Mixing performances of swirl flow and corrugated channel reactors

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Chemical Engineering Research and Design

j o u r n a l h o m e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / c h e r d

Mixing performances of swirl ﬂow and corrugated channel reactors

Akram Ghanem

^a

, Charbel Habchi

^b

, Thierry Lemenand

^a

, Dominique Della Valle

^a,c

, Hassan Peerhossaini

^d,∗

aLUNAM Université, Laboratoire de Thermocinétique de Nantes, CNRS UMR 6607, 44306 Nantes, France

bEnergy and Thermo-Fluids Group ETF, School of Engineering, Lebanese International University LIU, Beirut, Lebanon

cONIRIS, 44322 Nantes, France

dUniv 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 curvature, 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 compared 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.

Keywords: Mass transfer intensiﬁcation; Static mixer; Continuous multifunctional heat exchanger/reactor; Helical screw-tape insert; Herringbone-patterned pipe; Iodide/iodate chemical probe

1. Introduction

Industrial ﬂuid processes include mixing, reaction, and/or heat transfer operations that are in many cases concomitant and strongly coupled. The desire for elevated production rates, increased process safety and high product quality, while maintaining low manufacturing and operation costs, have led to the development of in-line continuous static mixers and multifunctional heat exchangers/reactors (MHER) as an alternative to batch production in stirred vessels. Stirred vessels are still used for most industrial applications, thanks to their versatility and monitoring ﬂexibility: easy temperature control with hydrodynamics and dilution of reactants, decoupling of residence time and shear rates, mixing of hard-to-pump viscous products, etc. Yet they present several drawbacks compared to continuous processing: space requirements, equipment operation and maintenance costs, broad

∗ Corresponding author. Tel.: +33 607 53 31 61.

E-mail address:hassan.peerhossaini@univ-paris-diderot.fr(H. Peerhossaini).

Received 6 August 2013; Received in revised form 6 January 2014; Accepted 11 January 2014 Available online 2 February 2014

residence-time distributions, poor selectivity due to localized mixing and large segregation zones, lack of isothermal operation especially in the central zones, undesirable byproducts, and reduced safety and process control due to large batch ﬂuid volumes, in addition to high power requirements by the mobile parts. These considerations have promoted the use of stationary mixing elements in chemical, pharmaceutical, polymer synthesis, food processing, pulp and paper, paint and resin, water treatment, petrochemical industries, etc.

Static mixers may be used for operations such as mixing of miscible ﬂuids, heat transfer and thermal homogenization, and liquid–liquid dispersion as well as gas–liquid dispersion or gas–gas mixing where similar, and sometimes better performances can be achieved at lower cost than other processing techniques (Anxionnaz et al., 2008; Thakur et al., 2003). Actu- ally, in static mixers, the energy cost comes from the external pumping power needed to propel materials through the mixer,

http://dx.doi.org/10.1016/j.cherd.2014.01.014

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and numerous works have shown that it reaches much lower levels than that of stirred tanks per fluid unit mass (Anxionnaz et al., 2008; Bayat et al., 2012; Ferrouillat et al., 2006a, 2006b; Shi et al., 2011; Thakur et al., 2003). Moreover, temperature control is easier because the system is smaller, allowing the use of more concentrated reactants. Better selectivity and effluent reduction are important assets in continuous-flow solutions (Poux et al., 2010; Ferrouillat et al., 2006c).

Fluid mixing is a unit operation aimed at achieving a homogeneous distribution of the fluid properties, since this is essential for product quality in numerous industrial processes. It is involved in various heat- and mass-transfer operations. Mixing by molecular diffusion is not appropri- ate in many industrial processes – a length of about 200 diameters would be needed to obtain a well-mixed fluid in a straight pipe. Static mixers are accordingly designed to gen- erate convective transfer, through modified surfaces or series of motionless inserts fixed in pipes that guide the fluid flow across the pipe section. This radial transfer is essential to achieve homogeneity within a short distance while keeping pressure losses moderate. The purpose of the inserts or geometrical modifications is to divide and redistribute the fluid streams sequentially or to trigger the formation of specific flow structures by passively manipulating the natural flow forces in order to intensify transverse particle displacement from the wall vicinity to the active core flow.

Numerous static mixer designs have been proposed on different scales, but only a few models are used in industry and comparative performance studies remain few (Anxionnaz et al., 2008; Barrué et al., 2001; Bourne et al., 1992b; Cybulski and Werner, 1986; Hessel et al., 2003; Meijer et al., 2012;

Pahl and Muschelknautz, 1982; Thakur et al., 2003; Theron and Le Sauze, 2011; Zhang et al., 2012). Among the highest- performance devices studied in the literature, the Kenics^TM mixer (Chemineer) has a privileged place in the ranking (Jaffer and Wood, 1998; Hobbs et al., 1998; Rahmani et al., 2005), as do the SMX^TM(Li et al., 1997; Streiff and Kaser, 1991; Talansier et al., 2013) or SMV^TM mixers (Sulzer) (Lobry et al., 2013;

Paglianti and Montante, 2013). Choosing a suitable mixer configuration for a given application needs the quantification of the mixing process, a fundamental requirement for modern processes seeking better energy efficiency and product quality, especially those including fast chemical reactions (Ehrfeld et 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 characteristic times smaller than the mixing time, mixing kinetics becomes a key parameter for selectivity and overall reaction yield. Mix- ing efﬁciency, especially on the molecular scale, is linked to the capacity of the ﬂow to provide fresh reactant faster than the reactant consumption by reaction, and determines both productivity and selectivity by hindering undesirable slower secondary reactions.

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

Macromixing is the homogenization at the scale of the whole vessel that determines the environmental concentrations in the flow domain. This large-scale fluid distribution is affected by the mean velocity field and thus by particle transport 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 to migrate from the device inlet to the outlet, an indicator of velocity-field uniformity (Castelain et al., 1997; Mokrani et al., 1997; Habchi et al., 2009a; Villermaux, 1986). RTD steepness can be improved by generating radial convective transfer, for instance 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,mesomixingis the coarse-scale exchange between the fresh feed fluid and its surroundings, governed either by fluctuations in turbulent flow or fractal structures in laminar flow (Baldyga et al., 1995). In turbulent flow, the mechanism of scale reduction is governed by the energy cascade in the inertial-convective subrange of the turbulent spectrum, with wave numbers ranging between the integral and Kolmogorov scales. Thus, mesomixing is a homogenization process by advection due to velocity fluctuations, basically governed by the particle random path (similar to a mesoscopic diffusion based on the turbulence statistics, instead of Boltzmann statistics). Governed by the turbulent field, mesomixing is related to the turbulent kinetic energy (TKE)k, the Prandtl length scaleLP, their combination in the turbulent diffusivity,Dt, or even to the Reynolds stress ten- sor (Habchi et al., 2010). In laminar flow, the chaotic path of the fluid particles, usually induced by periodic flow direction modifications, 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 and Peerhossaini, 2002; Muzzio et al., 1992; Ottino, 1989; Toussaint et al., 1995).

Finally, micromixing is the ultimate mixing scale including molecular diffusion. The selectivity of chemical reactions depends on micromixing because it determines molecular contact (Baldyga and Bourne, 1999). In turbulent flow, the prevailing mechanism takes place in the viscous- convective subrange, i.e. for wave numbers ranging between the Kolmogorov and Batchelor scales. Turbulent fluctuations vanish and laminar stretching accelerates the aggregate size reduction up to the molecular diffusion scale, which quickly dissipates the concentration variance (Batchelor, 1953; Baldyga and Bourne, 1989). The limiting mechanism in this process is engulfment in the small vortices near the Kolmogorov scale (Baldyga and Bourne, 1999). It can be characterized by a micromixing time that is linked to the turbulence energy dissipation rate (Baldyga and Bourne, 1999). Follow- ing the Hinze–Kolmogorov theory based on the idea of energy cascade, the drop breakup in multiphase flows is also characterized 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 the micromixing process, enhancing the selectivity of fast chemical reactions. In laminar flows, micromixing results from the reduction in striation thickness up to the diffusive scales, and can be approximated by the stretching efficiency model used here (Falk and Commenge, 2010). Consequently, micromixing is responsible for the global performance of the MHER when the mixing at large scale, namely macromixing, is not limiting.

Several qualitative and quantitative techniques have been devised to study the global effects of these mechanisms in the reactors; they include acid–base or pH indicator reactions,

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dilution of colored dyes or ﬂuorescent materials, reactions yielding colored species, monitoring species concentration, and competing consecutive or parallel reactions (Ghanem et al., 2013b). The latter technique, namely the chemical probe method, has been successfully used to evaluate micromixing in batch and open-loop reactors with different chemical sys- tems (Barthole et al., 1982; Bourne et al., 1992a; Brucato et al., 2000; Ehlers et al., 2000; Fournier et al., 1996a).

The methods that employ competitive-consecutive or competitive-parallel reactions are based on the result of the local injection of a reagent in stoichiometric deficit in the main flow. Two reactions competing for a common reactant are carried out (Habchi et al. 2013), one should be very fast, and therefore proceeds only if mixing is extremely rapid. The other reaction should be fast but slower than the first reaction, and takes place when there is an excess of the common reactant, meaning when mixing is too slow to renew the quantities of the common reagent needed by the faster reaction. The local chemical reaction thus results from a competition between mixing at micro-scales and the reaction kinetics.

Quantitative information can be obtained on the yield of the slower secondary reaction. Consequently, mixing performance is characterized by the amount of secondary product formed: the greater the yield of the secondary reaction, the poorer the mixing quality. Micro-mixer performances can be compared on the basis of the qualitative segregation index Xs. Subsequent quantitative treatment of the experimental data is based on kinetic models taking into consideration the sensitivity of the iodine-forming reaction to the mixing study conditions like the ionic strength, the concentration, or the injection volume.

The Villermaux/Dushman method for characterizing the extent of micro-mixing through examining the iodine yield gives qualitatively consistent and intelligible results. For identical reactive system characteristics, it is suitable to rank different mixers or different operating conditions (Bourne, 2008; Ghanem et al., 2013b). In this work the Viller- maux/Dushman (or iodide/iodate) chemical probe is adopted for the experimental study. Mixing performance is evaluated at the scale of the reactive volume; this requires special design of the chemical probe to ﬁt the reaction time to the residence time. Necessary for a thorough comprehension of the approach and the results presented in this work, a detailed description of the principles, the methodology, the adaptive

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

The studied geometries include an insert-type static mixer equipped with helical screw-tape elements, a plain tube taken as the reference case and two modiﬁed-surface-type devices chosen to present comparable processing capacities and energy expenditures. These are corrugated rectangular channels with different radii of curvature; the one with smooth curvature is called “wavy channel” and the other is called “zigzag channel”. The four devices are characterized using the iodide/iodate chemical probe method implemented under similar conditions.

The following section elaborates on the experimental procedure, the accompanying important factors, and the test section geometries. In Section3, measurement results are presented and the performance of the different geometries is discussed. The ﬁnal section gives concluding remarks on the method and the mixer geometries.

2. Experimental study

2.1. Hydraulic setup

A schematic diagram of the hydraulic loop is shown inFig. 1.

The 200-liter tank containing the main flow reagent solution (KI, KIO3, H3BO3, NaOH) has an immersed pump in order to homogenize the initial mixture driven in the hydraulic loop by a rotary gear pump. The flow rate is controlled by a fre- quency modulator on the electrical power of the circulation pump and is measured by rotary flowmeters with 2% preci- sion. The temperature in the system is kept constant at 298 K by an immersed helical heat exchanger whose temperature is controlled by a thermostat. The reactor is preceded by a straight-pipe preconditioner of length 1.5 m to ensure fully developed 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 Siemens resin < 5␮m) and stored in the main tank shown inFig. 1. The

Fig. 1 – Hydraulic loop and injection system.

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Fig. 2 – (a) Tubular reactor, (b) an elementary tube and (c) connection chamber.

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

Another important parameter considered in dosing the reactants is pH. The formation of I₂depends on the value of pH compared to pH*, the threshold for the natural formation of I2(Custer and Natelson, 1949; Pourbaix, 1963). Therefore, pH must be higher than pH*. The value of pH* is 7, and the ideal initial pH value is 8.5 < pH > 9.5. For more details on the pH- potential diagram, seeGuichardon and Falk (2000),Mohand Kaci et al. (2006)andHabchi 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-syringe system providing a range of precisely controlled injection volume ﬂow rates from 10␮L/min to 160 mL/min.

The sulfuric acid is injected into the test section by an injection needle of 0.5 mm internal diameter connected to the syringes by flexible tubes. The flow rate of the injected sulfuric acid could influence the results, since it can perturb the main flow. Therefore, an a posteriori study is made to determine the maximum flow rate of the injected sulfuric acid for which there is no influence on the measured values.

An important issue is the dissociation constant for H2SO4

in water as reported byKölbl and Kraut (2010)andKölbl et al.

(2013). This has been taken into consideration to avoid hav- ing the H⁺ in excess. As required by the method, H⁺ ions should remain in stoichiometric defect with concentrations calculated following the adaptive procedure (Ghanem et al., 2013b; Habchi et al., 2011) to ensure global measurements by imposing a reactive volume equivalent to that of the vessel.

2.4. Measurement of the iodate concentration

The I2and I⁻₃ concentrations are experimentally determined by spectrophotometry. According to the Beer–Lambert law (Eq.

(1)), light absorptionAis proportional to the I⁻₃ concentration resulting from the equilibrium reaction:

[I⁻₃]= A

l (1)

whereis the optical length andis the molar extinction coef- ﬁcient of I⁻₃ at 353 nm equal to= 2597±148 m²/mol (Palmer et al., 1984; Mohand Kaci et al., 2006).

Once [I⁻₃] is measured, the I2concentration can hence be obtained from the iodine mass balance (Fournier et al., 1996a, 1996b). The ﬁnal products of the chemical reaction system are continuously analyzed through a channel placed 300 mm downstream from the reactor outlet.

2.5. Test section geometries

Four configurations are investigated in this study. The first enhanced geometry is a static mixer type reactor with four parallel circular stainless steel tubes fitted with helical inserts, of 1 mm thickness and pitch 20 mm (Fig. 2). Two parallel tubes are used as inlet flow and two others as outlet. A mixing chamber connects the four tubes, each of which is 100 mm long with internal diameter 8 mm and hydraulic diameterDh4.32 mm.

The two other enhanced geometries are two modiﬁed- surface-type devices, these are corrugated rectangular channels with different radii of curvature: one corrugated with a smooth bend curvature, called “wavy channel”, and the other with a herringbone pattern, called “zigzag channel”, presented inFig. 3, with geometric characteristics summarized inTable 1.

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

In all cases, the main ﬂow enters through the main reactor inlet; sulphuric acid is injected through wall injectors, and the reaction kinetics allows achieving the conversion throughout the reactor length. At the outlets toward the waste reservoir, a part of the ﬂow is by-passed to the spectrophotometer.

Table 1 – Geometric characteristics of the corrugated channels.

Wavy channel

Zigzag channel

Cross section (mm²) 2×4 2×4

Hydraulic diameter (mm) 2.67 2.67

Curvature radius (mm) 10.5 1.5

Bend angle 90^◦ 90^◦

Number of bends 13 26

Linear distance between two consecutive bends (mm)

10 10

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Fig. 3 – Corrugated rectangular channels: (a) wavy channel and (b) zigzag channel.

3. Results

The experimental characterization of the reactor configurations studied is based on the tri-iodide ion concentrations, assessed under the same range of operating conditions. They are presented and compared in terms of segregation index, mixing time, and inverse diffusion coefficient. Energy expenditures are given in the experimental pressure drop or the dimensionless friction factor, which are presented first to per- mit energy efficiency analysis.

3.1. Energy consumption and friction factor

The pressure losses measured by the differential manometers are plotted inFig. 4. The pressure drop in the zigzag geometry considerably exceeds that in the wavy, probably due to the abrupt changes in flow direction (smaller radius of curvature) and the greater number of bends in the zigzag channel. The tube fitted with helical inserts produces lower head losses per unit length than the above configurations.

For a dimensionless representation, the Darcy friction fac- torfis calculated for the four reactor conﬁgurations using Eq.

(2)and is plotted against Reynolds number inFig. 5:

f= P

(L/Dh) ((W²/2) (2)

whereLis the total developed length of the reactor. Owing to the above considerations, the zigzag channel retains the highest friction factors, followed by the tube ﬁtted with helical inserts that account for the greater dissipation than the plain wavy channel with lower friction factors. Naturally, the friction factors measured in the plain tube are the lowest and they

0.0005 0.001 0.01 0.02

0.001 0.01 0.1 1 10 100 1000

m.aPk(htgneltinurepsessolerusserP-1)

Mass flow rate (kg.s^-1) Zigzag channel

Wavy channel Tube with helical inserts Plain tube (d=8mm)

Fig. 4 – Energy consumption expressed as head losses per mixer unit length versus mass ﬂow rate.

are in good agreement with the theoretical trend of straight plain tubes, wheref= 64/Rein the laminar zone regardless of the relative roughness of the tube.

The plot inFig. 5shows that the friction factors in the tube with helical inserts are 4–6 times greater than those in a plain tube.

3.2. Segregation index XS

Following the adaptive procedure described byHabchi et al.

(2011) and Ghanem et al. (2013b), the initial reactant concentrations inTable 2are chosen simultaneously to respect the chemical constraints discussed above and ensure a characteristic time for the Dushman reaction (Dushman, 1904) t_r2corresponding to complete consumption of the reactants within the reactor length. Different concentrations of H⁺vary- ing between 0.1 and 1 mol L⁻¹ are injected for a range of main-ﬂow Reynolds numbers between 100 and 4000.

The segregation index is a preliminary qualitative indicator of mixing and is closely related to the reagent concentrations, so that it cannot be used to compare different geometries characterized under different conditions. In a first step, the segregation indexXSis plotted versus the Reynolds number based on the hydraulic diameter inFig. 6which compares the swirl flow to the corrugated channel configurations, with the plain tube serving as the reference geometry. It is observed thatXSdecreases with the Reynolds number over the whole range, implying better selectivity and enhanced mixing for higher Reynolds numbers.

The helical inserts geometry presents the lowest levels of segregation and the straight tube the highest; two orders of magnitude greater than the other reactors. The inﬂuence of static mixer geometry on the mixing performance is clearly

200 1000 5000

0.01 0.1 1

Theoretical trend, f = 64/Re Zigzag channel

Tube with helical inserts Wavy channel Plain tube

Friction factor, f

Reynolds number, Re

Fig. 5 – Friction factor in the four conﬁgurations versus Reynolds number.

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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 three reactors are easily distinguished. In contrast, as the regime becomes more turbulent at high Reynolds numbers, the curves become confounded, meaning that mixing is governed by turbulence and not by the geometrical design. This can be understood by looking at the mechanisms responsible for radial convective transfer in these three geometries. For the helical inserts, the ﬂow is twisted by the wall forcing even in the creeping regime. The curvature of streamlines creates additional stirring and generates hydrodynamic instabilities when the inertial forces become sufﬁciently strong (Fellouah et al., 2006).

A similar phenomenon occurs in the bends of the “wavy”

and “zigzag” channels, but here a flow instability arises due to the local curvature. In fact, when a fluid flows through a curved pipe of any cross section it is subjected to a secondary flow 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 balance the centrifugal force on the fluid due to its curved trajectory;

the pressure being greatest at the pipe outer wall and least at the inner wall. The fluid in the top and bottom wall boundary layers of the pipe moves slower than that near the pipe center and therefore requires a smaller pressure gradient to balance its reduced centrifugal force. Consequently, a secondary flow occurs in which the fluid near the pipe top and bottom walls moves inwards toward the center of the pipe. At the same time the fluid near the center of the pipe moves outwards. This in turn modifies the axial velocity. As the centrifugal force increases, so does the pressure gradient; the faster-moving fluid near the pipe center pushes the fluid in the outer wall boundary layer to the top and bottom walls and then inwards along the top and bottom walls (where it is retarded due to its proximity to these walls) toward the inner wall. Faster moving fluid is therefore constantly transported to the outer wall and retarded fluid is carried to the inner wall. This phenomenon can be exploited to enhance radial mass transfer without the use of inserts.

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

100 1000 5000

0,0001 0,001 0,01 0,1 1

Segregation index, Xs

Reynolds number, Re Plain tube (d=8mm)

Tube with helical inserts Zigzag channel Wavy channel

Fig. 6 – Segregation indexX_Sversus Reynolds number in the four geometries.

motion through a pipe of circular cross section whose axis is bent to the form of a circular arc of several revolutions. Dean treated the velocity and pressure fields by a Taylor expansion of the pipe curvature radius, and showed that the fourth power of this parameter, defining the Dean numberDe(Eq.(3)), gov- erns the secondary flow, namely the Dean roll cells appearing in the channel cross section as shown inFig. 7.

De=Re

Dh

Rc (3)

whereReis the Reynolds number based onD_h, the hydraulic diameter, and Rc is the channel radius of curvature (Dean, 1927). In a channel with large curvature radius compared to the hydraulic diameter (Dean hypothesis), the secondary structures start to appear at a threshold value ofDe= 36. As the Dean number increases, the axial velocity peak is shifted from the channel center to the outer wall. Beyond a critical value of the Dean number, due to the ﬂow instability, two additional vortices appear, which are called the Dean vortices. These vortices, once generated, promote the homogenization of the temperature (Fellouah et al. 2006), the velocity and the concentration gradients, and thus contribute to mixing performance enhancement in the corrugated channel ﬂow.

Following the definition given in Eq.(3), the Dean number is 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 secondary flow and Dean vortices in its sharp bends accounting for the lower 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 freedom in the design of such static mixers.

3.3. Mixing time

Nevertheless, the segregation index remains case-speciﬁc.

Quantitative mixing times, independent of the chemical system, are presented in this section. As they are derived from the XS value, the same ranking would be expected versus the Reynolds number. A further step here is to highlight the energy cost of obtaining a mixing performance, and to that end mixing times are presented as a function of energy

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

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0,1 1 10 20 0,001

0,01 0,1 1 10

Plain tube (d=8mm) Wavy channel Zigzag channel Tube with helical inserts

Mixing time,

t m

(s)

Specific energy, E (J.kg )

Fig. 8 – Mixing timet_mversus the speciﬁc energy in the four studied conﬁgurations.

consumed per unit mass of processed ﬂuid Es, calculated from the inlet-outlet differential pressure drop as:

ES=P

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Following the E-model ofBaldyga and Bourne (1984, 1989), the mixing time is computed for each reactor and plotted in Fig. 8as a function of the speciﬁc energy. The baseline geometry (8 mm-diameter plain tube) makes it possible to isolate the effect of the geometry on mixing enhancement and additional pressure drop.

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

The better performance attested by the experiments can be inferred from flow mechanisms that are absent in the corrugated geometries. The disturbance of the main flow due to the inserts, through periodic disruptions of the viscous boundary layer, together with the stirring caused by swirling motion over the whole volume, prevents the formation of stagnant zones, especially in the wall vicinity. In addition, the metallic insert splits the inward flow into two layers, thus reducing the initial segregation level by halving the striation thickness, and fluid parcels in the lamellae produced swirl in opposite directions until encountering collisions further downstream. Unlike the corrugated geometries, where the stirring is localized in the bends, these flow features can explain the reduction in mixing times for the same energy input.

Moreover, in the laminar regime, mixing times in the helical insert tube are two orders of magnitude smaller than in the plain tube. In the turbulent zone, the gap is narrower yet the plain tube still shows mixing times that are one order of magnitude greater than those in the conﬁguration with inserts. In other words, to produce comparable mixing times, the energy expenditures in a plain tube are roughly 100 times those in the tube ﬁtted with helical inserts.

It is established (Ferrouillat et al., 2006a) that smaller hydraulic diameters allow greater turbulent dissipation and micromixing due to reduced stratiﬁcation. Nevertheless, mixing capacity in the 2 mm×4 mm rectangular channels with hydraulic diameter 2.67 mm appears lower than that in the tube ﬁtted with inserts of hydraulic diameter 4.32 mm. At their worst, the measured mixing times in the tube with helical inserts are half that of the closest of the two other geometries for a given level of energy dissipation.

200 400 600 800 10001000 1400 2000 4000

10 100 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 Inverse diffusion coefficient, tm / Dh2(s.m-2)

Laminar - 1

- 3/2

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

3.4. Inverse diffusion coefﬁcient

To take into account the different hydraulic diameters and geometries of the reactors tested, it is useful to introduce the inverse diffusion coefﬁcienttm/D_h²to provide a kind of uni- versal representation for all static mixers in the comparable operating Reynolds number range; the lower the values of this coefﬁcient, the better the mixing qualities of the device. This analysis also shows the superiority of the tube with helical inserts

The experimental results are compared in Fig. 9 to the theoretical trends derived from the dimensional analysis developed here, in an attempt to validate the experiments and calculate the mixing efficiency, as defined by the stretching efficiency model presented byFalk and Commenge (2010) based on the concept introduced byOttino et al. (1979). In the laminar regime, following the stretching efficiency model (Falk and Commenge, 2010), tmvaries as a function of W⁻¹ or Re⁻¹ by assuming a nearly constant efficiency for a given device. In the turbulent regime, the engulfment model relatestmtoW^−3/2or toRe^−3/2(given thatεis proportional to W³/D_h). The experiments are in good agreement with these theoretical trends as shown inFig. 9. The percentage error is estimated at 10% over the Reynolds number range studied. It should also be noted that all experimental data lie within the 30% accuracy interval characteristic of the chemical method, as set by Falk and Commenge (2010). As the basic flow hydrodynamics is altered by the modification of the geometry, the laminar-to-turbulent transition is shifted down from the classical value of 2000 (Reynolds number for channel flow) and is even difficult to detect by pressure loss measurements. The plot inFig. 9, however, marks the tran- sitional zone in which mixing times curves show a break in slope and start to follow theRe⁻^3/2 law, a signature of turbulence that seems to appear around a Reynolds number of 1400.

The mixing efficiencyfor each reactor geometry can be deduced by adjusting the values ofso that theRe⁻¹ laminar fitting curve matches the experimental values and joins the turbulent curve at the hypothetical transition point. The measured values show that the rectangular channels share an efficiency of= 5.2%, while an efficiency of= 6.7% is calculated for the tube fitted with helical inserts; a larger proportion

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

These values are of the same order of magnitude as those reported byFalk and Commenge (2010)(3% in micromixers and 1% in extruders, as calculated by Baldyga and coworkers) but are slightly higher due to better energy exploitation in the conﬁgurations investigated here.

4. Conclusions

An analytical and experimental investigation of micromixing quantiﬁcation using a chemical probe method in geometries of multifunctional heat exchangers/reactors and static mixers is presented.

A new adaptive chemical probe procedure is used that involves fitting the concentrations of the injected sulfuric acid for each flow conditions in order to control the reaction volume and ensure global measurements. The mixing time is calculated and compared to two models according to flow regime: theengulfment modelin turbulent flow and thestretch- ing efficiency modelin laminar flow.

The adaptive method is applied to four geometries: a tubular reactor equipped with helical inserts, two inline rectangular curved-channel types (namely wavy and zigzag channels) and a plain straight tube which represents the reference case. This method permits the comparison of different geometries of different hydraulic diameters and their ranking in terms of mixing performance. At ﬁrst glance, it might appear that comparing rectangular channels to circular tubes, or characterizing mixing produced by different physical phenomena is irrelevant. However, an application of this study may also be to optimize the processes in which these reactors are destined to serve. From a process engineering point of view, mixing quality, pumping power, manufacturing costs, maintenance, and the overall feasibility of a device must be taken into consideration simultaneously. In fact, the criteria for choosing these geometries for investigation are quite product/cost oriented. The associated physical phenomena and geometric characteristics are selected to make it possible to compare mixing qualities and energy expenditures in devices that can carry out similar operations with comparable production rates and manufacturing costs. Similarly, the choice of the hydraulic diameter is based on the expected performance of each geometry in light of the constraint of mixing enhancement and operating costs. In other words, the expected mixing enhancement of the swirling shear ﬂow in the tube with helical inserts is compensated for by the reduced diameter of the rectangular channels, and the additional losses produced by the helical inserts, compared to the plain channels, are bal- anced by the larger diameter of the base circular tube, which can reduce head losses.

Owing to the small radius of curvature, the Dean numbers of the sharp bends in the zigzag channel are higher, thus pro- moting a more vigorous secondary radial ﬂow and giving it a small 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 Dean numbers increase until reaching a critical value above which the mixing action of Dean vortices becomes weak compared to the turbulent diffusion; the two curved-channel type mixers are indistinguishable in the turbulent regime. Consequently, the zigzag channel seems clearly less interesting because its mixing performance resembles that of the wavy channel in

the turbulent range, and its mixing efﬁciency is quite identical in the laminar regime. However, the feasibility of this device is also related to its fabrication process, which starts from the less common rectangular channel as compared to circular tubes and requires delicate bending techniques to avoid parasitic deformations. Together with the higher probability of fouling in sharp angles and more difﬁcult cleaning, these previous considerations argue against the zigzag tube.

At this millimeter scale, superior mixing qualities are found in the tube with helical inserts, as reflected in shorter mixing times for a given energy input, an earlier transition to turbulence and greater mixing efficiency in laminar flow than the other devices. It should be noted that the scale of the reactor especially the hydraulic diameter is an important factor for the mixing 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 byFerrouillat et al. (2006a), higher turbulent kinetic energy dissipation rates are produced in these latter. This fact, together with reduced striation thickness, segregation, and stratification in the smaller configurations with dimension account for shorter mixing times. Hence, the ideal experimental study should only consider configurations with equal hydraulic diameters, yet, in the present case and due to the enhancement mechanism of the helical inserts, the configuration with larger hydraulic diameter shows a better performance. Based on the above analysis, it is highly expected that this configuration will con- tinue to show better mixing qualities compared to corrugated channels with matching hydraulic diameters. Thus, an experimental study investigating configurations of equal diameter seems 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 to characterize different types of open-loop components in a production chain, such as static mixers, heat exchangers and chemical reactors.

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