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School of Chemical and Mineral Engineering
CEMI479
Comparison between the Plitt model
and an artificial neural network in
predicting hydrocyclone separation
performance
Neil Zietsman
23379936
Supervisor: Mr. A.F. van der Merwe
North-West University
Potchefstroom Campus
Date of submission:
26 October 2015
School of Chemical and Minerals Engineering
Declaration| i
Declaration
L.N. Zietsman 23379936, hereby declare that:
the text and references of this study reflect the sources I have consulted and
sections with no source references are my own ideas, arguments and/or conclusions.
This declaration is for the report entitled CEMI479: Comparison between the Plitt model
and an artificial neural network in predicting hydrocyclone separation performance
submitted for the partial fulfilment of the requirements for the B.Eng. Chemical Engineering
degree at the North-West University, Potchefstroom Campus.
Signed at Potchefstroom on the day of ______ October 2015.
_______________________
L.N. Zietsman 23379936
School of Chemical and Minerals Engineering
Acknowledgements| ii
Acknowledgements
I would like to thank the following people for their help during the year with my project:
My God for giving me strength during the year to complete this project
Mr. A.F. van der Merwe, my study leader for his help and guidance.
Workshop personnel for their help with regard to the technical problems that occurred
during the course of the year.
Mrs. Sanet Botes for her help with ordering the needed items in the project
Miss. Sarita van Loggenberg, my colleague, who helped me perform the hydrocyclone
experiments.
Mr. Nico Lemmer for his help on the Malvern Mastersizer 2000
School of Chemical and Minerals Engineering
Abstract| iii
Abstract
The hydrocyclone is an invaluable process unit which is popular for its use in the mineral
processing industry. As all classifiers, the hydrocyclone is not capable of perfect separation.
The ability of the hydrocyclone to separate particles into the correct streams could be
represented by a curve, known as a partition curve.
Two important variables could be obtained from the partition curve – the cut size, d50c, and the
sharpness of separation. These two variables could be used to fully describe the separation
efficiency of the hydrocyclone. Optimal control of the hydrocyclone could be achieved if
accurate values of the d50c and sharpness of separation could be obtained.
Unfortunately, this is easier said than done. On-line instrumentation for direct analysis of these
variables are unheard of. Additionally, the complex flow inside the hydrocyclone makes it
impossible to determine these variables indirectly through first principle calculations. The
solution is inference sensors, which make use of easily measured variables, like the flowrate
and solids percentage to determine the d50c and sharpness of separation.
Two methods of inference sensoring was covered in this study, namely an empirical method
(Plitt model) and an artificial neural network.
The modified Plitt model was specifically used in this case where its fudging factors were
changed to fit experimental data. The Plitt model was only capable of predicting the d50c to a
certain extent, but failed to predict the sharpness of separation.
The artificial neural network was trained with the backpropagation algorithm. The more input
variables the artificial neural network had, the better its predicting capability became. The
addition of regularization and momentum terms further increased the prediction power of the
neural network.
Keywords: hydrocyclone; d50c; sharpness of separation; artificial neural network; Plitt model,
fine cut point, variable size spigot
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Attached documents| iv
Attached documents
Folder name File name Description
Experimental
error Experimental Error
Excel® spreadsheet containing the data
and calculations that were done to
determine the experimental error
Plitt model Plitt model
Excel® spreadsheet containing the
calculations performed on the experimental
data with the Plitt model
Artificial neural
networks
Neil'sANN.rev3-d50c -
Du;
Neil'sANN.rev3-d50c -
Du+phi+Q;
Neil'sANN.rev3-d50c -
Du+phi+Q+P+S;
Neil'sANN.rev3-m - Du;
Neil'sANN.rev3-m -
Du+phi+Q;
Neil'sANN.rev3-m -
Du+phi+Q+P+S
The macro enabled Excel® spreadsheets
contain the program with which the artificial
neural networks were trained and validated
Meetings Various files
This folder contains all the minutes and
agendas of each meeting in Microsoft
Word® format
Data
processing Data processing
Contains the Excel® spreadsheet with
which the data processing was done
MSDS MSDS – Silica flour This is a PDF document containing the
MSDS of silica flour
Gantt chart Gantt chart This folder contains a Gantt chart that is
both in PDF format and MS Project format
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Table of contents| v
Table of contents
Declaration ............................................................................................................................. i
Acknowledgements ................................................................................................................ii
Abstract................................................................................................................................. iii
Attached documents ............................................................................................................. iv
Table of contents .................................................................................................................. v
List of figures ....................................................................................................................... vii
List of tables .......................................................................................................................... ix
List of acronyms .................................................................................................................... x
List of symbols ...................................................................................................................... x
Chapter 1 - Introduction ........................................................................................................ 1
1.1 Background............................................................................................................. 1
1.2 Problem statement .................................................................................................. 1
1.3 Aim and objectives .................................................................................................. 2
1.3.1 Aim .................................................................................................................. 2
1.3.2 Objective .......................................................................................................... 2
1.3.3 Methodology .................................................................................................... 2
Chapter 2 - Literature study................................................................................................... 3
2.1 The hydrocyclone ................................................................................................... 3
2.2 Hydrocyclone control .............................................................................................. 6
2.2.1 Sensors used in hydrocyclone performance determination .............................. 6
2.3 Soft sensors ............................................................................................................ 7
2.3.1 Empirical models ............................................................................................. 8
2.3.2 Artificial neural networks ................................................................................ 10
Chapter 3 - Experimental procedure ................................................................................... 20
3.1 Overview ............................................................................................................... 20
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Table of contents| vi
3.2 Raw materials ....................................................................................................... 20
3.3 Equipment ............................................................................................................ 20
3.4 Experimental setup ............................................................................................... 20
3.5 Experimental procedure ........................................................................................ 23
3.5.1 Preparation .................................................................................................... 23
3.5.2 Sampling ........................................................................................................ 25
3.5.3 Analysing ....................................................................................................... 26
3.5.4 Experimental error ......................................................................................... 27
Chapter 4 - Model development .......................................................................................... 30
4.1 Overview ............................................................................................................... 30
4.2 The Plitt model ...................................................................................................... 30
4.2.1 Split flow ........................................................................................................ 30
4.2.2 Cut size – d50c ................................................................................................ 31
4.2.3 Sharpness of separation ................................................................................ 31
4.3 The artificial neural network .................................................................................. 31
4.3.1 Artificial neural network architecture .............................................................. 31
Chapter 5 - Results and discussion ..................................................................................... 34
5.1 Deviations in the feed PSD ................................................................................... 34
5.2 Plitt model ............................................................................................................. 36
5.2.1 Cut size – d50c ................................................................................................ 36
5.2.2 Sharpness of separation ................................................................................ 38
5.3 Artificial neural networks ....................................................................................... 39
5.3.1 Cut size – d50c ................................................................................................ 40
5.3.2 Sharpness of separation ................................................................................ 46
Chapter 6 - Conclusion and recommendations.................................................................... 53
6.1 Conclusion ............................................................................................................ 53
6.2 Recommendations ................................................................................................ 53
6.3 Further study ......................................................................................................... 54
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List of figures| vii
Bibliography ........................................................................................................................ 55
Appendix A Data processing ............................................................................................ I
Appendix B Data processing source code ...................................................................... IV
Appendix C Processed data ...................................................................................... XVIII
Appendix D Experimental error data ............................................................................ XXI
Appendix E ANN source code .................................................................................... XXII
Appendix F ECSA exit level outcomes ..................................................................... XXXII
Appendix G Hazard identification and risk assessment ............................................. XXXV
List of figures
Figure 2.1: Hypothetical flow inside the hydrocyclone viewed from the top of the hydrocyclone.
Adapted from Plitt (1976) ...................................................................................................... 4
Figure 2.2: Corrected and non-corrected partition curve adapted from Schneider (2001) ...... 5
Figure 2.3: Diagram of the computational nodes and weights of an artificial neural network
adapted from Jain (1996) .................................................................................................... 10
Figure 2.4: Supervised learning with reference to Hagan et al. (2002) ................................ 13
Figure 2.5: Polynomial of first order produces a bad fit for the data. Reproduced from Bishop
(2008:11) ............................................................................................................................ 14
Figure 2.6: Polynomial of high order producing something that looks like a good fit for all the
data points, but the predictive power of the polynomial is sacrificed. Reproduced from Bishop
(2008:12). ........................................................................................................................... 15
Figure 2.7: A lower order polynomial that has the capability to generalize well ................... 16
Figure 3.1: Diagram of the hydrocyclone setup ................................................................... 21
Figure 3.2: The experimental hydrocyclone setup ............................................................... 22
Figure 3.3: The Marcy scale ................................................................................................ 24
Figure 3.4: Malvern Mastersizer 2000 ................................................................................. 26
Figure 3.5: Experimental error of the d50c with a 95% confidence interval .......................... 29
Figure 3.6: Experimental error of the sharpness of separation with a 95% confidence interval
........................................................................................................................................... 29
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List of figures| viii
Figure 4.1: Experimental split flow values plotted with the predicted Plitt model split flow values
........................................................................................................................................... 31
Figure 4.2: Learning capability of one of the 6 developed artificial neural networks ............. 33
Figure 5.1: Particle size distribution of 25 different feed samples ........................................ 34
Figure 5.2: Example partition curve before justifications ...................................................... 35
Figure 5.3: Experimental vs. adjusted values of Rf .............................................................. 36
Figure 5.4: Experimental cut point plotted with the cut point predicted by the Plitt model .... 37
Figure 5.5: Plitt model predicted cut size vs. experimental cut size plotted over the y=x curve
........................................................................................................................................... 37
Figure 5.6: Experimental sharpness of separation plotted with the sharpness of separation
predicted by the Plitt model ................................................................................................. 38
Figure 5.7: Plitt model predicted m vs. experimental m plotted over the y=x curve .............. 39
Figure 5.8: Results of neural network 2 trained with a training speed of 0.2 and a maximum
amount of epochs of 8000 ................................................................................................... 40
Figure 5.9: Results of neural network 2 trained with a training speed of 0.2 and a maximum
amount of epochs of 8000 ................................................................................................... 41
Figure 5.10: Results of neural network 3 trained with a training speed of 0.2 and a maximum
amount of epochs of 8000 ................................................................................................... 41
Figure 5.11: Calculated d50c plotted with the experimental d50c values of neural network 3
trained with a maximum of 60000 epochs and a training speed of 0.02 .............................. 42
Figure 5.12: Predicted d50c vs. experimental d50c plotted over the y=x curve for the neural
network trained with a maximum of 60000 epochs and a training speed of 0.02 ................. 43
Figure 5.13: Calculated vs. experimental values of neural network 3 trained with a maximum
of 60000 epochs and a training speed of 0.02 with the addition of the momentum term ...... 44
Figure 5.14: Predicted d50c vs. experimental d50c plotted over the y=x curve for a neural network
trained with a maximum of 60000 epochs and a training speed of 0.02 .............................. 44
Figure 5.15: Calculated vs. experimental values of neural network 3 trained with a maximum
of 60000 epochs and a training speed of 0.02 with the addition of the regularization term .. 45
Figure 5.16: Predicted d50c vs. experimental d50c plotted on the y=x curve for the neural
network trained with a maximum of 60000 epochs and a training speed of 0.02 with the
addition of the regularization term ....................................................................................... 46
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List of tables| ix
Figure 5.17: Results of neural network 4 trained with a training speed of 0.5 and a maximum
amount of epochs of 20000 ................................................................................................. 47
Figure 5.18: Results of neural network 5 trained with a training speed of 0.5 and a maximum
amount of epochs of 25000 ................................................................................................. 47
Figure 5.19: Results of neural network 6 trained with a training speed of 0.2 and a maximum
amount of epochs of 8000 ................................................................................................... 48
Figure 5.20: Experimental and predicted values of neural network 6 trained with a maximum
of 60000 epochs and a training speed of 0.02 ..................................................................... 49
Figure 5.21: Predicted vs. experimental m plotted over the y=x graph for neural network 6
trained with a maximum of 60000 epochs and a training speed of 0.02 .............................. 49
Figure 5.22: Predicted and experimental values of neural network 6 trained with a maximum
of 60000 epochs and a training speed of 0.02 with the addition of the momentum term ...... 50
Figure 5.23: Predicted m vs. experimental values m plotted over the y=x line for neural network
6 trained with a maximum of 60000 epochs and a training speed of 0.02 with the addition of
the momentum term ............................................................................................................ 50
Figure 5.24: Predicted vs. experimental values of neural network 6 trained with a maximum of
60000 epochs and a training speed of 0.02 with the addition of the regularization term ...... 51
Figure 5.25: Predicted m vs. experimental m plotted over the y=x curve for neural network 6
trained with a maximum of 60000 epochs and a training speed of 0.02 with the addition of the
regularization term .............................................................................................................. 51
List of tables
Table 2.1: Sensors used in the on-line monitoring of hydrocyclone performance .................. 7
Table 3.1: Processed data used for the experimental error determination ........................... 27
Table 3.2: Values for substitution into the student's t equation ............................................ 28
Table 4.1: Different artificial neural networks that were programmed .................................. 32
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List of acronyms| x
List of acronyms
Acronym Description
ANN Artificial neural network
HIRA Hazard identification and risk assessment
MS Microsoft
MSDS Material safety data sheet
PPE Personal protective equipment
PSD Particle size distribution
List of symbols
Symbol Description
Al2O3 Aluminium oxide
K2O Potassium oxide
Fe2O3 Iron(III) oxide
CaO Calcium oxide
Na2O Sodium oxide
𝑑 Size of a particle in 𝜇𝑚.
𝑑50𝑐 Hydrocyclone corrected cut point. This is the particle size that has an
equal chance of either leaving through the underflow or the overflow. Its
unit is in 𝜇𝑚.
𝐷𝑐 Hydrocyclone diameter in 𝑐𝑚
𝐷𝑖 Inlet diameter in 𝑐𝑚
𝐷𝑜 Vortex finder diameter in 𝑐𝑚
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List of symbols| xi
𝐷𝑢 Underflow/apex/spigot diameter in 𝑐𝑚
𝛿𝑗𝑂 Error of the output node 𝑗.
𝛿𝑗𝐻 Error of the hidden node 𝑗.
𝐸 Error of a neuron’s output
�̃� Conditioned error of the neuron’s output
𝜂𝑣 Viscosity of the carrier fluid in 𝑐𝑝
𝐹𝑖 Fudging factor of the modified Plitt model where 𝑖 = 1,2,3…
ℎ Free vortex height in 𝑐𝑚
𝑘 Constant that takes into account the effect of the solids density on the
corrected cut size.
𝑚 Sharpness of separation. This is the slope of the partition curve that
indicates how well the classification is taking place inside the
hydrocyclone. The higher the value of m, the closer the hydrocyclone will
be to an ideal classifier.
𝑀 Momentum factor defined by the user
𝑚𝑠 Mass of silica sand that has to be added to the storage tank in 𝑘𝑔
𝑛 The amount of weights attached to node 𝑗
𝑛𝑢 Number of data points available in the set
Ω Penalty term
𝑃 Pressure over the hydrocyclone in 𝑘𝑃𝑎
𝜑 Percentage solids in the feed
𝑄 Volumetric feed flow rate in 𝑙𝑖𝑡𝑒𝑟𝑠
𝑚𝑖𝑛𝑢𝑡𝑒
𝑅 Regularization factor defined by the user
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List of symbols| xii
𝑅𝑓 Recovery of the carrier liquid to the underflow
𝜌𝑝 Density of the hydrocyclone feed slurry in 𝑔
𝑐𝑚3
𝜌𝑠 Density of the solid phase in 𝑔
𝑐𝑚3
𝑆 Split flow – The volumetric flow of the underflow divided by the volumetric
𝑆𝑡 Standard deviation in the data
𝜎𝑗𝐻 Output value of the transfer function of node 𝑗 in the hidden layer
𝜎𝑗𝑂 Output of the neuron in the output layer 𝑗.
𝑡𝑛−1(𝛼
2) Critical t value that could be obtained from the back cover of Devore and
Farnum (2005)
𝑇(95%) Critical t value for a 95% confidence.
𝜐 Parameter for controlling the importance of the bias term
𝑉𝑤 Volume of water in the storage tank in 𝑚3
𝑤𝑖 Value of weight 𝑖, where 𝑖 = 1,2,3…
𝑤𝑖𝑗 Value of the weight that goes from node 𝑖 to node 𝑗.
𝑤𝑖𝑗𝐻 Value of the hidden layer weight that goes from node 𝑖 to 𝑗.
∆𝑤𝑖𝑗𝐻 Value with which weight 𝑤𝑖𝑗
𝐻 has to be updated
𝑤𝑖𝑗𝐻(𝑡 − 1) The value of the previous weight 𝑤𝑖𝑗
𝐻
𝑤𝑖𝑗𝑂 Value of the output weight that goes from node 𝑖 to node 𝑗.
∆𝑤𝑖𝑗𝑂 Value with which the output weight, 𝑤𝑖𝑗
𝑂 has to be updated
𝑤𝑖𝑗𝑂(𝑡 − 1) Value of the previous weight, 𝑤𝑖𝑗
𝑂
𝑋 Average of a data set
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List of symbols| xiii
𝑥𝑖 Input value from weight 𝑖.
𝑥𝑗𝐻 Output of the hidden node 𝑗.
𝑥𝑗𝑂 Output of the output node 𝑗.
𝜉𝑗 Signal sent to node 𝑗, where 𝑗 = 1,2,3…
𝑦 Partition number. This is the value displayed on the partition curve’s y-
axis for a certain particle size, 𝑑.
𝑦′ Corrected partition number
𝑦𝑗 Desired output of output node 𝑗
𝑧 Amount of weights connected to the cell
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Introduction| 1
Chapter 1 - Introduction
1.1 Background
Hydrocyclones are very handy process units when it comes to classifying particles according
to size or density in the mineral processing industry. Unfortunately, it is very difficult for the
operator to monitor the performance of the hydrocyclone while on-line (Coelho & Medronho,
2000). Empirical models had to be developed in order to predict how the hydrocyclone will
perform under certain conditions by acting as inference sensors (Kraipech et al., 2005).
One popular empirical model used to predict the performance of a hydrocyclone is the Plitt
model (Flinthoff et al., 1987). L.R. Plitt developed this model to be robust by gathering a large
number of experimental data. The data was gathered by operating a wide range of
hydrocyclone geometries at different operating conditions (Plitt, 1976). This model is in general
not very accurate in predicting the performance, i.e. the separation efficiency of hydrocyclones
(Silva et al., 2009).
Artificial Neural Networks can be used to predict the performance of complex systems like the
hydrocyclone (Kutz, 2003). What makes this method so special is its ability to learn through
parallel processing (McMillan, 1999). Given a certain amount of experimental data, the ANN
can identify underlying patterns in the data which gives it the ability to predict the outcome,
given certain input parameters (Jain, 1996).
South Africa has a very large mining industry (Anglo American Platinum, 2013; Anglo Gold
Ashanti, 2013). Improving the performance of the hydrocyclone could possibly lead to the
growth in the South African economy, as mineral processing becomes more efficient. The use
of hydrocyclones are not limited to the mining industry. These process units are also globally
used in the petrochemical, environmental and food processing industries (Sripriya et al.,
2007). The improved use of the hydrocyclone could thus have a large impact globally in
various industries.
1.2 Problem statement
Inadequate control of the hydrocyclones on a mineral processing plant may lead to
inefficiencies in the downstream process units, ultimately leading to a loss in profit for the
company. Monitoring the on-line performance of a hydrocyclone is not a simple task. Inference
sensors1 that make use of empirical models or artificial neural networks are possible solutions
1 The terms inference sensors and soft sensors are used interchangeably in this study
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Introduction| 2
to this problem. A study is needed to determine which of these methods will be more
appropriate for predicting hydrocyclone performance.
1.3 Aim and objectives
1.3.1 Aim
Improve hydrocyclone efficiency by producing a soft sensor that has the ability to accurately
predict hydrocyclone performance.
1.3.2 Objective
Compare the predictive power of an empirical model, namely the Plitt model, with the
predictive power of an artificial neural network trained with the backpropagation algorithm by
making use of experimental hydrocyclone data.
1.3.3 Methodology
Do a literature study on the operation of the hydrocyclone, empirical models for
predicting hydrocyclone performance and artificial neural networks;
Do a HIRA study before sampling on the hydrocyclone commences;
Devise a procedure for obtaining representative samples from the hydrocyclone;
Obtain more than 100 samples from the hydrocyclone;
Gather data on the samples’ PSD by analysing the samples with the Malvern
Mastersizer 2000;
Process the data for it to be in a suitable form for inserting into an empirical model and
an artificial neural network;
Develop the artificial neural network from the literature study that was previously
conducted;
Find optimal architectures and parameters for the artificial neural network through trial-
and-error;
Substitute the processed data into the empirical model and compare its output (d50c
and sharpness of separation) with that of the experimental data;
Substitute the processed data into the trained artificial neural networks and compare
the output to the experimental data;
Compare the two soft sensors, the empirical model and the artificial neural network,
with each other and come to a conclusion over which is better for predicting
hydrocyclone performance
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Literature study| 3
Chapter 2 - Literature study
2.1 The hydrocyclone
Hydrocyclones are commonly used in the mineral industry for the classification of particles
after grinding (Flinthoff et al., 1987). It is usually installed in a closed circuit grinding unit where
it is used to separate the under size particles from the course particles (Kelly & Spottiswood,
1982:201). The course particles are returned to the grinder for further comminution while the
under size particles leave the circuit (Wills, 2006:224-225). Advantages of hydrocyclones
include simple design, low operational costs and the capability of handling large volumes of
pulp (Sripriya et al., 2007). Complex mechanical devices like spirals and rake classifiers have
been replaced by cyclones2, due to their simple structure that contains no moving parts
(Napier-Munn et al., 2005:309).
Its applications are however not limited to the mineral industry as it is also used in the chemical
industry, power generation industry, textile industry and more. By customising its structure,
the hydrocyclone can be used for specific applications like (Svarovsky, 1984:1):
Liquid clarification
Slurry thickening
Cleansing solid particles
Elimination of gasses from liquids
Classification of particles takes place due to the difference in settling velocities of the particles
being classified. The settling velocities can be a function of either particle size and/or particle
density, depending on whether a homogeneous or heterogeneous ore is classified (Kelly &
Spottiswood, 1982:199). A homogeneous ore contains particles of similar densities. Particles
of homogeneous ores will be classified according to their size (Flinthoff et al., 1987).
The feed enters the cylindrical section of the hydrocyclone tangentially where it forms a vortex
inside the cyclone’s cone shaped body. The fluid follows a helical path until it reaches the
spigot, also known as the apex, where a portion of the downward flow leaves through the
spigot as the underflow. The remaining downward flow follows an upward spiral, located on
the inside of the outer vortex, and leaves via the vortex finder (Svarovsky, 1984:30-31). The
reason for the formation of the upward spiral is not fully understood (Svarovsky, 1984:41).
Particles of similar density or size gather together due to the competition between the drag
forces and centrifugal forces acting on these particles (Napier-Munn et al., 2005:309-310). If
the density of the carrier liquid is lower than that of the solids being separated, the centripetal
2The terms, “hydrocyclone” and “cyclone” are used interchangeably in this study
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Literature study| 4
force on the solid particles will be larger than the centripetal force of the liquid. On the other
hand, the centripetal force acting on the particle will increase as the particle size increases for
homogeneous ores (Hibbeler, 2010:131). The centripetal force acting on the particles
dominates the drag force also acting on the particles in a radial direction. Larger particles thus
reach the boundary layer, formed between the liquid and the wall of the cyclone, with more
ease than the smaller particles. The particles in the boundary layer leave the cyclone via the
apex under ideal conditions. The finer particles that could not reach the boundary layer by the
time the apex is reached, is transported to the inner spiral where it leaves throught the vortex
finder (Svarovsky, 1984:41).
Random turbulence, hindered settling and the interaction between the carrier liquid and the
solid particles makes describing the flow inside the hydrocyclone very difficult. Determining
the separation performance of the hydrocyclone is thus not an easy task (Sripriya et al., 2007).
The performance of the hydrocyclone is defined as the ability to separate particles into the
desired size ranges (Kelly & Spottiswood, 1982:204). According to Svarovsky (1984) the
separation performance of the hydrocyclone could be determined if the corrected cut size,
𝑑50𝑐, and the sharpness of separation, m, could be calculated. This is done with the use of a
corrected partition curve. The grade efficiency curve, also called the partition curve or the
Tromp curve, is a plot of the particles in a certain size range, on the x-axis, vs. the fraction of
Figure 2.1: Hypothetical flow inside the hydrocyclone viewed from the top of the hydrocyclone. Adapted from Plitt (1976)
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Literature study| 5
these particles in the feed leaving the hydrocyclone through the underflow (Frachon & Cilliers,
1999) as can be seen on Figure 2.2. The grade efficiency curve cannot be approximated from
first principles and has to be determined by using experimental data (Svarovsky, 1984:17).
Figure 2.2: Corrected and non-corrected partition curve adapted from Schneider (2001)
The 𝑑50𝑐, also known as the cut size, is the particle size that has an equal chance to exit the
hydrocyclone through the vortex finder or through the underflow. The corrected cut size is
used instead of the real cut size, as this gives a better indication of the separation forces that
are present in the hydrocyclone. More information on the corrected partition curve follows
later. The sharpness of separation, m, indicates how well the classification is taking place in
the cyclone. The higher the value of m, the closer the hydrocyclone is to an ideal classifier
(Napier-Munn et al., 2005:311).
In practice, some of the particles, irrespective of their size, in the hydrocyclone bypasses the
classification. By controlling the operating conditions of the cyclone, these deviations from
ideal separation could be lowered, but never eliminated (Napier-Munn et al., 2005:310-311).
Two paths that could be followed for bypassing classification are mentioned below.
Small particles tend to stay suspended in the liquid which leaves the hydrocyclone through
the underflow. According to Frachon and Cilliers (1999), Plitt (1976) and Svarovsky (1984:20)
the fraction small particles bypassing to the underflow is directly proportional to the liquid
recovery to the underflow, Rf. A corrected partition curve is constructed to remove the effect
of the bypass to the underflow as can be seen on Figure 2.2. Another phenomenon that
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Literature study| 6
causes the undersize particles to leave though the underflow, is when the undersize particles
are trapped in the boundary layer by the larger particles. The corrected partition curve
constructed with the use of equation 2.1 might thus not be capable of taking into account all
of the undersize particles leaving via the underflow.
Another way in which classification could be bypassed is if particles near votex finder leaves
via the overflow (Svarovsky, 1984:40). No corrections are made on the partition curve to take
this effect into account, but this effect will however be held in mind when the results are
interpreted.
𝑦′ =
𝑦 − 𝑅𝑓
1 − 𝑅𝑓
2.1
2.2 Hydrocyclone control
If a hydrocyclone is not operated to produce the desired overflow and underflow, it could lead
to poor performance in downstream processes (Eren & Gupta, 1988). Fines in the underflow
lead to overgrinding, while coarse material in the overflow can cause downstream separation
problems (Aldrich et al., 2014). Slight changes in the operating conditions of the hydrocyclone
could markedly affect the performance of the hydrocyclone (Neesse et al., 2004). The operator
of a hydrocyclone might not always be aware of the cyclone’s underperformance and is in
addition frequently incapable of returning the cyclone to its optimal operation. There is thus a
need for methods to efficiently determine the performance of the cyclone while in operation
(Napier-Munn et al., 2005:309). Optimising the hydrocyclone is not an easy task, as the
variables are often interlinked with each other. Models that are reasonably accurate are
capable of finding the optimum operating conditions for the hydrocyclone even if the variables
like the split flow and pressure are for example dependent on each other (Napier-Munn et al.,
2005:320).
2.2.1 Sensors used in hydrocyclone performance determination
Variables like the pressure drop over the cyclone, the flow rates in and out of the cyclone and
the feed are commonly monitored while the cyclone is on-line. With all this information, the
operator might still not be able to control the performance of the cyclone effectively (Napier-
Munn et al., 2005; Aldrich et al., 2014). Numerous studies have been conducted to find a
suitable method to control the performance of the hydrocyclone, many of which has not been
widely used in the industry (Aldrich et al., 2014). Table 2.1 contains a list of current sensors
that have been developed for determining hydrocyclone performance.
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2.3 Soft sensors
One other way the performance can be monitored on-line is by developing a soft sensor, like
an artificial neural network (Napier-Munn et al., 2005). Soft sensors use operational data from
the plant to predict variables that are usually difficult and/or costly to measure on-line (Kadlec
Table 2.1: Sensors used in the on-line monitoring of hydrocyclone performance
Sensor Description
Acoustic Sensors An acoustic sensor was mounted externally on the
hydrocyclone and after a suitable model was found, could
accurately predict various parameters like the solids
concentration and the flow rate. Variables like the d50c and
sharpness of separation are however not determined with the
use of this method (Hou et al., 1998).
Videographic
Measurement
A video camera was used to monitor the discharge angle of
the hydrocyclone. The discharge angle of the cyclone is said
to be linked to the performance of the cyclone (Concha et al.,
1996; Neesse et al., 2004). Although this method has some
challenges, it is a cost effective way to determine the discharge
angle with good accuracy (Janse van Vuuren et al., 2011).
Photographic
measurement
Aldrich et al. (2014) used images and other experimental data
from the underflow of a experimental hydrocyclone setup to
develop a model that had the ability to identify the mean
particle size in the underflow. Instead of using the discharge
angle of the underflow like Janse van Vuuren et al. (2011), the
textural information that the images provided of the underflow
was utilised.
Measurement using a
laser beam
A laser beam is pointed at the underflow of the cyclone where
the reflection of the laserbeam is measured with a camera to
determine if the cyclone is in the spray or roping state (Neesse
et al., 2004).
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et al., 2009). Two possible soft sensors for the control of the hydrocyclone, empirical models
and artificial neural networks, will be discussed in this study.
2.3.1 Empirical models
Empirical models had to be developed in order to predict how the hydrocyclone will perform
under certain conditions (Kraipech et al., 2005). Although Flinthoff et al. (1987) states that
these models have been widely accepted, Chen et al. (2000) however states in his study that
these models are not reliable. Coelho and Medronho (2000), reasons that these models will
only work well if the cyclone is operated in the range that was used to obtain the data to fit the
models.
One popular empirical model that is used to predict the performance of a hydrocyclone, is the
Plitt model (Flinthoff et al., 1987). L.R. Plitt developed this model to be robust by gathering a
large number of experimental data. The Plitt model was designed to also take into account the
theories around the complex flow of the hydrocyclone (Plitt, 1976). These theories include the
residence time theory and the equilibrium orbit theory (Chen et al., 2000). The theories alone
are incapable of describing the hydrocyclone performance (Napier-Munn et al., 2005:312).
The data was gathered by operating a wide range of hydrocyclone geometries at different
operating conditions (Plitt, 1976). This model is in general not very accurate in predicting the
performance of hydrocyclones (Silva et al., 2009).
The Plitt model consists of four empirical equations. These equations are used to calculate
the corrected cut size, the flow split between the underflow and overflow, the sharpness of
separation and the pressure drop over the hydrocyclone (Plitt, 1976). Although the Plitt model
is designed to work without calibration, Flinthoff et al. (1987) recommends inserting empirical
constants, F1 – F4, that will take into account the unique conditions under which the cyclone
operates. Only one experimental data point is needed to tune these empirical constants. By
default, the values of these constants are all equal to 1.
𝑑50𝑐 = 𝐹1
39.7𝐷𝑐0.46𝐷𝑖
0.6𝐷01.21𝜂𝑣
0.5 exp(0.063𝜑)
𝐷𝑢0.71ℎ0.38𝑄0.45 (
𝜌𝑠 − 11.6 )
𝑘
2.2
𝑚 = 𝐹21.94 exp (−
1.58𝑆
1 + 𝑆)(
𝐷𝑐2ℎ
𝑄)
0.15
2.3
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𝑃 = 𝐹3
1.88𝑄1.78 exp(0.0055𝜑)
𝐷𝑐0.37𝐷𝑖
0.94ℎ0.28(𝐷𝑢2 + 𝐷𝑜
2)0.87
2.4
𝑆 =
𝐹4 (3.29𝜌𝑝0.24 (
𝐷𝑢𝐷𝑜
)3.31
ℎ0.54(𝐷𝑢2 + 𝐷𝑜
2)0.36𝑒0.0054𝜑)
𝐷𝑐1.11𝑃0.24
2.5
Where:
𝐷𝑐= Cyclone diameter in 𝑐𝑚
𝐷𝑖= Inlet diameter in 𝑐𝑚
𝐷𝑜= Vortex finder diameter in 𝑐𝑚
𝐷𝑢= Underflow/apex diameter in 𝑐𝑚
ℎ= Free vortex height in 𝑐𝑚
𝜌𝑝= Density of the cyclone feed slurry in 𝑔
𝑐𝑚3
𝜌𝑠= Density of the solid phase in 𝑔
𝑐𝑚3
𝜂𝑣= Viscosity of the carrier fluid in 𝑐𝑝
𝜑 = Percentage solids in the feed
𝑄 = Feed flow rate in 𝑙𝑖𝑡𝑒𝑟𝑠
𝑚𝑖𝑛𝑢𝑡𝑒
𝑑50𝑐 = Corrected cut size in 𝑚𝑖𝑐𝑟𝑜𝑛𝑠
𝑚 = Sharpness of separation which is dimensionless
𝑃 = Gauge pressure in 𝑘𝑃𝑎
𝑆 = Split flow. This is the volume of the underflow divided by the volume of the overflow and it
is a dimensionless quantity
According to Plitt (1976), the PSD of the feed slurry has a negligible effect on the outcome of
the d50c of the underflow.
After determining the d50c and m, with the Plitt model, these values can then be inserted into
the Rosin-Rammler equation, equation 2.6, to obtain the corrected partition curve.
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𝑦′ = 1 − exp(−0.693(
𝑑
𝑑50𝑐)𝑚
) 2.6
Where 𝑑 is the particle size in 𝑚𝑖𝑐𝑟𝑜𝑛𝑠 and 𝑦′ is the corrected volume of a certain particle size
that was recovered in the underflow.
2.3.2 Artificial neural networks
Various studies have been done on the use of artificial neural networks for the prediction of
hydrocyclone performance and have proven to be successful (Eren et al., 1997a; Eren et al.,
1997b; Karimi et al., 2010)
The human brain has powerful learning, generalization and parallel computing abilities. It is
desired to give computers the same abilities by copying the principle operation of brain cells
and developing artificial neural networks (ANN) (Jain, 1996). ANNs are not limited to soft
sensors. Awodele and Jegede (2009) reasons that ANN promises a wide range of new
applications in the areas such as education and medicine in the future. This is the reason why
research in this field has been booming in the past few decades (Gallant, 1994:1).
Figure 2.3: Diagram of the computational nodes and weights of an artificial neural network
adapted from Jain (1996)
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An artificial neural network consists of computational units called nodes3. These nodes are
located in sets called layers. Connections, called weights, connects the nodes of one layer to
the following layer. Information transported through the weights can only travel in one
direction. Figure 2.3 illustrates the computational nodes in a three-layer neural network4. The
arrows represent the weights and their direction.
Values, either positive or negative, are assigned to each of the weights. The magnitude of the
value assigned to the weight determines how large the effect of the data transported through
that weight will be on the neural network. The larger the magnitude of the weights, the larger
the effect. The input data travels through the weights to which they are connected. The data
traveling through that weight is multiplied by the value of that weight. When the data reaches
hidden layer 1 through the weights, an input value to the node is calculated. More information
on these calculations later. The input value is substituted into a function called an activation
function which calculates an output called an activation. The activation travels through the
weights to the next nodes and the same operation is performed. This is done until the ANN
produces its final output (Gallant, 1994:1).
Parameters that influence the output of the network include:
The number of layers
The number of nodes in the hidden layers
The activation function used in the nodes
The values of the weights
Number of input variables
2.3.2.1 Network topography
The network topology involves the arrangement of nodes and connections in the network.
These arrangements can be classified into 2 main categories: Feed-forward networks or
feedback networks.
In feed-forward networks, information can only be carried in one direction, from the input to
the output. This type of network is mainly used for pattern recognition purposes. Figure 2.3
illustrates a feed-forward neural network.
In a feedback or recurrent networks, the information can either travel in the forward direction
to the output or return in the input direction, i.e. make a loop (Awodele & Jegede, 2009).
For the purposes of this study, a feed forward structure will be used.
3 The words “nodes” and “neurons” are used interchangeably in this study. 4 The terms “neural networks” and “artificial neural networks” are used interchangeably in this study.
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2.3.2.2 Other artificial neural network parameters
2.3.2.2.1 Initial weights
Initial weight values between -0.1 and 0.1 are randomly chosen. Assigning non-random
weights could lead to weights that perform the same action and does not lead to sufficient
convergence. The weights need to be unique when initialising training to increase the chances
of identifying the pattern in the data (Gallant, 1994:213). Another, more complex approach
proposed by Gallant (1994:220), is to initialise the weights connected to a certain cell to a
random value between -2/z and 2/z, where z is the amount of weights connected to the cell.
2.3.2.2.2 Training speed
A large value for the training speed, 𝜇, gives a faster convergence. This convergence can
however only be maintained up to a certain point where the network will become unstable and
diverge. This is called overtraining. It is advised to choose a training speed that has a positive
value no larger than 0.1. Although this results in slow training, the neural network has a better
chance to find the local minimum (Gallant, 1994:220).
2.3.2.2.3 Momentum
Momentum is used to increase the training speed. The momentum term consists of the
change in weight at the previous iteration, multiplied by the momentum parameter. An
additional benefit of adding momentum is the removal of noise that might occur during weight
updating. The weight thus converges smoothly (Gallant, 1994:221).
2.3.2.2.4 Number of hidden neurons
It is very common for the backpropagation algorithm used in the industry to only contain one
hidden layer, the main reason being that networks with more hidden layers learn very slowly.
Neural networks with one hidden layer are known to be universal approximators. The only way
to determine whether a network with multiple or a single hidden layer should be used, is by
trail-and-error (Gallant, 1994:221).
2.3.2.3 Machine learning
In order for an ANN to produce better results, an algorithm has to be written that gives the
ANN the ability to adjust its self. This is called machine learning (Nag, 2010). The two main
types of machine learning are supervised and unsupervised learning.
In supervised learning, the ANN is given input data to produce an output. The output the ANN
produces for the given input data is evaluated with the desired output. If the output from the
ANN does not match the desired output, the necessary adjustments are made with the use of
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the learning algorithm (Gallant, 1994:6). The diagram in Figure 2.4 attempt to better describe
what is meant with supervised learning.
In contrast, unsupervised is not provided with the desired output. Instead, unsupervised
learning is used to adjust the ANN so that it can group data that show similar patterns (Gallant,
1994:7). Applications of unsupervised learning include finding the probability distribution of
data and identify groups of data that show similar properties and occur close together, i.e.
cluster identification (Bishop, 2008:10). In this study, supervised learning will be used, as the
experimental data from the hydrocyclone provide the ANN with input and the desired output.
2.3.2.4 Learning algorithms
There are a number of learning algorithms in existence that are used to adjust the neural
network in order to achieve the desired output. Some algorithms include the perceptron
learning algorithm, radial basis function algorithm and the Boltzmann learning algorithm (Jain,
1996). The question arises: “Which algorithms would be fit for a certain application?”
According to Jain (1996) the backpropagation algorithm, among others, are fit for use in control
systems. Gallant (1994:225), on the other hand reasons that trial-and-error has to be used to
find the appropriate algorithm. From his experience, he found that one should first try to use a
single-cell model before using a complex algorithm like the backpropagation algorithm.
A large problem that occurs in all systems is the presence of noise which commonly occurs in
real world applications. Noise is the introduction of erroneous data into the data set. It could
Figure 2.4: Supervised learning adapted from Hagan et al. (2002)
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either be that the data is false or absent (Gallant, 1994:9). Artificial neural networks, on the
other hand are capable of handling noise (Gallant, 1994:10).
2.3.2.5 Problems with artificial neural networks
2.3.2.5.1 Failure to generalise
The purpose of training an ANN is not so much as to reproduce the exact values of the training
data, but rather to develop a network that is capable is producing a general answer that would
be expected in the training data range (Zhang et al., 2003; Bishop, 2008:332).
To explain the difference between good and bad generalization of ANNs, Bishop (2008:9-12)
uses the analogy with the complexity of an ANN and the order of a polynomial (polynomial of
high or low order). Given a certain data set generated by adding random values to the output
of a known function, say 𝑦 = sin(𝑥). Two polynomials are used to fit the data. The one
polynomial is of a high order and the other of a low order. The results of the first order
polynomial that were fit to the data could be seen on Figure 2.5. This corresponds to a neural
network with only one hidden node that produces a bad fit to the data. A possible solution is
to increase the number of free parameters. In the case of a neural network, the number of
hidden nodes will be increased. As can be seen in Figure 2.6, the higher order polynomial
produces a good fit for the all the data points. It is however a bad representation of the sine
wave, as there are plenty of oscillations (Bishop, 2008:9-12).
Figure 2.5: Polynomial of first order produces a bad fit for the data. Reproduced from Bishop (2008:11)
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To address this problem of finding a suitable complexity for the ANN, two concepts, the
variance and bias (not to be confused with bias weights) are used. The bias is a measure of
the amount with which the overall average of the ANN output differs with that of the given data.
Figure 2.5 has a high bias value while Figure 2.6 has a low bias value. The variance is used
as a measure of how well the ANN output will fit to another data set with that does not include
the ANN training data. A low variance value can be expected in Figure 2.5, while a high
variance value can be expected in Figure 2.6. The variance and bias goes hand in hand – an
increase in the variance leads to a decrease in the bias and vice-versa. The goal is to decrease
the value of both the variance and the bias (Bishop, 2008:334-335).
2.3.2.5.2 Regularization
Over-fitting is the result of weights with high values. In order to suppress the weights from
obtaining large values, regularization is applied. In regularization, the error of the output is
conditioned in order to produce a smoother output. This is done by adding a penalty term, Ω,
to the error, 𝐸. The conditioned error, �̃�, can be calculated with the help of equation 2.7.
�̃� = 𝐸 + 𝜐Ω 2.7
Bishop (2008:338) provides two ways in which the penalty term can be calculated. One of the
two methods is the Tikhonov regularizers which will not be discussed in this study. Another is
Figure 2.6: Polynomial of high order producing something that looks like a good fit for all the data points, but the predictive power of the polynomial is sacrificed. Reproduced from Bishop (2008:12).
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the weight decay method. In this method, the penalty term is equal to the sum of squares of
all the weights and biases. The equation could be observed in equation 2.8.
Ω =
1
2∑𝑤𝑖
2
𝑖
2.8
The weight decay regularizer suppresses the weights from obtaining large values which will
cause over-fitting (Bishop, 2008:338-339).
2.3.2.5.3 Structural stabilization
Trial-and-error could be used to find a more suitable structure that has little complexity, but
produces good results. One way of doing this, is by varying the number of hidden nodes or by
adding bias weights to the network (Gallant, 1994:221; Bishop, 2008:332).
2.3.2.5.4 More data points
The number of training data points and the possible curves that can fit through these training
data points are inversely proportional to each other (Zhang et al., 2003). If one desires to train
a complex network for reasons such as more accurate results, one simply has to add more
training data to the network.
A neural network that has the ability to generalize should give output as displayed by the lower
order polynomial in Figure 2.7.
Figure 2.7: A lower order polynomial that has the capability to generalize well
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2.3.2.6 The backpropagation algorithm
As mentioned above, the backpropagation algorithm is one of the popular neural network
training algorithms that is suitable for use in process control environments. It was decided that
this training algorithm will be used for the neural network in this study. It should be noted that
this algorithm mentioned here is developed for a neural network that has a single hidden layer.
The sources used in the development of this artificial neural network include Jain (1996) and
Basheer and Hajmeer (2000). The steps are as follow:
1. Choose the amount of input, hidden and output nodes. This will then also tell you how
many weights there will be in the neural network architecture.
2. Assign random values to the weights.
3. Propagate the signal forward by multiplying the inputs to the neural network with the
with the weights that connect the inputs to the hidden neurons, then sum the results of
the weights that goes to each of the hidden nodes to produce the signal that is sent to
the specified node as can be seen in equation 2.9.
𝜉𝑗 =∑𝑥𝑖𝑤𝑖𝑗
𝑛
𝑖=0
2.9
Where:
𝜉𝑗=Signal sent to node 𝑗.
𝑛=The amount of weights attached to the node 𝑗.
𝑥𝑖=Input value from weight 𝑖
𝑤𝑖𝑗=Weight attached to the input node, 𝑖, and the hidden node 𝑗.
4. Substitute the input signal into the activation function. The sigmoid activation function
was chosen and can be seen in equation 2.10.
𝜎𝑗𝐻 =
1
1 + 𝑒−𝜉𝑗
2.10
Where 𝜎𝑗𝐻 is the output value of the transfer function of node 𝑗 in the hidden layer.
5. The output of node 𝑗, 𝜎𝑗, is then fed forward to the next layer of nodes, the output nodes
where equation 2.11 is applied.
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𝜉𝑗 =∑𝜎𝑖
𝐻𝑤𝑖𝑗
𝑛
𝑖=0
2.11
6. The signal to the output nodes, 𝜉𝑗, is again substituted into the sigmoid function in
equation 2.12 to produce the output of the output layer nodes.
𝜎𝑗𝑂 =
1
1 + 𝑒−𝜉𝑗
2.12
Where 𝜎𝑗𝑂 is the output of the output layer nodes. Note that 𝜎𝑗
𝑂 is equal to 𝑥𝑗𝑂 that will be
mentioned soon.
7. The error of the output neurons could then be calculated by comparing the output of
the output neurons with the desired output of the training data with the use of equation
2.13 (Gupta & Lam, 1998).
𝛿𝑗𝑂 = (𝑥𝑗
𝑂 − 𝑦𝑗)𝑥𝑗𝑂(1 − 𝑥𝑗
𝑂) 2.13
Where:
𝛿𝑗𝑂=Error of the output node 𝑗
𝑥𝑗𝑂=Output of the output node 𝑗. Again note that 𝑥𝑗
𝑂 is equal to 𝜎𝑗𝑂.
𝑦𝑗=Desired output of the output node 𝑗
8. The values with which the weights between the output layer nodes and the hidden
layer nodes are changed could now be calculated with equation 2.14 (Gupta & Lam,
1998).
∆𝑤𝑖𝑗
𝑂 = 𝜂𝛿𝑗𝑂𝑥𝑗
𝑂 −𝑀𝑤𝑖𝑗𝑂(𝑡 − 1) − 𝜂𝑅 × (
𝑤𝑖𝑗𝑂(𝑡 − 1)2
((1 + 𝑤𝑖𝑗𝑂(𝑡 − 1))
2)2)
2.14
Where:
∆𝑤𝑖𝑗𝑂=The value with which weight 𝑤𝑖𝑗
0 has to be updated
𝜂=Training speed defined by the user
𝛿𝑗𝑂=Error of the output node 𝑗
𝑥𝑗𝑂=Output of the output node 𝑗
𝑀=Momentum factor defined by the user
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𝑅=Regularization factor defined by the user
𝑤𝑖𝑗0 (𝑡 − 1)=The value of the previous weight 𝑤𝑖𝑗
𝑂
9. The new weight values can then be calculated with equation 2.15.
𝑤𝑖𝑗𝑂 =𝑤𝑖𝑗
0 (𝑡 − 1) − ∆𝑤𝑖𝑗𝑂 2.15
Where 𝑤𝑖𝑗𝑂 is the new value of the output weight that extends from node 𝑖 in the hidden
layer to node 𝑗 in the output layer.
10. The next step is to calculate the error of the hidden nodes with the help of equation
2.16 (Gupta & Lam, 1998).
𝛿𝑗𝐻 = 𝑥𝑗
𝐻 × (1 − 𝑥𝑗𝐻) × 𝑤𝑖𝑗
𝑂(𝑡 − 1) × 𝛿𝑗𝑂 2.16
Where
𝛿𝑗𝐻=Error of the hidden node 𝑗
𝑥𝑗𝐻=Output of the hidden node 𝑗
11. Now that the error of the hidden layer of nodes are known, the increment with which
the weights that extend from the input layer to the hidden layer has to change could
now be calculated with equation 2.17 (Gupta & Lam, 1998).
∆𝑤𝑖𝑗
𝐻 = 𝜂𝛿𝑗𝐻𝑥𝑗
𝐻 −𝑀𝑤𝑖𝑗𝐻(𝑡 − 1) − 𝜂𝑅 × (
𝑤𝑖𝑗𝐻(𝑡 − 1)2
((1 + 𝑤𝑖𝑗𝐻(𝑡 − 1))
2)2)
2.17
Where:
∆𝑤𝑖𝑗𝐻=The value with which weight 𝑤𝑖𝑗
𝐻 has to be updated
𝛿𝑗𝐻=Error of the hidden node 𝑗
𝑥𝑗𝐻=Output of the hidden node 𝑗
𝑤𝑖𝑗𝐻(𝑡 − 1)=The value of the previous weight 𝑤𝑖𝑗
𝐻
12. The new weights can then be calculated with the help of equation 2.18.
𝑤𝑖𝑗𝐻 =𝑤𝑖𝑗
𝐻(𝑡 − 1) − ∆𝑤𝑖𝑗𝐻 2.18
The above steps could be repeated with the data from a new sample. An epoch is completed
if the artificial neural network has gone through the entire set of training data. A new epoch is
started by again going through the training data set.
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Chapter 3 - Experimental procedure
3.1 Overview
Various slurries were prepared to be fed to the hydrocyclone. Different operating conditions
were imposed on the hydrocyclone. All necessary operating conditions, samples and other
data, were recorded on each run. A PSD analysis of the samples were carried out on the
samples. The gathered information could then be used to determine the d50c and the
sharpness of separation.
3.2 Raw materials
The solid particles that had to be separated was micron sized silica quartz particles, MQ15,
supplied by Micronized SA Limited. According to Tew (2012) the particles contain 98.50%
silica, with small amounts of Al2O3, K2O, Fe2O3, CaO and Na2O. The particles have a density
of 2650𝑘𝑔
𝑚3 and a d50c and m values of 20 𝑚𝑖𝑐𝑟𝑜𝑛𝑠 and 1.9 respectively.
The carrier fluid used in this case was municipal water from the Tlokwe municipality.
3.3 Equipment
3X 5 litre buckets
2X 20 litre buckets
1X water gun
1X Marcy scale
1X Doppler flow meter
50X poly tops
1X large syringe
1X spoon
3.4 Experimental setup
A diagram showing the experimental setup could be observed on Figure 3.1. The geometry of
the hydrocyclone that was used in this study is displayed on Table 3.1. The alphabetical
numbering in Figure 3.1 are explained below:
A: Slurry storage tank
B: Circulation pump
C: Main feed bypass valve
D: Feed fine tune bypass valve
E: Feed shutdown valve
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Experimental procedure| 21
F: Pressure gauge
G: Doppler flow meter
H: Hydrocyclone
I: Hydrocyclone overflow
J: Hydrocyclone underflow
K: Sample taken from the hydrocyclone overflow
L: Sample taken from the hydrocyclone underflow
M: Mixer
Figure 3.1: Diagram of the hydrocyclone setup
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Experimental procedure| 22
The mixer, mentioned above, consists of square tubing that transports the fluid from the
fine tuning bypass valve to the bottom of the storage tank. Holes were made at the end of
the square tubing in order for the slurry to be sprayed towards the sides of the storage
tank so as to promote better mixing.
Two sampling containers are located above the storage tank and below the overflow and
underflow outlets. As soon as the container of the underflow is pushed in under the
underflow, a mechanism pushes the overflow pipe in the overflow container, meaning that
the overflow and the underflow are sampled simultaneously. The experimental
hydrocyclone setup can be seen on Figure 3.2. The two containers that store the underflow
and the overflow are also indicated on this figure.
Figure 3.2: The experimental hydrocyclone setup
Where:
A: Hydrocyclone
B: Hydrocyclone overflow
C: Hydrocyclone underflow
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Experimental procedure| 23
D: Sampling container for the underflow
E: Sampling container for the overflow
F: Slurry storage tank
Table 3.1: Hydrocyclone geometry
Part Size
𝐷𝑐 10 𝑐𝑚
𝐷𝑖 3.03 𝑐𝑚
𝐷𝑜 3.4 𝑐𝑚
ℎ 53 𝑐𝑚
3.5 Experimental procedure
3.5.1 Preparation
3.5.1.1 Doppler flow meter calibration
The Doppler flow meter is installed on a suitable place where minimum noise will occur due to
turbulence in the piping. The storage tank was initially loaded with water only. After the pump
was turned on, one person read the value from the Doppler flow meter display, while the other
person fills the underflow and overflow containers with the water coming from the
hydrocyclone. The underflow and the overflow of the hydrocyclone is equal to the feed to the
hydrocyclone. The person filling the underflow and overflow buckets also has to keep track of
the time in which the containers are filled. From the volume of the water collected and the time
in which the water was collected, one can then calculate the real feed flowrate to the
hydrocyclone. The Doppler flow meter was calibrated accordingly. This procedure was
repeated until the error between the Doppler flow meter and that measured with the container
and watch method was small enough.
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Experimental procedure| 24
3.5.1.2 Marcy scale calibration
The Marcy scale is a handy tool that could be used to determine the density of a slurry mixture.
A picture of the Marcy scale could be observed on Figure 3.3. Before its use, it has to be
calibrated with the above mentioned municipal water. The density value is set to 1000𝑘𝑔
𝑚3 when
calibrated.
3.5.1.3 Slurry preparation
One of the variables that also has to be monitored is the volumetric percentage of solids in the
feed. The slurry tank (storage tank) was firstly filled with 200 𝑙𝑖𝑡𝑒𝑟𝑠 of municipal water. The
weight of silica sand that has to be added to the tank to obtain a certain volumetric solids
percentage is calculated with equation 3.1.
𝑚𝑠 =
𝜑 × 𝑉𝑤1𝜌𝑠
− 𝜑 ×1𝜌𝑠
3.1
Where:
𝑚𝑠=Mass of silica sand that has to be added to the storage tank
𝜑=Desired volume percentage of solids in the slurry
Figure 3.3: The Marcy scale
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Experimental procedure| 25
𝜌𝑠=Density of silica sand = 2650𝑘𝑔
𝑚3
𝑉𝑤=Volume of water in the tank = 200 𝑙𝑖𝑡𝑒𝑟 = 0.2 𝑚3
3.5.2 Sampling
Step-by-step instructions for obtaining samples from the rig5 are given in this section. These
steps could only be followed after the preparation mentioned in chapter 3.5.1 have been
completed:
1. Make sure that the valve between the pump opening and the storage tank exit is fully
opened;
2. Make sure that there are no objects in the storage tank that could cause pump failure.
3. Close the feed shutdown valve;
4. Close both of the valves from the overflow and underflow containers;
5. Fully open both the feed bypass valves;
6. Turn on the pump;
7. The slurry from the bypass valves will lead to plenty of turbulence in the storage tank.
It is however recommended that the storage tank also be mixed manually so as to
ensure that most of the silica particles are suspended in the slurry;
8. Fully open the feed shutdown valve;
9. Slowly close the feed bypass valves while keeping an eye on the pressure gauge. Stop
closing the bypass valves as soon as the required pressure is reached;
10. One person has to take note of the flow rate, while the other person has to push in the
underflow sampling container. This has to be done at the same time. The person
recording the flow rate from the Doppler flow meter also has to start and stop a
stopwatch when the containers are firstly inserted and pulled out again;
11. As soon as the underflow and overflow containers have been pulled out, the pump
may be stopped;
12. Separate buckets have to be inserted under the hoses that are connected to the outlet
valves of the underflow container and the overflow container;
13. Slowly open the outlet valves of the overflow and underflow containers and collect the
underflow and overflow samples in the separate buckets. The content of the underflow
and overflow containers have to be stirred well while the outlet valves are opened so
as to avoid silica sand from settling and remaining in the underflow or overflow
containers;
5 The terms “rig” and “hydrocyclone experimental setup” are used interchangeably in this study.
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Experimental procedure| 26
14. The buckets have to be weighed separately on scale. The scale should previously
have been reset with the mass of the buckets that are used. Similar buckets thus have
to be used. The mass of the content inside the buckets are recorded;
15. A smaller sample of overflow and the underflow are taken by mixing the slurry in the
buckets and filling a poly top with the content. The poly tops should be labelled
thoroughly.
16. The remaining content in the buckets are again stirred before the Marcy scale bucket
is filled with the slurry. The Marcy scale bucket is put on the Marcy scale to determine
the density of the slurry. The densities of both the slurries have to be determined this
way.
The buckets containing the remaining slurry are emptied into the slurry storage tank of the rig.
The above steps are then repeated for the next sample.
3.5.3 Analysing
The particle size distribution of the underflow samples are all determined with the use of the
Malvern Mastersizer 2000. The particles are circulated through the Mastersizer where they
eventually pass through a laser beam. The particles passing through the laser beam scatter
some of the radiation from the laser beam. The intensity of the backscattering of the laser light
from the particles are measured with special backscatter detectors. The angle at which the
light is scattered is inversely proportional to the size of the silica particles (Malvern
instruments, 2005). Figure 3.4 is a picture of the Malvern Mastersizer 2000.
Figure 3.4: Malvern Mastersizer 2000
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Experimental procedure| 27
3.5.4 Experimental error
For the experimental error determination, 4 random operating conditions were chosen from all
the experiments that were conducted. Six runs were completed on each of these operating
conditions. A total of 26 experiments were thus completed in order to determine the
experimental error. The conditions at which each of the sets were done, as well as the results
could be observed in Appendix D. All the calculations that were done in the determination of
the experimental error could be found in the electronically attached spreadsheet named
“Experimental Error”.
It is assumed that the data follows a normal distribution. Due to the small amount of available
data for each of the sets, the experimental error had to be determined using the student’s t
test (Devore & Farnum, 2005:313-318).
The experimental error could be determined with equation 3.2.
𝑡𝑛−1(
𝛼
2) ×
𝑆𝑡
√𝑛𝑢
3.2
Where:
𝑡𝑛−1(𝛼
2)=Critical t value that could be obtained from the back cover of Devore and Farnum
(2005)
𝑆𝑡=Standard deviation of the data
𝑛𝑢=Number of data points available in the set
A 95% confidence interval was used to obtain the experimental error. The processed
experimental data that were used for the determination of the experimental error could be
observed in Table 3.2.
Table 3.2: Processed data used for the experimental error determination
Data Set 1 Set 2 Set 3 Set 4
d50c m d50c m d50c m d50c m
1 21.71 1.35 28.41 2.03 17.72 1.13 29.58 1.44
2 16.59 1.37 24.51 1.92 21.91 1.21 30.83 1.66
3 18.63 1.49 29.57 1.89 22.02 1.21 30.99 1.69
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Experimental procedure| 28
4 - - - - 22.26 1.30 31.44 1.71
5 - - - - 23.41 1.34 31.66 2.00
Valid 18.44 1.29 26.03 1.97 21.82 1.16 29.74 1.60
Two of the data points in both sets 1 and 2 have been discarded due to their large deviation
with the rest of the data in the set. One data point in each set have been used as a validation
data point. The values that were substituted into equation 3.2 to calculate the experimental
error of each set could be observed on Table 3.3. The results of the d50c experimental error
for each data set and the results of the sharpness of separation error for each data set could
be observed on Figure 3.5 and Figure 3.6 respectively.
Table 3.3: Values for substitution into the student's t equation
Data Set 1 Set 2 Set 3 Set 4
d50c m d50c m d50c m d50c m
n 3 3 3 3 5 5 5 5
S 2.58 0.076 2.65 0.075 2.17 0.073 0.811 0.20
X6 18.98 1.40 27.50 1.95 21.46 1.24 30.90 1.70
T(95%) 4.303 4.303 4.303 4.303 2.776 2.776 2.776 2.776
6 X is the average of the data in the specific set
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Experimental procedure| 29
Figure 3.5: Experimental error of the d50c with a 95% confidence interval
Figure 3.6: Experimental error of the sharpness of separation with a 95% confidence interval
Large errors are observed for the d50c values in each of the sets in Figure 3.5. This could be
ascribed to the varying feed PSDs that will be dealt with later in this paper. The experimental
errors of the sharpness of separation as seen on Figure 3.6 are however acceptable.
5
10
15
20
25
30
35
40
45
0 1 2 3 4 5
d_5
0c
Set Number
D50c Average
Validation Data
0.5
1
1.5
2
2.5
3
0 1 2 3 4 5
m
Set Number
m Average
Validation Data
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Model development| 30
Chapter 4 - Model development
4.1 Overview
121 samples where processed to be put through the artificial neural network and the Plitt
model. Unfortunately, the raw data needed to be processed before it was fit to use in the
artificial neural network and the Plitt model. For more information on how the data was
processed, please refer to Appendix A.
4.2 The Plitt model
As mentioned before, the modified Plitt model with the fudging factors will be used in an
attempt to predict the d50c and the sharpness of separation of the hydrocyclone operated under
certain conditions. Of the 121 data points, 69 samples were used to fit the fudging factors with
the help of the Excel® add-in, Solver. The input parameters from the experimental data that
were not used in the tuning of the fudging factors were then substituted into the Plitt model.
The d50c and sharpness of separation results from Plitt model were then compared to the
corresponding experimental results. The Plitt model calculations could be found in the
electronically attached spreadsheet named “Plitt model”
4.2.1 Split flow
As mentioned before, this paper will only focus on predicting the d50c and the sharpness of
separation, m. The processed input data from Appendix C was inserted into the d50c and
sharpness of separation equations of the Plitt model.
For the split flow variable, 𝑆, of the d50c equation either the experimentally calculated 𝑆 or the
split flow calculated with one of the Plitt model equations given in equation 4.1 could be used.
4.1
The value of 𝐹4 was determined by minimizing the error between 69 of the experimental and
calculated split flow values with the help of the Excel® add-in, Solver. The resulting value of
𝐹4 was found to be 0.13. The remaining experimental values were then compared with
corresponding Plitt model values under the same operating conditions. The results of this
investigation are presented in Figure 4.1. Very small deviations from the experimental split
flow values are observed, meaning the split flow values from the Plitt model is suitable for
further use.
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Model development| 31
4.2.2 Cut size – d50c
The d50c value was calculated with equation 2.2. Just as with the split flow, 69 experimental
data points were used to adjust the value of 𝐹1. There are however another variable, 𝑘, that
could be adjusted in this equation. It was observed that solver could either vary 𝐹1 or 𝑘 to
obtain a minimum error. A value of 0.5 was arbitrarily chosen for 𝑘, while 𝐹1 was varied. The
resulting value for 𝐹1 is 64.9.
4.2.3 Sharpness of separation
The fudging factor of the sharpness of separation was determined the same way as the above
mentioned fudging factors.
4.3 The artificial neural network
The backpropagation algorithm will be used to train the neural network. A few modifications
were made to the ANN. This includes the addition of a regularization term and the addition of
a momentum term. Both these terms could be observed in equation 2.14 and 2.17. All artificial
neural networks that were constructed had various input variables and only one output
variable. The output variable was either the d50c or the sharpness of separation.
4.3.1 Artificial neural network architecture
Six different artificial neural networks have been written. The amount of neurons in each of
these networks could be varied between 1 and 20, while the input and output neurons cannot
be changed. Table 4.1 displays a list of all the ANNs that have been programmed. All these
programs could be found under the attached folder named “Artificial neural networks”.
Figure 4.1: Experimental split flow values plotted with the predicted Plitt model split flow values
0
0.5
1
1.5
2
2.5
0 10 20 30 40 50 60
Split
flo
w
Sample number
S experimental
S predicted
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Model development| 32
Table 4.1: Different artificial neural networks that were programmed
Neural network number Input variables Output variables
1 Du d50c
2 Du, 𝜑 and Q d50c
3 Du, 𝜑, Q, P, and S d50c
4 Du m
5 Du, 𝜑 and Q m
6 Du, 𝜑, Q, P, and S m
Separate neural networks for the d50c and sharpness of separation were constructed, as neural
networks that had both these variables as output, lacked the ability to learn.
It was decided that the first ANN of both the d50c output and sharpness of separation output
should only have the spigot diameter as input variable as it is known that this variable has the
largest effect on the hydrocyclone performance. In this study, the spigot diameter was
changed by switching off the pump and manually inserting a new spigot with a different
diameter. In industry, this would however be impractical. In a study conducted by Eren and
Gupta (1988), the spigot size could be adjusted pneumatically while the cyclone was on-line.
This study will thus be applicable to hydrocyclones which spigot size could be changed while
the cyclone is on-line.
The second set of neural networks contained the same inputs that are needed in the Plitt
model – the volumetric percentage solids in the feed 𝜑 and the feed volumetric flowrate 𝑄.
This neural network and the Plitt model are thus on equal grounds and could be compared
with one another.
For the third and last set of neural networks, the split flow and pressure drop over the cyclone
were added as inputs to test whether the predictive power of the neural network will improve.
Each neural network that were constructed had the ability to test 20 different architectures with
one click of a button. The networks could thus be run on multiple computers at the same time.
More neural networks could thus be tested in a shorter amount of time in comparison with
MATLAB®’s Neural Network Toolbox™.
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Model development| 33
Each of the neural networks were trained with roughly 75% of the experimental data. The
remaining 25% of the data was used as validation data. The validation data was used for all
the results that are displayed in chapter 5. None of the training data were thus used for
validation purposes.
To display the learning capability of the developed neural networks, a neural network that had
the spigot diameter as input parameter and the sharpness of separation as output parameter
was trained with 80 epochs and a training speed of 0.02. The results are displayed on Figure
4.2. The reader is referred to Appendix E for the source code of one of the artificial neural
networks.
Figure 4.2: Learning capability of one of the 6 developed artificial neural networks
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Results and discussion| 34
Chapter 5 - Results and discussion
After processing the data, it was found that the PSD of the feed varied considerably. An
alternative for calculating the feed PSD is dealt with in this section.
As mentioned before, the neural network was written in order to make it convenient for the
user to test multiple neural network architectures at once. This functionality was used to filter
out the more suitable neural network architectures for predicting the cut size and the
sharpness of separation. These filtered out neural networks were then further optimised. The
results as well as a discussion of these results are given in this section.
5.1 Deviations in the feed PSD
From the start of the sampling and analyses, it was assumed that the feed PSD remained
constant for all the slurry batches, as the same silica sand product from the same
manufacturer was used each time. It thus only seemed necessary to sample and determine
the PSD of the feed once and sample the underflow of each run, instead of sampling both the
underflow and the overflow of each run. This meant the total amount of PSD analyses could
be cut in half. The resulting partition curves that were produced had partition values that
exceeded 1 or was lower than 0. This means that the material balance did not solve. After
Figure 5.1: Particle size distribution of 25 different feed samples
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0 20 40 60 80 100 120 140 160
Vo
lum
e %
so
lids
Particle size [microns]
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Results and discussion| 35
taking samples of 25 different slurry mixtures7, it was found that the PSDs differed significantly
from each other as can be seen on Figure 5.1.
This meant that the partition curve could no longer be calculated from one feed PDS sample.
A solution to this problem was to calculate 25 different partition curves from the feed PSDs
that could be seen on Figure 5.1 for each of the underflow samples that were analysed.
One out of the 25 partition curves had to be chosen. The chosen partition curve had to fulfil
two criteria. Firstly, there may not be a value on the partition curve that exceeds 1. This would
mean that more of a certain size of particles exits the cyclone than have entered the cyclone.
According to the literature study, the correction made to the partition curve in order to obtain
the corrected partition curve is equal to the recovery of water to the underflow. The partition
curve thus also has to intersect the y-axis at a value that is close to the value of Rf. This is the
second constraint the partition curve has to meet.
Unfortunately, the Mastersizer was incapable of accurately measuring the particle sizes that
were smaller than 8.4 𝜇𝑚. The curve on Figure 5.2 shows the large fluctuations that occur at
particle sizes smaller than 8.4 𝜇𝑚. This phenomenon occurred in all the partition curves.
According to the results from the Mastersizer, the particles under 8.4 𝜇𝑚 amounted to 0.1%
of the total particles. The values of these particles will thus be neglected. The partition curve
value of the 8.4 𝜇𝑚 will thus be taken as the recovery of liquid to the underflow.
7 The word batch and slurry mixture are used interchangeably
Figure 5.2: Example partition curve before justifications
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Results and discussion| 36
A similar phenomenon was observed for particles larger than 95 𝜇𝑚. These particles
amounted to less than 0.05% of the total particles. It would thus also be a safe assumption to
ignore these particle sizes in further calculations.
After the partition curve that suited the description above was chosen, small changes were
made to the value of Rf so that it would be equal to the experimental Rf value. These small
changes could be observed on Figure 5.3.
5.2 Plitt model
5.2.1 Cut size – d50c
The d50c results of the modified Plitt model are displayed on Figure 5.4. The blue line connects
the experimental data points, while the orange line connects points that were predicted by the
Plitt model. The results are displayed in another form on Figure 5.5: where the predicted vs.
actual values are plotted over the 𝑦 = 𝑥 curve. To determine how well the data fits the 𝑦 = 𝑥
curve, a value called the coefficient of determination is calculated. This resulted in a 𝑅2 value
of 0.664.
Figure 5.3: Experimental vs. adjusted values of Rf
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Results and discussion| 37
Figure 5.5: Plitt model predicted cut size vs. experimental cut size plotted over the y=x
curve
Figure 5.4: Experimental cut point plotted with the cut point predicted by the Plitt model
0
5
10
15
20
25
30
0 10 20 30 40 50 60
d5
0c
[mic
ron
s]
Sample number
d50c experimental
d50c predicted
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Results and discussion| 38
From Figure 5.4, it is clear that the Plitt model was capable of predicting the cut point to a
certain extent. At the larger d50c values, the Plitt model tends to overpredict the d50c, while the
opposite is true for the smaller d50c values. The absolute error for the 44 validation data
points are 71.5 𝜇𝑚. The average error per predicted d50c value is thus 1.6 𝜇𝑚 which is
acceptable.
5.2.2 Sharpness of separation
The sharpness of separation results from the Plitt model could be observed on Figure 5.6.
Again, the experimental values for the sharpness of separation are connected by the blue line,
while the predicted values are connected by the orange line.
Figure 5.6: Experimental sharpness of separation plotted with the sharpness of separation
predicted by the Plitt model
0
1
2
3
4
5
6
0 10 20 30 40 50 60
Shar
pn
ess
of
sep
arat
ion
Sample number
m experimental
m predicted
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Results and discussion| 39
Figure 5.7: Plitt model predicted m vs. experimental m plotted over the y=x curve
From Figure 5.6 and Figure 5.7, it is clear that the Plitt model is incapable of predicting the
sharpness of separation. Deviations with an absolute value of 2 could easily be observed on
these figures.
5.3 Artificial neural networks
Various experiments were conducted in order to determine which neural network architecture
and parameters will be more suited for predicting the d50c and the sharpness of separation.
The same architectures and parameters were tested on both the d50c and the sharpness of
separation.
In the first series of tests, the number of epochs and the training speed was held constant
while the number of neurons in the hidden layer was varied between 3 and 20. Below 3 hidden
neurons, the neural network lacked the complexity to adequately predict the d50c and the
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Results and discussion| 40
sharpness of separation. All six neural networks mentioned in Table 4.1 were tested with these
architectures and parameters.
The architecture and parameters that were revealed to be the best out of those tested,
underwent further testing by increasing the amount of epochs by orders of magnitude and
decreasing the training speed so as to increase the chances of finding the global minimum.
The momentum and regularization terms were tested with the same architecture and
parameters as those used by neural network mentioned in the previous paragraph.
5.3.1 Cut size – d50c
5.3.1.1 Neural network screening
The results of training the neural network with only the spigot diameter, Du, as input are given
in Figure 5.8Error! Reference source not found.. Overtraining8 occurred at all of the tests
accept for the neural network that had 3 hidden neurons. It should be noted that the neural
networks stop training as soon as overtraining started.
The results in Figure 5.8 show that a simple neural network with no more than 6 hidden
neurons had the best prediction capabilities. Adding more hidden layers tends to
overcomplicate the network, leading to poorer results.
Figure 5.8: Results of neural network 2 trained with a training speed of 0.2 and a maximum amount of epochs of 8000
8 Overtraining takes place when the artificial neural network stops to converge to an answer and starts to diverge
50
55
60
65
70
75
80
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Co
mb
ined
ab
solu
te e
rro
r o
f 4
4 r
esu
lts
[mic
ron
s]
Number of hidden neurons
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Results and discussion| 41
By adding more input parameters to the neural network, even better results are achieved. The
results could be seen on Figure 5.9. All networks in this test were trained until overtraining
commenced. A maximum error just above 1.35 micron per validation data point was achieved
in this neural network.
Figure 5.9: Results of neural network 2 trained with a training speed of 0.2 and a maximum amount of epochs of 8000
Two more input parameters, the split flow and the pressure drop over the cyclone were
inserted. The addition of these two parameters produced better results than the previous tests.
No trend could be observed in the absolute error as the amount of neurons were increased.
The results are given on Figure 5.10.
Figure 5.10: Results of neural network 3 trained with a training speed of 0.2 and a maximum amount of epochs of 8000
52
53
54
55
56
57
58
59
60
61
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Co
mb
ined
ab
solu
te e
rro
r o
f 4
4 r
esu
lts
[mic
ron
s]
Number of hidden neurons
44.4
44.6
44.8
45
45.2
45.4
45.6
45.8
46
46.2
46.4
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Co
mb
ined
ab
solu
te e
rro
r o
f 4
4 r
esu
lts
[mic
ron
s]
Number of hidden neurons
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Results and discussion| 42
5.3.1.2 Enhancing the neural network
From the results in Figure 5.8, Figure 5.9 and Figure 5.10, it is clear that the predictive power
of the neural network increases with an increase in the number of inputs. It was thus decided
to further develop neural network 3 for predicting the d50c of the hydrocyclone.
The neural network was given 12 hidden neurons and firstly trained with a maximum of 60000
epochs and a training speed of 0.02. It was expected that the absolute error observed in Figure
5.10 would decrease, instead, the error increased with almost 1.6 microns to 47.19 𝜇𝑚. An
explanation for this phenomenon could be that this neural network just happened to step over
the local minimum that was found by the neural network in Figure 5.10. Another representation
of the results is given in Figure 5.12.
Although there are some neural network output values that differ with 2 𝜇𝑚, from the
experimental d50c values, Figure 5.11 shows that the neural network has adequate prediction
power.
Figure 5.11: Calculated d50c plotted with the experimental d50c values of neural network 3
trained with a maximum of 60000 epochs and a training speed of 0.02
13
15
17
19
21
23
25
27
0 10 20 30 40 50
d_5
0c
Validation sample number
Calculated d_50c
Experimental d_50c
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Results and discussion| 43
Figure 5.12: Predicted d50c vs. experimental d50c plotted over the y=x curve for the neural
network trained with a maximum of 60000 epochs and a training speed of 0.02
For the next enhancement, the momentum term will be used. The momentum constant was
given a value of 1 × 10−6. The other parameters and architecture of the neural network
remains unchanged. A significant reduction of more than 3 𝜇𝑚 was observed in the combined
error when compared to the previous test. The value of the combined error in this case is
44.15 𝜇𝑚. It can also be seen on Figure 5.14 that the 𝑅2 value decreased by 0.05 to 0.795.
When looking at the calculated and experimental graph on Figure 5.13, certain improvements
could be spotted. As an example, the last validation data point lies on the predicted d50c value.
This was not the case in Figure 5.11.
School of Chemical and Minerals Engineering
Results and discussion| 44
Figure 5.13: Calculated vs. experimental values of neural network 3 trained with a maximum of 60000 epochs and a training speed of 0.02 with the addition of the momentum term
Figure 5.14: Predicted d50c vs. experimental d50c plotted over the y=x curve for a neural network trained with a maximum of 60000 epochs and a training speed of 0.02
13
15
17
19
21
23
25
27
0 10 20 30 40 50
d_5
0c
Validation sample number
Calculated d_50c
Experimental d_50c
School of Chemical and Minerals Engineering
Results and discussion| 45
For the final neural network enhancement, the momentum term will be deactivated while the
regularization term will the activated. The regularization constant was set to a value of 1 ×
10−4. The regularization term only produced slight improvements when compared to the
initial neural network enhancement. The resulting combined absolute error was 46.45 𝜇𝑚.
The validation results are displayed on Figure 5.15. The predicted vs. experimental d50c
could be observed on Figure 5.16. Only slight differences are observed in the graphs of
Figure 5.13 and Figure 5.15.
Figure 5.15: Calculated vs. experimental values of neural network 3 trained with a maximum of 60000 epochs and a training speed of 0.02 with the addition of the regularization term
13
15
17
19
21
23
25
27
0 10 20 30 40 50
d_5
0c
Validation sample number
Calculated d_50c
Experimental d_50c
School of Chemical and Minerals Engineering
Results and discussion| 46
Figure 5.16: Predicted d50c vs. experimental d50c plotted on the y=x curve for the neural network trained with a maximum of 60000 epochs and a training speed of 0.02 with the
addition of the regularization term
5.3.2 Sharpness of separation
5.3.2.1 Screening of neural networks
Screening of the neural networks with the sharpness of separation as output was done the
same way as the screening of the d50c neural networks. The results of neural networks 4, 5
and 6 are given in Figure 5.17, Figure 5.18 and Figure 5.19 respectively. Most of the networks
were trained until overtraining commenced.
School of Chemical and Minerals Engineering
Results and discussion| 47
Figure 5.17: Results of neural network 4 trained with a training speed of 0.5 and a
maximum amount of epochs of 20000
Figure 5.18: Results of neural network 5 trained with a training speed of 0.5 and a
maximum amount of epochs of 25000
9
9.2
9.4
9.6
9.8
10
10.2
10.4
10.6
10.8
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Co
mb
ined
ab
solu
te e
rro
r o
f 4
4 r
esu
lts
[mic
ron
s]
Number of hidden neurons
8
8.5
9
9.5
10
10.5
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Co
mb
ined
ab
solu
te e
rro
r o
f 4
4 r
esu
lts
[mic
ron
s]
Number of hidden neurons
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Results and discussion| 48
Figure 5.19: Results of neural network 6 trained with a training speed of 0.2 and a maximum amount of epochs of 8000
The same phenomena that happened in neural networks 1, 2 and 3 were observed in neural
networks 4, 5 and 6. When the amount of inputs to the neural network was less than or equal
to 3, the predictive capability of the neural networks reached their peak when the amount of
hidden neurons were capped at 8. There again was no trend in the prediction power of the
neural network as the amount of hidden neurons were increased for the neural network that
had 5 inputs. An increase in the amount of inputs to the neural network also lead to an
improved predicting capability. It was thus decided that neural network 6 should be further
developed.
5.3.2.2 Enhancing the neural network
It was decided that neural network 6 should be given 13 hidden nodes, as good results were
obtained with this amount of hidden nodes as can be seen on Figure 5.19. The network was
trained with a maximum of 60000 epochs and a training speed of 0.02. After the first test, the
momentum term was added with a momentum constant of 1 × 105 and in the second test, the
momentum term was deactivated and the regularization term was inserted with a
regularization constant of 0.001. The results could be observed on Figure 5.20, Figure 5.22
and Figure 5.24. Predicted vs. calculated plots could be seen on Figure 5.21, Figure 5.23 and
Figure 5.25.
9.06
9.07
9.08
9.09
9.1
9.11
9.12
9.13
9.14
9.15
9.16
9.17
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Co
mb
ined
ab
solu
te e
rro
r o
f 4
4 r
esu
lts
[mic
ron
s]
Number of hidden neurons
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Results and discussion| 49
Figure 5.20: Experimental and predicted values of neural network 6 trained with a
maximum of 60000 epochs and a training speed of 0.02
Figure 5.21: Predicted vs. experimental m plotted over the y=x graph for neural network 6
trained with a maximum of 60000 epochs and a training speed of 0.02
2
2.5
3
3.5
4
4.5
0 5 10 15 20 25 30 35
m
Validation sample number
Experimental m
Predicted m
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Results and discussion| 50
Figure 5.22: Predicted and experimental values of neural network 6 trained with a
maximum of 60000 epochs and a training speed of 0.02 with the addition of the
momentum term
Figure 5.23: Predicted m vs. experimental values m plotted over the y=x line for neural
network 6 trained with a maximum of 60000 epochs and a training speed of 0.02 with the
addition of the momentum term
2
2.5
3
3.5
4
4.5
0 5 10 15 20 25 30 35
m
Validation sample number
Experimental m
Predicted m
School of Chemical and Minerals Engineering
Results and discussion| 51
Figure 5.24: Predicted vs. experimental values of neural network 6 trained with a maximum of 60000 epochs and a training speed of 0.02 with the addition of the
regularization term
Figure 5.25: Predicted m vs. experimental m plotted over the y=x curve for neural network
6 trained with a maximum of 60000 epochs and a training speed of 0.02 with the addition
of the regularization term
2
2.5
3
3.5
4
4.5
0 5 10 15 20 25 30 35
m
Validation sample number
Experimental m
Predicted m
School of Chemical and Minerals Engineering
Results and discussion| 52
Accept for the large outliers observed near validation sample number 20 and at validation
sample 13, the sharpness of separation was predicted with reasonable accuracy. When
comparing the graphs, Figure 5.20, Figure 5.22 and Figure 5.24, one observes that there are
no significant differences. The combined absolute error of the neural network that had no
regularization nor momentum term had a combined validation error of 9.11 for the sharpness
of separation, meaning that the sharpness of separation was out with an average value of
0.21 per validation sample. The momentum term further decreased the combined error to
9.04, while the addition of the regularization term significantly decreased the combined error
to 8.87, meaning that the average error per validation sample prediction was decreased to
0.2.
School of Chemical and Minerals Engineering
Conclusion and recommendations| 53
Chapter 6 - Conclusion and recommendations
6.1 Conclusion
In this study, the modified Plitt model and an artificial neural network trained with the
backpropagation algorithm were compared with one another to act as soft sensors for
predicting hydrocyclone d50c and sharpness of separation.
After conducting the experiments, and only analysing the PSD of the underflow samples, it
was found that the PSD of the feed varied considerably with each sample that was taken. An
alternative to calculate the PSD of the feed was devised.
The Plitt model was capable of predicting the d50c with a low accuracy. Its results could
however still be used to sufficiently control the d50c of a hydrocyclone, as the results in Figure
5.4 show that the experimental and predicted lines frequently fall on each other. The Plitt
model, however, had no predicting power when it came to predicting the sharpness of
separation.
The error in the results of the artificial neural network decreased when the number of input
neurons were increased. For a small number of input parameters (between 1 and 3) a simple
neural network achieved better results than the complicated neural networks that had more
than 6 hidden neurons.
The prediction of the cut size from the neural network that had only the diameter of the spigot
as input was more than 20% better than the prediction made by the modified Plitt model.
Improved prediction was observed when the number of input neurons to the neural network
was increased.
In comparison to the Plitt model that could not predict the sharpness of separation, the artificial
neural network was capable of predicting the sharpness of separation with an average error
of the sharpness of separation being 0.2, which is a significant error, but could still be used to
sufficiently control the hydrocyclone.
Artificial neural networks are superior to the Plitt model when it comes to predicting the
sharpness of separation and the cut size. Artificial neural networks are thus recommended for
hydrocyclone performance prediction. The Plitt model is however much more user friendly and
can predict the cut size with reasonable accuracy.
6.2 Recommendations
When an inference sensor is needed for predicting hydrocyclone performance, it is
recommended that the hydrocyclone be equipped with as much sensors as possible. This
School of Chemical and Minerals Engineering
Conclusion and recommendations| 54
means that the artificial neural network will have more inputs, resulting in a better prediction
of the hydrocyclone performance.
The experimental procedure and setup could be improved in the following ways:
Never rely on the density of the slurry that was calculated with the ratio of water to
silica sand added to the slurry batch. Even if the storage tank is thoroughly mixed,
there will always be some silica sand that settles.
Determine the density of the feed from the underflow and overflow densities obtained
from the Marcy scale.
Due to the importance of the Marcy scale in the experimental procedure, it is
recommended that a second Marcy scale be obtained and the densities of the
underflow and the overflow be determined with both Marcy scales.
Analyse the PSDs of both the underflow and the overflow. From this information, the
PSD of the feed could be obtained. Do not attempt to sample a slurry mixture in the
tank to determine the feed PSD as the results will be misleading.
Only construct neural networks that has one output variable.
6.3 Further study
There is little literature that deals with predicting the sharpness of separation of the
hydrocyclone. Without the sharpness of separation, the separation efficiency of the
hydrocyclone could not be determined. Further study is thus required.
A further study could also be conducted on the variable diameter spigot used in the study of
Eren and Gupta (1988).
Additional input that could be inserted to the artificial neural network include that angle of
discharge of the underflow and the recovery of water to the underflow.
School of Chemical and Minerals Engineering
Bibliography| 55
Bibliography
Aldrich, C., Uahengo, F.D.L. & Kistner, M. 2014. Estimation of particle size in hydrocyclone underflow streams by use of multivariate image analysis. Minerals engineering, 70:14-19.
Anglo American Platinum. 2013. Integrated report 2013. <http://www.angloamericanplatinum.com/~/media/Files/A/Anglo-American-Platinum/documents/aap-iar-2013.pdf> Date of access: 3 March 2015.
Anglo Gold Ashanti. 2013. Integrated report 2013. http://www.aga-reports.com/13/ir/review/regional-reviews/south-africa Date of access: 3 March 2015.
Awodele, O. & Jegede, O. 2009. Neural networks and its applications in engineering. Paper presented at the Informing science & IT education conference, Macon, United States, 12-15 June 2009.
Basheer, I.A. & Hajmeer, M. 2000. Artificial neural networks: fundamentals, computing, design and application. Journal of microbiological methods, 43:3-31.
Bishop, C.M. 2008. Neural networks for pattern recognition. Auckland: Oxford university press.
Chen, W., Zydek, N. & Parma, F. 2000. Evaluation of hydrocyclone models for practical applications. Chemical Engineering Journal, 80:295-303.
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Concha, F., Barrientos, A., Montero, J. & Sampaio, R. 1996. Air core and roping in hydrocyclones. International Journal of Mineral Processing, 44-45:743-749.
Devore, J. & Farnum, N. 2005. Applied statistics for engineers and scientists. 2nd Edition. United States: Curt Hinrichs.
Eren, H., Fung, C.C. & Wong, K.W. 1997a. An application of artificial neural network for prediction of densities and particle size distributions in mineral processing industry. Paper presented at the IEEE instrumentation and measurement, Ottawa, Canada, 19-21 May 1997.
Eren, H., Fung, C.C., Wong, K.W. & Gupta, A. 1997b. Artificial neural networks in estimation of hydrocyclone parameter d50c with unusual input variables. IEEE transactions on instrumentation and measurement, 46(4):908-912.
Eren, H. & Gupta, A. 1988. Instrumentaion and on-line control of hydrocyclones. Paper presented at the International conference on process control, Oxford, 13-15 April 1988.
Flinthoff, B.C., Plitt, L.R. & Turak, A.A. 1987. Cyclone modeling: a review of present technology. CIM Bulletin, 80(905):39-50.
Frachon, M. & Cilliers, J.J. 1999. A general model for hydrocyclone partition curves. Chemical Engineering Journal, 73:53-59.
Gallant, S. 1994. Neural network learning and expert systems. Cambridge: MIT Press.
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Hou, R., Hunt, A. & Williams, R.A. 1998. Acoustic monitoring of hydrocyclone performance. Minerals engineering, 11(11):1047-1059.
Jain, A.K. 1996. Artificial neural networks: a tutorial. Computer, 29(3):31-44.
Janse van Vuuren, M.J., Aldrich, C. & Auret, L. 2011. Detecting changes in the operational states of hydrocyclones. Minerals engineering, 24:1532-1544.
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Kelly, E.G. & Spottiswood, D.J. 1982. Introduction to mineral processing. 2nd Edition. New York: Wiley.
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Malvern instruments. 2005. Mastersizer 2000. http://www.ccm.usherbrooke.ca/fr/services/ccm/LCG/pdf/mastersizer2000.pdf Date of access: 15 October 2015.
McMillan, G. 1999. Process/Industrial Instruments and Controls Handbook. 5th Edition. New York: McGraw-Hill.
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Neesse, T., Schneider, M., Golyk, V. & Tiefel, H. 2004. Measuring the operating state of the hydrocyclone. Minerals engineering, 17:697-703.
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School of Chemical and Minerals Engineering
Data processing | I
Appendix A Data processing
The data of each sample collected from the hydrocyclone includes the following:
Volumetric feed flowrate 𝑙𝑖𝑡𝑒𝑟𝑠
𝑚𝑖𝑛𝑢𝑡𝑒
Mass of the underflow and the overflow collected in the underflow and overflow
containers
Density of the underflow
Density of the overflow
Pressure drop over the hydrocyclone
Percentage solids in the feed
Apex diameter
PSD of the underflow
From the above information, it is possible to calculate the following values that are required in
this study:
Spilt flow
Recovery of the feed liquid to the underflow, Rf
d50c
Sharpness of separation
The split flow could be calculated by substituting the densities and the mass of the underflow
and the overflow into equation A.1.
𝑆 =𝑉𝑢𝑉𝑜
=𝑚𝑢 ×
1𝜌𝑢
𝑚𝑜 ×1𝜌𝑜
A.1
The recovery of the water in the feed to the underflow has to be calculated next. In order to
calculate this, the volume of the underflow and the overflow, as well as their densities will be
used.
The volumetric solids percentage in the underflow could be calculated with equation A.2.
𝑥𝑢 =𝜌𝑢 − 𝜌𝑤𝜌𝑠 − 𝜌𝑤
A.2
The volumetric solids percentage of the overflow could be calculated with the same formula
as can be seen in equation A.3.
School of Chemical and Minerals Engineering
Data processing | II
𝑥𝑜 =𝜌𝑜 − 𝜌𝑤𝜌𝑠 − 𝜌𝑤
A.3
Where:
𝑥𝑢=Fraction of solids in the underflow
𝑥𝑜=Fraction of solids in the overflow
𝜌𝑢=Density of the underflow slurry in 𝑘𝑔
𝑚3
With these values now known, the recovery of water to the underflow can be calculated with
equation A.4.
𝑅𝑓 =
𝑉𝑢(1 − 𝑥𝑢)
𝑉𝑢(1 − 𝑥𝑢) + 𝑉𝑜(1 − 𝑥𝑜)
A.4
Next on the list to be calculated is the d50c and sharpness of separation, m. The PSD analysis
from the Malvern Mastersizer 2000 could now be used. In this case, the PSDs of the underflow
and the feed are known. When future analyses are done, the recommendation in chapter 6.2
should preferably be followed. In this recommendation the PSDs of both the underflow and
the overflow are used.
The partition curve could be calculated with A.5.
𝑃𝑖 =
𝑈𝑖𝐹𝑖
=𝑢𝑖𝑆𝑢𝑓𝑖𝑆𝑓
A.5
Where:
𝑃𝑖=The partition number or the recovery of particle size 𝑖 to the underflow
𝑈𝑖=Volume of particle size 𝑖 in the underflow
𝐹𝑖=Volume of particle size 𝑖 in the feed
𝑢𝑖=Volume fraction of particle 𝑖 in the underflow (excludes the carrier fluid)
𝑆𝑢=Volume of solids in the underflow calculated with equation A.6
𝑓𝑖=Volume fraction of particle 𝑖 in the feed (excludes the carrier fluid)
𝑆𝑓=Volume of solids in the feed calculated with equation A.7
School of Chemical and Minerals Engineering
Data processing | III
𝑆𝑢 = 𝑥𝑢𝑉𝑢 A.6
𝑆𝑓 = 𝑥𝑓𝑉𝑓 A.7
Where 𝑉𝑓 is the volumetric flowrate of the feed that could be calculated if the volumetric
flowrate of the overflow and the underflow are added together.
The corrected partition curve could now be calculated with
𝑃𝑖′ =
𝑃𝑖 − 𝑅𝑓
1 − 𝑅𝑓
A.8
Where 𝑃𝑖′ is the corrected partition curve number.
From the corrected partition curve, the d50c and the sharpness of separation, m, could now be
calculated by fitting the Rosin Rammler equation (equation 2.6) to the partition curve by
varying the sharpness of separation variable, m, and the d50c variables with the used of the
Excel® add-in, Solver.
To help with the processing of the large amount of data, a program in Excel® Visual Basic
had to be written. The program is stored in the file named “Data processing”. The source code
for this program could be observed in Appendix B.
School of Chemical and Minerals Engineering
Data processing source code | IV
Appendix B Data processing source code
Private Sub Calc1BTN_Click()
'Define the input variables
Dim Q As Single 'Stores the feed flow rate [l/min]
Dim P As Single 'Stores the pressure drop over the cyclone [kPa]
Dim Du As Single 'Stores the spigot inside diameter [cm]
Dim i As Integer 'Stores the current sample number
Dim Holder As String
Dim RightCounter As Integer
Dim Test As String
Dim WrongCounter As Integer
Dim ToDos As Integer
For i = 1 To 170 'This for loop goes through all the samples in the _
sheet "DataSheet"
Holder = Sheets("DataSheet").Cells(i * 4, 1)
Test = Sheets(Holder).Cells(1, 9)
If Test = "Right" Then
RightCounter = RightCounter + 1
ElseIf Test = "Wrong" Then
WrongCounter = WrongCounter + 1
Else
ToDos = ToDos + 1
Call SampleCalc1(i) 'The first calculations of the sample are _
performed in this sub
End If
School of Chemical and Minerals Engineering
Data processing source code | V
Next i
Sheets("DataSheet").Select
MsgBox ("Right: " & RightCounter & vbNewLine & "Wrong: " & WrongCounter _
& vbNewLine & "To Do: " & ToDos)
End Sub
Sub SampleCalc1(i As Integer) 'First calculations of the sample are _
performed
'Define the input variables
Dim MassUF As Single 'Stores the mass of the underflow [kg]
Dim MassOF As Single 'Stores the mass of the overflow [kg]
Dim DenUF As Single 'Stores the density of the underflow [kg/m^3]
Dim DenOF As Single 'Stores the density of the overflow [kg/m^3]
'Define calculation variables
Dim VolUF As Double 'Stores the volume of the underflow [m^3]
Dim VolOF As Double 'Stores the volume of the overflow [m^3]
Dim Splitflow As Double 'Stores the splitflow
Dim volfracsolUF As Double 'Stores the fraction of solids in the _
underflow
Dim volfracsolOF As Double 'Stores the fraction of solids in the _
overflow
Dim volsolUF As Double 'Stores the volume solids in the underflow
Dim volsolOF As Double 'Stores the volume solids in the overflow
Dim volwatUF As Double 'Stores the volume of water in the underflow
Dim volwatOF As Double 'Stores the volume of water in the overflow
Dim Rf As Double 'Stores the water recovery to the underflow
School of Chemical and Minerals Engineering
Data processing source code | VI
Dim persol As Double 'Stores the percentage solids in the feed
Dim TotalVolSol As Double 'Stores the total volume of solids in the _
feed
'Values from "DataSheet" are retrieved
MassUF = Sheets("DataSheet").Cells(i * 4 - 1, 4) / 1000 'Devided by 1000 _
to convert to kg
MassOF = Sheets("DataSheet").Cells(i * 4, 4) / 1000
DenUF = Sheets("DataSheet").Cells(i * 4 - 2, 7)
DenOF = Sheets("DataSheet").Cells(i * 4 - 1, 7)
'Volume of the overflow and underflow are calculated
VolUF = MassUF / DenUF
VolOF = MassOF / DenOF
'The splitflow is calculated
Splitflow = VolUF / VolOF
'The fraction of solids in the overflow and underflow are calculated _
where the density of water is taken as 1000kg/m^3 and that of silica _
as 2650kg/m^3
volfracsolUF = (DenUF - 1000) / (2650 - 1000)
volfracsolOF = (DenOF - 1000) / (2650 - 1000)
'The volume of solids in the overflow and underflow are calculated
volsolUF = VolUF * volfracsolUF
volsolOF = VolOF * volfracsolOF
TotalVolSol = volsolUF + volsolOF
'The volume of water in the overflow and the underflow is calculated
volwatUF = VolUF - volsolUF
volwatOF = VolOF - volsolOF
School of Chemical and Minerals Engineering
Data processing source code | VII
'The recovery of water to the underflow is calculated
Rf = volwatUF / (volwatUF + volwatOF)
'The percentage solids in the feed is calculated
persol = ((volsolUF + volsolOF) / (VolUF + VolOF)) _
* 100
'Export some of the variables to the other subroutines
Call SheetMaker(i, Splitflow, Rf, persol)
Call SheetCalc(i, TotalVolSol, volsolUF)
End Sub
Sub SheetMaker(i As Integer, Splitflow As Double, Rf As Double, persol As Double)
'In this subroutine, _
the new sheets that will store the information of each sample are created
'Declair the variables needed to make the sheets
Dim StringHolder As String 'Stores the name of the current sheet
Dim Tags As Worksheet 'The variable that will help with searching _
through the current sheets is created
StringHolder = Sheets("DataSheet").Cells(i * 4, 1) 'The sample name _
is retrieved
For Each Tags In Worksheets 'This for loop goes through all the _
sheets that has been created previously in this workbook
If Tags.Name = StringHolder Then 'This if statement checks _
whether the current sheet name matches that of the string _
held in "StringHolder". If it does, that sheet will be deleted
Application.DisplayAlerts = False
Sheets(StringHolder).Delete
School of Chemical and Minerals Engineering
Data processing source code | VIII
Application.DisplayAlerts = True
End If
Next
Sheets.Add(, ActiveSheet).Name = StringHolder 'A new sheet with the _
name stored in "StingHolder" is created and added after the sheet that _
containes this command button
'The headings of the new sheet are inserted
Sheets(StringHolder).Cells(1, 1) = "Size Interval"
Sheets(StringHolder).Cells(1, 2) = "Partition"
Sheets(StringHolder).Cells(1, 3) = "Corrected"
Sheets(StringHolder).Cells(1, 4) = "My Corrected"
Sheets(StringHolder).Cells(1, 5) = "Rosin"
Sheets(StringHolder).Cells(1, 6) = "Error"
Sheets(StringHolder).Cells(1, 12) = "Calc Rf"
Sheets(StringHolder).Cells(2, 12) = "My Rf"
Sheets(StringHolder).Cells(3, 12) = "D50c"
Sheets(StringHolder).Cells(4, 12) = "m"
Sheets(StringHolder).Cells(5, 12) = "P"
Sheets(StringHolder).Cells(6, 12) = "Feed sol%"
Sheets(StringHolder).Cells(7, 12) = "Du"
Sheets(StringHolder).Cells(8, 12) = "Q"
Sheets(StringHolder).Cells(9, 12) = "Split"
Sheets(StringHolder).Cells(1, 8) = "Status"
Sheets(StringHolder).Cells(6, 8) = "Rf_min"
Sheets(StringHolder).Cells(7, 8) = "Error"
Sheets(StringHolder).Cells(1, 13) = Rf 'The value of Rf is stored in _
the sheet for easy access
Sheets(StringHolder).Cells(2, 13) = Rf 'As an initial guess, the val _
ue of Rf is used in My Rf
School of Chemical and Minerals Engineering
Data processing source code | IX
Sheets(StringHolder).Cells(9, 13) = Splitflow 'The splitflow is stored
'Values from the sheet "DataSheet" are retrieved and stored in the _
selected sheet
Sheets(StringHolder).Cells(5, 13) = Sheets("DataSheet").Cells(i * 4, 7)
Sheets(StringHolder).Cells(7, 13) = Sheets("DataSheet"). _
Cells(i * 4 - 2, 10)
Sheets(StringHolder).Cells(8, 13) = Sheets("DataSheet").Cells(i * 4 - 2, 4)
'The calculated values from the previous sub routine is retrieved and _
stored in the new sheet
Sheets(StringHolder).Cells(6, 13) = persol
Sheets(StringHolder).Cells(1, 13) = Rf
'Values for d50c and m are guessed
Sheets(StringHolder).Cells(3, 13) = 25
Sheets(StringHolder).Cells(4, 13) = 1.2
End Sub
Sub SheetCalc(i As Integer, TotalVolSol As Double, volsolUF As Double)
Dim sizefracUF As Double 'Stores the volume fraction solids of a certain _
size in the underflow
Dim sizefracFEED As Double 'Stores the volume fraction solids of a _
certain size in the feed
Dim NameHolder As String 'Stores the name of the sheet that is focused _
on
Dim volsizeUF As Double 'Stores the total volume of solids in a specific _
size interval in the underflow [m^3]
Dim volsizeFEED As Double 'Stores the volume of solids in a certain size _
interval in the feed
School of Chemical and Minerals Engineering
Data processing source code | X
Dim Partition As Double 'Stores the recovery of solids to the underflow
NameHolder = Sheets("DataSheet").Cells(i * 4, 1) 'Retrieves the name of _
the sheet
For f = 1 To 43 'This for loop will insert the size interval to the sheet
Sheets(NameHolder).Cells(f + 1, 1) = Sheets("DataSheet").Cells _
(1, f + 11)
Next f
For t = 1 To 25 'This for loop will go through all the feed PSDs
Sheets(NameHolder).Cells(55, t) = t
For c = 1 To 43 'This for loop will go thorugh all the size intervals
sizefracUF = Sheets("DataSheet").Cells(i * 4, c + 11) / 100 'Cal - _
lates the volume fraction of solids in a certain size interval _
in fraction
volsizeUF = volsolUF * sizefracUF 'Calculates the total volume in a _
certain size interval in m^3
sizefracFEED = Sheets("FeedPSD").Cells(c + 1, t + 1)
volsizeFEED = TotalVolSol * sizefracFEED
Partition = volsizeUF / volsizeFEED
Sheets(NameHolder).Cells(55 + c, t) = Partition
Next c
Next t
'The corrected partition curve is calculated
Sheets(NameHolder).Range("C2").Select
ActiveCell.FormulaR1C1 = "=(RC[-1]-R1C13)/(1-R1C13)"
School of Chemical and Minerals Engineering
Data processing source code | XI
Sheets(NameHolder).Range("C2").Select
Selection.AutoFill Destination:=Sheets(NameHolder).Range("C2:C44")
'The Rosin Rammler equation is fitted to my corrected partition curve
Sheets(NameHolder).Range("E2").Select
ActiveCell.FormulaR1C1 = "= 1-EXP(-0.693*((RC[-4]/R3C13)^R4C13))"
Sheets(NameHolder).Range("E2").Select
Selection.AutoFill Destination:=Sheets(NameHolder).Range("E2:E44")
'The Error of the Rosin Rammler and the corrected partition curve is _
calculated
Sheets(NameHolder).Range("F2").Select
ActiveCell.FormulaR1C1 = "= ABS(RC[-2]-RC[-1])"
Sheets(NameHolder).Range("F2").Select
Selection.AutoFill Destination:=Sheets(NameHolder).Range("F2:F44")
'The "My Corrected" column is calculated
Sheets(NameHolder).Range("D2").Select
ActiveCell.FormulaR1C1 = "=(RC[-2]-R2C13)/(1-R2C13)"
Sheets(NameHolder).Range("D2").Select
Selection.AutoFill Destination:=Sheets(NameHolder).Range("D2:D44")
'Some of the values in the "My Corrected" column are made zero
For v = 2 To 18
Sheets(NameHolder).Cells(v, 4) = 0
Next v
'The absolute value of the first value of "My Corrected" is _
calculated
Sheets(NameHolder).Range("F45").Select
ActiveCell.FormulaR1C1 = "=SUM(R[-1]C:R[-43]C)"
School of Chemical and Minerals Engineering
Data processing source code | XII
'The sum of errors is calculated
Sheets(NameHolder).Range("I6").Select
ActiveCell.FormulaR1C1 = "=ABS(R19C4)"
'The error at the bottom of the page is copied up
Sheets(NameHolder).Range("I7").Select
ActiveCell.FormulaR1C1 = "=R45C6"
If Calc2Chkbox.Value = True Then
Call Calc2(NameHolder)
End If
End Sub
Sub Calc2(NameHolder As String) 'When activated, this will calculate _
the feed PSD that best fit the specific sample
Dim FirtZ As Single 'Stores the first value of the partition curve
Dim Rf As Single 'Stores the recovery of water to the underlflow
Dim Difz As Single 'Stores the difference between the first value of the _
partition number and the calculated Rf value
Dim maxz As Single 'Stores the maximum value of a range
Dim minz As Single 'Stores the minimum value of a range
Dim winner As Integer 'Stores the feed PSD number that has the best fit
For i = 1 To 25 'This for loop will go through all the feed PSDs
FirtZ = Sheets(NameHolder).Cells(73, i)
Rf = Sheets(NameHolder).Cells(1, 13)
Difz = Abs(FirtZ - Rf)
Sheets(NameHolder).Cells(53, i) = Difz
For t = 1 To 18 'This for loop goes through some of the size fractions
School of Chemical and Minerals Engineering
Data processing source code | XIII
maxz = Sheets(NameHolder).Cells(95, i)
If Sheets(NameHolder).Cells(95 - t, i) > maxz Then
maxz = Sheets(NameHolder).Cells(96 - t, i)
End If
Sheets(NameHolder).Cells(54, i) = maxz
Next t
Next i
minz = Sheets(NameHolder).Cells(53, 1)
winner = 1
For y = 2 To 25
If Sheets(NameHolder).Cells(53, y) < minz Then
minz = Sheets(NameHolder).Cells(53, y)
winner = y
End If
Next y
Sheets(NameHolder).Range("D2:D44").Select
With Selection.Interior
.Pattern = xlSolid
.PatternColorIndex = xlAutomatic
.Color = 15773696
.TintAndShade = 0
.PatternTintAndShade = 0
End With
School of Chemical and Minerals Engineering
Data processing source code | XIV
Sheets(NameHolder).Cells(1, 15) = "Winner"
Sheets(NameHolder).Cells(1, 16) = winner
Sheets(NameHolder).Cells(2, 15) = "Min"
Sheets(NameHolder).Cells(2, 16) = minz
Sheets(NameHolder).Cells(3, 15) = "Max"
Sheets(NameHolder).Cells(3, 16) = maxz
For r = 1 To 43
Sheets(NameHolder).Cells(r + 1, 2) = Sheets(NameHolder). _
Cells(r + 55, winner)
Next r
If Solv1.Value = True Then
Call Solver1(NameHolder)
End If
End Sub
Sub Solver1(NameHolder As String)
solverreset
Sheets(NameHolder).Select
SolverOk SetCell:="$Z$2", MaxMinVal:=2, ValueOf:=0, _
ByChange:="$M$2", Engine:=1, EngineDesc:="GRG Nonlinear"
SolverSolve (True)
solverreset
solverreset
Sheets(NameHolder).Select
SolverOk SetCell:="$F$45", MaxMinVal:=2, ValueOf:=0, _
ByChange:="$M$3:$M$4", Engine:=1, EngineDesc:="GRG Nonlinear"
School of Chemical and Minerals Engineering
Data processing source code | XV
SolverSolve (True)
solverreset
ActiveSheet.Shapes.AddChart2(240, xlXYScatterSmoothNoMarkers).Select
ActiveChart.SeriesCollection.NewSeries
ActiveChart.FullSeriesCollection(1).Name = "=""Corrected"""
ActiveChart.FullSeriesCollection(1).XValues = Sheets(NameHolder) _
.Range("$A$2:$A$44") '"=uf001new!$A$2:$A$44"
ActiveChart.FullSeriesCollection(1).Values = Sheets(NameHolder). _
Range("$D$2:$D$44") '"=uf001new!$D$2:$D$44"
ActiveChart.ChartArea.Select
ActiveSheet.Shapes("Chart 1").IncrementLeft 384
ActiveSheet.Shapes("Chart 1").IncrementTop -20.25
End Sub
Private Sub RetBtn_Click()
Dim Counter As Integer 'Holds track of the sample number
Sheets.Add(, ActiveSheet).Name = "Summary"
'The headers of the sheet "Summary" are created
Sheets("Summary").Cells(1, 1) = "\phi"
Sheets("Summary").Cells(1, 2) = "Q"
Sheets("Summary").Cells(1, 3) = "S"
Sheets("Summary").Cells(1, 4) = "P"
Sheets("Summary").Cells(1, 5) = "Du"
Sheets("Summary").Cells(1, 7) = "d50c"
Sheets("Summary").Cells(1, 8) = "m"
Sheets("Summary").Cells(1, 10) = "CalcRf"
Sheets("Summary").Cells(1, 11) = "MyRf"
School of Chemical and Minerals Engineering
Data processing source code | XVI
For t = 1 To 170 'This for loop goes through all the samples in the _
sheet "DataSheet"
Counter = Counter + 1
SName = Sheets("DataSheet").Cells(t * 4, 1) 'The name of the sample's
'sheet is retrieved
'If one of the sample sheets is marked "Wrong", then that sheet is _
skipped
If Sheets(SName).Cells(1, 9) = "Right" Then
Call StoreDat(Counter, SName)
Else
Counter = Counter - 1
End If
Next t
End Sub
Sub StoreDat(ByVal Counter As Integer, ByVal SName As String)
Dim phi As Double 'Feed solid %
Dim Q As Double 'Volumetric feed flow rate
Dim S As Double 'Split flow
Dim P As Double 'Pressure
Dim Du As Double 'Spigot diameter
Dim d50c As Double 'd50c
Dim m As Double 'Shapness of separation
Dim CalcRf As Double 'Calculated water recovery to the underflow
Dim MyRf As Double 'Adjusted recovery of water to the underflow
School of Chemical and Minerals Engineering
Data processing source code | XVII
'Retrieve the calculated data
phi = Sheets(SName).Cells(6, 13)
Q = Sheets(SName).Cells(8, 13)
S = Sheets(SName).Cells(9, 13)
P = Sheets(SName).Cells(5, 13)
Du = Sheets(SName).Cells(7, 13)
d50c = Sheets(SName).Cells(3, 13)
m = Sheets(SName).Cells(4, 13)
CalcRf = Sheets(SName).Cells(1, 13)
MyRf = Sheets(SName).Cells(2, 13)
'All flow rates are written in L/min
If Q < 20 Then
Q = Q * 60
End If
'Store the calculated data in a sheet
Sheets("Summary").Cells(Counter + 1, 1) = phi
Sheets("Summary").Cells(Counter + 1, 2) = Q
Sheets("Summary").Cells(Counter + 1, 3) = S
Sheets("Summary").Cells(Counter + 1, 4) = P
Sheets("Summary").Cells(Counter + 1, 5) = Du
Sheets("Summary").Cells(Counter + 1, 7) = d50c
Sheets("Summary").Cells(Counter + 1, 8) = m
Sheets("Summary").Cells(Counter + 1, 10) = CalcRf
Sheets("Summary").Cells(Counter + 1, 11) = MyRf
End Sub
School of Chemical and Minerals Engineering
Processed data | XVIII
Appendix C Processed data
Given Output
Run # Φ [%] Q [liters/min] S P [kPa] Du [cm] d50c [microns] m
1 1.69986 244.5 0.544962 92.00005 2.5 17.66313371 2.92202773
2 1.66544 213.66 1.29286 62 3 16.17171592 3.10376314
3 2.404552 222.66 0.140211 80.00003 1.5 25.17505927 2.52787572
4 2.087507 205.08 0.141425 73.5 1.5 23.77946671 2.27893268
5 2.600397 218.7 0.279845 67.99955 2 21.84222559 2.5688807
6 3.494631 228.06 0.596941 80 2.5 22.24664519 2.07061297
7 3.913908 222.6 1.61833 73.5 3.5 15.7337204 3.10273315
8 2.720839 251.04 0.522734 96.5 2.5 18.40977779 2.74669281
9 2.251916 246.66 0.5196 96.5 2.5 18.79550086 2.89213589
10 3.051449 225.6 0.542065 80.00003 2.5 19.77695069 2.83803783
11 1.329325 215.1 0.274854 73.5 2 20.26840629 2.43869745
12 2.258862 218.04 0.560153 73.5 2.5 17.97994585 2.74493577
13 2.133247 235.44 1.171561 73.5 3 16.33986556 2.94757378
14 1.365685 239.16 0.416858 92.00005 2.5 19.69173492 2.64548374
15 1.43384 218.28 0.535853 73.5 2.5 19.55908035 2.70848052
16 2.709411 225 0.279461 85 2 20.4132581 3.06952904
17 1.459614 229.26 1.056604 73.5 3 15.82208053 3.35056451
18 3.968196 245.52 1.742435 73.5 3.5 15.10384593 3.92868286
19 3.713268 244.68 0.514875 92.00005 2.5 21.40801222 2.18642919
20 1.286023 220.62 0.574188 73.5 2.5 18.50661932 2.91098016
21 3.056846 210.72 0.572225 67.99955 2.5 18.95271097 2.554091
22 3.018925 220.68 0.551918 73.5 2.5 18.35608964 3.06844869
23 2.26771 179.82 0.709792 50.5 2.5 19.00050366 2.56409372
24 1.685535 227.04 0.276303 85 2 25.90174433 1.93362332
25 2.512995 219.48 0.55922 73.5 2.5 19.62668094 2.61585898
26 2.797108 196.68 0.637743 56.00045 2.5 17.60385983 2.81246608
27 2.304915 196.68 0.298671 62 2 22.47328147 2.47890489
28 3.322398 197.94 0.64266 56 2.5 19.05721287 2.57342287
29 2.329241 236.64 0.55876 85 2.5 20.50854792 2.39299133
30 2.745848 189.12 0.663256 56.00045 2.5 18.15050956 2.64368276
31 2.886055 198.42 0.306488 62 2 21.90192579 2.38477985
32 3.706451 230.94 1.111032 73.5 3 16.60697902 3.26995851
33 1.313114 197.28 0.303064 62 2 21.45704641 2.58141132
34 3.021544 234.42 1.671793 67.99955 3.5 15.95754367 3.3828627
35 2.757227 229.68 0.970686 73.5 3 13.89502161 4.76739468
36 2.922461 237.42 1.654703 73.5 3.5 16.67944872 3.2686609
37 2.545808 216.72 0.627637 73.5 2.5 17.47772145 3.60222745
38 2.398254 182.94 0.334807 50.5 2 22.45328749 2.34980981
39 1.513598 248.64 0.51698 92.00005 2.5 17.65869522 2.74112357
40 2.708331 228.72 0.583455 73.5 2.5 17.98129083 2.84390874
School of Chemical and Minerals Engineering
Processed data | XIX
41 2.887919 199.92 0.331799 62 2 21.54445875 2.41125269
42 2.372339 191.58 0.30734 62 2 21.51785924 2.2480178
43 2.523559 211.56 0.279168 73.5 2 18.6589717 3.0168836
44 3.344824 203.76 1.864246 50.5 3.5 14.10703934 4.30396231
45 1.997808 270.42 1.627724 96.5 3.5 14.99876816 3.65766194
46 2.381545 267.36 0.986231 96.5 3 16.49542753 3.31490072
47 2.664498 249.66 1.044765 85 3 15.19231382 3.85450307
48 1.74529 228.54 0.540952 73.5 2.5 20.2553473 2.42722602
49 0.828973 210.72 0.618851 73.5 2.5 18.78090267 2.4953478
50 2.639677 250.5 1.043598 80.00003 3 17.07025127 2.9869346
51 1.890381 247.92 0.261902 96.5 2 20.77429575 2.45545381
52 3.282329 204.12 0.589184 62 2.5 17.88127358 2.75403292
53 0.878988 213.24 0.613246 73.5 2.5 17.18371551 2.95368686
54 2.025149 237.66 1.934818 73.5 3.5 13.98317424 4.52800458
55 2.621405 209.94 1.168482 62 3 16.67858579 3.07078049
56 2.452342 218.22 0.559194 73.5 2.5 19.42705478 2.69854736
57 2.866912 192.3 0.594211 56.00045 2.5 20.17155488 2.8169273
58 2.240576 241.8 0.533061 85 2.5 18.09833598 2.92491251
59 3.228658 220.26 0.563727 73.5 2.5 18.50623585 3.15418263
60 2.591063 244.98 0.545057 96.5 2.5 18.9750142 2.66325356
61 3.587327 235.2 1.107373 73.5 3 15.80830514 3.21404701
62 2.017803 218.58 0.283624 80.00003 2 19.25693874 2.81241486
63 2.643956 220.5 0.58515 73.5 2.5 19.12371243 2.67289938
64 1.310601 211.8 0.603282 73.5 2.5 20.93584615 2.37000244
65 4.199905 207.84 0.12938 73.5 1.5 24.9674299 1.9985633
66 1.658331 222.24 0.288898 85 2 20.16752147 2.62998717
67 2.492221 220.2 0.579805 73.5 2.5 19.42432178 2.67285009
68 0.897399 203.7 0.116874 73.5 1.5 21.31370199 2.44187801
69 1.527422 190.14 0.31806 62 2 21.31804583 2.84890422
70 3.414962 193.92 0.641569 56 2.5 18.18404235 2.80041182
71 3.702788 232.2 1.072705 73.5 3 16.578708 3.33839907
72 2.659981 199.5 0.598627 56.00045 2.5 19.75777072 2.58612282
73 0.552598 213.18 0.562195 73.5 2.5 21.56133592 2.12041595
74 2.024781 219 1.805619 60.045 3.5 13.8048085 4.77157728
75 2.326333 225.54 0.276579 85 2 19.47150797 2.58152233
76 2.642222 220.14 0.553937 73.5 2.5 21.06950699 2.27239309
77 2.362739 215.64 0.541087 67.99955 2.5 20.98519835 2.17103499
78 2.208887 242.52 0.527157 87.99 2.5 17.07470337 2.87460327
79 2.395077 220.2 0.557518 73.5 2.5 19.74300496 2.56354912
80 1.592669 246.3 1.083226 85 3 15.13884579 3.60111393
81 3.061191 215.46 1.204112 62 3 16.36779207 3.59398304
82 2.063873 203.28 1.269596 56.00045 3 16.46866041 3.30806363
83 3.645274 215.1 0.566624 73.5 2.5 18.32834616 3.11015966
84 1.958606 200.4 1.262059 56.00045 3 14.4961796 3.92853935
85 3.214574 226.86 1.102837 73.5 3 18.33841534 2.78762557
School of Chemical and Minerals Engineering
Processed data | XX
86 2.194286 216.36 0.135273 73.5 1.5 24.09940542 2.41116629
87 2.148065 217.14 1.158054 62 3 14.8066246 3.98846543
88 3.132617 246.72 1.083936 85 3 15.90521236 3.41669461
89 3.840857 245.82 1.79355 73.5 3.5 16.23496303 3.37440255
90 2.045328 175.14 0.153073 50.5 1.5 23.81013072 2.54459633
91 3.095857 215.76 1.206003 62 3 16.90405748 3.22866961
92 2.677514 237 0.278369 85 2 20.00006654 2.8609171
93 2.781939 204.6 0.278237 73.5 2 20.60448094 2.40739507
94 2.907741 232.08 1.123522 73.5 3 15.39401988 3.85351952
95 1.612302 193.32 0.310935 62 2 21.36381719 3.06585432
96 3.392846 229.14 1.116524 73.5 3 15.02961359 3.71865083
97 1.464103 243.12 1.08303 85 3 16.60096448 3.27680332
98 1.598391 216.42 0.292124 85 2 21.55786483 2.40778021
99 1.213497 213.48 0.298655 73.5 2 20.95678224 2.34819812
100 2.766897 254.22 1.012136 85 3 17.86673801 2.67751193
101 2.668857 213.72 0.548952 65.0015 2.5 21.25161767 2.29108849
102 2.01867 217.74 0.607982 67.99955 2.5 19.4653081 3.05822737
103 3.138117 252.42 1.090636 85 3 16.35565889 3.34711231
104 2.742736 211.98 1.160882 56.00045 3 14.36268237 4.06865538
105 1.446872 216.54 1.181585 62 3 16.12299352 3.38803078
106 2.357605 250.38 1.071753 85 3 14.78273842 3.56653929
107 3.58461 219.42 0.594438 73.5 2.5 17.79600264 2.77484605
108 2.839117 205.44 0.118157 73.5 1.5 24.13637863 2.22064247
109 2.19576 238.62 1.72273 73.5 3.5 17.01999127 2.87652474
110 2.212699 219 0.609548 73.5 2.5 17.15642707 3.53745683
111 2.379422 240.9 1.700301 73.5 3.5 14.97252646 3.77537739
112 2.124861 223.08 0.559162 73.5 2.5 18.83267088 2.96266872
113 2.037528 278.04 1.503341 96.5 3.5 17.31782995 2.71665116
114 3.376588 215.04 0.176226 80.00003 1.5 21.2911723 2.48259109
115 3.347966 240 0.527567 87.99 2.5 19.42920501 2.58904238
116 2.175198 232.2 0.564298 80.00003 2.5 17.16303467 2.91768219
117 2.288282 207.96 0.135709 68.003 1.5 24.87787051 2.41982415
118 3.141929 225.18 0.519934 80.00003 2.5 19.01813201 2.50098881
119 2.377447 273.42 1.599523 96.5 3.5 16.34973073 3.33707472
120 2.439948 235.62 0.574565 85 2.5 20.38595325 2.42744405
121 2.146811 217.08 0.727341 67.99955 2.5 17.25116705 2.76274276
School of Chemical and Minerals Engineering
Experimental error data | XXI
Appendix D Experimental error data
Name Pressure [kPa] sol %
Du [cm] m_OF [g] m_UF [g] 𝜌 _OF 𝜌_UF Q [l/min]
#1 - 1
68 0.625 2.5
10269.7 6203 1005 1025 210
#1 - 2 9708.3 5777.3 1006 1025 216
#1 - 3 10137.3 6013.1 1001 1029 216
#1 - 4 8680.9 5134.2 1005 1038 222
#1 - 5 10298 6104.2 1001 1028 222
#1 - 6 6056.9 3828.7 1028 1069 210
#2 - 1
62 1.25 2
7409.1 2316.1 1015 1095 210
#2 - 2 8087 2492.4 1001 1095 210
#2 - 3 6434.7 1974.5 1018 1090 204
#2 - 4 6861.1 2136.9 1012 1095 210
#2 - 5 6010.5 1868.6 1007 1085 210
#2 - 6 7549.6 2363.9 1020 1090 210
#2 Extra 2 6916.2 2249.1 1025 1100 192
#3 - 1
85 1.25 3
4971 5245.4 1020 1062 240
#3 - 2 4488.3 4563.3 1029 1060 240
#3 - 3 4293.8 4428.4 1034 1058 240
#3 - 4 4547.1 4867.1 1029 1060 240
#3 - 5 4010.7 4243.6 1031 1050 246
#3 - 6 3419.3 3475.1 1035 1068 240
#4 - 1
73.5 1.85 2
8535 2608.8 1035 1111 210
#4 - 2 7409.1 2224.3 1031 1095 210
#4 - 3 7268.8 2175.5 1030 1090 216
#4 - 4 6510 1969.8 1032 1105 216
#4 - 5 8218.4 2494 1028 1110 216
#4 - 6 11459.1 3331.9 1030 1112 216
School of Chemical and Minerals Engineering
ANN source code | XXII
Appendix E ANN source code
Sub Run_Click() 'When the run button is clicked, the code below executes
'The user input variables are defined:
Dim NumHidden As Integer 'Variable that stores the # of hidden nodes
Dim Speed As Single 'Variable that stores the learning rate
Dim Epochi As Integer 'Variable that stores the # epoch
Dim Epoch As Integer 'This variable is varied in the for loop
Dim Moment As Single 'Stores the momentum weight from the UI
Dim Reg As Single 'Stores the regularization weight from the UI
Dim Datap As Integer 'This is a variable that keeps track of the current
'datapoint that has to be executed
'Define the matrixes that have to be used
Dim InputWeights(19, 4) As Double 'A matrix is defined that will store the
'values of the weights going from the input nodes to the hidden nodes. The
'weights comming from each of the nodes are stored in a separate column
Dim OutputWeights(19, 0) As Double 'The output weights are stored
Dim HiddenInput(19, 0) As Double 'The calculated input values to the hidden
'nodes are calulated and stored in this matrix
Dim HiddenOutput(19, 0) As Double 'The outputs of the hidden nodes are stored
'here
Dim Checker As String 'This variable checks to see whether overtraining has
'occured or not
Dim a As Double 'Stores the value of the Hidden input
'The user inputs are retrieved from the sheet "UI"
Speed = Sheets("UI").Cells(2, 2) 'The training speed is retrieved
Epochi = Sheets("UI").Cells(1, 2) 'The number of epoch the user inserts is
'retrieved
Moment = Sheets("UI").Cells(3, 2) 'The momentum is stored in this variable
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Reg = Sheets("UI").Cells(4, 2) 'The regulazation weight that the user enters
'is stored here
Dim CellCounter As Integer 'This variable stores the amount of hidden nodes
'that the user wishes to test
CellCounter = NodeCount() 'This function counts the amount of hidden nodes
'the user wants to test
For Nodes = 1 To CellCounter 'This for loop is looped until all the hidden
'nodes that the user wants to test is tested
NumHidden = Sheets("UI").Cells(7 + Nodes, 1) 'The number of hidden nodes
'are retrieved from the sheet "UI"
Sheets("Calc").Range("F2:F80000").Clear 'The SSE values of the previous
'loop is cleared
For i = 0 To (NumHidden - 1) 'This for loop is used to populate the
'matrixes with random values. One is subtracted,as the elements in
'the matrix start at 0
InputWeights(i, 0) = Sheets("Data").Cells(i + 2, 23)
InputWeights(i, 1) = Sheets("Data").Cells(i + 2, 24)
InputWeights(i, 2) = Sheets("Data").Cells(i + 2, 25)
InputWeights(i, 3) = Sheets("Data").Cells(i + 2, 26)
InputWeights(i, 4) = Sheets("Data").Cells(i + 2, 27)
OutputWeights(i, 0) = Sheets("Data").Cells(i + 2, 23)
Next i
For Epoch = 1 To Epochi
For Datap = 4 To 74 'This loop will go through all the training
'data points
'The inputs and outputs of the ANN are defined
Dim phi_input As Double 'solids %
Dim Q_input As Double 'feed flowrate
Dim Du_input As Double 'diameter of spigot
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Dim P_input As Double 'pressure drop
Dim S_input As Double 'feed split
Dim d50_output As Double 'cutpoint
'''''''''''''''Feed Forward''''''''''''''''''
'The input values are retrieved from the
'sheet "Data"
phi_input = Sheets("Data").Cells(Datap, 13)
Q_input = Sheets("Data").Cells(Datap, 14)
Du_input = Sheets("Data").Cells(Datap, 5)
P_input = Sheets("Data").Cells(Datap, 16)
S_input = Sheets("Data").Cells(Datap, 15)
'The inputs are fed forward to the hidden layer
For k = 0 To (NumHidden - 1) 'Remember that we
'begin the counting of matrixes at 0
HiddenInput(k, 0) = InputWeights(k, 0) * _
phi_input + InputWeights(k, 1) * Q_input + _
InputWeights(k, 2) * Du_input _
+ InputWeights(k, 3) * P_input + _
InputWeights(k, 4) * S_input
a = HiddenInput(k, 0)
HiddenOutput(k, 0) = 1 / (1 + Exp(-a))
Next k
'Needed to calculate the inputs and outputs _
'to the hidden nodes are calculated
Dim Summer As Double 'Sums all the terms to _
the node
Dim InputLastNode As Double
'These two variables are initialized to avoid
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'previous loop values from intefering
Summer = 0
InputLastNode = 0
For y = 0 To (NumHidden - 1)
InputLastNode = Summer + HiddenOutput(y, 0) _
* OutputWeights(y, 0)
Summer = InputLastNode
Next y
d50_output = 1 / (1 + Exp(-InputLastNode))
'Retrieve the target values of d50
Dim d50_target As Double
d50_target = Sheets("Data").Cells(Datap, 19)
'Calculate the error for each of the outputs
Dim Errord50 As Double
Errord50 = d50_output * (1 - d50_output) * _
(d50_output - d50_target)
'Adjust the output weights
Dim deld50 As Double 'The change that has to _
take place in the d50 is stored here
For w = 0 To (NumHidden - 1)
deld50 = Speed * Errord50 * _
HiddenOutput(w, 0) + OutputWeights(w, 0) _
* Moment + Speed * Reg * _
(OutputWeights(w, 0) / ((1 + _
OutputWeights(w, 0) ^ 2) ^ 2))
OutputWeights(w, 0) = OutputWeights(w, 0) _
- deld50
Next w
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'Adjust input weights 1
Dim MultiM As Double
Dim ErrorHid(19, 0) As Double
For g = 0 To (NumHidden - 1)
MultiM = OutputWeights(g, 0) * Errord50
ErrorHid(g, 0) = HiddenOutput(g, 0) * _
(1 - HiddenOutput(g, 0)) * MultiM
Next g
'Adjust the input weights 2
Dim delphi As Double 'The change in the _
weight going from phi is stored
Dim delQ As Double 'The change in the weight _
going from Q is stored
Dim delDu As Double 'The change in the _
weights going from Du is stored
Dim delP As Double 'The change in the _
weights going from P is stored
Dim delS As Double 'The change in the _
weights going from S is stored
'In this for loop, the input weights are _
adjusted
For p = 0 To (NumHidden - 1)
delphi = Speed * ErrorHid(p, 0) * _
phi_input + InputWeights(p, 0) * Moment + _
Speed * Reg * (InputWeights(p, 0) / _
((1 + InputWeights(p, 0) ^ 2) ^ 2))
InputWeights(p, 0) = InputWeights(p, 0) _
- delphi
delQ = Speed * ErrorHid(p, 0) * Q_input _
+ InputWeights(p, 1) * Moment + _
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Speed * Reg * (InputWeights(p, 1) / ((1 _
+ InputWeights(p, 1) ^ 2) ^ 2))
InputWeights(p, 1) = InputWeights(p, 1) _
- delphi
delDu = Speed * ErrorHid(p, 0) * _
Du_input + InputWeights(p, 2) * Moment + _
Speed * Reg * (InputWeights(p, 2) / _
((1 + InputWeights(p, 2) ^ 2) ^ 2))
InputWeights(p, 2) = InputWeights(p, 2) _
- delDu
delP = Speed * ErrorHid(p, 0) * P_input _
+ InputWeights(p, 3) * Moment + _
Speed * Reg * (InputWeights(p, 3) / _
((1 + InputWeights(p, 3) ^ 2) ^ 2))
InputWeights(p, 3) = InputWeights(p, 3) _
- delP
delS = Speed * ErrorHid(p, 0) * S_input _
+ InputWeights(p, 4) * Moment + _
Speed * Reg * (InputWeights(p, 4) / _
((1 + InputWeights(p, 4) ^ 2) ^ 2))
InputWeights(p, 4) = InputWeights(p, 4) _
- delS
Next p
'The SSE is calculated
Dim SSE_d50 As Double
SSE_d50 = (d50_output - d50_target) ^ 2
Sheets("Calc").Cells(Datap - 1, 1) = SSE_d50
Next Datap
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Sheets("Calc").Cells(Epoch + 1, 6) = Sheets("Calc").Cells(2, 4)
'The value from the cell the calculates the total sse is stored in
'another cell
'If the previously calculated SSE is smaller than the current SSE _
then the variable "Checker" will have a value of "Yes" and all the _
calculated SSE's of this test is copied to the sheet "UI". The _
current for loop will also be exited.
If Sheets("Calc").Cells(Epoch, 6) < Sheets("Calc"). _
Cells(Epoch + 1, 6) Then
Checker = "Yes"
Sheets("Calc").Range("F2:F2002").Copy Destination:= _
Sheets("UI").Cells(36, Nodes)
Exit For
Else 'If the above is not true, "Checker" receives a value of "No"
Checker = "No"
End If
Next Epoch 'The next epoch of the test is executed
'The SSE values of the previous test is copied to the sheet "UI"
Sheets("Calc").Range("F2:F2002").Copy Destination:=Sheets("UI"). _
Cells(36, Nodes)
'The validation loop begins here
For Valid = 75 To 104 'This loop will go through all the _
validation data points
'The validation inputs are retrieved
phi_input = Sheets("Data").Cells(Valid, 13)
Q_input = Sheets("Data").Cells(Valid, 14)
Du_input = Sheets("Data").Cells(Valid, 5)
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P_input = Sheets("Data").Cells(Valid, 16)
S_input = Sheets("Data").Cells(Valid, 15)
'The inputs and outputs of the hidden layer are calculated
For k = 0 To (NumHidden - 1) 'Remember that we begin _
the counting of matrixes at 0
HiddenInput(k, 0) = InputWeights(k, 0) * _
phi_input + InputWeights(k, 1) * Q_input + _
InputWeights(k, 2) * Du_input + _
InputWeights(k, 3) * P_input + InputWeights(k, 4) _
* S_input
HiddenOutput(k, 0) = 1 / (1 + Exp(-HiddenInput(k, 0)))
Next k
'These values are initialized to avoid values from the _
previous loop from being carried over
Summer = 0
InputLastNode = 0
For y = 0 To (NumHidden - 1) 'This time, one is not subtracted, _
as we want the last value _
'of the summer to be added
InputLastNode = Summer + HiddenOutput(y, 0) * _
OutputWeights(y, 0)
Summer = InputLastNode
Next y
'The output d50c is calculated and copied to the sheet "Calc"
d50_output = 1 / (1 + Exp(-InputLastNode))
Sheets("Calc").Cells(Valid - 72, 9) = d50_output
'The target d50c is retrieved and stored in the sheet "Calc"
d50_target = Sheets("Data").Cells(Valid, 19)
Sheets("Calc").Cells(Valid - 72, 8) = d50_target
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Next Valid 'The next validation data point is put through the loop
'The sum of the absolute error of the validation data is copied to _
the sheet "UI"
Sheets("UI").Cells(7 + Nodes, 2) = Sheets("Calc").Cells(33, 10)
'If the "Checker" variable is equal to "Yes", then the epoch at _
which the over training occured is copied to the sheet "UI". _
If no overtraining occured however, it is indicated with a _
hifin
If Checker = "Yes" Then
Sheets("UI").Cells(7 + Nodes, 3) = Epoch
Else
Sheets("UI").Cells(7 + Nodes, 3) = "'-"
End If
Next Nodes 'The next test is conducted
End Sub
Function NodeCount()
'The variable "Counter" stores the amount of tests that the user _
wants to run
Dim Counter As Integer
t = 100
Do Until t = ""
Counter = Counter + 1
t = Sheets("UI").Cells(7 + Counter, 1)
Loop
NodeCount = Counter - 1 'One is subtracted, as the Do Until Loop _
adds an extra one.
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'As initialization, some of the cells in the sheets "UI" and _
and "Calc" are erased
Sheets("UI").Range("B8:C27").Clear
Sheets("Calc").Range("F2:F80000").Clear
End Function
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ECSA exit level outcomes | XXXII
Appendix F ECSA exit level outcomes
ELO 4: Investigation, experiments and data analysis
In order for the student to pass this ELO, the student has to prove that he or she can conduct
investigations and experiments. The student should also have the ability to design these
experiments.
Before the experiments were conducted on the hydrocyclone, an in-depth literature
study was done on the operation of the hydrocyclone, as well as on the Plitt model and
artificial neural networks;
From the knowledge of the literature study, a topic for the research, as well as a plan
for the investigation was formulated;
A routine was designed in which representative samples could be obtained from the
hydrocyclone. The developed routine could be observed on chapter 3 of this study;
The particle size distribution of the particles was determined with the use of the
Malvern Mastersizer 2000 with its accompanying software;
To help with the processing of the large amount of data, a program was written in
Excel® Visual Basic;
Referring back to chapters 4 and 5, interpretations were made from the processed data
and conclusions were drawn;
All the procedures followed in this study were recorded in this paper.
ELO 6: Professional and technical communication
This ELO is passed if the student can prove that he or she can communicate effectively, both
orally and in writing with engineers and non-engineers.
This report was structured in a logical way in order to effectively communicate the
experimental procedure and findings;
At the beginning of the year, a project proposal presentation was delivered orally to an
engineering audience;
Another oral presentation was delivered in order to convince the instructors that the
student will be able to finish the project;
Regular meetings were held during the course of the year with Mr. Frikkie van der
Merwe, the study leader. The agenda and minutes of these meeting are attached
electronically under the file named “Meetings”;
During the year, some technical problems occurred with the hydrocyclone. It was
necessary to communicate with workshop personnel to fix the problem;
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ECSA exit level outcomes | XXXIII
The master’s degree student that also worked on the rig was Miss Sarita van
Loggenberg. She was an electrical engineering student who sometimes needed advice
on the chemical engineering side of the project
ELO 8: Individual, team and multidisciplinary working
The student that passes this ELO is capable of working effectively as an individual and also
in teams. The student also has to be capable of working in multidisciplinary environments.
A literature study had to be done on the operation of the hydrocyclone and artificial
neural networks;
As mentioned above, Sarita van Loggenberg, who is an electrical engineer, helped
with the conducting of the experiments. She sometimes also needed chemical
engineering advice for her master’s project.
Neil Zietsman and Sarita van Loggenberg worked as a team to complete the
experiments before the deadline on which they decided;
Also mentioned above was that the workshop personnel were used to solve some of
the technical problems that appeared on the rig during the course of the year.
ELO 9: Independent learning ability
To pass this ELO, the student has to prove that he or she has the capability to use the skills
learned through the course of his or her undergraduate study to independently solve problems.
A literature study was done on the hydrocyclone and artificial neural networks;
Out of the literature study, an artificial neural network was independently constructed.
This neural network was programed with enhancements such as a regularization term
and a momentum term. The developed neural network also had the ability to test 20
different architectures at once;
Neil Zietsman devised a plant that produced more representative samples in a faster
time;
The experimental results had to be interpreted and conclusions had to be made at the
end of this report;
Trail-and-error had to be used in order to find the more appropriate neural network
architecture.
ELO 10: Engineering professionalism
To pass this ELO, the student has to prove that he or she acted professionally and ethically
while doing the project. The student also had to be capable of exercising his or her judgement
and take responsibility within own limits of competence.
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ECSA exit level outcomes | XXXIV
A HIRA study was conducted to identify possible hazardous conditions for those
working on the rig and also those working on other projects around the rig;
PPE was always worn while conducting experiments or analyses;
After a day’s work on the rig, the rig and the space around the rig was cleaned;
Each meeting held with the study leader was planned in advance. After the meeting,
a minutes was written. The minutes and agenda of each meeting could be found under
the attached folder named “Meetings”;
A good relationship was maintained with NWU personnel who were also involved in
this project.
ELO 11: Engineering management
For the student to pass this ELO, the student has to demonstrate knowledge and
understanding of engineering management principles and economic decision-making.
When something was required for the project, an order form was completed and signed
by the study leader;
Proof of payments with the study leader’s signature was handed in to the project
financial manager for further administrative processing;
The project spending did not exceed the budget of R2 200;
A Gantt chart was composed at the beginning of the year to plan all the project activities
that took place throughout the year. This could be observed in the folder named “Gantt
chart”;
Meetings were organised with the study leader and Miss Sarita van Loggenberg as
can be seen in the attached folder named “Meetings”
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Hazard identification and risk assessment | XXXV
Appendix G Hazard identification and risk assessment
Student: L.N. Zietsman (23379936)
Study Leader: Mr. F.A. van der Merwe
Brief Description of the Activity/Equipment
All activities and experiments will be conducted in the following locations at the North
West University of Potchefstroom’s Engineering Faculty:
a) Setting up the hydrocyclone – N3C-G67
b) Experiment preparation – N3C-G67
c) Gathering samples from the hydrocyclone – N3C-G67
d) Analysis of the samples – N1-Biolabs-G08
The following activities will be performed to obtain experimental data for the use in academic
papers. Please refer to the schematic of the hydrocyclone setup in Figure 3.2:
1. Install all the necessary equipment on the hydrocyclone. This involves working with
mechanical tools like spanners and screw drivers. Silicone might be used to attach
equipment like the flow meter to the pipes;
2. A slurry containing fine quartz and water will be prepared;
3. The area around the hydrocyclone is cleaned before the hydrocyclone is started;
4. The pump that circulates the slurry through the hydrocyclone will be started;
5. The valve which is used to control the pressure and the slurry feed rate to the
hydrocyclone has to be adjusted manually;
6. If the hydrocyclone is operating as desired, the overflow and the underflow will be
sampled manually by pushing the underflow container under the exit of the underflow;
7. The values displayed by the pressure gauge and flow meter has to be recorded while
the samples are taken;
8. The samples will be analysed with the Mastersizer 2000;
Significant Hazards Identified:
1. When working on wet floors at the hydrocyclone setup, it may be possible to slip and
fall;
2. The electric pump can be a hazard, as one’s clothing could get stuck in the rotating
parts of the pump;
3. Water is handled near electricity and may lead to electrocution;
4. One can lose one’s focus when sampling that could for example lead to one’s fingers
being pinched;
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Hazard identification and risk assessment | XXXVI
5. Swallowing silicon dioxide particles. This can lead to vomiting and abdominal pain.
Please refer to the electronically attached MSDS named “MSDS – Silica flour”;
6. Silicon dioxide coming into contact with eyes will irritate the eyes;
7. Inhaling small silicon dioxide particles that are smaller than 10 𝜇𝑚 can have severe
health problems, as the lungs cannot easily remove such small particles. This can also
cause cancer;
8. High pressure in the piping and the hydrocyclone that can cause equipment to explode;
9. In many cases when one cannot reach something on the hydrocyclone or take a
reading because the sensor is too high, one tends to climb on and over things to reach
the specific object or take the reading of a sensor. This might lead to the person falling
or straining him/herself.
Risk Assessment
Hazard Identified
Likelihood of
Occurrence
(grade 1-4)
(A)
Severity
(grade 1-4)
(B)
Risk (A x B)
Slip and fall on wet floor 3 2 6
Clothes getting stuck in rotating parts 1 4 4
Electrocution 1 4 4
Losing focus when sampling 3 1 3
Swallow silicon dioxide 1 2 2
Silicon dioxide coming into contact with
eyes
3 2 6
Inhaling silicon dioxide 3 3 9
High pressure danger 1 4 4
Strain or falling while reaching for
objects
4 2 8
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Hazard identification and risk assessment | XXXVII
Control Measures
Existing:
1. Emergency electrical shut down
2. Electrical isolation
3. Water draining channels
4. Eye cleanser
5. The dangerous mechanical parts of the pump are isolated
6. The hydrocyclone setup has been quality tested.
Procedural Controls:
1. A dust mask has to be worn when working with dry quartz dust;
2. Non-slip shoes should be worn when working with the hydrocyclone;
3. Make sure that there is enough open space to avoid falling over things;
4. Make sure that you are fully awake when conducting the experiments or analysing
samples with the Mastersizer 2000;
5. Use a ladder or stepping block when something cannot be reached on the
hydrocyclone.
6. Make use of the necessary personal protective equipment when handling the quartz.
This includes wearing a dust mask and safety goggles.
7. Clean wet floors immediately.
8. Keep a safe distance from the circulation pump
Signed: ……………………………………….. (Safety advisor/Risk assessor)
Date: ………………………………………
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