Dispersing of gases or immiscible liquids within another continuous liquid fluid phase
is a standard operation for the processing of multiphase systems in the industrial
areas like food, cosmetics pharmaceuticals and fine chemicals. The commonly used
dispersing tools are rotor-stator devices with tooth-/pin geometries arranged in
circular, concentric or axial, stacked disc configurations. Depending on the gap width
between the rotor-stator pins or tooth elements and the viscosity function of the twophase
system, laminar, turbulent or transitional flow conditions act in the dispersive
mixing procedure. – A new generation of dispersing devices are the static or dynamic
membrane devices in which the disperse drops or bubbles are formed once and
detached from the membrane surface by cross- or co-flowing fluid streams. This
procedure means significantly reduced mechanical treatment of the multiphase fluid
system and allows to tailor narrow disperse size distributions.
Rotor-Stator Dispersing Processes
Our recent work concerning Rotor-Stator (R/S) dispersing process developments
has mainly focused on laminar to transition domain flow conditions. We have
investigated bubble and drop break-up in some detail under steady and transient
shear or elongation as well as mixed flow conditions. Different types of R/S (model)
flow apparatus were applied comprising concentric  /eccentric  cylinder-, four
roller- , single-  and multi-toothed  geometries. From respective experiments
expanded maps of critical bubble-/drop break-up characteristics (critical Weber
(WeC) or critical Capillary (CaC) numbers as a function of viscosity ratio
λ, deformation rate G, total deformation D and flow type α have been derived.
Figure 1 exemplarily demonstrates CaC(α) for drop break-up under pure shear (α =
0), equibiaxial (α = -1), or planar (α = 1) elongation as well as for mixtures of these
flow types. Steady and transient drop break-up were investigated experimentally (a),
by numerical flow simulation using CFD (b) and by a non- equilibrium
thermodynamics modeling approach (c) /1-3/. Consistent results from these three
approaches (a-c) were received for surfactant free as well as for surfactant covered
drop interfaces.

Fig.1: Critical Ca- number for different flows

Fig.2: Surfactant distribution at drop
interface in shear flow

For surfactant covered drop interfaces a criterion to distinguish between diffusion
and convection driven interfacial coverage with surfactant molecules was defined as
the ratio of Peclet number (Pe) / Capillary number (Ca), denoted as α and
implemented into a convection diffusion equation which forms the bases for
respective CFD calculations. As a result of these calculations surfactant
concentration distributions along the interfacial contour of drops deformed in shear,
elongation and mixed flows were received and satisfying comparability with
experimental drop deformation data was found. Figure 2 shows such calculated
concentration distributions of surfactant at the deformed drop interface for different
Capillary numbers and a viscosity ratio λ of 4.
The impact of transient shear and elongation flows has been investigated within an
eccentric cylinder gap and transferred to a complex multi toothed rotor-stator
dispersing geometry. CFD based simulations applying a particle tracking procedure
along distinct particle flow tracks allowed us to quantify the transient drop
deformation history of selected drops along their paths through the dispersing
apparatus. Comparisons with respective experimental results demonstrated again
good agreement as demonstrated in Figure 3.

Fig. 3: left: Transient deformation and Ca-number; right: multi-toothed R/S geometry
Membrane / Micro-channel Dispersing Processes
In addition to rotor-stator flow devices we considered also channel / nozzle / pore
flows with respect to their dispersive mixing performance. New microfluidics devices
have been developed in our Laboratory at ETH Zürich in close collaboration with the
University of Queensland in Brisbane (Australia); Prof. J. Cooper-White. Within the
lasts two years we investigated drop formation in co-flow and cross-flow micro- and
macro channels. By means of micro particle imaging the velocimetry (Micro-PIV) we
accessed velocity fields around respective drops and used this information for
optimizing the dispersing channel flow geometries and to derive scale up criteria
(e.g. We = f (Re) characteristics) over several orders of magnitude like demonstrated
in figure 4 .
As a scaleable solution with application relevance, derived from micro channel cross
flow results, a Rotating Membrane Device (ROME) with Controlled Pore Distance
(CPD) was developed. The cross flow is generated by the rotational motion of a
membrane cylinder within a surrounding concentric housing through which the
continuous fluid phase is axially pumped. The disperse fluid or gas-phase enters

Fig 4: Micro-/macro channel co-flow dispersing; experimental data / M. Duxenneuner
through a hollow shaft into the rotating cylinder membrane body and forms disperse
liquid droplets or bubbles at the membrane surface, from which the cross flowing
continuous fluid phase flow detaches them as soon as a critical shear stress is
exceeded .

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.