Computers and Chemical Engineering 32 (2008) 2946–2957

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Computers and Chemical Engineering

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Finite pointset method for simulation of the liquid–liquid flowfield in an extractor

Christian Drumma, Sudarshan Tiwarib, Jorg Kuhnertb, Hans-Jorg Barta,∗

a Lehrstuhl f. Thermische Verfahrenstechnik, TU Kaiserslautern, Gottlieb-Daimler-Strasse, 67663 Kaiserslautern, Germanyb Fraunhofer Institut Techno- und Wirtschaftmathematik, Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany

a r t i c l e i n f o

Article history:Received 22 October 2007Received in revised form 20 March 2008Accepted 20 March 2008Available online 8 April 2008

Keywords:CFD

a b s t r a c t

Finite pointset method (FPM) is applied for the simulation of the single- and two-phase flow field in arotating disc contactor (RDC) type extraction column. FPM is a numerical method to solve fluid dynamicequations. This is a Lagrangian and meshfree particle method, where the particles move with fluid velocityand carry all information necessary for solving fluid dynamic quantities. The simulations are validatedby single- and two-phase 2D particle image velocimetry (PIV) measurements. In addition, the results arecompared to simulations of the commercial CFD code Fluent. The results show that FPM can predict theone- and two-phase flow field in the RDC, whereas the predicted velocities are in good agreement withthe experimental ones. FPM also bears comparison with the results of the commercial CFD code Fluent.

FPM

Meshfree methodExtraction columnLP

© 2008 Elsevier Ltd. All rights reserved.

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. Introduction

Liquid–liquid extraction is a separation process, which is basedn the different distribution of the components to be separatedetween two liquid phases. Liquid–liquid extraction processes areidely applied in chemical and biochemical industries. Classical

xtraction equipments are mixer-settler cascades and differentypes of counter-current extraction columns like the RDC type,hich is analyzed here. The simulation of these counter-current

iquid–liquid extraction columns still requires improvement. Today,he layout of an extractor is mainly based on manufacturer’s knowl-dge and simplified models, e.g. the HTU-NTU concept methods,here 1D flow models, such as the back flow or dispersion model,

re used to account for the non-idealities of the flow. In mostases these models are too simple to describe the real hydrody-amic behavior. That is why the design of extraction columns oftenequires pilot plant experiments and is time and money consum-

Abbreviations: BiCgStab, stabilized biconjugate gradient; CFD, computationaluid dynamics; FPM, finite pointset method; ILA, intelligent laser application; LDA,

aser doppler anemometry; PBM, population balance model; PIV, particle imageelocimetry; RANS, Reynolds averaged Navier–Stokes; RDC, rotating disc contactor;PH, smoothed particle hydrodynamics.∗ Corresponding author.

E-mail addresses: cdrumm@mv.uni-kl.de (C. Drumm), tiwari@itwm.fhg.deS. Tiwari), kuhnert@itwm.fhg.de (J. Kuhnert), bart@mv.uni-kl.de (H.-J. Bart).

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098-1354/$ – see front matter © 2008 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2008.03.009

ng. On the other hand, there is an ongoing claim by the industry forore straightforward, faster and money saving simulation meth-

ds.Computational fluid dynamics (CFD) are one possibility in this

irection, since CFD can predict the 3D flow field in an extractionolumn and also account for the non-idealities. In addition to theow field, CFD simulations can deliver single design parameters ashe residence time of the droplets (Modes & Bart, 2001). Currentesearch in the last years also focused on the link of CFD and theopulation balance models (PBM) (Vikhansky & Kraft, 2004). PBMan specify droplet movement, coalescence and breakage of the dis-ersed phase and consider the particulate behavior of the droplets

n an extraction column. For a successful combination of CFD andBM and a general application of CFD in the field of liquid–liquidxtraction columns, the performance of CFD simulations for extrac-ion columns should be tested. In recent years, only a few workingroups focused on the simulation of the hydrodynamics in counter-urrent stirred liquid–liquid extraction columns. The authors oftenelped themselves by only considering the single-phase flow in theolumn (Fei, Wang, & Wan, 2000; Modes & Bart, 2001). Two-phaseow fields were simulated and investigated only by Rieger, Weiss,

eigley, Bart, and Marr (1996), You and Xiao (2005), Vikhansky &

raft (2004) and Drumm and Bart (2006). Only the latter showed aomplete comparison of the simulated flow field with experiments.

In this paper we present the simulations of liquid–liquid two-hase flows in an extraction column. The primary (continuous)

C. Drumm et al. / Computers and Chemical

Nomenclature

�a vector�b vectorB Ball of radius hwith center �xCD drag coefficientdd droplet diameterdx difference of x-coordinate of i th particle to central

particlef scalar�F interfacial force�g gravity forceh radius of interaction of neighboring particle�H forceI identity matrixJ quadratic functionalm number of neighbor particleM geometrical matrixN total number of particleO(N) capital order of Np pressurepdyn dynamic pressurephyd hydrostatic pressureP set of neighbor particlesR� �-dimensional Euclidean spaceRe relative Reynolds numberS stress tensort time�v velocity vectorw weight functionW diagonal matrix�x position vector

Greek letters˛ volume fractionˇ constant� boundaryı Dirac delta function� constant factor for the Gaussian weight function� Lagrange multiplier� dynamic viscosity� dimension in Euclidean space� density� shape function for spatial derivative approximation a scalar function˝ computational domain

Subscript32 Sauter mean diameterl primary phased dispersed phaseN Neumann boundaryD Dirichlet boundaryi particle indexin inflowout outflow

pttt

(dtd1uaLtOdfa2llaO2

c&mwucettwcpdpkF

trwMmmcYpemapaiaIutoaclations was already described elsewhere (Drumm & Bart, 2006),

w wall

hase is modeled by the Navier–Stokes equations. The motion ofhe secondary (disperse) phase is calculated by solving the equa-ion of motion in which the inertia, drag and buoyancy forces areaken into account.

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Engineering 32 (2008) 2946–2957 2947

We solve both types of equations by the finite pointset methodFPM). FPM is a Lagrangian and meshfree method for solving fluidynamic equations. Meshfree methods are originally developedo simulate fluid dynamics problems. Smoothed particle hydro-ynamics (SPH) is referred to as the first meshfree method (Lucy,977; Monaghan, 1992). Since then it has been extended to sim-late the compressible Euler equations in fluid dynamics and waspplied to a wide range of problems (see Hu & Adams, 2006; Liu &iu, 2003; Monaghan, 1994; Morris, 2000). The implementation ofhe boundary conditions is the main problem of the SPH method.ur approach differs from the method of SPH in approximations oferivatives and the treatment of boundary conditions. In its originalormulation, SPH is easy for implementation, but it provides poorccuracy/convergence (Duarte & Oden, 1996; Griebel & Schweitzer,000). Further, variety of meshfree methods were proposed in the

ast decade for solving fluid dynamics and solid mechanics prob-ems. Detailed discussion on the development of meshfree methodsnd on their application, can be found, e.g., in Belytschko, Krongauz,rgan, Flemming, and Krysl (1996), Griebel and Schweitzer (2002,005, 2006), Liu. Liu, and Lam (2003) and other references there.

FPM is suitable to handle, for examples, flow problems withomplicated and rapidly changing geometry (Kuhnert, Tramecon,

Ullrich, 2000), free surface flows (Tiwari & Kuhnert, 2002) andultiphase flows (Tiwari & Kuhnert, 2007). In our previous workse have presented the simulations of two-phase immiscible flowssing FPM (Tiwari & Kuhnert, 2007), where we have considered alass of points, where the phases are distinguished by the color forach fluid particle and advect the color function which results inracking the interface accurately. The normal and the curvature ofhe interface can be computed from the color function. In this paper,e do not use the color function, instead we establish the separate

luster of points for each phase. Each of these separate cluster ofoints will act as a numerical grid to approximate the governingifferential equations for each phase. These point clouds are decou-led from each other, however they are able to communicate anyind of information among the clusters. A detailed description ofPM is presented in Section 3.

For the validation of the simulations particle image velocime-ry measurements are conducted. The PIV technique allows theecording of a complete flow velocity field in a plane of the flowithin a few microseconds (Kompenhans, Raffel, & Willert, 1998).ost publications of PIV velocity measurements concern measure-ents in single-phase flows. While literature concerning velocityeasurements in gas–liquid flow is also available, literature con-

erning liquid–liquid problems is seldom. For example, Bujalski,ang, Nikolov, Solnordal, and Schwarz (2006) conducted single-hase water CFD simulations and PIV measurements in a pulsedxtraction column, which is normally operated in a two-phaseode. In a common dense liquid–liquid system PIV measurements

re not possible because of the different refractive index of thehases. Laser Doppler anemometry (LDA) velocity measurementsre still possible but do not allow a comparison of the whole veloc-ty profile (Rieger et al., 1996). The problem can be overcome whenpplicate a refractive index matching technique (Budwig, 1994).n the present work, the refractive index matching technique assed by Augier, Masbernat, and Guiraud (2003) is adopted. Theechnique makes the measurement of the flow field in the aque-us phase possible and also allows for a phase discrimination. Inddition the simulations are also compared to simulations of theommercial CFD code Fluent. The approach of the Fluent simu-

herefore, it is not described in detail in this paper. We have usedhe same geometry for Fluent and FPM simulations.

In Section 2, we present the mathematical model. In Section 3 weresent the basic idea of FPM and FPM schemes for two-phase flows.

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n Section 4, the essential PIV setup is summarized shortly, whileection 5 contains results and comparisons of the single-phase asell as two-phase flows and some summary and conclusions areresented in Section 6.

. Governing equations and numerical scheme

.1. Governing equations

The governing equations for the continuous liquid phase of mul-iphase flows can be derived from the Navier–Stokes equations foringle-phase flow. In this paper we consider all equations in theagrangian form. The continuity equation for the primary liquidhase is

d˛l

dt= −˛l(∇ · �vl), (1)

here the index l stands for the liquid phase,˛l the volume fraction,�l the velocity vector for the liquid phase and d/dt is the mate-ial derivative. Similarly, the continuity equation for the secondaryhase is given by

d˛d

dt= −˛d(∇ · �vd), (2)

here the index d denotes the quantities for the secondary (dis-ersed) phase. In addition to (1) and (2) the volume fractions mustatisfy the following constraints

l + ˛d = 1. (3)

he conservation of momentum for the liquid phase is given by

d�vl

dt= −∇p

�l+ 1˛l�l

∇ · Sl + �g + 1˛l�l

�Fl (4)

here�l is the liquid density, p the pressure, which is same for bothhases, �g the gravitational force, �Fl the interfacial forces and Sl ishe stress tensor.

l = ˛l�l[∇�vl + (∇�vl)T − 1

3(∇ · �vl)I], (5)

here �l is the dynamic viscosity of the liquid. The inter-phasenteraction term consists of different momentum exchange mech-nisms. Only the drag force was taken into account, while the virtualass force and the lift force can be neglected for a liquid–liquid

nteraction as shown by Wang and Mao (2005) in a stirred tank. Thenterfacial momentum transfer (drag force) between two phases isiven by

�l = 3

4˛d�l

CD

dd|�vd − �vl|(�vd − �vl), (6)

here dd is the diameter of the droplets of the dispersed liquidhase and CD is given by Schiller and Naumann (1935)

D ={

24Re

(1 + 0.15Re0.687) if Re ≤ 1000

0.44 if Re > 1000

nd Re is the relative Reynolds number defined as

e = �l|�vd − �vl|dd

�l. (7)

imilarly, the momentum balance equation for the dispersed phases given by

d�vd

dt= −∇p

�d+(

1 − �l

�d

)�g + 1

˛d�d

�Fd, (8)

Engineering 32 (2008) 2946–2957

here �Fd = −�Fl. In both phases the particle positions are given by

d�xl

dt= �vl (9)

d�xd

dt= �vd. (10)

n order to fulfill the volume fractions constraint we renormalize˛lnd ˛d after computing from Eqs. (1) and (2) in every time step as

l = ˛l

˛l + ˛dand ˛d = ˛d

˛l + ˛d. (11)

.2. Numerical scheme for primary liquid phase

We solve the primary liquid phase with the FPM in combinationith Chorin’s pressure projection method (Chorin, 1968). In thisork we decompose the pressure into hydrodynamic and dynamicressures, i.e. p = phyd + pdyn. The scheme consists of two fractionalteps and is of first order accuracy in time. In the first step weompute the intermediate velocities �v∗ and in the second step weorrect the velocity with the constraint that velocity fulfills theontinuity equation. In our numerical scheme, we substitute thealue of ∇ · �vl which appeared in the stress strain tensor Sl Eq. (5).oreover, after some calculations, we obtain

· Sl = ∇ · (˛l�l ∇�vl) +(�vl, ˛l�l,∇) = ∇ · (∇�vl) +(�vl, ,∇)

(12)

here is some straight forward, but lengthy term and = ˛l�l.oreover, the operator ∇ · (∇�vl) can be re-expressed by

· (∇ �vl) = 12

[�( �vl) + � �vl − �vl�

]. (13)

his means, once we construct the shape functions for the Laplaceperator�, then the differential operator ∇ · (∇ �vl) can be approx-mated by combination of � applied to different functions. Hencehe momentum Eq. (4) can be re-expressed in the simple form

d�vl

dt= −∇p

�l+ 1˛l�l

[∇ · (∇�vl) +(�vl, ,∇)]

+ 34˛d

˛l

CD

dd|�vd − �vl|(�vd − �vl) + �g

= −∇p�l

+ 1˛l�l

∇ · (∇�vl)−Gl�vl+ �H, (14)

here Gl = (3/4)(˛d/˛l)(CD/dd)|�vd − �vl| and the vector �H consistsf all the forces as source term.

In the following we describe the projection scheme for contin-ous liquid flow in the FPM framework.

(i) Initialize:

pn+1dyn := pndyn (15)

(ii) Move the particles

�xn+1l := �xnl +�t�vnl (16)

(iii) Compute ˛n+1l implicitly by

˛n+1l = ˛nl

1 +�t(∇ · �vn)(17)

l

(iv) Compute the actual hydrostatic pressure pn+1hyd from

∇ ·(

1�l

∇pn+1hyd

)= ∇ · �g (18)

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C. Drumm et al. / Computers and Ch

with boundary condition

∂pn+1hyd

∂�n = �g · �n on �w and �in, pn+1hyd = p0

hyd on �out (19)

(v) Establish preliminary pressure

p = pn+1hyd + pndyn (20)

(vi) Compute implicitly the intermediate velocity �v∗l from[

(1 +�tGl)I −�t

˛l�l∇ · (∇)

]�v∗

l = �vnl − �t

�l∇p+�t �Hn (21)

(vii) Correct the velocity

�vn+1l = �v∗

l − �t

�l∇ n+1 (22)

with the constraint (continuity equation)

∇ · �vn+1l = − 1

˛nl

d˛l

dt= − 1

˛nl

˛n+1l − ˛nl�t

(23)

viii) Updating the dynamic pressure

pn+1dyn = pndyn + n+1 (24)

where n+1 is obtained from (which we obtain taking diver-gence on (22))

∇ ·(

1�l

∇ n+1)

= 1�t

[1˛nl

˛n+1l − ˛nl�t

+ ∇ · �v∗l

](25)

with boundary conditions similar to phyd given in (19).

We note that in the left-hand side of Eq. (21) the operator1 +�tGl)I − (�t/˛l�l)∇ · (∇) and the left-hand side of Eq. (25)ive rise to the construction of a big sparse matrices, each line ofhich containing the local, discrete approximation of the opera-

ors. On the right-hand sides appear a load vector. Hence Eqs. (18),21) and (25) represent big sparse linear systems. In summary, inach time step, we construct the approximation of Laplace opera-or, and with the help of this operator we solve the equations forll three velocity components and the pressure Poisson equations.n addition to that, we need to approximate the gradient vectors.n the next section, we demonstrate how to approximate gradientnd the Laplace operators by the finite pointset method.

.3. Numerical scheme for secondary phase

In order to solve the differential equations for each phase sepa-ately, we need to exchange the flow quantities from one phase tonother and vice versa. After solving the equations on the primaryhase particles, we interpolate the pressure, volume fraction andelocity on each particle in the secondary phase from the cluster ofhe primary phase. Since we have established the cluster of pointsor each phase separately, we have to interpolate the quantities atn arbitrary particle in one phase from the surrounding cluster ofarticles from the other phase. This can be easily achieved with theelp of least squares method, described in the next section.

The following steps are followed to compute the quantities onecondary phase.

(i) Move particles

�xn+1 := �xnd +�t�vnd (26)

d

(ii) Compute ˛n+1d implicitly by

˛n+1d = ˛nd

1 +�t(∇ · �vnd)(27)

vTipa

Engineering 32 (2008) 2946–2957 2949

and normalize the volume fractions in order to fulfill the con-sistency condition (3) as follows

˛n+1d = ˛n+1

d

˛n+1d + ˛n+1

l

and ˛n+1l = ˛n+1

l

˛n+1d + ˛n+1

l

(28)

iii) compute the velocity �vn+1d implicitly by

�vn+1d =

�vnd +�t[−(∇pn+1/�d) + (1 − (�l/�d))�g + Gd�vn+1l ]

1 +�tGd, (29)

where Gd = (3/4)(�l/�d)(CD/dd)‖�vnl − �vnd‖.

. Finite pointset method

The basis of the computations in FPM is a point cloud, whichepresents the flow field. The points of the cloud are referred to asarticles or numerical grids. They are carriers of all relevant physical

nformation. The particles have to completely cover the whole flowomain, i.e. the point cloud has to fulfill certain quality criteria (par-icles are not allowed to form “holes” which means particles haveo find sufficiently many neighbors; also, particles are not allowedo cluster; etc.). The point cloud is a geometrical basis, which allowsor a numerical formulation making FPM a general finite differencedea applied to continuum mechanics. That especially means, if theoint cloud would reduce to a regular cubic point grid, then FPMould reduce to a classical finite difference method. The idea of

eneral finite differences also means that FPM is not based on aeak formulation like Galerkin’s approach. Rather, FPM is a strong

ormulation which models differential equations by direct approx-mation of the occurring differential operators. The method useds a moving least squares idea which was especially developed forPM.

.1. Initialization of point cloud

The most important parameter in the FPM is the average dis-ance between the particles h. We allow this parameter as a userefined smoothed function in space h(�x) and it has a variation lesshan 1.

In general, our computational domain has a boundary withiecewise linear shell functions. For example, line segments in 2Dnd triangles or quadrilaterals in 3D. In the first step, we establishne particle in the center of each shell element. Then, these particlesre the sources for other new particles and we establish new par-icles which have minimum distance ah(�x) and maximum distance(�x), where a is a positive constant, for examples, a = 0.2. Then wese these established particles as the source of new particles as longs the boundary is filled and the minimum and maximum distanceatisfied.

To initialize the interior particles we start the boundary particless the source. We first construct one layer inside the domain. Thevailable particles are the sources for other new interior particlesnd then we successively insert particles till the center is reachednd above minimum and maximum distance conditions are satis-ed. After completion of initialization, we apply once more holelling and removing closed algorithm, which are described in the

ollowing sections.It is not necessary that our initial particle distribution has to

e placed in regular lattice. Since the particles move with fluid

elocity, the regularity will be destroyed within few time iterations.hen they may cluster or scatter. Therefore, we need to monitorn every two to three time steps to maintain the good quality ofarticle distribution with the help of adding and removing mech-nism.

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.2. Neighbor searching

Searching neighbors of particles is the most important andime consuming part of the meshfree method. After the ini-ialization, particles are numbered from 1 to N and for eachndex one has direct access to the position �xi. The funda-

ental operation to be done on the point cloud is to findeighbors for a point inside a ball with given radius h. In ourimulation we have assumed constant h all over the domain.We construct a voxel data structure which contains the com-

utational domain. The voxels form a regular grid of squaresespectively cubes in 2D and 3D with side length h. For each voxel,he indices of points are contained in it. Moreover, for each point its known, in which voxel it is contained and which are the pointsnside the radius h.

We establish three types of lists. The first list requires to loopver all points and computing its voxel. This is of complexity O(N).he second list can be obtained from the first list by sorting withespect to the voxel indices. This is of complexity O(N logN). Ofourse, in time dependent problems, one generates the lists justnce from scratch, and then in the following update from the pre-ious time step.

Hence, for a point �xi, all points inside the ball B(�xi, h) need to beound. The second list gives direct access to the voxel, where thisoint is contained in. Now all points in the voxel and its 8 (in 2D)nd 26 (in 3D) neighboring voxels are tested for being inside theall. The indices of the points in these voxels are given by the first

ist. Since each voxel contains O(1) points, this operation is of con-tant effort. Hence the total complexity of a neighborhood searchor every point is O(N). Finally, the neighborhood information isaved in the third list.

.3. Filling hole

Finding a hole in the point cloud is a more tricky task, sincerovided is only information where points are and not wherehere are none. An important fact is that the center of a voidircle of maximal radius must be a Voronoi point. The Voronoiell (Voronoi, 1907) of point �xi is the set of all points closero �xi than to any other point. If the points are in general posi-ion, �+ 1 Voronoi cells meet in one point, called the Voronoioint.

Finding all Voronoi points in 2D can be done in O(N logN)ime, e.g. by Fortune’s planesweep algorithm (Fortune, 1987).n 3D, the Voronoi points can be found in O(N2) time by con-erting the problem to finding a 4D convex hull. As for holelling the Voronoi points are only to be detected locally, subdi-ision algorithms can reduce the effort to O(N logN) (Shewchuk,999).

Obviously in 3D, but also in 2D, constructing the full Voronoiiagram can be too expensive for a large number of points. This

s in some sense satisfying, since the complexity of meshing aoint cloud in 3D is one major motivation for putting our focus oneshless methods. If the Voronoi diagram was easy to compute,

o would be the Delaunay triangulation. For the task of identifyingoles, the existing voxel (or octree) structure can be used to con-truct local, partially overlapping, Voronoi diagrams. If the pointloud is not too deformed, this approach successfully identifies theargest holes in O(N) time, since the number of points considered

ocally is of order O(1). Once the too largest holes are identified,ew points are inserted, which is for each new point of complexity(1). After the insertion of new particles we use the moving leastquares interpolation for approximation of fluid dynamic quanti-ies.

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Engineering 32 (2008) 2946–2957

.4. Removing points

Particles which are too clustered are thinned by merging pairsf close by points into a single one. By an iterative application, alsoarge clusters can be thinned out. The two closest points can beound in O(N) time by looping over all points and for each pointnding its closest neighbor by checking all points in its circulareighborhood. With the same procedure, one can find the k shortestistances, as well as all points closer than a given distance dmin.emoving a point is negligible effort. Introducing a new point isf complexity O(1), since it is one interpolation problem using alleighboring points.

One prescribes a minimum distance between two points.ssume two points closer than this distance are detected. Thenoth are removed and replaced by a new point, i.e. the two par-icles are merged into a new single one. Let the two particles havehe positions �xk and �xl, and let the data �uk and �ul be associated dataectors. The new particle is inserted in the center of mass of thewo particles, and the data is linearly interpolated.

� = (�xk + �xl)/2 �u = (�uk + �ul)/2. (30)

ote that the position of the new particle is actually not too impor-ant, since the two particles are close by anyhow (i.e. one could placehe new particle at a position of one of the two removed ones). Theata, however, must be interpolated equally. An alternative to theean value of the data vector, one can use the higher order moving

east squares approximations.

.5. Approximation of model equations by FPM

As we have already mentioned in the previous section we needo interpolate flow quantities from one cluster point to the otherne in order to perform two-way coupling. In addition to that weave to approximate the spatial derivatives in each phase and solvehe Poisson equation for the primary phase. In this section, weescribe the numerical scheme for solving the Poisson equationy FPM. For the details of FPM and its applications, we refer one ofur previous paper (Tiwari & Kuhnert, 2007) and references theren.

Consider the Poisson equation

= f in ˝∈R3 (31)

ith the boundary conditions

= g on �D and∂

∂�n = h on �N, (32)

here �D and �N denote the Dirichlet and Neumann boundaries,espectively. The numerical discretization is a generalized finite dif-erence method. The first idea on the generalized finite difference

ethod is presented in Liszka and Orkisz (1980), where the sparseinear system is directly derived from the Taylors series expansion.he constrained least squares method is presented in Tiwari anduhnert (2002), where the Poisson equation and the boundary con-ition are added in the Taylor series expansion as constraint. Theomparison of both methods is presented in Iliev and Tiwari (2002).t is found that the method presented by Tiwari and Kuhnert is moretable. We still encounter some instability problem during the sim-lation, where the distribution of neighbor particles are so bad thathe iterative method of the sparse linear system does not converge.herefore, we need to add extra strong conditions such that the

terative method converges.

Consider the finite set of points �xi for i = 1,2, . . . , N in ˝ ∪� .e want to construct the differential operators at every particle

osition from its neighboring cloud of points. Let �x be an arbi-rary particle position. We introduce a weight function such that

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he particles sitting closer to �x has higher influence than those sit-ing far from it. Moreover, in order to restrict the number of pointse consider a weight functionw = w(�xi − �x;h) with small compact

upport, where h determines the size of the support. The weightunction can be quite arbitrary, however it makes sense to chooseGaussian weight function of the form

i = w(�xi − �x;h) ={

exp(−� ‖�xi − �x‖2

h2), if

‖�xi − �x‖h

≤ 1

0, else,(33)

here � is a positive constant and is considered to be in the rangef 2–6, depends upon users. Let P(�x, h) = {�xi : i = 1,2, . . . ,m} behe set of m neighbor points of �x = (x, y, z) in a ball of radius h.

e note that the central particle �x is one element of the neighboret P(�x, h). For consistency reasons, some obvious restrictions areequired, for example, in 2D if we want the second order approxi-ation of derivatives at a arbitrary particle, there should be at least

ix particles including this central particle and they should neithere on the same line nor on the same circle. Hence we define theize of h such that the minimum number of neighbors is guaran-eed for the approximation of derivatives. Hence, new particles willave to be brought into play as the particle distribution becomesoo sparse or, logically, particles will have to be removed from theomputation as they become too dense.

As in the classical discretization method, we construct the shapeunctions �i, i = 1, . . . ,m such that

≈m∑i=1

�i i. (34)

n order to construct �i we consider m Taylor series expansions of( �xi) about �x

(�xi) = (�x) + dxi x + dyi y + dzi z + 12

dx2i xx + dxi dyi xy

+dxi dzi xz + 12

dy2i yy + dyi dzi yz

+12

dz2i zz + O(‖d�xi‖3), (35)

or i = 1, . . . ,m, where dxi = xi − x, dyi = yi − y, dzi = zi − z.Now we multiply (35) by �i and sum over 1 to m

m

i=1

�i i =(

m∑i=1

�i

) +

(m∑i=1

�i dxi

) x +

(m∑i=1

�i dyi

) y

+(

m∑i=1

�i dzi

) z +

(12

m∑i=1

�i dx2i

) xx

+(

m∑i=1

�i dxi dyi

) xy +

(m∑i=1

�i dxi dzi

) xz

+(

12

m∑i=1

�i dy2i

) yy +

(m∑i=1

�i dyi dzi

) yz

+(

12

m∑i=1

�i dz2i

) zz + O(‖d�xi‖3). (36)

rom the expression (36) we obtain the approximation of the

aplace operator

≈m∑i=1

�i i + O(‖d�xi‖3) (37)

haa

a

Engineering 32 (2008) 2946–2957 2951

f the following conditions satisfy

m∑i=1

�i = 0,m∑i=1

�i dxi = 0,m∑i=1

�i dyi = 0,

m∑i=1

�i dzi = 0,m∑i=1

�i dx2i = 2,

m∑i=1

�i dxi dyi = 0,

m∑i=1

�i dxi dzi = 0,m∑i=1

�i dy2i = 2,

m∑i=1

�i dyi dzi = 0,m∑i=1

�i dz2i = 2 (38)

t consists of 10 constraints (38) and we denote them byG1, . . . , G10.e observe in the classical finite difference approximation that

he sum of all coefficients equals to zero, where the first condi-ion in (38) also fulfills the same condition. Moreover, in order tobtain a stable linear system, we add one more condition that theoefficient on the central particle must be negative, similar to thelassical finite difference discretization. Therefore, we consider thedditional condition

(xi = x) = − ˇh2, (39)

here ˇ is a non-negative small number and the constant h is theize of compact support. Hence, we have 11 constraints (38 – 39)or the approximation of the Laplace operator.

Now, we minimize the functional

=m∑i=1

�2i

wi(40)

ith respect to constraints G1, . . . , G11. From the Euler–Lagrangeheorem, this constrained minimization is equivalent to

∂J

∂�i+

11∑j=1

�j∂Gj∂�i

= 0, i = 1, . . . ,m, (41)

here�1, . . . , �11 are the Lagrange multipliers. More explicitely wexpress

�i/wi + �1 + dxi�2 + dyi�3 + dzi�4 + dx2i �5 + dxi dyi�6

+ dxi dzi�7 + dy2i �8 + dyi dzi�9 + dz2

i �10 + �11ıix = 0, (42)

or i = 1, . . . ,m, where

ix ={

1 if xi = x0 else

he above system of equations can be written in the matrix form

�a = �b, (43)

here

=

⎛⎝ 1 dx1 dy1 dz1 dx2

1 dx1 dy1 dx1 dz1 dy21 dy1 dz1 dz2

1 ı1x

.

.

....

.

.

....

.

.

....

.

.

....

.

.

....

.

.

.1 dxm dym dzm dx2

m dxm dym dxm dzm dy2m dym dzm dz2

m ımx

⎞⎠ ,

� = [�1, . . . , �11]T, �b = −2[�1/w1, . . . , �m/wm]T. In general we

ave m > 11, therefore the linear system (43) is overdeterminednd we obtain the solution from the weighted least squares methodnd the solution is explicitely given by

� = (MTWM)−1

(MTW)�b, (44)

2952 C. Drumm et al. / Computers and Chemical Engineering 32 (2008) 2946–2957

i(

(

Hf1

�

Is

∑

∑

ai

Mrh

(

a

(

a

ost

t

∑

Iebbfor the Poisson equation gives the second order convergence (seeIliev & Tiwari, 2002).

Fig. 1. Experimental setup.

fMTWM is nonsingular, whereW = diag[w1, . . . ,wm]. The vectorMTW)�b can be written explicitely with the help of conditions (38)

MTW)�b = (−2)

(0 0 0 0 2 0 0 2 0 2 − ˇ

h2

)T

.

ence, we can compute the Lagrange multipliers �i, i = 1, . . . ,11rom (44) and substituting them in (42), the shape functions �i, i =, . . . ,m can be expressed by

i = −wi2

(�1 + dxi�2 + dyi�3 + dzi�4 + dx2i �5 + dxi dyi�6

+dxi dzi�7 + dy2i �8 + dyi dzi�9 + dz2

i �10 + ıix�11) (45)

f the particle �x belongs to the Neumann boundary, then the con-traints are given by

m

i=1

�i = 0 (46)

m

i=1

�i d �xi = �n (47)

nd the matrix M and the vector (MTW)�b can be computed accord-ngly.

Similarly, we can compute other derivatives, where the matrixremains the same, only the constraints will be different which

esult different vectors (MTW)�b. For examples, to compute x weave

MTW)�b = (−2)

(0 1 0 0 0 0 0 0 0 0 − ˇ

h2

)T

nd to compute xx we have

MTW)�b = (−2)

(0 0 0 0 2 0 0 0 0 0 − ˇ

h2

)T

nd so on.Once we compute the Lagrange multipliers as above, we then

btain the approximation (34) for the particle �x. Hence, if we con-ider �xj an arbitrary particle and �xji its neighbors of number m(j),hen we have the following sparse linear system of equations for

Fs

Fig. 2. Numerical grid.

he unknowns j, j = 1, . . . , N

m(j)

i=1

�ji ji = fj. (48)

n this sparse linear system the neighbor particle index gives thentry of the matrix components. This linear system can be solvedy any iterative solvers. In this paper, we have solved this by the sta-ilized biconjugate gradient (BiCgStab) method. The above scheme

ig. 3. FPM, simulated instantaneous velocity magnitude and vectors in 2D xz-plane,ingle-phase flow.

C. Drumm et al. / Computers and Chemical

Ffl

4

pepaau

Fp

dositats

uNbcsibSeCpgs

lopifor the two-phase measurements. The iso-optical liquid–liquidsystem is composed of n-heptane as the dispersed phase and awater–glycerine mixture (44 wt% water), which have the samerefractive index at a temperature of 25 ◦C. The heptane droplets do

ig. 4. FPM, average velocity magnitude and vectors in 2D xz-plane, single-phase

ow.

. Experimental PIV setup

A 1 m section of a 150 mm diameter column with 5 RDC com-artments as internals was used as the experimental basis. Thexperimental setup together with the dimensions of the RDC com-

artments is depicted in Fig. 1. The column can be operated in onend two-phase conditions, the aqueous phase entering at the topnd leaving at the bottom, while the second phase is fed into the col-mn through a distributor at the bottom. The distributor generates

ig. 5. FPM, averaged velocity magnitude and vectors in 2D xz-plane in one com-artment and planes for comparison, single-phase flow.

Fi

Engineering 32 (2008) 2946–2957 2953

roplets in the millimeter range (d23 = 2.5 mm). Variable rotationf the disc stirrers was warranted by an agitator. The optical mea-urement area (48 mm ×30 mm) in one extraction compartments the height of one compartment and the symmetry region fromhe shaft to the wall of the column. The RDC column was operatedt throughputs of 200 l/h for the single-phase flow and 100 l/h forhe aqueous phase and organic phase in the two-phase flow. Thetirrer revolution was varied between 150 and 300 rpm.

An ILA (Intelligent Laser Applications GmbH) PIV system wassed. The laser sheet is generated by two 25 mJ double pulsedd:Yag lasers (Minilite Continuum) operated at a frequencyetween 1 and 4 Hz. The laser sheet was expanded by a cylindri-al lens to a thickness of around 2 mm, while the height of the laserheet in the column was around 0.1 m. The wavelength of the laserss 532 nm. Each laser pulse is composed of two flashes separatedy a delay between 0.5 and 1 ms. A 12-bit digital CCD camera (PCOensicam) records pictures with a resolution of 1280 × 1024 pix-ls, whereas the measurement area used 970 × 600 pixels of theCD array, thus one pixel is 0.05 mm in real length scale. A squareerspex box filled with water was placed outside the cylindricallass column to minimize the effect of optical distortion. The ILAynchronizer synchronized the laser and the cameras.

In the single-phase scenario water was used. In a common denseiquid–liquid system PIV measurements are not possible becausef the different refractive index of the phases, where the dis-ersed second phase masks the measurement area. Therefore, an

so-optical system introduced by Augier et al. (2003) was applied

ig. 6. Comparison PIV–FPM, measured (top) and simulated (bottom) average veloc-ty magnitude and vectors in 2D xz-plane in one compartment, single-phase flow.

2954 C. Drumm et al. / Computers and Chemical Engineering 32 (2008) 2946–2957

nmsdidT(fcbwdf

5

Atfl

Fm

Fm

siFitbi&tpstapuFh

5

fl1tTdwt

Fig. 7. Fluent single-phase (top) and two-phase (bottom) flow simulations.

ot mask the measurement area and allow a clear view. Further-ore, a small quantity of the fluorescent dye Rhodamine 6G only

oluble in the aqueous phase is dissolved to enhance the phaseiscrimination. The dye emits a 560 nm wavelength orange light

n the continuous phase. Hollow spherical glass particles with aiameter of 10 �m and a density of 1100 kg/m3 seeded the flow.herewith almost neutral buoyancy is assured for both the single= 1000 kg/m3 for water) and two-phase scenario (= 1180 kg/m3

or glycerine–water mixture). The interrogation areas for the crossorrelation were 64 × 64 pixel with a 50% overlapping, followedy an 32 × 32 pixel adaptive cross correlation. Local median filtersere used to eliminate outliers, which were interpolated. One hun-red pairs of pictures were taken to filter the effects of turbulenceor the determination of ensemble-averaged velocity fields.

. Numerical results

Simulations were carried out on the geometry depicted in Fig. 2.s a matter of course the grid was removed for the FPM simula-

ions to release the vantages of a meshfree method. The volumeows, systems and stirrer revolutions in the single- and two-phase

ig. 8. Comparison PIV–FPM–Fluent, measured and simulated average velocityagnitude in 2D xz-plane at stirrer level, single-phase flow.

ccti

Fm

ig. 9. Comparison PIV–FPM–Fluent, measured and simulated average velocityagnitude in 2D xz-plane at upper half compartment height, single-phase flow.

imulations are the same as in the pilot plant column describedn Section 4. In this paper we focus on FPM and want to comparePM to PIV experiments and standard CFD approaches. Therefore,n the Fluent simulations just the standard k– model was used forhe closure of turbulence without considering more suitable tur-ulence models. A comparison of different closures for turbulence

n the framework of Fluent was done already elsewhere (DrummBart, 2006). Since only 2D PIV measurements were possible in

he extractor, the velocities are compared in the PIV measurementlane (xz-plane). The instantaneous velocities resulting from FPMimulations are shown in Fig. 3. To compare these results withhe average velocities of PIV and Fluent (RANS k– model) time-veraged velocities were computed using 50 instantaneous velocityrofiles after steady state was reached. Around 10–20 s were sim-lated in order to reach this steady state, whereas the time step inPM simulations was around 1.5E −3 and around 230,000 particlesave been used for each phase.

.1. Single-phase simulation results

In the shown results of the single-phase scenario, the volumeow of the aqueous phase is 200 l/h and the stirrer revolution is50 rpm. Contours of average velocity magnitude and vectors fromhe FPM simulations for the whole geometry are shown in Fig. 4.he velocities from FPM, PIV and Fluent are compared along threeifferent lines inside the compartment. The position of these lines,hich are at the stirrer level (0.045–0.075 m) and halfway between

he stirrer and the upper and lower stators of one extraction

ompartment (0.027–0.075 m), are shown in Fig. 5. A qualitativeomparison between the PIV measurements and the FPM simula-ions is visible in Fig. 6. It can be seen that the FPM simulations aren good agreement with the experiments. As expected, the highest

ig. 10. Comparison PIV–FPM–Fluent, measured and simulated average velocityagnitude in 2D xz-plane at lower half compartment height, single-phase flow.

C. Drumm et al. / Computers and Chemical Engineering 32 (2008) 2946–2957 2955

Fv

vcotaaRFnm(ht

Fm

Fm

aaaetimasRmw

5

opmutwpFAac

ig. 11. Comparison PIV–FPM, measured (top) and simulated (bottom) averageelocity magnitude and vectors in 2D xz-plane in one compartment, two-phase flow.

elocities are near the stirrer tip. From the velocity vectors and theontours, two big vortices can be estimated in the compartment;ne between the stators and the other between the stirrer. A smallhird vortex is visible above the stirrer because the flow field turnsround at this position towards the stirrer tip. All these phenomenare obvious in both the FPM simulations and experiments in Fig. 6.esults of the Fluent simulations are shown in the upper part ofig. 7. In contrast to FPM, the big vortex between the stirrers isot well developed in Fluent using the standard k– model. The

aximum velocity at the stirrer tip is also better predicted by FPM

Fig. 6), while Fluent under-predicts this value (Fig. 7). On the otherand, FPM tends to over-predict the velocities in the dead zones inhe middle of each big vortex. The comparisons for the velocities

ig. 12. Comparison PIV–FPM–Fluent, measured and simulated average velocityagnitude in 2D xz-plane at stirrer level, two-phase flow.

atctv

Fm

ig. 13. Comparison PIV–FPM–Fluent, measured and simulated average velocityagnitude in 2D xz-plane at upper half compartment height, two-phase flow.

long the different lines are shown in Figs. 8–10. For the velocitiest the stirrer level it is obvious that FPM over-predicts the velocitiest the stirrer level, while Fluent under-predicts these velocities. Anxplanation for the deviation near the stirrer could be the fact thathe field of maximum velocities near the stirrer tip is more broadn FPM. For the comparisons at the upper and lower half compart-

ent height, FPM can predict the velocity profile better than Fluent,lso because the vortices are better developed, again comprising alight over-prediction for FPM and an under-prediction for Fluent.ecapitulatory, one can say that FPM matches the overall experi-ental data and can describe all flow phenomena such as vortices,hereas at some points FPM tends to over-predict the velocities.

.2. Two-phase simulation results

In the shown results of the two-phase scenario, the volume flowf the aqueous phase is 100 l/h, the volume flow of the organichase is 100 l/h and the stirrer revolution is 150 rpm. The Sauterean diameter of the droplet distribution (d32 = 0.0025 m) was

sed in the drag term in Eq. (6) for both the FPM and Fluent simula-ions. The Fluent results, applying the standard k– model togetherith the Euler–Euler multiphase model, are depicted in Fig. 7(lowerart). Contours of average velocity magnitude and vectors from thePM simulations and the PIV measurements are shown in Fig. 11.s in the single-phase scenario, the FPM simulations are in goodgreement with the experiments (Fig. 11). Still, two big vorticesan be estimated in the compartment; one between the stators

nd the other between the stirrer. Because of the rising dropletshe vortices shift or turn around and are directed to the top of theolumn, so that the vortex between the stirrer moves to the posi-ion above the stirrer and vice versa. The small third vortex is nowisible under the stirrer because of the opposite rotating direction.

ig. 14. Comparison PIV–FPM–Fluent, measured and simulated average velocityagnitude in 2D xz-plane at lower half compartment height, two-phase flow.

2956 C. Drumm et al. / Computers and Chemical

Fv

Affvtsus2rxttvmWapa

6

rtcwtattwctl

flocmLsod

A

(

R

A

B

B

B

C

D

D

F

F

G

G

G

G

H

I

K

K

L

L

L

L

M

M

M

M

ig. 15. Comparison FPM–Fluent, simulated velocity magnitude including swirlelocity, two-phase flow.

gain FPM can predict all these flow phenomena. The velocitiesrom FPM, PIV and Fluent are again compared along the three dif-erent lines Figs. 11–14. FPM correctly predicts the shape of theelocity profile along the lines through the compartment. However,he same trends as in the single-phase case can be recognized; FPMlightly tends to overestimate the velocities while the velocities arendervalued in the Fluent simulation. The overall prediction is rea-onable. It has to be pointed out that especially the velocities in theD PIV measurement area are hard to predict since they are onlyesults of the shear stresses while the main velocity is normal to thez-plane. Although no experimental measurements in this direc-ion are possible, in Fig. 15, the simulated overall velocity includinghe velocity vertical to the 2D xz-plane is depicted. The maximumelocity near the stirrer is the track speed of the stirrer tip (v = 0.7/s). From the stirrer to the wall the velocity is decreasing to zero.hile the velocities at the boundaries are predicted correctly and

re the same in both FPM and Fluent, it is again obvious that FPMredicts a higher velocity in the y-direction between the stirrersnd from the stirrer to the wall.

. Summary and conclusions

Single- and two-phase simulations of a RDC extractor were car-ied out in the framework of the meshfree code FPM and comparedo 2D PIV measurements. In addition to that the results were alsoompared to Fluent simulations where the standard k– modelas applied. The results show that FPM can predict the one- and

wo-phase flow field in the RDC, whereas all flow phenomena suchs vortices can be described. FPM matches the overall experimen-al data and the predicted velocities are in good agreement with

he experimental ones. Both methods predict reliable flow fieldshereas FPM do not pale in comparison to the commercial CFD

ode. At some points FPM tends to overestimate the velocities whilehe velocities are undervalued in the Fluent simulation. Recapitu-atory, one can say that FPM was successfully applied to model the

R

S

Engineering 32 (2008) 2946–2957

ow fields in a stirred extraction column. In future, meshfree meth-ds like the FPM have to be considered as an attractive alternative tolassical mesh based methods like finite difference, finite elementethods in the field of chemical engineering. Since FPM is a fully

agrangian method, it is more suitable for the simulation of theecondary phase in chemical engineering problems: here the sec-ndary phase consists normally of moving bubbles, particles androplets.

cknowledgement

We would like to thank the Deutsche ForschungsgemeinschaftDFG) for the financial support of our research.

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