Hydrodynamic modeling of fluidized-bed crystallizers with use of the multiphase CFD method

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Chemical Engineering and Processing 48 (2009) 1527–1533

Contents lists available at ScienceDirect

Chemical Engineering and Processing:Process Intensification

journa l homepage: www.e lsev ier .com/ locate /cep

ydrodynamic modeling of fluidized-bed crystallizers with use of theultiphase CFD method

anusz Wójcik ∗, Roch Plewikepartment of Chemical Engineering, Silesian University of Technology, M. Strzody 7, 44-100 Gliwice, Poland

r t i c l e i n f o

rticle history:eceived 4 March 2009eceived in revised form 7 October 2009

a b s t r a c t

This article presents modeling of fluidized-bed crystallizers with use of the multiphase CFD method forthe first time. There are substantial differences in predictions of the model and one-phase simulations.

© 2009 Elsevier B.V. All rights reserved.

ccepted 11 October 2009vailable online 20 October 2009

eywords:rystallizationluidization

omputationluid mechanics

. Introduction

Ever since in the nineteen twenties the fluidized-bed crystal-izers (FBCs) have been introduced into the industrial practice, [1]hey are used in the situations where large crystals are required [2].he success of this technology would hardly be possible without aonstant progress in the design of FBCs and a better understandingf their performance resulting from the extensive investigations3,4]. But the FBC technology has still considerable potentialsor development and to achieve it more accurate modeling toolsre required. Most of the previous studies of FBCs made use ofhe ideal classifying bed model (cf. [5–9]), the inherent draw-ack of which is the inability to predict the polydispersity ofrystals.

To take into account particle segregation and their longitudinalixing in the bed Frances et al. [10] considered a series of idealixers, one upon another as one multistage crystallizer. Shiau and

u [3] used this concept for batch FBC. Basing on the earlier ideas ofennedy and Bretton [11] also the axial dispersion modeling con-ept was applied in the studies of FBC performance [12,13] to giveore accurate prediction of the real situation.CFD techniques were also used to analyse crystallization and

recipitation processes in a last decade (cf. [14–16]), however, forarticle concentrations smaller then 5%.

In the most recent studies of FBC [17,18] we used a one-phasend two-phase models of CFD to investigate the performance of a

∗ Corresponding author.E-mail address: [email protected] (J. Wójcik).

255-2701/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.cep.2009.10.006

conical apparatus equipped with a central tube and stator and alsobehaviour of monodyspersed fluidized-bed in such crystallizer. Theinlet velocity to the central pipe was a process parameter and thesizes of crystals hold in the bed were determined. The next fourcrystallizer configurations were investigated: conical end of thecentral tube, conical end of the central tube and stator, straight cen-tral tube, straight central tube and stator. The second one seemedto be the best because allowed for larger liquor velocities, which isequivalent to smaller apparatus. Another advantage of this config-uration is more piston like liquid profile at the outlet which allowfor smaller particles to be hold-up in the FBC.

The aim of these studies was to obtain even closer portrayal ofthe FBC operation, especially when the start-up solids concentra-tions are about 10%. To allow for that effect we used not a single-but multiphase CFD code. To our best knowledge this is a first suchstudy of FBCs.

2. Modeling

One of the most popular types of fluidized-bed crystallizers isOslo fluidized-bed crystallizer with classification leg, which is pre-sented in Fig. 1. Crystals are fluidized by means of fresh solutioncirculation, going to the bed from a central tube and coming outby an overflow. The smallest crystals are elutriated from the bedby streams of solution. The largest crystals, which sedimentation

velocity is larger than the solution velocity, pass into the classifica-tion leg. Smaller crystals are removed from it by an additional inletstream.

In [19] there are some industrial data concerning NaCl crys-tallization, which were used for preparation of a model. This

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1528 J. Wójcik, R. Plewik / Chemical Engineering

r(dtff

R

pu

etpF

2

ppSsadvcts

Unstructured three-dimensional mesh created in programme

Fig. 1. Scheme of Oslo fluidized-bed crystallizer with classification leg [29].

eport gives information about production of 3 t/h of 2–3 mm NaCl40 mass% in suspension) crystals from 230 m3 apparatus of 6 miameter. Simple calculations give equivalent porosity at the bot-om of classification leg equal to ε = 0.7328. This value is requiredor computation of crystals sedimentation velocity. Using modifiedormula of Todes

e = Arε5.75

18 + 0.6√

Arε4.75(1)

roper conditions of flow can be calculated. The working supersat-ration level is read from [20].

It is decided to diminish 10 times the dimensions of the mod-led crystallizer in order to optimize required computation timeogether with accuracy of the 3D model. The simulations wereerformed by commercial packet of computational fluid dynamicsluent 6.x.

.1. Model equations

For Eulerian multiphase calculations, FLUENT uses thehase coupled SIMPLE (PC-SIMPLE) algorithm [21] for theressure–velocity coupling. PC-SIMPLE is an extension of theIMPLE algorithm [22] to multiphase flows. The velocities areolved coupled by phases, but in a segregated fashion. The blocklgebraic multigrid scheme used by the density-based solverescribed in [23] is used to solve a vector equation formed by the

elocity components of all phases simultaneously. Then, a pressureorrection equation is built based on total volume continuity ratherhan mass continuity. Pressure and velocities are then correctedo as to satisfy the continuity constraint.

and Processing 48 (2009) 1527–1533

2.1.1. The pressure-correction equationFor incompressible multiphase flow, the pressure-correction

equation takes the form

n∑k=1

1�rk

{∂

∂t˛k�k + ∇ · ˛k�k ��′

k + ∇ · ˛k�k ��∗k

}= 0 (2)

where �rk is the phase reference density for the kth phase (definedas the total volume average density of phase k), ��′

kis the velocity

correction for the kth phase, and ��∗k

is the value of ��k at the cur-rent iteration. The velocity corrections are themselves expressed asfunctions of the pressure corrections.

2.1.2. Volume fractionsThe volume fractions are obtained from the phase continuity

equations. In discretized form, the equation of the kth volume frac-tion is

ap,k˛k =∑

nb

(anb,k˛nb,k) + bk = Rk (3)

aa,k is the relative velocity, bk represents the drift or the relativevelocity. In order to satisfy the condition that all the volume frac-tions sum to one,

n∑k=1

˛k = 1 (4)

2.1.3. Fluid–solid exchange coefficientThe fluid–solid exchange coefficient Ksl can be written in the

following general form:

Ksl = ˛s�sf

�s(5)

where f is defined differently for the different exchange coefficientmodels (as described below), and �s, the “particulate relaxationtime”, is defined as

�s = �sd2s

18�l(6)

where ds is the diameter of particles of phase s. All definitionsof f include a drag function (CD) that is based on the relativeReynolds number (Res). It is this drag function that differs amongthe exchange coefficient models.

For the model of Wen and Yu [24], the fluid–solid exchangecoefficient is of the following form:

Ksl = 34

CD˛s˛l�l| ��s − ��l|

ds˛−2.65

l(7)

where

CD = 24˛lRes

[1 + 0.15(˛lRes)0.687] (8)

and Res is defined by Eq. [25]

Res = �lds| ��s − ��l|�l

where the subscript l is for the lth fluid phase, s is for the sth solidphase, and ds is the diameter of the sth solid phase particles.

2.2. Grid optimization

Gambit for the whole volume of crystallizer was applied to cal-culations. The grid was created from mixed (tetrahedral/hybrid)elements. Quality of the mesh was assessed using EquiAngle Skewcriterion. It was not permitted to the worst element of grid to reach

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J. Wójcik, R. Plewik / Chemical Engineering and Processing 48 (2009) 1527–1533 1529

Fig. 2. The influence of the size of crystals on porosity of the fluidal-bed for ˛ = 10%; (a) d = 0.5 mm; (b) d = 1.5 mm; (c) d = 0.4–3 mm.

tIo

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wo

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he value of QESC = 0.8. In every case the worst element was located.f its coordinates were in the crucial zone of the apparatus, i.e.utside the central pipe, the mesh was adapted.

Preliminary checking calculations were conducted for structuraleshes in the range of turbulent flow. The grids of different number

f computational cells from 20,000 to 600,000 have been tested. Usef both thinner and denser computational meshes caused worseonvergence of calculations at the level of 10−4 of normalized resid-als. For the thinner meshes worse results were obtained becausef large differences in values of average speeds and momentumn neighbouring cells. When mesh was thick averaging of solidsoncentration was the problem together with calculation time. Weppreciate professional advice from Ansys Inc. to solve this issue.

To final calculations the grid with 54,019 computational cellsas chosen, because it has given convergence about 10−6 to 10−8

f normalized residuals and y+ > 11.225 for applied model.For one-phase modeling [17] standard k–ε model was used

or the simulations. In the case of multiphase flow, the Eulerianultiphase model with standard k–ε method added by k–ε mul-

iphase dispersed model has been used, respectively. To obtainteady state and convergence of 10−6 to 10−8 of normalized resid-als, about 1000–2000 first order upwind iterations were required

or one-phase model [17]. For the multiphase model calcula-ions were performed in an unsteady system until reaching nineesidence times. Then the convergence of 10−4 to 10−7 of normal-zed residuals was obtained for about 42,000 first order upwindterations.

2.3. Validation of the CFD model

Messing and Hofmann [19] had given industrial data for crystal-lization of NaCl in the fluidized-bed crystallizer with mean crystalproduct ca 3 mm. The crystallizer diameter was 6 m and workingvolume equal to 230 m3. From “normal”-vacuum crystallization theproduct is about 0.5 mm. If these crystals are used as seed they hadto be retended in the bed. From our simulations could be seen thatcrystals with the diameter 0.4 mm are almost completely stay inthe bed. Pulley [8] and Bransom [5] had found that ratio of seed andproduct crystals diameters is about 0.1–0.125. Our value is equal to0.133 by axial velocity of Messing and Hofmann [19]. It seems tobe an additional validation of the CFD model.

3. Results and discussion

In Fig. 2 the volumetric concentration of mother liquor (theporosity of the bed) is presented. It is worth to note, that 3D modelis considered, which is not axial symmetrical. Fig. 2a shows itfor fluidization of monodispersed 0.5 mm crystal particles. Fig. 2bdemonstrates it for 1.5 mm crystals, respectively. Fig. 2c presentsit for polydispersed bed consisted of 10 size classes of particles,

i.e. 0.4; 0.6; 0.9; 1.2; 1.5; 1.8; 2.1; 2.4; 2.7 and 3 mm. Solids totalvolumetric concentration in each case is equal to ˛ = 0.10. For thepolydispersed bed (Fig. 2c) the volumetric concentration of eachsize class is equal to ˛ = 0.01. The slabs on the left side of Fig. 2cshow in 3D perspective also the porosity of the bed. In Fig. 2b two

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1530 J. Wójcik, R. Plewik / Chemical Engineering and Processing 48 (2009) 1527–1533

Fig. 3. The distribution of size volume fractions for the polydispersed bed (a) d = 0.4 mm; (b) d = 0.9 mm; (c) d = 1.8 mm; (d) d = 3 mm.

cooItittootsispttwwtc

ross-sections (A-A and B-B) are marked for future use. One canbserve that there are no solids present in the neighbourhood of theutlet of central pipe and the upper outlet of the apparatus (Fig. 2c).t proves that the inlet velocity is properly chosen. It prevents set-ling and incrustation of crystals on the bottom of apparatus andn the proximity of the outlet of central pipe. Fig. 2a gives evidencehat expansion of the bed takes part in the whole working space ofhe apparatus for 0.5 mm grains. Bright strap visible near the upperutlet from apparatus marks that crystals are not totally washedut from the bed. Accumulation of grains could be seen outsidehe conical part of the central pipe. Such concentration profile ofolids proves existence of circulation loop of particles in the work-ng space of apparatus instead of ideal classification layers. Fig. 2bhows smaller expansion of the bed of 1.5 mm particles in com-arison to 0.5 mm crystals. Larger concentration of solids outsidehe conical part of the central pipe and on the outer wall of the

ank at top of the bed is caused by formation of circulation loop asell. Fig. 2c presents expansion of polydispersed bed in the wholeorking space of apparatus. Circulation loops could be seen too. On

he basis of earlier investigations [17] where next four crystallizeronfigurations were investigated: conical end of the central tube,

conical end of the central tube and stator, straight central tube,straight central tube and stator, it was confirmed, that the second isthe best internal configuration when considering the vertical speedin section A-A (Fig. 2b). It should be as small as possible and morepiston like, which prevents washing out of the smallest crystals(seed) from the bed.

Fig. 3 shows distribution of solids volume fraction of differentsize classes in polydispersed bed in the working space of the appa-ratus for the same case as at Fig. 2c. Four most interested picturesare chosen for presentation. The decrease of expansion of the bedwith increase of diameter of crystals could be seen. As larger theparticles the bigger their concentration at the bottom of the segre-gation pipe. The circulation loops could be seen even more clearlythan before. Fig. 3a presents that particles of 0.4 mm diameter arekept at top of apparatus. Such small crystals were washed out ofthe bed in the two-phase monodispersed system. Reason for such

behaviour could be influence of the polydispersed bed similar tohindered settling. Dark strap in the upper section of the crystallizerconfirms that phenomenon. Therefore crystals of the diameter of0.4 mm can be applied as seed, which is in good agreement with[19].

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J. Wójcik, R. Plewik / Chemical Engineering and Processing 48 (2009) 1527–1533 1531

˛ = 10

Fttsfltdao

tetrcw

pscip

Fig. 4. Vectorial liquid velocity profiles for (a) pure water; (b)

Fig. 4 presents vectorial liquid velocity profiles within the tank.ig. 4a shows it for one-phase model. Fig. 4b and c demonstratehem for two-phase monodispersed systems of 0.5 and 1.5 mm par-icles. Fig. 4d display it for the polydispersed system. Everyone canee clearly presence of the circulations loops. However, they are dif-erent in character for every case. For pure water there is one largeoop in the whole volume of the tank. In the monodispersed sys-ems separation of two loops could be seen. Polidispersed systememonstrates separation of two, differently directed loops, largert the bottom and smaller in the middle of the bed and stabilizationf flow in the upper part of the crystallizer.

Fig. 5 presents radial velocity profiles in the section B-B of theank for the same cases as in Fig. 4. For pure water the flow isvenly distributed in entire section (Fig. 5a). Monodispersed sys-ems exhibit disturbances in such flow (Fig. 5b and c) with someadial circulation loops, where the polidispersed system totallyhaotic flow. For monodispersed suspension such chaos increasesith the diameter of particles.

Vertical velocity profiles in the A-A (Fig. 2b)) cross-section is

resented in Fig. 6. The A-A section is very important for propereeds crystals hold-up in the bed. More piston like profile allows toatch smaller crystals more uniformly in the whole cross-sectionnstead of tunneling some of them in the peak position of therofile. It can be seen, that the solids cause velocity suppression

% d = 0.5 mm; (c) ˛ = 10% d = 1.5 mm; (d) ˛ = 10% d = 0.4–3 mm.

(e.g. line d) and diameter reduction of the circulation loops. It isinteresting to note, that predictions of the one-phase model aresimilar to that of the two-phase one for smaller particles. Unfor-tunately they differ much in comparison to the polydispersedsystem.

Comparing Figs. 2 and 4 it can be seen that the largest con-centration of solids is at peak of circulation loops. Probably it iscaused by the cyclonic action of the loops. It seems that it couldcause additional attrition of crystals. In the same time the loopsallow for better supersaturation relief within the crystallizer [26].The vortex liquid flow, generates additionally energy dissipation,which increases crystals destruction and erosion of the inner jacketsurface.

The results proof earlier observations from literature [16,18],concerning strong polydispersed particles concentration distribu-tions in both horizontal as well as in vertical cross-section of theapparatus. The character of liquid flow demonstrates existence oninner circulation loops in both mentioned directions (Figs. 4–6).These inner circulation loops doubtless affect the heterogeneous

behaviour of the bed. The shape of apparatus shell necessary forhydraulic classification of polydispersed suspension in the indus-trial conditions, as well as conical outlet from the central pipecauses disadvantageous axial and radial velocity distribution in theannular zone of the vessel (Figs. 4–6).

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1532 J. Wójcik, R. Plewik / Chemical Engineering and Processing 48 (2009) 1527–1533

Fig. 5. Radial velocity profiles in the B-B cross-section for (a) pure water; (b

Fw

4

aEbsm

mnt

ig. 6. Comparison of solution velocity profiles in the A-A cross-section for (a) pureater; (b) ˛ = 10% d = 0.5 mm; (c) ˛ = 10% d = 1.5 mm; (d) ˛ = 10% d = 0.4–3 mm.

. Conclusions

The applied method of multiphase CFD is a useful tool for thenalysis of fluidized-bed crystallization. For multiphase flow, theulerian multiphase (MP) model with standard k–ε method addedy k–ε (MP) dispersed model has been used respectively. There areubstantial differences in predictions of MP model and one-phaseodel, which was found in previous CFD studies by Ref. [27].

It would be very interesting to couple population balance with

ass transfer and crystallization kinetics to the present hydrody-amic model, requiring further work and a lot of computationalime, like in Ref. [28].

) ˛ = 10% l = 0.5 mm; (c) ˛ = 10% d = 1.5 mm; (d) ˛ = 10% d = 0.4–3 mm.

Collected data, other way not to receive, put new light on thesuspension flow behaviour in the annular zone of the FBC.

Appendix A. Nomenclature

Ar Archimedes numberRe Reynolds numberb represents the drift or the relative velocityd diameter (m)K the fluid–solid exchange coefficientm mass flow rate (kg/s)QESC coefficient of EquiAngle Skewy+ dimensionless distance from the wall˛ volume fractionε porosity of the bed� dynamic viscosity (Pa s)� density (kg/m3)� the “particulate relaxation time”�� overall velocity vector (m/s)

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