sil screen

Rotor-Stator Devices: The Role of Shear and the Stator

T. L. Rodgersa,∗, M. Cookea

aSchool of Chemical Engineering & Analytical Science, The University of Manchester, OxfordRoad, Manchester, M13 9PL

Abstract

High shear rotor-stator mixers are widely used in process industries including themanufacture of many food, cosmetic, pharmaceutical, and health care products.Many of these products involve emulsification where the dropsize distributionaffects the processing and the product properties. Therefore,an understanding ofthe mechanisms that breaks the drops is key for any design process. In rotor-statordevices there are two main mechanisms that can break drops, one due to the rotorand one due to the stator. For the invisid systems studied, this article shows thatwhen a rotor-stator device is used in a recycle loop the effective equilibrium dropsize is largely unaffected by the presence of the stator and is mainly dependant onthe rotor. The article also goes on to show that the effective equilibrium drop sizedata can be correlated on the agitator shear rate.

Keywords: Rotor-stator devices, Shear rate, Equilibrium drop size

1. Introduction1

High shear rotor-stator mixers are widely used in process industries including2

the manufacture of many food, cosmetic, pharmaceutical, and health care prod-3

ucts. Rotor-stator devices provide a focused delivery of energy, power and shear4

to accelerate physical processes such as mixing, dissolution, emulsification, and5

de-agglomeration. To reliably scale-up these devices we need to understand the6

relationship between rotor speed, flow rate, shear rate, andthe energy dissipated7

by these devices.8

∗Corresponding AuthorEmail address: [email protected] (M. Cooke)

Preprint submitted to Chemical Engineering Research and Design June 12, 2013

For emulsification, the drop size distribution affects the processing and the9

product properties. In a two phase process, the mass transfer rate between the10

phases is proportional to the interfacial area. This interfacial area changes with11

the drop size distribution which varies with the conditionsinside the vessel and12

time. Hence successful process design depends on developing a mechanistic un-13

derstanding of drop break-up in these reactors. There are two competing theories14

on drop break-up mechanisms. These are break-up due to turbulent eddies, i.e.15

energy dissipation rate, and break-up due to the agitator shear rate.16

Break-up due to turbulent eddies is generally based on the work of [7] and [6]17

which utilises the concept of eddy turbulence to define a limiting drop size. It is18

usually assumed that drop break-up occurs due to the interactions of drops with19

the turbulent eddies of sufficient energy to break the drop [9].20

Therefore, for a given fluid system the effective equilibrium drop size (this21

is the drop size after a sensible processing time, when the drop size reduction22

with time is very small and almost unmeasurable) is dependent on the energy23

per unit mass and thus should scale-up with this value when using geometrically24

similar vessels. For low viscosity dispersed phase dilute liquid-liquid systems,25

the drops are inviscid since the internal viscous stresses are negligible and only26

the interfacial tension surface force contributes to stability. The maximum stable27

equilibrium drop size,dmax can be related to the maximum local energy dissipation28

rate,ǫmax, by equation 1 for isotropic turbulence [8, 3].29

dmax = C1

(

σ

ρ

)3/5

ǫ−2/5max (1)

Figure 1 presents drop size data from previously published literature for a30

silicone oil and water system as a function of the energy dissipation rate,ǫ; the31

gradient of the line has been set to−2/5 in agreement with equation 1. The lack32

of correlation between the effective equilibrium drop size and energy dissipation33

rate seems to point to the fact that this may not be the correctmechanism. This34

is not surprising since this theory applies to isotropic turbulence in the universal35

equilibrium regime, whereas it is known that breakup occursclose to the agitator36

where the turbulence is both non-isotropic and intermittent.37

Break-up due to the agitator shear rate is based on a balance between the38

external viscous stresses and the surface tension forces [9]. If the break-up is39

due to the agitator shear rate then the effective equilibrium drop size is related to40

the maximum shear rate. This would mean that lower power number agitators can41

produce smaller drops than higher power number agitators, as low power number42

2

agitators may have a higher shear rate. This has been seen experimentally by43

[14]. [1] states when scale-up is performed on a constant energy dissipation rate,44

smaller drops are observed at larger scales. This is likely due to the shear rate45

increasing at larger scales when the energy dissipation rate is kept constant.46

The maximum shear rate is proportional to the agitator tip speed [11]; how-47

ever, this maximum shear rate constant is difficult to measure for all systems. It48

makes physical sense that this maximum shear rate constant is proportional to an49

average shear rate constant. Although, it is strictly only applicable in the laminar50

regime [4], the Metzner-Otto constant,KS , is a good measure of the average shear51

rate near the impeller. It has been claimed that for power lawfluidsKs varies with52

the power law index although for practical considerations this affect has found to53

be small. Tanguyet al. (1996) [12] concluded that for practical considerations a54

constant value ofKS can be considered for shear thinning and shear thickening55

fluids. KS has been found to vary linearly with the agitator flow number which is56

a function of Reynolds number [13]. However, in the turbulent regime the flow57

number is constant, so again we have a constant (though higher) value ofKS .58

This means the correlative shear rate used will beKS ND, i.e. the proportional59

constant multiplied by the tip speed. It should be noted herethat this shear rate60

technically has units of m s−1 instead of s−1, but as previously mentioned, this is61

just a representative value as the true value is proportional to KS ND, which means62

that this proportionality constant must have units of m−1.63

Figure 2 presents drop size data for a silicone oil and water system as a func-64

tion of the agitator blade shear rate,KS ND, the best fit line gradient is equal to65

−1.2. The fact that all the values lie on the same line points towards shear rate66

being the dominant break-up mechanism.67

Neither of these two mechanisms will predict the correct values for the drop68

size if the system undergoes coalescence. If there is coalescence within the system69

the bulk flow from the agitator is important as well, as this effects the circulation70

time, thus the time away from breakage. The systems studied within this paper71

are non-coalescing systems, which was checked over a periodof several days.72

2. Methodology73

The experimental rig (Figure 3) consists of an agitated mixing tank with an74

in-line Silverson 150/250 MS high shear rotor-stator mixer (Silverson Machines75

Ltd., Chesham, UK). The mixing tank has a 60 litre capacity with a diameter of76

0.420 m. To allow both analysis of equlibrium drop sizes and single pass drop77

size data the mixing tank was connected to the rotor-stator mixer via a 38.1 mm78

3

pipeline, with a second line used to feed the rotor-stator outlet to the drain for the79

single pass experiments. The inlet and outlet lines includetemperature probes so80

the temperature could be controlled to 25◦C for all the experiments. The Silverson81

rotor-stator mixer has a double concentric rotor with a diameter of 0.0635 m which82

sits within close fitting screens. The fitted screens are standard double emulsifier83

screens [2].84

Two experimental systems were used; a 1 wt% 350 cSt silicone oil and 0.5 wt%85

sodium laureth sulfate in water solution, which was passed through the Silverson86

both a single pass and equlibrium configuration, both with and without the screens87

as a range of rotor speeds and flow rates; and a 0.13 wt% 5 cSt silicon oil in water88

system as used by [10] which was passed through the Silversonin the equlibrium89

configuration both with and without the screens at a range of rotor speeds at a flow90

rate of 0.167 kg s−1..91

Samples of the solutions were collected just after the Silverson in-line mixer92

and analysed using a Mastersizer X long bed laser diffraction particle analyser93

(Malvern Instruments, Malvern, UK) to determine the drop size distribution, full94

details are given by [5].95

3. Effect of the Screens on Drop Size96

A 1 wt% 350 cSt silicone oil and 0.5 wt% sodium laureth sulfatein water solu-97

tion has been passed through a Silverson 150/250 in-line rotor stator mixer. This98

is carried out under a range of agitation rates and flow rates.First as a single pass99

and then as a recycle arrangement for both the standard double screen arrangement100

and with no screens.101

Figure 4 shows the results for these experiments. When the system is used for102

a single pass the screens reduce the drop size compared to a single pass with no103

screens. Whereas when the system is used in a recycle arrangement there is very104

little difference in drop size when there are screens or no screens.105

With no screens there is a large variation of the single pass drop size with flow106

rate, especially at low agitation rates. This is due to by-pass of some of the drops107

around the rotor. This is shown by the reduction in drop size for a single pass with108

the screens, and the more consistent drop size produced withvariation in flow rate.109

The screens also cause some re-circulation to the agitator which is why the drop110

size is slightly reduced.111

There is evidence that the stator has some effect on the break-up, especially112

at the higher flow rates. This is because the high flow rated3,2s are the smallest113

drop sizes for the single pass data with the screens, see Figure 4, and the largest114

4

Setup Po(turbulent) PoRe(laminar)KS k1

Standard dual screens 0.24 573.6 46.2 8.02No screens 0.21 276.2 6.6 7.05

Table 1: Summary of power and shear constants for the Silverson 150/250 [2].

size without the screens. This is due to the fact that as the flow rate increases the115

pressure drop through the screens is increased which increases the shear through116

the holes.117

As the system approaches the effective equilibrium this bypass and recircula-118

tion does not matter as material that my have bypassed on a single pass will even-119

tually get broken by the agitator on another pass. The drop size for recycle loop120

are smaller than for the single pass, this is because for the recycle loop the drops121

are approaching their effective equilibrium drop size for that shear rate whereas122

those after a single pass are not. This is because under the recycle arrangement123

the drops pass through the agitator more than once increasing the probability that124

the small drops produced will become stable.125

The power number of an in-line rotor-stator device can be given by equa-126

tion 2 [2]. Table 1 provides the power and shear constants forthe Silverson mixer127

from [2].128

Po= Poz + k1NQ (2)

129

This means that the power number for the above experiments vary from about130

0.25 for the lowest flow rate to 0.66 for the highest flow rate, over a 2.5 times131

increase in power. However, the equlibrium drop sizes are mostly the same for132

different flow rates. This provides evidence that the energy dissipation rate may133

not be a suitable correlator for drop sizes, especially for rotor-stator devices.134

4. Rotor-Stator Shear135

For comparison to the data in Figures 1 and 2, the 0.13 wt% 5 cStsilicon oil136

in water system used by [10] was used in the Silverson 150/250 inline mixer to137

an effective equilibrium drop size for both the standard screens and no screens.138

No real difference between the values for the two setups was discovered,again139

showing the small impact of the screens on the equilibrium drop size.140

Figure 5 presents the same data as Figure 1 but includes the authors data for141

the Silverson system both with and without the screens. The best fit line is given a142

slope of−0.4; however, it can be seen that this does not fit the data well. It is clear143

5

that although the exponents for all the agitators are around−0.4, the multiplier for144

each is different.145

In considering equilibrium drop size, it is typical to use the maximum local146

value of the rate of energy dissipation rather than the average value (as used in147

Figures 1 and 5). The best order of magnitude estimate of the maximum dissi-148

pation rate is based on using the swept volume of the impellerinstead of tank149

volume [11]. Therefore, Figure 5 has been modified to produceFigure 6. Figure 6150

has the same best fit line slope as Figure 5. It can be seen that Figure 6 does better151

at correlating agitators of the same type together, e.g. allthe Rushton turbine data152

is now almost collapsed onto one trend; however, it is still not able to correlate the153

drop size data from different agitator types.154

As the screens seem have little effect on the effective equilibrium drop size155

it can be surmised that the drops are broken mainly by the agitator. This means156

that the shear rate of importance is that from the agitator only, not the full rotor-157

stator shear rate. Therefore, for the data to line up with thedata from Figure 2 the158

Metzner-Otto constant that should be used is the one just forthe agitator, i.e. 6.6.159

As previously mentioned, the agitation rates can be converted to a representa-160

tive shear rate by multiplying by the agitator Metzner-Ottoconstant and its diame-161

ter. This means that the data for the Silverson can be plottedon Figure 2 producing162

Figure 7. It can be seen that using the agitator shear rate thedata fits much better163

than with the energy dissipation rate. If the total Metzner-Otto constant was used164

for the standard screens system the drop size would be under predicted.165

The fact that the screens provide very little advantage for effective equilibrium166

drop size and that the agitator is the limiting factor, then asystem like this can be167

attributed an apparent drop size Metzner-Otto constant, which is equal to the that168

of just the agitator.169

It should be noted that although the screens provide very little help for equilib-170

rium drop break up they play a part in other processes, such asdeagglomeration.171

It is also noted that since the Metzner-Otto constant is muchhigher when the172

screens are present, the apparent viscosity will be lower for shear thinning flu-173

ids. This will change the viscosity ratio which may impact onthe equilibrium174

drop size. There is also some evidence that at very high flow rates the increased175

velocity through the holes help to break up material that bypasses the agitation,176

resulting in a shorter time to the effective equilibrium.177

6

5. Discussion and Conclusions178

• The agitator shear rate correlates the drop size for a range of agitator types179

and scales for a particular fluid system,180

• For rotor-stator devices it is the rotor shear rate that is important for drop181

break-up not that from the stator, especially at low flow rates,182

• The stator helps single pass design as it reduces bypass around the agitator,183

• Design for drop break-up should focus on improving the rotoras this is184

dominant in drop break-up.185

6. Acknowledgments186

Tom Rodgers would like to thank The University of Manchester’s EPSRC187

CTA (Collaborative Training Account) and Unilever for financial support during188

his PhD. The authors would also like to thank the SCEAS workshop staff who189

helped with equipment modifications and construction.190

Nomenclature191

D Agitator diameter m192

d3,2 Volume to surface average diameter m193

dmax Maximum stable diameter m194

k1 Power flow constant −195

KS Metzner-Otto constant −196

N Agitation rate s−1197

P Power W198

Q Flow rate kg s−1199

ǫ energy dissipation rate W kg−1200

ρ Density kg m−3201

NQ Flow number =Q/ρND3202

Po Power number =P/(ρN3D5)203

Poz Power number at zero flow rate =P/(ρN3D5)204

Re Reynolds number =ρND2/µ205

7

References206

[1] J. Bałdyga, J.R. Bourne, A.W. Pacek, A. Amanullah, A.W. Nienow, Chemi-207

cal Engineering Science 56 (2001) 3377–3385.208

[2] M. Cooke, T.L. Rodgers, A.J. Kowalski, Journal of Chemical Engineeringin209

press (2010).210

[3] J.T. Davies, Chemical Engineering Science 42 (1987) 1671–1676.211

[4] D. Doraiswamy, R.K. Grenville, A.W. Etchells III, Industrial and Engineer-212

ing Chemisrty Research 33 (1994) 2253–2258.213

[5] S. Hall, M. Cooke, A. El-Hamouz, A.J. Kowalski, ChemicalEngineering214

Sciencein press (2011).215

[6] J.O. Hinze, Journal of the American Institute of Chemical Engineers 1216

(1955) 289–295.217

[7] A.M. Kolmogorov, Doklady Akademii Nauk 66 (1949) 825–828.218

[8] D.E. Leng, R.V. Calabrese, Handbook of Industrial Mixing: Science and219

Practice, Wiley-Interscience, 2004, pp. 639–753.220

[9] Y. Liao, D. Lucas, Chemical Engineering Science 64 (2009) 3389–3406.221

[10] M. Musgrove, S. Ruszkowski, Proceedings of the 10th European Conference222

on Mixing, Delft, Netherlands July 2-5 (2000) 165–172.223

[11] E.L. Paul, V.A. Atiemo-Obeng, S.M. Kresta (Eds.), Handbook of Industrial224

Mixing: Science and Practice, John Wiley & Sons Inc., New Jersey, USA,225

2004.226

[12] D.E. Tanguy, E. Thibault, E. Brito De la Fuente, Canadian Journal of Chem-227

ical Engineering 74 (1996) 222–228.228

[13] J. Wu, L.J. Graham, N.N. Mehidi, AIChE Journal 52 (2006)2323–2332.229

[14] G. Zhou, S.M. Kresta, Chemical Engineering Science 53 (1998) 2063–2079.230

8

Fig. 1. Variation of mean effective equilibrium drop size for a 5 cSt silicon oil231

and water system against the power per unit mass. Data taken from [14] and [10].232

Dotted lines are 20% from the best fit.233

234


and water system against the agitator blade shear rate. Datataken from [14] and236

[10]. Dotted lines are 20% from the best fit.237

238

Fig. 3. Schematic of the equipment used for the experiments. The Silverson239

is used in either a single or multiple pass arrangement.240

241

Fig. 4. Variation of the single pass and effective equilibrium drop size for a 1 wt%242

350 cSt silicone oil and 0.5 wt% sodium laureth sulfate in water. OP and EQ refer243

to a one pass and effective equilibrium run respectively. WS and NS refer to with244

screens and without screens respectively.245

246


and water system against the power per unit mass. Data taken from [14] and [10].248

*Data from current work in Silverson both with and without screens. Dotted lines249

are 20% from the best fit.250

251


and water system against the maximum energy dissipation rate (power per unit253

agitator swept mass). Data taken from [14] and [10]. *Data from current work in254

Silverson both with and without screens. Dotted lines are 20% from the best fit.255

256


and water system against the agitator blade shear rate. Datataken from [14] and258

[10]. *Data from current work in Silverson both with and without screens. Dotted259

lines are 20% from the best fit.260

9

10-2

10-1

100

101

ε / W kg-1

10-5

10-4

10-3

d 3,2 /

m

A310 T=0.24 m, D/T=0.35A310 T=0.24 m, D/T=0.55HE3 T=0.24 m, D/T = 0.25PBT T=0.24 m, D/T=0.25RT T=0.24 m, D/T=0.25RT T=0.29 m, D/T=0.5RT T=0.26 m, D/T=0.33RT T=0.17 m, D/T=0.5RT T=0.17 m, D/T=0.33

261

10

1 10 100K

SND / m s

-1

10-5

10-4

10-3

d 3,2 /

m

A310 T=0.24 m, D/T=0.35A310 T=0.24 m, D/T=0.55HE3 T=0.24 m, D/T=0.25PBT T=0.24 m, D/T=0.24RT T=0.24 m, D/T=0.25RT T=0.29 m, D/T=0.5RT T=0.26 m, D/T=0.33RT T=0.17 m, D/T=0.5RT T=0.17 m, D/T=0.33

262

11

To Drain

(for single pass)

(for multiple pass)

Sample

point

Silverson263

12

50 100100 150N / s

-1

5

10

15

20

25

303540

d 3,2 /

µm

OP NS 0.083 kg s-1

OP NS 0.278 kg s-1

OP NS 0.472 kg s-1

OP NS 0.667 kg s-1

OP WS 0.083 kg s-1

OP WS 0.278 kg s-1

OP WS 0.472 kg s-1

OP WS 0.667 kg s-1

EQ NS 0.083 kg s-1

EQ NS 0.278 kg s-1

EQ NS 0.472 kg s-1

EQ NS 0.667 kg s-1

EQ WS 0.083 kg s-1

EQ WS 0.278 kg s-1

EQ WS 0.472 kg s-1

EQ WS 0.667 kg s-1

264

13

10-2

100

102

ε / W kg-1

10-6

10-5

10-4

10-3

d 3,2 /

m

A310 T=0.24 m, D/T=0.35A310 T=0.24 m, D/T=0.55HE3 T=0.24 m, D/T = 0.25PBT T=0.24 m, D/T=0.25RT T=0.24 m, D/T=0.25RT T=0.29 m, D/T=0.5RT T=0.26 m, D/T=0.33RT T=0.17 m, D/T=0.5RT T=0.17 m, D/T=0.33Silverson 150/250 inline mixer*

265

14

10-2

10-1

100

101

102

εmax

/ W kg-1

10-6

10-5

10-4

10-3

d 3,2 /

m

A310 T=0.24 m, D/T=0.35A310 T=0.24 m, D/T=0.55HE3 T=0.24 m, D/T = 0.25PBT T=0.24 m, D/T=0.25RT T=0.24 m, D/T=0.25RT T=0.29 m, D/T=0.5RT T=0.26 m, D/T=0.33RT T=0.17 m, D/T=0.5RT T=0.17 m, D/T=0.33Silverson 150/250 inline mixer*

266

15

1 10 100K

SND / m s

-1

10-6

10-5

10-4

10-3

d 3,2 /

m

A310 T=0.24 m, D/T=0.35A310 T=0.24 m, D/T=0.55HE3 T=0.24 m, D/T=0.25PBT T=0.24 m, D/T=0.24RT T=0.24 m, D/T=0.25RT T=0.29 m, D/T=0.5RT T=0.26 m, D/T=0.33RT T=0.17 m, D/T=0.5RT T=0.17 m, D/T=0.33Silverson 150/250 inline mixer*

267

16

sil screen

Documents