Chemical Engineering Science 102 (2013) 76–86

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Chemical Engineering Science

0009-25http://d

n CorrE-m

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

An analytical model to describe the motion of a low concentrationof spherical particles within a Newtonian fluid

J.M. Wilms a,n, G.J.F. Smit b, G.P.J. Diedericks b

a Mechanical and Mechatronic Engineering, University of Stellenbosch, Private Bag X1, 7602 Matieland, South Africab Division of Applied Mathematics, University of Stellenbosch, Private Bag X1, 7602 Matieland, South Africa

H I G H L I G H T S

G R A P H I C A L A

� Closure of the fluid–solid drag with aRepresentative Unit Cellmodel (RUC).

� Derivation of a particle viscosityterm for modelling of particle–parti-cle interaction.

� Numerical and physical settling tubeexperiments for validation ofthe model.

09/$ - see front matter & 2013 Elsevier Ltd. Ax.doi.org/10.1016/j.ces.2013.07.043

esponding author. Tel.: +27 763034492.ail addresses: fine.wilms@gmail.com, fine@sun

B S T R A C T

a r t i c l e i n f o

Article history:Received 13 May 2013Received in revised form23 July 2013Accepted 27 July 2013Available online 2 August 2013

Keywords:Particle interactionTwo-phase flowFluid–solid dragClosure modellingSettling-tube

a b s t r a c t

In this paper a particle–particle interaction term is derived and incorporated into the two-fluid analyticalmodel of Smit et al. (2011). This model was developed for low concentration spherical particle motion ina Newtonian fluid and the inclusion of particle interactions is required for instances where particlescollide. Moreover, such a modification serves as a first step towards the modelling of higher particleconcentrations. A brief overview of the analytical derivation of the model by Smit et al. (2011) is includedfor clarity and a detailed derivation of the newly developed particle–particle interaction term is given. Inthis derivation, particle interaction is described using impulse mechanics with a collision sphere modelin a centre of mass reference frame for collision detection. The updated model is included into an existingFortran 95 program and validated with experimental data obtained by the authors from camera- andsettling tube procedures.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The motion of discrete particles within a Newtonian fluid isfrequently modelled by treating the combination of phases as asingle fluid phase or by evaluating the constituent phases asindividual fluids. Alternatively, two-phase motion is described bymodelling each of the discrete particles with Newton's second lawand then tracking their individual motions. The first two methodsconstitute the classical Euler approach and require the specifica-tion of an empirical particle viscosity which places a limitation on

ll rights reserved.

.ac.za (J.M. Wilms).

the physical analysis of the particle motion. The latter, DiscreteElement Method (DEM), has become more popular with theincrease in computational resources but is, however, expensivein this regard and often dependent on parallel computing overmultiple processors.

The empirical nature of the mixture- or the two-fluid approachand the computational cost of tracking schemes are avoided in themethod described here. The model derived by Smit et al. (2011)uses the Navier–Stokes mass- and momentum conservation equa-tions to impart the motion of the fluid but attempts to preservethe discrete nature of the particles by constructing their mass- andmomentum conservation equations on the basis of Newton'ssecond law. Instead of tracking the particles, the equations derivedfor a single particle are averaged over a Representative Elementary

J.M. Wilms et al. / Chemical Engineering Science 102 (2013) 76–86 77

Volume (REV). The REV is a volume which should contain bothcontinuum and particle phases, and should be representative ofthe entire domain. More information on the criteria to which anREV should adhere may be found in the work of Whitaker (1967)and Bachmat and Bear (1986). The original REV integrationprocedure, applied in the averaging of the fluid phase, is, however,substituted with summation in this work to account for theparticles' disjoint nature.

Averaging of the momentum conservation equations yieldssurface integral terms. The integral terms over the phase-separating face embody momentum transfer between the phases.They emerge in each of the phases' momentum conservationequations identically but with opposite sign, thus ensuring thatmomentum is conserved over the entire domain. These terms areclosed with a Representative Unit Cell (RUC) model, which is arectangular volume of minimum dimensions into which thegeometric properties of the REV may be embedded. The RUCwas first developed by Du Plessis and Masliyah (1988) and hasbeen modified here to incorporate the relative slip velocity.

Momentum transfer may, however, also occur between theparticles themselves, due to particle–particle collisions. Interactionof this type resides in the integral term over interfaces which areestablished when particles come into close proximity of eachother. It is consequently isolated to the particle phase momentumconservation equation. The aforementioned interaction was notincluded by Smit et al. (2011) and is described here using impulsemechanics with a collision sphere model in a centre of massreference frame for collision detection.

The model is coded in Fortran 95 and the numerical velocitydata, yielded, is validated with empirical data from settling-tubeexperiments.

2. Two-phase flow model

The two-phase flow model for low concentration sphericalparticle motion through a Newtonian fluid excluding particleinteraction was discussed in Smit et al. (2011). A brief overviewis given here of the averaging procedures used and a detaileddiscussion is presented for the derivation of a particle–particleinteraction term.

2.1. The fluid phase

The REV averaging procedure for fluid quantities, ψ , which areassumed to be finite, continuous and differentiable is given by

ψ f ¼1Uf

∭Ufψ dUf ; ð1Þ

where Uf is the volume of the fluid phase within the REV. Theapplication of the averaging procedure, given by Eq. (1), yields thefollowing averaged momentum conservation equation for the fluidphase:

ρf∂ϵf vf

∂tþ ρf∇ � ϵf vf vf

� �¼ ρf gϵf�ϵf∇pf þ μf∇ � ϵf∇vf

� �þ I fs; ð2Þ

where the drag force between fluids and solids is summarised in,I fs, as

I fs ¼1Uo

ZSfs

� ~pf 1 þ τf

� �� nf dS: ð3Þ

In Eqs. (2) and (3), ρf is the density of the fluid phase; vf and vs arethe average velocities of the fluid- and solid phases, respectively; gis the gravitational acceleration; and μf is the dynamic viscosity ofthe fluid. The average fluid pressure is denoted by pf and the fluidvolume fraction is the ratio of fluid-to-total volume, given by,

ϵf ¼Uf =Uo. The shear stress between the fluid and the solid isdenoted by τ

fand the integral is taken over the fluid–solid

interface, Sfs.The averaged mass conservation equation for the fluid phase is

given by

∂ϵf∂t

þ ∇ � ϵf vf ¼ 0: ð4Þ

2.2. The solid phase

The solid phase is composed of discrete, solid, rigid particleswhich are, apart from when they collide, completely surroundedby the fluid phase. In consideration of the solid phase's disjointnature, the averaging procedure of Eq. (1) is adapted to

γs ¼1Us

∑n

i ¼ 1γiνi; ð5Þ

where Us denotes the combined solid volume which consists of allsolid particle volumes, νi, within the REV whereas γi is a propertyassociated with particle i and is defined at its centroid. Theaveraged mass conservation for the solid phase is given by

∂ϵsρs∂t

þ ∇ � ϵsvs

� �¼ 0; ð6Þ

where ρs is the density of the solid phase, the solid volume fractionis given by ϵs ¼ Us=Uo, and the average solid velocity is denoted byvs.

The momentum conservation of a single particle is given by

midvi

dt¼mig þ∑F ; ð7Þ

where mi is the mass of particle i; vi denotes its velocity and thelast term in Eq. (7) represents all external forces apart fromgravity, g . Summation of Eq. (7) over all particles in an REV andsubsequent application of Eq. (5) yields the following for theaveraged solid phase momentum equation

ρs∂∂tϵsvs þ ρs∇ � ϵsvsvs ¼ ϵsρsg�I fs þ

1Uo

ZSss

si� ni dS; ð8Þ

The particle stress is denoted by si. Following Enwald et al. (1997),Crowe et al. (1998), Soo (1990), and Kleinstreuer (2003) theparticle stress is assumed to be a linear combination of stressinduced by the surrounding continuum, sf , and stress instigatedby neighbouring particles, sss

si¼ s

fþ s

ss: ð9Þ

The particle induced stress may in itself be decomposed in africtional and a kinetic-collisional component (Enwald et al., 1997;Dartevelle, 2003). For dilute flows it is assumed that the frictionalcomponent may be dropped and it follows that the particle stressmay be written as

si¼ s

fþ s

kc; ð10Þ

where skc denotes the kinetic-collisional shear stress. FollowingEnwald et al. (1997) and Dartevelle (2003) this is composed of apressure and a shearing component. For the purpose of the currentwork it is assumed that the small grain size and dilute concentra-tions yield a particle pressure which, when compared to the shear,may be considered negligible. It follows that the particle shearstress is given by

si¼�pf 1 þ τf þ τ

kc: ð11Þ

In Eq. (11) the fluid stress has been decomposed into a pressure, pf,and a shearing component, τ

f, whilst τ

kcdenotes the kinetic-

collisional particle shear stress. The latter stress can physically

Fig. 1. Elastic two-dimensional collision with specular reflection.

J.M. Wilms et al. / Chemical Engineering Science 102 (2013) 76–8678

only occur where particles come into contact with each other. Itfollows that such a term will only exist on a particle–particleinterface, Sss. Moreover, the continuum pressure, pf, and shear, τ

f,

are defined only at interfaces and within volumes where thecontinuum phase is present. It follows that the averaged solidphase momentum equation is equivalent to

ρs∂∂tϵsvs þ ρs∇ � ϵsvsvs ¼ ϵsρsg�I fs þ

1Uo

ZSssτkc� ni dS; ð12Þ

The particle interaction effect enters Eq. (12) through theintegral expression on the right-hand side of Eq. (12). Thecombined solid–solid interface, Sss, is equal to the sum of theindividual interfaces, SssðiÞ and it follows that this effect may bewritten as the sum of particle–particle interaction forces experi-enced by each particle

I s ¼1Uo

∑ZSssðiÞ

τkc� nsðiÞ dS: ð13Þ

The integrand denotes a force, F kcsðiÞ, parallel to an incremental

surface element, dS, on which it acts and it follows that anexpression for the particle induced momentum contribution isgiven by

I s ¼1Uo

∑N

i ¼ 1F kci ; ð14Þ

where F kci is the resultant force acting on the ith particle due to

multiple collisions with its neighbours. In the remainder of thissection such a resultant particle interaction force, experienced by asingle particle, is modelled.

2.2.1. The Centre of Mass (COM) reference frame and elasticityThe centre of mass possesses the property of having a constant

velocity, unaffected by the changing motion of its parts. To anobserver, placed at the centre of mass, the velocity will appear tobe zero. This is true independent of the dimensionality or elasticityof the system. It therefore holds that, in the absence of a netexternal force, the total momentum remains zero before and aftera collision in the COM frame of reference and the velocity of theCOM, vCOM , will be unchanged. Hence,

ΔP1;x ¼�ΔP2;x; ð15Þwhere P1;x and P2;x are the x-components of the momenta ofParticles 1 and 2, respectively, and ΔPi;x ¼ Pi;x�P′

i;x denotes thechange in momentum of Particle i due to the collision. Similarly,

ΔP1;y ¼�ΔP2;y: ð16ÞThe additional assumption of full elasticity, results in an unalteredspeed for each particle before and after impact with a separateparticle, i.e.

jv1j ¼ jv′1j: ð17Þ

In a similar manner it can be shown that,

jv2j ¼ jv′2j: ð18Þ

It follows that, in an elastic collision, the speed of the individualparticles does not change, though their directions may change,depending on the shapes of the bodies and the point of impact.

2.2.2. Relative mass and velocitiesFrom Eq. (15) it follows that

Δu1 ¼1m1

ΔP1;x and Δu2 ¼� 1m2

ΔP1;x;

where Δui denotes the difference between the pre- and post-collisional values. Subtraction, yields

Δu1�Δu2 ¼1mn

ΔP1;x; ð19Þ

where mn ¼ ðm1m2Þ=ðm1 þm2Þ is commonly known as the relativemass. From Eq. (19) it follows that the change in the x-componentof momentum for Particle 2 may be written in terms of relativevelocity and mass as

mnðurelÞ�mnðu′relÞ ¼ΔP1;x; ð20Þ

where urel ¼ u1�u2 and u′rel ¼ u′

1�u′2, respectively, denote the pre-

and post-collisional x-component of the relative velocity. Eq. (20)is ratification for the observation made by Fan and Zhu (1998), Soo(1990) and Clark (2009) that the collision between two movingparticles is equivalent to the case where a particle collides withanother which possesses the same relative mass and relativevelocity.

2.2.3. Specular reflection and x-directed motionFrom Eq. (18) and under the assumption that the collisions are

specular and that the initial velocity of Particle 2 is parallel to thex-axis (i.e. jv2j ¼ u2), as shown in Fig. 1, it follows that themagnitude of the x-component of Particle 2's outgoing velocity,v′2, is given by

ju′2j ¼ jv′

2j cos ð2θÞ ¼ ju2j cos ð2θÞ: ð21ÞFrom Fig. 1 it is seen that the direction of u′

2 is opposite to u2and it follows that

u′2 ¼�u2 cos ð2θÞ: ð22Þ

Similarly, the x-component of Particle 1's velocity is given by

u′1 ¼�u1 cos ð2θÞ: ð23Þ

Combination of Eqs. (22) and (23) then yields

u′rel ¼�urel cos ð2θÞ: ð24Þ

Eq. (24) is substituted into Eq. (20) to yield the following result forthe change in the x-component of the total momentum in terms ofrelative mass and velocity

ΔP1;x ¼ 2mnurel cos2 θ: ð25Þ

The x-component of the force exerted by Particle 2 onto Particle1 is given by

f kc ¼ dP1;x

dt�ΔPx

Δt: ð26Þ

In order to derive an expression for the average force exerted dueto multiple particles colliding with one another, it is necessary toaverage the forces over a collision sphere. A discussion on aver-aging techniques, which employs the concept of a collision sphere,follows in the next section.

2.2.4. The collision sphere: a control volume formulationWhen two particles of radii r1 and r2, possessing the same

relative mass -and velocity, collide, a collisions sphere with radiusrc ¼ r1 þ r2 may be constructed to average the force exerted by oneover the other, cf. Clark (2009). The collision sphere is a control

Fig. 2. Two-dimensional view of a collision sphere formed around Particle 1.

J.M. Wilms et al. / Chemical Engineering Science 102 (2013) 76–86 79

volume analogous to the representative elementary volume (REV)and is illustrated in Fig. 2.

The volume of the collision sphere, VCV , is indicated by thedashed line in Fig. 2. One particle is considered to be the centralparticle around which the collision sphere is centred and islabelled as a particle of Type 1. Type 1 particles are made up fromparticles with radii equal to r1 where-as Type 2 particles consist ofthose particles with radii equalling r2. In Fig. 2 the Type 1 particleis taken as the central sphere. Any Type 2 particle which crossesthe border of the collision sphere will inevitably make contactwith the central sphere (Clark, 2009).

The total force experienced by a particle of Type 1 if N2 particlesof Type 2 were to cross the boundary of the collision sphere isgiven by

f kc ¼N2ΔPΔt

: ð27Þ

The volume average of f kc over the collision sphere may then becalculated by integrating over the collision sphere and is given by

F kci ¼ 1

VCV

ZVCV

N2ΔPΔt

dV: ð28Þ

It is assumed that the flow is compliant to a simple shear regimeas illustrated in Fig. 3. The velocity therefore consists of anx-component only which is entirely dependent on the y-dimen-sion (i.e. vrel ¼ uðyÞreli). The force due to the simple shear collisionsof a total of N2 Type 2 particles on a centred Type 1 particle shouldalso only have an x-component and it is for this reason that thevector notation is dropped in calculations to follow.

Following Clark (2009), Soo (1990) and Fan and Zhu (1998), thevolume integral is then reduced to a surface integral1:Eq. (28) maythus be written in terms of the x-component of the momentumchange due to a collision, as

Fkci ¼ N2

VCV

ZS ?

ΔPxurel dS? : ð29Þ

Substitution of Eq. (25) into Eq. (29), yields

Fkci ¼ N2

VCV

ZS ?

2mnu2rel cos

2 θ dS? : ð30Þ

1 The relation between the incremental volume element, dV , and the corre-sponding surface element, dS, is given by Krause (2005) as

dV ¼ ðv � nÞdSΔt:

The incremental surface area, dS? , directed perpendicular to the x-direction is shown in Fig. 4 and is given in spherical coordinates as

dS? ¼ r2 sin θ cos θ dθ dϕ; ð31Þand the shear flow may be expressed as

urel ¼∂u∂y

y¼ ∂urel

∂yr sin θ cos ϕ; ð32Þ

where r, θ, and ϕ are as illustrated in Fig. 3. It follows that Eq. (30)may be written as

Fkci ¼ N2

VCV

ZS ?

2mnðr1 þ r2Þ4∂urel

∂y

� �2

sin 3 θ cos 2 ϕ cos 3 θ dθ dϕ:

ð33ÞReferring to Fig. 3 it is seen that only the upstream half of the tophalf of the collision sphere is subjected to particle collisions by asimple shear influx of Type 2 particles. It follows that the collisionsphere volume is given by VCV ¼ 2=3πðr1 þ r2Þ3 and that integra-tion should take place over the quarter sphere, subjected to theType 2 particles, hence

Fkci ¼ 2N2

2=3πðr1 þ r2Þ3Z π=2

0

Z π=2

02mnðr1 þ r2Þ4

∂urel

∂y

� �2

� sin 3 θ cos 2 ϕ cos 3 θ dθ dϕ: ð34ÞLet the number density of a Type 2 particle cloud be the number ofparticles of Type 2 divided by the volume over which they have animpact. For the case of particles colliding with the top half spherethe number density is given by

n2 ¼N2=ð2=3πðr1 þ r2Þ3Þ: ð35ÞSubstitution of the number density into Eq. (34) yields

Fkci ¼ 4n2

Z π=2

0

Z π=2

0mnðr1 þ r2Þ4

∂urel

∂y

� �2

sin 3 θ cos 2 ϕ cos 3 θ dθ dϕ:

ð36ÞIntegration of Eq. (36) yields the force exerted by the shear flow ofa particle cloud of Type 2 on a single particle of Type 1:

Fkci ¼ π

12n2mn ∂urel

∂y

� �2

ðr1 þ r2Þ4: ð37Þ

For the case of identical particles, the relative mass is given bymn ¼mi=2. Applying this relation and expressing the mass interms of density, ρi, and volume, νi, and then rewriting in termsof the particle volume fraction, ϵs ¼∑iνi=VCV , allows for Eq. (37) tobe expressed as

Fkci ¼ νiϵsρidi4

∂urel

∂y

� �2

: ð38Þ

If the force of Eq. (38) is projected onto the half-circle perpendi-cular to the y-axis, the shear stress exerted onto a single particleby its surrounding cloud is

τkc ¼ ϵsρid2i

12∂urel

∂y

� �2

: ð39Þ

Eq. (39) bares close resemblance to the equations given byBrennen (2005) for the shear stress term derived by Haff (1983),namely

τkcHaff ¼ gs ϵsð Þρsd2s∂urel

∂y

� �2

; ð40Þ

here gsðϵsÞ is a function of the solid void fraction. Haff (1983)required gðϵsÞ to tend towards zero as ϵs approaches zero. Thefunction for gsðϵsÞ for Eq. (39) is given by

gs ϵsð Þ ¼ ϵs12

; ð41Þ

Fig. 3. Sphere of Type 1 subjected to shear flow of cloud of Type 2 particles.

Fig. 4. Projected surface element, S? .

J.M. Wilms et al. / Chemical Engineering Science 102 (2013) 76–8680

and thus the limiting condition as proposed by Haff (1983) issatisfied.

2.2.5. Particle viscosityEq. (39) may be written in a similar form as Newton's law of

viscosity

τkc ¼ μs∂urel

∂y; ð42Þ

where the particle viscosity is given by, μs ¼ ϵsρid2i =12∂urel=∂y.

The particle phase therefore exhibits non-Newtonian fluidproperties since its viscosity is not constant but a function of thedeformation tensor.

This provides a basis for the two-fluid treatment of two-phaseflow, where the particle phase is not treated as discrete but as afluid from the beginning of the derivation of the momentumexpressions. In such cases the particle viscosity is, however,chosen from one of the many empirically derived viscosityexpressions available (Enwald et al., 1997).

Eq. (42) states that the stress, τkc , is proportional to the velocitygradient, ∂urel=∂y and the constant of proportionality is the particleviscosity, μs. The two-dimensional shear stress and particle inter-action force is derived in Appendix A and stated here as

F kci ¼ ϵsρidi

4∂u∂y

þ ∂v∂x

� �∂u∂y

þ ∂v∂x

� �i þ ∂u

∂yþ ∂v

∂x

� �j

� �νi: ð43Þ

Eq. (43) is substituted into Eq. (14) and it follows that

I s ¼� 1Uo

∑N

i ¼ 1

ϵiρidi4

∂u∂y

þ ∂v∂x

� �∂u∂y

þ ∂v∂x

� �i þ ∂u

∂yþ ∂v

∂x

� �j

� �νi: ð44Þ

For a constant particle diameter and density, this may beexpressed in terms of averaging notation introduced with Eq. (5)

I s ¼�ϵ2s ρsds4

∂u∂y

þ ∂v∂x

� �� �2sn̂ ; ð45Þ

It is assumed that Eq. (45) may be cast into the following form:

I s ¼�ϵ2s ρsds4

∂us

∂yþ ∂vs

∂x

� �2

n̂ : ð46Þ

Substitution of Eq. (46) into Eq. (12) yields the followingexpression for momentum conservation of the discrete phase

ρs∂∂tϵsvs þ ρs∇ � ϵsvsvs ¼ ϵsgρs�ϵs∇pf�

ϵ2s ρsds4

∂us

∂yþ ∂vs

∂x

� �2

n̂�I fs;

ð47Þwhich concludes the constitutive modelling procedure.

2.3. Closure of the interaction between phases: representative unitcell

It was shown by Smit et al. (2011) that the open integral, givenby Eq. (3), may be closed using a secondary modelling procedureby means of a Representative Unit Cell (RUC), cf. Du Plessis andMasliyah (1988). This particle-continuum surface integral repre-sents the momentum transfer between the two phases. For agranular porous medium the average geometrical properties of thesolid structure within the RUC may be resembled by a cube of solidmaterial as shown in Fig. 5. A two-dimensional schematic of theRUC is shown in Fig. 6.

The RUC has a volume of Uo and linear dimension, d. The widthand volume of the solid is respectively given by ds and Us. The fluidvolume is denoted by Uf and is divided into stream-wise, U J , andtransverse, U? , sections. The fluid–solid interface, Sfs, is dividedinto its stream-wise, S J , and transverse, S? , sections. The outwarddirected unit vectors for the fluid and solid phases are depicted bynf , and ns, respectively. Following Smit et al. (2011), Eq. (3), isgeneralised to account for both the low Reynolds number flow,where only viscous drag is present, and the Forchheimer regime of

Fig. 6. Two dimensional RUC schematic.

Fig. 7. Setup for vertical settling simulation.

Fig. 5. Representative Unit Cell (RUC).

J.M. Wilms et al. / Chemical Engineering Science 102 (2013) 76–86 81

higher Reynolds number (but still laminar) flow. Expressions werederived for each of the two regions of applicability and werematched with an asymptote matching technique (Churchill andUsagi, 1972). The equation is applicable to low- and high Reynoldsnumber flows (in the laminar regime) whilst it also holds for theentire range of porosities.

I tot ¼� μϵs

d2s

25:4ϵ2f ϵ1=3s

ð1�ϵ2=3s Þ2ð1�ϵ1=3s Þþ 18

!" #s24

þ 12cdρfds

ϵsϵ2f

ð1�ϵ2=3s Þ2J⟨vf ⟩

ff�⟨vs⟩

ss J

" #s#1=s⟨vf ⟩

ff�⟨vs⟩

ss

� �: ð48Þ

The values of the drag coefficient, cd, and the asymptoticmatching parameter, s, are unknown and calculated retrospec-tively following comparison with existing empirical data sets.

A cd-value of 1.9 is recommended for packed beds (ϵf � 0:4) byDu Plessis and Woudberg (2008). However, for the limiting valuesof ϵf-1, that is: for extremely dilute solutions, experimental datasuggests a drag coefficient corresponding to the Stokes dragcoefficient for a single particle, cd¼0.44 and a shifting parameter,s¼0.6.

3. Numerical implementation

The SIMPLE algorithm was adapted in order to solve both theparticle- and fluid time dependent momentum equations. Thecalculated velocities were used to solve the volume fractions. Thealgorithm was applied in Fortran 95 and designated Two PhaseMotion Simulation (2PMS).

3.1. Vertical motion

Simulations were done in order to predict the vertical settlingmotion of an evenly distributed 3.6 g sample of silicon particleswith a density of ρs ¼ 2500 kg=m3 through water with a density ofρf ¼ 1000 kg=m3. The depth and width of the water columnthrough which the particles fell were set to 1.7 m and 0.150 m,respectively and the setup for the simulation is illustrated in Fig. 7.

The vertical settling simulations done with 2PMS are verifiedagainst experimental data obtained from settling-tube experi-ments in Section 4. In the following sections the boundaryconditions applied for the setup of the simulation as well as itsstability with regard to the selection of grid size and time stepintervals are discussed.

3.2. Boundary conditions

At the left- and right boundaries of the setup shown in Fig. 7 itwas assumed that both the particles and the continuum were atrest (i.e. a no-slip boundary condition was applied). At the upperboundary it was assumed that a zero-gradient boundary conditionexisted and at the bottom it was assumed that the particles andthe continuum would be stationary to be representative of thesettling-tube experiments, discussed in Section 4.

The particle volume fraction, ϵs, was assumed to be zero at theupper edge of the setup and was assumed to have a zero-gradienton the left, the right and the base boundaries. The pressure at thetop boundary was set equal to zero. A zero-gradient pressurecondition was applied at all other boundaries.

The boundary conditions are summarised in Table 1.The initial guess for the pressure values was set to zero. The

input condition for the particle volume fraction proved proble-matic since the simulation had to represent the physical settling-tube experiments for which this value could not be measured. It

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10−4

Number of iterations

Per

cent

age

rela

tive

diffe

renc

e in

ave

rage

gro

up v

eloc

ity t = 1 st = 2 st = 3 st = 4 st = 5 st = 6 st = 7 st = 8 st = 9 st = 10 s

Fig. 8. Convergence within a time step.

0 2 4 6 8 100

2

4

6

8

10

12

14

Time,[s]

Ave

rage

vel

ocity

,[cm

/s]

Grid=85x15Grid=170x30Grid=340x60

ds=1.00mm

ds=0.75mm

ds=0.50mm

ds=0.30mm

Fig. 9. Grid analysis for vertical settling simulations.

Fig. 10. Time analysis for vertical settling simulations.

Table 1Boundary conditions for setup shown in Fig. 7.

Property Top Left and right Bottom

ϵs Zero gradient Zero gradient Zero gradientp 0 Zero gradient Zero gradientvf and vs 0 0 0

J.M. Wilms et al. / Chemical Engineering Science 102 (2013) 76–8682

was however known that for each of the physical experiments amass of 3.6 g particles was inserted in the form of a single layerinto the settling-tube. To make the simulations comparable tothese conditions, the particle volume fraction was changedaccording to the selection of the grid size to always ensure thata single layer entry would be representative of a particle mass of3.6 g. The following was used to determine the particle volumefraction:

ϵs ¼ms

ρsΔxΔyNx; ð49Þ

where ms and ρs denote the mass and density of the particles andΔx, Δy and Nx are the grid dimensions for a single cell in the x- andy-directions and the total number of grid nodes in the horizontaldirection, respectively.

The particles were released with a zero initial velocity and thefluid too was assumed at rest at initiation.

3.3. Grid geometry and time steps

The stability of 2PMS with regard to grid and time stepselection was analysed using three grid sizes of 85�15, 170�30,and 340�60 on a domain 170 cm� 15 cm in size. Three time stepintervals of 0.005 s, 0.01 s and 0.05 s were applied to each of thegrid allotments and simulations were performed for particles1 mm, 0.75 mm, 0.50 mm, and 0.30 mm in diameter.

3.3.1. Convergence within a time stepDuring each time step, the code was iterated until the percen-

tage relative difference between the average group velocities oftwo successive iterations was less than 0.1%. This was done bycalculating the average group velocity for each iteration as

uavgs jð Þ ¼∑usði; jÞϵsði; jÞ

∑ϵsði; jÞ; ð50Þ

which yielded the average velocity of each column of the grid, themean of which is the average group velocity. The percentagerelative difference between the average group velocity for twosuccessive iterations, I�1 and I, was then obtained as

%DIFF ¼ 2juavgs ðI; tÞ�uavg

s ðI�1; tÞjjuavg

s ðI; tÞ þ uavgs ðI�1; tÞj100: ð51Þ

In Fig. 8 an example of the relative difference criteria, given byEq. (51), at various time steps for a 1 mm-diameter particle and atime step of 0.01 s is illustrated.

3.3.2. Grid and time step stabilityFollowing Patankar (1980), the fully implicit scheme was used

to ensure that the result for a simulation is independent of the gridor the time step interval choices. Grid independence for a timestep of 0.01 s is illustrated in Fig. 9 and time step independence isshown in Fig. 10. Figs. 9 and 10 also show that a terminal groupvelocity is reached for each of the simulations after a period of10 s.

It was however found that numerical diffusion increasedrapidly as the grid was made coarser. It is clear that grid sizeplays an important role in the magnitude of the diffusion affectingthe convective solution. Therefore, one could, in principle, dimin-ish the influence of the diffusive terms in the solution by means ofgrid refinement.

3.3.3. Convergence of the average group velocity over timeResults show that the particles accelerate from their initial

stationary state under the influence of gravity. During this accel-eration period the particles spread out considerably. As theparticle group falls, the surrounding fluid is also set into motion.

Fig. 11. Average group velocities for vertical particle motion.

Table 2Particle sizes.

Size range (mm) Average (mm)

0.15–0.25 0.20.20–0.30 0.250.25–0.50 0.3750.50–0.75 0.6250.75–1.00 0.875

J.M. Wilms et al. / Chemical Engineering Science 102 (2013) 76–86 83

The fluid caught within the group moves faster than the fluid atthe front end of the group and, due to the nature of the equationsdiscussed previously, this results in the front end of the cloudbeing exposed to less drag than the internal sections, causing it tomove faster. The effect is an increase in the size of the cloud and adecrease in its concentration. However, numerical instabilities alsocontribute to the diffusion phenomenon. Fig. 11 shows how thegroup velocity of the particles tends to zero as the particles reachthe lower boundary of the simulation setup. The rate at which thevelocity decreases appears to be proportional to the size of thegroups' constituent particles.

Although the spread of the particle cloud in physical experi-ments may be ascribed to a difference in particle size and theinitial surface tension forces between the particles and the fluidmatter, this cannot be used to explain the diffusion seen in thenumerical experiments since the particles are assumed to be ofequal size and surface tension is not included in the expressionsused to simulate the motion.

At each time step the simulation data for the particle velo-cities and the particle volume fractions were captured andinserted into a Matlab routine to determine the average velocityof the cloud. This average was determined by multiplying eachgrid point velocity with its corresponding concentration anddividing by the sum of the concentrations. From Figs. 9 and 10it is apparent that for each particle size, the cloud reached aterminal velocity value after a lapse of 10 s. These terminalvelocity values are compared to those obtained via experimentswith a settling-tube.

3.3.4. Comparison between analytical and numerical resultsThe numerical results were compared to results obtained by

using Matlab's fzero procedure. The latter utilises a combination ofbisection, secant, and inverse quadratic interpolation methods toobtain the roots for an expression.

In deriving the solution to the particle momentum conserva-tion equation it was assumed that the pressure gradient may beapproximated with the buoyancy term:

∇p¼ ρf g : ð52Þ

The former assumption should only be applied to cases wherethe direction of the predominant pressure difference coincideswith that of the gravitational force acting on the particles. It wasfurthermore assumed that terminal velocity was reached. Appli-cation of the aforementioned assumptions yielded the following

for the terminal particle velocity:

0 ¼ ϵsg ρs�ρf� � μϵs

d2s

36ϵf ϵ1=3s

ð1�ϵ2=3s Þð1�ϵ1=3s Þþ 18

!" #s"

þ 12cdρfds

ϵsϵ2f

ð1�ϵ2=3s Þ2Jvf�vs J

" #s#1=svf�vs

� �: ð53Þ

Comparisons between the results obtained by 2PMS and thoseobtained by solving the terminal velocity expression, given byEq. (53), with Matlab's fzero algorithm, are illustrated in Fig. 15for asymptotic fitting parameters, s¼0.5, s¼0.6, and s¼0.7.From Fig. 15 it follows that an increase in the value of s,increases the magnitude of the group velocity with respect tothe particle diameter.

The relative error between the results determined with thefzero procedure in Matlab and those obtained via 2PMS wasdetermined by

%Error ¼ ValMatlab�Val2PMS

ValMatlab� 100; ð54Þ

for which the maximum error was determined as 0.248%. Withthis small relative error, the predictive capability of the code wasvalidated.

4. Physical experiments

In order to further verify the validity of the model, it wascompared to experimental results reported by Smit et al. (2011).They determined the terminal fall velocity of a group of silicaparticles with a settling-tube and by camera. Details regarding theexperimental setup and methodology may be found in Smit et al.(2011) and only the results obtained are discussed here.

4.1. Sample characteristics

Spherical glass beads, ranging from 0.15 mm to 1.0 mm indiameter, were used in the experiments and the average size ofeach sample was regarded as the representative sample size forthe purpose of comparison between experimental, simulated andanalytical measurements. The size categories are given in Table 2.

The silicon beads were supplied by the company, SigmundLindner (SiLi), which provided the chemical composition as 72.5%SiO2, 13% Na2O, 9.06% CaO, 4.22% MgO and 0.58% Al2O3. Thespecific weight of the beads was given as 2.50 kg/l (i.e. 2.50 g/cc)and the three larger samples are shown in Fig. 12.

4.2. Experimental results and processing

An example of the strain data for a 3.6 g sample of particles,ranging from 0.15 mm to 0.25 mm in diameter, is given in Table 3.

The percentages listed in the first column of the table areinterpreted as the percentage of the total amount of particleswhich remain in suspension and therefore have a correspondingvelocity equal to or slower than those given by the third column.

Fig. 12. Silicon beads used for the experiments. (a) 0.25–0.50 mm, (b) 0.50–0.75 mm and (c) 0.75–1.00 mm.

Table 3Output for strain data for a 3.6 g 0.15–0.25 mm size range sample.

Percentage in suspension Percentage on pan Velocity (m/s) Size (mm)

95 5 0.0308 0.22490 10 0.0301 0.22084 16 0.0295 0.21675 25 0.0289 0.21350 50 0.0269 0.20125 75 0.0245 0.18716 84 0.0234 0.18110 90 0.0218 0.1735 95 0.0200 0.163

Fig. 13. Settling-tube data for particle size range sample: 0.75–1.00 mm.

J.M. Wilms et al. / Chemical Engineering Science 102 (2013) 76–8684

The percentages listed in the second column indicate thoseportions of the total amount of particles that have settled onto theweighing pan and thus have a velocity equal to or faster than thevelocity values given by the third column.

The fourth column in Table 3 indicates the sizes of the particlesthat fall in the percentile categories listed in columns 1 and2 according to the Standard Relation Curve which is an empiricalcurve, developed by Fromme in 1977 and improved by Schoonees(Soltau, 2009), which relates particle size to settling velocity as

Dx ¼ 29730w2x þ 4173wx þ 67:38; ð55Þ

where Dx is the xth percentile grain size (in μm) and wx denotesthe xth percentile settling velocity (in m/s).

Since the particle sizes are known, the data given by the fourthcolumn were not used for this work. It is, however, apparent fromthe grain size results for the 0.15–0.25 mm sample, listed inTable 3 that the empirical curve provided a fairly accurate estimateof the particle sizes in that it yielded a size range of 0.16–0.22 mmfor the given range of 0.15–0.25 mm.

For each of the sample sizes a minimum of 5 experimental runswere made. The results were closely correlated and it was there-fore taken that the experimental procedure was successful andthat the experiments are repeatable.

Results obtained for four separate experimental runs forparticle sizes in the range of 0.75–1.00 mm are shown in Fig. 13.The percentage values indicated on the x-axis of Fig. 13 denote thepercentage of particles which fell slower than the correspondingvelocity value on the y-axis. The average velocity for each sampleset was calculated using the trapezium rule to obtain the entirearea underneath the graph for each run and dividing the said areaby the 90 units it spans on the x-axes. These results are illustratedin Fig. 15.

As discussed previously, digital images of the falling particleswere taken. The positions of a portion of the particles weredigitised relative to markings that have been made on the tube,and the speeds of the particles were calculated (using the PhotronFASTCAM viewer software). For various particles within a single

experiment, the distance traversed by a particle along with thetime required for the distance to be completed was recorded, asillustrated in Fig. 14. The average of the speeds obtained was thenassumed to denote the average speed of the group of particles forthat specific experimental run.

Various experiments were done for each of the particle sizeranges, listed in Table 2, the average of which was used forcomparative purposes in Fig. 15. Example photographs taken forthe 0.75–1.00 mm sample are shown in Fig. 14. The averagevelocity results for each sample is shown in conjunction withthe settling-tube experiments in Fig. 15 from which it follows thatclose correlation between photographic, settling-tube, numericaland analytical results were obtained.

4.3. Conclusions and discussion

Results obtained for vertical two-phase motion upheld thoseproduced with MATLAB's (2010) fzero method. These resultscorrespond well with physical data obtained through settling-tube experiments and the model is therefore regarded as a reliableprediction mechanism.

In Fig. 15, results given by 2PMS, the fzero Matlab procedure, aswell as camera- and settling-tube experimental procedures arecompared to each other. For each particle size range, used forexperiments and listed in Table 2, an average was calculated andassociated with the average obtained from all the experimentalresults for both the camera and settling-tube experiments. Theseaveraged values for the settling-tube and camera experiments areshown in Fig. 15 and the largest discrepancy was obtained for aparticle diameter of ds ¼ 0:2 mm.

From Fig. 15 it can be seen that the numerical results generatedby 2PMS for a fitting coefficient s¼0.6 correspond best to bothsettling-tube- and camera experimental values. As with the

Fig. 14. Particle positions. (a) Position 1 and (b) Position 2.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

14

16

18

Particle diameter [mm]

Term

inal

gro

up v

eloc

ity [c

m/s

]

2PMS with s = 0.5Matlab fzero method with s = 0.52PMS with s = 0.6Matlab fzero method with s = 0.62PMS with s = 0.7Matlab fzero method with s = 0.7Experimental with CameraSettling Tube

Fig. 15. Correlation between numerical simulations, analytical solution andexperiments.

J.M. Wilms et al. / Chemical Engineering Science 102 (2013) 76–86 85

comparison between the settling-tube and the camera experi-ments, the greatest difference was found for particles with anaverage particle diameter of ds¼0.2 mm.

It should be noted that the results obtained numerically arevery much dependent on the choice of s of which the physicalsignificance is yet to be determined. However, the numericaloutput from 2PMS yielded the correct trend when compared toexperimental data. Experiments with smaller particles provedmore difficult than those done for larger particles since thesewere influenced most by surface tension at the beginning of theexperiment and showed a much more diffusive nature as it spread

over nearly the total length of the settling-tube during theexperiments. This made it difficult to determine a value for theaverage group velocity since they did not show group behaviour.Experimental results obtained for larger particles are thus deemedmore accurate and correlated well with simulations.

Appendix A. Extension of collisional-kinetic force to twodimensions

In this appendix the particle–particle interaction force of Eq.(38) is extended to two dimensions.

The three dimensional form of the Newtonian law is given by

τ ¼ 2μD ; ðA:1Þ

where τ is the stress tensor, μ is a constant of proportionality andD is the rate of deformation- or rate of strain tensor which is relatedto the velocity gradient tensor, ∇v, but unlike ∇v, is symmetric.

The asymmetric velocity gradient tensor is divided into asymmetric rate of strain tensor, D , and an asymmetric vorticity-or spin tensor, S :

∇v ¼D þ S ; ðA:2Þ

where

D ¼ 12 ∇v þ ∇vT�

; ðA:3Þ

and

S ¼ 12 ∇v�∇vT�

: ðA:4Þ

The symbol T, which appears in Eqs. (A.3 and A.4), denotes thetranspose operation and the symmetric tensor may be expressed

J.M. Wilms et al. / Chemical Engineering Science 102 (2013) 76–8686

in matrix form as

12

∇v þ∇vT� ¼ i j k� � 2∂u

∂x∂v∂x þ ∂u

∂y∂w∂x þ ∂u

∂z∂u∂y þ ∂v

∂x 2∂v∂y

∂w∂y þ ∂v

∂z∂u∂z þ ∂w

∂x∂v∂z þ ∂w

∂y 2∂w∂z

0BB@

1CCA

i

j

k

0BB@

1CCA: ðA:5Þ

The spin tensor given in Eq. (A.4) does however not influence theviscosity and it follows that

τ ¼ 2μD

¼ μð∇v þ ∇vT Þ: ðA:6Þ

The two-dimensional force vector is thus given by

f rsavg

¼�n? � ϵsρsds4

∇v þ ∇vT ∇v þ ∇vT� νi:

ðA:7Þ

Let the shear rate tensor be denoted by

_γ ¼∇u þ∇uT ;

¼ ∂u∂y

þ ∂v∂x

� �ji þ ∂u

∂yþ ∂v

∂x

� �ij; ðA:8Þ

The magnitude of which is given by2

_γ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12Tr

0 ∂u∂y þ ∂v

∂x∂u∂y þ ∂v

∂x 0

0@

1A �

0 ∂u∂y þ ∂v

∂x∂u∂y þ ∂v

∂x 0

0@

1A

24

35

vuuut ;

¼ ∂u∂y

þ ∂v∂x: ðA:9Þ

The shear stress term is thus given by

τs¼ μs

∂u∂y

þ ∂v∂x

� �ji þ ∂u

∂yþ ∂v

∂x

� �ij

� �; ðA:10Þ

where the particle viscosity is given by

μs ¼ϵsρsd

2s

12∂u∂y

þ ∂v∂x

� �: ðA:11Þ

2 The magnitude of any tensor A is defined as

A ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12

A : AT� �r

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12Tr AT � A� �r

:

Eq. (A.7) may therefore be expressed as

F kci ¼ ϵsρsds

4∂u∂y

þ ∂v∂x

� �∂u∂y

þ ∂v∂x

� �i þ ∂u

∂yþ ∂v

∂x

� �j

� �νi: ðA:12Þ

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