Powder Technology 220 (2012) 122–137

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Open-source MFIX-DEM software for gas–solids flows: Part I—Verification studies

Rahul Garg a,b, Janine Galvin c, Tingwen Li a,b, Sreekanth Pannala d,⁎a National Energy Technology Laboratory, Morgantown, WV, 26507, USAb URS Corporation, Morgantown, WV, 26507, USAc National Energy Technology Laboratory, Albany, OR, 97321, USAd Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA

⁎ Corresponding author.E-mail addresses: rahul.garg@gmail.com (R. Garg), J

(J. Galvin), litingwen@gmail.com (T. Li), pannalas@ornl

0032-5910/$ – see front matter © 2011 Elsevier B.V. Alldoi:10.1016/j.powtec.2011.09.019

a b s t r a c t

a r t i c l e i n f o

Available online 19 September 2011

Keywords:Discrete Element Method (DEM)Computational gas–solids flowLagrangian–Eulerian (LE)Eulerian–Eulerian (EE)Computational fluid dynamicsMultiphase flows

With rapid advancements in computer hardware, it is now possible to perform large simulations of granularflows using the Discrete Element Method (DEM). As a result, solids are increasingly treated in a discreteLagrangian fashion in the gas–solids flow community. In this paper, the open-source MFIX-DEM softwareis described that can be used for simulating the gas–solids flow using an Eulerian reference frame for the con-tinuum fluid and a Lagrangian discrete framework (Discrete Element Method) for the particles. This methodis referred to as the continuum discrete method (CDM) to clearly make a distinction between the ambiguityof using a Lagrangian or Eulerian reference for either continuum or discrete formulations. This freely availableCDM code for gas–solids flows can accelerate the research in computational gas–solids flows and establish abaseline that can lead to better closures for the continuum modeling (or traditionally referred to as two fluidmodel) of gas–solids flows. In this paper, a series of verification cases is employed which tests the differentaspects of the code in a systematic fashion by exploring specific physics in gas–solids flows before exercisingthe fully coupled solution on simple canonical problems. It is critical to have an extensively verified code asthe physics is complex with highly-nonlinear coupling, and it is difficult to ascertain the accuracy of theresults without rigorous verification. These series of verification tests set the stage not only for rigorousvalidation studies (performed in part II of this paper) but also serve as a procedure for testing any newdevelopments that couple continuum and discrete formulations for gas–solids flows.

anine.Galvin@netl.doe.gov.gov (S. Pannala).

rights reserved.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction and background

Multiphase flows are ubiquitous in devices involved in gas–solidscontacting and processing that are found in many industries, such asthose in energy production, chemical processing and pharmaceuticals.Simulations of these systems can lead to a better understanding ofthe physical phenomena occurring in these multiphase flows. Withverified and validated models these applications may be optimizedto achieve higher efficiencies and become more environmentallyfriendly. From the numerical viewpoint, simulations that solve thecontinuum conservation equations for both phases have traditionallybeen called Eulerian–Eulerian (EE), or two fluid method (TFM) simula-tions. On the other hand, simulations that consider the carrier phase as acontinuumand the dispersed phase asmade up of discrete entities havetraditionally been called Lagrangian–Eulerian (LE) simulations. Whilemany acronyms are used for simulations that consider both phases ascontinua, here we will refer to them as TFM simulations. Furthermore,to clearly emphasize both the continuum and discrete representationsand to avoid confusion with the reference frameworks used to solve

these equations, we will refer to LE simulations as continuum discretemethod (CDM) simulations.

A number of open source and commercial codes are capable ofperforming both TFM and CDM simulations for dense gas–solids flows.For example, commercially available codes like Fluent, and open sourcecodes like CFDlib [1], OpenFoam, and MFIX [2,3], are all capable ofperforming TFM simulations for chemically reacting multiphase flows.Similarly, commercially available codes like Fluent and Barracuda,and open sources codes likeMFIX-DEM, KIVA [4], Fluent DPM (DiscreteParticle Method) and dense-phase DPM modules, and OpenFoam,are all capable of CDM simulations. In regard to TFM simulations,while all the above-mentioned codes solve for similar forms of thegoverning equations, they primarily differ in their closures for vari-ous submodels (such as solids stresses, interphase drag, etc.) andtheir numerical treatment.

In regard to CDM simulations, like the TFM simulations, all thecodes solve similar forms of the governing equations for the carrierphase (with differences in numerical treatment, closure models, etc.).For the dispersed phase, however, all publicly available codes exceptfor MFIX-DEM employ a parcel-based (also called as computational/notional/nominal particles based approach in literature) approach. Inthe parcel approach a finite number of parcels are tracked rather thanusing actual individual particles as dictated by the number density of

123R. Garg et al. / Powder Technology 220 (2012) 122–137

the solids-phase. Each parcel may represent either a fractional numberof real particles or many real particles grouped together to form a singleparcel. For example, in very dilute regions of spray applications, manyparcels are used to represent one real particle in order to mitigate thehigh statistical errors that would be associated with very few real parti-cles. On the other extreme, in very dense fluidized bed like applicationsor device-scale problems, many real particles are grouped together andare represented by a single parcel in order to reduce the high computa-tional cost with tracking the real number of particles. Since the particlesin this approach are represented by statistically weighted parcels, thecollisions between parcels (unlike collisions between particles) cannotbe directly resolved necessitating the use of indirect collision models.For example stochastic collision models, such as the droplet colli-sion algorithm of [5] (used in Fluent DPM and OpenFoam) or theless expensive (but similar) no time counter algorithm of [6], havebeen used for calculating collisions in CDM simulations of dilutegas–solids flows. For CDM simulations of dense gas–solids flows,the collisions between parcels have been modeled by an ad-hocsolids stress term that prevents the solids from over packing (e.g.,Fluent's dense-phase DPM and Barracuda). In the MFIX-DEM code(for CDM simulations) the dispersed phase is represented by actualindividual particles and the collisions are directly resolved using thesoft-sphere (based on a spring-dashpot model) approach of [7].While simulations using MFIX-DEM are limited to small problemsizes due to high computational cost incurred in the particle neigh-bor search algorithm, this approach (using actual particles) doesserve as a good tool to verify and develop new closures for varioussubmodels used in TFM simulations.

The above discussion focused on TFM and CDM simulations, whichare designed to model two-phase flows having both a carrier and adisperse phase. Advanced codes are also available for studying pureparticulate (or granular) flows in the absence of a carrier phase. Inall such codes, the solids-phase is represented by actual particles andcollisions are directly resolved. Such simulations are generally referredto as Discrete Element Method (DEM) simulations. Examples of DEMsimulation codes include open source codes, such as LAMMPS [8] andYade [9], and commercial codes, such as EDEM (http://www.dem-solutions.com/index.php) and Itasca [10]. Efforts to couple suchstandalone DEM codes to existing computational fluid dynamic (CFD)solvers have recently been undertaken with the goal to leverage thecombined abilities that were originally developed for each code indi-vidually. For example, the EDEM code provides users the ability tocouple its DEM modules with other CFD codes such as Fluent. Veryrecently OpenFoam has been coupled to Yade [11] and LAMMPS.However, these OpenFoam codes for CDM type simulations are stillin the beta stage and are unavailable to the broader CFD community.Although, EDEM, a commercial code, provides coupling hooks withother commercial CFD codes such as Fluent, the inability of users toreadily understand and modify the source code limits it to mostlyend/expert users.

The open source MFIX-DEM code can be used for DEM, CDM, andTFM simulations from a single source code. A basic structure for DEMand CDM simulations has existed in MFIX for several years. Despitethis time and even though MFIX provides an excellent opportunityto run different statistical descriptions from one software platform,MFIX has not been as widely used for CDM and DEM simulations asit has for TFM simulations due to lack of rigorous verification andvalidation.

An important step prior to the application of any model is veri-fication and validation of that model. Verification refers to the pro-cess of evaluating the numerical accuracy of a model [12] wherethe accuracy of the solution algorithm can be assessed by applyingthe model to problems for which the solution is already known (e.g.,via an analytical solution of a limiting case). This approach shouldreveal whether the code contains errors but does not guaranteethat it is completely correct. That is, the code may show agreement

with the solution for one test problem, but disagreement with thatof another untried test problem which may invoke different com-ponents of the code. Another obstacle in conducting verification isthat relatively few problems are available in multiphase flows inwhich an exact solution is available [12]. With this in mind, thecurrent effort included a series of test cases, of varying complexity,which were selected for their ability to test different aspects of thecode.

In the paper we are building on earlier MFIX-DEM developments,which are detailed in previous works [13–16]. The spring-dashpotmodel has been rigorously verified by performing a series of simpletests, such as a freely falling particle, two stacked particles compressedbetween two boundaries, and a particle sliding on a rough surface. Thepurpose of these simple tests is to verify independently each compo-nent of the spring-dashpot model. For the CDM simulations, qualitativeand quantitative analyses of particles in vortex flows has been per-formed to validate the accuracy of gas–solids coupling. This paperalso documents the current MFIX-DEM code along with pointers tothe code and a discussion on the theory. Part II of this paper willdetail a series of validation cases to complete the verification andvalidation of the MFIX-CDM code.

In the next section, the details of CDM and DEM simulationsare provided in a manner that is consistent with the MFIX-DEMimplementation.

2. Governing equations

In MFIX-DEM, the gas-phase governing equations for the massand momentum conservation are similar to those in traditionalsingle-phase CFD but with additional coupling terms due to dragfrom the solids-phase. The solids-phase is modeled using discreteparticles. The current implementation of MFIX-DEM is restrictedonly to hydrodynamics (no chemistry or heat and mass transfer).Work is under way to extend the DEM implementation to includethese additional physics. Below is a list of the governing equationsalong with the numerical implementation, including the couplingprocedure.

2.1. Gas-phase

The governing equations, implemented in MFIX [3], for the gas-phase continuity and momentum conservation in the absence of phasechange, chemical reactions, growth, aggregation, breakage phenomena,are:

∂ εgρg� �∂t þ∇· εgρgvg

� �¼ 0; ð1Þ

and

DDt

εgρgvg

� �¼ ∇· ¯Sg þ εgρgg− ∑

M

m¼1Igm: ð2Þ

In the above equation, εg is the gas-phase volume fraction, ρg is thethermodynamic density of the gas phase, vg is the volume-averagedgas-phase velocity, Igm is the momentum transfer term between thegas and the mth solids phase, and ¯Sg is the gas-phase stress tensorgiven by

¯Sg ¼ −Pg ¯I þ ¯Tg; ð3Þ

where Pg is the gas-phase pressure. Also, ¯Tg is the gas-phase shearstress tensor,

¯Tg ¼ 2μg ¯Dg þ λg∇·tr ¯Dg

� �¯I; ð4Þ

D

V i

V j

j

i

y

ij

x

Fig. 1. Schematic of two particles i and j having diameters Di and Dj in contact. Particleshave linear and angular velocities equal to Vi,Vj and ωi,ωj, respectively. Overlapδn=0.5(Di+Dj)−D. ηij is the unit vector along the line of contact pointing from parti-cle i to particle j.

124 R. Garg et al. / Powder Technology 220 (2012) 122–137

where ¯Dg ¼ 12 ∇vg þ ∇vg

� �Th iis the strain rate tensor, and μg and λg

are the dynamic and second coefficients of viscosity of the gas phase.Discussion and definition of the interphase momentum transfer termis reserved until Section 2.2.3.

2.2. Solids-phase: Discrete Element Method (DEM)

In the DEM approach, the mth solids-phase is represented by Nm

spherical particles with each particle having diameter Dm and densityρsm. Solids phases are differentiated based according to radii and den-sities. Accordingly, the diameter and density of themthsolids-phase isdenoted by Dm and ρsm, respectively. For M solids phases, the total

number of particles is equal to N ¼ ∑M

m¼1Nm. These N particles are

represented in a Lagrangian frame of reference at time t byX ið Þ tð Þ;V ið Þ tð Þ;ω ið Þ tð Þ;D ið Þ; ρ ið Þ; i ¼ 1;…;Nn o

, where X ið Þ tð Þ denotes

the ith particle's position,V ið Þ tð Þ and ω(i) denote its linear and angularvelocities, D(i) denotes its diameter, and ρ(i) represents its density. Itis implicit that if a particle belongs to mth solids-phase, then its diam-eter and density are, respectively, equal to Dm and ρsm (i.e., equal tothe diameter and density of the mth solids-phase). The mass m(i)

and moment of inertia I(i) of the ith particle are equal to ρ ið Þ πDið Þ3

6and

m ið ÞD ið Þ2

10, respectively. The position, linear and angular velocities

of the ith particle evolve according to Newton's laws as:

dX ið Þ tð Þdt

¼ Við Þ tð Þ; ð5Þ

m ið Þ dVið Þ tð Þdt

¼ Fið ÞT ¼ m ið Þ

gþ Fi∈k;mð Þd tð Þ þ F

ið Þc tð Þ; ð6Þ

I ið Þ dωið Þ tð Þdt

¼ Tið Þ ð7Þ

where g is the acceleration due to gravity, F i∈k;mð Þd is the total drag

force (pressure+viscous) on ith particle residing in kth cell and belong-ing to themth solids-phase,F ið Þ

c is the net contact force acting as a resultof contact with other particles, η is the outward pointing normal unitvector along the particle radius, T ið Þ is the sum of all torques acting onthe ith particle, and F

ið ÞT is the net sum of all forces acting on the ith par-

ticle. The next three subsections discuss in detail the calculation of thecontact and drag forces.

2.2.1. Contact forcesThe advantage of the DEM approach over that of solving continuum

equations for solids-phase lies in its explicit treatment of particle–particle collisions. In MFIX-DEM, the soft-sphere model (based onthe spring-dashpot model, first proposed by [7]) is used to modelthe particle–particle and particle–wall collisions. The details of thesoft-sphere model are given below.

As shown by the schematic in Fig. 1, consider two particles i and jin contact that have diameters equal to D(i) and D(j) and are located atX ið Þ and X jð Þ. The particle i is moving with linear and angular veloci-ties equal to V ið Þ and ω(i), respectively. Similarly, the particle j ismoving with linear and angular velocities equal to V jð Þ and ω(j),respectively. The normal overlap between the particles is calculatedas

δn ¼ 0:5 D ið Þ þ D jð Þ� �

− Xið Þ−X

jð Þ��� ���: ð8Þ

The unit vector along the line of contact pointing from particle i toparticle j is

ηij ¼X

jð Þ−Xið Þ

X jð Þ−X ið Þ�� �� ; ð9Þ

and the relative velocity of the point of contact becomes

Vij ¼ Við Þ−V

jð Þ þ L ið Þω ið Þ þ L jð Þω jð Þ� �

×ηij ; ð10Þ

where L(i) and L(j) are the distance to the contact point from the cen-ter of particles ith and jth, respectively. They are given by

L ið Þ ¼X

jð Þ−Xið Þ

��� ���2 þ r ið Þ2−r jð Þ2

2 X jð Þ−X ið Þ�� �� ; ð11Þ

and

L jð Þ ¼ Xjð Þ−X

ið Þ��� ���−L ið Þ

; ð12Þ

where r(i)=0.5D(i) and r(j)=0.5D(j) are the particle radii.The normalVnij and tangentialVtij components of contact velocity,

respectively, are

Vnij ¼ Vij·ηijηij ≡ Við Þ−V

jð Þ� �

·ηijηij ; ð13Þ

and

Vtij ¼ Vij−Vij·ηijηij : ð14Þ

The tangent to the plane of contact tij is

tij ¼Vtij

Vtij

�� �� : ð15Þ

In soft-sphere approach, the overlap between the two particlesis represented as a system of springs and dashpots (Fig. 2) in boththe normal and tangential directions. The spring causes the reboundoff the colliding particles and the dashpot mimics the dissipation ofkinetic energy due to inelastic collisions. The spring stiffness coeffi-cients in the tangential and normal directions are kt and kn, respec-tively. Similarly, the dashpot damping coefficients in the tangentialand normal directions are ηt and ηn, respectively. For a given parti-cle the spring stiffness and dashpot damping coefficients essentiallydepend on the solids-phase the particle belongs to. For example, if

1 Like for the spring stiffness and dashpot damping coefficients, the friction coeffi-cient μm‘ will also depend on the solids-phases the colliding particles belong to. How-ever, for the sake of clarity, the subscripts are omitted in favor of just μ.

Fig. 2. Schematic of the spring-dashpot system used to model particle contact forces insoft-sphere approach.

125R. Garg et al. / Powder Technology 220 (2012) 122–137

the ith particle belongs tomth solids-phase and the jth particle belongs to‘th solids-phase, then the spring stiffness coefficients are given by knm‘

and ktm‘. Similarly, the dashpot damping coefficients are given by ηnm‘

and ηtm‘. However, in order to keep the formulation simple, the sub-scripts m; ‘ð Þ are dropped and it is noted that the spring stiffness anddashpot damping coefficients will depend on the solids-phases the col-liding particles belong to.

The normal and tangential components of the contact force Fij , attime t, are decomposed into the spring (conservative) force FS

ij andthe dashpot (dissipative) force FD

ij as

Fnij tð Þ ¼ FSnij tð Þ þ F

Dnij tð Þ; ð16Þ

and

Ftij tð Þ ¼ FStij tð Þ þ F

Dtij tð Þ: ð17Þ

The normal spring force FSnij at any time during the contact is

calculated based on the overlap δn between the particles and isgiven by

FSnij ¼ −knδnηij : ð18Þ

For tangential spring force, a time history is maintained that be-gins once the contact initiates. At any time during the contact, thetangential spring force is given by

FStij ¼ −ktδt ð19Þ

where δt is the tangential displacement. At the initiation of the con-tact the tangential displacement is calculated as

δt ¼ Vtijminδnj j

Vij·ηij

;Δt

!: ð20Þ

while at any time (t+Δt) the tangential displacement is calculatedas

δt t þ Δtð Þ ¼ δt tð Þ þVtijΔt: ð21Þ

In the above expression the accumulated tangential displacementδt(t) at time t will not necessarily lie on the tangent plane at t+Δt.Therefore, the above expression for tangential displacement is furthercorrected to ensure that the tangential displacement lies in thecurrent tangent plane. The corrected tangential displacement is

obtained by subtracting the normal component of δt(t+Δt) fromδt(t+Δt) itself, which is given as

δt t þ Δtð Þ ¼ δt t þ Δtð Þ− δt t þ Δtð Þ·ηij

� �ηij : ð22Þ

For the case of finite Coulomb friction between particles,1 if thefollowing holds at any time during the contact,

Ftij

��� ��� N μ Fnij

��� ���; ð23Þ

then the sliding is assumed to occur and the tangential contact forceis given by

Ftij ¼−μ Fnij

��� ���tij if tij≠0

−μ Fnij

��� ��� δtδtj j if tij ¼ 0; δt≠0

0 otherwise:

8>>><>>>:

ð24Þ

It is important to note that the ith particle in the contact i–j pairexperiences a contact force equal toFij and the jth particle, accordingto Newton's third law of motion, experiences an equal and oppositecontact force (i.e. −Fij). Therefore, the total contact force F

ið Þc tð Þ at

any time on the ith particle is given as

Fið Þc tð Þ ¼ ∑

j ¼ 1j≠i

NF

Sij tð Þ þ F

Dij tð Þ

� �ð25Þ

and the total torque acting on ith particle is calculated by

Tið Þ tð Þ ¼ ∑

j ¼ 1j≠i

NL ið Þηij×Ftij tð Þ� �

: ð26Þ

2.2.2. Relationship between dashpot coefficients and coefficients ofrestitution

For collisions between particles belonging to the mth and ‘th

solids-phases, the normal dashpot damping coefficient ηnm‘ is relatedto the normal coefficient of restitution enm‘ [17,18] by

enm‘ ¼ exp −ηnm‘t

coln;m‘

2meff

!; ð27Þ

where meff ¼ mmm‘= mm þm‘ð Þ is the effective mass and tcoln;m‘ is thecollision time between m and ‘ solids phases. It is given by

tcoln;m‘ ¼ πknm‘

meff− η2

nm‘

4m2eff

!−1=2

: ð28Þ

From the above two expressions, ηnm‘ is obtained as

ηnm‘ ¼2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimeffknm‘

pln enm‘j jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

π2 þ ln2 enm‘

q ; ð29Þ

and a similar expression can be written for ηtm‘.The time step Δt is typically taken to be equal to one by fifty of

the minimum collision time (i.e., Δt ¼ min tcol;m‘=50� �

).

Fig. 3. Schematic for the free fall verification case: a smooth spherical particle fallingonto a fixed wall. The forces acting on the particle during contact are also presented.

126 R. Garg et al. / Powder Technology 220 (2012) 122–137

Specification of spring stiffness coefficients in DEM simulations isnot very straightforward. If values close to the real physical valuesare chosen, then the very small time step will make these simula-tions computationally prohibitive. Therefore a value of normalspring stiffness coefficient∼105 cm/s is usually specified. Although inEq. (27) the normal spring coefficient knm‘ is defined as pairwise, inpractice a constant value (i.e., knm‘ ¼ kn;∀m; ‘) is used. Followingthe approach of [8,17], by default the tangential spring stiffness coef-ficient is set equal to two-fifths of the normal stiffness coefficient(i.e., ktm‘ ¼ 2=5kn;∀m; ‘). Similar to [8], the tangential damping coef-ficient is by default taken to be half of normal damping coefficient(i.e., ηtm‘ ¼ 0:5ηnm‘;∀m; ‘).

This finishes the discussion on the contact model used in MFIX-DEM. In the next section, details of the gas–solids momentum transferterm, and also the two options available in MFIX-DEM to compute thesame, are given.

2.2.3. Estimation of gas–solids momentum transfer term IgmConsider the ith particle, belonging to mth solids-phase, that re-

sides in kth computational cell at time t. The drag force on this particleis represented as

Fi∈k;mð Þd ¼ −∇Pg X

ið Þ� �

Vm þ β i∈kð Þm Vm

εsmvg X

ið Þ� �

−Við Þ

� �; ð30Þ

where Pg Xið Þ

� �and vg X

ið Þ� �

are the gas-phase mean pressure Pg and

velocity vg fields at the particle location, Vm ¼ πD3m

6 is the particle vol-ume, and βm

(i∈ k) is the local gas–solids momentum transfer coefficientfor particle i residing in kth cell. An explicit functional form of βm

(i∈ k) isnot known theoretically, and therefore, different correlations de-duced from experimental and numerical studies are used to modelthis term. Nevertheless, a general parametrization for βm

(i∈ k) that sub-sumes different models can be written as

β i∈kð Þm ¼ β ρm;Dm; V

ið Þ−vg Xið Þ

� ���� ���; ρg; μg� �: ð31Þ

The gas–solids momentum transfer term Igm, at xk, that enters thegas-phase momentum conservation equation (Eq. (2)) is computed as

Ikgm ¼ 1

Vk∑Nm

i¼1F

i∈k;mð Þd K X

ið Þm ;xk

� �; ð32Þ

where K Xið Þm ;xk

� �is a generic kernel with compact support that

determines the influence of the particle force at Xið Þm on a grid node

located at xk, and Vk is the geometric volume of the kth grid cell.In MFIX-DEM, there are two methods available to calculate the

above drag force. In the first method, for a particle residing in kth cell,rather than computing themean gas-phase velocity at the particle loca-tion vg X ið Þ

� �, a cell-centered value of vg is used. Similarly, rather than

using the velocity of each particle V ið Þ, a local cell averaged velocity ofthe mth solids-phase vsm is used. With this simplification, the momen-tum transfer coefficient for all particles of mth solids-phase that residein cell k is constant and has the following functional form

β ∀i∈kð Þm ¼ β kð Þ

m ¼ ρm;Dm; vsm xkð Þ−vg xkð Þ��� ���;ρg; μg� �

; ð33Þ

where xk is the center of the kth cell. Therefore, the drag force on the ith

particle belonging to solids-phase m and residing in cell k is

Fi∈k;mð Þd ¼ −∇Pg xkð ÞVm þ β kð Þ

m Vm

εsmvg xkð Þ−vsm xkð Þ� �

: ð34Þ

Under this approximation of constant drag force on all particlesresiding in a particular cell, the gas–solids momentum transfer termIkgm is estimated in kth cell as

Ikgm ¼ −εsm∇Pg xkð Þ þ β kð Þ

m vg xkð Þ−vsm xkð Þ� �

: ð35Þ

In the second method to calculate gas–solids momentum transferterm, the mean gas-phase velocity is interpolated to the particle loca-tion. Using Eq. (32), the drag force on each particle is then projectedback onto to the Eulerian gas-phase grid. However, in order to avoidthe complexities in numerical algorithm that will arise as a result offorward and backward interpolation of the gas-phase pressure field,the pressure drag force term is evaluated at the cell center (resultingin equal pressure drag force on all particles residing in a particularcell). Therefore, the gas–solids momentum transfer term Ikgm is esti-mated in kth cell as

Ikgm ¼ −εsm∇Pg xkð Þ þ 1

Vk∑Nm

i¼1

β i∈kð Þm Vm

εsmvg X

ið Þm

� �−V

ið Þm

� �K X

ið Þm ;xk

� �:

ð36Þ

The computational implementation details of the MFIX-DEM codehave been documented in the form of a user manual [19]. However,for the sake of completion, some of the key components of the MFIX-DEM code, such as, time integration schemes, neighbor search algo-rithms, and interpolation schemes, are briefly discussed in Appendix A.

3. DEM verification tests

The MFIX-DEM code is extremely complex with the interactionbetween the fluid-solver, particle-solver, collision-algorithms, bound-aries, etc. In addition, the fluid-solver is on a staggered-grid with scalarquantities solved on the cell centers while the velocities are computedon the cell faces. With all the above complexities, limited verificationmay be performed by visually comparing the code segments to theequations being solved. A series of verification studies for pure granularflows aswell as gas-particle flows were performed to probe the accura-cy of each of the units of this complex model. Additional tests may beadded in the future as they become available or as new features areincorporated.

3.1. Free falling particle

In this case, a single smooth (frictionless) spherical particle freelyfalling under gravity from its initial position bounces upon collisionwith a fixed wall. For a schematic of the problem see Fig. 3. Thetranslational motion of the particle can be described in three stages:free fall, contact and rebound. Following the work of [20], an analyticexpression for particle motion during each stage is obtained (seeAppendix B for details).

time[s]

par

ticl

e ce

nte

r p

osi

tio

n y

[cm

]

par

ticl

e ve

loci

ty v

[cm

/s]

0 0.1 0.2 0.3 0.4 0.50

10

20

30

40

50

-300

-200

-100

0

100

200

300

AyAvDEM yDEM v

(a)

time[s]

0 0.2 0.4 0.6

-5

-4

-3

-2

-1

0

1

2

3

4

5kn=5*10 7,e n=1.0kn=5*10 7,e n=0.9kn=1*10 7,e n=0.9kn=5*10 7,e n=0.7

(b)

Fig. 4. (a) Comparison between analytic solution and DEM results for the single particle free fall case. (a) Particle position and velocity for system with kn=5×107 dyne/cm anden=0.9. (b) Relative percent error between analytic and DEM results for four different systems. In (a) and (b) the solid vertical line, labeled tc, refers to the time of collision,while in (a) the dotted vertical line, labeled tr, refers to the time of rebound.

127R. Garg et al. / Powder Technology 220 (2012) 122–137

The analytic solution for particle position and velocity, labeledwith (A), is compared to the results obtained from DEM simulation,labeled with (DEM), in Fig. 4(a) for a system with a particle-wallspring coefficient kn=5×107 dyne/cm and a particle-wall restitu-tion coefficient en=0.9. Note that ηn can be found using Eq. (29)and knowing kn, en, and the effective mass meff, which for particle-wall contact is simply taken as the particle mass mp. In all of theverification studies of the free fall system the following values areused: rp=10 cm, particle material density ρp=2.6 g/cm3, ho=50 cmand g=980.0 cm/s2. Differences between the DEM results andanalytic solution are difficult to discern in Fig. 4(a). Accordingly,the relative percent error in the prediction of particle position (�y)is presented in Fig. 4(b) for two values of the spring coefficient(5×107 and 1×107 dyne/cm) and three different values of the co-efficient of restitution (1.0,0.9,0.7). For any quantity Q, the relativepercent error �Q between the values predicted by DEM simulation(denoted by {Q}) and analytically expected values (denoted by QA)can be defined as

�Q ¼ 100×QA− Qf g

QA

��������: ð37Þ

For the four cases shown, the magnitude of the percent error isgenerally less than 1% during all three stages. The case characterizedby kn=1×107 dyne/cm and en=0.9 is the exception. In this particu-lar case, the particle center nearly touches the fixed boundary(y→0); as a result, the percent error in particle center position isrelatively large during the contact stage as the absolute values of yapproach zero. This case also has both the smallest spring constant(softest particle) and the largest time step for the DEM simulations.

A few reasons for some of the discrepancies observed between theDEM and analytic solution are discussed. In these DEM simulations,position is updated using a first order scheme (i.e., the error in posi-tion grows during each stage with each successive time step).Other, higher order schemes could be used to update the positionwith potentially more accurate results. Errors are also introduced atthe start and end of the collision. In the DEM simulation the particleposition will be advanced such that its edge will overlap with thewall before the contact (collision) is detected, and as a result, theparticle motion is still considered as freely falling even though it isin contact with the wall. Similarly, the particle is advanced a finitedistance beyond the wall while still being considered in contactwith the wall. Either of these errors may be mitigated by using

smaller time steps, which in the current DEM code is achieved byusing larger spring constants or restitution coefficients closer to 1.

Thus far the discussion has focused on comparing the simulationresults to the analytical solution from a soft-sphere collision model.This comparison also serves to verify the implementation of themodel in the MFIX code and reflects the accuracy of the integrationmethod. As evident by Fig. 4(b), the first-order time-stepping methodappears to be sufficient for this case (errors less than 1%).

In addition to the soft-sphere comparisons, the simulation resultsmay also be compared to the analytic solution from a hard-spherecollision model. The hard-sphere model does not involve a contactstage (collisions are assumed instantaneous). If the particle is droppedfrom an initial height ho, (recall ho= particle center position) then themaximum height it reaches after its first collision with the wall isen2(ho−rp)+ rp. A general expression for the maximum height of

the particle center attained after k collisions is

hmax;k ¼ ho−rp� �

e2kn þ rp: ð38Þ

Fig. 5(a) shows the evolution of hmax,k obtained from DEM simula-tion (denoted DEM) compared with the above analytical expression(denoted A) for different values of en. The problem setup is the sameas what was used in Fig. 4 with rp=10 cm, ρp=2.6 g/cm3, ho=50 cmand g=980.0 cm/s2. The relative percent error in hmax,k (�hmax, k

) isshown in Fig. 5(b). This “error” is really a reflection of the differencebetween the hard-sphere and soft-sphere collisionmodels (note thaterrors associated with time-stepping were already demonstrated inFig. 4(b) to be minimal in this case). In the limit of the hard-spheremodel (increasing the spring constant) the difference between thetwo models will decrease, as is demonstrated in Fig. 5(b) foren=0.9. However, to accurately capture collisions in such a limitrequires increasingly small time steps, and in turn, increased com-putational time. The error is minimal for the purely elastic case. Incontrast, for inelastic collisions the error may exhibit a local maxi-mum before the particle comes to a rest at which point the errorremains constant.

3.2. Two stacked particles compressed between two boundaries

This case study is based on the work of [20] and consists of a systemof two stacked particles placed between two fixed walls, as shown inFig. 6, so that they are compressed. The particles are of equal radiusbut may have differing densities. The lower wall is placed at y=0, the

kth bounce [-] kth bounce [-]

hm

ax,k

[cm

]

0 2 45

10

15

20

25

30

35

40

45

50

55

en=1.0 (A)en=0.9 (A)en=0.7 (A)en=0.5 (A)en= 1.0 (DEM)en= 0.9 (DEM)en= 0.7 (DEM)en= 0.5 (DEM)

(a)

erro

rin

hm

ax,k

[%]

0 2 4 6 8 10 12

0

2

4

6

8

kn=5*107, en=1.0kn=5*107, en=0.9kn=5*109, en=0.9kn=5*107, en=0.7kn=5*107, en=0.5

(b)

Fig. 5. Comparison between the analytic solution from a hard-sphere model and the DEM results for a freely falling particle under gravity. Evolution of (a) hmax, k (the maximumheight attained after k collisions with a wall) and (b) relative percent error (Eq. (37)) between analytic and DEM results for different values of normal coefficient of restitutionen (in (a), a constant value of kn=5×107 is used for all cases) and normal spring stiffness coefficient kn.

128 R. Garg et al. / Powder Technology 220 (2012) 122–137

upper wall at y=3.6rp=yw, and the initial position of the two particlesis y1o=y(t=0)=0.25yw and y2o=y(t=0)=0.75yw. In this setup theparticles and thewallswill remain in contact at all times so that the con-tact spring force will always be in compression. The differential equa-tion of motion for this system is examined and a numerical solutionobtained (see Appendix C for details) which is then compared withthe results from the DEM simulation.

In all of the verification studies of the two stacked particle sys-tem the following values are used: rp=0.05 cm, ρp1=20 g/cm3,ρp2=10 g/cm3, kn=knw=1×106 dyne/cm, and g=980.665 cm/s2.Thus, the lower particle is twice as dense as the upper particle(m1=2m2). Two coefficients of restitution are tested, a perfectlyelastic case (en=1) and a slightly inelastic case (en=0.8). (Recallthat the damping coefficients are determined using Eq. (29) andknowing the spring constants, the restitution coefficients, and theeffective masses).

The numerical solution for each particles position, labeled (A), iscompared to the results obtained from DEM simulation, labeled(DEM), in Fig. 7(a)–(d) for both restitution coefficients examined.Differences between the DEM results and numerical solution aredifficult to discern. Accordingly, the relative percent error in theprediction particle position (�y) is presented in Fig. 8. For the twocases shown the magnitude of the percent error is generally less

Fig. 6. Schematic for the two stacked particle system verification case: two smoothspherical particles stacked between two fixed walls so that the system is alwaysunder compression. The various forces acting on particle 1 and on particle 2 are alsoindicated.

than 0.2% indicating that the DEM results agree well with thenumerical solution.

The previous two test cases targeted verification of the normalspring-dashpot model for particle–wall and particle–particle collisions.Since both cases assume smooth surfaces, implementation of thetangential spring-dashpot model along with Coulomb friction wasnot verified. Therefore, the next test case targets verification of thetangential spring dashpot model and Coulomb friction by examininga spherical particle slipping on a rough surface.

3.3. Ball slipping on a rough surface

As shown by the schematic in Fig. 9, a spherical ball with finitetranslational velocity but zero angular velocity is released on a roughsurface. As a result of finite slip at the point of contact between theball and rough surface, sliding friction will act in the direction shownin Fig. 9. This sliding friction will reduce the translational velocity and,at the same time, generate an angular velocity until there is zero slipat the point of contact, i.e., v=ωR. After the zero slip condition isreached, sliding friction will cease to act and the solid ball will keepon moving with fixed translational and angular velocities.

From the force balance shown in the free body diagram, thenormal contact force Fn=W=mg, where W and m are, respectively,the weight and mass of the spherical ball, and g is the accelerationdue to gravity. The tangential contact force Ft, which is the forcedue to sliding friction, is equal to μmg. Therefore, the evolution equa-tions for translational and angular velocities become

dvxdt

¼ −μg; ð39Þ

and

dωdt

¼ μmgRI

; ð40Þ

respectively, where I=2/5mR2 is the moment of inertia of thespherical ball. The above equations can be integrated with the initialconditions vx;ωf gt¼0 ¼ v0;0f g, where v0 is the initial translationalvelocity of the ball. From the evolution equations for vx and ω, thetime ts at which the ball ceases to slip (i.e., vx=ωR) can be calculatedas

ts ¼2v07μg

: ð41Þ

par

ticl

e 1

cen

ter

po

siti

on

[cm

]

0 0.2 0.4 0.6 0.8 10.041

0.042

0.043

0.044

0.045

0.046

0.047

0.048 particle 1 (DEM)particle 1 (A)

(a)

par

ticl

e 2

cen

ter

po

siti

on

[cm

]

0 0.2 0.4 0.6 0.8 10.133

0.134

0.135

0.136

0.137

0.138

0.139

0.14 particle 2 (DEM)particle 2 (A)

(b)

time*0.001 [s]

par

ticl

e 1

cen

ter

po

siti

on

[cm

]

0 0.2 0.4 0.6 0.8 10.041

0.042

0.043

0.044

0.045

0.046

0.047

0.048 particle 1 (DEM)particle 1 (A)

(c)

time*0.001 [s]

par

ticl

e 2

cen

ter

po

siti

on

[cm

]

0 0.2 0.4 0.6 0.8 10.133

0.134

0.135

0.136

0.137

0.138

0.139

0.14 particle 2 (DEM)particle 2 (A)

(d)

Fig. 7. Comparison between analytic solution and DEM results for two stacked particle system. Panels (a) and (b) correspond to system with en=1.0. Panels (c) and (d) correspondto system en=0.8. The y- position for the center of particle 1 is given in (a) and (c) while the y- position for the center of particle 2 is given in (b) and (d).

129R. Garg et al. / Powder Technology 220 (2012) 122–137

The non-dimensional translational and angular velocities at ts are

v′x;ω′�

t¼ts¼ vx

v0;ωRv0

�t¼ts

¼ 57;57

�ð42Þ

Fig. 10 shows the comparison of t′=μgts/v0 (left axis), andv′x;ω′�

t¼ts(right axis) obtained from DEM simulation with the an-

alytic values for four different values of coefficient of friction with

time*0.001 [s]

par

ticle

erro

rin

y,

y (%

)

0 0.2 0.4 0.6 0.8 1-0.15

-0.1

-0.05

0

0.05

0.1

0.15particle 1particle 2

(a)

Fig. 8. Relative percent error in particle position (�y) between the numeric solution and DEMto systems with en=1.0 and en=0.8, respectively.

μ=0.2, 0.4, 0.6, and 0.8. The relative error (not shown) is alwaysless than 0.1%, indicating that the MFIX DEM code is able to accu-rately capture the sliding friction force.

The above three test cases were limited to pure DEM simulationsso that each component of the spring-dashpot model could be veri-fied independently without additional complications. In the followingsection the test cases include both gas and solids. The objective ofthese cases is to verify the gas–solids coupling mechanism. As before

par

ticle

erro

rin

y,

y (%

)

time*0.001 [s]0 0.2 0.4 0.6 0.8 1

-0.15

-0.1

-0.05

0

0.05

0.1

0.15particle 1particle 2

(b)

results corresponding to the systems presented in Fig. 7. Panels (a) and (b) correspond

Fig. 9. Schematic of the second verification problem. A spherical ball with finite trans-lational velocity and zero angular velocity is placed on a rough surface. Forces acting onthe ball are shown by the free body diagram on the right.

Fig. 11. Schematic of the advection of the particles on a circle in an oscillating vortexfield.

130 R. Garg et al. / Powder Technology 220 (2012) 122–137

the complexity of these test cases gradually increases. The aim of thefirst test case is verification of the procedure used to interpolate theEulerian gas-velocity field to the particle location wherein the accura-cy of the interpolation procedure can be quantified. An analytic formfor the gas-phase velocity field set as a time evolving single vortexfield is employed. Massless or non-interial particles are placed alonga circle (2D) or sphere (3D) and act as tracer particles that movewith the gas velocity field which is interpolated to the particle loca-tion. For all coupled cases the default method, a second-order accu-rate Lagrange polynomial (see Appendix A.3), is used to interpolatethe gas-phase velocity to the particle location. With the interpolationprocess verified, the second test case seeks to verify the implementa-tion of the drag force computation on the particles. This second caseinvolves a fixed (non-evolving) analytic form for the gas-phase veloc-ity field which is set as a 2D Taylor-Green vortex field. In this case,the particles have finite mass and the effect of particle inertia ongas–solids interaction is studied by varying the Stokes number (de-fined as the ratio of particle response time to fluid time scale). Sincethis case is not amenable to an analytic solution, the qualitative predic-tions fromMFIX-CDM are compared with the results from experimentsand past numerical studies. The two cases given above consist of one-way coupling wherein the gas-phase velocity field is not affected bythe presence of solid particles (i.e., Igm ¼ 0 in Eq. (2)). To assess theimplementation of the drag force on the gas-phase (i.e., nonzeroIgm), the last test case involves a fully coupled system. In this finalcase, the terminal velocity of a small spherical particle freely fallingunder gravity through a gas-phase is examined.

3.4. Advection of a circle and sphere in an oscillating vortex field

The objective of this test case is to quantify the accuracy of in-terpolating gas-phase velocity field at the particle location. Mass-less particles are arranged in a circle or sphere and subject to an

µ

t’

0.2 0.3 0.4 0.5 0.6 0.7 0.80.285

0.2855

0.286

0.2865

0.287

0.2875

0.71424

0.71426

0.71428

0.7143

0.71432

t’ (DEM)vx’ (DEM)

’ (DEM)t’ (A){vx, ’ (A)}

{vx, ,

, }

Fig. 10. Comparison of t′ ¼ μgtsv0

(left axis), and v′x;ω′�

t¼ts(right axis) obtained from

DEM simulation with the analytic values for four different values of the coefficient offriction.

off-centered oscillating vortex field. The particles become distortedfrom their original arrangement but then return after one cycle.This case serves to isolate the error arising from the gas velocityinterpolation routines in arbitrary directions from other possibleerrors introduced during the drag calculation. Moreover, thissetup has also been by others used to test advection algorithms[22–24].

For the 2D (circle) case a single vortex velocity field, like those fortesting algorithms for interface tracking [24] with temporal deforma-tion [22], is used. The particles are seeded in a circle of radius 0.15 off-centered at (0.5, 0.75) in a unit square box (see the schematic inFig. 11). The gas-phase vortex field is created by setting the x and ycomponents of the gas velocity as

u ¼ 2sin2 πxð Þsin 2πyð Þcos πt=Tð Þ;v ¼ −sin 2πxð Þsin2 πyð Þcos πt=Tð Þ:

ð43Þ

The particles in the vortex are sheared from their circular arrange-ment in arbitrary directions (because the center of the particles isoff-center to that of the vortex). The degree of deformation will de-pend on the value of T, the oscillation period/cycle. In this test caseT is set to 0.25 so that the particles will undergo small deformations.A qualitative assessment is made by comparing the graphical resultsover several periods. Fig. 12 shows the particles along with gas ve-locity vectors at t=0,T/5,2T/5,3T/5,4T/5,T, 2T, and 16T. As can beseen in this case, the particles are deformed at t=T/5,2T/5,3T/5,and 4T/5 and restored to original location at t=T, 2T, and 16T asexpected.

The error is evaluated by comparing differences in data observedbetween t=0 and at even multiples of T (i.e., t=T, 2T, 16T). TheL∞(|x|∞=max|xi|) error as a function of the number of cycles is plot-ted in Fig. 13. The error is under 2.5×10− 5 after 15 cycles indicatingthat the interpolation schemes work well in 2D. Although not per-formed here, a similar analysis could also be conducted in 3D.

Since massless/tracer particles were used, implementation of thedrag force computation on the particles was not tested. Therefore,the goal of the next test case is to verify the drag force computation.To this end, the effect of particle interia on gas–solids interaction isinvestigated and the results from MFIX-CDM are compared withthose from past experiments and numerical studies.

(a) t = 0 (b) t =T/5 (c) t = 2T/5

(d) t = 3T/5 (e) t = 4T/5 (f) t =T

(g) t = 2T (h) t = 16T

Fig. 12. Figure showing the gas velocity vectors along with particle locations at t=0,T/5,2T/5,3T/5,4T/5,T, 2T, and 16T.

131R. Garg et al. / Powder Technology 220 (2012) 122–137

3.5. Particle motion in vortex

To qualitatively verify the gas–solids coupling mechanism this casestudies the motion of particles with finite mass in a 2D vortex gasfield. A non-evolving 2D Taylor-Green vortex flow field is employed

ug ¼ −cos kxxð Þsin kyy� �

;

vg ¼ sin kxxð Þcos kyy� �

;ð44Þ

Fig. 13. L∞ error as a function of the number of cycles of the imposed oscillating vortexfield.

where kx and ky are the wavenumbers of Taylor-Green vortices. Note,the gas-phase velocity field is constant and not affected by the presenceof solid particles.

In the absence of gravity, the extent of gas–solids interaction isquantified by the particle Stokes number St, which is defined as theratio of the particle response/relaxation time to the fluid flow timescale (time available for particle–fluid interaction) or

St ¼T p

T g: ð45Þ

In the above equation, T p is the particle relaxation/response timedefined as

T p ¼ρpdp

2

18μg; ð46Þ

and T g is the flow time scale which is defined as

T g ¼ LU; ð47Þ

where L and U are the characteristic length and characteristic velocityof the flow, respectively.

A very small particle Stokes number implies a very small particleresponse time. In this case, the drag force has a dominant role andparticles instantaneously relax to the carrier gas-phase velocity field.At the other extreme, a very large particle Stokes number implies highly

132 R. Garg et al. / Powder Technology 220 (2012) 122–137

inertial particles and the drag force does not affect the particles trajecto-ries. As a result, particles having high Stokes number do not deviatemuch from their original trajectories. In the context of the previoustest case with massless particles, the particle Stokes number was zero.In this test case, the dependence of the gas–solids interaction on theparticle Stokes number is assessed, albeit qualitatively.

For the simulation setup, a unit square square domain with periodicboundaries is used. Particles with diameter and density of 100 μm and1800 kg/m3, respectively, are uniformly seeded throughout the compu-tational domainwith zero initial velocities. The initial solid volume frac-tion is equal to 0.05. Different particle Stokes numbers are obtained byvarying the gas viscosity, which in turn, changes the particle responsetime T p (see Eq. (46)). The drag force on particles is computed usingthe Wen and Yu drag correlation [25].

Fig. 14 shows snaphots of the flow streamlines and the solid parti-cles in the Taylor-Green vortex field for four different particles Stokesnumbers. At the lowest Stokes number of 0.002 (Fig. 14(a)) the particlespatial distribution remains nearly uniform, which again, is because ofthe instantaneous response of the particles to the gas velocity field forSt=0.002. So that the particles act like tracer particles and followalong the flow streamlines. As the particle Stokes number approachesone (i.e., St∼O 1ð Þ, Figs. 14(b) and (c)) the particles are observed topreferentially concentrate around the edge of the vortices. This be-havior can be explained by referring back to the Stokes number def-inition in Eq. (45). Specifically, the gas flow time scale is smallest atthe vortex core and highest at the edge so that the local particleStokes number decreases from core to the edge of the vortex. There-fore, particles are too inertial to follow the small scale vortices near

(a) St = 0.002

(c) St = 2

Fig. 14. Snapshot of solid particles in Taylor-Green vortex for different Stokes numbers.

the core. At the edges, however, the flow time scales are approxi-mately equivalent to the particle response time (St∼O 1ð Þ) and highcentrifugal forces inside the vortex push the particles outwardwhich then accumulate at the vortex edges.

At the highest particle Stokes number considered, St=20, thespatial distribution of particles remains uniform and unchanged fromthe initial distribution (Fig. 14(d)). However, unlike St=0.002 casewhere the particles behave as tracers andmove along flow streamlines,here the particle velocities remain nearly unchanged from the initialcondition of rest. These observations are similar to those made in pastexperimental [26] and numerical studies [27–29].

In this test case the drag force computation on solid particleswas qualitatively verified. Recall, however, that the gas-phase velocityfield was not influenced by the solid particles. In typical CDM simula-tions the gas-phase is affected by the presence of solid particles. There-fore, the next test case targets the implementation of the drag forceacting on the gas-phase.

3.6. Particle terminal velocity

In this verification test case the implementation of the gas–solidscoupling (through interphase drag force) is examined. The velocityof a very small spherical particle falling downward under gravitythrough a gas-phase flowing in the upward direction evolves by

dvp

dt¼

g ρp−ρg� �

ρp−3

4

ρg vp−vg

��� ���2dpρp

Cd; ð48Þ

(b) St = 0.2

(d) St = 20

Solid lines represent the gas-flow streamlines and dots represent the solid particles.

133R. Garg et al. / Powder Technology 220 (2012) 122–137

where dp is the particle diameter,g is the gravitational acceleration, ρpand ρg are the densities of particle and gas, respectively, and Cd is thedrag coefficient. The drag coefficient Cd is estimated from the [30]drag correlation for single particle in an unboundedmedium, which is

Cd ¼ 24Re

1þ 0:15Re0:687� �

ð49Þ

where Re is the Reynolds number based on the slip velocity between

the particle and the gas-phase, and is defined as Re ¼ρg vp−vg

��� ���dpμg

.

Note that in the equation for the evolution of the particle velocity, avery small particle size is assumed so that the pressure from thedrag force may be neglected.

In this example, the particle diameter (dp) is 100 μm, the densityρp is 2000 kg/m3 and the initial particle velocity is set equal to zero.The gas-phase is set to flow upward at ug=0.4 m/s with densityρg=1.2 kg/m3, and viscosity μg=1.8×10−5 Pa·s. Given the initialconditions and properties of the particle and gas flow, Eq. (48) canbe solved numerically (using the above form for Cd) to obtain the par-ticle velocity at any time. When the weight of the particle is exactlybalanced by the upward buoyancy and drag forces, the terminal ve-locity is reached which is obtained from the numerical solution ofEq. (48).

Two different scenarios are considered in this verification case. Inthe first case (referred to as decoupled case), the gas-phase velocity isfixed and does not evolve. In the second case (referred to as coupledcase), the gas-phase velocity also evolves and is affected (throughinterphase drag force) by the presence of freely falling particle.The evolution of particle velocity obtained from MFIX-CDM for thetwo cases (decoupled and coupled) are compared with the numeri-cal solution of Eq. (48) in Fig. 15. The particle velocity obtained fromMFIX-CDM compares well with the numerical solution for bothcases considered. While this test is very simple and limited due tothe weak interphase drag force which does not cause any significantchanges to the gas-phase velocity field, it does help to more fullyverify the gas–solids coupling implementation in MFIX-CDM.

4. Conclusions

Multiphase flows are prevalent in a range of natural phenomenonand various industrial processes. While these flows are practically

Time (s)

Par

ticl

e ve

loci

ty (

m/s

)

0 0.2 0.40

0.01

0.02

0.03

0.04

0.05

0.06

0.07

num. sol.DecoupledCoupled

Fig. 15. Comparison of the particle velocity evolution obtained from MFIX-DEM withthe numerical solution of Eq. (48). Properties of the system are ρg=1.2 kg/m3,μg=1.8×10−5Pa·s, ug=0.4 m/s, ρp=2000 kg/m3, and dp=100 μm.

important, they are also extremely complex and this is largely dueto their multiscale nature: span multiple time and length scales. Asdiscussed in the introduction, many Computational Fluid Dynamic(CFD) codes have been developed that attempt to predict the hydro-dynamics and related characteristics of multiphase flows in order toprovide insights into the systems. These codes may differ in theirmathematical modeling approaches and/or solution techniques. Thefocus of this paper has been on the continuum discrete methodologyof the open-source codeMFIX, that is, MFIX-DEM. The underlying theo-ry (e.g., governing equations and physical models) was presented first,followed by the numerical implementation and a series of verificationtests.

We conducted detailed and systematic studies targeting the im-plementation of the normal collision model and the time steppingalgorithm (see Cases 1 and 2 involving a freely falling particle andtwo stacked particles). Case 3 (ball slipping) targeted implementa-tion of the tangential force model. The next two test cases wereagain more complex and were designed to target the interpolationroutines, which are used when the particles and the fluid are coupled.The final case (terminal velocity) was slightly more complex than theprevious cases and served as a relatively simple test of the drag force.For this case, the code was invoked both with and without coupling tothe fluid phase. All of these cases demonstrated fairly good agreementwith the corresponding analytic solution (when available) or yieldedthe anticipated behavior for the problem.

Practically speaking, full verification is not possible (one cannotprove that numerical formulation and corresponding code is free ofbugs), however, an acceptable level of confidence in the CDM modelwas pursued. Additional test cases may be developed in the futureand applied to test other aspects of the code. Moreover, as the codemay continue to evolve and as new features are added, aptly designedtest cases will be needed to verify the new code.

While verification has been the primary focus, validation is anoth-er important step prior to using the model for physical insights. Vali-dation refers to the process of assessing the ability of a (verified)model to accurately predict the physical phenomena observed ex-perimentally [12]. Good validation involves testing the modelagainst data for a wide range of conditions. Context, however, is alsoimportant in the validation process as different applications may re-quire different degrees of validation. For example, a model may be val-idated on the basis of showing correct trends but not for purposes ofengineering design. Like verification, full validation is not practicallypossible as some future experiment may show deficiencies in themodel which had previously gone undetected [12]. The current effortdid not include any validation studies and that is performed in part IIof this paper.

Acknowledgments

This technical effort was performed in support of the National Ener-gy Technology Laboratory's ongoing research in advanced numericalsimulation of multiphase flow under the RES contract DE-FE0004000.Theworkwas partly performed at the Oak Ridge National Laboratory,which is managed by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725. The authors thank Drs. Sofiane Behyahia, Tom O'Brien,Chris Guenther and Madhava Syamlal at National Energy TechnologyLaboratory, and Dr. Shankar Subramaniam at Iowa State University forhelpful discussions. RG and TL were supported in part by an appoint-ment to the U.S. Department of Energy (DOE) Postgraduate ResearchProgram at the National Energy Technology Laboratory administeredby the Oak Ridge Institute of Science and Education (ORISE).

Appendix A. Computational details

In this section, we will give you a short overview of the computa-tional implementation of the physics given above in the MFIX-DEM

134 R. Garg et al. / Powder Technology 220 (2012) 122–137

code. This information serves as a starting point for understandingthe numerical methods, the code structure, and the implementation.A more detailed Doxygen output can be downloaded from the MFIXwebsite for those who wish to dig deeper into the code (https://mfix.netl.doe.gov/documentation/dem_refman.pdf). In addition, anHTML version of Doxygen output can be found at https://mfix.netl.doe.gov/ members/develop/doxygen_docs/dem/main.html so that onecan browse through the code easily.

The details of the numerical methods employed for the continuumpart of the MFIX-DEM are not discussed here as they are widely cov-ered elsewhere [2]. The important computational aspects of the DEMmethod for coupled gas-granular flows include time-integration,neighbor search algorithms, and interpolation of the continuumquantities to the discrete particle locations and vice-versa. More de-tails on the time integration method are provided below while theneighbor search algorithm is discussed in Appendix A.2.

Appendix A.1. Time integration

Time integration in DEM is one of the most widely researched areain the broad area of molecular dynamics [31–33]. A stable, efficientand energy preserving time-integration scheme is desirable. Limitedtime-integration options are currently available in MFIX-DEM andthis is open to further extension in the future. The default time inte-gration scheme is a first-order technique. In the first-order scheme,the translational velocity, particle center position, and the angularvelocity at time t+Δt are obtained from values at time t by

Við Þ t þ Δtð Þ ¼ V

ið Þ tð Þ þ Fið ÞT tð Þm ið Þ Δt; ðA:1Þ

Xið Þ t þ Δtð Þ ¼ X

ið Þ tð Þ þVið Þ t þ Δtð ÞΔt; ðA:2Þ

and

ω ið Þ t þ Δtð Þ ¼ ω ið Þ tð Þ þTið Þ tð ÞI ið Þ Δt; ðA:3Þ

respectively, where Fið ÞT and T

ið Þ are the total force and torque actingon the particle (cf. Eqs. (6) and (7)).

Recall that the time-step in MFIX-DEM's soft-sphere approach isbased on spring stiffness coefficient, so it is usually very small (smallparticle–particle overlap). Subsequently, the first-order time steppingscheme is generally a good option, as it is fast and has less memoryrequirements than other schemes [34]. It should be noted that thefirst-order scheme can suffer from poor energy conservation. How-ever, energy conservation is not as crucial of an issue in gas–solidssystems as it is in molecular or pure granular systems due to thepresence of non-conservative forces such as inter-phase drag forceand inelastic collisions [35].

Appendix A.2. Neighbor search algorithm

One of the most important and time consuming component of anyparticle-based simulations is the neighbor search algorithm. Variousneighbor search algorithms are implemented in MFIX-DEM codewhich can be turned on by specifying appropriate flags in the inputfile. The users can choose between N-square, Quadtree, Octree andCell-linked list neighbor search algorithms. The complete details ofthe neighbor search algorithm can be found in the MFIX-DEM docu-mentation [19].

Appendix A.3. Interpolation algorithms

As discussed earlier, in MFIX-DEM the drag force on particles iscomputed either by interpolating the mean gas-phase velocity at

the particle location (Eq. (30)) or by using the cell centered value ofthe mean gas-phase velocity (Eq. (34)). While the latter approach istrivial, the interpolation of gas-phase velocity field at particle locationis further explained in this section. For the Eulerian representation ofgas-phase, MFIX uses a staggered grid finite volume scheme [36], asshown in Fig. A.16 for a 2D grid. In this schematic, the outer cellsenclosed by dashed lines represent the ghost cells. Horizontal andvertical arrows show the x- and y- components of the velocity field,respectively. Solid dots represent the scalar grid where scalar quanti-ties, such as pressure field, gas voidage, etc., are computed.

In the staggered grid scheme, each velocity component has itsown grid. Therefore, interpolation of the gas-phase velocity at theparticle location requires calculating a basis function (or spatialweight) for each component of the gas-phase velocity field resultingin two separate calculations in 2D or three calculations in 3D. Thisprocedure becomes computationally very expensive when scaledwith the total number of particles in the system. A much less expen-sive approach involves first computing all components of the gas-phase velocity to the same grid, referred to as the interpolation grid.This approach is taken in MFIX-DEM. Specifically, the gas-phase ve-locity components are first calculated to the interpolation grid asshown by crosses in the 2D schematic in Fig. A.16. and again in thezoomed in schematic for a single cell in Fig. A.17. Assuming a uniformgrid, the x- and y- components of gas-phase velocity (represented asμg and vg) at location (I, J) in Fig. A.17 are computed as

μg I; Jð Þ ¼μg i−1; jð Þ þ μg i−1; j−1ð Þ

2;

vg I; Jð Þ ¼vg i; j−1ð Þ þ vg i−1; j−1ð Þ

2:

ðA:4Þ

Similar expressions can be written for 3D. The notation ug(i, j)implies the x- component of gas velocity at the east face of cell(i, j) (see Fig. A.17), while ug(i−1, j) refers to the x- componentof gas velocity at the east face of cell (i−1, j) (cell not shown).Likewise, vg(i, j) represents the y- component of gas velocity atthe north face of cell (i, j) (cell not shown), and vg(i, j−1) the y-component of gas velocity at the north face of cell (i, j−1). Thegas-phase velocity from the interpolation grid (crosses) is then in-terpolated at the particle location. This last step follows a verystandard procedure and the interested reader is referred to [37]for more details. While various interpolation schemes are availablein MFIX-DEM, the default scheme is a second-order accurate La-grange polynomial.

Appendix B. Analytical solution: freely falling particle case

Appendix B.1. Stage I: free fall

The expression describing the particle motion during the free fallis obtained from the force balance on the particle as

y ¼ −g ðB:1Þ

y ¼ ∫ ydt ¼ −gt ðB:2Þ

y ¼ ∫ ydt ¼ ho−12gt2 ðB:3Þ

with the initial conditions y t ¼ 0ð Þ ¼ 0 and y(t=0)=ho. In theseequations g is the acceleration due to gravity, ho is the initial distanceof the particle center from the wall, y is the particle's center positionwith respect to the wall, y is the velocity, and ÿ is the acceleration.

1 2 53 4 7

1

2

3

4

5

6

6 8 9 10

7

8

9

10

Interpolation grid

Scalar grid

u grid

v grid

Fig. A.16. 2-D Schematic of the staggered grid scheme used for discretizing the gas-phase field in MFIX.

135R. Garg et al. / Powder Technology 220 (2012) 122–137

Appendix B.2. Stage II: contact

The time at which the particle contacts the fixed wall (tc) signifiesthe end of the free fall stage and the beginning of the contactstage. This time corresponds to the particle center position equal tothe particle radius (i.e., y(t= tc)=rp) and its value can be found viaEq. (B.3):

tc ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ho−rp� �

=g

r: ðB:4Þ

Using the expression for tc and Eq. (B.2), the velocity just prior tocontact can be described as

vc ¼ −ffiffiffiffiffiffiffiffiffiffiffi2gho

q¼ −

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2g yo−rp� �r

: ðB:5Þ

In this case, the particle–wall collision is treated using a soft-sphere approach, specifically the linear spring-dashpot model

(i,j

(I,J) S

W

Fig. A.17. Schematic of 2D staggered grid

discussed earlier. Accordingly, the expression for particle accelerationduring contact is given by

y ¼ −g− knmp

y−rp� �

− ηn

mpy: ðB:6Þ

For convenience the terms β ¼ ηn=2ffiffiffiffiffiffiffiffiffiffiffiffiknmp

pandωo ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffikn=mp

pare

introduced so that Eq. (B.6) can be rewritten and rearranged as

yþ 2βωo yþω2oy ¼ ω2

orp−g: ðB:7Þ

The solution to this equation depends on the value of β. For βb1(under damped system) the expression describing particle motionduring contact is

y ¼

gω2

ocos

ffiffiffiffiffiffiffiffiffiffiffiffiffi1−β2

qωot

� þ

−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2g ho−rp� �r

þ βgωo

ωo

ffiffiffiffiffiffiffiffiffiffiffiffiffi1−β2

p sinffiffiffiffiffiffiffiffiffiffiffiffiffi1−β2

qωot

�

2666664

3777775exp −βωotð Þþ rp−

gω2

o

� ;

ðB:8Þ

)E

N

scheme zoomed in to the cell level.

136 R. Garg et al. / Powder Technology 220 (2012) 122–137

with the initial conditions y t ¼ 0ð Þ ¼ vc and y(t=0)=rp. The particlevelocity is

y ¼

−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2g ho−rp� �r

cosffiffiffiffiffiffiffiffiffiffiffiffiffi1−β2

pωot

� �þ

βωo

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2g ho−rp� �r

−g

ωo

ffiffiffiffiffiffiffiffiffiffiffiffiffi1−β2

p sinffiffiffiffiffiffiffiffiffiffiffiffiffi1−β2

qωot

�

2666664

3777775exp −βωotð Þ: ðB:9Þ

Appendix B.3. Stage III: rebound

The time at which the particle is no longer in contact with thefixed wall (tr) signifies the end of the contact stage and the beginningof the rebound stage. This time corresponds to when the particle cen-ter position is equal to the particle radius (i.e., y(t= tr)=rp) and itsvalue can be found via Eq. (B.8). Using the value of tr and Eq. (B.9),the velocity at the end of the contact stage (vr) can be found. For par-ticle motion during rebound the same starting equation is used as forfree fall, but the initial conditions differ:

y ¼ ∫ ydt ¼ −gt þ vr ðB:10Þ

y ¼ ∫ ydt ¼ rp þ vrt−12gt2 ðB:11Þ

with initial conditions y t ¼ 0ð Þ ¼ vr and y(t=0)=rp.Eqs. (B.3), (B.8) and (B.11) are used to solve for the particle posi-

tion versus time and the values of tc and tr are used to stitch the threestages (free fall, contact, and rebound) together.

Appendix C. Analytical solution: two stacked particles case

Appendix C.1. Motion of particle 1: lower particle

Referring to Fig. 6, a general expression for the acceleration of par-ticle 1 ( y1) is as follows

y1 ¼ F1b þ F1kw þ F12k þ F1dw þ F12d; ðC:1Þ

where the terms on the right-hand-side represent the various forcesacting on particle 1, specifically, F1b= gravity force, F1kw= particle 1-wall spring force, F1dw= particle 1-wall damping force, F12k =particle 1-particle 2 spring force, F12d = particle 1-particle 2 dampingforce. The expressions for each of these forces are shown below:

F1b ¼ −g; F1kw ¼ − knwm1

y1−rp� �

; F1dw ¼ −ηn1w

m1y1;

F12k ¼ − knwm1

2rp− y2−y1ð Þ� �

and F12d ¼ −ηn12

m1y1− y2ð Þ;

ðC:2Þ

where g is the acceleration due to gravity, knw is the particle–wall springcoefficient, F12k is the particle–particle spring coefficient, ηn1w is theparticle–wall damping coefficient for particle 1, ηn12 is the particle–particle damping coefficient between particles 1 and 2, m1 is themass of particle 1, rp is the particle radius, y1 is the y position of thecenter of particle 1 with respect to the lower wall, ˙y1 is the velocityof particle 1 and similarly, y2 is the y position of the center of particle2 and y2 is the velocity of particle 2.

Appendix C.2. Motion of particle 2: upper particle

A general expression for the acceleration of particle 2 ( y2) is asfollows

y2 ¼ F2b þ F2kw þ F21k þ F2dw þ F21d; ðC:3Þ

where the terms on the right-hand-side are the various forces actingon particle 2, specifically, F2b= gravity force, F2kw= particle 2-wall spring force, F2dw = particle 2-wall damping force, F21k=particle 1-particle 2 spring force, F21d= particle 1-particle 2 dampingforce. The expressions for each of these forces are shown below:

F2b ¼ −g; F2kw ¼ − knwm2

rp− yw−y2ð Þ� �

;

F2dw ¼ −ηn2w

m2y2; F21k ¼ −m1

m2F12k and F21d ¼ −m1

m2F12d;

ðC:4Þ

where ηn2w is the particle–wall damping coefficient for particle 2,m2 isthe mass of particle 2, and the other quantities are as defined earlier.

An analytic expression for themotion of each particle can readily beobtained for perfectly elastic particles (F1dw=F2dw=F12d=0) of equalmass (m1=m2). For particles of unequal mass (m1≠m2) and inelastic-ity, the problem becomes more complicated. Accordingly, a numericalsolution is found using numerical methods, specifically, using theLsode function with the default options as implemented in GNU Octave[21]. This function is designed to solve a set of differential equationswith the form dy

dt ¼ f y; tð Þ with y(to)=yo. Therefore, the two second-order differential equations describing the system (Eqs. (C.1) and(C.3)) are re-written as a set of four first-order differential equations.The four coupled first-order differential equations are then solvedusing Lsode with the initial conditions y1 t ¼ 0ð Þ ¼ 0, y2 t ¼ 0ð Þ ¼ 0,y1(t=0)=0.25yw and y2(t=0)=0.75yw.

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