Comparison between the Plitt model and an artificial neural network in predicting hydrocyclone...

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


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


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


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;


Du+phi+Q;


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


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


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


5.3 Artificial neural networks ....................................................................................... 39

5.3.1 Cut size – d50c ................................................................................................ 40


Chapter 6 - Conclusion and recommendations.................................................................... 53

6.1 Conclusion ............................................................................................................ 53

6.2 Recommendations ................................................................................................ 53

6.3 Further study ......................................................................................................... 54


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


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 5.11: 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


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


List of tables| ix


amount of epochs of 20000 ................................................................................................. 47


amount of epochs of 25000 ................................................................................................. 47



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


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


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


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


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


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


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


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


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


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)


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


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.


Literature study| 7

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).


Literature study| 8

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


Literature study| 9

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


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.


Literature study| 10

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)



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.



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



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)



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)



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).



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



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.



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



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.


Experimental procedure| 20

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



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



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



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.



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



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.



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



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



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



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


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.



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



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™.



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


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]



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



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


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



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



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.


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



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



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.


5.3.1.1 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



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.


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.


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]


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]




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



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.



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


Calculated d_50c

Experimental d_50c



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


Calculated d_50c

Experimental d_50c



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


5.3.2.1 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.



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

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[mic

ron

s]


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

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4 r

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[mic

ron

s]





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]




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


Experimental m

Predicted m



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


Experimental m

Predicted m



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


Experimental m

Predicted m



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.


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


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.


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.

Coelho, M.A.Z. & Medronho, R.A. 2000. A model for performance prediction of hydrocyclones. Chemical Engineering Journal, 84:7-14.

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.

Gupta, A. & Lam, S.M. 1998. Weight decay backpropagation for noisy data. Neural netoworks, 11:1127-1137.

Hagan, M.T., Demuth, H.B. & De Jesus, O. 2002. An introduction to the use of neural networks in control systems. International journal of robust and nonlinear control, 12(11):959-985.

Hibbeler, R.C. 2010. Engineering Mechanics Dynamics. 12th Edition in SI units. Singapore: Pearson.

http://www.angloamericanplatinum.com/~/media/Files/A/Anglo-American-Platinum/documents/aap-iar-2013.pdf

http://www.angloamericanplatinum.com/~/media/Files/A/Anglo-American-Platinum/documents/aap-iar-2013.pdf

http://www.aga-reports.com/13/ir/review/regional-reviews/south-africa

http://www.aga-reports.com/13/ir/review/regional-reviews/south-africa


Bibliography| 56

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.

Kadlec, P., Gabrys, B. & Strandt, S. 2009. Data-driven soft sensors in the process industry. Computers and chemical engineering, 33:795-814.

Karimi, M., Dehghani, A., Nezamalhosseini, A. & Talebi, S. 2010. Prediction of hydrocyclone performance using artificial neaural networks. The journal of the Southern African institute of mining and metallurgy, 110:207-212.

Kelly, E.G. & Spottiswood, D.J. 1982. Introduction to mineral processing. 2nd Edition. New York: Wiley.

Kraipech, W., Chen, W., Dyakowski, T. & Nowakowski, A. 2005. The performance of the empirical models in industrial hydrocyclone design. International Journal of ineral Processing, 80:100-115.

Kutz, M. 2003. Biomedical Engineering and Design Handbook. 2nd Edition. Vol. 1. New York: McGraw-Hill Professional.

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.

Nag, A. 2010. Biosystems engineering. 1st Edition. New York: The McGraw-Hill Companies, Inc.

Napier-Munn, T.J., Morrel, S., Morrison, R.D. & Kojovic, T. 2005. Mineral comminution circuits: their operation and optimisation. 3rd Edition. Queensland: Julius Kruttschnitt Mineral Research Centre.

Neesse, T., Schneider, M., Golyk, V. & Tiefel, H. 2004. Measuring the operating state of the hydrocyclone. Minerals engineering, 17:697-703.

Plitt, L.R. 1976. A mathematical model of the hydrocyclone classifier. CIM Bulletin:114-123.

Schneider, C. 2001. Technical notes 3: Hydraulic classifiers. http://www.mineraltech.com/MODSIM/ModsimTraining/Module2/TechnicalNotes3.pdf Date of access: 28 February 2015.

Silva, C.L.Q., Penna, W., Araujo, A.C.B., Brito, R.P., Vasconcelos, L.G.S. & Alves, J.J.N. 2009. Model fine tuning for the prediction of hydrocyclone performance - An industrial review. International Journal of Mineral Processing, 92:34-41.

Sripriya, R., Kaulaskar, M.D., Chakraborty, S. & Meikap, B.C. 2007. Studies on the performance of a hydrocyclone and modeling for flow characterization in presence and absence of air core. Chemical Engineering Science, 62:6391-6402.

Svarovsky, L. 1984. Hydrocyclones. 1st Edition. London: Technomic.

Tew, J. 2012. MQ15. http://www.micronized.com/pdfs/TDS_MQ_15_2012_Sep_-_2.pdf Date of access: 16 October 2015.

Wills, B.A. 2006. Wills' Mineral Processing Technology. 7th Edition. Oxford: Butterworth-Heinemann.

http://www.ccm.usherbrooke.ca/fr/services/ccm/LCG/pdf/mastersizer2000.pdf

http://www.mineraltech.com/MODSIM/ModsimTraining/Module2/TechnicalNotes3.pdf

http://www.micronized.com/pdfs/TDS_MQ_15_2012_Sep_-_2.pdf


Bibliography| 57

Zhang, S., Liu, H.-X., Gao, D.-T. & Wang, W. 2003. Surveying the methods of improving ANN generalization capability. Paper presented at the 2003 international conference on machine learning and cybernetics, Xi'an, China, 2-5 November 2003.


Data processing | I

Appendix A Data processing

The data of each sample collected from the hydrocyclone includes the following:

Volumetric feed flowrate 𝑙𝑖𝑡𝑒𝑟𝑠


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.


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


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.


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


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


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


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


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


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


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)"


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])"


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


ActiveCell.FormulaR1C1 = "=SUM(R[-1]C:R[-43]C)"


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


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


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"


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"


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


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


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


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


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


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


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


ANN source code | XXIII

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)





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


ANN source code | XXIV

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


ANN source code | XXV

'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


ANN source code | XXVI

'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 + _


ANN source code | XXVII

Speed * Reg * (InputWeights(p, 1) / ((1 _

+ InputWeights(p, 1) ^ 2) ^ 2))


- delphi

delDu = Speed * ErrorHid(p, 0) * _

Du_input + InputWeights(p, 2) * Moment + _


((1 + InputWeights(p, 2) ^ 2) ^ 2))


- delDu

delP = Speed * ErrorHid(p, 0) * P_input _



((1 + InputWeights(p, 3) ^ 2) ^ 2))


- delP

delS = Speed * ErrorHid(p, 0) * S_input _



((1 + InputWeights(p, 4) ^ 2) ^ 2))


- 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


ANN source code | XXVIII

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)


ANN source code | XXIX

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


ANN source code | XXX

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.


ANN source code | XXXI

'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


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;


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.


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”


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;


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


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: ………………………………………

Comparison between the Plitt model and an artificial neural network in predicting hydrocyclone...

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