mixing

Comparison of a Networks-of-Zones Fluid Mixing

Model for a Baed Stirred Vessel with

Three-Dimensional Electrical Resistance

Tomography

T. L. Rodgers1, F. R. Siperstein1, R. Mann1, T. A. York2,

and A. Kowalski3

1School of Chemical Engineering and Analytical Science, University of Manchester,

M60 1QD, UK

2School of Electronic and Electrical Engineering, University of Manchester, M60

1QD, UK

3Unilever R & D Port Sunlight, Quarry Road East, Wirral, CH63 3JW, UK

Abstract. Reliable models for the simulation of mixing vessels are important for

the understanding of real life mixing problems. To achieve these models information

about the mixing in the system must be measured to compare with the predicted

values. Electrical resistance tomography has the capability to measure spacial and

temporal changes within a vessel in three-dimensions even in optically inaccessible

environments.

This paper discusses the creation of a network-of-zones model for the prediction of

mixing within a vessel with a Cowles disk type agitator. Solving of the network-of-zones

simplified transport equations for the vessel predicts the concentration distribution

of an inert tracer added to the vessel. The change in this distribution with time

is calculated and compared with visual inspection of the vessel. The concentration

corresponding author

E-mail address: [email protected] Deceased at time of writing

Comparison of NoZ with ERT 2

distribution inside the vessel is also measured using electrical resistance tomography

and shows good agreement with the predicted values.

Keywords: Tomography, Mixing, Network-of-Zones, Cowles Disk, Modelling

Submitted to: Meas. Sci. Technol.


1. Introduction

Reliable models for the simulation of mixing vessels are important for the understanding

of real life mixing problems. These models should have a balance between a very

realistic set of complex equations that take a high amount of computer power and a

more simplistic set that only require a small amount. The maximum efficiency for

calculations is given by a model that is as simple as possible, represents the physical

situation well, and involves the minimum number of adjustable parameters, which are

easy to evaluate numerically.

Commonly, computational fluid dynamics is used to evaluate and quantify fluid

mechanics, especially for baed vessels [1, 2]; the unbaed, heavily swirling case is less

well characterised [3]. These simulated results often agree very well with experimental

results. However, until recently computational fluid dynamics only provided information

on fluid mechanics, due to the heavy computational load. Mixing curves involving

the evolution of concentration fields and liquid-phase reactions using the reagent

concentration fields have started to be investigated [4, 5, 6].

The network-of-zones (NoZ) model requires far less computer power as it is a very

simple model, but still provides a realistic description of real processes. The model is

created by dividing the vessel volume into a number of perfectly mixed zones which

are connected to each other using convective, diffusional, and swirl flows. NoZ models

have been used to simulate one-phase liquid systems [7], reactive liquid systems [8],

precipitation reactions [9], gas-liquid systems [10], and solid-liquid systems [11].

Electrical resistance tomography (ERT) is one of the few measurement methods that

is able to investigate an environment fully when both significant spatial and temporal

variation in property distributions are coupled with optical inaccessibility. Currently

there is a large amount of work being carried out with ERT within mixing and in other


areas of research both within the process industries and outside [12, 13, 14, 15].

2. Methods

2.1. The Network-of-Zones Model

2.1.1. Model Design An example 3D network is shown in figure 1 which is for a

16 32 100 configuration of zones, comprising 51200 elements. In this configuration

there are 100 tangential elements and 16 16 axial/radial planes above and below the

impeller. The elements are designated positions with i, j and k coordinates representing

there radial, axial, and tangential positions respectively, where i N, i [1, n];

j N, j [1, 2n]; k N, k [1, m]. n is the number of radial zones and m

is the number of tangential zones. The number of radial zones can be any number,

but it is common to have an even number for symmetry: n 2N. The number of

tangential zones can also be any number but it is also common for this to be an even

number for symmetry; also this should be chosen to make the zones as cubic as possible:

m 2N, m 2pin.

The axial/radial plane is shown by figure 2; each zone in the network experiences

several different flows. The main one of these is the agitator circulation flow, q, which

flows out from the impeller in the radial direction to the vessel wall, where it evenly

splits into up and down flows, which loop around back to the agitator, as seen in figure 2.

Each zone also experiences an isotropic turbulence, which is represented by a fraction

of the agitator flow loop, , through the sides in the axial/radial plane not experiencing

the main flow loop. In the tangential direction each zone experiences a swirl motion

from the agitator, which is represented by a fraction of the agitator flow loop, L for

the clockwise swirl and R for the anti-clockwise swirl. For the circled zone in figure 2

these flows can be represented as in figure 3. In the agitator region these interchanges


Figure 1: A network for a 16 32 100 configuration with 51200 equal volume zones,

only the surface elements can be seen.

are intensified to a locally higher proportion of the agitator flow loop, by multiplying

by an additional factor A.

The transfer flows in figure 3 can be represented by the amounts given in equation 1.

The change in concentration of an element (circled zone in figure 2) is given by the sum

of the transfer flows, which can be simplified to equation 2 for an inert tracer, where CI


Figure 2: Zone assembly slice in the k plane showing the agitator circulation flow loop

(black) and the turbulent exchange (grey).

is the concentration of inert tracer in the cell and V is the volume of the cell.

(A) = qCI,i1,j,k

(B) = qCI,i,j,k

(C) = qCI,i,j1,k

(D) = qCI,i,j,k

(E) = qCI,i,j,k


C(i,j1,k)

C(i,j,k)

C(i,j+1,k)

C(i,j,k1)

C(i1,j,k)

C(i+1,j,k)

C(i,j,k+1)

(A)

(B)

(C)(D)

(E)

(F)

(G)

(H)(I)

(J)

Clockwise swirl flow

Agitator circulation loopAnti-clockwise swirl flowExchange flow

Figure 3: Mass flows for a general zone, with flows given by equation 1.

Vi,j,kdCI,i,j,k

dt= q

CI,i1,j,k

(1 + 2 + L + R)CI,i,j,k

+ (CI,i,j1,k + CI,i,j+1,k)

+LCI,j,k1

+RCI,i,j,k+1

(2)

A large number of NoZ simulations are carried out without the modelling of baes

in the vessel [10, 11, 16]. However, a large number of vessels used in production have

baes, which means that these should be modelled in the simulation. To do this the

swirl flow can be stopped at the bae and passed around the bae with the agitator

circulation flow rate, so the mass balance is satisfied. An extra set of zones will be left

behind the bae to represent the dead zone seen experimentally behind baes [17].

This will be linked to the other zones with the exchange flow rate as shown by figure 4.


Swirl Direction

Figure 4: Flow to represent a bae in the vessel, arrows refer to figure 3.

2.1.2. Determination of Parameters In a large number of NoZ simulations the

parameters used have been fitted so that the images look like the experimental images.

This can result in parameters and a model that works well for one particular situation

but cannot be applied easily to others. A better way is to try to produce a robust model

that can be applied to different situations with the parameters taken from experimental

data.

The total agitator circulation flow rate, Q, can be calculated by use of the total

flow number, NQT, the agitator rate, N , and the agitator diameter, D, as shown by

equation 3.

Q = NQTND3 (3)

However, this is the total flow from the agitator, but as can be seen from figure 2

this flow is assumed to be in loops. There are n/2 loops in each half of the vessel in

each 2D plane, and m pseudo 2D planes, this means that the agitator circulation flow

rate per loop, q, can be given by equation 4.

q =NQTND

3

nm(4)


The total flow number is the flow number calculated from taking into account

the whole flow from the agitator, i.e. the velocity profile integrated from the agitator

centre to the point of flow reversal. Kumaresan and Joshi (2006) [18] provide a

large number of total flow numbers for different pitch blade turbines and hydrofoils.

Costes and Couderc (1988) [19] use laser doppler anemometry to calculate the total

flow number for a Rushton turbine to be 3.4 0.4

As well as the agitator circulation flow loops, the exchange flow between these

loops, QEF, has been examined [20] and can be given by equation 5, where NEF is total

exchange flow number.

QEF = NEFND3 (5)

Again, this is the total exchange flow, but as can be seen from figure 2 this flow

is split between the zones. There are n zones exchanging between the loops in the

axial direction (same analysis is true in the radial direction) in each 2D plane, and m

pseudo 2D planes, this means that the exchange flow per element, q, can be given

by equation 6; which for simplification of the model, means that can be given by

equation 7.

q =NEFND

3

nm(6)

=NEFNQT

(7)

Vasconcelos et al. (1995) [20] experimentally determined that the total exchange

flow number for a Rushton turbine was (0.2360.009) (T/D). For a half tank diameter

agitator this means that the value of would be approximately 0.14 which agrees well

with the value predicted by Ying (1993) [16].

The swirl factors can be given by the same style of relationship as the exchange


factor, equations 8 and 9.

L =NSLNQT

(8)

R =NSRNQT

(9)

The flow multiplier around the agitator can be determined from the difference in

the magnitude of the velocity of the agitator to the rest of the vessel. This difference

can be a couple orders of magnitude, 10-100 [21].

Values predicted from these relationships for a Rushton turbine agree well with

values determined by fitting from Ying (1993)[16] [18, 19, 20]. Use of these values

provides data that fits well with experimental data, Figure 5.

Currently there is no data in the open literature that contains these same values for

a Cowles disk. Therefore, the values have been calculated by a best fit comparison with

the overall mixing time based on experimental data. These values have been compared

to the values for a Rushton turbine to help produce sensible values, i.e. it is known that

a Cowles disk has a lower flow number than a Rushton turbine and that a Cowles disk

0 10 20 30 40 50 60N / rpm

0

20

40

60

80

100

120

95

/ s

CorrelationERTNoZ

Figure 5: Comparison of NoZ model mixing time prediction for a Rushton turbine with

experimental data from Stephenson et al. (2005)[22] and the mixing time correlation

from Grenville and Nienow (2004)[23].


has very little overall swirl flow it is just the actual number that is experimentally

unknown. The values used for the Cowles disk model are given in Table 1 [24].

2.2. Electrical Resistance Tomography Measurements

2.2.1. Experimental Equipment and Procedures Experiments were carried out in a

0.914 m diameter (T ), shallow dished-based, stirred tank reactor. The vessel is a

Perspex cylindrical vessel with 4 standard baes of width w = T/10 and thickness

0.01 m. The cylindrical section is 1.5 m high and is fitted inside a square jacket through

which water can be circulated for temperature control. The square jacket provides

distortion free viewing 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, with a diameter of 0.05 m, is continuous and fits into a PTFE

bearing located in the centre of the dished base.

The vessel is fitted with a Cowles disk agitator (CD) with 32 angled teeth (16 up

and 16 down) of diameter DCD = T/3 and a clearance of c = T/3. The vessel was

filled to a liquid level of H = T . Figure 6 illustrates a schematic diagram of the vessel

arrangements.

A 100 ml aliquot of 50 g l1 salt solution (conductivity 11.5 S m1) was added to

Parameter Value

NQT 2.0

NEF 1.5

NSL 0.5

NSR 0.5

n 16

m 100

A 100

Table 1: Values of the parameters used in the NoZ simulations for a Cowles disk

agitator.


Figure 6: Schematic representation of the stirred tank reactor geometry for the

experiments.

the surface of the vessel, half way between 2 of the baes.

The vessel is equipped with an array of 128 steel electrodes, 5 cm high and 3 cm

wide, set in 8 rings of 16 equally spaced electrodes. The electrodes are mounted on a

bae cage, constructed of 16 thin wall-flush plastic ducts, each with 8 electrodes. The

bottom plane of electrodes sits 0.5 cm above the join between the cylindrical part of the

vessel and the dished base. To use the top plane of electrodes requires a fluid height of

1.1 m (H/T = 1.17). The bottom 6 planes of electrodes were used, giving good coverage

of the vessel as shown in Figure 7(a), as the seventh and eight planes were above the

liquid height.

2.2.2. Tomography Settings and Reconstruction The ITS P2000 was chosen for the

experiments presented in this paper as it is the best performing ERT instrument,

available to us, for experiments requiring high temporal resolution and is capable of

successfully monitoring homogeneity [25]. The excitation frequency is the frequency of

the alternating current injected into the electrodes, and was selected to be 9.6 kHz; using


lower frequencies slows the sampling rate which means that it is too slow for monitoring

the mixing, and using higher frequencies increase the noise in the data which means

good quality data cannot be collected. The sampling time interval is the time taken for

the ITS P2000 to take one plane of measurements, and was selected to be 40 ms; using

a longer sampling time means that it is too slow for monitoring the mixing, and using

a shorter sampling time means that the voltages are not collected over a long enough

time which means good quality data cannot be collected. The delay cycles is the time

between each cycle of measurements, i.e the sampling time interval times the number

of delay cycles, which was selected to be 3; using a larger number of delay cycles means

that the sampling is too slow for monitoring the mixing, using a smaller number means

good quality data cannot be collected as the equipment is trying to work to quickly and

can become saturated.

The number of samples per frame is selected to be 1; taking more samples per

frame allows averaging, but means that the sampling is too slow for monitoring the

mixing. The frames per download is set equal to the number of measurement frames,

which means that the data is not downloaded from the ITS P2000 box to the computer

until the end of the experiment; this allows the equipment to run slightly faster allowing

a faster sampling rate. The injection current was set to be 50 mA; if more current is

used the collected voltages can saturate the collectors meaning that good quality data

cannot be collected, and using a lower voltage means that the noise in the measurements

is proportionally larger also meaning that good quality data cannot be collected. The

sampling strategy used was the normal adjacent, which means that the input current is

applied over pairs of adjacent electrodes and the voltage measured on pairs of adjacent

electrodes on the same plane; this is currently the only strategy available with the ITS

P2000.


Table 2 summarises the optimal data acquisition settings employed during the

experiments and shows the sampling rate for the collection of one frame of data (i.e. all

six planes). Using these optimal data acquisition settings resulted in a signal-to-noise

ratio (SNR) of approximately 59 dB at a reactor charge conductivity of 0.01 S m1.

The SNR is the ratio of the mean of one voltage reading over a number of frames, n, to

the standard deviation of the reading, averaged over the number of readings in a frame,

equation 10. In this case the SNR was taken over 30 frames to produce a good average.

SNR = 20 log[ meanstandard deviation

]= 10 log

[n

ni=1 V

2i

(n

i=1 Vi)2 1

](10)

An accurate geometric model of the vessel, including the exact electrode positions,

was developed using constructive solid geometry (Figure 7(a)). The baes and the

agitator were also modelled as they strongly affect the measured voltage signals [26]

and so are requisite for the finite element model accuracy. Figure 8 shows the reference

set of voltage measurements for plane 3 (near the agitator) and a calculated set with

the baes and the agitator modelled and without. The average error for the model

with the modelled baes and agitator is 10.6 %, and it can be seen from Figure 8 that

Setting Value

Excitation frequency 9.6 kHz

Sampling time interval 40 ms

Delay cycles 3

Samples per frame 1

Frames per download Equal to the number of measurement frames

Injection current 50 mA

Sampling strategy Normal adjacent

Sampling rate 3.06 fps

Table 2: ITS P2000 data acquisition settings.


the shape of the voltages fit well. The average error for the model without the interior

modelled is 29.0 % and it can be seen from Figure 8 that the shape is not that close

to the measured voltages. The geometric model was meshed (using an advancing front

surface mesh and Delaunay techniques) to give 8198 tetrahedral elements, k, using the

Netgen mesh generator [27], as shown by Figure 7(b). The collected voltage data was

reconstructed using the generalised singular value decomposition (gsvd) algorithm [28]

based on the finite element model. 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. Conversely, the inclusion of a high number of singular values yields

(a) (b)

Figure 7: Representation of the stirred tank for the tomography reconstruction, (a)

electrode configuration, (the top planes of electrodes are not shown as they are above

the water level). (b) finite element model (only surface elements visible).


0 10 20 30 40 50 60 70 80 90 100Measurement

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0V

olta

ge /

V

Measured VoltageCalculated with fittingsCalculated without fittings

Figure 8: Measured voltages for plane 3 (near the agitator) compared to the calculated

voltages with and without the modelling of the interior fittings (i.e. baes and agitator).

an image with potentially higher spatial resolution which is less robust to measurement

noise.

In EIT, the aim of the problem is to calculate a new admittivity (conductivity for

collection of the real part of the voltage only) distribution from a change in the boundary

value voltage. The change in the admittivity from a reference state can be given by the

least square solution to a first order assumption between the change in measured voltages

to the change in admittivity, equation 11 [26], where J is the Jacobian matrix given as

the differential of the boundary voltage with respect to the admittivity distribution at a

reference state and L is the regularisation matrix (a matrix approximation to a partial

differential operator) [29].

= argmin

J V 2 + 2 L2 (11)


For the matrices in equation 11 there exists matrices UJ Rmk, UL R

kk with

UTJ UJ = I0, UTLUL = I0, and a nonsingular X R

kk which give equation 12.

UJ 00 UL

T JL

X =

0M 0

(12)

With = diag (1, . . . , k) Rkk and M = diag (1, . . . , k) R

kk such that

0 1 . . . k 1, 1 . . . k 0, and 2i +

2i = 1 for

i = 1, . . . , k. The solution to equation 11 can then be given by equation 13 [26]

with F = diag (f1, . . . , fk) Rkk with diagonal elements given by equation 14

= XF1UTJ V (13)

fi =(i/i)

2

(i/i)2 + 2

, i = 1, . . . , k (14)

The regularisation parameter in the algorithm, was determined frame by frame to

be the value obtained by an analysis of the Discrete Picard Condition [30]. The Discrete

Picard Condition compares the generalised singular values (representing the change in

the data, i/i) with the Picard coefficients (representing the noise in the data,uTJ,i V ),

and gives the value where they are equal. Values of the generalised singular values

greater than the Picard coefficients contain recoverable data and should be utilised,

which occurs if the regularisation parameter is set to this equality, equation 15.

=j+1j+1

where j = max[i], i

(ii

)1 uTJ,i V 1, i [1, , k] (15)

Figure 9(a) shows an example comparison of the generalised singular values with

the Picard coefficients; the point where they cross gives the value of the regularisation

parameter. Figure 9(b) shows the variation of the regularisation parameter with the


frame number; during the mixing the regularisation parameter increases as the sharp

changes in conductivity create more noise in the data. The automatic identification of

optimum algorithm parameters, based on measured data, is a novel development and

contrasts strongly with common practice where parameters are selected based on best

fit with known or anticipated solutions.

The data from the tracer addition can be compared to the NoZ model by placing a

high concentration of material in one zone of the NoZ model. The model is appropriate

if the mixing time, tracer images, and local tracer concentrations predicted are similar

to those measured experimentally.

For comparison, the ERT data collected needs to be converted into a salt

concentration. This was achieved by adding known concentrations of salt solution to the

ERT vessel and measuring the reconstructed conductivity, giving a simple correlation.

The fit of this correlation is shown in Figure 10, and shows that as the concentration

gets over about 0.2 g l1, a small change in concentration produces a large change in the

measured conductivity, and that at concentrations less than 0.001 g l1, the ITS P2000

struggles to differentiate the concentration change.

0 100 200 300 400 500 600 700Index

10-15

10-10

10-5

100

105

1010

1015

Mag

nitu

de

Generalised singular valuesPicard Coefficients

(a)

0 50 100 150 200 250 300 350 400 450 5000

0.0001

0.0002

0.0003

0.0004

0.0005

Reg

ulris

atio

n pa

ram

eter

0 100 200 300 400 500Frame number

(b)

Figure 9: Discrete Picard Analysis. (a) Comparison of the generalised singular values

with the Picard coefficients. (b) Change of the regularisation parameter with the frame

number.


0.01 0.012 0.014 0.016 0.018 0.02Conductivity / S m-1

0.001

0.01

0.1

1

Conc

entra

tion

/ g l-

1

C=0.820+0.124ln(-0.0087)

Figure 10: Comparison of ERT average measured conductivity with the concentration

of salt solution in the vessel.

3. Results and Analysis

Figure 11 shows the variation of the mixing time predicted by the NoZ model for the

Cowles disk compared to the times measured by ERT and the prediction of the mixing

time from the correlation by Grenville and Nienow (2004)[23]. It can be seen that the

NoZ and the ERT predict very similar values of the mixing time which are very similar

to the currently used correlation.

Figure 12 presents the global average conductivity trace for the ERT experiment

at 100 rpm and the resulting mass balance generated when 5 g of salt were added to

the vessel after 20 s (the 100ml tracer). From about 35 seconds the global average

conductivity trace is still varying, but the mass balance is producing a relatively constant

value. This is due to the fact that there are no extreme values of the conductivity being

measured so the correlation from Figure 10 is in the most reliable region (conductivities

from 0.0102 to 0.012 Sm1). The mass balance also takes into account the volume of


0 5 10 15 20 25 30 35 40N / s-1

0

5

10

15

20

95

/ sCorrelationERTNoZ prediction

Figure 11: Comparison of NoZ model mixing time prediction for a Cowles

disk with experimental data from ERT and the mixing time correlation from

Grenville and Nienow (2004)[23].

each finite element to work out the mass in each element from the element conductivity,

while the overall conductivity simply takes an average of the conductivity measurements,

ignoring the element volume even though they are of different volumes due to the

meshing technique used.

The unrealistic spikes in the total mass added (above 5 g) are probably due to

the ERT over predicting the size of the salt tracer plume initially. This is due to the

high localised values of conductivity created when the initial high conductivity tracer

is added. ERT struggles to accurately measure these sharp boundaries and ends up

smearing the boundaries, producing a wider plume. This result in a large area of

high concentration being calculated as the concentration is almost independent of the

conductivity at these high conductivities.

Figure 13 compares the concentration of salt predicted by the NoZ model to that

measured by ERT for 4 different points in the vessel. The location of these points is


(a) Average global conductivity (b) Total mass balance

Figure 12: The average global conductivity, (a), which when scaled using the

concentration calibration gives the total mass of salt in the vessel, (b).

shown in Figure 14. It can be seen from Figure 13 that the traces given by the NoZ

data agree well with that taken from the ERT data.

0

0.02

0.04

0.06

0.08

0.1

Conc

entra

tion

/ g l-

1

ERT dataNoZ data

0 10 20 30 40 50 60Time / s

0

0.02

0.04

0.06

0.08

Conc

entra

tion

/ g l-

1

0 10 20 30 40 50 60 70Time / s

Point 1 Point 2

Point 3 Point 4

Figure 13: Concentration profiles produced by NoZ simulation compared to those

produced by ERT at the four points in the vessel shown in figure 14. The dotted line is

the true vessel final concentration, and the dashed line is the addition time.


Figure 14: Position of the comparison points between the NoZ and the ERT

concentrations given in figure 13.

Figure 15 and Figure 16 show the images produced from the visualisation, the NoZ

model, the ERT data with the threshold taken as the final conductivity minus the final

experimental noise, and the ERT data concentration calibration. It can be seen that the

NoZ simulation gives a tracer plume that looks like the real case and the ERT data. The

only difference in the images is that the ERT data over predicts the size of the initial

plume (images for 3 and 5 s) and that the plume is slightly offset counter clockwise.

This over prediction is due to a smearing of the plume boundaries as discussed above,

and the slightly offset plume is due to a slight interference in the reconstruction due to

the front right bae as the addition tube in the experiments was not perfectly centred.

4. Future Work

This method could be extended by developing a correlation between the conductivity

and the concentration that varies with position in the vessel. This would reflect the

Compariso

nofNoZwith

ERT

23

(a) Before (b) Feed (c) 3 s (d) 5 s (e) 8 s (f) 11 s (g) 35 s (h) 60 s

Figure 15: Visualisation of the vessel for the addition of a brine tracer, from above - top row, and from the side - second row; and

the network-of-zones model, from above - third row, and from the side - bottom row. (a) Before addition, (b) Feed point, (c) 3 s ,

(d) 5 s, (e) 8 s, (f) 11 s, (g) 35 s, and (h) 60 s at 100 rpm.

Compariso

nofNoZwith

ERT

24

(a) Before (b) Feed (c) 3 s (d) 5 s (e) 8 s (f) 11 s (g) 35 s (h) 60 s

Figure 16: Tomography reconstruction for the addition of a brine tracer, from above - top row, and from the side - second row;

and the salt concentration prediction, from above - third row, and from the side - bottom row. (a) Before addition, (b) Feed point,

(c) 3 s , (d) 5 s, (e) 8 s, (f) 11 s, (g) 35 s, and (h) 60 s at 100 rpm.


fact that the sensitivity of the measurements are highest around the outside of the

vessel and lowest in the middle. Ideally, a correlation of this sort would not be needed

if a measurement strategy is used that has much more uniform sensitivity; however,

currently no such strategy has been developed for industrial sized mixing vessels.

The calculation of the total mass could be incorporated into the reconstruction

to iteratively fix the reconstruction based on the mass. However, there are several

methods that could be incorporated and extensive study would be required to asses

the best method (e.g. reduce all the conductivities evenly, reduce all the conductivities

less than the a given percentage of the plume, real-time re-mesh around each plume

estimation for a better reconstruction[31], etc.).

The reconstruction process carried out produces about 30 frames per second, which

with the current data collection rate would be able to be used as an on-line sensor

system, which could give the mass of material added during an addition.

5. Discussion and Conclusions

Using a NoZ model based on experimental data provides a more useful and robust model

which can be adapted to other situations, e.g. suspension of solids and emulsions[24].

The parameters used for the Cowles disk fit the experimental data well, and produce

images similar to the photos. Calibration of the ERT data to the salt concentration

gives concentration data that fits the NoZ model very well, and provides a good mass

balance on the amount of salt. The only problem is the slight over prediction of the

tracer plume size at high concentrations by the ERT measurements. This calculation of

the total mass could be incorporated into the reconstruction method to help drive the

reconstruction to a more accurate result.


Nomenclature

C Concentration

c Agitator clearance m

D Agitator diameter m

D Agitator diameter

H Liquid height m

i Radial element number

J Jacobian matrix

j Axial element number

k Tangential element number

k Number of finite elements

L Regularisation matrix

m Number of measurements

m Number of tangential elements

N Agitation rate

n Number of axial elements

n Number of frames

NEF Total exchange flow number

NQT Total flow number

q Agitator circulation flow

T Vessel diameter m

V Element volume

V Voltage V

w Bae width m

Turbulent flow factor


A Agitator turbulent flow multiplier

L Clockwise swirl flow factor

R Anti-clockwise swirl flow factor

Admittivity S m1

CD Cowles disk agitator

gsvd Generalised singular value decomposition

NoZ Network-of-zones

SNR Signal to noise ratio dB

Regularisation parameter

6. References

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7. Acknowledgements

Tom Rodgers would like to thank The University of Manchesters EPSRC CTA

(Collaborative Training Account) and Unilever for financial support during his PhD.

The authors would like to thank the SCEAS workshop staff who helped with equipment

modifications and construction.

IntroductionMethodsThe Network-of-Zones ModelModel DesignDetermination of Parameters

Electrical Resistance Tomography MeasurementsExperimental Equipment and ProceduresTomography Settings and Reconstruction

Results and AnalysisFuture WorkDiscussion and ConclusionsNomenclatureReferencesAcknowledgements

mixing

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