Mass transfer and emulsification by chaotic advection

Mass transfer and emulsification by chaotic advection

Thierry Lemenand

^a

, Charbel Habchi

^b

, Dominique Della Valle

^a,c

, Jérôme Bellettre

^a

, Hassan Peerhossaini

^d,⇑

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

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

cONIRIS, Géraudière, 44322 nantes, France

dUniv. Paris Diderot, Sorbonne Paris Cité, Institut des Energies de Demain (IED), CNRS UMR 8236, Paris, France

a r t i c l e i n f o

Article history:

Received 28 May 2013

Received in revised form 8 December 2013 Accepted 10 December 2013

Keywords:

Chaotic advection Drop-breakup diagram Drop-size distribution Emulsification Liquid/liquid dispersion

Multifunctional heat exchanger-reactor

a b s t r a c t

This study characterizes a new mixing process based on chaotic advection for the production of water/oil (w/o) emulsified engine fuel. At low and intermediate Reynolds numbers, in curved pipes, Dean roll-cells induce radial convective mass transfer, a mechanism exploited in the technology of helical static mixers.

The succession of bends of alternating curvature planes produces in addition spatially chaotic flow trajectories that enhance the mixing over and above that in the reference helically coiled geometry made of coplanar bends. This feature is assessed in the present work by comparison of the droplet sizes obtained in helical (regular) and alternating (chaotic) static mixers of the same cross section and the same tube length. The coils are assembled from 90°bends, and the chaotic configuration is obtained by turning each bend by ±90°with respect to the previous one.

To cover a large range of Reynolds numbers [30–350] and capillary numbers [0.1–1], a lipophilic solvent (Butanol) is added to the continuous phase (oil) to decrease the viscosity and a surfactant (Tween 20) is used beyond the critical concentration (covering the whole w/o interface) to decrease the interfacial tension. Drop-size measurements at the exit of the mixer in the chaotic advection configura- tion show substantial drop-size reduction and tightening in drop-size distribution. Moreover, it is shown that mixing intensification by chaotic advection is almost independent of Reynolds number in the studied range, with a mixing efficiency enhancement around 30%.

Ó2013 Elsevier Ltd. All rights reserved.

1. Introduction

The mixing of immiscible fluids in industrial processes, for phase dispersion or emulsification, is a complex issue, since the breakup mechanisms are difficult to understand and quantify [1–3]. Homogeneous mixing of stress-sensitive or viscous fluids is a difficult process: human blood, which is a suspension of fragile biological structures, can be easily damaged by shear stresses, or liquid chocolate, which contains various fats, may crystallize under moderate shear. Obviously, for such products that need to be processed under soft conditions, or for very viscous fluids, mixing cannot be carried out in turbulent regime, and therefore coiled pipe mixer can offer an appropriate solution. The transverse motion developed in bends by Dean roll-cells (and Dean vortices) favors momentum, heat and mass radial transfers[4]. This mecha- nism is the cause of the principle phenomenon occurring in the coiled tubes mixers, that become even more efficient when the chirality of the successive bends is alternated, namely the ‘‘twisted pipe configuration’’[9].

In laminar flow through curved pipes, centrifugal forces create a secondary flow in the radial direction consisting of a pair of counter-rotating roll-cells known as Dean roll-cells[5,6]. For high Dean numbers, a centrifugal instability appears close to the outer concave wall of the tube and generates another pair of counter- rotating vortices called Dean vortices. The main difference between Dean vortices and Dean roll-cells is the mechanism that gives rise to them. Dean roll-cells are present even at the lowest Dean num- bers and are due to the imbalance between centrifugal and viscous forces (similar to a box with differentially heated side walls). Dean vortices, on the other hand, are due to an instability phenomenon that appears only after an instability threshold is crossed (similar to Rayleigh–Bénard convection). The other major difference is the space domain in which the cellular structures appear: the large Dean roll-cells occupy almost the whole cross-section, while the smaller Dean vortices occupy only a small part of it. More detailed studies of the hydrodynamics of Dean flow are reported in[7].

The advantages of using a chaotic-advection twisted pipe-flow as a mixer and/or heat exchanger have been established in previ- ous studies [8–12] when compared with a straight or helically coiled pipe. The geometric perturbation in the twisted-pipe config- uration generates three-dimensional chaotic trajectories in the 0017-9310/$ - see front matterÓ2013 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.12.015

⇑Corresponding author. Tel.: +33 6 07 53 31 61.

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

Contents lists available atScienceDirect

International Journal of Heat and Mass Transfer

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 / i j h m t

secondary Dean flow induced by the wall curvature. Chaotic advec- tion in such geometries produces efficient macro-mixing and heat- transfer intensification in the laminar regime in the convenient range of Dean number. In a curved pipe, the secondary flow is superimposed on the axial flow and plays the role of internal agi- tators. At each bend rotation of opposite chirality, the Dean cells deeply reorganize a phenomenon that reduces the boundary layer thickness. This behavior is completely different from that in the helically coiled tube, where a continuous thickening of the bound- ary layer in the flow direction is observed. Consequently, the geo- metrical perturbation generated in the twisted configuration (chaotic) increases the heat and mass transfers[8–11,13].

The heat transfer in bends is significantly enhanced in a partic- ular range of Dean numbers (36–120), without significant increase in the pressure drop[7]. Habchi et al.[9], studying the dispersion in helically coiled and chaotic twisted pipe flows, showed that an increase in flow rate decreases drop size. Both Eulerian and Lagrangian analytical studies showed that chaotic advection causes fluid particles to visit regions of high shear and elongation rates, both intensifying emulsification (smaller droplets) and improving the homogeneity (smaller RMS). Habchi et al.[14]have also shown that the liquid/liquid dispersion is improved for higher external phase viscosities.

The effect of the continuous-phase viscosity and the interfacial tension on droplet size is investigated here in order to address the practical issue of optimization for this type of emulsification. The droplet-size distribution is in fact a key parameter, for example in the emulsified fuels[15], which are very efficient in pollutant emission reduction for biofuel combustion[16]. The study by Hab- chi et al.[14]was limited to low Reynolds numbers (below 70). In order to investigate higher Reynolds numbers (up to 350), in some of the present experimental runs the continuous-phase viscosity is decreased by addition of a Butanol fraction, avoiding an increase in the flow rate but without having a significant effect on the interfa- cial tension. In other experiments reported here, the interfacial tension between oil and water was lowered, by adding the surfac- tant Tween 20, without changing the mixture viscosity. The global effect of the reduction in viscosity and interfacial tension is to

produce competitive effects that can be described as ‘‘positive’’ if they work toward a smaller droplet size and ‘‘negative’’ otherwise.

A negative effect of the reduction in external phase viscosity is the weakening in the external forces able to deform and split the drops. An associated positive effect is the increase in both the Rey- nolds and Dean numbers, leading to the intensification of the sec- ondary flow and associated strain rates. The critical capillary number governing breakup in laminar flow is also decreased by reduction of the external/internal viscosity ratio in the viscosity range of interest here. In light of these complex competitive effects, the experimental results are explained using theoretical insights provided by Grace–Taylor laminar breakup analysis[17,18].

We report on a dispersion process achieved in an open-loop reactor consisting of simple curved pipe segments assembled in two configurations: helically coiled and chaotic twisted pipe. The same number of bends is used to construct both coils, as they are assembled in a continuous way to form a succession of either heli- cal or alternate bends. They are of the same length, hydraulic diam- eter and cross-sectional area.

This paper is organized as follows: in Section2the experimen- tal setup and methods for the water–oil dispersion are presented.

In Section3, some theoretical issues about Dean flow and disper- sion theory are briefly recalled. Section4is devoted to the results, and the effects of the physical properties – viscosity and interfacial tension – are discussed for the two geometries. Conclusions are drawn in Section5.

2. Experimental setup and methods 2.1. Hydraulic loop and test sections

Fig. 1shows a schematic diagram of the hydraulic loop used in the experiments. Experiments are carried out for two-phase flow using immiscible fluids. The working fluids (tap water, vegetal oil) are stored in two separate tanks. The oil is driven by a centrif- ugal pump and the water is supplied by a constant-level feed tank connected to the water supply. Flow rates are controlled by valves

Flow meters

Centrifugal pump Visualization box

Setting tank

Water tank

Oil tank Test section

Flow meters

Fig. 1.Schematic diagram of the hydrodynamic loop for fuel-like emulsification.

and measured with two flowmeters. Water is injected at the test section inlet by a needle of inside diameter 2 mm, designed to min- imize flow disturbance and avoid additional breakup of the dis- persed phase. For accurate evaluation of breakup performance, the volume fraction of the dispersed aqueous phase is very low (less than 1%) so that coalescence is negligible in both the main flow and the visualization box. Because of the heating due to the lighting system, the fluids undergo a temperature change during the experiments (between 25°C and 35°C) that must be taken into account because of the high temperature sensitivity of the oil vis- cosity. The oil temperature is controlled by a chromel/alumel (type K) thermocouple located in the oil admission circuit.

The two test sections are composed of a succession of 90°bends with constant radius of curvature. Both geometries have 25 bends of 8 mm inside diameterD, the same unfolded length (L= 2 m) and the same bend curvature radius (RC= 44 mm). The test section dimensions are summarized inTable 1. The same apparatus can be arranged in two configurations: helically coiled pipe or twisted pipe. In the latter case, the chaotic advection pipe is achieved by alternating the bend curvature plane by ±90°, as shown inFig. 1.

2.2. Working fluids

The working fluids are vegetal oil for the continuous phase and water for the dispersed phase. The oil-phase viscosity is measured using a Mettler™ RM180 rheometer. In order to investigate the external phase viscosity effect, another continuous phase is pre- pared by adding 15% Butanol, which decreases the oil viscosity twofold. As the oil is very thermosensitive, the temperature effect is taken into account by Arrhenius’s law:

l

_c¼

l

_c;T0exp E R

1 T1

T0

ð1Þ whereEis the activation energy,

l

c,T0is the continuous-phase vis- cosity at reference temperatureT0, andR= 8.314 J K¹mol¹. The value of the activation energy Eis found to be 26.15 kJ for pure oil and 23.79 kJ for the mix oil + 15% Butanol.

Tween 20, dissolved in water, is used as surfactant to decrease the interfacial tension; it is generally used in solution in the water phase at a mass concentration ranging between 0.2% and 0.5%. The quantity judged adequate is necessarily greater than the interface saturation to provide constant interfacial tension in all tests. It is

important to note that this surfactant is of ‘‘small molecule’’ type, so that the created interface very quickly reaches thermodynamic equilibrium.

Interfacial tension is measured with a Krüss™ tensiometer (K12) by the ring method. Measurements carried out with Tween 20 percentages ranging between 0% and 1% showed that the inter- facial tension remains constant starting from a concentration of 0.2%, the value retained for the experiments. The physical properties of the fluids are summarized inTable 2.

2.3. Data acquisition

The droplet visualization device seen inFig. 1is a rectangular Plexiglas™ window fixed on top of a parallelepiped box and directly plugged at the exit of the test section with conic/rectangu- lar connections. The diffuser/distributor is designed first in order to maintain, as much as possible, the shear stresses at the same level as in the test section, and second, with a 7°angle, to avoid flow recirculation and thus prevent drop coalescence at the pipe exit.

Moreover, the box depth is small enough (13 mm = 1.625D) to pre- vent drop overlap. In fact, overlap did not occur for the range of flow rates in these experiments, but may occur for higher flow rates.

Pictures of the emulsion are taken with a high-frequency digital Canon™ camera placed vertically above the visualization window with its optical axis perpendicular to the window plane. The dispersed flow in the visualization box is lighted from below by an intense diffused white light. For given operating conditions, a sequence of independent images is selected and recorded; this se- quence constitutes the statistical sample of the drops. Drop diam- eters are measured from the recorded images using IMAQ Vision Builder 6 software. At the end of the analysis, a table of diameters of at least 200 drops for each run is obtained. This limiting number was obtained after a sampling campaign determining the minimal number of measured diameters that allowed convergence of the size distribution.

The experimental size distributions are fitted with a log-normal law. By taking 99% of the cumulative volume curve, a reproducible value for the maximum diameterdmax can be determined. The polydispersity of the distribution is characterized by the standard deviation factor; it is also of high interest for the quality of the emulsification process.

The Sauter mean diameterd32, defined in Eq.(2), represents the mean surface diameter:

d32¼

R‘³fð‘Þd‘

R‘²fð‘Þd‘ ð2Þ

wheref(‘) is the size probability density function. Generally, the typical Sauter diameter is found to be about half of the maximum diameterdmax.

Table 1

Test section dimensions.

Radius of circular pipe 4 mm

Bend curvature radius 44 mm

Curvature angle in bend plane 90°

Number of bends 25

Total curved length 1.80 m

Total straight length between bends 0.20 m

Total length 2.00 m

Table 2

Operating conditions for the set of experiments for helical and chaotic configurations at 298 K.

Continuous phase

Dispersed phase Continuous phase viscosity (Pa s)

Interfacial tension (N/m)

Dispersed phase volume fraction (%)

Total flow rate (mL/s)

Reynolds number

Oil Water 0.052 0.041 0–2 14–69 50–240

Water + 0.2%

Tween 20

0.052 0.025 0–2 11–39 30–110

Oil + 15%

Butanol

Water 0.025 0.035 0–2 11–39 70–300

Water + 0.2%

Tween 20

0.025 0.018 0–2 11–45 70–350

2.4. Reproducibility

Reproducibility testing was carried out to check the effect of a new operator and new trial on the measured diameters of the final size distribution: runs were repeated three times on different days for both geometries and each fluid. The maximum standard devia- tion based on Sauter mean diameter was found to be less than 7%.

3. Theoretical grounds

3.1. The Grace–Taylor theory of dispersion

In the emulsification process, one phase is dispersed into small droplets of various diameters. The drop size distribution is decisive for the properties of the final product, for instance emulsion texture and stability in a food product, or the result of a chemical reaction at the interface. The immiscible phase dispersion in the main flow is due to the strain rates and the resisting Laplace forces.

Grace [18] underlines the dispersion mechanisms for the two strain modes, shear and extensional, whose effects on breakup are quite different.

In order to characterize the balance between the viscous and capillary forces, a dimensionless parameterCa, the capillary num- ber, is defined by the ratio:

Ca¼d

s

2

r

^ð3Þ

wheredis the drop diameter,

r

the interfacial tension and

s

the vis- cous stress. Grace[18] has established that drop breakup occurs when the capillary number exceeds some critical value depending on the viscosity ratio p that reflects the drop’s internal viscous resistance:

p¼

l

_d

l

_c ^ð4Þ

where

l

cand

l

d are respectively the continuous and dispersed- phase viscosities.

The theoretical approach leads to the determination of an equi- librium size corresponding to the critical value of the capillary number. The dimensional values of the maximum diameter that can withstand an effective stress

s

can thus be expressed by the following equation:

dmax¼2

r

s

^{Cacr ð5Þ}

The critical capillary number is determined by the Grace curve for shear and elongational flows. The droplet’s maximum diameter can then be deduced from the maximal value of the shear stress in the flow.

3.2. Shear and elongational stresses in a Dean flow

The Dean number is a dimensionless number that characterizes the ratio between the viscous forces and the centrifugal forces in curved channels; it is defined as:

De¼WD³⁼²

m

R¹⁼²_C ð6Þ

whereDis the tube inner diameter,Wthe fluid axial velocity,

m

the fluid kinematic viscosity andRCthe bend curvature radius. This can also be expressed with the Reynolds numberReas:

De¼Re ffiffiffiffiffiffi

D RC

s

ð7Þ

with Re¼WD

m

^ð8Þ

The maximum droplet diameter capable of withstanding the flow stresses is governed by the maximal value of the strain rates.

For a straight pipe this is given by the wall shear rate:

c

_straight tubemax¼4W

D ð9Þ

The maximal value of the viscous stress must be determined for bends. For this purpose, the kinematic field was computed from the theoretical formulae for Dean flow[5,6]. These maximal values of the shear and elongation rates in the bend,

c

_bendmaxand

e

_bendmax, as provided in Habchi et al.[9]:

c

_bendmax¼4^W_D 1þ₃₆¹ _R^D

CDe

h i0:5

e

_bendmax¼1:1^W_D 1þ210⁷_R^D

CDe³

h i0:7

8>

<

>: ð10Þ

allow determination of the maximum droplet diameterviathe cap- illary numbers, Eq. (3), and then the corresponding maximum diameter by using Eq.(5).

4. Results and discussion

Experiments are run for different flow rates corresponding to Reynolds numbers between 30 and 350, i.e. Dean numbers be- tween 13 and 150. The parameters characterizing the dispersion process are the flow configurations (helical or chaotic), the flow rate, the continuous-phase viscosity and the interfacial tension.

The drop size distributions are measured for the four pairs of con- tinuous-dispersed phase fluids summarized inTable 2.

4.1. Chaotic effect on drop-size distribution

The droplet size distribution for a flow with pure oil and water dispersion is shown inFig. 2in which the measurement points are fitted by normal law curves. The distribution obtained in the cha- otic flow is shifted towards lower diameters and the peak is also sharper. The standard deviation of the drop-size distribution is about 20% lower for the twisted pipe than for the helically coiled pipe. This trend appears for Dean numbers at which the chaotic

0.5 1.0 1.5 2.0 2.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Droplet size distribution

Droplet diameter (mm)

Helically coiled pipe Chaotic twisted pipe

Fig. 2.Droplet size distribution for helically coiled and chaotic twisted pipe flows with pure oil and water emulsification forDe= 50 (i.e.Re= 115).

behavior begins to produce effects comparable to shear flow. The standard deviation seems minimal at low Dean numbers, where the mixing process is globally homogeneous over the whole test section but has the lowest mixing efficiency. Increasing the Dean number increases the standard deviation; the inhomogeneity of the observed emulsion is increased because of the poor radial dis- tribution of the fluid in the mixer cross section until an effective Dean number is achieved.

Fig. 2clearly shows that the chaotic configuration on the one hand causes more effective drop breakup than the helical configu- ration because lower droplet diameters are observed. On the other hand, that the drop distribution is tighter showing that the drops visit flow zones in which stretching stresses are greater in the cha- otic than in the helical flow.

4.2. Effect of external phase viscosity on drop size

Butanol is added to the continuous phase to decrease the vis- cosity of the external phase

l

c, so that the 15% volume fraction makes it possible almost to halve the oil viscosity. For identical operating conditions, the continuous-phase viscosity reduction also produces smaller drops size. Actually, the Dean number is in- creased by a factor of two (Eq.(6)), and then the external viscous strains (Eq.(10)) are enhanced for the same flow rate, something that appears to favor the breakup.

The Sauter drop diameters are plotted versus Dean number in Figs. 3 and 4. The drop diameters in the chaotic flow are systemat- ically smaller than those in the helically coiled flow. The ratio be- tween the helical and chaotic measured diameters is quite constant for the cases with oil as continuous phase: inFig. 3(a) the droplet sizes are about 21% smaller in the chaotic twisted flow than in the helically coiled pipe for the case of water as dispersed phase, and the difference remains almost constant at about 26% for the case of water + 0.2% of surfactant.

The three-dimensional chaotic trajectories generated by the recombination of the Dean cells in the twisted pipe allow statisti- cally all drops to ‘‘visit’’ the high-shear regions in the tube cross section, while this is not possible in the helically coiled geometry where the drops are trapped in the cells for their whole residence time. For higher Reynolds numbers, it is expected that the differ- ence between the two geometries would be lessened. The advanta- ges of the Dean roll-cells (and of the chaotic trajectories) decrease when the Reynolds numbers approach the transition to turbulence.

In this regard, it was shown in[7]that the relative effectiveness for heat transfer of the chaotic flow is maximal for Reynolds numbers around 200.

In the case of Butanol added to oil, inFig. 3(b), the benefits of chaotic intensification is less marked and vanishes for Dean num- bers around 150: in this range, no difference in droplet breakup can

0 20 40 60 80 100 120

0.00 0.05 0.10 0.15 0.20 0.25 0.30

d32 / D

Dean number,De Helically coiled pipe

O / W

O / W + 0.2% Tween 20 Chaotic twisted pipe

O / W

O / W + 0.2% Tween 20

(a)

(b)

0 20 40 60 80 100 120 140 160

0.00 0.05 0.10 0.15 0.20 0.25 0.30

d32 / D

Dean number,De Helically coiled pipe

O + 15% Butanol / W

O + 15% Butanol / W + 0.2% Tween 20 Chaotic twisted pipe

O + 15% Butanol / W

O + 15% Butanol / W + 0.2% Tween 20

Fig. 3.Effect of surfactant on Sauter diameter for (a) oil and (b) oil + 15% Butanol, for helically coiled and chaotic twisted pipe flows.

0 20 40 60 80 100 120 140 160

0,00 0,05 0,10 0,15 0,20 0,25 0,30

d32 / D

Dean number, De

Helically coiled pipe O / W

O + 15% Butanol / W Chaotic twisted pipe

O / W

O + 15% Butanol / W

(a)

(b)

0 20 40 60 80 100 120 140 160

0,00 0,05 0,10 0,15 0,20 0,25 0,30

d32 / D

Dean number, De Helically coiled pipe

O / W + 0.2% Tween 20

O + 15% Butanol / W + 0.2% Tween 20 Chaotic twisted pipe

O / W + 0.2% Tween 20

O + 15% Butanol / W + 0.2% Tween 20

Fig. 4.Sauter diameter comparison of oil with or without addition of 15% Butanol and (a) water and (b) water mixed with 0.2% Tween 20, for helically coiled and chaotic twisted pipe flows.

be seen between the chaotic and helical flow configurations. In fact, the highest Dean numbers are 150, which correspond to Rey- nolds numbers equal to 350. However, the relative effectiveness of the chaotic flow compared to the helical flow is greater at low Dean numbers and decreases for higher Dean numbers. This feature is analogous to the results in[7]which showed that heat-transfer intensification by chaotic advection decreased from 27% to 13%

when the Dean number increased from 36 to 120.

The effect of the external viscosity on the size distribution is highlighted in Fig. 4: it is positive without surfactant (Fig. 4(a)) and negative with surfactant (Fig. 4(b)). This is probably due to the opposing effect of Butanol addition. In fact, on the one hand, the flow rates and the velocity gradients are increased for a given Dean number; on the other hand, the viscosity is lowered and the shear stress may be decreased. As previously remarked, the dif- ference is damped for the higher Dean numbers. However, one can note that the Sauter diameters are systematically smaller for the less viscous external fluid at a given flow rate.

4.3. Effect of interfacial tension on drop size

Tween 20 is added to the dispersed phase to decrease the inter- facial tension. The physical properties of the fluids given inTable 2 show that the addition of 0.2% of Tween 20 makes it possible to cut the interfacial tension almost in half. For identical operating condi- tions (same flow and same capillary numbers), the interfacial ten- sion decrease produces smaller drops.

InFig. 3(a) the Sauter diameters are compared for pure oil with or without 0.2% Tween 20, in both helically coiled and chaotic twisted pipe flows. It is observed that droplet diameters are re- duced by a factor 2 in similar hydrodynamic conditions: this trend is consistent with the hypothesis that breakup is governed by the capillary number that is also increased in the same proportion by the interfacial tension in Eq.(3). In the case of the less viscous external phase,Fig. 3(b) oil + Butanol, the effect of the surfactant is damped as the diameter reduction is only of a few percent.

4.4. The drop-breakup diagram

The results presented so far do not collapse on a master curve as a function of Dean number for a given geometry. This suggests that, in addition the Dean number, other control parameters may be in- volved in the breakup process. The theoretical development sug- gests that at least four dimensionless parameters are possible candidates:

– the capillary number for the dispersion mode,

– the viscosity ratio for the critical value of the capillary number, – the Dean number for the flow strains,

– the Reynolds number for the laminar/inertial/turbulent transitions.

This makes global understanding and analysis of the measure- ments rather complex, since variations in the interfacial tension and the viscosity influence several control parameters simulta- neously. In this section we attempt to interpret the results in the light of the Grace–Taylor breakup theory, and then to plot the crit- ical capillary number versus the viscosity ratio.

The critical capillary number is determined by Eq.(3)in which the viscous stress

c

_bendmaxis given by Eq.(10). These values are re- ported inFig. 5, which plots the critical capillary numbers for the- oretical simple shear and pure elongational flows as a function of viscosity ratio; the real curve should lie between the two cases depending on the elongational efficiency.

In Fig. 5the critical capillary numbers are computed for all the experimental runs and are consistently plotted on the

drop-breakup diagram. The values are merely arranged in two blocks according to viscosity ratiop: in the rangep< 1 the capillary number decreases whenpincreases, a trend actually observed in Fig. 5(a). The effect of Butanol is both positive and negative: it low- ers the breakup limit, as previously explained, but also weakens the external stress. Globally, the resulting action of Butanol is favorable for dispersion in that case. The effect of Tween 20 is more visible inFig. 5(b) and appears to provide lower drops groups: as expected, the decrease in interfacial tension contributes to better breakup efficiency.

The relative location of the results with respect to the critical curves for pure shear and pure elongational flows, i.e.following the Cacr-coordinate, is more subtle. The first point to consider is the theoretical feature that this coordinate is linked to the elonga- tional efficiency of the flow; the second is related to the fact that the critical capillary curves are obtained for homogenous flows.

Regarding the first point, it can be deduced from the set of Eq.

(10)that the elongational component increases ‘‘faster’’ with Dean number than the shear component, so that all the parameters that favor the Dean number must lower the critical capillary number for a given flow. Note that for the viscosity effect (lowering the

‘‘cloud of measured points’’ for the highest p), and inside each

‘‘cloud’’, the Dean number is also increased due to the increase in the flow rates, explaining the dispersion of the measurements be- tween the two curves. The lower position of the chaotic ‘‘cloud’’

may be explained by the more homogeneous flow generated by the twisted pipe, providing better exposure of the liquid/liquid mixture to the more active zones of the pipe section. Globally, the average value of the capillary number in the chaotic case is

10^-3 10^-2 10^-1 10⁰ 10¹

10^-2 10^-1 10⁰ 10¹ 10²

10^-2 10^-1 10⁰ 10¹ 10²

without Butanol

with Butanol

Simple shear flow Elongational flow Helically coiled pipe Chaotic twisted pipe

Critical Capillary number, Cacr

Viscosity ratio, p

(a)

0,01 0,02 0,04 0,06 0,08 0,1

5x10^-2 10^-1 10⁰ 5x10⁰

Hollow symbol : Hellically coiled pipe Full symbol : Chaotic twisted pipe

Simple shear flow Elongationnal flow Critical Capillary number, Cacr

Viscosity ratio, p

O / W

O / W +0.2% Tween O + 15% Butanol / W

O + 15% Butanol / W+0.2% Tween

(b)

Fig. 5.(a) Drop breakup diagram. Shear flow curve is adapted from [18]and elongational curve from[22]. (b) Zoom showing the all couples of continuous- dispersed phase fluids.

lower than for the helical case by 20% to 30% in the range of Dean numbers [13–150] as shown inFig. 6; moreover, this trend inten- sifies for decreasing Dean numbers, according to a linear law represented with dashed line inFig. 6:

CaCr Chaotic=CaCr Helical¼0:70þ710⁴De ð11Þ

The present results thus seem consistent with the theoretical trends. However, it should be remarked that inFig. 5the theoret- ical curves overestimate the drop diameters with our computing hypothesis. This can be attributed to the choice of the maximum strain rate in determining the capillary number. In fact, in the inho- mogeneous flow, it would be more accurate to construct the capil- lary number using an ‘‘apparent strain rate’’, more representative of process conditions, that is by definition lower than the maxi- mum value and would lead to slightly higher capillary numbers.

This would be time-consuming because a Metzner–Otto process would be necessary, experimentally or by CFD.

4.5. Mixing efficiency

The energy cost of the helically coiled and chaotic twisted pipe mixers is compared with that of existing inline mixers reported in the literature[19–21,23–25]. The interfacial contact areaAis given by the Sauter diameter:

A¼6U d32

ð12Þ whereUis the volume fraction of the dispersed phase.

The energy consumptionEis calculated from the pressure drop DP:

E¼DP

q

^ð13Þ

The pressure dropDPis obtained from the experimental correlation in[26]for the flow through helically coiled tubes.Fig. 7shows that the helically coiled and chaotic twisted pipes are located in the small-energy consumption zone (around 10–30 J kg¹) with good creation of interfacial area (around 100–300 m²m³). The chaotic twisted pipe exhibits systematically higher interfacial area than the helically coiled pipe for quite similar energy consumption. The Sulzer and Walker mixers seem to have the highest interfacial area but lie in the range of higher energy consumption. The Lightnin mixer provides the same order of magnitude of diameters as the coiled pipes, but with greater energy consumption. The Kenics static

mixer exhibits similar energy consumption but with a lower interfacial area.

The performances of the helical and chaotic mixers are very comparable to the other static mixers since they exhibit low power consumptions in order to create relatively large interfacial areas.

They operate efficiently at small Reynolds numbers, with Dean number in the range [1–1000]. Moreover, the use of twisted cha- otic pipe as a static mixer depends on the pipe length (here 2.0 m, see Table 1). The final droplet-breakup size is certainly reached before the exit of the chaotic mixer, and an optimized length should be able to create the same granulometric distribu- tion and consume less energy. This optimization study will be carried out in future work. The relatively low energy cost and the high interfacial area show that helically coiled and chaotic twisted pipe flows can have good impact in the industrial applications, in moderate shear rates and viscosities emulsification conditions.

5. Conclusions

The liquid/liquid dispersion achieved in the chaotic twisted- pipe geometry is confirmed as more efficient than in the helically coiled pipe. The basic mechanism exploited in both systems is mix- ing by Dean roll-cells generated in the bends. Moreover, the twisted-pipe configuration is shown to profit from the chaotic advection.

Beyond the advantage of the chaotic geometry, the effects of functional additives in the working fluids are shown to improve emulsification for the same operating conditions. The presence of a small Butanol fraction (about 15%) reduces the drop size by about 10%, averaged over all measurements, and the presence of a surfac- tant also improves the emulsification process by about 20%. The attempt to interpret the variation in drop size as dependent on the operating conditions shows trends consistent with the theoret- ical results.

After further investigation to optimize the process conditions – pipe diameter and length, additive fractions, flow rates, tempera- ture – versus a given productivity and pressure drop, the ‘‘chaotic’’

mixer will, we believe, be attractive in producing emulsions such as emulsified fuel for novel engineering applications.

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20 40 60 80 100 120 140 160

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

W / O

W + 0.2% Tween 20 /O W / O + 15% Butanol

W + 0.2% Tween 20 /O + 15% Butanol CacrChaotic / CacrHelical

Dean number,De

Fig. 6.Ratio of capillary numbers for chaotic and helical flow (at identical flowrates).

1 10 100 1000

10 100 1000 10000

Walker

Sulzer

Lightnin

Kenics

Interfacial area, A (m2/m3)

Energy consumption, E (J/kg) Helically coiled pipe

Chaotic twisted pipe

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