Chemical Engineering Science 65 (2010) 6282–6291

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Chemical Engineering Science

0009-25

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/ces

Electromagnetically induced chaotic mixing in a pipe mixer

Sugilal Gopalakrishnan a,b, Andre Thess a,�

a Institute of Thermodynamics and Fluid Mechanics, Ilmenau University of Technology, P.O. Box 100565, 98684 Ilmenau, Germanyb Nuclear Recycle Group, Bhabha Atomic Research Centre, Mumbai 400085, India

a r t i c l e i n f o

Article history:

Received 11 December 2009

Received in revised form

6 September 2010

Accepted 10 September 2010Available online 18 September 2010

Keywords:

Chaos

Computation

Homogenisation

Mixing

Static mixer

Three-dimensional chaotic flow

09/$ - see front matter & 2010 Elsevier Ltd. A

016/j.ces.2010.09.016

esponding author. Tel.: +49 3677 692445; fa

ail address: thess@tu-ilmenau.de (A. Thess).

a b s t r a c t

A numerical investigation is carried out to study the mixing by chaotic advection in an electromagnetically

driven pipe mixer. It consists of a pipe with two inner electrodes which are energized alternatively. An

externally applied magnetic field along the flow direction interacts with the local electric currents inducing

a spatially varying electromagnetic body force in the model fluid. The system is an extension of Aref’s

blinking vortex model to three dimensions. The Lagrangian motion of passive tracer particles is numerically

simulated to quantify the degree of stirring. The model predictions indicate that the chaotic mixing is

strongly dependent on the modified Hartmann number, the electrode switching frequency and the

electrode separation distance. A comparison with numerical simulation results obtained using a

dimensional model with glass melt shows that the chaotic mixing behavior of the glass melt is very

close to the non-dimensional model predictions. The computational results presented here will be useful for

developing efficient glass melt homogenisation systems.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

High energy laser (HEL) applications require glass windowswith a very high optical quality (Bayya et al., 2006). Defects likebubbles and crystalline particles in the glass window give rise toscattering and refractive index inhomogeneity inducing opticaldistortion, which is highly undesirable for such applications. HELglass windows, particularly of low melting and low viscosityglasses, are made by stirring the molten glass during coolingwithout the use of a mechanical stirrer within the glass, byrotating the mold or crucible in which the glass is cooling (Tran,2009). Electromagnetically induced chaotic mixing in a pipemixer can be a promising substitute for making high qualityoptical glasses as it eliminates the complex rotating system. Sucha device can be well adapted for a continuous production. It is alsosuperior to conventional static mixers which lead to pressuredrops that are so high that the process becomes impractical forhighly viscous glass melts.

Chaotic advection plays a significant role in a variety of mixingapplications in engineering, especially in systems with lowReynolds numbers. In chaotic mixing, various layers of fluid arebent and folded repeatedly so as to bring initially distant partsclose together and separate initially close parts exponentiallyfarther. As a result, the interfacial area between the fluid-layersincreases, whereas the thickness of each layer or striationdecreases. These complementary actions improve the mixing

ll rights reserved.

x: +49 3677 692411.

process substantially. The theory of chaotic mixing has beenwell developed due to the pioneering work of Aref and co-workersas well as Ottino and co-workers (Phelan et al., 2008). A thoroughreview of the key concepts in chaotic mixing is given by Ottino(1989).

In Aref’s well-known blinking vortex model, the flow isinduced by two corotating point vortices, separated by a fixeddistance (2b), that blink on and off periodically with a constantperiod (T) in a two-dimensional, inviscid, incompressible fluid in acircular domain of a given radius (a) (Aref, 1984). For the first halfof the period, the vortex with a strength G resides at (b,0) and forthe remainder of the period, it remains at (�b,0) where boa. Thedynamics of the blinking vortex flow are governed by twodimensionless parameters: m¼GT=2pa2, a dimensionless periodof the oscillation, and b¼ b=a, a dimensionless amplitude of theoscillation. For a given b, chaotic advection is localized when m issmall enough. As m increases, the chaotic region grows and theextent of homogenisation increases. Mixing is greatly enhanced asthe transverse intersection of the time-dependent streamlinesincreases.

Efficient mixing can be produced in pipe flows by spatialchanges along the axis of the pipe. Two successive streamlineportraits, say at z and zþDz for spatially periodic flows, whensuperimposed, should show intersecting streamlines whenprojected onto x�y plane to create the chaotic advection (Ottinoand Wiggins, 2004). The classical partitioned-pipe mixer (Khkharet al., 1987) and the standard Kenics static mixer (Hobbs andMuzzio, 1997) are typical examples for the spatially periodic pipemixers. In these mixers, a cross-sectional motion is induced eitherby the rotation of the pipe, in the case of the partitioned-pipe

S. Gopalakrishnan, A. Thess / Chemical Engineering Science 65 (2010) 6282–6291 6283

mixer, or by the static mixing element with a helical twist, in thecase of the Kenics mixer, whereas the axial flow is caused by apressure gradient. The main difference between the two mixers isthat the direction of rotation of the cross-sectional flow is same inthe adjacent elements of the partitioned-pipe mixer while it isopposite in the static mixer. The mixing strength in a partitioned-pipe mixer can be expressed in terms of the product of thecharacteristic strain rate of the cross-sectional flow and theaverage residence time based on the axial velocity. The analogousparameters in a Kenics static mixer are the pitch and aspect ratioof each helical element.

The present study has been motivated by the experimentalworks reported in the literature to investigate the influence of theLorentz forces on homogenisation of electrically heated glassmelts (Hulsenberg et al., 2004; Krieger et al., 2007). These studiesdemonstrate the capability of the Lorentz forces imposed by anexternally applied magnetic field to effectively stir the glass meltin which an electric field is maintained by means of a pair ofimmersed electrodes. This permits the possibility of developing aspatially periodic blinking vortex flow model. The goal of thepresent work is to systematically investigate electromagneticallyinduced chaotic advection in such a spatially periodic blinkingvortex flow model.

The paper is organized as follows. The mathematical modelused for investigating the present problem is presented inSection 2 and implementation of the numerical solution isdiscussed in Section 3. Computational results and their discussionare given in Section 4, which is followed by final conclusions inSection 5.

2. Mathematical model

The problem to be considered here is defined in Fig. 1, whichillustrates the configuration of the electromagnetically drivenpipe mixer. It consists of an outer pipe, with diameter D andlength L, and two inner rod electrodes, with diameter d and length

Fig. 1. Perspective view of the electromagnetically driven pipe mixer.

L, which are separated by a radial distance b. The inner electrodesare energized alternatively while the outer pipe is kept at zeroelectric potential invariably. The potential difference between theouter pipe and the inner electrode gives rise to an electric currentin the plane perpendicular to the axis of the pipe. When a steady,uniform external magnetic field is applied along the flowdirection, the interaction between the magnetic field and thecurrent density field results in an electromagnetic body force fieldin the electrically conducting fluid. This Lorentz force field iscapable of twisting the flow around the inner electrodes. Thetime-periodic energization of the inner electrodes can lead totransverse intersection of the successive streamline portraitswhen projected onto the x�y plane suggesting the occurrence ofthe chaotic advection.

The model fluid is assumed to be Newtonian. The time-dependent flow inside the pipe mixer can be computed by solvingthe unsteady Navier–Stokes equation using the Boussinesqapproximation,

r @V

@tþrV � rV¼�rpþr � ½ZðrVþrVT

Þ�þr0gbðT�T0ÞþJ� B, ð1Þ

together with the incompressibility condition,

r � V¼ 0, ð2Þ

where V is the velocity vector; J is the electric current density; B isthe externally applied magnetic flux density; p is the pressure; g isthe acceleration due to gravity; r, b, Z are the density, coefficientof thermal expansion and viscosity, respectively. The ratio ofthe electric body force to the electromagnetic body force canbe expressed as ðersrjÞ=ðs2B0Þ, where e is the electricpermittivity; s is the electrical conductivity and j is the electricscalar potential. Eq. (1) assumes that the electric body force½ersðrjÞ2=s� is negligible compared to the electromagnetic bodyforce ½srjB0� as ½ðersrjÞ=ðs2B0Þ�51 for the present problem.

The temperature T in the model fluid can be obtained bysolving the energy conservation equation:

rcP@T

@tþrcPV � rT ¼r � ðkrTÞþ

J2

s, ð3Þ

where cP is the specific heat; k is the thermal conductivity; s is theelectrical conductivity and the term J2=s is the volumetric heatsource on account of Joule heating. The viscous heating isneglected as the Brinkman number, Br¼ Z0V2

0 =k0DT, satisfies thecondition Br51.

The electric current density J can be expressed by thegeneralized Ohm’s law (Moreau, 1990):

J¼ s �rjþV � B�@A

@t

� �, ð4Þ

where j is the electric scalar potential and A is the magneticvector potential. The first term ð�srjÞ results from the spatialdistribution of the electric scalar potential. The second term ðsV �BÞ is the induced current density due to the motion of theelectrically conducting fluid in a magnetic field. For a very smallmagnetic Reynolds number ðRem ¼ m0s0V0D, where m0 is themagnetic permeability) the induced current density due to themotion of the fluid can be neglected. The third term ð�s@A=@tÞ isthe induced current density due to a time varying magnetic fieldand is zero for a steady (static) magnetic field. Thus, theexpression for the electric current density becomes J¼�srjfor the present system. The continuity of the electric currentstates that r � J¼ 0. Therefore, the electric current density in themodel fluid can be obtained by solving the Laplace equation,

r � ðsrjÞ ¼ 0, ð5Þ

S. Gopalakrishnan, A. Thess / Chemical Engineering Science 65 (2010) 6282–62916284

together with the simplified Ohm’s law

J¼�srj ð6Þ

(Choudhary, 1986; Giessler and Thess, 2009; Gopalakrishnanet al., 2010).

For time-periodic energization of the inner electrodes, the three-dimensional flow in the pipe mixer becomes time-periodic and itsnumerical simulation requires a substantial amount of computationaltime and resources. However, if spatially periodic, time-independentboundary conditions are used to describe the conditions of the innerelectrodes, a steady three-dimensional flow field can be obtainedeasily. The Lagrangian particle motion in the steady three-dimen-sional flow can be numerically simulated to study the influence ofvarious governing parameters on the mixing behavior of this flow.The results, thus obtained, can be useful to understand the mixingbehavior of the time-periodic three-dimensional flows. Therefore, asteady three-dimensional model is used here to describe the flow inthe electromagnetically driven pipe mixer. Following non-dimen-sional equations govern the electromagnetically driven pipe mixeroperating under isothermal conditions:

V � rV¼�rpþ1

Rer2Vþ

M

ReðJ� BÞ, ð7Þ

r � V¼ 0, ð8Þ

r2j¼ 0, ð9Þ

J¼�rj: ð10Þ

All quantities used in Eqs. (7)–(10) are dimensionless. The non-dimensionalization was effectuated by using D, V0, D=V0, j0, s0j0=D

and B0 as reference scales for length, velocity, time, electric potential,current density and magnetic flux density, respectively. Here V0 is theaverage axial velocity, j0 is the electrode potential and B0 isthe externally applied magnetic flux density in the direction of flow.The dimensionless numbers Re and M are, respectively, the Reynoldsnumber and the square of the modified Hartmann number which willbe defined below. Re is the representative of the ratio of an inertialforce to viscous force and M is the representative of the ratio of theLorentz force to viscous force. Using the above scales, Re (the Reynoldsnumber) can be expressed as

Re¼DV0

n , ð11Þ

and M (the square of the modified Hartmann number) can beexpressed as

M¼s0j0B0D

rnV0: ð12Þ

It should be noticed that the modified Hartmann numberð ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis0j0B0D=rnV0

pÞ differs from the conventional Hartmann

number ð ¼ B0Dffiffiffiffiffiffiffiffiffiffiffiffis=rn

pÞ in that the former describes the ratio

between the imposed Lorentz forces and the viscous forces,whereas the latter describes the ratio between the inducedLorentz forces and the viscous forces. The conventional Hartmannnumber is negligibly small in the present application. Unlike inthe Stokes flow, the inertial term in Eq. (7) cannot be neglected inthe present problem for which ReZ1 and 0rMr1000.

In the present study, L¼10D and d¼D=5 are used. Thedimensionless radial separation distance between the left andright electrodes is

s¼b

D: ð13Þ

All velocity components were set to zero at the solid boundarieswhile (Vx¼0, Vy¼0,Vz¼1) was specified at the inlet boundary and

p¼0 was specified at the outlet boundary. The inlet and outletboundaries were treated as electrically insulated. The modelincludes an entrance zone and an exit zone, each having a unitlength, in which both the pipe and inner electrode potentials wereset to zero. This implies that the Lorentz force can exist only in thezone 14z49. In this zone ð14z49Þ, the electric potential on thepipe wall was set to zero while j¼max½0,sinð2pNsðz�1ÞÞ� wasspecified on the left electrode wall and j¼max½0,�sinð2pNsðz�1ÞÞ�was specified on the right electrode wall. Here Ns is the dimension-less switching frequency at which the left and right electrodes areenergized to produce a spatially periodic electric field. In otherwords, 1=Ns is the dimensionless periodicity length of the spatiallyvarying electric field.

3. Numerical solution

The model equations, Eqs. (7)–(10), were solved using thecommercial computational fluid dynamic software FLUENTTM.A GAMBIT-generated non-uniform structured grid with 910 400cells was used in this study. Although symmetry (Cortelezzi andMezic, 2009) and mapping (Anderson and Meijer, 2000) can beused to obtain the Poincare sections relatively faster in steadythree-dimensional incompressible flows with rotational symme-try, local stretch ratios and progressive snapshots of the spread oftracer particles were used in the present study to compare theextent of mixing under different operating conditions of the pipemixer. In order to study the chaotic advection in the pipe mixer,the Lagrangian motion of 25 600 passive tracer particles weretracked employing the discrete phase model of FLUENTTM.These particles, initially distributed uniformly between points(x¼�0.005, y¼0, z¼1) and (x¼0.005, y¼0, z¼1), were trackedusing the Runge–Kutta scheme with a time step of Dt¼ 0:001.

The box counting method (Liu et al., 1994) was utilized toquantify the particle dispersion in the pipe mixer. In this study,positions of the Lagrangian particles at a given time are projectedonto a given horizontal plane. (This is analogous to a snapshot ofthe particles taken by a camera at one end of the pipe with anilluminating lamp on the other end.) As the box size is related tothe number of particles tracked, the box size and the number ofparticles are chosen such that 98% of the boxes would contain atleast one out of the 25 600 randomly distributed particles. Thedegree of mixing can be quantified in terms of a stirring index, e,which is defined as

e¼ 1

K

XK

i ¼ 1

oi, ð14Þ

where K is the total number of boxes used and oi is a weightingfactor for the ith box (Kim and Beskok, 2007). The weightingfactor oi can be defined in terms of the number of particles in thebox (ni) and number of particles per box for a homogeneousmixing state ðnmax ¼ 25 600=KÞ as

oi ¼ni

nmaxif nirnmax,

oi ¼ 1 if niZnmax:

4. Results

4.1. Effect of the modified Hartmann number

In order to illustrate the effect of the modified Hartmannnumber on chaotic mixing in an electromagnetically driven pipe

Fig. 3. Streamtraces of 20 points uniformly distributed between (x¼0, y¼�0.4,

z¼0) and (x¼0, y¼0.4, z¼0) obtained for different values of M with Re¼1, Ns¼12

and s¼0.5. Panels shown are (a) M¼0; (b) M¼100; (c) M¼500; (d) M¼1000.

S. Gopalakrishnan, A. Thess / Chemical Engineering Science 65 (2010) 6282–6291 6285

mixer, a non-uniform distribution of a passive scalar wasintroduced into the pipe inlet and its behavior in an Eulerianframework at high Peclet number was monitored along the pipelength. A similar approach was reported by Rodrigo et al. (2003).The concentration of the passive scalar inside the mixer wasobtained by solving numerically the scalar transport equation:

V � rC ¼1

Per2C, ð15Þ

where C is the species concentration and Pe is the Peclet number.Here the Peclet number is defined as Pe¼ V0D=D0, where D0 is themass diffusivity. Since the present study emphasizes mixing byadvection, a high value of Pe¼1000 was used for the numericalsolution of Eq. (15) to keep the mass transport by diffusion muchsmaller than that by advection. The pipe wall was assumed to benon-diffusive. The inlet concentration was specified as a functionof the x- and y-coordinates, C0¼x2+y2, which has a standarddeviation sC ¼ 0:0727 for the given geometry with s¼0.5.Standard deviations of the species concentration across the pipecross-section along the pipe length for different values of M withRe¼1, Ns¼12 and s¼0.5 are compared in Fig. 2. For M¼0, the sC

at the pipe exit (i.e. z¼10) is 0.0328. Fig. 2 shows that sC � 10�4 atz¼3 when M¼103. Thus, Fig. 2 clearly illustrates the mixing effectas a function of M.

In order to visualize the flow behavior as a function of M,streamtraces of 20 points uniformly distributed between (x¼0,y¼�0.4, z¼0) and (x¼0, y¼0.4, z¼0) obtained for differentvalues of M with Re¼1, Ns¼12 and s¼0.5 are compared in Fig. 3.For M¼0, the flow is purely axial and all the streamtraces arevertical lines as seen in panel (a). The Lorentz force imposed bythe externally applied magnetic field results in twisting of thestreamtraces and the extent of twisting increases with increase inM (panel (b)). As M increases further, the streamtraces turnaround the left and right electrodes alternatively while travelingaxially (panels (c)–(d)). This can lead to transverse intersection ofthe successive streamline portraits when projected onto the x�y

plane suggesting the occurrence of the chaotic advection.To evaluate the extent of chaotic mixing in the pipe mixer, the

Lagrangian motion of 25 600 passive tracer particles was trackedby integrating the Eulerian velocity field ðV¼ dX=dtÞ. Theseparticles were initially distributed uniformly between points(x¼�0.005, y¼0, z¼1) and (x¼0.005, y¼0, z¼1) to form a

Fig. 2. Standard deviations of species concentration across the pipe cross-section

along the pipe length for different values of M with Re¼1, Ns¼12 and s¼0.5. The

inlet concentration is C0¼x2+y2 with standard deviation, sC ¼ 0:0727.

Fig. 4. Time evolution of the material line formed by 25 600 tracer particles

initially distributed uniformly between points (x¼�0.005, y¼0, z¼1) and

(x¼0.005, y¼0, z¼1) for M¼1000, Re¼1, Ns¼12 and s¼0.5. Panels shown are

at times (a) t¼1; (b) t¼2; (c) t¼3; (d) t¼4; (e) t¼5.

material line. Time evolution of this material line for M¼1000,Re¼1, Ns¼12 and s¼0.5 is illustrated in Fig. 4. It clearly suggestsa non-linear stretching of the passive material line. Snapshots of

S. Gopalakrishnan, A. Thess / Chemical Engineering Science 65 (2010) 6282–62916286

spread of passive tracer particles at t¼4 projected onto the x�y

plane for different values of M (¼0, 100, 500, 1000) with Re¼1,Ns¼12 and s¼0.5 are presented in Fig. 5. The M¼0 case (Panel (a)),as expected, does not exhibit any significant stretching or folding.However, the stretching and folding improve as M increases and

Fig. 5. Snapshots of spread of passive tracer particles at t¼4 projected onto the

x�y plane for different values of M with Re¼1, Ns¼12 and s¼0.5. Panels shown

are (a) M¼0; (b) M¼100; (c) M¼500; (d) M¼1000.

Table 1Slope of the linear fit of ln(l/l0) for 0otr0:5 for different values of M with Re¼1,

Ns¼12 and s¼0.5.

Modified Hartmann

number (M)

Slope of the linear

fit of ln(l/l0)

0 0.94

100 1.28

500 3.89

800 6.30

1000 7.61

Fig. 6. Time evolution of local stretch ratio (a) and stirring index

leads to a globally chaotic flow at M¼1000 (Panel (d)). Table 1presents slopes of the linear fits of ln(l/l0) for 0otr0:5 fordifferent values of M. It indicates that the rate of mixing increaseswith increase in M. Time evolution of local stretch ratio (l/l0) andstirring index ðeÞ for different values of M with Re¼1, Ns¼12and s¼0.5 are presented in Fig. 6(a) and (b), respectively. Table 1and Fig. 6 clearly show that both the stretch ratio and the stirringindex strongly depend on the modified Hartmann number for agiven axial flow.

In a partitioned-pipe mixer, the product of the characteristicstrain rate of the cross-sectional flow and the average residencetime in the pipe is a measure of cross-sectional stretching (Khkharet al., 1987). For the present problem with a constant aspect ratioðL=D¼ 10Þ, the dimensionless number M defined by Eq. (12) is ameasure of the cross-sectional stretching. Therefore, the modifiedHartmann number is expected to govern the cross-sectionalstretching irrespective of the Reynolds number. In order to verifythis aspect, local stretch ratio calculations were carried out forM¼500 and M¼1000 with Re¼2, Ns¼12 and s¼0.5. These resultsare compared with those for Re¼1 in Fig. 7(a), which shows agood agreement between them. This was further substantiated bycomparing the slope of the linear fit of ln(l/l0) for 0otr0:5. Theresults are presented in Fig. 7(b), which supports the inferencemade from Fig. 7(a).

4.2. Effect of electrode switching frequency

For the present study, the pipe is invariably kept at zeropotential while the left electrode potential is given byj¼max½0,sinð2pNsðz�1ÞÞ� and the right electrode potential isgiven by j¼max½0,�sinð2pNsðz�1ÞÞ� for 14z49. Here Ns is thedimensionless switching frequency at which the left and rightelectrodes are energized spatially. For Ns¼0, there exists noLorentz force. In the present model, the parameter Ns isresponsible for the blinking action and, therefore, controls thetransverse intersection of the successive streamline portraits in agiven system. As a result, the chaotic advection in the electro-magnetically driven pipe mixer is expected to be dependent on Ns.

The effect of Ns on the chaotic mixing inside the pipe mixer isillustrated in Fig. 8, which presents snapshots of spread of passivetracer particles at t¼4 projected onto the x�y plane for differentvalues of Ns (¼1, 2, 4, 12) with M¼1000, Re¼1 and s¼0.5. Panel(a), which corresponds to Ns¼1, shows that the material lineundergoes rotational stretching due to the Lorentz forces. As Ns

increases, the repeated stretching and folding result in better

(b) for different values of M with Re¼1, Ns¼12 and s¼0.5.

Fig. 7. Comparison of l/l0 (a) and slope of the linear fit of ln(l/l0) for 0otr0:5 (b) for different values of M and Re with Ns¼12 and s¼0.5.

Fig. 8. Snapshots of spread of passive tracer particles at t¼4 projected onto the

x�y plane for different values of Ns with M¼1000, Re¼1 and s¼0.5. Panels shown

are (a) Ns¼1; (b) Ns¼2; (c) Ns¼4; (d) Ns¼12.

S. Gopalakrishnan, A. Thess / Chemical Engineering Science 65 (2010) 6282–6291 6287

mixing (see Panels (b)–(d)). Time evolution of the local stretchratio (l/l0) and the stirring index ðeÞ for different values of Ns,ranging from 1 to 20, with M¼1000, Re¼1, and s¼0.5 arepresented in Fig. 9(a) and (b), respectively. Fig. 9 clearly showsthat there exists an optimum Ns away from which the chaoticmixing effect starts deteriorating. Fig. 10 shows that the optimummixing occurs at Ns¼12 for M¼1000, Re¼1 and s¼0.5.

The effect of the switching frequency (Ns) can be reflected on M

by multiplying it with a factor ½1�expð�Ns=2Þ� if NsrNoptimum. Inorder to illustrate the combined effect of M and Ns, the stirringindex is plotted as a function of M½1�expð�Ns=2Þ� for differentvalues of Ns, ranging from 1 to 12, with M¼1000, Re¼1 and s¼0.5in Fig. 11.

4.3. Effect of electrode separation distance

Another parameter which significantly affects the performanceof the electromagnetically driven pipe mixer is the electrodeseparation distance, s. In order to investigate the effect of

the electrode separation distance, numerical simulations werecarried out for four different values of s (¼ 0.3, 0.4, 0.5 and 0.6).Fig. 12 compares the snapshots of spread of passive tracerparticles at t¼2 projected onto the x�y plane for different valuesof s with M¼1000, Re¼1 and Ns¼12. Panel (a), which corre-sponds to s¼0.3, shows the presence of a regular region near thepipe wall. This can be attributed to the low electric currentdensity near the pipe wall for s¼0.3 case. Panel (d), whichcorresponds to s¼0.6, shows a substantial reduction in thematerial stretching. This is on account of the reduction in thetransverse intersection of the streamline portraits as the Lorentzforce is very weak in the central core of the pipe mixer. Table 2presents slopes of the linear fits of ln(l/l0) for 0otr0:5 fordifferent values of s. It indicates a higher slope for s¼0.3 duringthe initial period. This is on account of the fact that the initiallocation of the material line lies in the region of a high electriccurrent density and, therefore, a high Lorentz forces. Fig. 13presents the time evolution of local stretch ratio and stirringindex for different values of s with M¼1000, Re¼1 and Ns¼12. Itclearly shows that s¼0.5 gives rise to the best mixing among thevarious cases studied. These results show that both localstretching and global mixing (long term) behavior are requiredto compare the performance of the pipe mixer. Based on themodel predictions, s¼0.5 is recommended as the optimum valueof the electrode separation distance.

4.4. Comparison with glass melt homogenisation

In order to validate the applicability of the results obtainedusing the non-dimensional model to glass melt homogenisation,numerical simulations were carried out employing a dimensionalmodel with glass melt as the model fluid. A 1 m long pipe with0.1 m diameter was considered for this purpose. Two innerelectrodes with a diameter of 0.02 m separated with a radialdistance of 0.05 m were used for the numerical simulation. Thetemperature-dependent physical properties of the glass melt usedfor the simulation are given in Table 3 (Giessler and Thess, 2009;Gopalakrishnan et al., 2010). The Joule heat generation and thegravitational body forces in the molten glass were also taken intoaccount. The dimensional form of the model equations weresolved numerically using FLUENTTM.

A uniform inlet fluid temperature (1525 K) and an isothermalpipe wall (1525 K) boundary conditions were enforced such thatthe dimensional model is close to the isothermal, non-dimen-sional model. An axial velocity of 0.0025 m/s chosen based on amechanically stirred industrial glass melt homogenizer with a

Fig. 9. Time evolution of local stretch ratio (a) and stirring index (b) for different values of Ns with M¼1000, Re¼1 and s¼0.5.

Fig. 10. Effect of electrode switching frequency (Ns) on the stirring index for

M¼1000, Re¼1 and s¼0.5.

Fig. 11. Effect of electrode switching frequency (Ns) on the stirring index for

M¼1000, Re¼1 and s¼0.5.

Fig. 12. Snapshots of spread of passive tracer particles at t¼2 projected onto the

x�y plane for different values of s with M¼1000, Re¼1 and Ns¼12. Panels shown

are (a) s¼0.3; (b) s¼0.4; (c) s¼0.5; (d) s¼0.6.

Table 2Slope of the linear fit of ln(l/l0) for 0o tr0:5 as a function of s with M¼1000,

Re¼1 and Ns¼12.

Electrode separation

distance (s)

Slope of the linear

fit of lnðl=l0Þ

0.3 9.73

0.4 6.92

0.5 7.61

0.6 6.75

S. Gopalakrishnan, A. Thess / Chemical Engineering Science 65 (2010) 6282–62916288

diameter of 100 mm and a length of 250 mm. The glass pull ratethrough this dynamic mixer is 550 g/min of a phosphate glasswith a density of 3520 kg/m3. An axial velocity which isapproximately 10 times higher than that of the dynamic mixerwas used in order to achieve a higher throughput. An axialmagnetic flux density of 0.7232 T was used for the simulation. Themaximum electrode potential ðj0Þ was 10 V. These conditionscorrespond to M¼1000 and Re¼1. Other parameters were Ns¼12

Fig. 13. Time evolution of local stretch ratio (a) and stirring index (b) for different values of s with M¼1000, Re¼1 and Ns¼12.

Table 3Physical properties of glass melt used for the numerical simulations.

Density (r, kg/m3) 3328.7

Coefficient of expansion (b, 1/K) 10�4

Specific heat (cP, J/kg K) 1285

Thermal conductivity (k, W/m K) 3.4538�10�8T3+2�10�6T2

�0.0032T+2.68

Viscosity (Z, Pa s) 0.0992 e1046.2/(T�1033.41)

Electrical conductivity (s, S/m) 1.74�106 e(�20 300/T)

Fig. 14. Comparison of the local stretch ratios obtained for the glass melt with the

results of the non-dimensional model. The parameters are M¼1000, Re¼1, Ns¼12

and s¼0.5.

Fig. 15. Snapshots of spread of passive tracer particles projected onto the x�y

plane obtained for time-periodic electrode switching with M¼1000, Re¼1, Nt � 12

and s¼0.5. Panels shown are (a) t¼0.25; (b) t¼0.5; (c) t¼0.75; (d) t¼1; (e)

t¼1.25; (f) t¼1.5.

S. Gopalakrishnan, A. Thess / Chemical Engineering Science 65 (2010) 6282–6291 6289

and s¼0.5. Fig. 14 compares the local stretch ratios obtained forthe glass melt with the results of the non-dimensional model. Itshows that the chaotic mixing behavior of the glass melt is veryclose to the non-dimensional model predictions. The maximumand minimum temperatures observed in the glass melt are 1526.4and 1525 K, respectively. The numerical results are practicallyconsistent with the assumption of isothermal condition of thenon-dimensional model. The pressure drop calculated for theelectromagnetically driven pipe mixer with M¼1000, Re¼1,Ns¼12 and s¼0.5 is 19.5 Pa.

4.5. Comparison with time-periodic electrode switching

As the spatially periodic electrode switching of the two innerelectrodes is not physically realizable, a time-periodic electrodeswitching was considered in which the two inner electrodeswere energized alternatively with a time period of T’s which

Fig. 16. Comparison of the local stretch ratio obtained for spatially periodic

electrode switching (M¼1000, Re¼1, Ns¼12, s¼0.5) and time periodic electrode

switching (M¼1000, Re¼1, Nt¼12, s¼0.5).

S. Gopalakrishnan, A. Thess / Chemical Engineering Science 65 (2010) 6282–62916290

corresponds to a dimensionless switching frequency of Nt givenby the expression Nt ¼ 0:8L=V0T. (This expression was obtainedbased on the fact that the Lorentz force does not exist in theentrance zone and exit zone, each having a length of 0.1 L.)

The dimensional model described in Section 4.4 was modifiedby replacing the spatially periodic electrode switching with time-periodic electrode switching. The time-periodic boundary condi-tion for the left electrode was j¼ 10�max½0,sinð2pNt tÞ� andthat for the right electrode was j¼ 10�max½0,�sinð2pNt tÞ�.A time period T¼27 s, which is equivalent to Nt � 12, was used forthe simulation. Other dimensionless parameters of the simulationare M¼1000, Re¼1 and s¼0.5.

Evolution of the spread of passive tracer particles obtained forthe time-periodic electrode switching case is given in Fig. 15.Panels shown in Fig. 15 show that the material line is stretched bythe left and right electrodes alternatively. As the time proceeds,repeated stretching and folding result in a well-mixed glass meltexcept near the boundaries. Fig. 16 compares the local stretchratios obtained for the time-periodic electrode switching withthat of the spatially periodic electrode switching. It shows thatmaterial stretching in both cases are in good agreement. Thisclearly suggests that the electromagnetically driven pipe mixerwith physically realizable time-periodic electrode switching caneffectively homogenize the glass melt flow through it. Desireddegree of mixing can be achieved by selecting appropriate pipelength. Multiple units with different electrode orientations arealso possible to obtain the required extent of mixing.

5. Conclusions

An electromagnetically driven pipe mixer comprising of a pipewith two inner electrodes was conceptualized to homogenizehighly viscous and weakly electrically conducting liquids. Onepossible application is glass melt homogenisation to produce highquality optical glasses. The mixer works based on the principle ofAref’s blinking vortex model. The inner electrodes are energizedalternatively while the pipe is kept at zero electric potentialinvariably. An externally applied magnetic field along the flowdirection interacts with the local electric currents inducing aspatially varying electromagnetic body force in the flowing fluid.

These Lorentz forces make the flow to turn around the left andright electrodes alternatively while traveling axially. This givesrise to transversely intersecting streamline portraits, which inturn lead to the chaotic advection.

The Lagrangian motion of passive tracer particles wasnumerically simulated using an isothermal, non-dimensionalmodel to study the effect of various governing parameters onthe chaotic mixing inside the electromagnetically driven pipemixer. The present study shows that the modified Hartmannnumber, the electrode switching frequency and the electrodeseparation distance are the governing parameters characterizingthe chaotic mixing in the system. For the present problem, themodified Hartmann number is a measure of the cross-sectionalstretching and, therefore, strongly influences the chaotic mixing.In the present model, the electrode switching frequency isresponsible for the blinking action and, therefore, controls thetransverse intersection of the successive streamline portraits in agiven system. There exists an optimum electrode switchingfrequency away from which the chaotic mixing effect startsdeteriorating. Based on the model predictions, an electrodeseparation distance of s¼0.5 is recommended as the optimumvalue of the electrode separation distance.

In order to validate the applicability of the results obtainedusing the non-dimensional model to glass melt homogenisation,numerical simulations were carried out employing a dimensionalmodel with glass melt as the model fluid. Results obtained showthat the chaotic mixing behavior of the glass melt is very close tothe non-dimensional model predictions. A time-periodic electrodeswitching, in which the two inner electrodes were switchedalternatively, was numerically simulated using the dimensionalmodel. Model predictions show that the material stretching in thespatially periodic and the time-periodic switching cases are ingood agreement. The numerical predictions are expected to beuseful for developing an efficient glass melt homogenisationsystem.

Acknowledgements

This work was supported by the Deutsche Forschungsge-meinschaft under Contract number TH497/26-1. The authors aregrateful to Ulrich Lange and Gunter Weidmann for the usefuldiscussions and Henning Schwanbeck for the valuable supportfrom the computer centre, TU Ilmenau.

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