Mixing in agitated tanks represents a fundamental part of
the chemical, petrochemical, pharmaceutical, food and
paper industries. This unit operation is commonly carried
out under steady flow conditions with the impeller
centered with respect to the vessel and rotating at constant
speed in one direction only. Mixing has been a popular
subject for the last years. Most studies have been focused
on power consumption, measurements of mixing times
and visualization of flow patterns. A number of techniques
have been developed for visualizing flow patterns in
stirred vessels. Mavros (2001) reviewed the use of very
sophisticated techniques for observing the flow behaviour
in stirred vessels including particles image velocimetry
(PIV), planar laser induced fluorescence (pLIF) and
thermal pulses, among others.
Mixing operations are usually carried out in some
applications at low to moderate Reynolds number, in
which the turbulent regime cannot be achieved. The
presence of important segregated regions and dead zone
generated with open impellers at low Reynolds number
has been extensively reported in the literature (Solomon et
al., 1981; Lamberto et al., 1996). Such pathologies persist
even if baffles and mixed flow impellers are used resulting
in very long mixing times, high energy consumption and
undesirable by-products due to poor reaction selectivity
can be generated. The most common way to vanish and
eliminate these regions consists of increasing the
rotational speed, however, this leads to an energy
consuming process, long mixing times, and in some cases
it may be detrimental to the final product especially when
mixing shear sensitive media.
It has been demonstrated that the flow structures generated
under steady flow conditions can be destroyed if the flow
is continuously perturbed if the impeller is lightly
displaced from tank centreline in the radial direction as
well as by using time-dependent revolutions. The works
reporting the use of spatial and temporal conditions
promoting chaotic flows in stirred vessels are reviewed in
this paper.
STEADY-STATE FLOW
Mixing operations are usually carried out at low to
moderate Reynolds number. The presence of important
segregated regions and dead zone generated with open
impellers at low Reynolds number has been extensively
reported in the literature (Solomon et al., 1981; Lamberto
et al., 1996). Those pathologies persist even if baffles and
mixed flow impellers are used resulting in very long
mixing times, high energy consumption and undesirable
by-products due to poor reaction selectivity can be
generated. The most common way to enhance mixing by
minimizing the effect of such non-homogeneities consists
of using wide impeller or increasing the rotational speed.
However, the latter option leads to high-energy levels and
in some cases it could be detrimental for materials
sensitive to shear.
Solomon et al (1981) reported for the first time the
observations of well-mixed regions around the impellers
surrounded by stagnant fluid, which are known as caverns
in the case of yield stress fluids. Considering a cavern as a
sphere with center on a Rushton turbine they proposed
empirical expressions for determining the geometrical
characteristics of such flow structures. The observations of
these flow structures were later confirmed by Alvarez et al
(2002) and Ascanio et al (2002).
UNSTEADY-STATE FLOW
The most common way to enhance mixing by minimizing
the effect of flow structures such caverns (well-mixed
regions) and isolated regions (static or quasi-static zones)
consists of using wide impellers or by increasing the
rotational speed. However, the latter option leads to highenergy
levels and in some cases it could be detrimental for
materials sensitive to shear. Therefore, other alternatives
based on the theory of chaos principles have been
developed and reported in the literature. They consist
basically on the use of spatial and temporal scenarios as
described below.

Eccentric impellers
Different alternatives based on pioneering studies on
chaotic laminar mixing have been reported in the
literature, in which it was demonstrated that
homogenization could be quickly achieved by using
eccentric cylinders rotating alternatively in both directions
during short times (Ottino et al 1988; Swanson and Ottino
1990; Muzzio et al 1991; Muzzio et al 1992). One of the
main reasons of using eccentric impellers is that they
provide the best mixing action where a vortex is not
required or not desirable.
Alvarez et al (2002) proposed the use of spatial conditions
based on the radial position of the impeller with respect to
the tank centreline. They showed that a minimal radial
displacement of the impeller can enhance the mixing
performance in terms of the mixing times. Similar results
were found by Ascanio et al (2002) by using different one
or two off-centered impellers in the stirred vessel.
Dynamic perturbations
Lamberto et al 1996 proposed the use of time-dependent
stirring speed to destroy the caverns generated under
steady mixing conditions. They demonstrated that the
mixing of Newtonian fluids could be clearly enhanced if
the fluid dynamics was continuously perturbed. This
method was shown to prevent the formation of segregated
regions close to the impeller, confirming the theoretical
results of Aref (1984). Using time-dependent rotation
directions changes conditions, Ascanio et al (2002)
demonstrated that segregated regions can be gradually
vanished and finally destroyed if the flow is continuously
perturbed.
It is clear from the above contributions based on the
fundamental concepts of chaos theory that nonconventional
mixing strategies appear to be promising to
enhance mixing in stirred tanks in laminar regime. During
the symposium a general overview of the works reporting
scenarios based spatial and temporal conditions will be
given for preventing the formation of coherent segregated
regions in the vicinity of the impeller resulting in energyefficient
processes.

Pulp fibre suspensions display non-Newtonian rheology,
including a yield stress. Under certain mixer operating
conditions this creates caverns (regions of active mixing)
around the impellers with the cavern size affecting the
extent and quality of mixing attained. Due to the opacity
of pulp suspensions it is not possible to measure cavern
size with direct optical techniques, like photography.
Consequently two non-invasive techniques suitable for use
in opaque media were evaluated for determining the
cavern dimensions: electrical resistance tomography
(ERT) and ultrasonic Doppler velocimetry (UDV). The
agitation of several pulp suspensions in a 38 cm diameter
cylindrical vessel was studied using these methods over a
range of operating conditions. ERT is a non-invasive
technique that images differences in conductivity between
regions in the mixer using voltage measurements made at
the vessel periphery. Cavern measurement by ERT is very
rapid (data are collected within a few seconds) but it
suffers from poor spatial resolution (approximately 5 to
10% of the vessel diameter – from 1.9 to 3.8 cm in our
case). Two methods were evaluated for creating the
conductive environment imaged by ERT – the injection of
saline solution or the addition of small metallic tracer
particles to the region surrounding the impeller. UDV was
used to determine the cavern boundary by measuring the
locations at which suspension velocity fell to zero for
multiple linear paths through the vessel. While UDV
provided better spatial resolution of the cavern than ERT
(about 2 mm), multiple measurements (and consequently
significant time) were needed to build up the profile of the
cavern boundary.
Cavern size as a function of impeller rotation speed is
reported for a range of pulp suspension mixing conditions
(hardwood and softwood pulps, suspension mass
concentrations from 2 to 4%, two impeller offsets from the
wall, and two suspension height-to-chest diameter ratios)
in the 38-cm diameter cylindrical chest. A scaled version
of a commercially available axial flow impeller designed
for use in pulp suspension agitation (the Maxflo,
Chemineer Inc.) was used in the standard side-entering
configuration used for pulp stock chests. Measured cavern
diameters were compared against the axial force model
developed by Ammaullah et al. (1998) for predicting
cavern diameters in non-Newtonian fluids. The
discrepancy between the experimental data and model
predictions were fairly large, although they decreased with
increasing yield stress Reynolds number. The discrepancy
was attributed to the proximity of the impeller to the
vessel walls in the side-entering configuration studied. An
alternative correlation is presented for predicting the
cavern volume in pulp suspensions in this mixing
configuration based on the suspension yield stress

increasing yield stress Reynolds number. The discrepancy
was attributed to the proximity of the impeller to the
vessel walls in the side-entering configuration studied. An
alternative correlation is presented for predicting the
cavern volume in pulp suspensions in this mixing
configuration based on the suspension yield stress

A new type static mixer composed of σ-shaped element
was developed, in which multilamination of fluid layers
proceeds through systematic splitting and inverse
recombination. The number of elements required for
complete mixing, n, was measured by conducting the
decolourising reaction of iodine with sodium thiosulfate
for various total flow rates of two fluids to be mixed. n
increased with Re when Re is less than 10, but it decreased
with Re at larger Re. When Re exceeds this critical value,
CFD analysis shows that a larger deformation and stretch
of the fluid interface take place due to the bending and
winding channel structure of σ-shaped element as Re
increases. This flow behaviour accelerates mixing rate,
resulting in the considerable decrease in the number of
elements for complete mixing. In addition, an analysis for
Figure
residence time distribution of fluid particles demonstrates
that the flow in the mixer approaches the plug flow with
increasing the Reynolds number and the number of
elements.
NOMENCLATURE
a largest width of channel of mixing element
b depth of channel of mixing element
n number of mixing elements for complete mixing
Re Reynolds number = ρuava/μ
uav cross-sectional average velocity = Q/ab
Q total flow rate of two fluids fed to mixer
MIXER STRUCTURE
Figure 1 shows channel geometry of a unit element of σ-
type plate static mixer (Hirata, 2006), in which the dotted
circles represent the cross-sections of inlet and outlet for
the fluids to be mixed. Channels with rectangular crosssection
were grooved in a plate to conform this geometry
and each grooved element was aligned in a row. A
packing sheet or plate with circular holes is sandwiched
by two plates with a row of unit elements prepared in this
way, to one of which a Y-shaped channel was connected
for introducing two fluids to be mixed as shown in Figure
2. Each hole of the sandwiched sheet or plate serves as a
channel connecting the outlet of a unit element on a plated
to the inlet of the following unit element on the other plate.
In this way channels for mixing fluids can be constructed.
We call this type of mixer σ-type plate static mixer since
the shape of the unit element resembles σ in Greek
character.

this way, static mixing with splitting and inverse
recombination progresses in the mixer.
VISUALIZATION OF MIXING PTROGRESS
Mixing progress was visualized by using the decolourising
reaction of iodine with sodium thiosulphate. An example
of the visualized images is shown in Fig.3, which were
taken at Re=1.2 using the square channel with a=b=3mm.
At low Reynolds numbers, the static mixing progresses
identically by splitting and inverse recombination as
shown in the figure. As the Reynolds number increases,
mixing progress deviates from the ideal static mixing
because of the secondary flows generated in the threedimensionally
bent portions in the mixer. The occurrence
of secondary flows has been confirmed by CFD
calculation.
NUMBER OF ELEMENTS REQUIRED FOR
COMPLETE MIXING
The numbers of elements required for completing the
decolourising reaction, n, which were obtained for a
square channel with a=b=1mm, is plotted against
Reynolds number in Fig. 4. n increases with Reynolds
number at Re < 10. This is due to that the molecular
diffusion is limited to narrow regions adjacent to the
interface of two liquids because the residence time in the
element is decreased with increasing the flow velocity. At
larger Reynolds number, n decreases with Re. The
decrease in n is mainly caused by the secondary flows
generated in the three-dimensionally bent portions in the
mixer. At Reynolds number greater than 102, the number
of elements required for complete mixing is less than 10.
RESIDENCE TIME DISTRIBUTION
F-curves in a unit element are shown for various Reynolds
numbers in Fig. 5. Using the three-dimensional velocity
data obtained by CFD calculation, F-curves were obtained
by tracking 105 fluid particles that were uniformly
distributed on the mid-plane of the interconnecting
circular channel at a time. Although this curve does not
represent the normal F-curve obtained by a step change in
concentration, reactor performance of σ-type mixer may
be discussed based on it. As shown in the figure, F-curve
for fluid particles, which are sharp compared with that in
the laminar pipe flow, approach the distribution of plug
flow as Reynolds number increases. It has also been
confirmed that the flow in the mixer tends to approach the
uniform residence time distribution with increasing the
number of mixing elements. You can even create solid metal hitch covers using this method.