technical paper - mixing analysis of a newtonion fluid
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7/21/2019 Technical Paper - Mixing Analysis of a Newtonion Fluid
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chemical engineering research and design 8 6 ( 2 0 0 8 ) 1434–1440
Contents lists available at ScienceDirect
Chemical Engineering Research and Design
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c h e r d
Mixing analysis of a Newtonian fluid in a 3D planetary
pin mixer
Robin Kay Connelly a,∗, James Valenti-Jordan b
a Department of Food Science, University of Wisconsin-Madison, 1605 Linden Dr., Madison, WI 53706, USAb Campbell Soup Company, 1 Campbell Place, Mail Stop 210, Camden, NJ 08103, USA
a b s t r a c t
The mixograph is a planetary pin mixer that has been used for decades to evaluate the mixing tolerance and large
strain rheology of hydrated flour. In this work, computational fluid dynamics (CFD) has been used to gain greater
understanding of the mixing action of this mixer by evaluating both local and global measures of mixing using
particle tracking. In this study, mixing of a highly viscous, Newtonian corn syrup is simulated. Segregation scale,
length of stretch and efficiency are used to evaluate the mixer. It is shown that this planetary pin mixer does not
experience as much axial mixing as cross-sectional mixing over the same time span. Additionally, it is observed that
some pin positions are more efficient than others. These results are being used to compare this mixer with other
mixers used for similar purposes in the food industry.
© 2008 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Keywords: Planetary pin mixer; Mixograph; Computational fluid dynamics; CFD; Mixing analysis; Mixing efficiency
1. Introduction
The mixograph (National Instruments, Lincoln, NE, USA) is a
planetary pin mixer used to evaluate the mixing tolerance
of hydrated wheat flour dough in the food industry. It is a
four-pin planetary pin mixer with three stationary pins, where
the moving pins travel on an epitrochoid path. Two separate
groups have looked at the motion of the mixograph inde-
pendent of the response curve (Buchholz, 1990; Steele et al.,
1990). In this study, the mixing profile of the mixograph-style
reomixer (Reomix Instruments, Lund, Sweden) is looked at
using a Newtonian corn syrup. The reomixer is used as analternative to the 10g of flour mixograph and differs from
the mixograph only in the specific spacing of the pins and
walls.
Theresultsprovided by this instrument consist of thereac-
tion torque on thebaseof themixing bowl causedby the forces
that have been applied to the stationary pins and walls of the
bowl by the dough as it is moved by the planetary pins. These
results are used to determine the point of peak development
and the mixing tolerance of the dough, which is related to
the development of the gluten protein structure during hydra-
tion and mixing. A more fundamental understanding of the
∗ Corresponding author. Tel.: +1 608 262 8033; fax: +1 608 262 6872.E-mail address: rkconnelly@wisc.edu (R.K. Connelly).Received20 August 2008; Accepted 27 August 2008
results is possible when types of forces acting on the dough
within the mixing chamber are known. One tool that is help-
ful in resolving these forces is computational fluid dynamics
(CFD). Previous studies have looked at a 2D cross-section of
the mixograph ( Jongen, 2000; Jongen et al., 2003). This study
continues into 3D where the effect of the angled bottom of
the bowl is incorporated and the full 3D mixing profile is
analysed.
CFDmixing studies withother mixing geometrieshavecov-
ered topics such as comparison of mixers through numerical
simulation (Rauline et al., 1998, 2000; Jongen, 2000; Jongen et
al., 2003; Connelly and Kokini, 2007a,b), assessment of mix-ing effectiveness (Tanguy et al., 1992, 1996, 1997; Bertrand et
al., 1994; Wong and Manas-Zloczower, 1994; Yang and Manas-
Zloczower, 1994; Yang et al., 1994; Connelly and Kokini, 2004,
2006) and design and optimization of improved mixing ele-
ments (Liuetal.,2006). During these studies,various measures
of mixing have been used or developed to help assess how
well the mixing has taken place. In this study, we will be mak-
ing use of some of these mixing assessment tools (Connelly
and Kokini, 2007b), which are available in the finite element
method (FEM) CFDprogramPolyflow (Fluent Inc.,Lebanon, NH,
USA).
0263-8762/$ – see front matter © 2008 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.cherd.2008.08.023
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chemical engineering research and design 8 6 ( 2 0 0 8 ) 1434–1440 1435
Nomenclature
c average concentration
c j, c
j concentration of the points in the jth pair
D dimension
D rate of strain tensor
e efficiency of mixing F deformation gradient tensor
j index variable
Ls scale of segregation
ˆ m current orientation unit vector
M number of pairs of material points
N number of material points
r distance between a pair of points
R(|r|) Eulerian coefficient of correlation
S2 sample variance
t time
v velocity vector
x position of a fluid element at time, t
X initial position of a fluid element
Greek letters
distance beyond which there is no correlation
length of stretch
arithmetic mean of the length of stretch
motion as function of time and initial position
2. Model description
2.1. Geometry, pin motion, mesh and boundary
conditions
The mixer geometry is modelled as filled to 50 ml and the
meshes are shown in Fig. 1. The mesh elements are triangular
prisms. In the standard reference frame, the darker pins are
the moving pins, which follow an epitrochoid (planetary) pin
path, while the lighter pins are the stationary pins. The pin
motion is accounted for in the simulations using the mesh
superposition technique (Avalosse, 1996; Avalosse and Rubin,
2000). In order to simplify the pinpaths so that themesh could
be created to provide a consistent pin shape and volume at
every time step with this approach, the simulation is con-
Fig. 1 – Geometry and mesh where the pins with the light
grey tops are stationary in the standard reference frame.
ducted in the rotating reference frame (RRF). The wall of the
vessel and the “stationary” pins were set to rotate counter-
clockwise around the central vertical axis at the rate of the
primary gear (88 rpm), while the “moving” pins were set to
rotate clockwise around their vertical planetary axis at the
rate of the secondary gear (66 rpm). This approach also allows
the true epitrochoid (planetary) pin path to be modelled more
easily and the mesh to be better refined in the areas near thepins.
The 2D horizontal cross-section mesh selected as a start-
ing point for the 3D mesh was created such that the pin shape
remained a nearly constant hexagon made up of six trian-
gular elements with a time step of 0.0227 s that moved the
pins one element along the pin path. The 2D mesh was also
subjected to a mesh discretization and time step study using
quadratic FEM interpolation for velocity and linear continuous
FEM interpolation for pressure, where it was found that the
difference in the solution was only 2–3% along the pin paths
when themeshdensity wasincreased 16 times andonly about
0.14% if a four times smaller time step is used. Therefore, this
mesh was deemed adequate for use in this study. Also, analy-sis of the difference between the initial time steps on start-up
of the simulation and the results when those pin positions
were repeated showed that the results from the first 0.1 s of
the simulation should not be used for steady-state analysis,
due to start-up effects ( Jordan, 2006).
Globe Corn Syrup 011420 (Corn Products, Westchester,
IL), was selected as the test fluid because the single
proportionality constantof Newtonian viscositywas lesscom-
putationally difficult to simulate. Also, its viscosity lies in the
range of observed viscosities exhibited by developing dough
(Dhanasekharan et al., 1999). This corn syrup has a Newto-
nian viscosity of 5400 cP and a density of 1.409g/cm3 at 49 ◦C.
A no slip boundary condition was used at the walls and on thepins. The top surface was given a full slip boundary condition
but was otherwise constrained to maintain a flat shape.
The flow and mixing simulation was done using the FEM
CFD package Polyflow 3.10.2 (Fluent Inc., Lebanon, NH) with
a mesh created using Gambit 2.3 and post-processed using
Fieldview 10F (Intelligent Light, Rutherford, NJ). The simula-
tion was run on an IBM IntelliStation Z Pro with dual Intel
Xeon 3.4 GHz–64 bit processors and 6 GB RAM running Red
Hat Enterprise Linux WS 3. The 3D simulation was done using
mini-element velocity and linear pressure FEM interpolations.
With these FEM interpolation types, the problem size was
1.4 GB and took a CPU time of 12.68 h.
2.2. Particle tracking and visualizations
In orderto analyse theabilityof thereomixer to fully distribute
the fluid over the entire domain, the fate 10,000 initially ran-
domly distributed material points were studied over 360 time
steps for a total simulation time of 8.182s. In order to track
these points over time, Polyflow and its mixing post-processor,
Polystat, were usedto track the positionsof the 10,000 material
points over all the time steps using the flow profile solutions.
The trajectories of the massless material points or particles
were calculated by the time integration of the equation x = v
using a fourth order explicit Runge-Kutta scheme within an
element with local rather than global coordinates. The time
step was sized such that the final position in crossing an ele-
ment is always on the element boundary so that the element
coordinates maybe transformed to the localcoordinatesof the
next element to be crossed before continuing the integration.
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1436 chemical engineering research and design 8 6 ( 2 0 0 8 ) 1434–1440
Fig. 2 – Horizontal mixing analysis: data at (a) 0, (b) 120, (c) 240, and (d) 360 time steps.
The sum of the steps when transformed to global coordinates
gives the successive positions of the particles in real space
(Debbaut et al., 1997; Ishikawa et al., 2001). The default of
an average of three steps to cross an element was chosen
for the time integration, with the particle positions and kine-
matic parameters recorded at the same time intervals as used
between pin positions. The results were visually inspected to
illustrate the flow profile within the mixer, as well as quanti-
tatively analysed to provide standard mixing indicators over
the entire flow field.
One analysis studied the mixing in the horizontal plane.
The randomly distributed material points were initially
divided along the x = 0 plane and arbitrarily assigned a con-centration value of 1 (light grey) or 0 (dark grey). A pictorial
representation of this division is shown in Fig. 2a. Comple-
tion of mixing is visually indicated by a random distribution
of light and dark grey points over the entire flow domain. In
a second analysis, several cutting planes were used to alter-
nate between zones of randomly distributed points initially
assigned a concentration value of 1 (light grey) or 0 (dark grey)
in order to determine the extent of vertical mixing because
low vertical mixing was expected in this geometry. The cut-
ting planes are located at a height of 1, 3, 5, 7, 9, 11, 13, 15,
17, and 19.698 mm, which created 11 sub-regions in the flow
domain, as shown in Fig. 3a.
2.3. Measures of mixing
The information generated from producing the visual depic-
tions of the mixing above were also analysed statistically. The
scale of segregation is a statistical measure of the mean dis-
tance from a point atwhich it is equally probable to find a light
grey point as a dark grey point. The positions of the particles
at any given time were used to calculate the value of the scale
of segregation, which is defined as (Dankwertz, 1952; Tadmore
and Gogos, 1979; Brodkey, 1985; Chella, 1994):
Ls =
0
R(|r|) d|r|,
Fig. 3 – Vertical mixing analysis: results at (a) 0 and (b) 360
time steps.
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where
R(|r|) =
M
j=1(c j − c) · (c
j − c)
MS2 .
R(|r|) is the Eulerian coefficient of correlation between con-
centrations of pairs of points in the mixer separated by |r|
where R(0)=1 for points having the same correlation andR( )=0atlarge |r| where there is no correlation.The number of
pairs is M = N(N−1)/2 where N is the number of points and S2
is the sample variance. The concentration of the points in the
jth pair is c j
and c j , while c is the average concentration. The
minimum value occurs when the initially segregated particles
become randomly distributed and is a function of the number
of particles tracked and the size of the flow domain. Calcula-
tion of thescaleof segregationwas done at each recorded time
step in orderto track theevolutionof this parameterovertime.
If there are dead spots or faults in the flow that create areas
of the mixer where parts of the initially segregated material
cannot reach, this parameter will not be able to reduce to the
minimum value. In addition, the segregation scale is a globalaverage value that cannot pinpoint the exact location, size or
number of local flow defects (Dankwertz, 1952; Tadmore and
Gogos, 1979; Brodkey, 1985; Chella, 1994).
The 10,000 points were also used to follow the lamellar
mixing parameters, including length of stretch and efficiency
(Ottino et al., 1979, 1981; Ottino andChella, 1983; Ottino, 1989).
Given a motion x= (X, t) where initially X= (X, 0) for an
infinitesimal material line segment dx=F·dX located at posi-
tionx at time t andthe deformation tensoris F=, thelength
of stretch of a material line is defined as = |dx| /|dX|. The
arithmetic mean of the length of stretch, , has been shown
to be directly related to the geometric mean striation thick-
ness and is a measure of the growth of the interfacial area(Alvarez et al., 1998; Muzzio et al., 2000; Zalc et al., 2002). An
exponential increase in the length of stretch over time is a
necessary requirement for effective mixing (Ottino, 1989). The
local or instantaneous efficiency of mixing for isochoric flows
is defined as
e =/
(D : D)1/2 =
−D : ˆ m ˆ m
(D : D)1/2 =
D ln /Dt
(D : D)1/2 ,
where D is the rate of strain tensor, ˆ m is the current orienta-
tion unit vector and (D:D)1/2 is the limit of −D : ˆ m ˆ m according
to Cauchy–Schwarz’s inequality (Ottino et al., 1979, 1981). The
efficiency canbe thought of as the fraction of the energy dissi-
pated locally that is used to stretch a fluid element at a given
instant in a purely viscous fluid (Ottino, 1989) and falls in the
range {−1, 1}. A value of −1 would indicate that all the energy
dissipated was used to shorten the length of the material line,
in effect completely unmixing it, while a value of 1 indicates
that all the energy dissipated was used to stretch the material
line. The time-averaged efficiency is defined as (Ottino, 1989):
e = 1t
t0 e dt.
Typical behaviour of the time averaged mixing efficiency
ranges from the decay of the efficiency with time as t−1 for
flows with no reorientation such as the simple shearing flows,to flows with some periodic reorientation but still decaying
on average with time as t−1, to the optimum case for mixing,
which is flows with strong reorientation that have a positive,
constant average value of the efficiency (Ottino, 1989).
3. Results
3.1. Topographical mixing analysis and scale of
segregation
Fig. 2a shows the initial conditions of the 10,000 data points
discussed above. Fig. 2b and c shows the progression of the
mixing as bulk rotational flow has started in the mixer andareas where the pins have travelled are showing significantly
more blending of the differently shaded points. In Fig. 2d, the
mixing has approached completion at 360 timesteps, based on
optical verification. The main effect holding back the comple-
tion of the mixing is that the points at the walls have not been
displaced much. This lack of movement is due to fluid bound-
ary layers near the wall where there is a no slip boundary
condition, and has been seen experimentally in similar mix-
ers (Gouillart et al., 2007; Thiffeault et al., 2008). This has been
shown to reduce the overall mixing effectiveness for similar
mixers from exponential to power law (Gouillart et al., 2007;
Thiffeault et al., 2008).
When the mixing is analysed using the segregation scale,
the results in Fig. 4 show that most of the mixing takes
place early in the simulation. When a single dark grey par-
ticle invades a large volume of light grey particles, the average
distance for the light grey points to the nearest dark grey
point drops drastically. Therefore, the initial large drop in
segregation scale is expected and observed in Fig. 3. Over-
all, the segregation scale reduces to a nearly constant value
of ∼0.8mm by 270 time steps, which is near the minimum
possible with the level of discrimination available with 10,000
points.
Next, the vertical mixing capability of the reomixer is anal-
ysed. The 11 zones of alternating colour at time zero are
shown in Fig. 3a. As mixing progresses, only minimal mixing
is observed. In Fig. 3b, the points are just beginning to show
blending after 360 time steps. These results show that there is
relatively little vertical mixing in comparison to the amount
of horizontal mixing in the reomixer occurs during the 8.182s
simulated here. However, the slow vertical mixing seen in the
simulation should still provide significant vertical distributive
mixing in the normal run time of this mixer, which is on the
order to 10 min.
The initial segregation scale value for the axial mixing case
shown in Fig. 5 was much lower than initial value for the hor-
izontal mixing analysis due to the effect of the initial multiple
layers. However, the initial slope was much lower, so that the
Fig. 4 – Segregation scale of vertically divided particles for
analysis of horizontal mixing.
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1438 chemical engineering research and design 8 6 ( 2 0 0 8 ) 1434–1440
Fig. 5 – Segregation scale of horizontally divided particles
for analysis of axial mixing.
Fig. 6 – Log of the length of stretch.
value after 360 time steps was higher than that of the hori-
zontal mixing analysis case. There was also little indication of
levelling off to a constant value within the timeframe of the
simulation.
Fig. 5 also shows that thesegregation scale data was mildly
oscillating along its decline. The period of the oscillations was
every 20 time steps, which was the same frequency that the
mixer takes to produce a geometrically similar orientation.
This indicates some periodic reorientation or “un-mixing” of
the material being mixed.
The oscillations were also present in the results shown in
Fig. 4, but are much easier to see with this narrow segrega-tion scale range. These results were not unexpected since the
straight vertical pins did not provide any vertical pumping
and the simplification of a fixed, flat shape for the top sur-
face used in the simulation further constrains vertical motion
of the fluid.
3.2. Length of stretch and efficiency
The purpose of tracking the length of stretch is to under-
stand how much the material is being stretchedin a particular
region. Thelength of stretch is a measure of how mucha mate-
rial point, conceptualized as an infinitesimal line, is stretched
over time. The length of stretch was tracked over time and
recorded in Fig. 6 as the logarithm of the value of the stretch.
It showed good mixing represented by a logarithmic increase
in the length of stretch (Avalosse and Rubin, 2000). The stan-
dard deviation also increased with time because there were
Fig. 7 – Length of stretch distribution at 360 time steps.
some points that did not experience much stretch, especially
along the walls as shown in Fig. 7.
The oscillation observed in Fig. 6 implies that the plane-tary nature of the path of the pins impacts the progression of
stretch. This oscillation repeats every 20 time steps with the
maximum increaseoccurring approximately halfwaybetween
the repetition at 10, 30, 50, etc. time steps. The pin configura-
tion and velocity vectors at one of these odd multiples of ten
are shown in Fig. 8.
This pin configuration was referred to as the efficient
configuration because it causes the highest increase in the
amount of stretch. When at the efficient configuration, the
stretch increased the quickest because the manner by which
the moving pins approach and straddle a stationary pin cre-
ated a high elongational situation in the flow as the moving
pin pulls away from the stationary pin.The instantaneous efficiency is a powerful tool because it
tells an investigator where/what time step to look for the most
efficient mixing. In Fig. 9, the instantaneous efficiency mean
andstandard deviation data were shown to repeat foreach set
of 20 time steps. The most efficient time step was found pre-
cisely in the middle of the bounds of the 20 time steps. This
Fig. 8 – Pin position and velocity vectors at an efficient
configuration (130 time steps).
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chemical engineering research and design 8 6 ( 2 0 0 8 ) 1434–1440 1439
Fig. 9 – Mean instantaneous efficiency.
result agrees with the conclusion based on the analysis of the
length of stretchresults. The pin position in Figs.8and10b was
themost efficient time step with respectto instantaneous effi-
ciency, while theleast efficient position was where thepin and
wall were closest as shown in Fig. 10a. The standard deviationof the data in Fig. 9 indicates a broad spectrum of data, which
means that theefficiency values for some material points may
be much higher.
In Fig. 10, the areas of high instantaneous efficiency,
denoted in red, were shown to be located near the stationary
pins and in areas of high moving pin velocity in the starting
pin orientation. High efficiency was noted around the sta-
tionary pins because the inertia in the fluid was causing the
stationary pins to incite mixing by being a disturbance in the
Fig. 10 – Instantaneous efficiency at (a) 120 and (b) 130 time
steps.
Fig. 11 – Mean time averaged efficiency.
translational flow pattern. Fig. 8 supports this claim as a picto-
rial representation of the velocities at this pin position, which
shows the bulk of the fluid moving with some inertial veloc-
ity and high gradients in some areas of high efficiency seen inFig. 10b.
It is important to note the presence of low values as well in
Fig. 10, as these contribute to the standard deviation in Fig. 9.
The standard deviation in Fig. 9 also reveals another trend in
thedata, because theoscillationof thestandard deviation was
out of phase with the instantaneous efficiency. Occasionally,
the instantaneous efficiency curve produces “steps” at 5 or
15 time steps after the repetition around 13% efficiency. The
orientation of the bottom pins at five steps after the repetition
was the same as the efficient orientation, but the value of the
efficiency maxed out 10 steps after. The most efficient form
of mixing in this mixer was when passing a stationary pin
between two moving pins without having either pass insideof the stationary pin.
The time averaged efficiency is used for describing the net
efficiency of a mixer as time passes. In Fig. 11, the time aver-
aged efficiency was shown to increase quickly and then level
off to a value of 10% in an oscillating fashion. This value was
significant because it represented the net efficiency of mixing
over time. Since the standard deviation falls to a level of 6%,
it can be assumed that the bulk of the fluid is experiencing
mixing.
4. Conclusion
By analysing the results from the 3D simulation, informationwas gathered about the intensity and quality of mixing tak-
ing place within the reomixer. This information provided new
insight into the mechanics of how the reomixer, and subse-
quently all planetary pin mixers, achieve mixing.
In summary, the results show that most of themixing takes
place horizontally in the bulk of the fluid and not along the
walls or near the floor. Vertical mixing does take place slowly,
but may be significantly limited in the simulation by the non-
deforming top boundary condition. A particular orientation
of the seven pins within the mixer was shown to produce
faster/better mixing in the segregation scale, stretch, and effi-
ciency analyses. The likely cause of the effectiveness of the
mixing at this pin position is the effect of inertial velocity
taking place within the bulk of the mixer flow domain.
Future studies of this mixer will attempt to analyse the
impact of the deforming top surface on the vertical flow. In
addition, rheological models for more complex fluids will be
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1440 chemical engineering research and design 8 6 ( 2 0 0 8 ) 1434–1440
used in thefuture to observe the effects on themixing of prop-
erties such as shear thinning or yield stress that are common
in food materials such as dough.
Synergy exists between this work and that of Connelly and
Kokini (2006) on the twin sigma blade farinograph, another
common flour mixing tolerance evaluation instrument. When
that work is combined with this, as well as experimental and
future research on dough mixing, the nature of the effect of mixing on dough development may be unlocked.
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