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chemical engineering research and design 9 2 ( 2 0 1 4 ) 2213–2222

Contents lists available at ScienceDirect

Chemical Engineering Research and Design

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

ixing performances of swirl flow and corrugatedhannel reactors

kram Ghanema, Charbel Habchib, Thierry Lemenanda,ominique Della Vallea,c, Hassan Peerhossainid,∗

LUNAM Université, Laboratoire de Thermocinétique de Nantes, CNRS UMR 6607, 44306 Nantes, FranceEnergy and Thermo-Fluids Group ETF, School of Engineering, Lebanese International University LIU, Beirut,ebanonONIRIS, 44322 Nantes, FranceUniv Paris Diderot, Sorbonne Paris Cité, Institut des Energies de Demain (IED), CNRS-UMR 8236, Paris, France

a b s t r a c t

Four different geometrical solutions for tubular reactors are compared for transfer intensification in fluid processes:

(1) a compact multi-tube with helical screw-tape inserts, (2) a plain corrugated channel with a smooth bend curva-

ture, called “wavy channel”, (3) a plain corrugated channel with a herringbone pattern, called “zigzag channel”, and

(4) a plain straight pipe serving as the reference case. The single-phase mixing abilities of these four devices are com-

pared by the chemical probe method (Villermaux/Dushman iodide/iodate system) for a range of main-flow Reynolds

numbers between 100 and 4000. The chemical probe method is used here to investigate the global mixing time in the

entire reactor volume, as deduced from the segregation index by a phenomenological model. Experimental results

reveal better mixing performance and reduced energy expenditures in the helical-insert tube, in both the laminar

and turbulent regimes.

© 2014 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

Keywords: Mass transfer intensification; Static mixer; Continuous multifunctional heat exchanger/reactor; Helical

screw-tape insert; Herringbone-patterned pipe; Iodide/iodate chemical probe

ally, in static mixers, the energy cost comes from the external

. Introduction

ndustrial fluid processes include mixing, reaction, and/oreat transfer operations that are in many cases concomitantnd strongly coupled. The desire for elevated productionates, increased process safety and high product quality,hile maintaining low manufacturing and operation costs,ave led to the development of in-line continuous staticixers and multifunctional heat exchangers/reactors (MHER)

s an alternative to batch production in stirred vessels. Stirredessels are still used for most industrial applications, thankso their versatility and monitoring flexibility: easy tempera-ure control with hydrodynamics and dilution of reactants,ecoupling of residence time and shear rates, mixing ofard-to-pump viscous products, etc. Yet they present severalrawbacks compared to continuous processing: space require-

ents, equipment operation and maintenance costs, broad

∗ Corresponding author. Tel.: +33 607 53 31 61.E-mail address: hassan.peerhossaini@univ-paris-diderot.fr (H. PeerhReceived 6 August 2013; Received in revised form 6 January 2014; AcceAvailable online 2 February 2014

263-8762/$ – see front matter © 2014 The Institution of Chemical Engittp://dx.doi.org/10.1016/j.cherd.2014.01.014

residence-time distributions, poor selectivity due to localizedmixing and large segregation zones, lack of isothermal oper-ation especially in the central zones, undesirable byproducts,and reduced safety and process control due to large batchfluid volumes, in addition to high power requirements by themobile parts. These considerations have promoted the useof stationary mixing elements in chemical, pharmaceutical,polymer synthesis, food processing, pulp and paper, paintand resin, water treatment, petrochemical industries, etc.

Static mixers may be used for operations such as mixingof miscible fluids, heat transfer and thermal homogenization,and liquid–liquid dispersion as well as gas–liquid dispersionor gas–gas mixing where similar, and sometimes better per-formances can be achieved at lower cost than other processingtechniques (Anxionnaz et al., 2008; Thakur et al., 2003). Actu-

ossaini).pted 11 January 2014

pumping power needed to propel materials through the mixer,

neers. Published by Elsevier B.V. All rights reserved.

2214 chemical engineering research and design 9 2 ( 2 0 1 4 ) 2213–2222

and numerous works have shown that it reaches much lowerlevels than that of stirred tanks per fluid unit mass (Anxionnazet al., 2008; Bayat et al., 2012; Ferrouillat et al., 2006a, 2006b; Shiet al., 2011; Thakur et al., 2003). Moreover, temperature con-trol is easier because the system is smaller, allowing the useof more concentrated reactants. Better selectivity and effluentreduction are important assets in continuous-flow solutions(Poux et al., 2010; Ferrouillat et al., 2006c).

Fluid mixing is a unit operation aimed at achieving ahomogeneous distribution of the fluid properties, since thisis essential for product quality in numerous industrial pro-cesses. It is involved in various heat- and mass-transferoperations. Mixing by molecular diffusion is not appropri-ate in many industrial processes – a length of about 200diameters would be needed to obtain a well-mixed fluid ina straight pipe. Static mixers are accordingly designed to gen-erate convective transfer, through modified surfaces or seriesof motionless inserts fixed in pipes that guide the fluid flowacross the pipe section. This radial transfer is essential toachieve homogeneity within a short distance while keepingpressure losses moderate. The purpose of the inserts or geo-metrical modifications is to divide and redistribute the fluidstreams sequentially or to trigger the formation of specific flowstructures by passively manipulating the natural flow forcesin order to intensify transverse particle displacement from thewall vicinity to the active core flow.

Numerous static mixer designs have been proposed ondifferent scales, but only a few models are used in industryand comparative performance studies remain few (Anxionnazet al., 2008; Barrué et al., 2001; Bourne et al., 1992b; Cybulskiand Werner, 1986; Hessel et al., 2003; Meijer et al., 2012;Pahl and Muschelknautz, 1982; Thakur et al., 2003; Theronand Le Sauze, 2011; Zhang et al., 2012). Among the highest-performance devices studied in the literature, the KenicsTM

mixer (Chemineer) has a privileged place in the ranking (Jafferand Wood, 1998; Hobbs et al., 1998; Rahmani et al., 2005), asdo the SMXTM (Li et al., 1997; Streiff and Kaser, 1991; Talansieret al., 2013) or SMVTM mixers (Sulzer) (Lobry et al., 2013;Paglianti and Montante, 2013). Choosing a suitable mixer con-figuration for a given application needs the quantification ofthe mixing process, a fundamental requirement for modernprocesses seeking better energy efficiency and product qual-ity, especially those including fast chemical reactions (Ehrfeldet al., 1999; Guo et al., 2013; Hsiao et al., 2014; Panic et al.,2004; Schonfeld et al., 2004; Stankiewicz and Moulijn, 2000;Wang et al., 2012). When the reactions have characteristictimes smaller than the mixing time, mixing kinetics becomesa key parameter for selectivity and overall reaction yield. Mix-ing efficiency, especially on the molecular scale, is linked tothe capacity of the flow to provide fresh reactant faster thanthe reactant consumption by reaction, and determines bothproductivity and selectivity by hindering undesirable slowersecondary reactions.

From a physical point of view, mixing is a multi-scalephenomenon. Three parallel mechanisms at different scales,namely macromixing, mesomixing, and micromixing can bedistinguished (Baldyga and Bourne, 1999).

Macromixing is the homogenization at the scale of thewhole vessel that determines the environmental concentra-tions in the flow domain. This large-scale fluid distributionis affected by the mean velocity field and thus by particletransport between high- and low-momentum regions in the

heat exchanger/reactor volume (Baldyga and Bourne, 1999).Macromixing is generally characterized by the residence time

distribution (RTD), the time taken by the fluid particles tomigrate from the device inlet to the outlet, an indicator ofvelocity-field uniformity (Castelain et al., 1997; Mokrani et al.,1997; Habchi et al., 2009a; Villermaux, 1986). RTD steepnesscan be improved by generating radial convective transfer, forinstance by trailing vortices downstream of vortex generators,or by using baffles that perturb the fluid path (Ajakh et al., 1999;Ferrouillat et al., 2006a; Fiebig, 1995; Ghanem et al., 2013a;Habchi et al., 2010, 2012a, 2012b; Lemenand et al., 2003, 2005,2010; Mohand Kaci et al., 2009, 2010; Momayez et al., 2004,2009, 2010; Mutabazi et al., 1989; Toe et al., 2002).

At the intermediate scale, mesomixing is the coarse-scaleexchange between the fresh feed fluid and its surroundings,governed either by fluctuations in turbulent flow or fractalstructures in laminar flow (Baldyga et al., 1995). In turbu-lent flow, the mechanism of scale reduction is governed bythe energy cascade in the inertial-convective subrange of theturbulent spectrum, with wave numbers ranging betweenthe integral and Kolmogorov scales. Thus, mesomixing is ahomogenization process by advection due to velocity fluctua-tions, basically governed by the particle random path (similarto a mesoscopic diffusion based on the turbulence statistics,instead of Boltzmann statistics). Governed by the turbulentfield, mesomixing is related to the turbulent kinetic energy(TKE) k, the Prandtl length scale LP, their combination in theturbulent diffusivity, Dt, or even to the Reynolds stress ten-sor (Habchi et al., 2010). In laminar flow, the chaotic path ofthe fluid particles, usually induced by periodic flow directionmodifications, can be characterized by the Lyapunov expo-nent (Castelain et al., 2001; Ghanem et al., 2013c; Habchi et al.,2009a, 2009b; Le Guer and Peerhossaini, 1991; Lemenand andPeerhossaini, 2002; Muzzio et al., 1992; Ottino, 1989; Toussaintet al., 1995).

Finally, micromixing is the ultimate mixing scale includ-ing molecular diffusion. The selectivity of chemical reactionsdepends on micromixing because it determines molecu-lar contact (Baldyga and Bourne, 1999). In turbulent flow,the prevailing mechanism takes place in the viscous-convective subrange, i.e. for wave numbers ranging betweenthe Kolmogorov and Batchelor scales. Turbulent fluctua-tions vanish and laminar stretching accelerates the aggregatesize reduction up to the molecular diffusion scale, whichquickly dissipates the concentration variance (Batchelor,1953; Baldyga and Bourne, 1989). The limiting mechanismin this process is engulfment in the small vortices near theKolmogorov scale (Baldyga and Bourne, 1999). It can be charac-terized by a micromixing time that is linked to the turbulenceenergy dissipation rate (Baldyga and Bourne, 1999). Follow-ing the Hinze–Kolmogorov theory based on the idea of energycascade, the drop breakup in multiphase flows is also char-acterized by the turbulence kinetic energy dissipation rate(Hinze, 1955; Lemenand et al., 2013; Streiff et al., 1997). Thus,an increase in turbulence kinetic energy dissipation favors themicromixing process, enhancing the selectivity of fast chem-ical reactions. In laminar flows, micromixing results from thereduction in striation thickness up to the diffusive scales, andcan be approximated by the stretching efficiency model usedhere (Falk and Commenge, 2010). Consequently, micromixingis responsible for the global performance of the MHER whenthe mixing at large scale, namely macromixing, is not limit-ing.

Several qualitative and quantitative techniques have beendevised to study the global effects of these mechanisms in

the reactors; they include acid–base or pH indicator reactions,

chemical engineering research and design 9 2 ( 2 0 1 4 ) 2213–2222 2215

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clmatrtmtcm

tmfpcXdsci

egir(mfaodad

ilution of colored dyes or fluorescent materials, reactionsielding colored species, monitoring species concentration,nd competing consecutive or parallel reactions (Ghanemt al., 2013b). The latter technique, namely the chemical probeethod, has been successfully used to evaluate micromixing

n batch and open-loop reactors with different chemical sys-ems (Barthole et al., 1982; Bourne et al., 1992a; Brucato et al.,000; Ehlers et al., 2000; Fournier et al., 1996a).

The methods that employ competitive-consecutive orompetitive-parallel reactions are based on the result of theocal injection of a reagent in stoichiometric deficit in the

ain flow. Two reactions competing for a common reactantre carried out (Habchi et al. 2013), one should be very fast, andherefore proceeds only if mixing is extremely rapid. The othereaction should be fast but slower than the first reaction, andakes place when there is an excess of the common reactant,

eaning when mixing is too slow to renew the quantities ofhe common reagent needed by the faster reaction. The localhemical reaction thus results from a competition betweenixing at micro-scales and the reaction kinetics.Quantitative information can be obtained on the yield of

he slower secondary reaction. Consequently, mixing perfor-ance is characterized by the amount of secondary product

ormed: the greater the yield of the secondary reaction, theoorer the mixing quality. Micro-mixer performances can beompared on the basis of the qualitative segregation index

s. Subsequent quantitative treatment of the experimentalata is based on kinetic models taking into consideration theensitivity of the iodine-forming reaction to the mixing studyonditions like the ionic strength, the concentration, or thenjection volume.

The Villermaux/Dushman method for characterizing thextent of micro-mixing through examining the iodine yieldives qualitatively consistent and intelligible results. Fordentical reactive system characteristics, it is suitable toank different mixers or different operating conditionsBourne, 2008; Ghanem et al., 2013b). In this work the Viller-

aux/Dushman (or iodide/iodate) chemical probe is adoptedor the experimental study. Mixing performance is evaluatedt the scale of the reactive volume; this requires special designf the chemical probe to fit the reaction time to the resi-ence time. Necessary for a thorough comprehension of the

pproach and the results presented in this work, a detailedescription of the principles, the methodology, the adaptive

Fig. 1 – Hydraulic loop an

procedure, and the phenomenological models used to obtainthe intrinsic mixing time by the iodide/iodate chemical probemethod can be found in papers recently published by theauthors (Ghanem et al., 2013b; Habchi et al., 2011).

The studied geometries include an insert-type static mixerequipped with helical screw-tape elements, a plain tubetaken as the reference case and two modified-surface-typedevices chosen to present comparable processing capacitiesand energy expenditures. These are corrugated rectangu-lar channels with different radii of curvature; the one withsmooth curvature is called “wavy channel” and the other iscalled “zigzag channel”. The four devices are characterizedusing the iodide/iodate chemical probe method implementedunder similar conditions.

The following section elaborates on the experimental pro-cedure, the accompanying important factors, and the testsection geometries. In Section 3, measurement results are pre-sented and the performance of the different geometries isdiscussed. The final section gives concluding remarks on themethod and the mixer geometries.

2. Experimental study

2.1. Hydraulic setup

A schematic diagram of the hydraulic loop is shown in Fig. 1.The 200-liter tank containing the main flow reagent solution(KI, KIO3, H3BO3, NaOH) has an immersed pump in order tohomogenize the initial mixture driven in the hydraulic loopby a rotary gear pump. The flow rate is controlled by a fre-quency modulator on the electrical power of the circulationpump and is measured by rotary flowmeters with 2% preci-sion. The temperature in the system is kept constant at 298 Kby an immersed helical heat exchanger whose temperatureis controlled by a thermostat. The reactor is preceded by astraight-pipe preconditioner of length 1.5 m to ensure fullydeveloped flow at the reactor inlet, and is followed by a post-conditioner of length 0.5 m.

2.2. Solution preparation and pH considerations

The buffer solution is first achieved with H3BO3 and NaOH,

which are dissolved in deionized water (on a Siemensresin < 5 �m) and stored in the main tank shown in Fig. 1. The

d injection system.

2216 chemical engineering research and design 9 2 ( 2 0 1 4 ) 2213–2222

enta

a part of the flow is by-passed to the spectrophotometer.

Table 1 – Geometric characteristics of the corrugatedchannels.

Wavychannel

Zigzagchannel

Cross section (mm2) 2 × 4 2 × 4Hydraulic diameter (mm) 2.67 2.67Curvature radius (mm) 10.5 1.5Bend angle 90◦ 90◦

Number of bends 13 26Linear distance between two 10 10

Fig. 2 – (a) Tubular reactor, (b) an elem

temperature is maintained constant at 298 K since the reactionkinetics are very sensitive to this parameter.

Another important parameter considered in dosing thereactants is pH. The formation of I2 depends on the value ofpH compared to pH*, the threshold for the natural formationof I2 (Custer and Natelson, 1949; Pourbaix, 1963). Therefore, pHmust be higher than pH*. The value of pH* is 7, and the idealinitial pH value is 8.5 < pH > 9.5. For more details on the pH-potential diagram, see Guichardon and Falk (2000), MohandKaci et al. (2006) and Habchi et al. (2013).

2.3. Acid injection

The sulfuric acid H2SO4 injection system consists of a reg-ulated stepper system connected to a multi-push-syringesystem providing a range of precisely controlled injection vol-ume flow rates from 10 �L/min to 160 mL/min.

The sulfuric acid is injected into the test section by aninjection needle of 0.5 mm internal diameter connected to thesyringes by flexible tubes. The flow rate of the injected sulfuricacid could influence the results, since it can perturb the mainflow. Therefore, an a posteriori study is made to determinethe maximum flow rate of the injected sulfuric acid for whichthere is no influence on the measured values.

An important issue is the dissociation constant for H2SO4

in water as reported by Kölbl and Kraut (2010) and Kölbl et al.(2013). This has been taken into consideration to avoid hav-ing the H+ in excess. As required by the method, H+ ionsshould remain in stoichiometric defect with concentrationscalculated following the adaptive procedure (Ghanem et al.,2013b; Habchi et al., 2011) to ensure global measurements byimposing a reactive volume equivalent to that of the vessel.

2.4. Measurement of the iodate concentration

The I2 and I−3 concentrations are experimentally determinedby spectrophotometry. According to the Beer–Lambert law (Eq.(1)), light absorption A is proportional to the I−3 concentrationresulting from the equilibrium reaction:

[I−3 ] = A

�l(1)

where � is the optical length and � is the molar extinction coef-

ficient of I−3 at 353 nm equal to � = 2597 ± 148 m2/mol (Palmeret al., 1984; Mohand Kaci et al., 2006).

ry tube and (c) connection chamber.

Once [I−3 ] is measured, the I2 concentration can hence beobtained from the iodine mass balance (Fournier et al., 1996a,1996b). The final products of the chemical reaction systemare continuously analyzed through a channel placed 300 mmdownstream from the reactor outlet.

2.5. Test section geometries

Four configurations are investigated in this study. The firstenhanced geometry is a static mixer type reactor with fourparallel circular stainless steel tubes fitted with helical inserts,of 1 mm thickness and pitch 20 mm (Fig. 2). Two parallel tubesare used as inlet flow and two others as outlet. A mixing cham-ber connects the four tubes, each of which is 100 mm long withinternal diameter 8 mm and hydraulic diameter Dh 4.32 mm.

The two other enhanced geometries are two modified-surface-type devices, these are corrugated rectangular chan-nels with different radii of curvature: one corrugated with asmooth bend curvature, called “wavy channel”, and the otherwith a herringbone pattern, called “zigzag channel”, presentedin Fig. 3, with geometric characteristics summarized in Table 1.

The fourth geometry is an 8 mm-diameter plain tube serv-ing as the baseline geometry. With a length of 400 mm,equivalent to the total length of the four elementary tubes ofthe helical insert configuration, this reference permits assess-ment of the enhancement produced by the inserts on mixingand their influence on the pressure drop.

In all cases, the main flow enters through the main reactorinlet; sulphuric acid is injected through wall injectors, and thereaction kinetics allows achieving the conversion throughoutthe reactor length. At the outlets toward the waste reservoir,

consecutive bends (mm)

chemical engineering research and design 9 2 ( 2 0 1 4 ) 2213–2222 2217

s: (a

3

Ttaamidm

3

Tacaatu

t(

f

wthiwt

Fm

Fig. 3 – Corrugated rectangular channel

. Results

he experimental characterization of the reactor configura-ions studied is based on the tri-iodide ion concentrations,ssessed under the same range of operating conditions. Theyre presented and compared in terms of segregation index,ixing time, and inverse diffusion coefficient. Energy expend-

tures are given in the experimental pressure drop or theimensionless friction factor, which are presented first to per-it energy efficiency analysis.

.1. Energy consumption and friction factor

he pressure losses measured by the differential manometersre plotted in Fig. 4. The pressure drop in the zigzag geometryonsiderably exceeds that in the wavy, probably due to thebrupt changes in flow direction (smaller radius of curvature)nd the greater number of bends in the zigzag channel. Theube fitted with helical inserts produces lower head losses pernit length than the above configurations.

For a dimensionless representation, the Darcy friction fac-or f is calculated for the four reactor configurations using Eq.2) and is plotted against Reynolds number in Fig. 5:

= �P

(L/Dh) (�(W2/2)(2)

here L is the total developed length of the reactor. Owingo the above considerations, the zigzag channel retains theighest friction factors, followed by the tube fitted with helical

nserts that account for the greater dissipation than the plain

avy channel with lower friction factors. Naturally, the fric-

ion factors measured in the plain tube are the lowest and they

0.0005 0.001 0.01 0.020.001

0.01

0.1

1

10

100

1000

m.aPk(htgneltinurep

sessolerusserP

-1)

Mass flow rate (kg.s-1)

Zigzag channelWavy channelTube with helical insertsPlain tube (d=8mm)

ig. 4 – Energy consumption expressed as head losses perixer unit length versus mass flow rate.

) wavy channel and (b) zigzag channel.

are in good agreement with the theoretical trend of straightplain tubes, where f = 64/Re in the laminar zone regardless ofthe relative roughness of the tube.

The plot in Fig. 5 shows that the friction factors in the tubewith helical inserts are 4–6 times greater than those in a plaintube.

3.2. Segregation index XS

Following the adaptive procedure described by Habchi et al.(2011) and Ghanem et al. (2013b), the initial reactant con-centrations in Table 2 are chosen simultaneously to respectthe chemical constraints discussed above and ensure a char-acteristic time for the Dushman reaction (Dushman, 1904)tr2 corresponding to complete consumption of the reactantswithin the reactor length. Different concentrations of H+ vary-ing between 0.1 and 1 mol L−1 are injected for a range ofmain-flow Reynolds numbers between 100 and 4000.

The segregation index is a preliminary qualitative indicatorof mixing and is closely related to the reagent concentrations,so that it cannot be used to compare different geometriescharacterized under different conditions. In a first step, thesegregation index XS is plotted versus the Reynolds numberbased on the hydraulic diameter in Fig. 6 which compares theswirl flow to the corrugated channel configurations, with theplain tube serving as the reference geometry. It is observedthat XS decreases with the Reynolds number over the wholerange, implying better selectivity and enhanced mixing forhigher Reynolds numbers.

The helical inserts geometry presents the lowest levels ofsegregation and the straight tube the highest; two orders of

magnitude greater than the other reactors. The influence ofstatic mixer geometry on the mixing performance is clearly

200 1000 50000.01

0.1

1

Theoretical trend, f = 64/Re

Zigzag channelTube with helical insertsWavy channelPlain tube

Fric

tion

fact

or, f

Reynolds number, Re

Fig. 5 – Friction factor in the four configurations versusReynolds number.

2218 chemical engineering research and design 9 2 ( 2 0 1 4 ) 2213–2222

Table 2 – Reagent concentrations for global mixing time measurements.

Reagents H3BO3 NaOH KIO3 KI− H+

Concentrations (mol L−1) 0.0005 0.0005 0.0003 0.0015 Variable

seen in the low-Reynolds number range, where the threereactors are easily distinguished. In contrast, as the regimebecomes more turbulent at high Reynolds numbers, the curvesbecome confounded, meaning that mixing is governed byturbulence and not by the geometrical design. This can beunderstood by looking at the mechanisms responsible forradial convective transfer in these three geometries. For thehelical inserts, the flow is twisted by the wall forcing evenin the creeping regime. The curvature of streamlines createsadditional stirring and generates hydrodynamic instabilitieswhen the inertial forces become sufficiently strong (Fellouahet al., 2006).

A similar phenomenon occurs in the bends of the “wavy”and “zigzag” channels, but here a flow instability arises dueto the local curvature. In fact, when a fluid flows through acurved pipe of any cross section it is subjected to a secondaryflow which occurs in planes perpendicular to the pipe axis.In fluid flows with curved streamlines (such as curved pipe)there must be a pressure gradient across the pipe to balancethe centrifugal force on the fluid due to its curved trajectory;the pressure being greatest at the pipe outer wall and least atthe inner wall. The fluid in the top and bottom wall boundarylayers of the pipe moves slower than that near the pipe centerand therefore requires a smaller pressure gradient to balanceits reduced centrifugal force. Consequently, a secondary flowoccurs in which the fluid near the pipe top and bottom wallsmoves inwards toward the center of the pipe. At the same timethe fluid near the center of the pipe moves outwards. Thisin turn modifies the axial velocity. As the centrifugal forceincreases, so does the pressure gradient; the faster-movingfluid near the pipe center pushes the fluid in the outer wallboundary layer to the top and bottom walls and then inwardsalong the top and bottom walls (where it is retarded due to itsproximity to these walls) toward the inner wall. Faster movingfluid is therefore constantly transported to the outer wall andretarded fluid is carried to the inner wall. This phenomenoncan be exploited to enhance radial mass transfer without theuse of inserts.

The first theoretical analysis of this secondary flow wasgiven by Dean (1927) for an incompressible fluid in steady

100 1000 50000,0001

0,001

0,01

0,1

1

Seg

rega

tion

inde

x, X

s

Reynolds number, Re

Plain tube (d=8mm)Tube with helical insertsZigzag channelWavy channel

Fig. 6 – Segregation index XS versus Reynolds number inthe four geometries.

motion through a pipe of circular cross section whose axis isbent to the form of a circular arc of several revolutions. Deantreated the velocity and pressure fields by a Taylor expansionof the pipe curvature radius, and showed that the fourth powerof this parameter, defining the Dean number De (Eq. (3)), gov-erns the secondary flow, namely the Dean roll cells appearingin the channel cross section as shown in Fig. 7.

De = Re

√Dh

Rc(3)

where Re is the Reynolds number based on Dh, the hydraulicdiameter, and Rc is the channel radius of curvature (Dean,1927). In a channel with large curvature radius compared to thehydraulic diameter (Dean hypothesis), the secondary struc-tures start to appear at a threshold value of De = 36. As theDean number increases, the axial velocity peak is shifted fromthe channel center to the outer wall. Beyond a critical value ofthe Dean number, due to the flow instability, two additionalvortices appear, which are called the Dean vortices. Thesevortices, once generated, promote the homogenization of thetemperature (Fellouah et al. 2006), the velocity and the concen-tration gradients, and thus contribute to mixing performanceenhancement in the corrugated channel flow.

Following the definition given in Eq. (3), the Dean numberis greater for a smaller curvature radius; the “zigzag” channel,with substantially higher Dean numbers, appears more effi-cient than the “wavy” one due to the accentuated secondaryflow and Dean vortices in its sharp bends accounting for thelower levels of segregation in this configuration. In addition,as the Dean vortices vanish a short distance from the bends,the distance between two bends can be a degree of freedomin the design of such static mixers.

3.3. Mixing time

Nevertheless, the segregation index remains case-specific.Quantitative mixing times, independent of the chemical sys-tem, are presented in this section. As they are derived fromthe XS value, the same ranking would be expected versusthe Reynolds number. A further step here is to highlight

the energy cost of obtaining a mixing performance, and tothat end mixing times are presented as a function of energy

Fig. 7 – Secondary flow in a curved rectangular duct.

chemical engineering research and design 9 2 ( 2 0 1 4 ) 2213–2222 2219

0,1 1 10 200,001

0,01

0,1

1

10

Plain tube (d=8mm)Wavy channelZigzag channelTube with helical inserts

Mix

ing

time,

t m (s

)

Specific energy, E (J.kg )

Fig. 8 – Mixing time tm versus the specific energy in thef

cf

E

tFtep

maTigiltesspucbt

cppnoet

hmihtwif

200 400 600 800 10001000 1400 2000 400010

100

1000

10000

100000

Experiments Theoretical trends

Plain tube (d=8mm) ~W - 1

(Falk and Commenge, 2010)

Wavy channel

Zigzag channel ~W - 3/2

(Baldyga and Bourne, 1989)

Tube with helical inserts

Reynolds number, Re

Turbulent

Inve

rse

diffu

sion

coe

ffici

ent,

t m /

Dh2

(s.m

-2)

Laminar

- 1

- 3/2

Fig. 9 – Inverse diffusion coefficient versus Reynoldsnumber for the four configurations.

our studied configurations.

onsumed per unit mass of processed fluid Es, calculatedrom the inlet-outlet differential pressure drop as:

S = �P

�(4)

Following the E-model of Baldyga and Bourne (1984, 1989),he mixing time is computed for each reactor and plotted inig. 8 as a function of the specific energy. The baseline geome-ry (8 mm-diameter plain tube) makes it possible to isolate theffect of the geometry on mixing enhancement and additionalressure drop.

The reactor fitted with helical inserts exhibits the highestixing performance over the whole Reynolds number range,superiority that is more marked in the laminar regime.

he better performance attested by the experiments can benferred from flow mechanisms that are absent in the corru-ated geometries. The disturbance of the main flow due to thenserts, through periodic disruptions of the viscous boundaryayer, together with the stirring caused by swirling motion overhe whole volume, prevents the formation of stagnant zones,specially in the wall vicinity. In addition, the metallic insertplits the inward flow into two layers, thus reducing the initialegregation level by halving the striation thickness, and fluidarcels in the lamellae produced swirl in opposite directionsntil encountering collisions further downstream. Unlike theorrugated geometries, where the stirring is localized in theends, these flow features can explain the reduction in mixingimes for the same energy input.

Moreover, in the laminar regime, mixing times in the heli-al insert tube are two orders of magnitude smaller than in thelain tube. In the turbulent zone, the gap is narrower yet thelain tube still shows mixing times that are one order of mag-itude greater than those in the configuration with inserts. Inther words, to produce comparable mixing times, the energyxpenditures in a plain tube are roughly 100 times those in theube fitted with helical inserts.

It is established (Ferrouillat et al., 2006a) that smallerydraulic diameters allow greater turbulent dissipation andicromixing due to reduced stratification. Nevertheless, mix-

ng capacity in the 2 mm × 4 mm rectangular channels withydraulic diameter 2.67 mm appears lower than that in theube fitted with inserts of hydraulic diameter 4.32 mm. At theirorst, the measured mixing times in the tube with helical

nserts are half that of the closest of the two other geometriesor a given level of energy dissipation.

3.4. Inverse diffusion coefficient

To take into account the different hydraulic diameters andgeometries of the reactors tested, it is useful to introduce theinverse diffusion coefficient tm/Dh

2 to provide a kind of uni-versal representation for all static mixers in the comparableoperating Reynolds number range; the lower the values of thiscoefficient, the better the mixing qualities of the device. Thisanalysis also shows the superiority of the tube with helicalinserts

The experimental results are compared in Fig. 9 to thetheoretical trends derived from the dimensional analysisdeveloped here, in an attempt to validate the experiments andcalculate the mixing efficiency �, as defined by the stretch-ing efficiency model presented by Falk and Commenge (2010)based on the concept introduced by Ottino et al. (1979). Inthe laminar regime, following the stretching efficiency model(Falk and Commenge, 2010), tm varies as a function of W−1

or Re−1 by assuming a nearly constant efficiency � for agiven device. In the turbulent regime, the engulfment modelrelates tm to W−3/2 or to Re−3/2 (given that ε is proportional toW3/Dh). The experiments are in good agreement with thesetheoretical trends as shown in Fig. 9. The percentage erroris estimated at 10% over the Reynolds number range stud-ied. It should also be noted that all experimental data liewithin the 30% accuracy interval characteristic of the chem-ical method, as set by Falk and Commenge (2010). As thebasic flow hydrodynamics is altered by the modification ofthe geometry, the laminar-to-turbulent transition is shifteddown from the classical value of 2000 (Reynolds number forchannel flow) and is even difficult to detect by pressure lossmeasurements. The plot in Fig. 9, however, marks the tran-sitional zone in which mixing times curves show a break inslope and start to follow the Re−3/2 law, a signature of tur-bulence that seems to appear around a Reynolds number of1400.

The mixing efficiency � for each reactor geometry can bededuced by adjusting the values of � so that the Re−1 lami-nar fitting curve matches the experimental values and joinsthe turbulent curve at the hypothetical transition point. Themeasured values show that the rectangular channels share

an efficiency of � = 5.2%, while an efficiency of � = 6.7% is calcu-lated for the tube fitted with helical inserts; a larger proportion

2220 chemical engineering research and design 9 2 ( 2 0 1 4 ) 2213–2222

(+30%) of the dissipated energy is used for mixing in this latterconfiguration.

These values are of the same order of magnitude as thosereported by Falk and Commenge (2010) (3% in micromixersand 1% in extruders, as calculated by Baldyga and coworkers)but are slightly higher due to better energy exploitation in theconfigurations investigated here.

4. Conclusions

An analytical and experimental investigation of micromixingquantification using a chemical probe method in geometriesof multifunctional heat exchangers/reactors and static mixersis presented.

A new adaptive chemical probe procedure is used thatinvolves fitting the concentrations of the injected sulfuric acidfor each flow conditions in order to control the reaction vol-ume and ensure global measurements. The mixing time iscalculated and compared to two models according to flowregime: the engulfment model in turbulent flow and the stretch-ing efficiency model in laminar flow.

The adaptive method is applied to four geometries: atubular reactor equipped with helical inserts, two inlinerectangular curved-channel types (namely wavy and zigzagchannels) and a plain straight tube which represents thereference case. This method permits the comparison of dif-ferent geometries of different hydraulic diameters and theirranking in terms of mixing performance. At first glance, itmight appear that comparing rectangular channels to circulartubes, or characterizing mixing produced by different physicalphenomena is irrelevant. However, an application of this studymay also be to optimize the processes in which these reac-tors are destined to serve. From a process engineering pointof view, mixing quality, pumping power, manufacturing costs,maintenance, and the overall feasibility of a device must betaken into consideration simultaneously. In fact, the criteriafor choosing these geometries for investigation are quite prod-uct/cost oriented. The associated physical phenomena andgeometric characteristics are selected to make it possible tocompare mixing qualities and energy expenditures in devicesthat can carry out similar operations with comparable produc-tion rates and manufacturing costs. Similarly, the choice of thehydraulic diameter is based on the expected performance ofeach geometry in light of the constraint of mixing enhance-ment and operating costs. In other words, the expected mixingenhancement of the swirling shear flow in the tube with heli-cal inserts is compensated for by the reduced diameter of therectangular channels, and the additional losses produced bythe helical inserts, compared to the plain channels, are bal-anced by the larger diameter of the base circular tube, whichcan reduce head losses.

Owing to the small radius of curvature, the Dean numbersof the sharp bends in the zigzag channel are higher, thus pro-moting a more vigorous secondary radial flow and giving it asmall advantage over the wavy channel in the laminar regime.For higher specific energies, and thus for higher flow rates,higher velocities, and higher Reynolds numbers, the Deannumbers increase until reaching a critical value above whichthe mixing action of Dean vortices becomes weak compared tothe turbulent diffusion; the two curved-channel type mixersare indistinguishable in the turbulent regime. Consequently,

the zigzag channel seems clearly less interesting because itsmixing performance resembles that of the wavy channel in

the turbulent range, and its mixing efficiency is quite iden-tical in the laminar regime. However, the feasibility of thisdevice is also related to its fabrication process, which startsfrom the less common rectangular channel as compared to cir-cular tubes and requires delicate bending techniques to avoidparasitic deformations. Together with the higher probabilityof fouling in sharp angles and more difficult cleaning, theseprevious considerations argue against the zigzag tube.

At this millimeter scale, superior mixing qualities are foundin the tube with helical inserts, as reflected in shorter mixingtimes for a given energy input, an earlier transition to turbu-lence and greater mixing efficiency in laminar flow than theother devices. It should be noted that the scale of the reactorespecially the hydraulic diameter is an important factor for themixing process. For the same value of the Reynolds number,mean flow velocities are higher in plain channels with rel-atively smaller diameters, and, as established by Ferrouillatet al. (2006a), higher turbulent kinetic energy dissipationrates are produced in these latter. This fact, together withreduced striation thickness, segregation, and stratification inthe smaller configurations with dimension account for shortermixing times. Hence, the ideal experimental study should onlyconsider configurations with equal hydraulic diameters, yet,in the present case and due to the enhancement mechanismof the helical inserts, the configuration with larger hydraulicdiameter shows a better performance. Based on the aboveanalysis, it is highly expected that this configuration will con-tinue to show better mixing qualities compared to corrugatedchannels with matching hydraulic diameters. Thus, an exper-imental study investigating configurations of equal diameterseems redundant in light of the theoretical expectation vis-à-vis the mixing performance ranking.

This modified chemical probe method, beyond the spe-cific study presented here, is shown to be a versatile tool tocharacterize different types of open-loop components in aproduction chain, such as static mixers, heat exchangers andchemical reactors.

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