1

Turbulent Statics of Flow Fields using Large Eddy Simulations in Batch High

Shear Mixers

Vikash Vashisth, Vimal Kumar1

Department of Chemical Engineering, Indian Institute of Technology Roorkee, Uttarakhand – 247667, INDIA

Abstract

Large eddy simulations (LES) have been carried out for the first time to characterize the flow

and turbulent characteristics in three different batch high shear mixers (HSMs) at a constant rotor

speed of 4000 rpm. Three HSMs having different stator heads such as circular, square and

inclined stator head with 6, 92 and 210 numbers of stator holes, respectively have been

investigated. Dynamic Smagorinsky model with sliding mesh method is used for the sub-grid

scale stresses at a Reynolds number of 52,000, to overcome the prediction of RANS models.

Numerical methodology is validated, in terms of Power number with the available numerical and

experimental studies and found in fairly good agreement. The velocity flow patterns and

fluctuations at different planes are predicted and vortexes have been observed within the stator

holes and bulk fluid. It is observed that velocity magnitude fluctuation is a function of rotor

rotations and stator holes size, and fluctuations in one jets emerging from stator holes affect the

fluctuations in other plane jets. Further, it is found that smaller the stator holes size, greater is the

energy distribution and hence greater will be the drop size distribution in the mixer. Therefore,

the inclined stator head HSM can be used for the uniform size distribution in application to de-

agglomeration and dispersion. It is found that the energy spectrum of Kolmogorov is followed

over the entire length scale for all HSMs and LES provided the richer flow and turbulent

information as compared to RANS model.

Keywords: Large eddy simulation; RANS; High shear mixing; Rotor-stator mixer;

Computational fluid dynamics.

1 Corresponding author: Dr. Vimal Kumar; Email: vksinfch@iitr.ac.in

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1. Introduction

High shear mixers (HSMs) are characterized by small gap (Karbstein and Schubert, 1995), high

rate of energy dissipation and tip speed ranging from 20,000 to 100,000 s-1 and 10 to 50 m/s,

respectively (Atiemo-obeng and Calabrese, 2004). High dissipation rate and tip speed of high

shear mixer provide the intense rate of turbulence and uniform rate of mixing, therefore they are

used for wide ranges of industrial processes, such as emulsification, dispersion, de-

agglomeration (Bałdyga et al., 2008; Hall et al., 2011; Jasińska et al., 2014; Kamaly et al., 2017;

Padron et al., 2008; Padron and Özcan-Taşkın, 2018), and typical application can be seen in

Zhang et al. (2012). High shear mixer operates in batch and in-line mode, and high shear stresses

are generated due to smaller gap between the rotor and stator that further resulted into intense

lateral mixing.

Various attempt has been made to study the flow characteristics and power draw in HSMs by

both experimental and numerical approach. Padron (2001) studied the flow characteristics and

power draw in different assembly of batch HSMs and found that power number was independent

of Reynolds number in turbulent regime and inversely proportional in laminar regime. Similar

conclusions were drawn by Doucet et al. (2005). In last few year different models were proposed

for the scale-up of batch and in-line HSMs. Schönstedt et al. (2015) proposed a new model to

scale-up the power consumption in an in-line HSM for high accuracy and throughput. Recently,

James et al. (2017b, 2017a) developed a model for the scale-up of batch HSMs based on power

constant, mixing time, surface aeration and emulsification.

Design of high shear mixers are very complicated and sometime for better mixing and

dispersion, HSMs require high power. Flow inside the tank having HSMs are very complex due

to high shear rate and that makes more difficult to understand the flow and mixing characteristics

at industrial scale. Computational fluid dynamics (CFD) tools are popular and less time

consuming to understand the flow filed inside the mixing tanks. These tools are very powerful

with less restriction to operating conditions to understand the design parameters, to conduct the

diagnosis at less cost. The flow characteristics and power drawn in HSMs in single phase

medium using k- turbulence model have been studied by number of researchers (Calabrese et

al., 2002; Özcan-Taşkin et al., 2011; Pacek et al., 2007; Utomo et al., 2009, 2008). Recently,

Minnick et al. (2018) studied the flow behavior in a tow-staged axial flow HSM using realizable

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k-ε turbulence model at different back pressure conditions. An attempt has been made to explain

the hydrodynamic differences between batch and in-line HSMs (Håkansson et al., 2017). It was

reported that for same geometries and rotor tip speeds, the batch and in-line HSMs had different

emulsification efficiency. Further, Zhang et al. (2017) predicted the flow patterns and power

draw in an in-line HSM using viscous fluid. An ultra-fine teethed HSM was used and

furthermore, the guidance to scale-up the ultra-fine teethed HSM was provided.

Recently, Vikash et al. (2017) studied the hydrodynamics and mixing in a batch HSM using CFD

modeling approach. A RANS based k-ε turbulence model with sliding mesh method was used to

predict the flow and mixing characteristics. CFD approach using k-ε turbulence model

underestimate the turbulent kinetic energy due to its inherent weakness. Mortensen et al. (2018)

validated this argument by comparing k-ε turbulence model with sliding mesh and multi-

reference frame methods with PIV measurement. They found that RANS models underestimated

the dissipation rate of turbulent kinetic energy (TKE). Further, it was reported that the detailed

investigation of flow characteristics required the large eddy simulation (LES) or fully resolved

method. According to Zhang et al. (2012), large eddy simulations (LES) approach provided a

more accurate information regarding the flow pattern and turbulent quantities as LES

successfully utilized to predict the same in stirred tanks, turbomachineries and cyclones

(Brennan, 2006; Byskov et al., 2003; Derksen and Van Den Akker, 1999; Hartmann et al., 2006;

Kato et al., 2003; Murthy and Joshi, 2008; Slack et al., 2000; Yeoh et al., 2004; Zadghaffari et

al., 2010).

A limited work is carried out on LES study in in-line HSMs (Xu et al., 2014, 2013) and they

have validated their work using LDA measurement technique and found a good agreement.

However, no attempt has been made for LES study in batch HSMs. In the present study, for the

first time, a LES study is carried out to predict the flow characteristics and turbulent quantities in

batch HSMs. 3D models of HSMs were used to carry out the numerical simulations. The flow

pattern, rate of energy dissipation, and turbulent kinetic energy analysis for different HSM

geometries were reported at a constant shear rate.

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2. Modeling approach

2.1. High shear mixer geometry

The flow and turbulent characteristics were investigated with the existing commercial rotor-

stator mixer, as discussed by Utomo et al. (2009) for circular and square stator heads, and on a

novel HSM (Vikash et al., 2017) with inclined stator head. The stator diameter was 8 mm,

2.6×2.4 mm (width to depth ratio) and 1.8 mm for circular, square and inclined heads,

respectively. The batch HSMs were having a 4 rotor blades with 28.2 mm diameter. Number of

holes in circular, square and inclined heads considered were 6, 92 and 210 with 1, 4 and 6

number of rows, respectively. Diameter of the implied tank was 150 mm with 150 mm height.

Detail dimensions can be seen in the above work from Utomo et al. (2009) and Vikash et al.

(2017).

2.2. Simulation

The governing equations in the master Cartesian coordinate system with a control volume finite

difference (CVFDM) and sliding mesh methods were solved using Ansys, Fluent 16.2. Standard

k- turbulence model with enhanced wall function at a rotor speed of 4000 rpm was used for

initial convergence at steady state. SIMPLE method was used for pressure and velocity coupling

equations and QUICK scheme was used for the discretization of time and space. The HSMs

geometries have been divided into different regions due to complexity in design such as rotor

fluid, stator or holes, interface (gap between rotor and stator) and inner-outer regions. A total of

8.5 millions hybrid cells have been used in the present study for all different HSMs for LES.

Around 1.5 millions hexahedral cells have been adopted for the rotor fluid region in all the

HSMs geometry. The gap between rotor and stator was divided into 10 hexahedral mesh points

for better investigation of turbulent characteristics. The boundary conditions on solid surface for

the solutions of governing equations were considered as no slip boundary, i.e. ux = uy = uz =0.

Water was taken as working fluid and the rotor speed for all the HSMs was kept constant at 4000

rpm (Re = 52,000). For all HSMs (circular head, square head and inclined head), transient

simulations of LES was started with a converged solution of steady-state k- turbulence model.

SIMPLE method was also used in LES simulations for pressure and velocity coupling equations.

Bounded central differencing scheme was used for the discretization of momentum equation and

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second order implicit scheme for transient formulations. 1/120 of rotor revolutions time was used

as the time step to carry out the transient simulations using sliding mesh method. Simulations

were carried out for 100 rotations for the fully development of the flow and turbulent

characteristics. Further, the solutions were zeroed out to initialize the time statistics flow field

information and data sampling was enabled with the LES. Furthermore, 20 rotor rotations were

performed to static flow field information at the same rotor speed and time step.

2.3. Methodology Validation

In the present numerical study, (Utomo et al., 2009) numerical predictions were used for the

methodology validation. They have used a L4RT square head mixer model having 4 blade rotor

with 4 alternate stator holes with each row having 23 holes and the k- turbulence model with 1.3

millions cells was used for numerical predictions of flow field information. According to them, if

the cells are more than 1 million for RANS based simulations than there is negligible change in

numerical solutions of hydrodynamics in HSMs. Rotor speed was kept at 4000 rpm and water

was taken as working fluid.

The present numerical predictions were also compared with the experimental findings of Padron

(2001), and findings are reported in Table 1. It was found that Power number and flow rates are

in good agreement with the available results and it can be seen that value is very closed to the

experimental results of Padron (2001). Power number was obtained as 2.2 as compared to the

numerical prediction of 2.05 (Utomo et al., 2009) which showed an improvement using LES

turbulent model. Similar improvement can be seen in the flow rate. Therefore, it can be observed

that LES turbulent modelling is an effective approach for the numerical investigation in complex

channels with highly turbulent flows.

Table 1 Comparison of different parameter for square holes stator head at 4000 rpm.

Stator

Power Number

(Present

prediction using

LES turbulence

model)

Power Number

(Utomo et al.,

2009) with k- turbulence

model)

Power Number

(Padron (2001)

experimental

work)

Flow Rate

(kg/s) Present

prediction with

LES

turbulence

model)

Flow Rate

(kg/s) (Utomo

et al., 2009)

with k- turbulence

model)

Square

head at

4000 rpm

2.2

2.05

2.3

0.418

0.389

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3. Results and discussions

3.1 Flow patterns in the jets for different HSMs

Velocity magnitude flow patterns were reported in Figures 1, 2 and 3 for different HSMs having

circular, square and inclined stator heads, respectively at a constant rotor speed of 4000 rpm. The

velocity magnitude flow patterns were analyzed after 120 rotor rotations. Different planes were

drawn to study the jets velocity patterns in different HSMs, one plane for circular stator head

(Fig. 1 (i)), two planes for square stator head (Fig. 2 (i)) and three for inclined stator head (Fig. 3

(i)) as number of rows were one, four and six, respectively. The net flow is inside the mixer head

from the bottom opening and out of the stator holes. Several vortex formations were observed

near the jets and away from the jets due to the high mass flow from the bulk as compared to the

stator holes mass flow. However, from Fig. 1 (ii), it can be seen that for circular stator holes

HSM, the jets were larger as compared to square head (Fig. 2 (ii)) and followed by inclined

stator head (Fig. 3 (ii)). Similar, observations were made by Utomo et al. (2009) for batch HSM,

Calabrese et al. (2002) for in-line HSM and Vikash et al. (2017) for inclined stator head HSM.

However, these authors reported the vortex formation inside the stator head only, while in the

present work vortex formations can be seen in the bulk fluid too. Therefore, it can be implied

that mixing also occur outside the stator.

From Fig. 2 (ii), it can be seen that jets were larger at plane z = 10.8 mm as compared to the

plane z = 6.7 mm, however for inclined HSM, larger jets were for plane z = 9.14 as compared to

other planes. Therefore, it can be suggested that jets would be larger with larger stator holes and

also inline to the horizontal direction. Velocity magnitude fluctuations with respect to rotor

rotation can be seen in Fig. 4 for circular stator head HSM. It can be seen that fluctuation is a

function of rotation and changes with respect to number of rotations. Velocity fluctuations were

measured at a distance of r = 30 mm. Velocity magnitude is high at the lower jets as compared to

the upper jets in square stator head HSM (Fig. 5) and starts to decrease with distance. Similar

observations was made for inclined stator head HSM (Fig. 7).

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(i)

(ii)

Fig. 1 (i) Plane ‘a’ considered for velocity patterns at z = 7.75 mm, and (ii) flow patterns of the

jets emerging from circular stator head HSM at a rotor speed of 4000 rpm.

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(i)

(ii)

Fig. 2 (i) Planes a and b considered for velocity patterns at z = 6.7 and z = 10.8 mm,

respectively, and (ii) flow patterns of the jets emerging from square stator head HSM at a rotor

speed of 4000 rpm.

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(i)

(ii)

Fig. 3 (i) Plane a, b and c considered for velocity patterns at z = 4.69 mm, z = 9.14 mm, and z =

10.8 mm, respectively. (ii) Flow patterns of the jets emerging from inclined stator head HSM at

rotor speed of 4000 rpm.

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Fluctuation in velocity magnitude is high at the lower plane and increases with an increase in

rotation, however opposite can be seen at the upper plane (Fig. 6) for square stator head HSM.

However, velocity fluctuation decreases with an increase in distance. This can be validated with

the flow patterns from Fig. 2 (ii) for square stator head HSM. Further, for inclined stator head

HSM, velocity fluctuation is high as compared to the circular and square stator head HSMs over

a distance of r = 30 mm. Furthermore, in inclined stator head HSM, fluctuations increase with an

increase in rotor rotation at plane z = 4.69 mm, and decrease with rotor rotation at z = 10.8 mm

and both increase and decrease in fluctuation can be seen with respect to rotor rotation for plane

z = 9.14 mm (Fig. 8). Therefore, it can be observed that jets emerging from one plane also affect

the fluctuation of other jets emerging from other plane with respect to rotor rotation. Further, it

was also observed that fluctuations are also a function of stator holes size.

Fig. 4 Variation in velocity magnitude at different rotor revolution with respect to the r = 30 mm

at the same plane, as shown in Fig. 1 (i), in circular stator head HSM at a rotor speed of 4000

rpm.

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Fig. 5 Variation of velocity magnitude with respect to the distance r = 30 mm at the same plane,

as shown in Fig. 2 (i), in square stator head HSM at a rotor speed of 4000 rpm.

(i)

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(ii)

Fig. 6 Variation in velocity magnitude at different rotor revolution with respect to the distance r

= 30 mm (i) at plane a, and (ii) at plane b in square stator head HSM at a rotor speed of 4000

rpm. Planes were drawn similar to the Fig. 2 (i).

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Fig. 7 Variation of velocity magnitude with respect to the distance r = 30 mm at the same plane,

as shown in Fig. 3 (i), in inclined stator head HSM at a rotor speed of 4000 rpm.

(i)

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(ii)

(iii)

Fig. 8 Variation in velocity magnitude at different rotor revolution with respect to the distance r

= 30 mm (i) at plane a, (ii) at plane b, and (iii) at plane c in inclined stator head HSM at a rotor

speed of 4000 rpm. Planes were drawn similar to the Fig. 3 (i).

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3.2 Turbulent kinetic energy and dissipation rate for different HSMs

Iso-contours of turbulent kinetic energy in different HSMs are shown in Fig. 9 at the rotor speed

of 4000 rpm after 120 rotor rotations. It can be seen that the jets are larger for larger stator holes

size, however more uniform distribution of energy can be seen near the smaller stator holes size.

Utomo et al. (2009) and Vikash et al. (2017), reported that the higher energy dissipation occurs

at the leading edge of the stator holes, hence it is not discussed in the present work. The focused

was more on to observe the uniform dissipation in the mixers. Further, it can be seen that more

uniform distribution occurs in the inclined stator head, than the square stator head and further

followed by the circular stator head HSM. Therefore, it can be suggested that smaller the size of

the stator holes more uniform will be the energy distribution. Similar observation can be seen in

the distribution of energy dissipation rate in different HSM (Fig. 10). The dissipation rate of

turbulent kinetic energy can alternatively be used in models of de-agglomeration.

3.3 Energy spectrum for different HSMs

The fast Fourier transformation (FFT) method was used to analyze the power spectral density of

turbulence with respect to the frequency in HSMs (Fig. 11) at a rotor speed of 4000 rpm. Results

were analyzed after the 100 rotations or fully developed turbulent flow in all the three HSMs for

another 10 rotations at the interface (gap between rotor and stator). From Fig. 11, it can be seen

that inertial subrange of universal equilibrium is achieved in the entire length scale as it follows

the -5/3 slope in the entire range. This is due to the fact that analysis was done after the fully

developed flow and only smaller eddies are present in the mixer. Further, it can be seen that

fluctuations in circular and square stator head HSM is more as compared to the inclined stator

head HSM in each time step. Therefore, it can be assumed that more uniform distribution of

energy can be obtained in inclined stator head HSM as compared to the circular and square stator

head HSM. It also proves that smaller the size of stator holes, higher will be the drop size

distribution.

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(i)

(ii)

(iii)

Fig. 9 Turbulent kinetic energy iso-contours (normalized by ) in (i) circular, (ii) square, and

(iii) inclined stator head HSMs at a rotor speed of 4000 rpm [(a) isometric view, and (b) top

view].

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(i)

(ii)

(iii)

Fig. 10 Turbulent energy dissipation rate iso-contours (normalized by ) in (i) circular, (ii)

square, and (iii) inclined stator head HSMs at a rotor speed of 4000 rpm [(a) isometric view, and

(b) top view].

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(i)

(ii)

19

(iii)

Fig. 11 Power spectral density as function of frequency at the interface between rotor and stator

for (i) circular, (ii) square, and (iii) inclined stator head HSMs at a rotor speed of 4000 rpm.

4. Conclusions

Large eddy simulations (LES) were carried out in batch HSMs for the first time in three different

design of HSMs such as circular, square and inclined stator heads. At first the modelling

approach was validated with the numerical and experimental results in terms of Power number in

square stator head HSM and found in improved and good agreement. Further, flow patterns and

velocity magnitude fluctuation at different planes and rotor rotations were analyzed. Several

vortex formations were observed within the stator holes and also in the bulk region. It was also

observed that velocity magnitude fluctuation is a function of jets length and stator holes size. In

inclined stator head HSM, fluctuation was more over a distance of r = 30 mm as compared to the

circular and square stator heads HSM. However, jets were longer for circular stator head HSM.

Further, it was found that fluctuation in velocity varies according to the height of stator holes, as

it increased with an increase in rotor rotation at lower plane and decreased at the higher plane.

The iso-contours of turbulent kinetic energy and dissipation rate were analyzed and found that

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smaller the holes size, more will be the drop size distribution. Further, energy spectrum of

turbulent flow was studied at the interface of rotor and stator and found a very good agreement

with the Kolmogorov energy spectrum over the entire range of turbulent length scale. Further, it

can be concluded that LES modelling provides a richer flow and turbulent characteristics around

the mixer and can be prefer over the RANS models. This study can be used for the good

estimation of turbulent kinetic energy and further in the application of de-agglomeration and

dispersion processes.

Acknowledgements

One of the authors (Vikash) gratefully acknowledges the financial support of the Ministry of

Human Resource Development (MHRD), Government of India in the form of a research

fellowship and Institute Computer Center (ICC), Indian Institute of Technology Roorkee for the

usage of computational facilities.

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