13th European Conference on Mixing
London, 14-17 April 2009
NUMERICAL AND EXPERIMENTAL ANALYSIS OF PARTICLE
STRAIN AND BREAKAGE IN TURBULENT DISPERSIONS
S. Maaßa*, S. Wollnyb, R. Sperlingb, M. Kraumea
Chair in Chemical Engineering, TU Berlin, Straße des 17. Juni 136,
10623 Berlin, Germany, e-mail: Sebastian.firstname.lastname@example.org
Department of Fluid Mechanics, Anhalt University of Applied Science, Bern-
burger Straße 55, 06366 Köthen, Germany
Abstract: This study deals with particle stress and breakage in turbulent flow. To estimate the particle
stress, experimental set-up and computational fluid dynamics (CFD) are combined. The experimental
set-up is used to examine the single drop breakage around a stirrer blade. The configuration of the
breakage cell permits a study of particle break-up under variation of the mother droplet size and/or the
flow velocity. CFD is used to investigate the flow pattern in this breakage cell. The CFD results are
used to verify the flow field in the breakage cell as an acceptable model of a stirred tank. The results
of the experimental investigations show that particle breakage takes place behind the stirrer blade.
Referring to CFD simulations this region is clearly dominated by the highest energy dissipation rates.
Key words: single drop breakage, CFD, stirred vessel, Rushton turbine
High energy dissipation rates are necessary for fast and high quality turbulent mixing in a
stirred vessel. In many biochemical, pharmaceutical and food industries processes is also a
need for gentle mixing. The key factor for optimizing such processes is to fulfil both needs.
Therefore the prediction of particle strain is of major importance for these industries.
Computational fluid dynamics (CFD) is a useful tool to obtain the three-dimensional ve-
locity fields in agitated vessels. This analysis can be used to predict flow patterns like turbu-
lent shear rate, turbulent kinetic energy and turbulent energy dissipation rates.
It is broadly known that different stirrer types create different particle strain for the same
energy input (P/V). Primarily the differences between axial and radial stirrers are obvious and
well described in literature [1, 2, 3, 9, 10, 11]. The reason for these differences has not been
analysed yet. Therefore Langer et al.  carried out investigations in a self developed measur-
ing section to analyse the different phenomena of shear and strain. Their results show that
clearly for the same energy input. The particles become much smaller by strain forces then by
shear forces. The same results are expected to occur in a stirred vessel. To verify these estima-
tions a distinction is necessary between shear flow and strain flow resulting from different
Concluding from 2D particle image velocimetry – measurements, Wille et al.  assume
that the ratio of strain to shear forces is higher for axial agitation compared with radial agita-
tion. In this work the different shear and strain forces created by a Rushton turbine shall be
simulated and analysed using CFD, Ansys-CFX 11.0 so that better understanding of particle
breakage is presented. The second aim of the CFD-simulations is to calculate the mean veloc-
ity field in the breakage cell and to compare the location of the particle break-up of the ex-
perimental results with numerical parameters like velocity and energy dissipation.
2. MATERIAL AND METHODS
2.1 Experimental investigations
For systematic analysis and quantitative understanding of particle strain and single drop
breakage a special experimental set-up for the investigation of such events was developed and
operated . A single blade representing a section of a Rushton turbine (d 0.08 m) is fixed in
a rectangular channel (□ 30 x 30 mm). A single droplet of a certain diameter between 300 –
3000 µm is introduced into a continuous water flow (see Fig. 1 – left) by a Hamilton® dosing
stirrer blade 4 mm
V stirrer disk
Fig. 1 – Experimental set-up: single drop breakage cell (left) and high-speed photography of a breaking 1 mm
mother droplet, original photos and post processed images (right)
Pictures of the breakage event are taken with a high-speed camera using frame rates up to
1000 frames per second (fps). Automated image recognition (commercial software ImagePro-
Plus®) is used to analyse the drop pictures with the output of number, size and centre of mass
position of all objects on a picture. The first step of picture analysis is the subtraction of a
reference picture from the picture with drops. To get a binary picture a threshold is set. The
final picture only contains black particles (see Fig. 1 – right). Automatic data treatment is
necessary to obtain statistically relevant results. More than 1000 events are recorded for one
parameter set-up (drop diameter, flow velocity).
y – coordinate [mm]
x – coordinate [mm]
0.0 5.5 16.5 33.0 63.25 m/s
Fig. 2 – Subdivision of the images for space analysis of single drop breakage events (left) and CFD flow simu-
lation of the single drop breakage cell (right - mean velocity and exemplary 1 mm droplet)
An area of 30.25 x 64.25 mm is resolved by 286 x 608 Pixels. The whole breakage event can
be analysed systematically by using such a high time (1000 fps) and space resolution. Thus
place and time of the breakage can be determined precisely. For space analysis the image is
divided into squares with a side length of 2.75 mm corresponding to 26 Pixels (see Fig. 2 –
left). To avoid interferences with the stirrer blade, the point of origin for the y-axis starts at
image pixel 10. Thus the area of interest is divided in 23 x 11 squares with an edge length of
2.75 mm, representing 253 different spatial magnitudes. All of these magnitudes have 4 coor-
dinates (xstart, ystart; xend, yend) describing the mounted area of this specific magnitude. As an
example the blanked magnitude in Fig. 2 – left is described in pixel and mm-coordinates:
[(234, 36; 260, 62) all in pixel] and [(24.75, 2.75; 27.5, 5.5) all in mm]. The relative velocity
between blade and liquid flow is approximately 1.0 to 3.0 m/s, a range typical for the flow
field around the stirrer.
2.2 Numerical investigations
By using computational fluid dynamics (CFD) the simplification of the flow field in the
breakage cell compared to a stirred tank and the drop breakage itself should be investigated.
The commercial software package ANSYS-CFX, Release 10 and the implemented shear-
stress-transport-turbulence model (SST ) was used for flow analyses. The main advantage
of this turbulence model is the combination of the advantages of the k-ε-model (for high Rey-
nolds number in the bulk flow) and the k-ω-model (low Reynolds number near to the wall).
This means that close to the wall the SST-turbulence model switches automatically from k-ε
to k-ω. Therefore the checking of the dimensionless wall distance (y+) is not necessary.
For the numerical investigations the presented breakage cell geometry was divided into
three regions. The first block in front of the stirrer blade and the second block behind the stir-
rer blade were resolved by nearly 285,000 hexahedral elements. The third block, a region near
to the stirrer blade (30 mm x 30 mm x 55 mm), was resolved by 1,064,113 tetrahedral ele-
ments ( 5·10-5 mL per element or mean edge length 0.7 mm). Summarised more than
1,440,000 elements or 550,000 nodes were used for the whole simulated region. For example
the comparison of a droplet (dP = 1 mm) and the fine resolved mesh is shown in Fig. 2 – right.
3.1 Comparison of breakage cell flow pattern and flow field in a stirred tank
The aim of the CFD-simulations was to calculate the mean velocity field in the breakage cell
and to compare the location of the particle break-up of the experimental results with numeri-
cal parameters like velocity and energy dissipation.
Due to the simulated flow-pattern around a stirrer blade in the breakage cell, first of all the
numerical results of the breakage cell were compared with the flow field in a stirred tank.
Considering Stoots and Calabrese  the breakage cell represents a rotating frame of refer-
ence (that means fixed stirrer). So this specific flow is characterized by a wake flow behind
the stirrer blade. The flow direction of this wake flow is against the normal flow direction in
the breakage cell. That leads to two typically vortices behind the stirrer blade in 2D plot (see
Fig. 2 - right). Normally stirred tanks are considered by a fixed frame of view (that
means rotating stirrer). To transform the breakage cell from a rotating to a fixed frame of view
the local velocities were subtracted by stirrer tip speed wTip. In this case the maximum abso-
lute velocity for a Rushton turbine is much higher than the stirrer tip speed. Stoots and
Calabrese measured a peak value for the mean tangential velocity of 1.43·wTip which corre-
sponds with 1.3·wTip in the breakage cell.
Anyway CFD-results include also the simulated energy dissipation rate and the turbulence
kinetic energy. By increasing flow velocity the local energy dissipation increases by
εloc/∆w³ = const. too. The maximum energy dissipation is simulated to εmax/∆w³ ≈ 8 m-1. For
example a flow velocity of ∆w = 1.5 m/s in the breakage cell should simulate a rotating speed
of N = 550 rpm in a stirred tank. This leads to a specific power input of ε = 0.92 m²/s³ in a
stirred tank and the energy dissipation ratio can be calculated to εmax/ε = 29 (see (1)). This
energy dissipation rate is well known for a radial impeller [4, 7].
ε max ε 8 ⋅ ∆w 3 27 m 2 s 3
= 8 → max = = = 29 (1)
∆w 3 ε ε 0.92 m 2 s 3
It is also well known that for turbulent flows drop breakage is usually related to the energy
dissipation rate [see also Fig. 6]. Nevertheless the deformation rates based on gradients of
mean velocity are also helpful to understand such processes. In Fig. 2 – right are shown a
1 mm droplet, the fine mesh resolution and the simulated mean velocities for a flow velocity
∆w = 1.5 m/s which represents wTip = 2.3 m/s (N = 550 rpm). For instance the shear-gradient
based on mean velocity close to the droplet is nearly γ = 1.420 s-1 (∆w = 1.42 m/s and
∆x ≈ dP = 1 mm). This situation leads to a typical drop-breakage-event which is shown in Fig.
1. The calculated shear-gradient mentioned above is equal to γ /N = 153 or a shear stress of
1.4 Pa. Stoots and Calabrese measured the mean velocity field in a turbulent stirred tank by
LDA and calculated deformation rates based on gradients of mean velocity. They locate the
maximum deformation rate on the blade edges and quantify it to γ /N = 150.
It is also possible to integrate the kinetic energy over the whole volume of the droplet.
This leads to an input energy ( Ekin = 1.1·10-7 J) which is equal to the surface energy of the
droplet (dP = 1 mm; σ = 32 mN/m EP = 1.0·10-7 J). At least the simulated parameters for
the mentioned droplet in Fig. 2 are listed in table 1.
table 1 – listing of the simulated parameters in the breakage cell
maximum velocity [m/s] 1,74
minimum velocity [m/s] 0,32
Volume averaged energy dissipation [m²/s³] 21
Volume averaged kinetic energy [m²/s²] 0,12
Volume averaged deformation rate based
on mean velocity
All explanations in this section should clarify the very good agreement between the simulated
parameters (velocity, deformation rates, energy dissipation) in the breakage cell and the
known experimental data in a stirred tank. Accordingly the flow pattern in the breakage cell
and the flow field in a stirred tank look very similar by using the same frame of reference.
This fact is very important for interpreting the experimental results.
3.2 Single drop breakage results
Firstly the locality of the breakage events in the single drop breakage cell is analysed. To rate
the local breakage allocation it is necessary to analyse the entrance allocation of the particles.
Therefore all breaking droplets are classified by their entry point into the area of interest (see
Fig. 2 – left) which is defined as the x-coordinate where the centre of mass is detected at y-
coordinates below 36 pixels, equal to 4 mm. Fig. 3 shows the results for the so developed
relative entrance probability. The relative frequency calculated by the entering number of
droplets is plotted against the pixel magnitudes over the width of the images. The investiga-
tions include two diameters (644 and 1000 µm) and three flow velocities (1.0, 1.5 and 2.0
m/s) of toluene droplets in a continuous water flow.
All three investigated flow velocities and both diameters show basically the same behav-
iour. Many more droplets are entering the analysed area at the right and at the left border than
in the centre of the channel, because the water flow is forced around the stirrer blade to the
sides. A second difference between relative entrance probabilities of the left and the right side
becomes obvious on closer inspection. The sum of the number and the relative frequency of
entering droplets of the first three pixel magnitudes is always higher than the sum of the last
three magnitudes (see Fig. 3). This phenomenon is especially significant for the experiments
with the small diameter of 644 µm and a flow velocity of 1.5 m/s. These permanent errors
have to be taken into account for the following analysis of drop breakage places.
flow velocity toluene
1,0 m/s droplets
1,5 m/s pH = 7
Relative frequency [-]
1,5 m/s (650 µm)
0 26 52 78 104 130 156 182 208 234 260 286
x-coordinate Pixel magnitudes [Pixel]
Fig. 3 – Relative entrance probability (y-coordinate smaller than 36 Pixels / 4 mm) for all breaking toluene
droplets under variation of mother drop diameter (644 and 1000 µm) and flow velocity (1.0 - 2.0 m/s)
The following plots in Fig. 4 and in Fig. 5 show the space analysis of the breaking droplets.
For all counted breakage events, the coordinates of the location of the drop breakages are col-
lected and classified after the presented pixel magnitudes (see Fig. 2 – left).
A direct comparison for two different velocities (1.0 and 2.0 m/s) is presented in Fig. 4.
Both diagrams show the results for 1000 µm toluene droplets. As presented in chapter 2, the
allocation of the breaking droplets into the different pixel magnitudes, starts for the y-axis at
10.0 pixels. The x-axis starts at 0.0, and therewith the first pixel magnitude with an edge
length of 26 pixels starts in the upper left corner from (10; 0) mounted to (36; 26). All length
scales have then been translated from pixels to mm. The relative frequency of breaking drop-
lets is now plotted against this rasterised 2D-area of the range around the stirrer model. The
same example as in Fig. 2 (blanked magnitude) for the lower velocity: The relative frequency
of the investigated breaking droplets for the magnitude [(24.75, 2.75; 27.5, 5.5) all in mm] is a
local maximum of approximately 2.2 percent. Such an analysis shows three main results.
diameter 1000 µm diameter 1000 µm
flow velocity 1.0 m/s flow velocity 2.0 m/s
Relative frequency [-]
0.0 x [mm] 0.0 x [mm]
33.0 y [mm] 30.25 33.0 30.25
& 63.25 & y [mm]
V V 63.25
Fig. 4 – Comparison of the location of breakage for toluene drops with a diameter of 1 mm at two different
flow velocities (left: 1.0 m/s; right: 2.0 m/s)
Firstly both diagrams show that the main number of breakages is occurring in the centre of the
stirrer range. No breakages occur, due to the wall grip, near the channel walls for magnitudes
with y-coordinates lower than 2.75 or higher than 27.5 mm. This means that, for closer dis-
tances than 2.75 mm from the wall, breakage is strongly hindered.
Secondly the diagrams show one major difference. Breakage ends mainly after a distance
of 500 pixel of the y-coordinate for the lower velocity. That means that 55 mm (0.65·d) be-
hind the stirrer blade no breakage is occurring anymore due to the flow patterns. For the
higher velocity (Fig. 4 – right diagram) the gradients of the flow are still able to break drop-
lets. So the allocation of breakage events is more homogeneous for the higher than for the
Thirdly the error shown and discussed in Fig. 3 can also be recognized in the breakage
plots. All diagrams including the following figures show slightly higher values for the smaller
x-Pixel classes than for the symmetric higher ones.
diameter 644 µm diameter 1000 µm
flow velocity 1.5 m/s flow velocity 1.5 m/s
Relative frequency [-]
0.0 y [mm] 0.0 y [mm]
& 33.0 x [mm] 30.25 & 33.0 x [mm] 30.25
V 63.25 V 63.25
Fig. 5 – Comparison of the place of breakage for toluene drops with two different diameters (644 and
1000 µm, left and right) for a flow velocity of 1.5 m/s.
In Fig. 5 the comparison of the breakage places for two different drop diameters (644 and
1000 µm, left and right) are presented for a third flow velocity (1.5 m/s). Both diagrams show
the same basic behaviour as the analyzed events in Fig. 4. On closer examination the distribu-
tion of the right diagram for the 1000 µm droplets with the flow velocity of 1.5 m/s lies ex-
actly between the 1.0 and 2.0 m/s data, presented in Fig. 4.
Analyzing the different results for the different diameters, it becomes obvious that the lar-
ger droplets break more homogeneously over the investigated range. This can be explained
with the surface forces of the droplets. The same flow field with the same gradients resulting
by the same flow velocity should stress both diameters equally. The smaller diameter is more
stable than the larger one and so breakage occurs only in a close range to the stirrer blade.
These results shall now be compared with the flow analysis of the CFD results.
3.2 Comparison of experimental and numerical results
Fig. 6 compares the experimental and numerical results. It shows that the area of high energy
dissipation (εloc > 20 m²/s³) is quite similar to the experimental breakage region with the high-
est number of breakage events. As mentioned above breakage ends mainly after a distance of
nearly 50 mm behind the stirrer blade. The region of high energy dissipation shown in Fig. 6
– right reaches a distance of 50 mm behind the stirrer blade too.
diameter 1000 µm
flow velocity 1.5 m/s
0.0 y [mm]
& 33.0 x [mm] 30.25
Fig. 6 – Comparison of experimental (left) and numerical results (right – isovolume of energy dissipation
above 20 m²/s³) for a flow velocity of 1.5 m/s ( wTip = 2,3 m/s)
In this work the experimental set-up to investigate the single drop breakage and particle strain
close to a stirrer blade was discussed. The developed breakage cell simplifies the turbulent
flow field near the stirrer blade. Accordingly numerical investigations were carried out to
judge the simplification of this application. The comparison of the numerical results in the
breakage cell and the experimental data in stirred tanks  show a good agreement for the
typically vortices behind the stirrer blade, the maximum velocity, the deformation rate based
on mean velocity and the energy dissipation ratio. Also the region of high energy dissipation
is similar to the experimentally detected region of single drop breakage.
We gratefully acknowledge the financial support from the Bayer Technology Services GmbH
and from the Federal Ministry of Education and Research by the AiF–project FKZ 1733X06.
d [m] - impeller diameter
dP [m] - particle / drop diameter
fps frames per second
N [m s] - stirrer speed
w [m s] - speed
w Tip [m s] - tip speed of the stirrer blade
x start; end [m] - space coordinate
y start; end [m] - space coordinate
ε loc; max [m² s ³] - global and local shear strain tensor
γ [1 s] - shear gradient
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