger than the size of anindividual particle (Van der Hoef et al., 2008) leading to thenecessity to specify a closure for the fluid–particle drag.

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.

0 20 40 60 80 100 120 1400.00

0.05

0.10

0.15

0.20

0.25

αp (W/m2.K)

f (-)

Fig. 11. Particle configuration (left) and distribution of the local fluid–particle heat transfer coefficient (right) defined by Eqs. (38) and (39) for b¼4 at Rep¼60 and voidage

of 0.6984. The calculation was carried out for 3000 particles using a 150�150�400 computational grid. The average fluid–particle heat transfer coefficient corresponding

to the distribution shown equals 49.25 W/(m2 K).

Table 9Data used for the pseudo 2D fluidized bed of spheres.

Parameter Value Unit

Computational grid 400�12�800¼3,840,000 (�)

Grid size 0.0005 m

Time step 2 �10�4 s

Particle radius 0.0025 m

Initial sphere positions Uniform lattice of (36�36)¼1296

spheres in bottom half of the domain

(�)

Fluid density 1216 kg/m3

Fluid viscosity 0.1 kg/(m s)

Fluid thermal conductivity 1.0 W/(m K)

Fluid heat capacity 10 J/(kg K)

Superficial liquid velocity 0.08 m/s

Fluid inlet temperature 293 K

Solid density 8000 kg/m3

Particle temperature 393 K

Collision parameters en, et, mt 0.90, 0.30, 0.10 (–)

Wall temperature 293 K

N.G. Deen et al. / Chemical Engineering Science 81 (2012) 329–344340

Due to the high specific fluid–particle surface area (ap¼776 m�1)rapid thermal saturation occurs in the bottom zone of the bed asevident from Fig. 14 showing the evolution of the axial profiles of thecup-mixing temperature of the fluid. After approximately 2 s thethermal wave caused by the step change in the fluid inlet tempera-ture arrives at the outlet of the system whereas after approximately3 s the fluid outlet temperatures becomes (nearly) constant and equalto the (fixed) surface temperature of the particles.

Fig. 15 shows the distribution of the local fluid–particle heattransfer coefficient defined by Eqs. (39) and (40) for b¼4 as afunction of the lateral position at t¼3 s. The average fluid–particleheat transfer coefficient corresponding to the distribution shownequals 789.0 W/(m2 K).

From the bed expansion as well as the occupancy of theparticles in the bottom zone of the fluidized bed a (nearlyconstant) average voidage of 0.80 can be calculated whichcorresponds to an average fluid–particle heat transfer coefficientof 970.0 W/(m2 K) according to the empirical correlation pro-posed by Gunn Eq. (37).

6. Conclusions

In this paper a new simulation technique has been presentedfor the direct numerical simulation of dense particle-laden flows.Particle-fluid and particle–particle interaction are taken intoaccount, respectively, via an implicit, second order method anda hard sphere discrete particle approach.

In the presented method there is no need for using an effectivediameter. Hence, no calibration procedure is required, which weview as a significant advantage over other methods. Dissipativeparticle–particle and/or particle-wall collisions are accounted viaa hard sphere DP approach using a three-parameter particle–particle interaction model accounting for normal and tangentialrestitution and tangential friction.

In the current model, we assume a uniform temperature withinparticles, which limits the application of the model to problems witha low Biot number or in other words to either small particles or solidswith a high thermal conductivity. In order to extend the methods totemperature gradients within the particles, a thermal equation has to

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0.25 s 0.50 s 0.75 s 1.00 s

1.25 s 1.50 s 1.75 s 2.00 s

2.25 s 2.50 s 2.75 s 3.00 s

Fig. 12. Evolution of bed structure as a function of time for fluidization of 1296 spheres at superficial velocity of 8 cm/s. Incipient fluidization velocity 1.17 cm/s; terminal

velocity 39.6 cm/s.

N.G. Deen et al. / Chemical Engineering Science 81 (2012) 329–344 341

be solved for the solids phase and coupled to the model for the fluidphase. This can be done by following the same approach used toenforce the boundary conditions for the fluid phase, which will leadto equations very similar in spirit as Eqs. (23) and (26). However thesurface temperature of the particle is unknown and hence anadditional equation has to be used. This equation follows from thecontinuity of the heat flux at the particle surface. By equating the heatflux for the fluid phase with the heat flux for the solid phase theunknown surface temperature is obtained in terms of the fluid andsolid temperatures in the vicinity of the particle surface.

Following a detailed verification of the method involvingstationary and moving particles, several multi-particle systemswere studied in more detail.

The first system considered was a stationary array of particlesfor which the velocity and temperature distributions were

calculated for several particle Reynolds numbers. From thecomputed velocity and temperature distributions first the axialprofiles of the cup-mixing fluid temperature were calculated andfitted against a simple one-dimensional interpretation model toobtain the bed-averaged fluid–particle heat transfer coefficients.These coefficients agreed reasonable well with results obtainedfrom the empirical correlation due to Gunn (1978). In additionlocal fluid–particle heat transfer coefficients were computedusing a driving force calculated from the locally averaged fluidtemperature. It was found that the effect of the size of the cubicaveraging volume (with its center coinciding with the center ofthe (reference) particle) became relatively small provided that asize of five times the particles is used.

Subsequently the model was used to study the fluidization of1296 spheres in a pseudo two-dimensional fluidized bed. Due to

Page 14: Chemical Engineering Science - ssu.ac.irssu.ac.ir/.../Articles/EN/1-s2.0-S0009250912004204-main.pdf · of granular flow (model 2), the discrete particle model (model 3), the immersed

Fig. 13. Particle configuration (left) and velocity map at the central plane for

a small sub-domain of the bed at t¼2 s. Incipient fluidization velocity 1.17 cm/s;

terminal velocity 39.6 cm/s.

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40290300310320330340350360370380390400

t=0.50 st=1.00 st=1.50 st=2.00 st=2.50 st=3.00 s

T g (K

)

z (m)

Fig. 14. Evolution of axial profile of the cup-mixing fluid temperature computed

from Eq. (33).

0.00 0.05 0.10 0.15 0.200

200

400

600

800

1000

1200

1400

1600

1800

a p (W

/(m2 .

K))

x (m)

Fig. 15. Distribution of the local fluid–particle heat transfer coefficient defined

by Eqs. (38) and (39) for b¼4.0 as a function of the lateral position at t¼3 s.

The average fluid–particle heat transfer coefficient corresponding to the distribution

shown equals 789.0 W/(m2 K).

N.G. Deen et al. / Chemical Engineering Science 81 (2012) 329–344342

the relatively high superficial velocity considerable (stable) bedexpansion was obtained with a considerable degree of non-uniformity in the flow field at the level of individual particles.For the expanded bed the distribution of the fluid–particle heattransfer coefficients was calculated as well revealing significantspatial variation.

Notation

ap Specific fluid–particle heat transfer surface (m�1)Cp Heat capacity (J/(kg K))d Channel diameter (m)dp Particle diameter (�)en Normal restitution coefficient (�)et Tangential restitution coefficient (�)F Dimensionless force (�)g(r) Kernel for averaging (�)Ip Moment of inertia (kg m2)mp Particle mass (kg)Nu Nusselt number (�)Pr Prandtl number (�)p Pressure (N/m2)Rp Particle radius (m)Re Reynolds number (�)Sp External particle surface (m2)t Time (s)T Temperature (K)Tf-s Torque exerted by fluid on the particle (N m)U0 Superficial velocity (m/s)V Volume (m3)w1 Terminal (steady state) sedimentation velocity (m/s)

Greek letters

ap Fluid–particle heat transfer coefficient (W/m2 K)b Parameter for averaging (�)Dt Time step (s)dt Thermal boundary layer thickness (m)e Voidage (�)Ff-s Heat flow rate from fluid to particle (W)l Thermal conductivity (W/(m K)m Dynamic viscosity (kg/(m s))r Density (kg/m3)x Dimensionless distance (�)

Vectors

Cm Net convective momentum flux (N/m3)Ch Net convective heat flux (W/m3)Ff-s Force exerted by fluid on the particle (N)g Gravitational acceleration (m/s2)n Unit normal vectorr Position vector (m)u Velocity (m/s)wp Particle translational velocity (m/s)op Particle rotational velocity (s�1)

Tensors

t Viscous stress tensor (N/m2)

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N.G. Deen et al. / Chemical Engineering Science 81 (2012) 329–344 343

Subscripts and superscripts

d Dragf Fluid phaseg Gravityp Particles Solid phasew Wally Localz z-direction

Operators

@

@tPartial time derivative (s-1)

r Gradient operator (m-1)rU Divergence operator (m-1)r

2 Laplace operator (m-2)

Table A1Computed instantaneous Nusselt numbers for different spatial resolutions N

where N is the ratio of the particle radius and the grid size. The relative errors

are given between brackets.

t (s) Nu (N¼4) Nu (N¼8) Nu (N¼16) Nu analytical

0.001 26.47 (þ7.74%) 25.96 (þ5.67%) 24.93 (þ1.47%) 24.568

Acknowledgements

The authors would like to thank the European ResearchCouncil for its financial support, under its Advanced InvestigatorGrant scheme, contract number 247298 (MultiscaleFlows) and itsStarting Investigator Grant scheme, contract number 259521(Cutting Bubbles).

0.002 19.75 (þ9.98%) 18.56 (þ3.35%) 18.06 (þ0.57%) 17.958

0.004 14.27 (þ7.42%) 13.51 (þ1.70%) 13.31 (þ0.20%) 13.284

Table A2Data used for the simulation of creeping flow through a simple cubic array of

cylinders at packing fraction of 0.5. For all simulations a 5�5 array of cylinders

was used and 6 computational cells in the direction of the axis of the cylinders.

Computational grid 40�40

80�80

160�160

Grid size 0.00400 m

0.00200 m

0.00100 m

Time step 0.00001 s

Packing fraction 0.500 (�)

Particle diameter 0.025532305 m

Fluid density 100.0 kg/m3

Fluid viscosity 0.050 kg/(m s)

Superficial fluid velocity 0.005 m/s

Table A3Computed dimensionless drag (defined by Eq. (A5)) acting on the central cylinder

for different computational grids. For a packing fraction of 0.500 Sangani and

Acrivos reported a value of 532.55 for the dimensionless drag defined by Eq. (A5).

Grid Fsim Relative error (%)

40�40 414.44 �22.2

80�80 520.48 �2.27

160�160 531.93 �0.12

Appendix A. Spatial convergence of the method

In this appendix we present more detailed results on thespatial convergence of the method. As a first test case we considerthe unsteady state heat diffusion from a hot sphere contained in a(very) large pool of quiescent fluid. At t¼0 the surface tempera-ture of the sphere is suddenly raised from T0¼293.0 K (initialfluid temperature) to T1¼393.0 K. The equation governing theunsteady heat conduction is given by:

@T

@t¼

a

r2

@

@rr2 @T

@r

� ðA1Þ

where a represents the thermal diffusivity and T the fluidtemperature. The following initial and boundary conditionsapply:

t¼ 0 : T ¼ T0

r¼ Rp : T ¼ T1 ðA2Þ

r¼1 : T ¼ T0

The final boundary condition is valid as long as the tempera-ture front does not reach the confining walls (focus is on earlytime behaviour). The analytical solutions for the temperaturedistribution in the liquid and the associated instantaneous Nus-selt number are given by:

Tðr,tÞ�T0

T1�T0¼

Rp

r1�erf

r�Rpffiffiffiffiffiffiffiffi4atp

� � �ðA3Þ

and:

NuðtÞ ¼aðtÞdp

ll¼ 2þ

2ffiffiffiffipp

at

Rp

� �0:5

ðA4Þ

In Table A1 we present the results of simulations using increasingspatial resolution N where N is the ratio of the particle radius Rp andthe grid size. For all these simulations a sufficiently small time step of

10�5 s was used to suppress temporal discretisation errors. Thethermal diffusivity was set to 10�3 m2/s and the particle radius to0.02 m. As evident form Table A1 very accurate results are obtainedfor the early time solution (corresponding to steep temperatureprofiles near the particle surface) with excellent spatial convergence.

As a second test case we consider creeping flow through asimple cubic array of cylinders at (surface) packing fraction of0.500. For this particular situation very accurate numerical results(obtained with a boundary integral method) have been publishedby Sangani and Acrivos (1982) in terms of the dimensionless dragFSA acting on a single cylinder in a periodic array defined by:

FSA ¼F

mU0ðA5Þ

here F is the force per unit length acting on a single cylinder, m theviscosity of the fluid and U0 the superficial velocity. In Table A2 wesummarize the data which have been used to assess the spatialconvergence of our method for this test case. The results of thesimulations are presented in Table A3 revealing again that veryaccurate results are obtained and excellent spatial convergence.

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