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Chemical Engineering Science 56 (2001) 4887}4899

Computational analysis of regular and chaotic mixingin a stirred tank reactor

D. J. Lamberto, M. M. Alvarez, F. J. Muzzio*

Department of Chemical and Biochemical Engineering, Rutgers University, P.O. Box 909, Piscataway, NJ 08855-0909, USA

Received 9 October 1998; received in revised form 29 November 1999; accepted 10 December 1999

Abstract

Mixing in an unba%ed stirred tank equipped with a 6-blade radial#ow impeller is examined computationally. Results demonstrate

that the #ow generated by constant impeller speed is partially chaotic. Under these conditions, the stretching experienced by #uidelements located within segregated toroidal regions of the #ow increases slowly at a linear rate characteristic of regular (non-chaotic)

#ows. Within the bulk #ow region, since the #ow is chaotic, stretching increases at the expected exponential rate. The use of dynamic

#ow perturbations enhances mixing; when time-dependent RPM are applied, a globally chaotic #ow is generated. We investigate

computationally the e!ect on mixing performance of protocols in which the agitation speed oscillates between two steady values.

Di!erent frequencies of speed change and several RPM settings are compared. Under variable RPM schemes, the segregated torii are

periodically relocated and #uid elements have the opportunity to abandon the regular regions. Under these conditions stretching

increases exponentially throughout the entire #ow domain. In general, mixing protocols with higher frequency of speed #uctuation

produce the largest increase in stretching rates. Counter-intuitively, at a given RPM #uctuation frequency, stretching rates were

higher for the lower RPM settings, either per revolutions or per unit of energy spent. 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Laminar mixing; Stirred tank; Chaos; CFD

1. Introduction and background

The development of e$cient laminar mixing techno-

logy continues to be critical to many industrial opera-

tions. For the processing of low-viscosity liquids, e$cient

mixing is frequently achieved by operating under turbu-

lent conditions. However, turbulence is precluded when

high-viscosity #uids are mixed, or must be avoided in

order to protect delicate shear-sensitive materials sus-

pended in the #ow (Sefton, Dawson, Broughton, Blys-

niuk, & Sugamori, 1987; Backer, Metzger, Slaber, Nevitt,

& Boder 1988; Cherry & Papoutsakis, 1998; Croughan,

Hamel, & Wang, 1987, 1988; Beck, Stiefel, & Stinnett,

1987; Papoutsakis, 1993; Tanaka, Semba, Jitsufuchi,

& Harada, 1988; Blynsniuk & Sefton, 1991; Omurtag,

Stickel, & Chevray, 1996). In such cases chaotic laminar

#ows can provide a practical alternative. Chaotic mixing

* Corresponding author. Tel.:#1-732-445-3357; fax:#1-732-445-

2421.

E-mail address: [email protected] (F. J. Muzzio).

has been found to occur in completely deterministic

systems devoid of any #uctuations of the #ow "eld.

Extensive work has been performed for several two-

dimensional chaotic #ows (e.g. the blinking vortex, the

cavity #ow, the journal bearing, the sine #ow), and both

computational and experimental techniques have been

rigorously developed for the quantitative characteriza-

tion of mixing processes in such systems (Aref, 1984;

Swanson & Ottino, 1990; Muzzio, Swanson, & Ottino,

1991; Muzzio & Liu, 1996; Alvarez, Muzzio, Cerbelli,

Adrover, & Giona, 1998; Muzzio et al., 2000). Unfortu-

nately, we are still far from achieving the same level of

detail in the analysis of 3-D #ows. In fact, only a few 3-D

relevant industrial systems have been analyzed numer-

ically using dynamical systems tools (e.g. Hobbs & Muz-

zio, 1998a,b,c; Rauline, Tanguy, Le BleHvec, & Bousquet

1997).

Mixing in a stirred tank is chosen as a case study

because although this is the most widely used type of

mixing equipment used for large-scale industrial opera-

tions, laminar mixing in stirred tanks is poorly under-

stood. Scaling correlations and mixing characterizations

have fallen short of accurately representing these systems

0009-2509/01/$- see front matter 2001 Elsevier Science Ltd. All rights reserved.

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Table 1

System dimensions and #uid properties

Tank

Tank diameter, DR

20.0 cm

Fluid depth, H 22.0 cm

Impeller radius, rG

3.81 cm

Impeller height, xG 7.08 cmBlade height, H@

1.46 cm

Blade width, =@

0.07 cm

Shaft diameter, DQ

0.79 cm

Fluid (glycerin)

Density, 1.26 g/cm

Viscosity, 707 cP

due in part to complex geometry and di$culty in

measuring speci"c parameters inside most stirred tanks,

and in part to an overall lack of understanding of mixing

processes in 3-D #ows.

In the last "ve years, the laminar #ow structure in

stirred tanks has been examined mainly through Eulerian

analysis of experimental and computational velocity"elds. Representative examples are the papers by Harvey

and Rogers (1996), Ranade (1997), Harvey, Wood, and

Leng (1997), and Lamberto, Alvarez, and Muzzio (1999).

However, the Eulerian approach does not tell us much

about mixing performance; indeed, velocity "eld in-

formation is only the starting point to investigate mixing.

While Lagrangian methods provide a much more com-

plete picture, examples of Lagrangian approaches to the

study of mixing in tanks are scarce: short-time computa-

tional tracer dispersion experiments have been used to

reveal #ow circulation patterns (see for example Tanguy,

Thibault, Brito-De La Fuente, Espinosa-Solares,

& Tecante 1998); Lyaponuv exponents have been used as

a gross estimator to assess mixing e$ciency in stirred

tanks (de la VilleHon et al., 1998).

In the laminar regime, a stirred tank agitated at con-

stant speed is characterized by the presence of toroidal

segregated regions that limited mixing performance. In

a previous paper (Lamberto, Muzzio, Swanson, & Ton-

kovich, 1996) we demonstrated experimentally that such

segregated regions were e!ectively destroyed by using

time-dependent impeller speed protocols, greatly enhanc-

ing mixing performance (these experimental results were

later con"rmed and extended by Yao, Sato, Takahashi,

& Koyama (1998)). In a second paper, we showed thatthe location, size, and shape of such segregated regions

could be predicted by CFD calculations (Lamberto et al.,

1999).

In this paper, such computations are considerably ex-

tended using Lagrangian analysis (tracer dispersion

simulations, PoincareH sections, stretching "elds, and

stretching distributions) to characterize in detail laminar

mixing performance in an unba%ed stirred tank reactor

equipped with a single 6-blade radial #ow Rushton im-

peller (the system dimensions are listed in Table 1). To

the best of our knowledge, this paper is the "rst to

address mixing in stirred tank geometries using a detailed

Lagrangian approach (with emphasis on stretching cal-

culations). It is numerically con"rmed that steadily

agitated stirred tanks are partially chaotic systems (as

supported by experimental observations), and that vari-

able RPM protocols improve the mixing performance of

stirred tanks. Computations are used to quantify such

improvements. Section 2 o!ers a description of the

computational #uid dynamic (CFD) techniques used to

determine the #uid velocity and to conduct a Lagrangian

investigation of mixing performance. In Section 3, com-

putations are used to analyze the #ow in the stirred tank

operated at constant impeller speeds, and in Section 4,

the use of dynamic #uctuations in the impeller speed is

analyzed. It is shown that widespread chaos is generated

under such conditions. Quanti"cation of the reduction in

the mixing time as a function of RPM #uctuation ampli-

tude and #uctuation frequency is discussed.

2. Computational methods

As a starting point, the three-dimensional velocity "eld

inside the tank is determined using the CFD software

package FLUENT2+. First, the geometry of the system is

constructed and discretized into cylindrical}polar quad-

rilateral cells. The grid used in the computations present-

ed in this paper contained over 150,000 nodes to describe

one-sixth of the tank geometry. Due to the symmetry ofthe system, for low-inertia #ows, a reasonable starting

point is to model one-sixth of the tank containing one

impeller blade. Once this task is completed, FLUENT2+

solves the conservation equations using a "nite-volume

iterative method to determine the velocity of the #uid at

the centroid of each cell. Velocity data are obtained for

Reynolds numbers in the range 8(Re(70. Speci"c

details about the mesh geometry generation and re"ne-

ment, and an extensive experimental validation of the

computational velocity "elds obtained by this approach,

have been presented elsewhere (Lamberto et al., 1999).

Particle tracking software was developed to perform

a Lagrangian investigation of laminar mixing for the

stirred tank system. Motion of passive tracers (point

particles) was tracked throughout the #ow domain to

simulate the mixing process and to reveal regions of

chaotic and non-chaotic (regular) #ow. The tracking soft-

ware used a directed search algorithm to determine par-

ticle positions within the computational cell domain and

a tri-linear interpolation scheme to determine particle

velocities from the known cell values obtained from CFD

simulations (Lamberto, 1997). Details will not be re-

ported here in the interest of brevity. Particles are placed

in pre-determined locations in the #ow domain and their

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Fig. 1. Vector plots of the axial and radial components of the velocity

within a vertical plane located at "303: (a) Re"8.64 and (b)

Re"69.12.

positions are tracked by integrating the equation of

motion:

dx(t)/dt"v(x, y, z). (1)

The time-independent Eulerian velocity "eld in a rota-

tional reference frame attached to the impeller was used

in Eq. (1) to yield constant boundary conditions; in otherwords, the tank walls move while the impeller remains

still. The integration of Eq. (1) is not a trivial task. In

chaotic systems the error in tracking calculations grows

exponentially. However, a good representation of mixing

events and patterns (consistent with experimental evid-

ence) can be achieved when a high-quality velocity "eld

description is combined with an adequate mesh resolu-

tion and an accurate integration algorithm (as an

example see Hobbs et al., 1997; Hobbs & Muzzio,

1998a,b,c). Souvaliotis, Jana, and Ottino (1995) investi-

gated the performance of"rst-, second-, and third-order

numerical integration methods in a 2-D mixing system

from which analytical solutions for particle trajectoriescan be determined. Their "ndings suggest that for real

systems, higher-order methods are preferable. Here,

a fourth-order Runge}Kutta scheme is used (Press, Flan-

nery, Teukolsky & Vetterling, 1986). This integration

method o!ers a good compromise between accuracy and

computational expense for this particular application

case. Di!erent integration time steps were used for the

di!erent Reynolds number cases analyzed. For each case,

di!erent runs were performed with decreasing time step

lengths until a consistent output was found in terms of

size and shape of the segregated regions as portrayed by

PoincareH sections. The velocity gradients fed into theintegration subroutine were obtained from the interpo-

lated velocity "elds. The resulting particle trajectories,

x(t), were back-transformed to the stationary frame by

adding the solid-body rotation of the reference frame.

Additionally, the amount of stretching experienced by

the #uid was calculated. Since the amount of inter-

material surface area generated by the mixing process is

directly proportional to the amount of stretching experi-

enced by #uid elements present at the location (Muzzio

et al. (1991); or in more detail Muzzio et al. (1999); for an

earlier articulation of these ideas see Ottino, Ranz,

& Macosko (1981)); an estimate of overall e$ciency of

the mixing process can be obtained by determining local

rates of stretching throughout the #ow domain. Stretch-

ing is calculated by considering a #uid element represent-

ed by an in"nitesimal vector l. This vector is located at an

arbitrary position in the #ow domain and it is stretched

as it is convected by the #ow. The evolution of this vector

is described by the equation

dlR/dt"(v)2 ) l

R(2)

Stretching () is de"ned as the ratio of the magnitude of

the vector l at time t to its initial magnitude.

Stretching,""lR"/"l

"""l

R". (3)

As with Eq. (1), Eq. (2) is integrated using fourth-order

Runge}Kutta. Stretching values are calculated for many

vectors (O(10)) throughout the #ow domain in an at-

tempt to reveal regions of non-chaotic and chaotic #ow.

3. Results and discussion

3.1. Computational velocity xeld

The three-dimensional velocity "eld is determined for

the stirred tank system using CFD. Figs. 1a and 1b are

two-dimensional vector plots for Re"8.64 and 69.12.

The "gures depict one half of a vertical cross section of

the stirred tank with the shaft and impeller shown on the

left. A detailed analysis of numerical accuracy, mesh

optimization, and an experimental validation of the velo-

city "elds are presented elswere (Lamberto et al., 1999).

To better reveal the underlying structure of the #ow, only

the axial and radial velocity components are displayed in

the "gure. Secondary circulation loops within the #ow

are revealed both above and below the impeller. These

circulation regions are the primary cause of segregation

in the system and their positions depend on impeller

speed (see for example Lamberto, 1996, 1999; Yao et al.,

1998). To investigate this behavior further, particle track-

ing simulations are performed.

3.2. Particle tracking simulations and PoincareH sections

To illustrate the motion of individual #uid particles in

the #ow, four #uid particles are tracked for 2 min of

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Fig. 2. Chaotic and regular trajectories in a stirred tank system. (a) Four particle pathlines obtained from simulation of mixing at 200 RPM

(Re"34.56) for 400 periods. Two of these particles, initially placed near the upper and lower segregated regions, trace out the surface of torii identical

in shape to those observed experimentally (blue trajectories). The other two particles, placed in the discharge stream of the impeller (one above and the

other below the impeller mid-plane), move in an apparently random manner throughout a much larger volume of the tank (red pathlines). However,

neither one crosses the mid-plane of the impeller. (b) Shape of the torus surfaces traced out by visualization experiments.

simulated mixing time at Re+35. Initially, two particles

are placed near the upper and lower segregated regions.

Particle 1 is placed at the same height xN

as the center of

the upper segregated region (xAS

) displaced radially by

1 cm: (xN

, rN

)"(xAS

, rAS#1 cm). Particle 2 is placed

similarly relative to the lower region: (xN

, rN

)"

(xAJrAJ#1 cm). The other two particles (denoted 3 and 4)are both located in the discharge stream of the impeller at

a radial position 3 cm in front of the tip of the blade.

Particle 3 is placed 1.5 mm above the mid-plane of the

impeller while particle 4 is placed 1.5 mm below the

mid-plane (xN"x

G#1.5 mm and x

N"x

G!1.5 mm).

The pathlines of these particles are shown in Fig. 2a.

The particles placed in the discharge of the impeller (red

particles) follow pathlines that move through a large

portion of the volume of the tank; their trajectories ap-

pear random. However, one of them always remains

above the mid-plane of the impeller while the other

always remains below the impeller, revealing symptoms

of vertical segregation that have often been observed by

practitioners. The blue particles trace out the surface of

torii that are essentially identical in shape to the seg-

regated regions observed in #ow visualization experi-

ments (Fig. 2b).

PoincareH sections are used as a diagnostic tool to

reveal regular regions of the #ow and to examine details

of the #ow structure. PoincareH sections are generated in

this paper by tracking a small number of particles

(O(100)) in the #ow for a large number of periods

(O(1000)) and marking the intersection of the particle

trajectories with a vertical plane attached to impeller

(note that tracking is performed using the rotating framevelocity "eld and therefore this vertical plane, although it

is attached to the impeller, is stationary). This typically

generates a two-dimensional picture of the #ow structure

that reveals chaotic #ow regions as a `random-lookingacloud of points and regular #ow regions as collection of

closed particle paths.

Fig. 3a shows the initial position of 77 particles placed

in the #ow in a cross-shaped pattern centered at the

positions of the segregated regions (since the posi-

tions of the segregated regions depend on Reynoldsnumber, slightly di!erent initial conditions are used

for each Reynolds number). In this case (100 RPM,Re"17.28) the particles are tracked for 400 periods of

the #ow where 1 period is de"ned as 1 revolution of the

impeller. In the PoincareH section shown in Fig. 3b, two

types of behavior are apparent; a random-looking cloud

of points starting at the discharge of the impeller (sugges-

ting chaotic #ow), and two islands that contain ellipti-

cal-shaped closed loops (suggesting non-chaotic #ow).

Fig. 3c reveals the structure within the lower segregated

region.

Neither the experimental observations presented else-

where (Lamberto et al., 1996; Yao et al., 1998) nor the

qualitative analogies made in this section constitute

proof of the presence of chaos in the stirred tank. To

achieve such proof, we must consider the rate of stretch-

ing of material elements throughout the #ow; sustained

exponential stretching in a bounded domain demon-

strates that the #ow is chaotic.

3.3. Stretchingxelds, distributions, and rates

Stretching "elds are generated by tracking the stretch-

ing experienced by &10,000 #uid particles uniformly

placed on a vertical plane located at "303 (i.e. mid-plane between impeller blades). Figs. 4a}4c show the

stretching "elds relative to the initial particle position

after mixing for 200 periods at 100, 200, and


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Fig. 3. PoincareH section of the stirred tank #ow (Re"17.28): (a) initial

positions; (b) after 400 periods (impeller rotations); (c) close-up of the

lower segregated region.

Fig. 4. Stretching "eld for +10,000 particles placed uniformly in

a vertical cross section of the tank. Stretching is plotted versus the

initial particle position and the colors represent stretching values that

increase from black to red. Particles are tracked for 200 periods at (a)

Re"17.28, (b) Re"34.56, and (c) Re"67.12. In all cases the poorly

mixed segregated regions appear as islands of low stretching sur-

rounded by a chaotic `seaa.

400 RPM (Re"17.28,34.56, and 69.12). The stretching

value is denoted by the color of the points in the plots

and increase in the usual order: black, dark blue, light

blue, green, yellow, magenta, and red. The segregated

regions are clearly evident in the "gures as two large oval

islands of low stretching values that appear above and

below the impeller. These islands are surrounded by

a `seaa of high stretching values increasing exponentially

with time and corresponding to the chaotic region of the

#ow. Even within the chaotic region, an extremely wide

distribution of stretching values is revealed by the com-

putations, indicating that laminar mixing in stirred tanks

causes a very wide distributions of micromixing inten-

sities, a characteristic of chaotic #ows (Muzzio et al.,

1991, 1999). As the Reynolds number increases, the

stretching "eld remains highly non-uniform (and as

a consequence also the intermaterial area density), al-

though it continues to span several orders of magnitude.

This observation has deep implications; for example,

transport-controlled chemical reactions will proceed at

extremely non-uniform rates in such #ows. For systems

with multiple reactions, wherein the product distribution

is often a!ected by mixing performance, such wide distri-

bution of micromixing rates would result in considerable

overproduction of waste products (Muzzio & Liu, 1996;

Zalc & Muzzio, 1999). Similar observations have been

reported for turbulent #ows by many authors; for a com-

prehensive reference see Baldyga and Bourne (1999).

The distribution of behaviors is examined statistically

by computing the probability density function (PDF) of

the logarithm of stretching values:

HL

(log

)"1

N

dN(log

)

dlog

, (4)

where N is the total number of #uid particles and

dN(log

) is the number of particles that have values

between log

and log

#d log

. The stretching

distributions for &40,000 particles placed uniformly

throughout the entire #ow domain are estimated for 100,

200, and 400 RPM and are plotted for the same number


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Fig. 5. Stretching distributions for+40,000 particles placed uniformly

throughout the tank after 200 periods of mixing at Re"17.28, 34.56,

and 67.12 (from left to right). The distributions resemble those obtained

in other #ows for partially chaotic conditions.

Fig. 6. Geometric mean of the stretching plotted versus time for the

bulkand segregated#ow regions. Stretching increases at an exponential

rate for the particles in the bulk #ow region while only at a linear rate in

the poorly mixed islands.

of impeller rotations (Fig. 5). The distributions broaden

with increasing Reynolds number and they all

display a large hump of low stretching values corre-

sponding to the segregated islands. They look qualitat-

ively similar to distributions computed for other #ows

with large non-chaotic islands (Muzzio et al., 1991; Liu,

Peskin, Muzzio, & Leong, 1994c; Hobbs & Muzzio,

1998).

To further analyze and quantify mixing in the system,

spreading of tracer was simulated by introducing 1000

#uid particles into the #ow. Each particle is locatedrandomly within a small circle of 1 cm radius centered

either at the discharge stream of the impeller or inside the

segregated regions. The radius of the circle was deter-

mined by screening several values and comparing

simulated dye streaks with dye streaks observed experi-

mentally; a value of 1 cm was large enough to simulate

dye injection but not so large for the particles to leave the

segregated regions.

The particles were mixed for 10 min at

100 RPM (Re+17) and their stretching values were

recorded. The geometric mean values of the stret-

ching experienced by the 1000 particles within the

bulk #ow region and the 1000 particles in the islands are

plotted versus time in Fig. 6. The stretching values

increase exponentially in the bulk #ow region but

only linearly in the segregated regions of regular

#ow. Similar trends are observed for all Reynolds

numbers studied. Since the #ow is laminar (Re+69),

the cause of the exponential growth in stretching is

the presence of chaos in the #ow. Thus, laminar stirred

tank #ows can be deemed partially chaotic and the slope

of the curve for particles outside the segregated regi-

ons is an estimate of the Lyapunov exponent,

+0.16 (Re+17).

3.4. Injection locations and impeller ewects

Tracking computations are performed to simulate

mixing of a small spherical drop of dye solution into the

bulk of the #uid. The e!ects of the initial position (`injec-

tion locationa) of the droplet on the resulting mixing

structures are examined and stretching rates are deter-

mined for a droplet that moves near the impeller. Com-

putationally, the droplet consists of 822 particles

uniformly distributed on the surface of a sphere with

a radius equal to 1 cm, `injecteda into three regions of the

#ow: (1) at the axial position of the impeller mid-plane,

1 cm in front of the tip of the impeller blade, and inter-

secting the separation plane at the impeller mid-plane; (2)at an axial position equal to twice the height of the

impeller and a radial position equal to the outer radius of

the impeller, and (3) at the center of the upper segregated

region for each Reynolds number. Simulation results for

a droplet in the lower region are analogous to results

obtained for the upper region and will not be presented

here.

Figs. 7a}c show the "nal position of the particles after

mixing for 400 periods at Re"34.56 (200 RPM). These

"gures clearly show the isolating e!ect of the islands and

the separatrix originated by the impeller mid-plane. The

particles within drop 1 cover much of the bulk #ow

region and experience exponential growth in stretching.

Note that they mostly populate the outer layers of the

segregated regions (corresponding to the portion of drop

1 that was initially inside the region). The particles dis-

perse above and below the impeller, since the initial

location of the blob straddles the impeller plane. The

particles in drop 2 experience similar stretching; however,

these particles remain con"ned to the region above the

impeller mid-plane, since their initial location is above

the separatrix. Drop 3 remains entirely within the upper

segregated region and experiences only a linear growth in

stretching as shown before (Fig. 6b).


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Fig. 7. Particle positions after 400 periods of mixing at 200 RPM for three droplets of particles. (a) The particles of droplet 1 (initially located at the

impeller discharge) spread throughout the tank and experience relatively high stretching values, except at the edge of the segregated region where they

experience low stretching. (b) The particles of droplet 2 (originally located above the impeller mid-plane in the chaotic region) also experience high

stretching and are not only isolated from the segregated region, but also from the region of the tank below the impeller. (c) Finally, the particles of

droplet 3 (initially located in the upper segregated region) remain isolated from the chaotic region and experience low stretching.

A closer examination of the evolution of drop 2 is

presented in Fig. 8. Averages of short-time stretching

values, 12, and single-period average stretching

multipliers (Muzzio et al., 1991) are plotted in Figs. 8a

and b; multipliers are de"ned as the ratio of attime t#t to the value of at time t (here t is equal to

0.1 s). The long-time plot is given in Fig. 8c. Since the

amount of intermaterial surface between components

being mixed increases proportionally to 12, the stretch-

ing multiplier indicates the global rate of the mixing

process in the #ow (for a full discussion, see Muzzio et al.,

1999).

A sharp increase in the stretching values (see Fig. 8a),

and a maximum peak in the multiplier plot (see Fig. 8b),

occur when particles pass between the blades of the

impeller and for a brief period after discharge from the

impeller. At t&2.3 s, as the stretching rate goes through

a minimum, the curve in Fig. 8a becomes almost #at and

the average multiplier is minimum. At this time the

particles are at their maximum distance from the impel-

ler. Subsequently, particles move slowly until t&4 s

when they are again drawn toward the impeller and their

stretching rate again increases. However, since the par-

ticles are more dispersed, not all of them experience an

increase in the stretching rate at the same time and

therefore the slope is not as steep as the initial increase.

After t&5 s the particles are distributed throughout the

chaotic region of the #ow, the stretching multiplier be-

comes roughly a constant (Fig. 8b), and the long-time

plot of the logarithm of acquires a constant slope

(Fig. 8c).

4. Time-dependent RPM

A method for eliminating segregation and enhancing

mixing can be developed based on several studies which

have shown that e$cient mixing in laminar #ows can be

achieved by generating a time-dependent chaotic #ow

(Aref, 1984; Aref & Balachandar, 1986; Beigie, Leonard,

& Wiggins, 1991, 1993; Franjione, Leong, & Ottino,

1989; Leong & Ottino, 1989; Liu, Muzzio, & Peskin,

1994a,b; Muzzio et al., 1991; Ott & Antonsen, 1988;

Ottino, Leong, Rising, & Swanson, 1988; Swanson & Ot-

tino, 1990). A similar approach has been used here to

enhance mixing in the stirred tank. Since the positions of

the segregated regions have been found to depend on the

impeller rotation rate (Lamberto et al., 1996, 1999),

a #uctuation of the rotation rate should lead to wide-

spread mixing by preventing the formation of stable torii.

Then, mixing times can be signi"cantly reduced by #uc-

tuating the impeller speed between two settings (Lam-

berto et al., 1996; Yao et al., 1998).

As the rotation rate changes from settingC1 to setting

C2, some of the #uid in the well-mixed region becomes

part of the newly formed segregated region. Since the

center of the new torus is displaced, rotation within the

torus forms a swirl pattern. When the rotation rate is


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Fig. 8. Evolution of the geometric mean stretching for droplet 2: (a) short-time average of stretching, (b) stretching multipliers, and (c) the long-time

stretching of the particles (2000 periods).

Our experimental observations derived from laser-induced #uor-

escent tracer experiments (not presented here) suggest that the duration

of transient periods is controlled by the ramp of the motor (typically 5 s)

and not by the response of the #uid (less than 30 s for the Re conditions

studied here). The contribution of these transient intervals to mixing is

di$cult to establish experimentally. An alternative is to reverse the

rotation of the impeller without changing the RPM value. In such

a condition, the segregated torii are only relocated during the transient.

Results of such experiments recently reported by Yao et al. (1998),

suggest that the role of transients is not signi"cant. Only minimum

enhancements of mixing were achieved by #ow reversals. Our own

experiments to be discussed in a future publication are consistent with

this observation.

changed back to setting C1 some of the #uid trapped

inside the segregated region at settingC2 is now released

into the well-mixed region and quickly becomes disper-

sed throughout the tank. Within the segregated region,

#uid previously found outside the segregated region

continues to swirl around #uid originally located in thechaotic region, getting better mixed. This process is con-

ceptually similar to kneading bread dough where the

dough is stretched and folded repeatedly to promote

good mixing. By using variable speed, stable segregated

regions do not form, and intermaterial area is generated

at an increased rate as observed by the striation patterns

formed in #ow visualization experiments.

In this section the e!ects of time-dependent RPM are

examined computationally. The most general approach

to computationally model the e!ect of variable RPM

protocols should consider the two constant velocity"elds

corresponding to the two velocity settings, and also the

transient between them. However, the simulation of tran-

sients in 3-D #ows is not a trivial task. In order to track

the position of#uid particles during transients, a series of

sucesive interpolations between steady velocity "elds

needs to be performed. Moreover, while the convergence

of a numerical scheme towards a steady-state #ow is

a relatively robust process amenable of experimental

validation, experimental veri"cation of numerical results

for a 3-D transient #ow is very di$cult, bringing into

question the practicality of such an approach. As a "rst

approximation, here we merely examine the e!ect of

iteration between steady-state conditions, without con-

sidering the added contribution of transients. Therefore,

time-dependent RPM is simulated by periodically alter-

nating two velocity "elds (v

(x, y, z) and v

(x, y, z)) into

Eqs. (1) and (2); each velocity "eld corresponding to

a di!erent impeller speed. The software is initialized by

reading in the node positions and velocity values for both"elds. Particles are tracked for a predetermined amount

of time using velocity "eld C1, followed by tracking

using velocity "eldC2 for the same amount of time. As

shown below, simulated and experimental results agree

quite well in spite of the simpli"cation incurred in ne-

glecting the transients. In the remainder of this paper,

mixing schemes are referred to by the "rst speed setting

(always the higher of the two) followed by the second

speed setting and the duration of each setting in seconds.

For example, if a mixing cycle consists of mixing at 100

RPM for 10 s followed by mixing at 50 RPM for 10 s, it is

referred to as 100/50/10.


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Fig. 9. Comparison of constant and time-dependent RPM mixing protocols. From left to right, the stretching distributions after 133, 267, 400, and 533periods are shown for (a) 400 RPM and (b) 200/100/30.

Fig. 10. E!ects of#uctuation frequency on the stretching distributions

for 200/100 RPM settings, for various #uctuation frequencies. All the

curves are practically identical.

4.1. PoincareH sections and stretching distributions

Mixing improvement due to variable RPM protocols

is manifested in PoincareH sections as an absence of seg-

regated structures (not shown here). It is also re#ected in

stretching distributions, which become more symmetric

as stretching values increase exponentially throughout

the entire #ow domain.Stretching distributions generated for the mixing

scheme 200/100/30 are contrasted with those generated

by constant-speed mixing at 400 RPM. Fig. 9a shows the

distributions for 400 RPM plotted for 20, 40, 60, and 80 s

of mixing. Each one has a large peak of low stretching

values, which corresponds to the segregated regions, fol-

lowed by a long high-stretching tail, which corresponds

to the bulk or chaotic region of the #ow. In contrast,

Fig. 9b shows distributions for 200/100/30 at times corre-

sponding to the exact number of periods as each of the

curves for the 400 RPM case. More desirable mixing

conditions are achieved through the use of time-depen-

dent RPM; the sharp peak of low stretching values has

almost entirely been eliminated and the distributions are

also narrower indicating that the mixing intensity is more

uniform throughout the stirred tank.

4.2. Ewects of impeller speedyuctuation frequency

Stretching distributions are used to examine the e!ects

of impeller #uctuation frequency. The impeller is #uc-

tuated between 200 and 100 RPM every 30, 15, 10, and

5 s. These frequencies were chosen based on preliminary

#ow visualization experiments. These #uctuation fre-

quencies are clearly unrealistic for the large tanks used in

most industrial operations. However, such tanks are typi-

cally operated at much lower speeds, and under such

conditions the corresponding #uctuation frequencies are

substantially lower. The stretching distributions are plot-

ted in Fig. 10, after 400 #ow periods (400 impeller revol-

utions) of mixing. Their shapes are almost identical for

each duration time, except for a small `humpa at lower

values of stretching that corresponds to a (relatively)

slow-mixing region that is created where the torii used to

be. This `humpa moves to higher values of stretching

as the #uctuation frequency increases between 2 speed


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Fig. 11. Positions and stretching values of the particles of droplet

3 after 2 min of mixing at (a) constant 200 RPM, (b) 200/100/30, (c)

200/100/15, (d) 200/100/10, and (e) 200/100/5.

changes per minute (i.e., every 30 s) to 12 changes per

minute (i.e., every 5 s), indicating that mixing may be

Fig. 12. Stretching versus number of revolutions for the conditions

depicted in Fig. 11a}e (from bottom to top). Stretching values increase

at an exponential rate for all the time-dependent RPM mixing schemes

while they increase only linearly for constant RPM mixing.

more e!ective for schemes that #uctuate impeller speed

more rapidly (consistent with experimental observations

by Yao et al., 1998).

To further investigate the e!ects of #uctuation fre-

quency, let us again consider the mixing of a droplet of

dye solution and focus on drop 3 (see previous section),

which is located inside the upper segregated region.

Fig. 11a shows the particle positions and stretching

values after mixing for 400 periods at 200 RPM

(Re"34). Figs. 11b}e are generated by #uctuating the

impeller speed between 200 and 100 RPM every 30, 15,

10, and 5 s, respectively. The corresponding stretching

rates are plotted in Fig. 12 (the stretching rate for con-

stant speed mixing at 200 RPM is included for compari-son). The stretching values obtained for time-dependent

RPM schemes are orders of magnitude greater than the

rates for constant speed. The #uid inside the previously

segregated regions now stretches at an exponential rate

and in general the greatest increases in the rate of stretch-

ing are observed for the higher #uctuation frequencies.

This agrees with Figs. 11d and e, which show that the

particles are most widely dispersed, and therefore better

mixed with the surrounding bulk #uid, for protocols that

#uctuate impeller speed every 5 and 10 s. It should be

noted that, although the stretching rates of the particles

in the segregated regions increase exponentially, these

rates are still considerably lower than those experienced

by particles in the bulk #ow region. This di!erence is the

source of the small `humpa observed in the stretching

distribution plots in Fig. 9.

Several mixing schemes employing di!erent pairs of

RPM settings have been examined for each #uctuation

frequency (every 5, 10, 15, and 30 s). Frequencies were

identi"ed corresponding to the maximum increase in

stretching for each pair of RPM settings. Results are

plotted in Fig. 13a. These results can now be compared to

determine the `most e$cienta mixing scheme. The reader

should bear in mind however, that the de"nition of


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Fig. 13. Comparison of the optimum mixing schemes for di!erent RPM settings: 100/50/10 (dashed line), 200/100/5 (dashed}dotted line), 400/50/5

(dotted line), and 400/300/10 (continuous line). (a) Setting 200/100/5 had the highest stretching rate per revolution; (b) 400/300/10 had the highest

stretching rate per unit time; and (c) 100/50/10 gave the highest rate per unit of power requirement.

e$ciency is somewhat arbitrary and that di!erent results

can be obtained for various de"nitions. For example, one

can choose stretchingper revolution as a basis for compari-son. This is a reasonable choice because for laminar #ows

in stirred tanks, the time scale is determined by the impel-

ler speed, and the dimensionless mixing time is given by

the number of revolutions. Therefore, laminar mixing op-

erations should be scaled up using the total number of

revolution as a basis. When compared in this manner, it is

observed that the #uid experiences the highest amount of

stretching per revolution for the mixing scheme 200/100/5,

followed by 100/50/10, and 400/50/5. The lowest is the

400/300/10 scheme. The fact that the lower RPM settings

outperformed the higher may seem counter-intuitive at

"rst since the size of the segregated regions has been

shown to be larger at the lower RPM settings (Lamberto

et al., 1996; Yao et al., 1998). However, the positions of the

segregated regions have also been shown to change most

drastically with changes in speed at lower Reynolds num-

bers (lower RPM settings) and therefore, since mixing

enhancements are caused by continuously displacing the

segregated regions, the largest enhancements are obtained

at the lowest speeds.

However, in cases where an intrinsic time scale is impor-

tant (as is the case, for example, for systems where the

characteristic time of a chemical reaction determines the

necessary mixing performance) one might be interested

in the amount of stretching per unit of clock time. Such

a comparison, shown in Fig. 13b, reveals that the `mixing

e$ciencya order has changed. The mixing schemes400/50/5, 400/300/10, and 200/100/5 all generate stretch-

ing values several orders of magnitude higher than the

100/50/10 scheme on a per-minute basis because they

apply more revolutions (i.e., more #ow periods) per unit

time.

In many other applications, a useful measure of the

mixing e$ciency must consider, in addition to mixing

time, the power requirements of the system. The

power dissipation per unit volume of each scheme

is estimated as the average of the power for const-

ant speed mixing at each RPM setting, where the

power per unit volume is calculated as P"2(D : D)

where D is the rate-of-strain tensor. The ratios of stretch-

ing to P are plotted for each condition in Fig. 13c. The

scheme 100/50/10 is now the most e$cient mixing method.

5. Conclusions

Lagrangian particle tracking software has been de-

veloped to perform an in-depth analysis of laminar mix-

ing in a stirred tank. Path lines of#uid particles placed

initially near the positions of the segregated regions

trace out the surfaces of three-dimensional torii. Their


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shapes are identical to the shapes of the poorly mixed

regions observed experimentally and the particles remain

in these regions inde"nitely. PoincareH sections of the

#ow reveal a complex internal structure of such regions.

The stretching distributions experienced by #uid

elements display a `humpa of low stretching values

and a long tail of high stretching values characteristicof a partially chaotic #ow. Further, stretching values

of #uid elements inside of the segregated regions

increase only linearly while the stretching values

experienced by #uid elements placed in the bulk

#ow increase at an exponential rate, characteristic of

a chaotic #ow.

A `globally chaotica #ow was generated through the

use of dynamic #ow perturbations. The e!ect of time-

dependent RPM on stretching values was numerically

examined for various mixing schemes; pairs of RPM

settings (consistent with the Reynolds numbers used

throughout this paper) and several frequencies of speed

changes were examined. Upon application of such

schemes, the `humpa previously observed in the stretching

distributions, corresponding to a slow-mixing region, was

reduced in size, the distributions were more symmetric,

and stretching increased at an exponential rate through-

out the #ow domain. Several speed #uctuation frequencies

were examined. In general, #uctuating impeller speed

every 5 and 10 s produced the largest increase in the

stretching rate, while increases where the speed was #uc-

tuated every 15 and 30 s produced the lowest rates. Rates

were higher for the lower RPM settings when plotted

either per revolutions or per unit of energy consumed, and

higher for the higher RPM settings when plotted as a func-tion of time. While at the laboratory scale considered in

this study energy considerations are inconsequential, at

the industrial scale, the expense of run time versus power

requirements must be carefully considered when applying

these techniques.

Notation

D rate-of-strain tensor

DQ

shaft diameterD

R

tank diameter

F(vH) probability density function ofv

H

H #uid level height

H@

height of impeller blade

l vector representing #uid element

N number of sample points used to determine

PDFN

Npumping coe$cient

N

impeller speed, rev/s

PDF probability distribution function

Re impeller Reynolds number (N

DG

/)

RPM impeller speed, rev/min

rG

impeller radius

v

velocity vector (vV

, vP, v

F)

vV

, vP, v

Faxial, radial, and tangential velocity compo-

nents

v velocity gradient tensor

xG

impeller mid-plane axial position

=@

width of impeller blade

Greek letters

stretching

#uid viscosity #uid density

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