DiMixing and Dissolution Times for a Cowles Diskssolution Paper

Mixing and Dissolution Times for a Cowles Disk

Agitator in Large-Scale Emulsion Preparation

Thomas L. Rodgers,, Michael Cooke, Flor R. Siperstein, and Adam Kowalski

School of Chemical Engineering and Analytical Science, University of Manchester, M60 1QD,UK, and Unilever R & D Port Sunlight, Quarry Road East, Wirral, CH63 3JW, UK

E-mail: [email protected]

Abstract

Emulsions are often found in formulated products. During the preparation of emulsions, it

is often desirable to dissolve highly viscous fluids, such as surfactants or other additives, into an

aqueous phase. In this work we measure dissolution times for a non-Newtonian highly viscous

surfactant (sodium laureth sulphate) in water using vessels of different diameter (D = 0.9 andD = 0.15 m) with high shear disk agitators (D/T = 1/3 and 0.62). To monitor the dissolutiontimes we use a volume averaged conductivity obtained by 3D electrical resistance tomography.

We compare the validity of the method and highlight the difference between dissolution times

for highly viscous fluids and blending times for tracers with the same viscosity as water.

Introduction

Many of the things that make our life more comfortable, safer, and that make us feel better are for-

mulated products. They contain the appropriate quantity of several compounds, mixed, arranged,University of ManchesterUnilever R & D Port Sunlight

1

Thomas L. Rodgers et al. Cowles Disk Dissolution

and structured in the correct way to perform a specific application. The performance of a for-

mulated product depends not only on its chemical composition (the ingredient), but also on thepreparation method. A wide range of formulations are emulsion based, such as margarine, creams,

paints, and pharmaceuticals, where a liquid is dispersed as small droplets into an immiscible liquid

that remains as a continuous phase.

Emulsion preparation often requires the addition of an emulsifier agent, such as a surfactant

or protein. The blending and dissolution mechanisms of the emulsifier can affect the properties of

the final product, and understanding such mechanisms is important for the control of the process

and product properties, as well as for the design and scale-up of new processes. These emulsifiers

generally have very different viscosities and densities to the bulk liquid to which they are added,

and the homogenization of such systems can often be very different to blending operations. As both

these processes involve mixing of miscible fluids it is often assumed that this is simply a blending

operation. It is however in reality a two stage process consisting of dissolution (breakdown of theinitial agglomerates) and miscible blending, both of which involve mixing.

Flooding experiment with sodium lauryl sulphate (SLES) clearly suggests that in this systemthere is a concentration gradient as there is a phase change, Figure 1. Due to the high viscosity

the SLES can be regarded as a soft solid, thus the surface tension is not important. As the SLES

dissolves into the water it forms a micellar solution with the same viscosity of water.

In this work we compare blending and dissolution rates in agitated vessels for fluids with

different viscosities, and show the factors needed for a general correlation that can be used for the

estimation of blending and dissolution times.

We used Cowles disk style agitators which are commonly used in the preparation of emulsions

due to their ability to produce a high shear rate, which is ideal for drop break-up.1 They also

have the ability to be used over a large range of emulsion viscosities and handle large viscosity

differences between the dispersed and continuous phases.2 Nevertheless, little information that

would be necessary for process scale-up, such as blending times and power numbers, are available

in the literature.

2


Figure 1: View of 70 wt% SLES in contact with water, showing a thin hexagonal phase boundarylayer. Real image width is 1 m.

Blending Times

The difference between blending and dissolution for the addition of liquid to liquid can be due

to a viscosity, density or surface tension difference between them. This difference can create an

interface between the liquids which has characteristic concentration gradients with diffusion as the

added liquid dissolves into the bulk. However, it is not yet fully understood how this divide can be

quantified and what mechanisms define the difference between blending and dissolution.

Currently there are no correlations which can predict the time taken to produce a homogeneous

mixture when a tracer is added to a bulk solution of a different viscosity.

A large amount of work has been carried out on the blending of liquids with similar physical

properties within the turbulent, transitional, and laminar regimes for Newtonian, shear thinning,

and visco-elastic fluids.3 It has been shown that the blending time depends on the specific power

input (P/V ), the tank geometry and the agitator Reynolds number (Re). It has been common placein the literature to correlate the dimensionless mixing time (95N) against other dimensionlessvalues which represent the above variations, i.e. the agitator power number (Po), the impeller totank diameter ratio (D/T ), the liquid height to tank diameter ratio (H/T ), and the agitator Reynolds

3


number. For blending in the turbulent regime the dimensionless mixing time is independent of the

Reynolds number and has been correlated to equation 1.3 It is worth commenting here that the

data for this correlation was mainly collected on vessels with H/T 1. Cooke et al.(1988)4 hasshown that for H/T >> 1 the exponent on H/T increases considerably due to zoning. However,

the correlating equation is of the same form and does not effect the general conclusions drawn.

Therefore, this correlation should be used with caution for large aspect ratios.

95N = 5.2Po13

(DT

)2(HT

) 12

(1)

Equation 1 can be rearranged using the definition of the power number to give equation 2, where

is the liquid density, V is the liquid volume in the tank given by equation 3 where is the

ratio of a dished base height to the tank diameter for a half ellipsoid base (this is very similar to atorispherical base but simpler to consider, 0 for a flat base to 0.5 for a hemispherical base).

95 = 5.64(

PV

) 13 (DT

) 13T

23

(HT

) 12(

HT

3

) 13(2)

V =pi

4T 3

(HT

3

)(3)

Equation 2, which only applies to the turbulent regime reveals that:

Maintaining constant tank dimensions, agitators of the same diameter have the same mixing

time if they have the same power input,

Increasing the agitator diameter will decrease the mixing time at constant power input,

Increasing the fluid hight will increase the mixing time,

The mixing time is independent of the fluid viscosity at a constant power input,

At a constant power per unit mass the mixing time is slightly longer in a dished base tank

compared to a flat bottom vessel,

4


When scaling-up at constant power per unit mass and geometry the mixing time will increase

with the tank diameter.

The boundary between the turbulent and the transitional regime can be given by equation 4.3

Blending in the transitional regime is not independent of the Reynolds number like the turbulent

regime, therefore equation 5 has been proposed.3 The blending is much slower in the transitional

regime and the variability on equation 5 are much larger than for the turbulent regime, this is due

to the uncertainties in the transitional regime.

Po13 ReT T = 6370 (4)

95N = 33489Po23 Re1

(DT

)2(5)

As equation 1 was rearranged in terms of the power, equation 5 can be rearranged to become

equation 6.

95 = 39340.7(

1

3

) 23 ( PV

) 23 (

)(DT

) 23T

23 (6)

Equation 6 reveals that:

Agitators of the same diameter have the same mixing time if they have the same power input

in the same fluid (if in a Newtonian fluid),

Increasing the agitator diameter will decrease the mixing time at constant power input,

The mixing time is proportional to the fluid viscosity and inversely proportional to the fluid

density at a constant power input,

At a constant power per unit mass the mixing time is longer in a dished base tank compared

to a flat bottom vessel (over 10 % longer for a hemispherical base),

When scaling-up at constant power per unit mass and geometry the mixing time will decrease

with the tank diameter.

5


The first two conclusions are the same as in the turbulent regime. The third is different and

shows the dependence of the mixing time on the fluid that is being mixed. The fourth is similar to

the turbulent regime, apart from the effect of a dished base is more pronounced. The fifth is the

opposite to what happens in the turbulent regime, this is due to when scaling up at constant power

per unit mass, the Reynolds number increases and the mixing time is inversely proportional to the

Reynolds number. It is therefore critical to know what regime you are working in when you are

trying to scale a process.

The boundary between the transitional and the laminar regime can be given by equation 7.3

Po13 ReT L = 183 (7)

Some correlations are available for the calculation of the blending time in the laminar regime, but

these are only for specific agitator systems; e.g. anchor type5 or ribbon type;6 or dont offer a

transferable correlation away from the system used.7

Most of the work upon which the correlations 1 and 5 are based rely on blending of fluids

with similar viscosity. For fluids of different viscosities these correlations will not always be

appropriate, as for some conditions mixing times can be much longer than those predicted. For the

blending of fluids of different viscosities the modified Reynolds number, Re, has been proposed

to characterize the different flow regimes,8 equation 8.

Re = Re(b

a

)=

bND2a

(8)

This has lead to the determination of two possible regions;

Stirrer controlled regime (Re > 100) where the mixing time does not depend on the added liq-uid properties, the dimensionless mixing time for a given system is constant, and equation 1

predicts the mixing time

Added liquid controlled regime (Re < 100) where the viscosity of the added liquid and the ag-

6


itation speed effect the mixing and where the added liquid can maybe said to be dissolving,

rather than simply mixing.

For the blending of fluids of different density the modified Richardson number, Ri, has been

suggested, in a similar manner to the modified Reynolds number,9 but this also takes into account

the added volume and not just the density difference, equation 9.

Ri = Ri(

VaVb

)=

(ab)gHVabN2D2Vb

(9)

Again two main regions have been defined;

Stirrer controlled regime (Ri < 0.025) where the mixing time does not depend on the addedliquid properties and equation 1 can be used.

Added liquid controlled regime (Ri > 0.05) where the density and volume of the added liquidand the agitation speed effect the mixing.

The range between Ri = 0.0250.05 seems to depend on the agitator as different authors have

obtained different critical values.9,10 This seems to indicate that there are other factors that need to

be taken into account.

There is some evidence11 that when mixing a tracer of a high viscosity that a smaller agitator

gives a smaller dimensionless mixing time (by several orders of magnitude) when run at the samepower as a larger agitator. This is the converse to what we would expect for blending a brine tracer

into water and from equations 1 and 5.

Table 1 shows a comparison of the pumping rate and the shear rate of a 45 4 blade pitch

blade turbine for two different diameters with equal power input, based on data available in Hem-

rajani and Tatterson (2004).12 Although the pumping rate is lower for the smaller agitator, theshear rate is higher. No matter what concept is used to define the shear in the agitator region, the

smaller agitator produces more shear at an equal P/V due to its greater speed. Therefore, when

mixing a tracer with high viscosity, where dispersion effects dominate, an agitator that gives a high

7


shear rate will perform better. The opposite occurs in blending operations where pumping rate

determines the mixing time.

Table 1: Comparison of pumping rate and shear rate for a 45 4 blade pitch blade turbine. Powernumbers and Flow numbers from Hemrajani and Tatterson (2004).12

Size T / m D / m N / rpm Po P / W NQ Q / m3s1 1/ s1 2/ s1T2 0.91 0.46 300.0 1.27 3165.0 0.61 0.291 2.29 750.0T3 0.91 0.30 589.7 1.27 3165.0 0.81 0.225 3.00 1474.21Value based on ND2Value based on 150N 12

It has also been noted that at lower speeds tracers of higher viscosity and density can settle

on the bottom of the tank.13 This means that the mixing time for viscous tracers will be vastly

different for tracers that are lifted by the agitator and those that are not. This critical agitation

speed, NCD, might be predicted by equation 10,14 where the mean density and viscosity are given

by equations 11 and 12. The very small exponents on the viscosity and surface tension show that

the NCD is fairly insensitive to these physical properties. Note the similarity to the Zwietering

equation (equation 13)15 with different parameter relating to solids rather than a second liquid.

NCD = Sg0.410.410.09m

D0.7250.545m 0.045 0.05 (10)

m = a +(1)b (11)

m =b

1(

1+ 1.5aa + b)

(12)

NJS = Sg0.450.450.1b

D0.850.55bd0.2p X0.13p (13)

Most previous work has either looked at fluids of similar viscosity in a range of regimes, or

looked at different viscosities with both materials having Newtonian like behaviour. From these

studies two regimes have been seen for solutions of different viscosities, the stirrer controlled

8


regime and the added liquid controlled regime. This work uses this analysis to examine the mixing

of a viscous shear thinning surfactant when added to water using a typical high shear agitator.

Methods and Materials

Equipment

Two vessels were used in the experiments, a 0.914 m (3 ft) and a 0.152 m (6 inch) diameter vessels.The 0.914 m vessel is a Perspex cylindrical vessel with a shallow dished base and 4 standard baffles

of width T/10. The cylindrical section is 1.5 m high and is fitted inside a square jacket throughwhich water can be circulated for temperature control. The square jacket provides distortion freeviewing windows for flow visualisation. The agitator rotational speeds are monitored using a

Ferro-magnetic proximity sensor coupled to a COMPACT MICRO 48 tachometer. The shaft is

continuous and fits into a PTFE bearing located in the centre of the dished base. The agitator

power is measured using a dual torque strain gauge system bonded to the shaft, designed to cancel

out bending moments. The strain gauge is wired to an ASTECH Strain Gauge Bridge fitted with

a telemetry readout, which is connected to a LABVIEW data acquisition unit. The temperature

of the vessel contents is monitored using a Pico PT100 temperature probe inserted through an

analysis block. The analysis block runs all the way up the back of the vessel allowing probes to

be inserted through 3/4 inch BSP tappings. Through another of these ports the inlet tube for the

salt injections into the agitator region was inserted. The vessel is also fitted with 8 rings of 16equally spaced EIT electrodes in a baffle cage configuration which are connected to a ITS P2000

tomography measurement system. Figure 2 shows a schematic of this vessel.

The agitator used is depicted in Figure 3 (a): this is a 0.305 m diameter Cowles disk with 32angled teeth (16 up and 16 down) which was rotated clockwise (angled edge first).

The 0.152 m vessel is a geometrically scaled down version of the 0.914 m vessel, without the

square jacket. The two agitators used in this vessel are depicted in Figure 3 (b) and Figure 3 (c):these are a 0.0956 m diameter Torrance TF/150 saw-tooth disk with 24 angled teeth (12 up and 12

9


Figure 2: Schematic of the 0.914 m vessel.

down) and a 0.051 m diameter Esco-Labor V4A high shear disk with 12 angled teeth (6 up and 6down), both were rotated clockwise (angled edge first). These agitators are the closest to Cowlesdisks that were available to us at the necessary scale.

(a) (b) (c)

Figure 3: Agitators used: (a) 0.305 m Cowles disk (b) 0.0956 m Torrance TF/150 saw-toothagitator (c) 0.051 m Esco-Labor V4A high shear disk.

Methods

Experiments were carried out on the 0.914 m tank using both 70 w/w% sodium lauryl sulphate

(SLES) in water (TEXAPON N701 supplied by Unilever, density of 1050 kg m3) and a brine

10


solution (100 g table salt dissolved per liter of tap water). The brine solution was added to thevessel in 100 ml aliquots either on the surface, using a funnel placed less than 1 cm from the

surface in a fixed position 2/3 of the radius from the agitator shaft and between two baffles; or

injected into the outlet flow of the agitator, using a pump. The energy input of both additions wassmall compared to that given by the agitator. The SLES was added to produce a final solution of

0.5 w/w% (5.73 kg for H/T = 1.42 (800 l) and 3.93 kg for H/T = 1 (550 l)) to the surface of thevessel in large lumps (approximately 0.1 m diameter) in under 30 seconds.

Experiments were carried out on the 0.152 m tank using 70 w/w% SLES in water and the brine

solution. The brine solution was added to the vessel in 2 ml aliquots on the surface, using a small

syringe at less than 1 cm from the surface in a fixed position 2/3 of the radius from the agitator

shaft and between two baffles. The SLES was added to produce a final solution of 0.5 w/w% (2.6 gfor H/T = 1.42 (3.7 l) and 1.8 g for H/T = 1 (2.5 l)) to the surface of the vessel either by a syringe(mouth radius of 1.5 mm) or in a single lump. The SLES was also added in twice the volume forthe double SLES runs. The energy input of the additions was small compared to that given by the

agitator. The experiments were repeated 5 times to produce an average value.

Electrical impedance tomography (EIT) is commonly used to monitor processes which requiregood temporal and spacial resolution. It has been used to measure mixing times in industrial sin-

gle phase and multi-phase systems.16 The ITS P2000 was chosen for the experiments as it is the

best performing EIT instrument for experiments requiring high temporal resolution and the abil-

ity of successfully monitoring homogeneity.17 Table 2 summarises the data acquisition settings

employed during the experiments. The signal-to-noise ratio (SNR) was checked to ensure that novoltage measurements were saturating the analogue to digital converter. If this occurs the recorded

voltages read a constant value for the given reading thus making the image reconstruction unre-

liable for calculating the conductivities inside the vessel. Therefore a SNR over 100 dB should

be avoided. The optimum injection current was found to be 50 mA resulting in a SNR ratio ofapproximately 52 dB for water at the start of the experiment.

The turbulent power number was calculated on the 0.914 m vessel using the calibrated dual

11


Table 2: ITS P2000 data acquisition settings.

Setting ValueExcitation frequency 9.6 kHzSampling time interval 40 msSamples per frame 1Frames per download Equal to the maximum number of framesDelay cycles 3Injection current 50 mASampling strategy Normal adjacent

strain gauge system mentioned above for the Cowles disk agitator. The power number of the

Torrance agitator was calculated over a range of Reynolds numbers using the same strain gauge

system on a smaller vessel. The low Reynolds number values were produced using different vis-

cosity glucose (Cerestar 01632) syrup solutions (the viscosity was varied by temperature change).This method was also repeated using the Torrance agitator on the shear thinning 2 w/w% Carbopol

940 solution (BF Goodrich). This allowed calculation of the Metzner-Otto constant as outlinedlater.

The viscosity of the solutions were measured using a Haake RV 20 viscometer with a MV2 cup

and bob modality. The shear profile is set so that there is an increase from 0.1 to 450 s1 in steps,

and then decreases back down to 0.1 s1. Each decade of shear rate has 5 even steps each for 2

minutes, with the 100 to 450 s1 having 3 even steps again each for 2 minutes. The temperature is

controlled using a water circulation jacket.As both the solutions were power-law fluids, and in fact shear thinning, they can be fitted to

the standard power-law viscosity relationship, equation 14. This gave the values given by Table 3.

As a sense of the magnitude of the viscosities of the solutions, at a shear rate of 1 s1 the viscosity

of the 70 w/w% SLES is 34.1 Pa s and that of the 2 w/w% Carbopol is 180 Pa s.

= kn1 (14)

12


Table 3: Properties of the non-Newtonian fluids at 25 C.

Solution n k/(Pa s)n70 w/w% SLES 0.412 34.1

2 w/w% Carbopol 0.134 180

Data Processing

The voltage measurements collected by the ITS P2000 were reconstructed into a true-3D image

using the generalised singular value decomposition (GSVD) method18 based on a finite elementmodel comprising 7344 elements. This approach decomposes the image into individual spatial

frequency components and affords the ability to control the number of generalised singular values

incorporated into the solution. The inclusion of a low number of singular values in the solution

yields an image with lower spatial resolution but which is robust to measurement noise. Con-

versely, the inclusion of a high number of singular values yields an image with potentially higher

spatial resolution which is less robust to measurement noise. The regularisation parameter in the

algorithm was selected frame by frame using the Discrete Picard Condition19 to produce the re-

sults with the most information. The Discrete Picard Condition compares the generalised singular

values (representing the change in the data) with the Picard coefficients (representing the noisein the data), and gives the value where they are equal. Values of the generalised singular valuesgreater than the Picard coefficients contain recoverable data and should be utilised, which occurs

if the regularisation parameter is set to this equality.

The values of the conductivity for the 7344 finite elements were combined together using the

root mean squared conductivity to produce an analysis of the mixing time, equation 15.

ln [RMS] =12

ln[

1N

N

i=1

(t,i0,i

inf,i0,i1

)2](15)

In equation 15, RMS is the root mean squared conductivity, N is the number of elements, t,i is

the conductivity of element i at the given time step, 0,i is the initial mean conductivity for element

i and inf,i is the final mean conductivity for element i. This approach gives a good calculation of

13


the mixing time as it weights the answer towards the elements that take the longest to mix.

The vessel can be considered 90 % mixed if the value of RMS drops below 2.3, which is

equivalent to (1/2) ln[(0.91)2

]. The 90 % mixing time is then the time at which the value never

becomes greater than2.3 again. The 90 % mixing time is used as there is about 8 % noise on the

final mean conductivity.

As the whole vessel is in the turbulent regime then this time can be scaled to a 95 % mixing

time for comparison with literature correlations by use of equation 16.

95 = 90ln [10.95]ln [10.9] (16)

Results and Analysis

Power Numbers and Metzner-Otto Constants

The power of an agitator can be calculated from the definition of the power number or from the

torque. Rearranging these definitions for the torque gives equation 17. In the turbulent regime,

the power number is a constant which means that the torque is proportional to the agitation speed

squared, as shown in Figure 4.

= PobD5

2piN2 (17)

From this method it can be determined that the Power number for the Cowles disk agitator in

the turbulent regime is 0.52 and that of the Torrance agitator is 0.21.

Figure 5 presents the values the power number for the Cowles disk and Torrance agitators over

a range of the Reynolds number. It can be seen that in the laminar regime both agitators have

a constant value of PoRe, which in this case is equal to 65 for the Cowles disk and 55.7 for the

Torrance agitator.

As the addition material is shear-thinning, it is important to know the shear rate so that the

14


0 5 10 15 20 25 30N2 / s-2

0

1

2

3

4

5

6

7

/ N

m

= 0.219N2R2 = 0.999

Figure 4: Plot of the torque, , against the agitation speed squared, N2, for a Cowles disk agitatorin water. The slope of the line is proportional to the power number, Po.

100 102 104 106Re or Repl

0.01

1

100

Po

Torrance NewtonianTorrance non-NewtonianCowles NewtonianCowles non-Newtonian

PoRe = 55.7

Po = 0.21

PoRe = 65

Po = 0.52

PoRepl = 5.8PoRepl = 10

Figure 5: Plot of the Power number against the agitator Reynolds number for the Cowles disk andTorrance agitators (Figure 3 (a), (b)). Laminar Newtonian and non-Newtonian data for the Cowlesdisk are taken from Foucault et al. (2005).20

15


apparent viscosity can be calculated. The average shear rate in the agitator region can be given by

the Metzner-Otto constant, Ks, multiplied by the agitation speed,21 equation 18.

= KsN (18)

In the case of a power-law fluid a power-law Reynolds number, Repl, can be defined as equa-

tion 19.22 The Power number for the Cowles disk and Torrance agitators in shear thinning fluids

over a range of power-law Reynolds numbers are shown in Figure 5.

Repl =Nn2D2

k (19)

In the laminar regime both agitators also have a constant value of PoRepl, which in this case is

equal to 10 for the Cowles disk and 5.8 for the Torrance agitator. As the true Reynolds number of

the agitators in the laminar regime is given by equation 20, rather than equation 19, then a simple

comparison of the laminar power constants for the non-Newtonian and the Newtonian fluids yields

the Metzner-Otto constant, as in equation 21.

Re =Nn2D2

kKn1s(20)

Ks =(

PoReplPoRe

) 1n1

(21)

Using the viscosity parameters for the Carbopol (Table 3) and knowing that the shear thinningfluid used for the Cowles disk non-Newtonian power measurements by Foucault et al. (2005)20

was 1.5 w/w% carboxymethyl cellulose solution with k = 60 (Pa s)0.22 and n = 0.22 then the

Metzner-Otto constants can be calculated to be 11 for the Cowles disk and 13.7 for the Torrence

agitator.

16


Same Viscosity Liquid-Liquid Blending Times

This section describes the blending of a brine tracer over a series of different agitation speeds in the

0.914 and 0.152 m vessels. Figure 6 shows an example of the reconstructed tomograms for a brine

tracer added to the 0.914 m vessel at an agitation rate of 100 rpm. For these three-dimensional

images, the colour scales are constant across all images, the minimum value used is the starting

conductivity of the tap water (0.01 S m1) and the maximum value used is the maximum of theglobal mean conductivity trajectory (0.0112 S m1 for the 100 rpm run). Four isosurfaces havebeen selected, at one-quarter intervals across the conductivity range starting at the first quartile

ending at the maximum of the colour scale, i.e. 25 %, 50 %, 75 % and 100 % of the range. The

opacity of the isosurfaces has been graded such that the lower conductivity isosurfaces are more

transparent than those of higher conductivity, allowing a clear visualisation.

Figure 6 (a) shows the initial homogeneity of the vessel at its initial conductivity of 0.01 S m1

and (b) shows the start of the feed of the higher conductivity brine feed (11.5 S m1). Figure 6 (c)is 4 seconds into the mixing and it can be seen that the high conductivity tracer has moved down

the vessel and out towards the wall due to the flow of the agitator. As the tracer hits the wall it

starts to move up and down the vessel walls forming a highly conducting area at the front of the

vessel, Figure 6 (d).By 16 seconds into the mixing, Figure 6 (e), the highly conducting area is still at the front the

vessel but starting to tangentially swirl in the direction of the agitator. This shows that the Cowles

disk gives only a small amount of tangential flow, with the main flow from the disk being limited

to the radial direction. After 32 seconds of mixing, Figure 6 (f), the most unmixed areas of thevessel are towards the back, away from the initial feed position. This is still the case after 60

seconds, Figure 6 (g), but by this time there is only a small area behind the back baffle that has asignificantly different concentration of tracer.

Figure 6 (h) shows the final homogeneous end point of the mixing, showing a conductivityof about 0.0102 S m1, it can be seen that the end point is quite homogeneous, with only small

amounts of variation.

17

Tho

masL

.Rodg

erset

al.

Cow

lesDisk

Dissolutio

n

(a) Before (b) Feed (c) 4 s (d) 8 s

(e) 16 s (f) 32 s (g) 60 s (h) 100 s

Figure 6: Tomography images for a surface feed salt tracer in the 0.914 m vessel at 100 rpm with 95 = 84 s (see Figure 7; (a) beforethe feed, (b) initial feed point, (c) 4 s into the mixing, (d) 8 s into the mixing, (e) 16 s into the mixing, (f) 32 s into the mixing, (g) 60 sinto mixing, (h) homogeneous end point (100 s).

18


Figure 7 shows the evolution of ln [RMS] over the mixing times for an example of surface

brine blending in the 0.914 m vessel. The pre addition value can be seen to around 0 with small

variation, this corresponds to the inital conductivity of 0.01 S m1. When the trace is less than

2.3 this corresponds to the final conductivity of about 0.0102 S m1. It is clear from Figure 7

that the mixing seems to occur in two stages, the first as a decay in the average conductivity, and

the second as secondary loop which reinforces the areas of high conductivity from the first decay.

It can also be clearly seen that the lower agitation speed takes much longer to mix and that the

average conductivity stays above the final value for much longer. Using Figure 7 the mixing time

0 10 20 30 40 50 60 70 80time (t) / s

-8

-6

-4

-2

0

2

ln(

RM

S)

100 rpm200 rpm328 rpm90% mixed

Figure 7: Example traces for the surface brine mixing in the 0.914 m vessel. The arrows show thetime at which the vessel can be considered to be 90 % mixed, the vertical line shows the time ofaddition.

can be calculated. The arrows show the time at which the system can be considered mixed and

the vertical line shows the time of addition, this means the mixing time is the difference. Figure 8

shows the mixing times for the brine tracers in the 0.914 and 0.152 m vessels for H/T = 1 and

1.42 and with both a surface and agitator feed for the 0.914 m vessel.

It can be seen from Figure 8 that there is little difference between the mixing time constant

19


104 105 106Re

0

5

10

15

20

Nt95

Po1/

3 (D/T

)2 (H

/T)-1

/2

0.914m H=1.42T (surface feed)0.914m H=1.42T (agitator feed)0.914m H=T0.152m H=1.42T0.152m H=T

Figure 8: Mixing time for brine tracers in the 0.914 m and 0.152 m vessels. The error bars showthe standard deviation of the results, and the horizontal line shows the mixing time constant in themixing correlation of Grenville and Nienow (2004)3 (5.2).

between the two heights in the 0.152 m vessel. Also the mixing time constant is very similar

to the value predicted by Grenville and Nienow(2004).3 However, the mixing time constant forthe 0.914 m vessel does not seem to be the same for the two liquid heights. For H/T = 1 the

mixing time constant is similar to the value predicted by Grenville and Nienow (2004),3 thoughslightly higher. This is possibly due to the turbulent eddies scaling with agitator size, thus taking

longer for the energy to dissipate into small eddies for the micro mixing with a larger agitator.23

For H/T = 1.42 the mixing time constant is almost twice that of H/T = 1. This could be due

to poor circulation in the large vessel from the Cowles disk, whereas this is not such a problem

in the smaller vessel. It can also be seen that surface feed takes slightly longer than the agitator

feed. This shows that the flow from the Cowles disk is weakest towards the surface of the vessel,

delaying the homogenation of the fluid in the vessel.

20


SLES Dissolution

This section describes the dissolution of SLES in the 0.914 and 0.152 m vessels and compares this

with the brine mixing times for different methods of addition at different liquid heights. Figure 9

shows the results for the dissolution times for the SLES addition with a syringe at H/T = 1 and

1.42 and with agitators of D/T = 1/3 and 0.62 in the 0.152 m vessel. It also shows the times for

the addition of SLES in one lump for agitators of D/T = 1/3 and 0.62 in the 0.152 m vessel and

the addition of SLES in lumps for agitator of D/T = 1/3.

102 103 104 105 106

Re100

101

102

103

104

Nt95

Po1/

3 (D/T

)2 (H

/T)-1

/2

0.914m SLES D/T=1/30.152m SLES lump D/T=0.620.152m SLES lump D/T=1/30.152m SLES syringe D/T=0.620.152m SLES syringe D/T=1/30.914m brine H=1.42T0.914m brine H=T0.152m brine

Figure 9: Comparison of mixing and dissolution times for salt tracers and SLES in both the 0.914and 0.152 m vessels. For the 0.152 m brine and the SLES additions, both the liquid heights areshown together as there is no discernible difference in the results. The black lines act as a guide tothe eye.

It can be seen from Figure 9 that when the SLES is added by syringe changing the liquid

height or the agitator speed does not affect the mixing time constant appreciably. However, the

agitator size does seem to affect the mixing time constant in a way that would not be expected for a

21


mixing mechanism dominated process, i.e the larger agitator seems to mix slower than the smaller

agitator. This demonstrates that when the SLES is in small pieces, as introduced by the syringe,

the dissolution of these pieces controls the mixing process as the mixing time constant is around

10 times higher than the brine mixing in the same vessel.

If the SLES is added as one lump then the mixing time constant is no longer constant with

agitation speed and seems to be affected by the agitator size, which suggests that it is not a mixing

mechanism. Also the smaller agitator gives a faster dissolution time, especially at lower agitation

rates, which would not be expected for a mixing process. This suggests that the dissolution is the

dominant effect and that increasing the agitation rate helps to increase this. At the higher agitation

rates the larger agitator produces mixing times which are almost as quick as the syringe addition

times.

Figure 9 also shows the dissolution time constant for the SLES addition into the 0.914 m vessel.

It can be seen that the dissolution time constant is greatly affected by the agitation rate, decreasing

by an order of magnitude when the agitation rate is increased by a factor of 3. We suggest that

the large lumps of SLES added are difficult to break up with the low shear rates of the slower

agitation speeds. Also the lower agitation rates have problems suspending the larger lumps of

SLES meaning that some sit on the base of the vessel diminishing the dissolution rates even more.

For dissolution of the SLES the change in liquid height does not affect the dissolution time as

the blending time is at least an order of magnitude less than the dissolution time, therefore the bulk

can almost be taken as perfectly mixed from the view point of the SLES.

Effect of the Modified Reynolds Number

One of the main issues with SLES, and other surfactants, is that the viscosity of the material varies

with the shear rate that it experiences. This means that an average value of the SLES viscosity can

be calculated for each agitation speed inside of the vessels if the shear rate is known within the

vessel. However, calculation of the shear rate in an agitated vessel is not a clear concept, especially

as it varies throughout the vessel.

22


In the turbulent regime, the turbulent stress can be calculated throughout the vessel which

could be used with a rheological equation to calculate the local viscosity of the material throughout

the vessel. This would provide a very complicated view of the process and would be difficult to

correlate as what average viscosity do you take. However, from observation of the breakup process,

the main breakup of the SLES occurs in the agitator region and little occurs elsewhere in the vessel.

This leads to the thought of using a local viscosity in the agitator region as this is where the main

processes happens. Figure 10(a) shows the 70 w/w% SLES approaching the agitator but with noreal break up, whereas Figure 10(b) shows the SLES right in the agitator region, with small piecesbeing sheared from it. This is typical of the whole process and reveals that this is not an emulsion.

(a) Before contact (b) At contact

Figure 10: Images of SLES in the agitator region. (a) SLES approaching the agitator, (b) Smallpieces of SLES being sheared from the main lump in the agitator region (within circle).

A convenient and commonly used method for determining the average viscosity in this agitator

region is by means of the Metzner-Otto equation, equation 18, and a rheological equation, equa-

tion 14. The Metzner-Otto constants are determined from measurements in the laminar regime and

are only strictly applicable in this regime. However, this approach is valid as the substance which

the Metzner-Otto constant is used on (SLES) gives a laminar Reynolds number and both the powerand mixing are independent of viscosity in the fully turbulent regime (i.e. the bulk water).

The modified Reynolds number (Re) is calculated from equation 8 using this viscosity. Thisnow allows the dissolution times to be plotted against the modified Reynolds number as shown by

Figure 11. From Figure 11 it can be seen that the trends do not change from Figure 9. The slope

23


100 102 104 106

Re*100

101

102

103

104

Nt95

Po1/

3 (D/T

)2 (H

/T)-1

/2

0.914m SLES D/T=1/30.152m SLES lump D/T=0.620.152m SLES lump D/T=1/30.152m SLES syringe D/T=0.620.152m SLES syringe D/T=1/30.914m brine H=1.42T0.914m brine H=T0.152m brine

Figure 11: Comparison of mixing and dissolution times for salt tracers and SLES in both the 6 inchand 3 ft vessels against the modified Reynolds number, shown with best fit lines. For the 0.152 mbrine and the SLES additions, both the liquid heights are shown together as there is no discernibledifference in the results.

24


of the curve in Figure 11 seem to scale well the agitator size. The equations of the SLES added

as lumps to a concentration of 0.5 w/w% all fit to the general equation 22 with values given in

Table 4. [Nt95Po

13

(DT

)2(HT

) 12]= A0ReA1 (22)

A1 represents the slope of the curve in Figure 11 and its dependence with the diameter of the

Table 4: Constants for equation 22.

D / m A0 A10.0508 820.30 -0.580.0956 5212.22 -0.970.3048 2.991011 -4.43

agitator is given by equation 23. A0 varies exponentially with the agitator diameter, as described

by equation 24. The fact that these two equations are dependant on the agitator diameter and not

the ratio of agitator to tank diameter shows the dependency of scale on this operation.

A1 = 0.3515.57D (23)

A0 = 6.4185e80.04D (24)

Being able to predict the values of A0 and A1 allows the dissolution time constant to be predicted.

Effect of Tracer Volume

Figure 12 shows that the mixing time is different when the volume of SLES added is larger. When

added from the syringe the mixing time constant is independent of Re, but higher than that of

the 0.5 w/w% solution. Also if the SLES is added as a single lump, the mixing time constant

is similar between the 1 and 0.5 w/w% solutions for high agitation speeds, but diverges as the

agitation speed decreases. This could be an indication that the SLES is dissolving; because the

concentration difference is less between the 1 w/w% solution and the added SLES giving a lower

mass transfer rate, which is exacerbated if the SLES is added in a lump at a low agitation speed.

25


100 101 102

Re*

101

102

103

Nt95

Po1/

3 (D/T

)2 (H

/T)-1

/2

double lumpsingle lumpdouble syringesingle syringe

Figure 12: Comparison of dissolution times for SLES additions of different volumes in the 0.152 mvessel against the modified Reynolds number, shown with best fit lines (D/T = 1/3).

26


Also, there is more SLES to dissolve.

The use of the modified Reynolds number does not take into account the volume of the tracer,

which can be seen from the different trends in Figure 12. The modified Richardson number how-

ever does involve a volume term, and in fact a liquid height term as well. This means it might

be an important parameter in trying to predict the dissolution for a change in the tracer volume.

Figure 13 shows the variation of the modified Fourier number in the 0.152 m vessel for both liquid

heights and both added tracer amounts. The modified Fourier number represents a dimensionless

form of the mixing time and is given by equation 25.

Fo = Fo(ab

ba

)=

a95aT 2

(25)

It can be seen from Figure 13 that the use of the Richardson number seems to collapse the lump

1e-05 0.0001 0.001 0.01 0.1Ri*

0.1

1

10

100

1000

Fo*

single lump D/T=1/3double lump D/T=1/3single lump D/T=0.62single syringe D/T=1/3double syringe D/T=1/3single syringe D/T=0.62double syringe D/T=0.62

Figure 13: Comparison of dissolution times for SLES additions of different volumes in the 0.152 mvessel against the modified Richardson number.

addition and the syringe addition into two separate trends independent of tracer volume, agitator

27


size and liquid height. This shows that a volume term is needed in the correlation, and the modified

Richardson number could be this term. The two lines could be collapsed on to each other by looked

at the surface area to volume ratio, like in equation 26. For low agitation speeds the value of

seems to need to be about 1, where as at high agitation speeds the value of needs to be much less

than 1. This fits with the idea that the higher shear rates of higher agitation speeds split the lumps

up more creating less effect of the lump size.

Fo ( s

V

)Ria (26)

Discussion and Conclusions

Experiments have been undertaken in 0.914 and 0.152 m diameter vessels to look at the mixing

and dissolution times of brine and high viscosity material. Mixing times in the turbulent regime

for materials of similar viscosities can already be predicted for many different agitators, and this

work shows that the correlation can also be used for high shear disk agitators, however, with some

caution for larger vessels. Materials with large viscosity differences however take much longer

to mix which needs to be understood to help development of processes. The time taken for high

viscosity fluids to mix seems to be dependent on the operating conditions and the scale of operation.

Previous work with similar viscosity liquids has shown that for agitators of different D/T

increasing this value decreases the mixing time, however this work shows that for high viscosity

differences this is not the case with smaller values of D/T giving smaller dissolution times. It has

also been shown for 0.5 w/w% SLES that the dissolution time scales with the modified Reynolds

number according to equation 27.

Nt95 Re(0.3515.57D) (27)

28


It has also been shown that the volume of the tracer has an effect on the dissolution time of SLES,

with it having a greater effect at low agitation speeds for one lump. This helps to show that the

main effect is a dissolution effect and not a mixing effect. It has also been shown that the modified

Richardson number can take the dissolution times of the SLES of different volumes and collapse

these to one trend, producing equation 28, where the value of varies with agitation speed.

Fo ( s

V

)Ria (28)

Nomenclature

D Agitator diameter m

dp Particle diameter m

g Acceleration due to gravity = 9.81 m s2

H Liquid height m

n Consistency index (Pa s)n

Ks Metzner-Otto constant -

N Agitation speed s1

N Number of readings -

n Power-law index -

NCD Critical dispersion agitation speed s1

NJS Just suspended agitation speed s1

P Power W

Q Flow m3 s1

s Surface area m2

T Vessel diameter m

t Time s

V Volume m3

29


Xp Percent weight solids per weight of liquid %

Ratio of dished base height to tank diameter -

Torque N m

Shear rate s1

Viscosity Pa s

Volume fraction of dispersed phase - Density kg m3

Conductivity S m1

Surface tension N m1

RMS Root mean squared conductivity S m1

95 95 % mixing time s

Fo Modified Fourier number , see equation (25) -Fo Fourier number = b95/bT 2

NQ Flow number = Q/ND3

Po Agitator power number = P/N3D5

Re Modified agitator Reynolds number , see equation (8) -Repl Power-law Reynolds number = Nn2D2/k

Re Agitator Reynolds number = ND2/

Ri Modified Richardson number , see equation (9) -Ri Richardson number = (ab)gH/bN2D2

SNR Signal to noise ratio dB

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Acknowledgement

Tom Rodgers would like to thank The University of Manchesters EPSRC CTA (CollaborativeTraining Account) and Unilever for financial support during his PhD. The authors would like tothank the SCEAS workshop staff who helped with equipment modifications and construction.

33

Nomenclature

DiMixing and Dissolution Times for a Cowles Diskssolution Paper

Documents

dissolution mechanisms

dissolution rates

dissolution timesfor

miscible blending

viscous fluids

viscosity of water

miscible fluids

continuous phase