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Hybrid neural network first-principles approach to process modeling

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Title:
Hybrid neural network first-principles approach to process modeling
Creator:
Gupta, Sanjay
Publication Date:
Language:
English
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x, 111 leaves : ill. ; 29 cm.

Subjects

Subjects / Keywords:
Bubbles ( jstor )
Diameters ( jstor )
Flow velocity ( jstor )
Mathematical constants ( jstor )
Minerals ( jstor )
Modeling ( jstor )
Neural networks ( jstor )
Parametric models ( jstor )
Phosphates ( jstor )
Velocity ( jstor )
Chemical Engineering thesis, Ph.D ( lcsh )
Dissertations, Academic -- Chemical Engineering -- UF ( lcsh )
Flotation -- Equipment and supplies ( lcsh )
Phosphate industry ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
Keyword: Flotation columns
Thesis:
Thesis (Ph.D.)--University of Florida, 1999.
Bibliography:
Includes bibliographical references (leaves 108-110).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Sanjay Gupta.

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University of Florida
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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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Full Text







HYBRID NEURAL NETH% ORK FIRST-PRINCIPLES APPROACH
TO PROCESS MODELING















By

SANJAY GUPTA


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1999






















This dissertation

is dedicated

to

my parents













ACKNOWLEDGE ENTS


I would like to take this opportunity to thank my ad isor Dr. Spyros A. Svoronos

for his continuing guidance, encouragement and support throughout the course of my

Ph.D. He not only guided me to learn new techniques, he was also helpful in showing me

the right course in some of the problems in my personal life.


I wish to thank Dr. Hassan El-Shall for his valuable inputs in the chemistry aspect

of this project I would also like to thank my other committee members, Dr. Richard

Dickinson, Dr. Oscar Crisalle, and Dr. Ben Koopman, for kindly reviewing my

dissertation and serving on my committee


The friendship and assistance of my colleagues, Pi-Ilsin Liu, Robert Bozic,

Rajesh Sharma, Dr. Cheng, Dr. Nagui, Rachel Worthen, and Lav Agarwal, will always be

valued


My respect for my parents, brother, and sister for having stood by me and for

giving me moral support always kept me motivated to complete this work.












TABLE OF CONTENTS



pace

ACKNOWLEDGMENTS .................... ............ ...... ..........ii

LIST OF FIGURES ................... ................................... ......... vi

ABSTRACT ....... .................................. .............ix

CHAPTERS

1 INTRODUCTION ..................... .... .. ........................ ..............1

2 ONE-LEVEL HYBRID MODEL ......................................................6

2.1 Introduction .............. ................. ................... .......... ...6
2.2 First-Principles M odel ......... ...... ...... ......... ......... .......... .. .......10
2.2.1 Boundary Conditions........................ .... ..........................13
2.2.2 Calculation of Recovery and Grade................................ .............16
2.2.3 M odel Parameters ................................... .......................... 17
2.3 The Hybrid M odel .............................. .............. ....... ........... 22
2.4 Materials and Methods ........... ....... ...... ...........................23
2.4.1 Experimental Setup and Procedure ...........................................23
2.4.2 Experimental Conditions.............. ................. ................... 27
2.4.3 Neural Network and Training ................................... ..............27
2.5 Results and Discussions .................................... ......................... 28
2.6 Conclusions ............................ .................. ........... .............. 39

3 TWO-LEVEL HYBRID MODEL ....................................... ...... ...........40

3.1 Introduction .............................. ....... ...... ............. 40
3.2 First-Principles M odel ............. ...... .... ..... .... .......... ... .... ...... 44
3.3 Calculation of M odel Parameters ......... .................. .... ..............50
3.4 The Hybrid M odel ....... .... ...... ...... ....... .... ..... ............. ...... 52
3.5 Materials and Methods ..................... ............. ............... 54
3.5.1 Experimental Setup and Procedure ................................... 54
3.5.2 Neural Network and Training ................................... .............56
3.6 Results and Discussions ............. .............. .. .................... ..........58
3.7 Conclusions .............. ................... ...... .. .................... 72











4 OPTIMIZATION PERFORMANCE MEASURES AND FUTURE WORK ............73

4.1 The Performance M measures .............. ................. ........ ...... ..........74
4.1.1 Selectivity ................ ...... ........ .......... .... ....... 74
4.1.2 Separation Efficiency................... ............. ... ... ..... .... 75
4.1.3 Economic Performance Measure......... ...............................75
4.2 The Optimization Algorithm .......................... ...........79
4.3 Initial Scattered Experiments........ ........ ..... ............. ........ 81
4.4 Results and Discussions ................. ................... ....... ................ 82
4.5 Future W ork ................................. ...... ............ .. ... ........ 86

APPENDICES

A CODE FOR THE FIRST PRINCIPLES MODEL FOR ONE LEVEL.................90

B CODE FOR THE FIRST PRINCIPLES MODEL FOR TWO LEVELS..............99

REFERENCES ......... .......... ....... ......................... 108

BIOGRAPHICAL SKETCH .............. ... ... ...... ... ...... ............... .......111













LIST OF FIGURES


Figure page


2.1 Flotation rate constants for phosphate and gangue are calculated by
using a one-dimensional search to invert the first-principles model ......19

2.2 Recovery of phosphate (%) as a function of flotation rate constant for
phosphate (kp)............ .............................................. 20

2.2 Recovery of gangue as a function of flotation rate constant for
gangue (kg)................. ............. ................... ..........21

2.4 Overall structure of the hybrid model ......................... ............. 23

2.5 A schematic diagram of the experimental setup ............................25

2.6 Performance ofNNI: Model versus experimental flotation rate
constant for phosphate (kp) ................. ............... ......... ...........29

2.7 Performance of NNII: Model versus experimental flotation rate
constant for gangue (kg)............ ...................................... 31

2.8 Performance of NNIII: Model versus experimental air holdup for
brother CP-100 ............. .......................................32

2.9 Performance of the overall hybrid model: Predicted versus
experimental recovery (%) for coarse feed size distribution...............33

2.10 Performance of the overall hybrid model: Predicted versus
experimental grade (%BPL) for coarse feed size distribution ............34

2.11 Performance of the overall hybrid model: Predicted versus
experimental recovery (%) for fine feed size distribution.................35

2.12 Performance of the overall hybrid model Predicted versus
experimental grade (%BPL) for fine feed size distribution..................36








2.13 Performance of the overall hybrid model: Predicted versus
experimental recovery (%) for unsized feed size distribution .............37

2.14 Performance of the overall hybrid model: Predicted versus
experimental grade (%oBPL) for unsized feed size distribution.............38

3.1 Schematic diagram of column for phosphate flotation.......................45

3.2 Overall structure of the hybrid model........................................53

3.3 Performance ofNNII. Model bubble diameter versus bubble
diameter inferred from experimental data when CP-100 was the
frother.................... ............................. .. .......... 59

3.4 Performance of NNIII: Model bubble diameter versus bubble
diameter inferred from experimental data when F-507 was the
fro th er...................................... ................. ... ......6 0

3.5 Performance of NNIII: Model bubble diameter versus bubble
diameter inferred from experimental data when OB-535 was the
brother. ............. ............ .................... ....... 61

3.6 Performance of NNIII: Model bubble diameter versus bubble
diameter inferred from experimental data when F-579 was the
brother. ....... ....... .................................. 62

3.7 Performance of NNIV: Model versus experimental air holdup for
brother CP-100. ..... .......... ... ................ ...........63

3.8 Performance of NNIV: Model versus experimental air holdup for
brother F-507................... ................................... ........ 64

3.9 Performance of NNIV: Model versus experimental air holdup for
brother OB-535.......... .............. ........... 65

3.10 Performance of NNIV: Model versus experimental air holdup for
brother F-579........ ........... ................ .. ........ .66

3.11 Performance of NNI: Model versus experimental flotation rate
constant for phosphate (kp)....... ...... ......... ........ ..............67

3.12 Performance of NNII: Model versus experimental flotation rate
constant for gangue (kg) ............ ... ........... .. .. .... ......... 69

3.13 Performance of the overall hybrid model: Predicted versus
experimental recovery (% ') for the four frothers..........................70









3.14 Performance of the overall hybrid model Predicted versus
experimental grade (0%BPL) for the four frothers............. ..............71

4.1 Value of phosphate rock as a function of %BPL ............. ..............78

4.2 The run-to-run optimization algorithm ................... ..............80

4.3 Neural network versus experimental flotation rate constant for
phosphate (kp)................... .......................... .. ... ......... 84

4.4 Neural network versus experimental flotation rate constant for
gangue (kg)........................................................ 85

4.5 Model versus experimental air holdup for brother F-507 ...................87













Abstract of Dissertation Presented to the Graduate School
Of the University of Florida in Partial Fulfillment of the
Requirements for Doctor of Philosophy



HYBRID NEURAL NETWORK FIRST-PRINCIPLES APPROACH
TO PROCESS MODELING

By

Sanjay Gupta

May 1999


Chairman: Dr. Spyros A. Svoronos
Cochairman Dr. Hassan El-Shall
Major Department Chemical Engineering


A hybrid model for a flotation column is presented which combines a first-

principles model with artificial neural networks. The first-principles model is derived by

making material balances on both phosphate and silica particles in the slurry phase.

Neural networks are used to relate the model parameters with operating variables such as

particle size, superficial air velocity, brother concentration, collector and extender

concentration, and pH. One-level and two-level hybrid modeling structures are compared

and it is shown that the two-level structure offers significant advantages over the other.

Finally, a sequential run-to run optimization algorithm is developed which combines the

hybrid model with an optimization technique. The algorithm guides the changes in the

manipulated variables after each experiment to determine the optimal column conditions.













Designed experiments were performed in a lab scale column to generate data for the

initial training of the neural networks.













CHAPTER 1
INTRODUCTION


Since the beginning of 1980s, the industrial application of flotation technology

has experienced a remarkable growth due to active theoretical and experimental research

and development. Flotation columns are slowly being accepted in the mineral processing

industry for the advantages they offer over conventional flotation equipment including

grade improvement, lower operating cost, and superior control. The ability of flotation

columns to produce concentrates of superior grade at similar recovery is derived from the

improved selectivity it offers.


Unlike conventional mechanical cells, flotation columns do not use mechanical

agitation to suspend particles Another distinct feature of the flotation column principle

is countercurrent contact between feed particles and air bubbles. The lack of moving

parts and lower reagent consumption results in a lower operating costs. The lower capital

cost for the equipment is attributed to its high capacity leading to the use of less units for

the same production rate


The current flotation practice in Florida phosphate industry involves the use a

two stage process with mechanical cells, ,where the feed is subjected to rougher flotation

in which fatty acids and fuel oil are used as collectors to separate the phosphate from

most of the sand. The rougher concentrate is then scrubbed by sulfuric acid to remove

the fatty acids and oil. The scrubbed material has to be washed with fresh water to








achieve a neutral pH. The scrubbed and washed material is then subjected to cleaner

flotation in which amine together with kerosene is used as collector to float sand This

stage of flotation is sensitive to impurities in water; thus, fresh water is used in most of

the plants as make up water. However, the fatty acid circuit uses recycled water. This

process has become less cost effective due to high cost of reagents and increasing

concentration of contaminants.


To prepare the phosphate feed, the mined phosphate ore (matrix) is washed and

de-slimed at 150 mesh. The material finer than 150 mesh is pumped to clay settling

ponds. The rock coarser than 150 mesh is screened to separate pebbles (-3/4 +14 mesh)

which are of high phosphate content. Washed rock (-14, +150 mesh) is sized into a fine

(usually 35 x 150 mesh) and a coarse flotation feeds (usually 14 x 35 mesh) which are

treated in separate circuits. Flotation of phosphates from the fine feed (35 X 150 mesh)

presents very few difficulties and recoveries in excess of 90% are achieved using

conventional flotation cells. On the other hand, recovery of phosphate values from the

coarse feed is much more difficult and flotation by itself usually yields recovery of 60%

or less.


The density of the solid, turbulence, stability and height of the froth layer, depth

of the water column, viscosity of the froth layer are known to effect the flotation process

in general (Boutin and Wheeler, 1967). However, the exact reasons for low recovery of

coarse particles in conventional flotation is not very well understood. There are several

hypotheses about the flotation behavior of coarse particles For instance, the floatability

of large particles could be due to the additional weight that has to be lifted to the surface








under the heavy turbulence conditions, and the difficulty to transfer and maintain these

particles in the froth layer. Some efforts towards improving the flotation of coarse

particles through stabilization of the froth layer, minimizing the froth height, and addition

of an elutriation water stream at the bottom of the column have been undertaken


The equipment used by the phosphate industry in flotation process are not

selective enough to take full advantage of new reagents and operating schemes, to

recover phosphate from the coarse feed or to optimize results with existing reagents. The

best way to increase the selectivity of phosphate flotation is to improve upon the design

of flotation equipment. Particularly the new equipment should improve the recovery of

coarse particles, while still providing the high selectivity of fine particles


It has been found both theoretically and practically that flotation columns have

better separation performance than conventional mechanical cells (Finch and Dobby,

1990). The use of flotation columns can not only help overcome some of the problems

related to coarse phosphate flotation but it has several other advantages as mentioned

above. Spargers or bubble generating systems are the single most important element in

the flotation columns They are generally characterized in terms of their air dispersion

ability. Frothers are the chemicals that help in controlling and stabilizing bubble size by

reduction of surface tension. Thus both of them play an important role in the overall

performance of flotation columns. Their interaction can be a crucial factor in the success

of flotation column


Flotation columns have been used predominantly in the coal beneficiation

industry. However, their application in other mineral industries, such as the phosphate, is








not very well studiedd Unlike other minerals, phosphate flotation deals with a

considerably larger size of particles (0.1-1mm) and therefore the operation of phosphate

flotation in a column is different from that of other minerals. High recovery and grade

and low operating cost depend largely on the optimal selection of operating variables

such as the air flow rate, the brother type and concentration, and the elutriation water rate.

The search of the optimal conditions can considerably benefit by the availability of a

model that can predict the effects of different operating conditions on column behavior.

Finch and Dobby (1990) and Lutrell and Yoon (1993) developed a one-phase axial

dispersion model in which particle collection is viewed as a first order net attachment rate

process. Sastry and Loftus (1988) considered both the slurry and air phases and they

used two separate first order rate constants for attachment and detachment of the

particles. However, these models cannot predict the effects of certain operating

conditions such as particle size, brother concentration, collector and extender

concentration, and pH on the flotation performance.


In this work, a mathematical model is developed that for the first time predicts the

effects of particle size, brother concentration, collector and extender concentration, and

pH on the flotation behavior. This is a hybrid model that combines a first-principles

model with artificial neural networks (ANNs). The first-principles model is derived by

making a material balance on solid particles in the slurry phase. First order reaction rate

constants are assumed for the attachment of the solid particles to the air bubbles. Single

output feedforward backpropagation neural networks are used to correlate the model

parameters with the operating variables.








Two hybrid modeling approaches are presented Chapter 2 describes a one-level

hybrid model that uses three different neural networks to predict the flotation rate

constant for phosphate, the flotation rate constant for gangue, and air holdup. Chapter 3

presents a two-level hybrid mode! in which neural networks are structured in two levels.

Two neural networks are used in the top-level to predict bubble diameter and air holdup.

The bubble diameter is used as an input in the neural networks of the bottom-level which

predict the flotation rate constants for phosphate and gangue The inherent advantages

and disadvantages of the two hybrid modeling approaches are also discussed in these

chapters.


In chapter 4, the hybrid model developed is combined with an on-line

optimization algorithm to determine the optimal conditions for column operation. The

algorithm guides successive changes of the manipulated variables such as air flow rate,

brother concentration, and pH, after each run to achieve optimal column operating

conditions Designed experiments were performed to generate data for the initial training

of the neural networks. The trained neural network is then used to guide the direction of

the new experiments.












CHAPTER 2
ONE-LEVEL HYBRID MODEL


Flotation is a process commonly employed for the selective separation of phosphate

from unwanted mineral. Column flotation is slowly gaining popularity in the mineral

processing industry, including the phosphate industry, due to its ability to improve

selectivity, lower operating cost, lower capital cost, and superior control. In this work, a

hybrid model is developed that combines a physicochemical model with artificial neural

networks. This model for the first time incorporates the effect of collector concentration,

extender concentration, and pH on the flotation performance. The physicochemical

model is based on axial dispersion with first order collection rates. Three basic

parameters are required in this model: flotation rate constant for phosphate, flotation rate

constant for gangue, and air holdup. Artificial neural networks are used to predict these

parameters. The model also takes into account the particle size distribution and predicts

grade and recovery for each particle size range. The model is validated against

laboratory column data.


2.1 Introduction

Even though the concept of column flotation was developed (Wheeler, 1988) and

patented (Boutin and Wheeler, 1967) in the early 1960s, its acceptance for the processing

and beneficiation of phosphate ores is relatively recent. The majority of the phosphate

plants employ mechanical cells. However, column flotation has simpler operation and

6








7
provides superior grade/recovery performance. For these reasons column

flotation is gaining increasing acceptance for the processing and beneficiation of

phosphate ores. Although it has been successfully employed for the selective separation

of phosphate from unwanted mineral, a totally predictive model still remains unavailable

for industrial use.


Flotation is a process to separate hydrophobic particles from hydrophilic particles.

The hydrophobic material has a tendency to attach to the rising bubbles and leaves from

the top of the column. The hIdrophilic material settles down and leaves from the bottom

of the column In this way, the phosphate containing material (frankolite or apatite) is

separated from gangue (mostly silica). The phosphate ore is first pretreated with fatty

acid collector and fuel oil extender. Fatty acid and fuel oil adsorb on the phosphate-

containing particles rendering them hydrophobic The flotation process is then used to

separate phosphate particles from gangue minerals.


A flotation column consists of three flow regimes a cleaning or froth zone, a lower

collection zone, and pulp-froth interface zone. The froth zone is the region extending

upward from the pulp-froth interface to the column interface. The collection zone is the

region extending downward from the pup-froth interface to the lowest sparger. A mineral

particle is recovered by a gas bubble in the collection zone of the column by particle-

bubble collision followed by attachment due to the hydrophobic nature of the mineral

surface. Since phosphate particles are considerably larger in size (0.1-1 mm), an

elutriation water stream from the bottom is added to maintain a positive upward flow

(negative bias) to aid lifting the particles upward.








8
The particle collection process in a column is considered to follow first order

kinetics relative to the solids particle concentration with a rate constant. Finch and

Dobby (1990) and Lutrell and Yoon (1993) used a one-phase axial dispersion model in

which particle collection is viewed as a first order net attachment rate process. Sastry and

Loftus (1988) considered both the slurry and air phases and they used two separate first

order rate constants for attachment and detachment of the particles Luttrell and Yoon

(1993) relate the particle net attachment rate constant to some operating variables using a

probabilistic approach. However, their approach cannot be used to predict the effect of

certain operating conditions such as brother concentration, collector concentration,

extender concentration, and pH.


For the model to be predictive, the functional dependence of the net attachment rate

constant (kp or kg) on the key operating variables needs to be determined. The functional

relationship of model parameters on the operating conditions is difficult to determine via

physicochemical reasoning. In our approach, we use neural networks to determine these

functional relationships. Artificial neural networks are a powerful tool, inspired by how

the human brain works, that can learn from examples any unknown functional

relationship. Their ability to approximate any smooth nonlinear multivariable function

arbitrarily well (Hornik et al., 1989) and their simple construction have led to great

interest in using neural networks.


Existing modeling strategies can be divided into white-box, black-box, and gray-box

(hybrid) strategies, depending on the amount of prior knowledge that is used for

development of the model. White-box modeling strategies are mainly knowledge driven.








9
Black-box modeling strategies are mainly data driven and the resulting models often

do not have reliable extrapolation properties. Black-box strategies have been applied to

many chemical processes, especially since convenient black-box modeling tools like

neural networks have become available (Bhat and McAvoy, 1990; Psichogios and Ungar,

1992a). Gray-box or hybrid modeling strategies are potentially very efficient if the

black-box and white-box components are combined in such a way that the resulting

models have good interpolation and extrapolation properties


There are two types of gray-box modeling approaLhes in %which a neural network is

combined with a black-box model: the parallel and the serial approach In the parallel

approach, the neural network is placed parallel with a white-box model In this case, the

neural network is trained on the error between the output of the white-box model and the

actual output. Su el al. (1992) demonstrated that the parallel approach resulted in better

interpolation properties than pure black-box models Johansen and Foss (1992) also used

a parallel structure where the output of the hybrid model was a weighted sum of a first-

principles and a neural network model


In the serial hybrid modeling strategy, the neural network is placed in series with the

first-principles model. Various researchers (Psichogios and Ungar, 1992a; Thompson

and Kamer, 1994) have shown the potential extrapolation properties of serial hybrid

models Psichogios and Ungar (1992b) used this approach for parameters that are

functions of the state variables and manipulated inputs Liu et al. (1995) developed a

serial hybrid model for a periodic wastewater treatment process by using ANNs for the








10
bio-kinetic rates of a first-principles model. Cubillo and Lima (1997) also used this

approach to develop hybrid model for a rougher flotation circuit


In this work, we employ a serial approach to integrate an approximate model,

derived from first-principles considerations, with neural networks which approximates

the unknown kinetics. The first-principles model is inverted to calculate two model

parameters for each set of measured recovery and grade. The neural networks are then

trained on the errors of calculated model parameters instead of the errors of the output of

the first-principles model as is the case with the above referenced works. Also, unlike

most other cited work, we employ experimental data instead of simulated data.


2.2 First-Principles Model


The basic equations representing the flotation of solid particles in a flotation column

can be written by making a material balance for the solid particles in the slurry phase.

This results in the following partial differential equations for the section above and below

the feed point, respectively:



j U C a2C
P= --- U l + D kp(d,)C, (2.1)


BC U2
l 1sl -z C'z1 p2


P2 +Uj ac 2 +DJ--kPd Cj (2.2)
-t- = + +D OZ2 p (d)C (2.2)
t 1-eg 8z








whe


re

Cj = Phosphate concentration ofjth mesh size particles for the section above the
feed point
Ci = Phosphate concentration ofjth mesh size particles for the section below the
P2
feed point
U = Superficial liquid velocity above the feed point
= Qp/A
Ut = Superficial liquid velocity below the feed point
= (Q -Qe)/ A


D = Dispersion coefficient
Qp = Product volumetric flow rate
Qt = Tailings volumetric flow rate
Qe = Elutriation volumetric flow rate
Ac = Cross-sectional area of the column
USi = Slip velocity ofjth mesh size particles
Eg = Air holdup
kp(d ) = Flotation rate constant for phosphate for jt mesh size particles

The follow ing assumptions are made in deriving the above equations:

1) The concentration of solid particles in the slurry phase is a function of height, z

only, and variations of the concentration in racial and angular directions can be

neglected

2) The air holdup is constant throughout the column

3) All the air bubbles in the system are of a single size.

4) Rate of detachment is either negligible or is a function of conditions in the slurry

phase. This assumption allows to treat the net attachment rate with just one

floatation rate constant.








12
The slip velocity is calculated using the expression of Villeneuve et al. (1996):

Sgd (ps- p,)(1 -_ )2.7
=i (2.3)
18., (1 + 0.15RJ, )

where the particle Reynolds number is defined as

d' U1 p (- ,)
Rep d1-) (2.4)
ep


where

g = Acceleration due to gravity (m/s2)

.L = Water viscosity (kg/ms)

pl = Water density (kg/m3)

p, = Solid density (kg/m3)

Cs = Volume fraction of solids in slurry

d = Particle diameter (m)

Since RJ is a function of Ui, an iterative procedure is used to calculate the slip

velocity. The procedure starts with an initial guess for U, and corresponding value of

RQp is plugged in Equation 2.3 and new value of UJ, is found. This new value is then

used in Equation 2.2 and this procedure is continued till convergence is achieved. The

axial dispersion coefficient is calculated by a modified expression of Finch and Dobby

(1990):


D= 0.063 (1-e )d, (2.5)
1.6)








where

dc = column diameter (m)

Jg = superficial air velocity (cm/s)


Equations 2.1 and 2.2 can be solved analytically for the concentration profile of the

solid particles at steady state. The resulting analytical expressions for the concentration

profile are

ie a2 1 2
Cj =K exp{--(ai Va +4bJ )z}+KK exp{--(aJ + aj +4bJ )z} (2.6)
Ci 2 1

S=K exp{- (d -VdJd2+4bj)z)+K exp{--(dj + dJ 4b )z} (2.7)
2 2

where


S1-g U M an-d d k (dJ)(1-g)
a= ;bJ- ; and dW=
D D D

Kj, K, K and KJ are the constants of integration to be determined by using

appropriate boundary conditions.



2 2 1 Boundary Conditions

A material balance at the top layer of the column (z = L) gives the following

equation:

dCj U Ci -Cp
AcAz P' U i C +A P2 PAD' -k(dJ)AcAzCJ
dt 1- l g C2 Az
J}


(2.8)








14
in the limit as Az -- 0, the above equation reduces to the following boundary

condition:


dC'
PI
dz
d z=L


Continuity of the concentration profile at the feed location gives

C = C
PI z=L= P2 z=Lf


(2.9)







(2.10)


A similar material balance at the feed inlet gives for the solid particles in the slurry

phase


QfC= Ae U C
where 1-) =L



where


+A+ + U C +
1-8 ) 2.=L
(2.11)


z=Lf


Phosphate feed concentration ofj1h mesh size particles

Feed volumetric flow rate

Feed location


At the bottom of the column (z = 0), due to the elutriation flow, the derivative of the

concentration profile reduces to the following expression:


dC
D P2
dz
z=0


Qe C
(1- g)Ac P


(2.12)


2 Z=0


C =


Qf =

Lf =










The four boundary conditions can be solved in conjunction with Equations 2.6 and

2.7 for K KJ, Ki, andK,. The resulting expressions for the constants of integration

are given by the following equations:



J I (QfcfJ A,D)
4
m (aj -oaJ)pj exp{af Lf )+(dj -yJ)qJ exp{yL f }+m (a -PJ)exp{ JLf }+ (dJ J)exp{J L,}


(2.13)

K = qKJ (2.14)

KJ =mJK (2.15)

K =pjmJKJ (2.16)



where

a' 1 aj2
j =- +_- a + 4b (2.17)
2 2


pJ-=---- J aJ +4b' (2.18)
2 2

j 1 2
y =- +1 dJ +4bJ (2.19)
2 2

6-dJ- dJ2 +4b (2.20)
2 2










pj=i -C- (2.21)
SADa +aj exp(aoL)
ACD

( I Qt -dJ-SJ
( .+dj-54
q AD (2.22)
t A -d+y
AcD

mj = q exp(y L)+ exp( f) (2.23)
pJ exp(a'Lf )+exp(P3'Lf)



The algorithm for solving the first-principles model is given in Appendix A.



2.2.2 Calculation of Recovery and Grade


Recovery (%) is defined as the ratio of the weight of the phosphate in the concentrate

stream to the weight of the phosphate in the feed stream. The recovery of the phosphate

particles of the jt mesh size can be expressed in terms of the feed and tailings flow rates

and concentration as

QC [Q.+A,(1-,6)U c g CP z
R = -f S I o.* 100 (2.24)
QfCV 9




Grade, a measure of the quality of the product, is defined as the ratio of the weight of the

phosphate to the total weight recovered in the concentrate stream. Grade is reported as %

Bone Phosphate of Lime (% BPL) which is the equivalent grams of tricalcium phosphate







17
Ca3(P04)2 in 100g of sample. Grade can be calculated as the ratio of the weight of

phosphate to the sum of the weight of the phosphate and gangue in the concentrate

stream:

Q +A- Q,+A(l g)Ui )U
GJ Q= fC +A(1 )U1 k z Iz = 73.3

(QffC'- [Qt +Ac (1-g)U g]C j )+(QrfC -[Q +Ac(1-g)U, ]Cj



(2.25)



where C' is the gangue concentration of the jth particle size and Cj is the gangue feed
g2 r9

concentration ofj'h particle size. The multiplication factor is 73.3 instead of 100, because

pure Florida phosphate rock measures at about 73.3 %BPL.



2 23 Model Parameters


The above model formulation has only two model parameters, namely, the

flotation rate constants for phosphate and gangue. The experimental analysis in the

industry is usually available in terms of grade and recovery of phosphate. The recovery

of gangue can then be readily calculated from grade and recovery of phosphate using the

following relationship:

R= G-(73.3-G )
RJ =-G (2.26)
G'(73.3-GJ)


where G| is the grade of the feed material.









The recovery of phosphate R, is only a function of the flotation rate constant

for phosphate, kp, and air holdup, Eg. Similarly, the recovery of gangue RJ is only a

function of flotation rate constant for gangue, kg, and air holdup, eg. Since air holdup is

measured, we can invert the model to determine the value of kp that results in the

measured recovery of phosphate R, and the value of kg that yields the measured

recovery of gangue R As shown in Figure 2.1, a one-dimensional search is performed

to determine the values of flotation rate constants when supplied with the recovery of

phosphate and gangue, respectively. This algorithm allows determination of the flotation

rate constants for each run, given the operating conditions and the performance of the

column in terms of grade and recovery. The algorithm requires two initial guesses of the

flotation rate constants which yield errors in the corresponding R, of opposite sign, and

then the program uses the method of false position (Chapra and Canale, 1988) to

determine the correct set of flotation rate constants.


Recovery of phosphate increases monotonically with flotation rate constant for

phosphate, kp. This is verified by calculating recovery for different values of flotation

rate constant and recovery was plotted against flotation rate constant. From the graph

shown in Figure 2.2, it is concluded that there is only value of floatation rate constant for

a given recovery. Similarly, from Figure 2.3, it is concluded that recovery of gangue

increases monotonically with flotation rate constant for gangue, kg.








Experimental grade and
recovery of phosphate


Recovery of
Said insolubles |
(if available)


Experimental
recovery
of phosphate


One-dimensional
search


g
Flotation rate constant
for gangue


kJ

Flotation rate constant
for phosphate


Figure 2. 1: Flotation rate constants for phosphate and gangue are calculated by using a one-
dimensional search to invert the first-principles model











100 *

90

44 80
~80
S70

t 60
o
a. 50
40
o 40

0
0 20

10
0
0 2 4 6 8 10
Flotation rate constant for phosphate (kp)

Figure 2.2: Recovery of phosphate (%) as a function of flotation rtae constant for
phosphate (kp)










100

90
80

4 70
c 60

.- 50
o
W 40
o 30
20

10 *
0
0 0.2 0.4 0.6 0.8
Flotation rate constant for gangue (kg)


Figure 2.3: Recovery of gangue (%) as a function of flotation rtae constant for
gangue (kd











2 3 The Hybrid Model


The overall structure of the hybrid model is shown in the Figure 2.4. The hybrid

model utilizes backpropagation neural networks (Rumelhart and McClelland, 1986) to

predict the values of parameters flotation rate constants, kp and kg, and air holdup, eg.

The factors that affect kp and kg are particle diameter, superficial air velocity, brother

concentration, collector concentration, extender concentration, and pH. The air holdup,

Eg, is mainly affected by superficial air velocity and brother concentration.


The hybrid model of Figure 2.4 integrates the first-principles model with three

artificial neural networks. Neural network, NNI, correlates the flotation rate constant for

phosphate, kp, with phosphate particle size, superficial air velocity, brother concentration,

collector concentration, extender concentration, and pH. Similarly, neural network, NNII

correlates the flotation rate constant for gangue, kg, with gangue particle size, superficial

air velocity, brother concentration, collector concentration, extender concentration, and

pH. Neural network NNIII correlates the air holdup, eg, with superficial air velocity and

brother concentration.


In this structure, all three neural networks are specific to the type of brother or

sparger used. This necessitates generation of new data and retraining of the neural

networks each time the brother or the sparger are changed.












Phosphate
particle
size


Frother
concentration


Collector
concentration


pH


Gangue
particle
size


Superficial
air velocity


Frother
concentration


Figure 2.4: Overall structure of the hybrid model










2.4 Materials and Methods

2.4.1 Experimental setup and Procedures


The experimental setup is shown in Figure 2.5. It includes an agitated tank

(conditioner) for reagentizing the feed and a screw feeder for controlling the rate of

reagentized feed to the flotation column. The agitated tank was 45 cm in diameter and 75

cm high. It was equipped with an impeller of two axial type blades (each 28 cm diameter)

The impeller rotation speed was fixed at 465 rpm. The impeller had about 3.8 cm

clearance from the bottom of the tank. The feeder with 2.5 cm diameter screw delivered

the conditioned phosphate materials to the column. The feed rate was controlled by

adjusting the screw rotation speed. Flotation tests were conducted using a 14.5 cm

diameter by 1.82 m high plexiglass flotation column. The feed inlet was located at 30 cm

from the column top. The discharge flow rate was controlled by a discharge valve and an

adjustable speed pump. Three flowmeters were used to monitor the flow rates for air,

brother solution, and elutriation water.


Three different feed sizes obtained from Cargill were used in the flotation

experiments: coarse feed with narrow distribution (14X35 Tyler mesh), fine feed with

wide size distribution (35X150 Tyler mesh), and unsized feed which is a mixture of the

above two (14X150 Tyler mesh). For each run, 50 kg of feed sample was added in the

pre-treatment tank and water was added to obtain 72% solids concentration by weight.

The feed material was then agitated for 10 seconds. 10 % soda ash solution was added to

the pulp to reach pH of about 9.4 and agitated for 10 seconds. Subsequently a mixture of



















Manometers











Sparger
Pressure
Gauge


Froth


"" Air
I----X-J-^----o- i, --. i
Pressure Filter 1'
Re'tulator


A schematic diagram of the experimental setup


Figure 2.5


ci
i i
i E







26
fatty acid (obtained from Westvaco) and fuel oil (No. 5 obtained from PCS

Phosphate) with a ratio of 1:1 by weight was added to the pulp. The total conditioning

time was 3 minutes. The conditioned feed material (without its conditioning water) was

loaded in the feeder bin located at the top of the column.


The brother selected for this study was CP-100 (sodium alkyl ether sulfate obtained from

Westvaco). Frother-containing water and air were first introduced into the column through

the sparger eductorr) at a fixed flowrate and brother concentration, and then the discharge

valve and pump were adjusted to get the desired underflow and overflow rates. Air holdup

was measured for the two-phase (air/water) system using a differential pressure gauge. After

every parameter was set and the two-phase system was in a steady state, the phosphate

material was fed to the column using the screw feeder. Water was also added to the screw

feeder to maintain the steady flow of the solids to the column at 66 % solids concentration.

To achieve steady state, the column was run for a period of three minutes with phosphate

feed prior to sampling. Timed samples of tailings and concentrates were taken. The collected

samples were weighed and analyzed for %BPL according to the procedure recommended by

the Association of Florida Phosphate Chemists (AFPC Analytical Methods, 1980). These

measurements were then used to calculate recovery of acid insolubles. These values were

then averaged with the values obtained from Equation 2.26 to obtain the RI used to
g


determine the flotation rate constants ofgangue.





27
2.4.2 Experimental Conditions

For the brothers investigated. 35 three-phase experiments were conducted. Seven

different levels of brother concentration (5, 6.6, 10, 15, 20, 23.4, and 25 ppm) was studied in

designed experiments Five different levels of collector and extender concentration (0.27,

0.41, 0.54, 0.64, and 1.7 kg/t) were used pH was varied from 8.2 to 9.9 at five different

levels (8.2, 8.5, 9.0. 9.5, and 9.9). Two superficial air velocities (0.46 and 0.7 cm/s) were

used for the designed experiments.


The particle size depended on the type of feed used. For coarse feed, the particle size

varied from 417 to 991 microns. For fine feed, the particle size varied from 104 to 417

microns whereas for the unsized feed distribution, the size ranged from 104 to 991 microns.



2.4.3 Neural Network Structure and Training


Single output feedforward backpropagation neural networks are used with a single

layer of hidden nodes. A unit bias is connected to both the hidden layer and the output layer.

Both the hidden layer and the output layer used a logistic activation function (Hertz et al.,

1992) and the input and the output values were scaled from 0 to 1.


During the training mode, training examples are presented to the network. A training

example consists of scaled input and output values. For NNI and NNII, the output values are

the flotation rate constants calculated from one-dimensional searches for phosphate and

gangue, respectively. For NNIII, the output value is the experimentally measured air holdup.


The training process is started by initializing all weights randomly to small non-zero

values. The random number is generated between -3.4 and +3.4 with standard deviation of

1.0 following the procedure recommended by Masters (1993). The optimal weights were





28
determined using simulated annealing (Kirkpatrick et al., 1983) and a conjugate gradient

algorithm (Polak, 1971). There are two approaches towards updating the weights. In one

approach, the input-output examples are presented one at a time and after each presentation

the weights are updated using rules such as the delta rule (Rumelhart and McClelland, 1986).

This method is attractive for its simplicity but is restricted to rather primitive optimization

algorithms. In contrast, the batch training approach allows use of powerful methodology for

nonlinear optimization. It processes each input-output example individually but updates the

weights only after the whole set of input-output examples has been processed. In this case,

the gradient is cumulated for all presentations, then the weights are updated, and finally the

sum of the squared errors is calculated.


The simulated annealing algorithm is used for eluding local minimum. It perturbs the

independent variables (the weights) while keeping track of the best (lowest error) function

value for each randomized set of variables. This is repeated several times, each time

decreasing the variance of the perturbations with the previous optimum as the mean. The

conjugate gradient algorithm is then used to minimize the mean-squared output error. When

the minimum is found, simulated annealing is used to attempt to break out of what may be a

local minimum. This alteration is continued until networks can not find any lower point. We

then hope that the local minimum is indeed the global minimum.



2.5 Results and Discussion


The performances of the three ANNs are shown in Figures 2.6-2.14. Figure 2.6

compares the flotation rate constants for phosphate (kp) determined from one-dimensional

searches with those predicted by NNI. As shown in this figure, NNI captures the dependence

of the flotation rate constant on particle size, superficial air velocity, brother concentration,











10
I-
4-0 9




o a.
6

4o .


c:3

2


0

0 1 2 3 4 5 6 7 8 9 10

Experimental flotation rate constants for phosphate (kp)

Figure 2.6 Performance of NNI: Model versus experimental flotation rate constant for
phosphate (k,)
phosphate (kp)





30
collector and extender concentration, and pH. Similarly, Figure 2.7 compares flotation rate

constant for gangue (kg) determined from one-dimensional searches with those predicted by

NNII. As shown, NNII successfully predicts the flotation rate constant for gangue. Figure

2.8 presents the air holdup (g) predicted using NNIII against those measured experimentally.

A satisfactory match is seen.


The hybrid model integrates NNI, NNII, and NNIII as shown in Figure 2.4.

Predictions of the hybrid model are shown in Figure 2.9-2.14. Figures 2.9 and 2.10 compare

the experimental recovery (%) and grade (%BPL) with those predicted by the hybrid model,

respectively, for the coarse feed size distribution (14X 35 Tyler mesh). As shown in these

figures, the hybrid model successfully predicts both recovery and grade. Figures 2.11 and

2.12 compare the experimental recovery (%) and grade (%BPL) with those predicted by the

hybrid model, respectively, for the fine feed size distribution. As seen from these figures, the

hybrid model fails to successfully predict both recovery and grade. This is attributed to the

fact that fine feed has a very wide size distribution (35X150 Tyler mesh size) and only the

overall recovery and grade were measured experimentally. It is therefore necessary to utilize

narrow ranges of feed size and to analyze for recovery and grade according to each size range

instead of just one recovery and grade for the entire particle size distribution. This was

implemented for the unsized feed size which has even a wider size distribution (14X150

Tyler mesh). Figures 2.13 and 2.14 compare the experimental recovery (%) and grade

(%BPL) predicted by the hybrid model, respectively, for the unsized feed after it has been

sized and grade and recovery was determined for each size. As can be seen from these

figures, the hybrid model successfully predicts both recovery and grade.











0.8

S0.7

0.6
o0
0 ,
J4 0.5

0 0.4

o 0.3

3 0.2

0 0.1
ci.
0 *
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Experimental flotation rate constants for gangue (kg)

Figure 2.7 Performance of NNII: Model versus experimental flotation rate constant for gangue
(kg)














25


o
0s
CL
I.



a-
(1


0 5 10 15 20 25
Experimental air hold up (%)

Figure 2.8: Performance of NNIII: Model versus experimental air holdup for brother CP-100









100



98



G 96
>
0



o

92



90 .......
90 92 94 96 98 100
Experimental Recovery (%)


Figure 2.9 Performance of the overall hybrid model: Predicted versus experimental recovery
(%) for coarse feed size distribution











70


66


6 62


% ~58
o0


54


50
50 55 60 65 70
Experimental Grade (%)

Figure 2.10: Performance of the overall hybrid model: Predicted versus experimental grade
(%BPL) for coarse feed size distribution









100



80



0 60
60
o



o
40
*0

20
20



0
0 20 40 60 80 100

Experimental Recovery (%)

Figure 2.11: Performance of the overall hybrid model: Predicted versus experimental recovery
(%) for fine feed size distribution












70


60


50


40


30


20
20 30 40 50 60 70
Experimental Grade (%)

Figure 2.12: Performance of the overall hybrid model: Predicted versus experimental grade
(%BPL) for fine feed size distribution










100



99



S98
O
0

97
0


96



95
95 96 97 98 99 100
Experimental Recovery (%)

Figure 2.13: Performance of the overall hybrid model: Predicted versus experimental recovery (%)
for the unsized feed after it has been sized.











70 --

65

60

55

50

45

40

35

30 -- --T-_-,--
30 35 40 45 50 55 60 65
Experimental Grade (%)

Figure 2.14: Performance of the overall hybrid model: Predicted versus experimental grade
(%BPL) for the unsized feed after it has been sized.





39
2.6 Conclusions


In this work, we have demonstrated that a one-phase first-principles model can

effectively be coupled with the artificial neural networks for predicting the grade and

recovery of a phosphate flotation column with negative bias. Artificial neural networks are

used to predict the flotation rate constants and air holdup. Experimental data from a lab-scale

column were used to train the neural networks. The hybrid model successfully predicts the

effects of particle size, superficial air velocity, brother concentration, collector concentration,

extender concentration, and pH.













CHAPTER 3
TWO-LEVEL HYBRID MODEL


A new model for phosphate column flotation is presented which relates the effects of

operating variables such as brother concentration and air flow rate on column

performance. This is a hybrid model that combines a first-principles model with artificial

neural networks. The first-principles model is obtained from material balances on both

phosphate particles and gangue (undesired material containing mostly silica). First order

rates of net attachment are assumed for both. Artificial neural networks relate the

attachment rate constants to the operating variables. Experiments were conducted in a 6"

diameter laboratory column to provide data for neural network training and model

validation. The model is shown to successfully predict the effects of brother

concentration, particle size, air flow rate, and bubble diameter on grade and recovery.


3.1 Introduction


Flotation is a process in which air bubbles are used to separate a hydrophobic from a

hydrophilic species. The majority of the hydrophobic material gets attached to the

bubbles and leaves with the froth from the top of a cell or column separator, while the

hydrophilic material leaves from the bottom. This process is commonly used in the

minerals industry, including the phosphate industry, in which case the phosphate

containing rock (frankolite or apatite) is to be separated from gangue (mostly silica).

Flotation is also used to remove oil from wastewater and to remove ink from paper pulp.








In anionic phosphate flotation the mineral is first treated with fatty acid collector and fuel

oil extender. At proper concentrations these mostly adsorb on the phosphate-containing

particles rendering them hydrophobic. Then the phosphate-containing particles are

separated from gangue via the flotation process. The majority of the phosphate plants

employ mechanical cells. However, column flotation has simpler operation and provides

superior grade/recovery performance. For these reasons column flotation is gaining

increasing acceptance for the processing and beneficiation of phosphate ores.


Column flotation is frequently employed for the recovery of other minerals (e.g., coal,

copper, nickel, gold). In such applications the column can be divided into three zones: an

upper froth zone, a lower collection zone, and an intermediate interface zone. An

additional "wash water" stream is usually added from the top of the column. Phosphate

flotation deals with considerably larger particles of size 0.1-1 mm. As a result, instead of

wash water from the top, elutriation water from the bottom is added. Furthermore,

columns are typically operated with negligible froth and interface zones. This

considerably simplifies the modeling effort, as the only the collection zone needs to be

accounted for.


Particle transport in the collection zone is usually modeled as axial convection

coupled with axial dispersion. The Peclet number (Pe), or its inverse, the dispersion

number, governs the degree of mixing. Most models only consider the slurry phase

(Finch and Dobby, 1990; Luttrell and Yoon, 1993), in which case particle collection is

viewed as a first order net attachment rate process. A model that considers both slurry

and air phase was developed by Sastry and Loftus (1988). In this case particle








attachment and detachment are modeled separately with first order rates. Luttrell and

Yoon (1993) used a probabilistic approach to relate the particle net attachment rate

constant to some operating variables (e.g., air flow rate). However, their approach

involves empirical parameters and it cannot be used to predict the effect of certair

operating variables such as the brother and collector concentrations.


In this work, we use neural networks to determine the dependence of the phosphate

and gangue flotation rate constants on the operating variables. Artificial neural network!

have the ability to approximate any smooth nonlinear multivariable function arbitrarily

well (Hornik et al., 1989). This approach can be used to determine the dependence of the

performance of a flotation column (i.e., grade and recovery) on any operational variable

We demonstrate it in this work by developing a hybrid model that predicts the effect ol

brother concentration, air flow rate, feed rate and loading, elutriation flow rate, tailings

flow rate, and particle size distribution.


The idea of developing a hybrid model by combining a first-principles model (FPM]

with artificial neural networks (ANNs) is not new. Johansen and Foss (1992) and Su ej

al. (1992) proposed parallel structures where the output of the hybrid model is a weighted

sum of the first-principles and ANN models. Kramer et al. (1992) proposed a parallel

arrangement of a default model (which could be a first principles model) and a radial

basis function ANN. An alternative approach is to combine ANNs with a FPM in a serial

fashion, by using the ANNs to develop expressions for the FPM parameters or rate

expressions. Psichogios and Ungar (1992a, 1992b) proposed this scheme for parameters

that are functions of the state variables and manipulated inputs, and trained the neural







networks (i.e. determined the neural network parameters) on the error of the output of the

first-principles model. A similar approach was followed by Reuter et al. (1993) to model

metallurgy and mineral processes. Liu et al. (1995) developed a hybrid model for a

periodic wastewater treatment process by using ANNs for the bio-kinetic rates of a first-

principles model. The Psichogios and Ungar (1992a, 1992b) approach was used by

Cubillo et al. (1996) to model particulate drying processes, and by Cubillo and Lima

(1997) to develop a hybrid model for a rougher flotation circuit. Thompson and Kramer

(1994) combined the parallel and serial hybrid modeling approaches.


As in the Psichogios and Ungar (1992a, 1992b) approach, the hybrid model presented

here uses backpropagation ANNs for certain parameters of a FPM. However, instead of

training these ANNs on the errors of the measured outputs of the FPM (grade and

recovery), it inverts the FPM for each set of measurements to calculate corresponding

parameter values, and trains the ANNs on the errors of the calculated parameter values.

Another innovation of the present hybrid model is that it involves two levels of neural

networks. This structure has the advantage that if certain factors that affect the process

like the type of brother or air sparger used are changed, only the top level neural networks

need to be retrained. These only require experimental data that can be easily obtained

with short experiments that do not involve rock, and the large database of past grades and

recoveries is still valid and does not need to be replaced. Finally, in contrast to the above

referenced works, the hybrid model presented here is developed with experimental data

instead of simulated data.








The next section presents the first-principles model. The subsequent section deals

with the calculation of model parameters from measured outputs. This is followed by a

discussion of the artificial neural networks and their integration with the first-principles

model to develop a hybrid model. The fourth section describes the experimental setup,

materials used, experimental procedure, and the methodology used to train the neural

networks. The final section presents results and compares the model predictions of grade

and recovery to experimentally measured grade and recovery.


3.2 First-Principles Model


The FPM is obtained from material balances on both phosphate and gangue. It

neglects radial dispersion and changes in the air holdup. Following Luttrell and Yoon

(1993) the particle to bubble attachment and detachment rates are combined in one net

attachment rate, and this rate is assumed to be first order with respect to particle

concentration in the slurry.


The model subdivides the column into n layers as shown in Figure 3.1. Feed

containing both the desired (phosphate) and undesired (gangue) particles enters in a

slurry in layer k. An additional inlet stream is the elutriation water that enters in the

bottom of the column (layer n). Most of that flow is due to water that enters with the air

sparger, as most of the popular spargers are two-phase and introduce a considerable

amount of water. There are two outlet streams: the tailings stream through the bottom of

the column (layer n) that contains mostly gangue, and the product (concentrate) stream

that leaves from the top of the column.















Qf


Qe


Figure 3.1: Schematic diagram of column for phosphate flotation.








The particles are subdivided into size ranges according to the standard Tyler mesh

screens. Particles of a certain mesh are considered to have diameter the geometric mean

of the lower and upper limits. As the attachment rate constants and particle slip velocities

depend on particle size, a separate material balance is written for each mesh size.

Material balances at each layer yield the following equations for the phosphate particles:

Layer 1 (top)


+ D P -k(d )Cj
Azi' -k(d P

+D P "' k(dp)Cp
Az


U
if U, <
1 E

if U,>
~8g


(3.1)


Layer 2 to k-1: k = feed layer


C' -2CI +CJ
+ D Pi -kp,(dp)C
Az P) P"
C' -2CJ +CJ
+D p"' P P k, (d )CJ,
Az 2


if US p
l-8
Up
if U, >
1- 6
8


(3.2)


Feed Layer = k


(Qf/A)Cj Uj )C -( + U)C ,
1-gg


Az



(Q /A )Cj +(U1 C -( UJ)CJ
1- g (1- g l P


C' -2C + C
Pk+l Pk Pk-I
Az2

if U, 1-Es


C' 2C + C
Pk+I Pk Pk-I
Az2
U
if U'L >


(3.3)


dCJ
t
dt


dC
dt


dCJ
dt
dt


k,(dj )Cj






kp(dJ )CJ
)pPk


+D


+D








Layer k+1 to n-1

dCJ U CJ C CJ -2C _+C
-- =+ U p- -D _C"' pk k(dJ)C (3.4)
dt 1-E 1 Az Az2


Layer n (bottom)

Ut(+U Qt D+U P
-- +U 11 (1- po )A + -U
I --1"-- (1 C )AC --CD -- C
1 c ( )D P.- P _k, (d )CJ

dCC U
PdC if Uj >-
dt s 1-el

t + U C Q +U C
1- (1- P)A1 P" Ci -Ci
S(D ,_I Pn" -k, (dp)CJ
Az Az 2
U,
if U, <-
1--tB

(3.5)

where
AC = Cross-sectional area of the column
C = Phosphate feed concentration ofj'h mesh size particles
Cj = Phosphate concentration ofjth mesh size particles in the ith layer
Qf = Feed volumetric flow rate
Q, = Tailings volumetric flow rate
Qe = Elutriation volumetric flow rate
Qp = Product volumetric flow rate
Up = Superficial liquid velocity above the feed point
= Qp/A,
U, = Superficial liquid velocity below the feed point
= (Q, -Q)/A,
UJ, = Slip velocity of jh mesh size particles
E, = Air holdup
kp(dp) = Flotation rate constant for phosphate forjh mesh size particles








The slip velocity is calculated using the expression of Villeneuve et al. (1996):

gd (p-p,)(1-)27
U0si = -- (3.6)
18 ,(l +0.15R( )

,where the particle Reynolds number is defined as

d) Up ( )
R, s(3.7)
l1-

where

g = Acceleration due to gravity (m/s2)

ti = Water viscosity (kg/ms)

pl = Water density (kg/m3)

p, = Solid density (kg/m3)

(bs = Volume fraction of solids in slurry

dp = Particle diameter (m)

As the right hand side of Equation 3.7 is a function of U,,, the slip velocity is obtained by

solving Equations 3.6 and 3.7 iteratively as described in Chapter 2.


The axial dispersion coefficient is calculated by a modification of the Finch and

Dobby (1990) expression:

IJ Y3
D = 0.063 (1-g)dc (3.8)


where

dc = column diameter (m)

Jg = superficial air velocity (cm/s)








Equations analogous to 3.1-3.8 are valid for the gangue particles, but with a considerably

lesser effective flotation rate constant kg (d ).


In the limit as Az -- 0 the above difference equations become

-C U C 8ac C
-=~ U + D -kp(d )CJ (3.9)
at 1-6 & z &z2 p p

for the section above the feed and

P= +Uj +D -kp(d )C (3.10)
SSJ J) a D C & 2 (3.10)


for the section below the feed.


Recovery (%) is defined as the ratio of the weight of the phosphate in the concentrate

stream to the weight of the phosphate in the feed. The recovery for phosphate particles of

the jo, mesh size can be expressed in terms of the feed and tailings flow rates and

concentrations as



QCf A(Q,+Ao( -e)UjC C
R Q fC -[Q c- *100 (3.11)
= QfCj



Grade, a measure of the quality of the product, is defined as the ratio of the weight of the

phosphate to the total weight recovered in the concentrate stream. Grade is usually

reported as % Bone Phosphate of Lime (%BPL) which is the equivalent grams of

tricalcium phosphate, Ca3(P04)2, in 100 g of sample. For the typical Florida rock,








mineral that contains no gangue is 73.3 %oBPL. Grade can be obtained as the ratio of

phosphate to the sum of phosphate and gangue in the product:


QC_ -rt Qf[Q+Alc(1-s,)Uj
(QfC'-[Qt +Ac(l- g)Us ]Cn )+(QfC'-[Qt +A,(1-e,)UlC]C )'


(3.12)

where Cg is gangue concentration of jth particle size in the nth layer and C' is the

gangue feed concentration of jth particle size. The algorithm for solving the first-

principles model is given in Appendix B.


3.3 Calculation of Model Parameters


Since air-holdup eg is measured experimentally, the above FPM has only two

unmeasured model parameters for each particle size, namely, the flotation rate constants

for phosphate (kp) and for gangue (k). The experimental analysis usually available in

industrial flotation columns is in terms of grade and recovery of phosphate. Let WI

denote the weight of j'" size phosphate particles in the feed, W the weight of jth size

gangue in the feed, and Wg the weight of jth size gangue in the product. The grade of

feed is then

W'
G' =73.3 Wp (3.13)
Wfp + Wfg

and GJ is given by an analogous expression. The recovery of gangue can be readily

calculated from measurements of grade and recovery of phosphate using the following

relationship:








WJ RJ Gi(73.3-GJ
RJ =-g =-'- t (3.14)
SWf Gj (73.3-G')



In some cases direct measurements of the majority of gangue as acid insolubles may be

available. Then more reliable estimates of Rg can be obtained by averaging values

calculated from measurements of acid insolubles with values calculated from Equation

3.14. This was done in this work.


From the FPM equations follows that the recovery of phosphate depends only on kp,

while the recovery of gangue depends only on kg. This can be exploited to easily invert

the steady-state version of the model to determine from experimental measurements of

RI and G' corresponding kp and kg. As shown in Figure 2.1, this is accomplished with

one-dimensional searches. The search for kp is initialized with two values that yield

errors in the corresponding recovery R, of opposite sign. Since typically 0 < kp < 10

minl' the values of 0 and 100 min'1 are used. Then the method of false position (Chapra

and Canale, 1988) is used to iterate until the magnitude of the error in R, drops to less

than 10"3. It is possible that the calculated recovery has a higher value than the

experimental even for kp = 0. In these cases kp is set equal to zero. The above procedure

is also used to determine kg, except that the high initial value is set to 10 min-. Recovery

for both phosphate and gangue increases monotonically with respective flotation rate

constants as discussed in Chapter 2.










3.4 The Hybrid Model


The main factors affecting the air hold up eg are the superficial air velocity Jg and the

brother concentration Cfroth,. Several factors affect the flotation rate constants, k, and kg,

including particle diameter, superficial air velocity, brother concentration, collector

concentration, extender concentration, and pH. In this study we have conducted

experiments varying particle size, brother type, brother concentration, and superficial air

velocity, and develop a hybrid model that portrays the effect of these factors on the

performance of the column. The hybrid model utilizes backpropagation ANNs

(Rumelhart and McClelland 1986) to predict the values of the parameters Eg, kp, and kg.


The straightforward approach is to develop an ANN for each of the three parameters.

The inputs to the ANNs that predict kp and kg would be dp, Jg, and Cfrothe, while the inputs

of the ANN that predicts sg would be Jg, and Cfother. Each of the ANNs in this structure

would then depend on the brother and sparger used. A change in type of brother would

mean that the previously trained ANNs are no longer applicable and would necessitate

collection of a new set of training data and retraining of the networks. As changes in

brother or sparger are not uncommon, this is a disadvantage.


The main reason Jg and Cfroth, as well as the type of brother and sparger, affect the

flotation rate constants, is because they significantly affect the bubble size. An

alternative hybrid model architecture is shown in Figure 3.2. The neural networks are

structured in two levels. The first level consists of the ANNs for predicting kp (NNI) and








Superficial
air velocity


Frother
concentration


1
Phosphate Inferred
particle Bubble
size diameter


Gangue
particle
size


Specific to
frother/sparger
type


air
holdup


Diffusion
coefficient
(Finch and Dobby, 1990)


Recovery


Figure 3.2 : Overall structure of the hybrid model


Grade








kg (NNII) and receives as an input the inferred bubble size. This is the output of one of

the ANNs of the second (top) level, NNIII. The second level also includes NNIV, which

predicts air holdup. The advantage of this structure is that NNI and NNII are

independent of the type of brother and sparger used, and therefore would not need

retraining if these change.


As bubble size is not measured in industry, we infer it from the two-phase (air/water)

air holdup, Jg, and Ut using the well-known Drift-flux analysis (Yianatos et al., 1988).

The output required to train NNIV is the (two-phase) air holdup. Air holdup is relatively

easy to obtain, so after a change of brother or sparger the hybrid model of Figure 3.2 can

become functional in a short interval of time.



3.5 Materials and Methods

3.5.1 Experimental Setup and Procedures


Two types of experiments were conducted: two-phase (air/water) experiments to train

neural networks NNIII and NNIV, and three-phase experiments to train NNI and NNII

and to test the performance of the hybrid model.


The experimental setup for the three-phase experiments is shown in Figure 2.5. It

included an agitated tank (conditioner) for reagentizing the feed, a screw feeder for

controlling the rate ofreagentized feed, and a flotation column. The agitated tank was 45

cm in diameter and 75 cm high and was equipped with an impeller with two axial blades

(each 28 cm diameter). The impeller had about 3.8 cm clearance from the bottom of the

tank and its rotation speed was fixed at 465 rpm. The feeder with a 2.5 cm diameter








screw delivered the conditioned phosphate materials to the column. The feed rate was

controlled by adjusting the screw rotation speed. The flotation column was constructed

of plexiglass and had 14.5 cm diameter and 1.82 m height. The feeding point was located

at 30 cm from the column top. The discharge flow rate was controlled by a discharge

valve and an adjustable speed pump. Three flowmeters were used to monitor the flow

rates for air, brother solution, and elutriation water.


Phosphate feed (14X150 Tyler mesh) from Cargill was used as the feed material. For

each run, 50 kg of feed were introduced to the pre-treatment tank and water was added to

obtain 72 % solids concentration by weight. The tank was then agitated for 10 seconds.

10 % soda ash solution was added to the pulp to reach pH of about 9.4 and the slurry was

agitated for another 10 seconds. Subsequently, a mixture of fatty acids (a mixture of

oleic, palmetic, and linoleic acid obtained from Westvaco) and fuel oil (No. 5 obtained

from PCS Phosphates) with a ratio of 1:1 by weight was added to the pulp and the slurry

continued to be mixed. The total conditioning time was 3 minutes. The conditioned feed

material (without its conditioning water) was subsequently loaded to the feeder bin

located at the top of the column.


Four brothers were used, two commonly employed in industry, F-507 (a mixed

polyglycol by Oreprep) and CP-100 (a sodium alkyl ether sulfate by Westvaco), and two

experimental, F-579 (also a mixed polyglycol by Oreprep) and OB-535 (by O'Brien).

Frother-containing water and air were first introduced into the column through the

sparger (an eductor) at a fixed water flow rate and brother concentration (0 30 ppm),

and the superficial air velocity ranged from 0.24 0.94 cm/s. Then the discharge valve








and pump were adjusted to get the desired underflow and overflow rates. Air holdup was

measured using a differential pressure gauge. After the water/air system reached steady

state, the screw feeder was started. To achieve steady feed rate to the column, water was

added to the screw feeder at the rate that reduced the solids concentration to

approximately 66% by weight. The column was run for a period of three minutes with

phosphate feed prior to sampling. Timed samples of tailings and concentrates were

taken. The collected product samples, as well as feed samples, were dried, sieved using

Tyler meshes, weighed and analyzed for %BPL following the procedure recommended

by the Association of Florida Phosphate Chemists (AFPC Analytical Methods, 1980). In

addition, gangue content (as % acid insolubles) of the feed, tailings, and concentrate

streams was measured (AFPC Analytical Methods, 1980). These measurements were

then used to calculate recovery of acid insolubles. Subsequently these values were

averaged with the values obtained from Equation 3.14 to obtain the R used to determine

the flotation rate constants for gangue.


The two-phase experiments were identical to the three-phase experiments, except that

no solid feed was introduced to the column and the experiments were terminated when

the water/air system reached steady state.



3.5.2 Neural Network Structure and Training


NNI, NNII, NNIII, and NNIV of Figure 3.2 were feedforward backpropagation

artificial neural networks with a single layer of hidden nodes between the input and

output layers and a unit bias connected to both the hidden and the output layers. Inputs








and outputs were scaled from 0 to 1. The hidden and output layer nodes employed logistic

activation functions (Hertz et al., 1992).


For each of the four brothers investigated, 28 two-phase experiments were conducted

(full factorial design with 7 brother concentrations and 4 superficial air velocities). These

were used to train (19 data points) and to validate (9 data points) the top level neural

networks (NNIII and NNIV), a different pair for each brother. Three-phase runs yielded

28 experimental grades and recoveries, which were used to train (19 data points) and to

validate (9 data points) NNI and NNII. To set the number of nodes in the hidden layer of

each network, the number was increased until the sum of the absolute errors of the

training and validation outputs started increasing. In this manner an appropriate number

of hidden nodes was determined to be three for all the neural networks.


The training process started by initializing all weights randomly to small non-zero

values. The random numbers were generated in the range -3.4 to +3.4 with a standard

deviation of 1.0 following the procedure recommended by Masters (1993). The optimal

weights were determined by combining simulated annealing (Kirkpatrick et al. 1983) with

the Polak-Ribiere conjugate gradient algorithm (Polak, 1971). Simulated annealing

randomly perturbed the independent variables (the weights) and kept track of the best

(lowest error) function value for each randomized set of variables. This was repeated

several times, each time decreasing the variance of the perturbations with the previous

optimum as the mean. Then the conjugate gradient algorithm was used to minimize the

mean-squared output error. When the minimum was found, simulated annealing was

used to attempt to break out of what may be a local minimum. This alternation was








continued until a lower point could not be found. This approach improves the likelihood

of convergence to the global optimum.


3.6 Results and Discussion


The performance of the network for predicting bubble diameter (NNIII), the network

for predicting air holdup (NNIV), the network for predicting the phosphate flotation rate

constant (NNI) and the network for predicting the gangue flotation rate constant (NNII) is

shown in Figures 3.3-3.14. Figure 3.3 compares the NNIII output to the inferred bubble

diameter using experimental data when the brother was CP-100. The solid circles are for

the data used for training while the open squares are for the data used for validation.

Figures 3.4, 3.5, and 3.6 show the performance of NNIII when F-507, OB-535, and F-

579, respectively, were the brothers.


As these figures show, NNIII successfully predicts the inferred bubble diameter.

Figure 3.7 compares the air holdup predicted by NNIV to the experimental values

measured by a differential pressure cell when CP-100 was used as the brother. Figures

3.8, 3.9, and 3.10 show the performance of NNIV when F-507, OB-535, and F-579,

respectively, were used as brothers. As shown in these figures, NNIV successfully

predicts the air holdup for all brothers.


Figures 3.11 and 3.12 show the performance of NNI and NNII, respectively. Figure

3.11 presents the predicted flotation rate constants for phosphate (kp) against those

determined from one-dimensional searches using experimental data. As shown in this

figure, NNI does accurately predict low and high values of flotation rate constants.








1.2


0-- 1 -

L.
0.8-
O DO

0 0.6


m 0.4


2 0.2 --
*Training data
O Validation data
0 .--i I I
0 0.2 0.4 0.6 0.8 1 1.2
Inferred Bubble Diameter (mm)

Figure 3.3: Performance of NNIII: Model bubble diameter versus bubble diameter inferred from
experimental data when CP-100 was the brother











1.2


1


0.8


0.6


0.4


0.2


0


0 0.2 0.4 0.6 0.8 1 1.2
Inferred Bubble Diameter (mm)

Figure 3.4: Performance of NNIII: Model bubble diameter versus bubble diameter inferred from
experimental data when F-507 was the brother








1.2 -

E 1

a 0.8

I 0.6

) 0.4

0.2
Training data
0 ,E Validation data

0 0.2 0.4 0.6 0.8 1 1.2
Inferred bubble diameter (mm)
Figure 3.5: Performance of NNIII: Model bubble diameter versus bubble diameter inferred
from experimental data when OB-535 was the brother










1.3


1.2

1.1 O]

E 1

S 0.9
E
. 0.8

0.7

0.6

0.5 Training data
0 Validation data
0.4
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3
Inferred bubble diameter (mm)
Figure 3.6: Performance of NNIII: Model bubble diameter versus bubble diameter inferred from
experimental data when F-579 was the brother






























OLI~


STraining data
O Validation data


Experimental Air Holdup (%)


Figure 3.7:


Performance of NNIV: Model versus experimental air holdup for brother CP-100


.4 --


I

I


I










LU
z-u ----------------------------
18 -

16


O
b 14

- 12
0
-I 10


5 6 EO

4
Training data
2 D Validation data


0 5 10 15 20
Experimental Air Holdup (%)

Figure 3.8: Performance of NNIV: Model versus experimental air holdup for brother
F-507









30


25
Q \~
120
o a
0
15


. 10

CL 5
5 *eTraining data
_O Validation data
0
0 5 10 15 20 25 30
Experimental Air Hold-up (%)


Figure 3.9: Performance of NNIV: Model versus experimental air holdup for brother OB-535










25


20 --
20


0
' 15
I

10 171


5
Training data
0 Validation data
0
0 5 10 15 20 25
Experimental Air Hold-up (%)


Figure 3.10: Performance of NNIV: Model versus experimental air holdup for brother F-579





























0 2 4 6 8
Experimental flotation rate constants for Phosphate (kp)


Figure 3.11: Performance of NNI-Model versus experimental flotation rate constant for phosphate
(kp)








Figure 3.12 presents the flotation rate constants for gangue (kg) predicted using NNII

against those determined from experimental data. A very good match is seen.


The hybrid model integrates NNI, NNII, NNIII, and NNIV with the FPM as shown in

Figure 3.2. Predictions of the hybrid model are shown in Figures 3.13 and 3.14. Figure

3.13 presents the predicted recovery (%) against the experimental recovery for brother

CP-100 (square points), F-507 (circles), OB-535 (triangles), and F-579 (diamonds).

Similarly, Figure 3.14 compares the predicted grade (%BPL) against the experimental

grade for CP-100, F-507, OB-535, and F-579. It can be seen from these figures that

predicted recovery and grade from the hybrid model match closely the experimental

values, with the exception of one grade for OB-535. The root mean squared errors in

predicted recovery were 0.1%, 0.2%, 1.5%, and 0.4% for CP-100, F-507, OB-535, and F-

579, respectively. The root mean squared errors in predicted grade were 3.2 %BPL, 1.5

%BPL, 7.5 %BPL, and 1.5 %BPL for CP-100, F-507, OB-535, and F-579, respectively.


An alternative to the present modeling approach is to develop a pure neural-

networks model. This would, however, require a large number of inputs: not only

superficial air velocity, brother concentration, and particle size, but also feed flow rate,

feed concentration, elutriation flow rate, tailings flow rate, and solids loading. This

increase in number of inputs to eight would increase the number of weights (model

parameters) needed and therefore the number of three-phase data required for training.

Furthermore, as with an in-series hybrid model that uses one level of neural networks, a

change in brother or sparger would require generation of a new set of data and retraining

of all the networks. The hybrid model presented here with the two levels of neural











0 0.9
0.8

o 0.7

*a 0.6



0.3 '0.3

a. 0.2
0. 0 1 *Training data
D Validation data

0 0.2 0.4 0.6 0.8 1
Experimental flotation rate constants for gangue (kg)

Figure 3.12 Performance of NNII: Model versus experimental flotation rate
constant for gangue (k)










100

95

90


100


Experimental Recovery (%)


Figure 3.13: Performance of the overall hybrid model: Predicted versus experimental recovery (%) for
the four brothers


0O Frother CP-100
0 Frother F-507
A Frother OB-535
0 Frother F-579


85

80

75

70

65

60

55

50


60


70


80


.--I-I-I-I----------









80

70 -A

I60



OO
50 -

o 40
0

30 0 Frother CP-100
a 0 0 Frother F-507
20 A Frother OB-535
O Frother F-579
10
10 20 30 40 50 60 70 80
Experimental Grade (%BPL)
Figure 3.14: Performance of the overall hybrid model: Predicted versus experimental grade
(%BPL) for the four brothers








networks involves a relatively low number of inputs in the artificial neural networks, does

not require new three-phase data if a brother or sparger is changed, and gives very good

predictions of both grade and recovery.



3.7 Conclusions



A hybrid neural network modeling approach was presented and used to model a

flotation column for phosphate/gangue separation. This hybrid model is comprised of

two parts, a first-principles model and two levels of neural networks that serve as

parameter predictors of difficult-to-model process parameters. Experimental data from a

laboratory column were used to train and validate the neural networks, and it is shown

that the hybrid model captures the dependence of column performance on particle size,

brother concentration, and superficial air velocity.













CHAPTER 4
OPTIMIZATION PERFORMANCE MEASURES AND FUTURE WORK



High recovery and grade and low operating cost depend largely on the optimal

selection of operating variables. The search of the optimal conditions can considerably

benefit by the availability of a model that can relate the operating conditions to the

column performance. The hybrid model developed for the flotation column provides a

mathematical relationship between the operating variables and column performance.

This hybrid model can be combined with an optimization algorithm to determine the

optimal operating conditions for the flotation column.


We propose an algorithm that leads to the sequential optimization of a flotation

column. This algorithm guides successive changes in the manipulated variables after

each experiment to achieve optimal column operating conditions. Selectivity, which

combines recovery and grade, can be used as the performance measure of the column.

The hybrid model builds a relationship between the process manipulated variables and

the performance measure. The optimization algorithm dictates the changes in the

manipulated variables between successive runs. At each run manipulated variables are

set at their predicted optimal values. After the run is completed, the collected samples

should be collected and analyzed for recovery and grade. Then the new input-output data

are added to the neural network of the hybrid model and the network should be retrained.








New optimal manipulated variable values are predicted which set the conditions for the

subsequent run. This procedure should be repeated until convergence is obtained.


4.1 Performance Measures


The performance of a flotation column is affected by both recovery (%) and grade

(%BPL). To guide optimization it is necessary to combine the two outputs (grade and

recovery) in a single performance measure. Several performance measures are possible,

and some are presented below.


4.1.1 Selectivity


One way to achieve this is to use selectivity as the performance measure.

Selectivity is defined as


S= R /- (4.1)
Rb Rtb

where

R = Recovery of phosphate in the product stream.
Rb = Recovery of gangue in the product stream.
Rt = Recovery(or Rejectability) of phosphate in the tailings stream.
Rtb = Recovery(or Rejectability) of gangue in the tailings strea


We developed the following expression that relates selectivity to the recovery and

the grade of the product stream









--G(1-R)-R

(1-G)(1 -R)


\%here


G = Grade (%BPL) of phosphate in the product stream.

Gf = Grade (BPL) of phosphate in the feed.



4.1.2 Separation Efficiency


Separation efficiency is defined as follows:


E=R -Rb


(4.3)


In this case, the efficiency varies between -100 to 100.



4.1.3 Economic Performance Measure


The selectivity function or the separation efficiency does not include any

economic input such as cost of the reagents. Therefore an alternate performance measure

was developed which includes recovery, grade, and the reagent prices. A scheme for

penalizing lower grade rock has been developed. This scheme deducts differential costs,

relative to 66% BPL, for transportation and acidulation. The acidulation scheme assumes

soluble P205 losses increase in direct proportion to the amount of phosphogypsum. Thus,

the procedure requires an estimate of the quantity of phosphogypsum that is produced.


(4.2)








This performance measure is only applicable to plants, and can not be used with a lab-

scale flotation column. The procedure for this scheme is outlined below:


Assumptions

1. The price of rock of 66% BPL = $22.00
2. Zero insol %BPL = 73.33
3. Transportation cost = $2.50 per ton.
4. Soluble P205 losses = 1.00%
5. Insoluble P205 losses = 6.00%
6. Increase in soluble P205 losses is proportional to the amount of phosphogypsum
produced.

Transportation Penalty


Base case:


66% BPL rock (dry basis)
Freight cost per BPL ton = $2.5/0.66 = $3.79


12.50 3.79
BL /100


per BPL ton


S2.50
B L/100


B3.7erton
- 3.7 B- per ton
9)100


Where, BL = %BPL when grade < 66%

Acidulation Penalty


Base case:


66% BPL rock (30.21% P205, CaO:P205 = 1.49)


SB
Acid insol =100 1 BL
73.33


Calculation of the amount of Phosphogypsum:

Phosphogypsum components


= 1 ton rock x( B L
i'73.33J


Penalty:


Transportation penalty =


Acid insol









Unreacted = 1 ton rock x L x 0.06
73.33)
(B /100) 0
Dihydrate = 1 ton rock x L ) x 1.49 x (172/56) x (- 0.06)
2.184

Total amount of phosphogypsum = Acid insol + Unreacted + Dihydrate

(% so lub le PO, losses)/100
Soluble P205 losses = $300.0 x (% soper ton
2.184

= $1.37 per ton

(Total amount of phosphogypsum)
Acidulation Penalty = $ 62.0 x -1.37
BL

Sales value = Price of 66 %BPL rock (BL/66)15

Adjusted sales value = Sales value Transportation penalty Acidulation penalty

The adjusted value of the phosphate rock as a function of %BPL is shown

in Figure 4.1. Let

Feed solid flow rate = F, ton per year
Product solid flow rate = P, ton per year
Feed grade = Gf, %
Concentrate grade = G, %
Product recovery = R, %
Adjusted sales value of feed = Cf, $ per ton
Adjusted sales value of product = Cp, $ per ton
Reagent-i price = Cri, $/lb
Reagent-i usage = Ui lb/ton feed


The feed flow rate and the product flow rate can be related as:

P=F (Gf OOJ0 (4.4)
1010) G













40


20


0


-20


-40


-60


-80


BL (%BPL)


Figure 4.1: Value of phosphate rock as a function of%BPL


0
4-*
0

0
(0
*o
0i
**-
w,



13








Performance measure = CpP CrF F UjCn, /year (4.5)


4.2 The Optimization Algorithm


The idea behind the sequential optimization is to iterate between experimentation

towards the optimum and model identification until the optimum is reached. The

procedure is as follows:



(1) Initial experiments are performed and their results are analyzed.

(2) The neural networks are trained and the hybrid model is used to determine the

optimal factor values. If these are within the convergence limit of previous

experimental values, the procedure stops.

(3) Otherwise, an experiment at the calculated optimal value is performed and

analyzed.

(4) The data are added to the neural network training set, and the procedure returns to

step (2).



Figure 4.2 shows a more detailed description of the algorithm. After some

initialization runs have been completed, the samples are analyzed and the neural

networks are trained with the input-output data. Subsequently, using the standard Nelder-

Meade algorithm (Himmelblau, 1972), the values of manipulated variables that maximize

the selectivity are determined. If these values correspond to an interior point then the





80


START


Initial runs


ICP analysis


Performance measure


Train neural network


Determine position
of maximum


YES

I
Experimental run at
predicted maximum
I --


Is maximum in
interior of range?


ICP analysis


NO


Experimental run at
half-way point
-4 I


Performance measure


Train neural network


Determine position
of maximum


Difference between two consecutive maximum
less than pre-decided limits ?


Figure 4.2: The run-to-run optimization algorithm


NO E


YES

STOP


ih


I








next run will take place at these manipulated variable values; if on the other hand,

maximum selectivity is at an exterior point, the next run will be performed at the

midpoint between the last run and the predicted optimum values. After completion of the

next run, samples are measured for grade and recovery. The new data are subsequently

added to the training database and the neural network is retrained. The Optimization

algorithm is again used to calculate the new optimal values. These guide the next run,

and so on, until convergence is obtained.


The Nelder-Meade method (nonlinear Simplex) can be used to determine the

value of the manipulated variables at optimal performance. For three manipulated

variables, an initial simplex is defined with four points. This method then takes a series

of steps, moving the point of the simplex where the function is lowest through the

opposite faces of the simplex to a higher point. These steps are called reflections, and

they are constructed to conserve the volume of the simplex. The method expands the

simplex in one direction to take larger steps. When it reaches a lower point, it contracts

in the traverse direction. This is continued till the decrease in the function value

(selectivity) is smaller than some tolerance (1E-3).



4.3 Initial Scattered Experiments


Scattered experiments according to a factorial design were performed to generate

data for the initial training of the neural networks. Superficial air velocity, brother

concentration, and elutriation water flow rate were selected as the manipulated variables.

F-507, which is a non-ionic surfactant, was used as the brother in these experiments.








Experiments were performed with five different levels of superficial air velocity (0.24,

0.42, 0.60, 0.78, and 0.96 cm/s) and brother concentration (5, 10, 15, 20, 15 ppm), and

three levels ofelutriation water flow rate (9, 10, and 11 gallons per min.).


The design of experiments is shown in Table 4.1. The experiments were designed

so as to generate 13 data points, which is the minimum required for training the neural

networks, which have three hidden nodes. Experiments were performed according to the

design while keeping all other variables constant. After each experiment, three samples

from the tailings stream were collected. The samples were then analyzed for %BPL

content following the procedure recommended by the Association of Florida Phosphate

Chemists (AFPC Analytical Methods, 1980). Since the grade of the feed is known, grade

of the concentrate stream can easily be calculated by making a material balance around

the column.



4.4 Results and Discussions


The three neural networks of the hybrid model were trained using 13 data points

obtained from the designed experiments. The performance of these neural networks is

shown in Figures 4.3- 4.5. Figure 4.3 presents the predicted flotation rate constants for

phosphate (kp) against those determined from one-dimensional searches using

experimental data as described in chapter two and three. As shown in this figure, the

neural network satisfactorily captures the dependence of the flotation rate constant on the

selected manipulated variables. Similarly, Figure 4.4 presents the predicted flotation rate

constants for gangue (kg) against those determined from one-dimensional searches using








Table 4.1: Operating conditions


Frother Superficial air Elutriation water
Concentration velocity Flow rate
1 -1 -1 -1

2 -1 +1 -1

3 +1 -1 -1

4 +1 +1 +1

5 -1 0 +1

6 +1 0 -1

7 0 -1 +1

8 0 +1 +1

9 0 0 0

10 0 +0.5 0

11 +0.5 0 0

12 0 -0.5 0

13 -0.5 0 0
.1 __________________________________


for the factorial design














0.9

0 0.8
0
o "r
o9 0.7

0.6
0 a. 0.5
0o
M 0.4 -
o j.
0.3 -
S0.2

0.1

0
0 0.2 0.4 0.6 0.8 1

Experimental flotation rate constants for phosphate (kp)


Figure 4.3: Neural network versus experimental flotation rate constant for phosphate (kp)








0.1
S 0.09

(n 0.08
8 0.07
". 0.06
o 0.05


t "" 0.03
0.02

S 0.01 -
0

0 0.02 0.04 0.06 0.08 0.1
Experimental flotation rate constants for gangue (kg)

Figure 4.4: Neural network versus experimental flotation rate constant for gangue (k)








experimental data. Again, a very good match is seen Figure 4.5 compares the predicted

air holdup to the experimental values measured by a differential pressure cell. As shown

in this figure, neural network successfully predicts the air holdup.


Table 4.2 shows the results of the 13 designed experiments. As can be seen from

this Table, feed flow rate and the %solids content varied significantly. The screw feeder

operation was erratic and therefore we were unable to feed at the same rate in each run.

Feed flow rate was calculated based on the product flow rate and the tailings flow rate. A

specified volume of product and tailings were taken over a period of time (-20 s) and the

samples were dried and the weight was taken. In this way, solids flow rate in product and

tailings stream were obtained. An overall material balance on the column then gives the

feed flow rate. Similarly, an overall material balance on the water phase gives the water

flow rate in the feed stream. Solids feed flow rate and the water flow rate in the feed then

can be used to obtain the % solids in the feed. Unfortunately, the inability to control feed

flow rate and % solids content means that a meaningful run-to-run optimization cannot be

conducted.


4.5 Future Work

First, the screw feeder needs to be repaired or replaced. After this has been

accomplished, the hybrid model obtained from the designed experiments (Figures 4.3-

4.5) should be used with the Nelder-Meade algorithm to determine the experimental

conditions of the first optimization run. The results of the run should be analyzed for

grade and recovery and these data should be added to the neural network training sets.

The networks should then be retrained and the updated hybrid model used to determine










30

28

26

24
0.
-a 22
0
I 20

5 18

6 16

14

12

10
10 15 20 25 30
Experimental Air Holdup (%)


Figure 4.5: Model versus experimental air holdup for brother F-507









Table 4.2: Results of the runs from the factorial design
Frother Air Feed Tailings E'utria- Solids Grade Recovery
conc. flow flow feed tion content (%BPL) (%)
(ppm) rate rate flow flow (%)
(scfm) (gpm) rate rate
(gpm) (gpm)
1 5 0.0928 0.198 2.014 2.410 59.37 55.95 68.31
2 5 0.3711 0.418 1.779 2.351 35.11 55.36 40.04
3 25 0.0928 0.126 1.432 2.423 48.53 40.07 34.60
4 25 0.3711 0.284 2.062 2.919 35.03 61.86 45.90
5 5 0.2319 0.376 1.897 2 893 49.75 51.04 67.72
6 25 0.2319 0.284 1.650 2378 47.02 39.59 55.92
7 15 0.0928 0.264 2.355 2.922 36.43 62.68 47.49
8 10 0.3711 0340 2.275 2.927 41.65 53.99 48.53
9 15 0.2319 0.463 2.173 2.619 38.58 45.63 16.87
10 15 0.3015 0.370 1.838 2.645 42.65 46.26 54.10
11 20 0.2319 0.261 1.694 2.661 49.18 37.50 55.19
12 15 0.1624 0.281 1.853 2.634 42.36 68.40 47.92
13 10 0.2319 0.259 1.758 2.631 41.13 68.03 52.14





89


the conditions for the next run. This should be repeated with the algorithm of Figure 4.2

until convergence is obtained.










APPENDIX A
CODE FOR THE FIRST PRINCIPLES MODEL FOR ONE LEVEL


#include
#include
#include
#include
#define GQT 2.5372
#define GQF 0.6133
#define GQE 3.038
#define CS 66.0
#define BPL 24.9
#define ROS 2.6
#define DP 122.5
#define Eg 0.05762
#define Dia 0.5
#defineL 6.0
#define Ku 2.217556
#define KGu 0.999965
#define Qg 0.1778
#define FNF 1.0
#define LfL-FNF


//Tailings Flow rate (gallons/min)//
//Feed Flow rate (gallons/min)//
//Elutriation Flow rate (gallons/min)//
//% Solid in the feed (lb S/lb T)//
//% BPL of the feed (lb P/lb S) //
//Specific Gravity of solids in the feed//
//Particle size in microns //
//air hold up //
//Diameter of the column (ft.)//
//Height of the column (ft.)//
//Flotation rate const. for Phosphate (1/min.)//
//Flotation rate const. for Gaunge (1/min.)//
//Air flow rate(scfm)//
//Feed Location from the top (ft.)//


void main 1 (double, double, double[]);


void main l(double CF, double k, double B[])
{

double QF,QE,QT ,QT,QP,Area,UP,UT,UF,D,DP ,PHIS,USLi,REP,USL,diff,
double a,b,d,alpha,beta,gamma,delta,p,q,m;

QF=0.1336541 *GQF;
QE=0.1336541*GQE;
QT1=(0.1336541*GQT);
QP=QF-QT1+QE;
QT=QT1-QE;
Area=0.7853981 *Dia*Dia;
UP=QP/Area;
UT=QT/Area;
UF=QF/Area;




Full Text
17
Ca3(P04)2 in lOOg of sample. Grade can be calculated as the ratio of the weight of
phosphate to the sum of the weight of the phosphate and gangue in the concentrate
stream:
f QfC{ -j
Q.
+ Ac(1-6,)U.
CJ
r p2
\
z =0
v(QfCj-[Qt+Ac(l-6g)U]C;2
)+(QfCj -
z =0 *s
[Qt+Ac(l-sg)U,]CJg2
z =0 ^ y
(2.25)
where is the gangue concentration of the j'h particle size and Ci is the gangue feed
52 xg
concentration of jlh particle size The multiplication factor is 73.3 instead of 100, because
pure Florida phosphate rock measures at about 73.3 %BPL.
2.2.3 Model Parameters
The above model formulation has only two model parameters, namely, the
flotation rate constants for phosphate and gangue. The experimental analysis in the
industry is usually available in terms of grade and recovery of phosphate. The recovery
of gangue can then be readily calculated from grade and recovery of phosphate using the
following relationship:
Rj _RJpGf(73 3~GJ)
GJ(73.3-Gi)
(2.26)
where GJf is the grade of the feed material.


Predictedl flotation rate constants
for gangue (kg)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Experimental flotation rate constants for gangue (kg)
Figure 2.7 Performance of NNII: Model versus experimental flotation rate constant for gangue
(kg)


82
Experiments were performed with five different levels of superficial air velocity (0.24,
0.42, 0.60, 0.78, and 0.96 cm/s) and frother concentration (5, 10, 15, 20, 15 ppm), and
three levels of elutriation water flow rate (9, 10, and 11 gallons per min.).
The design of experiments is shown in Table 4.1 The experiments w'ere designed
so as to generate 13 data points, which is the minimum required for training the neural
networks, which have three hidden nodes. Experiments were performed according to the
design while keeping all other variables constant. After each experiment, three samples
from the tailings stream were collected. The samples were then analyzed for %BPL
content following the procedure recommended by the Association of Florida Phosphate
Chemists (AFPC Analytical Methods, 1980). Since the grade of the feed is known, grade
of the concentrate stream can easily be calculated by making a material balance around
the column.
4.4 Results and Discussions
The three neural networks of the hybrid model were trained using 13 data points
obtained from the designed experiments. The performance of these neural networks is
shown in Figures 4.3- 4.5. Figure 4.3 presents the predicted flotation rate constants for
phosphate (kp) against those determined from one-dimensional searches using
experimental data as described in chapter two and three. As shown in this figure, the
neural network satisfactorily captures the dependence of the flotation rate constant on the
selected manipulated variables. Similarly, Figure 4.4 presents the predicted flotation rate
constants for gangue (kg) against those determined from one-dimensional searches using


This dissertation was submitted to the Graduate Faculty of the College of
Engineering and to the Graduate School and was accepted as partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
May 1999
Winfred M. Phillips
Dean, College of Engineering
M. J. Ohanian
Dean, Graduate School


107
{
/y******************** Steady sate Rec0very calculation*******************
Pe=(((QT/Area)+USL)*(L))/(D*(l-Eg));
//printf("\n Pe=%lf',Pe);
Tp=((L)*(l-Eg))/((QT/Area)+USL);
//printf("\n Tp=%lf',Tp);
a=sqrt( 1 +(4*K*Tp/Pe));
//printf("\n a=%lf',a);
//printfC'Xn K*Tp=%lf',K*Tp);
bl=exp(a*Pe/2);
b2=exp(-a*Pe/2);
REQNP=( 1 -((4*a*exp(Pe/2))/((( 1 +a)*( 1 +a)*b 1)-((1 -a)*( 1 -a)*b2))))* 100;
R=( 1 -((1 -(REQNP/100)) ((QT+Area U SL)/QF)))*100;
c=sqrt( 1 +(4*KG*Tp/Pe));
dl=exp(c*Pe/2);
d2=exp(-c*Pe/2);
REQNG=(l-((4*c*exp(Pe/2))/(((l+c)*(l+c)*dl)-((l-c)*(l-c)*d2))))*100
RG=( 1 -((1 -(REQNG/100))*((QT+Area*SL)/QF)))* 100;
//printfT\n REQNP=%. llP/o",REQNP);
printf"\n Recovery=%. 1 lfyo",R);
//printf("\n REQNG=%lf%",REQNG);
Gradeqn=((QF*CF-(QT+Area*USL)*( l -(REQNP/100))*CF)/((QF*CF-
(QT+Area*USL)*( 1 -(REQNP/100))*CF)+(QF*CFG-(QT+Area*USL)*( 1 -
(REQNG/100))*CFG)))* 100.0;
Selecteqn=R-RG;
printf("\n GRADEQN=%. 1 lf %",Gradeqn);
printf("\n Sep_eT=%. 1 lf ",Selecteqn);
getch();
}
//*********Steady state Recovery calculation ends ***********
}


10
bio-kinetic rates of a first-principles model. Cubillo and Lima (1997) also used this
approach to develop hybrid model for a rougher flotation circuit.
In this work, we employ a serial approach to integrate an approximate model,
derived from first-principles considerations, with neural networks which approximates
the unknown kinetics. The first-principles model is inverted to calculate two model
parameters for each set of measured recovery and grade. The neural networks are then
trained on the errors of calculated model parameters instead of the errors of the output of
the first-principles model as is the case with the above referenced works. Also, unlike
most other cited work, we employ experimental data instead of simulated data.
2 2 First-Principles Model
The basic equations representing the flotation of solid particles in a flotation column
can be written by making a material balance for the solid particles in the slurry phase.
This results in the following partial differential equations for the section above and below
the feed point, respectively:
cC
j f
Pi
ct
cCJ
Pi
ct
Up
-us]
J
dCl
Pi
+ D
02CJ
Pi
l-*t
dz
dz2
u
>
dCi
d2C]
+ UJ,
P2
-+D
Pi
10.
dz
dz
(2.1)
(2.2)


92
else
p=((-(-beta+a))*exp((beta)*(L)))/((-alpha-*-a)*exp((alpha)*(L)));
q=-(delta-(QE/(Area*D)))/(gamma-(QE/(Area*D))),
m=(q*exp((gamma)*(Li))+exp((delta)*(Lf)))/(p*exp((alpha)*(Lf))+exp((beta)*(Lf)));
B[4]-(UF*CF/D)/((m*p*(a-
alpha)*exp((alpha)*(Lf)))+((d+gamma)*q*exp((gamma)*(Lf)))+(m*(a-
beta)*exp((beta)*(Lf)))+((d+delta)*exp((delta)*(Lf))));
//B[3]=(UF*CF/D)/((a+alpha+d+gamma)*exp((gamma)*(Lf)));
B[2]=m*B[4];
B[l]=p*m*B[4];
B[3]=q*B[4];
//B[l]=(exp((gamma-alpha)*Lf))*B[3];
//B[2]=0.0;
//3[4]=0.0;
}
void main(void)
{
double
QF,QE,QT 1 ,QT,QP,Area,DP 1 ,PfflS,USLi, REP, USL,diff, Frank, CF, CFG, K, KG, C,CG;
double RO,ROG,Grade,B[5],BG[5];
QF=0.1336541 *GQF;
QE=0.1336541 *GQE;
QT1 =(0.1336541 *GQT);
QP-QF-QT1+QE;
QT=QT1-QE;
Area-0.7853981 *Dia*Dia,
printf("\n UP=%lf',QP/Area);
//************** Calculation of slip velocity **************
DPI-DP/1000.0;
PHIS=(CS/100)/((CS/100)+((1-(CS/100))*ROS)),
USLi-0.0;
do
{
REP-5.12*DPl*USLi*ROS*(l. 0-PHIS);
USL=108.233*DP1*DP1 *(ROS-l)*pow((l-PHIS),2.7)/(l+0.15*(pow(REP,0.687)));
diflf-USL-USLi,
if(difF<0.0)


2
achieve a neutral pH. The scrubbed and washed material is then subjected to cleaner
flotation in which amine together with kerosene is used as collector to float sand. This
stage of flotation is sensitive to impurities in water; thus, fresh water is used in most of
the plants as make up water. However, the fatty acid circuit uses recycled water. This
process has become less cost effective due to high cost of reagents and increasing
concentration of contaminants.
To prepare the phosphate feed, the mined phosphate ore (matrix) is washed and
de-slimed at 150 mesh. The material finer than 150 mesh is pumped to clay settling
ponds. The rock coarser than 150 mesh is screened to separate pebbles (-3/4 +14 mesh)
which are of high phosphate content. Washed rock (-14, +150 mesh) is sized into a fine
(usually 35 x 150 mesh) and a coarse flotation feeds (usually 14 x 35 mesh) which are
treated in separate circuits. Flotation of phosphates from the fine feed (35 X 150 mesh)
presents very few difficulties and recoveries in excess of 90% are achieved using
conventional flotation cells. On the other hand, recovery of phosphate values from the
coarse feed is much more difficult and flotation by itself usually yields recovery of 60%
or less.
The density of the solid, turbulence, stability and height of the froth layer, depth
of the water column, viscosity of the froth layer are known to effect the flotation process
in general (Boutin and Wheeler, 1967). However, the exact reasons for low recovery of
coarse particles in conventional flotation is not very well understood. There are several
hypotheses about the flotation behavior of coarse particles. For instance, the floatability
of large particles could be due to the additional weight that has to be lifted to the surface


43
networks (i.e determined the neural network parameters) on the error of the output of the
first-principles model. A similar approach was followed by Reuter et al. (1993) to model
metallurgy and mineral processes. Liu et al. (1995) developed a hybrid model for a
periodic wastewater treatment process by using ANNs for the bio-kinetic rates of a first-
principles model. The Psichogios and Ungar (1992a, 1992b) approach was used by
Cubillo et al. (1996) to model particulate drying processes, and by Cubillo and Lima
(1997) to develop a hybrid model for a rougher flotation circuit. Thompson and Kramer
(1994) combined the parallel and serial hybrid modeling approaches.
As in the Psichogios and Ungar (1992a, 1992b) approach, the hybrid model presented
here uses backpropagation ANNs for certain parameters of a FPM. However, instead of
training these ANNs on the errors of the measured outputs of the FPM (grade and
recovery), it inverts the FPM for each set of measurements to calculate corresponding
parameter values, and trains the ANNs on the errors of the calculated parameter values.
Another innovation of the present hybrid model is that it involves two levels of neural
networks. This structure has the advantage that if certain factors that affect the process
like the type of frother or air sparger used are changed, only the top level neural networks
need to be retrained. These only require experimental data that can be easily obtained
with short experiments that do not involve rock, and the large database of past grades and
recoveries is still valid and does not need to be replaced. Finally, in contrast to the above
referenced works, the hybrid model presented here is developed with experimental data
instead of simulated data


80
70
60
50
40
30
20
10
Experimental Grade (%BPL)
gure 3 14: Performance of the overall hybrid model: Predicted versus experimental grade
BPL) for the four frothers


25
Figure 2.5 A schematic diagram of the experimental setup


52
3.4 The Hybrid Model
The main factors affecting the air hold up eg are the superficial air velocity Jg and the
frother concentration Cfrother- Several factors affect the flotation rate constants, kp and kg,
including particle diameter, superficial air velocity, frother concentration, collector
concentration, extender concentration, and pH. In this study we have conducted
experiments varying particle size, frother type, frother concentration, and superficial air
velocity, and develop a hybrid model that portrays the effect of these factors on the
performance of the column. The hybrid model utilizes backpropagation ANNs
(Rumelhart and McClelland 1986) to predict the values of the parameters sg, kp, and kg.
The straightforward approach is to develop an ANN for each of the three parameters.
The inputs to the ANNs that predict kp and kg would be dp, Jg, and Cfrother, while the inputs
of the ANN that predicts eg would be Jg, and Cfrother. Each of the ANNs in this structure
would then depend on the frother and sparger used. A change in type of frother would
mean that the previously trained ANNs are no longer applicable and would necessitate
collection of a new set of training data and retraining of the networks. As changes in
frother or sparger are not uncommon, this is a disadvantage.
The main reason Jg and Cfrother, as well as the type of frother and sparger, affect the
flotation rate constants, is because they significantly affect the bubble size. An
alternative hybrid model architecture is shown in Figure 3.2. The neural networks are
structured in two levels. The first level consists of the ANNs for predicting kp (NN1) and


2 13 Performance of the overall hybrid model: Predicted versus
experimental recovery (%) for unsized feed size distribution 37
2.14 Performance of the overall hybrid model: Predicted versus
experimental grade (%BPL) for unsized feed size distribution 38
3.1 Schematic diagram of column for phosphate flotation 45
3.2 Overall structure of the hybrid model 53
3.3 Performance ofNNIII: Model bubble diameter versus bubble
diameter inferred from experimental data when CP-100 was the
frother 59
3.4 Performance ofNNIII: Model bubble diameter versus bubble
diameter inferred from experimental data when F-507 was the
frother 60
3 5 Performance ofNNIII: Model bubble diameter versus bubble
diameter inferred from experimental data when OB-535 was the
frother 61
3.6 Performance ofNNIII: Model bubble diameter versus bubble
diameter inferred from experimental data when F-579 was the
frother 62
3.7 Performance of NNIV: Model versus experimental air holdup for
frother CP-100 63
3 8 Performance of NNIV Model versus experimental air holdup for
frother F-507 64
3.9 Performance of NNIV: Model versus experimental air holdup for
frother OB-535 65
3.10 Performance of NNIV: Model versus experimental air holdup for
frother F-579 66
3.11 Performance of NNI: Model versus experimental flotation rate
constant for phosphate (kp) 67
3.12 Performance ofNNII Model versus experimental flotation rate
constant for gangue (kg) 69
3.13 Performance of the overall hybrid model Predicted versus
experimental recovery (%) for the four frothers 70
vii


81
next run will take place at these manipulated variable values; if on the other hand,
maximum selectivity is at an exterior point, the next run will be performed at the
midpoint between the last run and the predicted optimum values. After completion of the
next iun, samples are measured for grade and recovery. The new data are subsequently
added to the training database and the neural network is retrained. The Optimization
algorithm is again used to calculate the new optimal values. These guide the next run,
and so on, until convergence is obtained.
The Nelder-Meade method (nonlinear Simplex) can be used to determine the
value of the manipulated variables at optimal performance. For three manipulated
variables, an initial simplex is defined with four points. This method then takes a series
of steps, moving the point of the simplex where the function is lowest through the
opposite faces of the simplex to a higher point. These steps are called reflections, and
they are constructed to conserve the volume of the simplex. The method expands the
simplex in one direction to take larger steps. When it reaches a lower point, it contracts
in the traverse direction. This is continued till the decrease in the function value
(selectivity) is smaller than some tolerance (1H-3).
4 3 Initial Scattered Experiments
Scattered experiments according to a factorial design were performed to generate
data for the initial training of the neural networks. Superficial air velocity, frother
concentration, and elutriation water flow rate were selected as the manipulated variables.
F-507, which is a non-ionic surfactant, was used as the frother in these experiments.


46
The particles are subdivided into size ranges according to the standard Tyler mesh
screens. Panicles of a cenain mesh are considered to have diameter the geometric mean
of the lower and upper limits. As the attachment rate constants and particle slip velocities
depend on particle size, a separate material balance is written for each mesh size.
Material balances at each layer yield the following equations for the phosphate particles:
Laver 1 (top)
dC
Pi
dt
U.
V1-8g
f
-U,
rj _pj rJ _rJ
+ D -Pi -kp(dMCj
Az Az- p p D
pi
TJJ _
'si
u.
V
1 -e
C
pi
Az
+ D
CJ -C
P2 Pi
Az
~kn(do)C
p v p / p,
u
if u <-ll
1-6,
if u>
u
1-8,
Laver 2 to k-1: k = feed layer
(3.1)
d_^
dt
U,
^-88
ui
CJ -CJ CJ -2CJ + CJ
^ + D ^-kp(dp)C(
r
U l
u.
Az
Az
1-E
8
Cl -C' Cl -2CI +CI
-P-l+D p"' -k_(dJp)C
Az Az2 p p p'
UD
1-6,
¡fu L>
u.
1-6,
Feed Laver = k
(3.2)
dCJ
Pk
(Qf / Ao)ci (-Li- U )C (-- + uj, )c;
1-E,
Pk 1-6,
Az
pk CJ 2CJ + CJ
+ D
Az
-MWk
if U <
u.
1 6,
dt
(Qf/A.)CUu--)CL1-(^- + U)C|k c, _
Az
-+D
Pk+1
2CJ + CJ
Pk Pk-1
Az"
if ui > LA
1 e.
(3.3)


3
under the heavy turbulence conditions, and the difficulty to transfer and maintain these
panicles in the froth layer. Some efforts towards improving the flotation of coarse
particles through stabilization of the froth layer, minimizing the froth height, and addition
of an elutriation water stream at the bottom of the column have been undertaken
The equipment used by the phosphate industry in flotation process are not
selective enough to take full advantage of new reagents and operating schemes, to
recover phosphate from the coarse feed or to optimize results with existing reagents. The
best way to increase the selectivity of phosphate flotation is to improve upon the design
of flotation equipment. Particularly the new equipment should improve the recovery of
coarse particles, while still providing the high selectivity of fine particles.
It has been found both theoretically and practically that flotation columns have
better separation performance than conventional mechanical cells (Finch and Dobby,
1990). The use of flotation columns can not only help overcome some of the problems
related to coarse phosphate flotation but it has several other advantages as mentioned
above. Spargers or bubble generating systems are the single most important element in
the flotation columns. They are generally characterized in terms of their air dispersion
ability. Frothers are the chemicals that help in controlling and stabilizing bubble size by
reduction of surface tension. Thus both of them play an important role in the overall
performance of flotation columns. Their interaction can be a crucial factor in the success
of flotation column.
Flotation columns have been used predominantly in the coal beneficiation
industry. However, their application in other mineral industries, such as the phosphate, is


TABLE OF CONTENTS
pane
ACKNOWLEDGMENTS iii
LIST OF FIGURES vi
ABSTRACT ix
CHAPTERS
1 INTRODUCTION 1
2 ONE-LEVEL HYBRID MODEL 6
2.1 Introduction 6
2.2First-Principles Model 10
2.2.1 Boundary Conditions 13
2.2.2 Calculation of Recovery and Grade 16
2.2.3 Model Parameters 17
2.3 The Hybrid Model 22
2.4 Materials and Methods 23
2.4.1 Experimental Setup and Procedure 23
2.4.2 Experimental Conditions 27
2.4.3 Neural Network and Training 27
2.5 Results and Discussions 28
2.6 Conclusions 39
3 TWO-LEVEL HYBRID MODEL 40
3.1 Introduction 40
3.2 First-Principles Model 44
3.3 Calculation of Model Parameters 50
3.4 The Hybrid Model 52
3.5 Materials and Methods 54
3.5.1 Experimental Setup and Procedure 54
3.5.2 Neural Network and Training 56
3.6 Results and Discussions 58
3.7 Conclusions 72
IV


Experimental Recovery (%)
Figure 3.13: Performance of the overall hybrid model: Predicted versus experimental recovery (%) for
the four frothers


94
#defme
#defme
#define
#define
#define
^define
#define
#define
#define
#defme
^define
^define
#defme
GQE 2.6948
//Elutriation Flow rate (gallons/min)//
CS 66.0
//% Solid in the feed (lb S/lb T)//
BPL 24.9
//% BPL of the feed (lb P/lb S) //
ROS 2.6
//Specific Gravity of solids in the feed1!
DP 122.5
//Particle size in microns //
Eg 0.115
//air hold up //
Dia 0.5
//Diameter of the column (ft.)//
L 6.0
//Height of the column (ft.)//
KGu 0.0
//Flotation rate const, for Gaunge (1/min.)//
Qg 0 1778
//Air flow rate(scfm)//
FNF 1.0
LfL-FNF
ES IE-3
//Feed Location from the top (ft.)//
void main l(double,double,double[]);
double model(double);
void main(void)
{
double kl,yl,gl,ku,yu,gu,kr[3],yr,gr,R[3],a,EA,Test,x;
double Grade,Grade_feed,Grade_prod;
int i;
R[l]=70.915;
Grade=25.8;
Grade_feed=BPL/73.3;
Grade_prod=Grade/7 3.3;
R[2]=R[l]*Grade_feed*(l-Grade_prod)/((l-Grade_feed)*Grade_prod);
printrV'\nComponent[ 1 ]-phosphate");
printf("\nComponent[2]=gangue");
for(i=l;i<=2;i-H-)
{
kl=0.0;
yl=model(kl);
gl=R[i]-yl;
if(gl>0.0)
{
do
{
printf"\nEnter an initial guess for flotation rate constant for component[%d]",i);
printfi"\nku=");
scanif"\n %lf', &ku);


5
Two hybrid modeling approaches are presented. Chapter 2 describes a one-level
hybrid model that uses three different neural networks to predict the flotation rate
constant for phosphate, the flotation rate constant for gangue, and air holdup Chapter 3
presents a two-level hybrid model in which neural networks are structured in two levels.
Two neural networks are used in the top-level to predict bubble diameter and air holdup.
The bubble diameter is used as an input in the neural networks of the bottom-level which
predict the flotation rate constants for phosphate and gangue. The inherent advantages
and disadvantages of the two hybrid modeling approaches are also discussed in these
chapters.
In chapter 4, the hybrid model developed is combined with an on-line
optimization algorithm to determine the optimal conditions for column operation. The
algorithm guides successive changes of the manipulated variables such as air flow rate,
frother concentration, and pH, after each run to achieve optimal column operating
conditions. Designed experiments were performed to generate data for the initial training
of the neural networks. The trained neural network is then used to guide the direction of
the new experiments.


Experimental Recovery (%)
Figure 2.13: Performance of the overall hybrid model: Predicted versus experimental recovery (%)
for the unsized feed after it has been sized.


68
Figure 3.12 presents the flotation rate constants for gangue (kg) predicted using NNII
against those determined from experimental data. A very good match is seen.
The hybrid model integrates NN1, NNII, NNIII, and NNIV with the FPM as shown in
Figure 3.2. Predictions of the hybrid model are shown in Figures 3 13 and 3.14. Figure
3.13 presents the predicted recovery (%) against the experimental recovery for frother
CP-100 (square points), F-507 (circles), OB-535 (triangles), and F-579 (diamonds).
Similarly, Figure 3.14 compares the predicted grade (%BPL) against the experimental
grade for CP-100, F-507, OB-535, and F-579 It can be seen from these figures that
predicted recovery and grade from the hybrid model match closely the experimental
values, with the exception of one grade for OB-535. The root mean squared errors in
predicted recovery were 0.1%, 0.2%, 1.5%, and 0.4% for CP-100, F-507, OB-535, and F-
579, respectively. The root mean squared errors in predicted grade were 3.2 %BPL, 1.5
%BPL, 7.5 %BPL, and 1.5 %BPL for CP-100, F-507, OB-535, and F-579, respectively.
An alternative to the present modeling approach is to develop a pure neural-
networks model. This would, however, require a large number of inputs: not only
superficial air velocity, frother concentration, and particle size, but also feed flow rate,
feed concentration, elutriation flow rate, tailings flow rate, and solids loading. This
increase in number of inputs to eight would increase the number of weights (model
parameters) needed and therefore the number of three-phase data required for training.
Furthermore, as with an in-series hybrid model that uses one level of neural networks, a
change in frother or sparger would require generation of a new set of data and retraining
of all the networks. The hybrid model presented here with the two levels of neural


30
28
26
24
22
20
18
16
14
12
10
Figure 4.5: Model versus experimental air holdup for frother F-507


APPENDIX B
CODE FOR THE FIRST PRINCIPLES MODEL FOR TWO LEVELS
#include
#include
#include
#include
#define
#define
#define
#define
#define
#define
#define
#define
#define
#define
#define
#define
#defme
#define
^define
#define
#define
#define
GQT 1.5249
//Tailings Flow rate (gallons/min)//
GQF 0.8858
//Feed Flow rate (gallons/min)//
GQE 2.5102
//Elutriation Flow rate (gallons/min)//
CS 67 2
//% Solid in the feed (lb S/lb T)//
BPL 18.2
//% BPL of the feed (lb P/lb S) //
ROS 2.6
//Specific Gravity of solids in the feedII
DP 208.25
//Particle size in microns //
Eg 0.1487
//air hold up //
Dia 0.5
//Diameter of the column (ft.)//
L 6.0
//Height of the column (ft.)//
Ku 6.042797
//Flotation rate const, for Phosphate (1/min.)//
KGu 0.048094
//Flotation rate const, for Gaunge (1/min.)//
Qg 0.2706
//Air flow rate(scfm)//
N 15
FNF 1.0
//Feed Location from the top (ft.)//
DELT 0.1
n N+l
al 1.0
//a 1 =0>explicit;a 1 = 1 >implicit//
void main 1 (double,double,double,doublet]);
void mainl(double CF,double K,double D,double C[])
{
static double A[n][n],V[n],S[n];
double A1,A2,A3,A4,A5,A6,A7,A8,A9,UP,UT,UT1,UF,DELZ,QP,QF,QT,QE,QT1;
double A10,A11,A12,A13,A14,
int 0[n];
int i,j,k,ii,Pivot,IDummy,NF;
double Big, Dummy,factor,Sum, Area,USLi,USL,REP,diff,PLUS,DPI;
QF=0.1336541*GQF;
QE=0.1336541 *GQE;
99


95
yu=model(ku);
gu=R[i]-yu;
a=gl*gu;
}
while(a>=0);
EA=1.1*ES;
while(EA>ES)
{
kr[i]=ku-(gu*(kl-ku))/(gl-gu);
x=kr[i];
if((kl+ku)!=0)
EA=fabs((ku-kl)/(kl+ku))* 100;
yr=model(x);
gr=R[i]-yr;
Test=gl*gr,
if(Test=0.0)
EA=0;
else if(Test<0.0)
ku=kr[i];
else if(Test>0.0)
kl=kr[i];
printfT,\nkr[%d]=%lf,.i,kr[i]);
}
printf("\nFlotation rate constant for component[%d]=%lf',i,kr[i]);
getchO;
}
else
printfT\nkr[%d]=0.0",i);
}
}
void main 1 (double CF, double k, double B[])
{
double QF,QE,QT1,QT,QP,Area,UP,UT,UF,D,DPI,PHIS,USLi,REP,USL,difF;
double a,b,d,alpha,beta,gamma,delta,p,q,m;
QF=0.1336541*GQF;
QE=0.1336541 *GQE;
QT1=(0.1336541 *GQT);


41
In anionic phosphate flotation the mineral is first treated with fatty acid collector and fuel
oil extender At proper concentrations these mostly adsorb on the phosphate-containing
particles rendering them hydrophobic. Then the phosphate-containing particles are
separated from gangue via the flotation process. The majority of the phosphate plants
employ mechanical cells. However, column flotation has simpler operation and provides
superior grade/recovery performance. For these reasons column flotation is gaining
increasing acceptance for the processing and beneficiation of phosphate ores.
Column flotation is frequently employed for the recovery of other minerals (e.g., coal,
copper, nickel, gold). In such applications the column can be divided into three zones: an
upper froth zone, a lower collection zone, and an intermediate interface zone. An
additional wash water stream is usually added from the top of the column. Phosphate
flotation deals with considerably larger particles of size 0.1-1 mm. As a result, instead of
wash water from the top, elutriation water from the bottom is added. Furthermore,
columns are typically operated with negligible froth and interface zones. This
considerably simplifies the modeling effort, as the only the collection zone needs to be
accounted for.
Particle transport in the collection zone is usually modeled as axial convection
coupled with axial dispersion. The Peclet number (Pe), or its inverse, the dispersion
number, governs the degree of mixing. Most models only consider the slurry phase
(Finch and Dobby, 1990; Luttrell and Yoon, 1993), in which case particle collection is
viewed as a first order net attachment rate process. A model that considers both slurry
and air phase was developed by Sastry and Loftus (1988). In this case particle


Experimental Recovery (%)
Figure 2 9 Performance of the overall hybrid model: Predicted versus experimental recovery
(%) for coarse feed size distribution


102
A[NF][i]=0.0;
//**************Defmition ofRow=NF+l toN-1 *************
for(i=NF+1 ;i {
A[i][i-l]=-(al*DELT*A10);
A[i][i]=1.0-(al*DELT*All);
A[i][i+l]=-(al*DELT*A12);
for(j=l;j A[i][j]-0.0;
for(i=i+2;j<=N;j++)
A[i][j]=0.0;
}
//************Definition 0f Row=N ***********************
A[N][N-1 ]=-(a 1 *DELT*A13);
A[N] [N]=1 0-(a 1 *DELT*A 14);
for(i=l;i A[N][i]=0.0;
//**********Row Definition ends *****************************
//************* Defifnition of column vector *******************
V[l]=(1.0+(1.0-al)*DELT*Al)*C[l]+(1.0-al)*DELT*A2*C[2];
//printf("\n V[ 1 ]=%lf \n",V[ 1 ]);
//getch();
for(i=2;i V[i]=(1.0+(1.0-al)*DELT*A4)*C[i]+(1.0-al)*DELT*A3*C[i-l]+(1.0-
al)*DELT*A5*C[i+l];
//printfT'Nn V[%d]=%lf \n",i, V[i]);
//getch();}
V[NF]=(DELT*A6)+(1.0+(1.0-al)*DELT*A8)*C[NF]+(1.0-al)*DELT*A7*C[NF-
1 ]+(l ,0-al)*DELT*A9*C[NF+l ];
//printT\n V[NF]=%lf \n",V[NF]);
//getch();
for(i=NF+l;i V[i]=(1.0+(1.0-al)*DELT*Al l')*C[i]+(1.0-al)*DELT*A10*C[i-l]+(l .0-
al)*DELT*A12*C[i+l];
//printf("\n V[%d]=%lf\n",i,V[i]);
//getch();}
V[NH1.0+(1.0-al)*DELT*A14)*C[N]+(1.0-al)*DELT*A13*C[N-l];
//printf("\n V[N]=%lf \nM,V[N]);
//getch();
//*********^*****£)efjn|tjon 0pcojurnn vector ends **********
/y**************** ORDERING ******************************


CHAPTER 2
ONE-LEVEL HYBRID MODEL
Flotation is a process commonly employed for the selective separation of phosphate
from unwanted mineral. Column flotation is slowly gaining popularity in the mineral
processing industry, including the phosphate industry, due to its ability to improve
selectivity, lower operating cost, lower capital cost, and superior control. In this work, a
hybrid model is developed that combines a physicochemical model with artificial neural
networks. This model for the first time incorporates the effect of collector concentration,
extender concentration, and pH on the flotation performance. The physicochemical
model is based on axial dispersion with first order collection rates. Three basic
parameters are required in this model: flotation rate constant for phosphate, flotation rate
constant for gangue, and air holdup. Artificial neural networks are used to predict these
parameters. The model also takes into account the particle size distribution and predicts
grade and recovery for each particle size range. The model is validated against
laboratory column data.
2.1 Introduction
Even though the concept of column flotation was developed (Wheeler, 1988) and
patented (Boutin and Wheeler, 1967) in the early 1960s, its acceptance for the processing
and beneficiation of phosphate ores is relatively recent. The majority of the phosphate
plants employ mechanical cells. However, column flotation has simpler operation and
6


ACKNOWLEDGMENTS
I would like to take this opportunity to thank my advisor Dr. Spyros A Svoronos
for his continuing guidance, encouragement and support throughout the course of my
Ph D He not only guided me to learn new techniques, he was also helpful in showing me
the nght course in some of the problems in my personal life.
I wish to thank Dr. Hassan El-Shall for his valuable inputs in the chemistry aspect
of this project. I would also like to thank my other committee members, Dr. Richard
Dickinson, Dr. Oscar Crisalle, and Dr Ben Koopman, for kindly reviewing my
dissertation and serving on my committee.
The friendship and assistance of my colleagues, Pi-Hsin Liu, Robert Bozic,
Rajesh Sharma, Dr. Cheng, Dr Nagui, Rachel Worthen, and Lav Agarwal, will always be
valued.
My respect for my parents, brother, and sister for having stood by me and for
giving me moral support always kept me motivated to complete this work.
iii


93
difr=-diff;
USLi=USL;
}
while(diff>=0.0001);
USL=(l-Eg)*USL;
printf("\n USL=%lf',USL);
y/***************** USL calculation ends ***************
Frank-BPL/0.733;
CF-(Frank/100.0)*(CS/100.0)*ROS*62 418I8/((CS/100.0)+(1,0-(CS/100.0))*ROS);
CFG=( 1 0-(Frank/100.0))*(CS/100.0)*ROS*62.41818/((CS/l 00.0)+( 1.0-
(CS/100.0))*ROS);
K=Ku,KG=KGu;
mainl(CF,K,B);
C-B[3]+B[4];
RO=(((QF*CF)-((QTl+Area*USL)*C))/(QF*CF))* 100.0;
mainl(CFG,KG,BG);
CG=BG[3]+BG[4],
ROG=(((QF*CFG)-((QTl+Area*USL)*CG))/(QF*CFG))* 100 0;
Grade=((QF*CF-(QTl+Area*USL)*C)/((QF*CF-(QTl+Area*USL)*C)+(QF*CFG-
(QTl+Area*USL)*CG)))*100.0,
Grade=Grade *0.733,
printf("\n C=%lf',C);
printTVn CG=%lf',CG);
printf("\n CF=%lf',CF);
printf("\n CFG=%lf',CFG);
printf("\n Overall Recovery=%. llf %",RO);
printfT"\n ROG=%. llf %",ROG);
printt("\n GRADE=%. llf %",Grade),
}
//include
//include
#include
//include
//define GQT 2.2551
//define GQF 0.9318
//Tailings Flow rate (gallons/min)//
//Feed Flow rate (gallons/min)//


72
networks involves a relatively low number of inputs in the artificial neural networks, does
not require new three-phase data if a frother or sparger is changed, and gives very good
predictions of both grade and recovery
3.7 Conclusions
A hybrid neural network modeling approach was presented and used to model a
flotation column for phosphate/gangue separation. This hybrid model is comprised of
two parts, a first-principles model and two levels of neural networks that serve as
parameter predictors of difficult-to-model process parameters. Experimental data from a
laboratory column were used to train and validate the neural networks, and it is shown
that the hybrid model captures the dependence of column performance on particle size,
frother concentration, and superficial air velocity.


4
not very weil studied. Unlike other minerals, phosphate flotation deals with a
considerably larger size of particles (0.1 -1mm) and therefore the operation of phosphate
flotation in a column is different from that of other minerals. High recovery and grade
and low operating cost depend largely on the optimal selection of operating variables
such as the air flow rate, the frother type and concentration, and the elutriation water rate
The search of the optimal conditions can considerably benefit by the availability of a
model that can predict the effects of different operating conditions on column behavior.
Finch and Dobby (1990) and Lutrell and Yoon (1993) developed a one-phase axial
dispersion model in which particle collection is viewed as a first order net attachment rate
process. Sastry and Loftus (1988) considered both the slurry and air phases and they
used two separate first order rate constants for attachment and detachment of the
particles. However, these models cannot predict the effects of certain operating
conditions such as particle size, frother concentration, collector and extender
concentration, and pH on the flotation performance.
In this work, a mathematical model is developed that for the first time predicts the
effects of particle size, frother concentration, collector and extender concentration, and
pH on the flotation behavior. This is a hybrid model that combines a first-principles
model with artificial neural networks (ANNs). The first-principles model is derived by
making a material balance on solid particles in the slurry phase. First order reaction rate
constants are assumed for the attachment of the solid particles to the air bubbles. Single
output feedforward back propagation neural netw orks are used to correlate the model
parameters with the operating variables.


where
11
Up
Ut
D
Qp
Q.
Qe
Ac
kPW
= Phosphate concentration ofjth mesh size particles for
feed point
= Phosphate concentration of jlh mesh size particles for
feed point
= Superficial liquid velocity above the feed point
= Qp/ac
= Superficial liquid velocity below the feed point
= (Q, -Qe)/Ac
= Dispersion coefficient
= Product volumetric flow rate
= Tailings volumetric flow rate
= Elutriation volumetric flow rate
= Cross-sectional area of the column
= Slip velocity ofjlh mesh size particles
= Air holdup
= Flotation rate constant for phosphate for mesh size particles
the section above the
the section below the
The following assumptions are made in deriving the above equations:
1) The concentration of solid particles in the slurry phase is a function of height, z
only, and variations of the concentration in radial and angular directions can be
neglected.
2) The air holdup is constant throughout the column.
3) All the air bubbles in the system are of a single size.
4) Rate of detachment is either negligible or is a function of conditions in the slurry
phase. This assumption allows to treat the net attachment rate with just one
floatation rate constant.


Model diameter (mm)
Figure 3.5: Performance of NNIII: Model bubble diameter versus bubble diameter inferred
from experimental data when OB-535 was the frother


89
the conditions for the next run This should be repeated with the algorithm of Figure 4.2
until convergence is obtained


Flotation rate constant for phosphate (kp)
Figure 2.2: Recovery of phosphate (%) as a function of flotation rtae constant for
phosphate (kp)


83
Table 4.1: Operating conditions for the factorial design
Frother
Concentration
Superficial air
velocity
Elutriation water
Flow rate
1
-1
-1
-1
2
-1
+ 1
-1
3
+ 1
-1
-1
4
+ 1
+ 1
+ 1
5
-1
0
+ 1
6
+ 1
0
-1
7
0
-1
+ 1
8
0
+ 1
+ 1
9
0
0
0
10
0
+0.5
0
11
+0.5
0
0
12
0
-0.5
0
13
-0.5
0
0


57
and outputs were scaled from 0 to 1. The hidden and output layer nodes employed logistic
activation functions (Hertz et a/.. 1992).
For each of the four ffothers investigated, 28 two-phase experiments were conducted
(full factorial design with 7 frother concentrations and 4 superficial air velocities). These
were used to train (19 data points) and to validate (9 data points) the top level neural
networks (NNIII and NNIV), a different pair for each frother. Three-phase runs yielded
28 experimental grades and recoveries, which were used to train (19 data points) and to
validate (9 data points) NNI and NNII. To set the number of nodes in the hidden layer of
each network, the number was increased until the sum of the absolute errors of the
training and validation outputs started increasing In this manner an appropriate number
of hidden nodes was determined to be three for all the neural networks.
The training process started by initializing all weights randomly to small non-zero
values. The random numbers were generated in the range -3.4 to +3.4 with a standard
deviation of 1.0 following the procedure recommended by Masters (1993). The optimal
weights were determined by combining simulated annealing (Kirkpatrick et al. 1983) with
the Polak-Ribiere conjugate gradient algorithm (Polak, 1971). Simulated annealing
randomly perturbed the independent variables (the weights) and kept track of the best
(lowest error) function value for each randomized set of variables. This was repeated
several times, each time decreasing the variance of the perturbations with the previous
optimum as the mean. Then the conjugate gradient algorithm was used to minimize the
mean-squared output error. When the minimum was found, simulated annealing was
used to attempt to break out of what may be a local minimum. This alternation was


CHAPTER 4
OPTIMIZATION PERFORMANCE MEASURES AND FUTURE WORK
High recovery and grade and low operating cost depend largely on the optimal
selection of operating variables. The search of the optimal conditions can considerably
benefit by the availability of a model that can relate the operating conditions to the
column performance. The hybrid model developed for the flotation column provides a
mathematical relationship between the operating variables and column performance.
This hybrid model can be combined with an optimization algorithm to determine the
optimal operating conditions for the flotation column.
We propose an algorithm that leads to the sequential optimization of a flotation
column. This algorithm guides successive changes in the manipulated variables after
each experiment to achieve optimal column operating conditions. Selectivity, which
combines recovery and grade, can be used as the performance measure of the column.
The hybrid model builds a relationship between the process manipulated variables and
the performance measure. The optimization algorithm dictates the changes in the
manipulated variables between successive runs. At each run manipulated variables are
set at their predicted optimal values. After the run is completed, the collected samples
should be collected and analyzed for recovery and grade. Then the new input-output data
are added to the neural network of the hybrid model and the network should be retrained.
73


96
QP=QF-QT1+QE;
QT=QT1-QE;
Area=0.7853981*Dia*Dia;
UP=QP/Area;
UT=QT/Area;
UF=QF/Area;
D=12.4*Dia*pow((0.3175*Qg/Area),0.3);
//************** Calculation of slip velocity **************
DP 1=DP/1000.0;
PHIS=(C S/100)/((C S/100)+(( 1 -(C S/100))*ROS));
USLi=0.0;
do
{
REP=5.12*DP1 *USLi*ROS*(l .0-PH1S);
USL= 108.233 *DP 1 *DPl*(ROS-l)*pow((l-PHIS),2.7)/( 1+0.15 *(pow(REP,0.687)));
diff=USL-USLi;
if(diff<0.0)
difF=-diff;
USLi=USL;
}
while(diff>=0.0001);
USL=(l-Eg)*USL;
//************* USL calculation ends ***************
a=(UP-USL)/D;
d=(UT+USL)/D;
b=k*(l-Eg)/D;
if^((a*a+4*b)<0.0)||((d*d+4*b)<0.0))
b=0.0;
alpha=(a/2)+(sqrt(a*a+4*b))/2;
beta=(a/2)-(sqrt(a*a+4*b))/2;
gamma=(-d/2)+(sqrt(d*d+4*b))/2;
delta=(-d/2)-(sqrt(d*d+4*b))/2;
//printf^"\n alpha=%lf",alpha);
//printf("\n beta=%lf\beta);
//printf^"\n gamma=%lf',gamma);
//prir.tf("\n delta=%lf',delta);
//printfT\n beta*L=%lf',(beta)*(L));
//printf("\n alpha*L=%lf',(alpha)*(L));
//printf(\n gamma*Lf=%lf',(gamma)*(Li));
//printf("\n delta*Lf=%lf',(delta)*(Lf));


25
20
15
10
5
0
igure 2 8:
Performance of NNIII: Model versus experimental air holdup for frother CP-100
to


Model Bubble Diameter (mm)
Figure 3.3: Performance of NNIII: Model bubble diameter versus bubble diameter inferred from
experimental data when CP-100 was the frother


Superficial
air velocity
Frother
concentration


30
collector and extender concentration, and pH. Similarly, Figure 2.7 compares flotation rate
constant for gangue (kg) determined from one-dimensional searches with those predicted by
NNII As shown, NNII successfully predicts the flotation rate constant for gangue. Figure
2.8 presents the air holdup (Sg) predicted using NNIII against those measured experimentally.
A satisfactory match is seen.
The hybrid model integrates NNI, NNII, and NNIII as shown in Figure 2.4.
Predictions of the hybrid model are shown in Figure 2.9-2.14. Figures 2.9 and 2.10 compare
the experimental recovery (%) and grade (%BPL) with those predicted by the hybrid model,
respectively, for the coarse feed size distribution (14X 35 Tyler mesh). As shown in these
figures, the hybrid model successfully predicts both recovery and grade. Figures 2.11 and
2.12 compare the experimental recovery (%) and grade (%BPL) with those predicted by the
hybrid model, respectively, for the fine feed size distribution. As seen from these figures, the
hybrid model fails to successfully predict both recovery and grade. This is attributed to the
fact that fine feed has a very wide size distribution (35X150 Tyler mesh size) and only the
overall recovery and grade were measured experimentally. It is therefore necessary to utilize
narrow ranges of feed size and to analyze for recovery and grade according to each size range
instead of just one recovery and grade for the entire particle size distribution. This was
implemented for the unsized feed size which has even a wider size distribution (14X150
Tyler mesh). Figures 2.13 and 2.14 compare the experimental recovery (%) and grade
(%BPL) predicted by the hybrid model, respectively, for the unsized feed after it has been
sized and grade and recovery was determined for each size. As can be seen from these
figures, the hybrid model successfully predicts both recovery and grade.


103
for(i=l;i<=N;i++)
{
0[i]=i;
S[i]=abs( A[i] [ 1 ]);
for(j=2,j<=N;j++)
{
if(abs(A[i][j])>S[i])
S[i]=abs(A[i][j]);
}
}
//
***********
*** Ordering ends
******************************
II
***************
Gauss Elimination
*************************
for(k=l ;k {
//
** **
Pivoting
*********
//
Pivot=k;
Big=abs(A[0[k]][k]/S[0[k]]);
for( i i=k+1; i i <=N; i i++)
{
Dummy=abs(A[0[ii]][k]/S[0[ii]]);
ifDummy>Big)
{ Big=Dummy;
Pivot=ii;
}
}
IDummy=0[Pivot],
0[Pivot]=0[k];
0[k]=IDummy;
II*** End Pivoting*******//
for( i=k+1; i <=N ;i++)
{
factor=A[0[i]][k]/A[0[k]][k];
for(]=k+l ;j<=N;j++)
{
A[0[i]][j]=A[0[i]][j]-(factor*A[0[k]]lj]);
}
V[0[i]]=V[0[i]]-(factor*V[0[k]]);
}
}
//***************Qauss£iimination ends *******************


Designed experiments were performed in a lab scale column to generate data for the
initial training of the neural networks.


77
Unreacted
Dihydrate
= 1 ton rock x
= 1 ton rock x
v 73.33;
(Bl/100)
2.184
0.06
x 1.49 x (172/56) x (1-0.06)
Total amount of phosphogypsum = Acid insol + Unreacted + Dihydrate
Soluble P2O5 losses = $ 300.0 x
(% solubleP;05 losses)/!00
2.134
per ton
= $1.37 per ton
(Total amount of phosphogypsum)
Acidulation Penalty = $62.0x- 1.37
B,
Sales value = Price of 66 %BPL rock (Bl/66)
1.5
Adjusted sales value = Sales value Transportation penalty Acidulation penalty
The adjusted value of the phosphate rock as a function of %BPL is shown
in Figure 4.1. Let
Feed solid flow rate = F, ton per year
Product solid flow rate = P, ton per year
Feed grade = Gf, %
Concentrate grade = G, %
Product recovery = R, %
Adjusted sales value of feed = Cf, S per ton
Adjusted sales value of product = Cp, $ per ton
Reagent-i price = Cn, $/lb
Reagent-i usage = U, lb/ton feed
The feed flow rate and the product flow rate can be related as:
f GA
( K~]
100>!
P = F

0
0
O
O
1 G J
(4.4)


110
Dytiamics and Control of Chemical Reactors, Distillation Columns, and Batch Processes,
DYCORD+, (1992).
Thompson, ML., and M. A. Kramer, Modeling Chemical Processes Using Prior
Knowledge and Neural Networks, AIChEJ., 40(8): 1328-1340 (1994).
Villeneuve, J., M.-V Durance, C. Guillaneau, A.N. Santana, R.V.G. da Silva, and MAS.
Martin, Advanced Use of Column Flotation Models for Process Optimization, in
COLUMN'96, Eds. Gomez, C O. and J A. Finch, The Metallurgical Society of the
Canadian Institute of Mining, Metallurgy and Petroleum, 51-62 (1996).
Wheeler D.A., Historical View of Column Flotation Development, in Column
Flotation88, Ed. K.V.S. Sastry, SME-AIME, Littleton, Colorado, 3-4 (1988).
Xu, M., and J A. Finch, The Axial Dispersion Model in Flotation Column Studies,
Mineral Engineering, 4: 553-562 (1991).
Yeager, D., C.L. Karr, and D A. Stanley, Column Flotation Model Tuning Using a
Genetic Algorithm, SME Annual Meeting, Preprint 95-204 (1995).
Yianatos, J.Q., J.A. Finch, G.S. Dobby, and M. Xu, Bubble Size Estimation in a Bubble
Swarm, J. Colloid Interface Sci., 126(l):37-44 (1988).
Yoon, R.H., G.H. Luttrell, and G.T. Adel, Advances in Fine Particle Flotation, in
Challenges in Mineral Processing, Eds. K.V.S. Sastry and M.C. Fuerstenau, SME-
AIME, Littleton, Colorado, 487-506 (1989).
Yoon, R.H., G.H. Luttrell, G.T. Adel, and M.J. Mankosa, Recent Advances in Fine Coal
Flotation, in Advances in Coal and Mineral Processing Using Flotation, Eds. S.
Chandler and R.R. Klimpel, SME-AIME, Littleton, Colorado, 211-218 (1988).


16
PJ=-
Qr
4-aJ ~PJ [>exp(PJL)
ACD
Qp
ACD
(2.21)
- a^ +aj ^exp(aJL)
qJ =
^l- + dJ-5J
ACD
i_^* 4. yJ
[ACD
(2.22)
j_ qJexp(yjLf) + exp(5JLf)
m :
pJ exp(aJLf ) + exp(pJLr)
(2.23)
The algorithm for solving the first-principles model is given in Appendix A.
2.2.2 Calculation of Recovery and Grade
Recovery (%) is defined as the ratio of the weight of the phosphate in the concentrate
stream to the weight of the phosphate in the feed stream. The recovery of the phosphate
particles of the jl mesh size can be expressed in terms of the feed and tailings flow rates
and concentration as
rp=
QfCj [q, +Ac(l-eg)U'|
C!
P2
z =0
QfCf
*100
(2.24)
Grade, a measure of the quality of the product, is defined as the ratio of the weight of the
phosphate to the total weight recovered in the concentrate stream. Grade is reported as %
Bone Phosphate of Lime (% BPL) which is the equivalent grams of tricalcium phosphate


3.14 Performance of the overall hybrid model: Predicted versus
experimental grade (%BPL) for the four frothers 71
4.1 Value of phosphate rock as a function of %BPL 78
4.2 The run-to-run optimization algorithm 80
4 3 Neural network versus experimental flotation rate constant for
phosphate (kp) 84
4.4 Neural network versus experimental flotation rate constant for
gangue (kg) 85
4.5 Model versus experimental air holdup for frother F-507 87
vi ii


47
Laver k+1 to n-1
dC
p,
dt
U.
-+U
Ci -Cl
Pi-1
Az
C]D
-D ^
-2 Cl +Ci
Pi P'-i
Az
~kD(dJD)CJD
(3-4)
Laver n (bottom)
dC
Pn
dt
U
v1-6*
' + U
' Q
c _
Pn- (l-eg)Ac
y
+ U C
sl Pn
Az
U ^ f Q
'+u\ a -i ^
*' Pn '
1 8
J
0-Bg)At
Az
ji
J
-+D
CJ CJ
~ kp(dJp)Cin
Az:
if u s--L-
1-e,
Ui !CP.
CJ -CJ
+D- ; => -kp(di)c;n
/Az
if U <-
u.
1-e,
(3.5)
where
Ac = Cross-sectional area of the column
C\ = Phosphate feed concentration of j mesh size particles
C1 = Phosphate concentration of jlh mesh size particles in the ith layer
Qf = Feed volumetric flow rate
Q, = Tailings volumetric flow rate
Qe = Elutriation volumetric flow rate
Qp = Product volumetric flow rate
Up = Superficial liquid velocity above the feed point
= Qp/ac
U, = Superficial liquid velocity below the feed point
= (Q,-Qe)/Ae
Uj, = Slip velocity of jth mesh size particles
eg = Air holdup
kp(dp) = Flotation rate constant for phosphate for j'h mesh size particles


54
kg (NNII) and receives as an input the inferred bubble size. This is the output of one of
the ANNs of the second (top) level, NNII1. The second level also includes NNIV, which
predicts air holdup. The advantage of this structure is that NNI and NNII are
independent of the type of brother and sparger used, and therefore would not need
retraining if these change.
As bubble size is not measured in industry, we infer it from the two-phase (air/water)
air holdup, Jg, and Ut using the well-known Drift-flux analysis (Yianatos et a/., 1988).
The output required to train NNIV is the (two-phase) air holdup. Air holdup is relatively
easy to obtain, so after a change of frother or sparger the hybrid model of Figure 3.2 can
become functional in a short interval of time.
3.5 Materials and Methods
3.5.1 Experimental Setup and Procedures
Two types of experiments were conducted: tv/o-phase (air/water) experiments to train
neural networks NNIII and NNIV, and three-phase experiments to train NNI and NNII
and to test the performance of the hybrid model.
The experimental setup for the three-phase experiments is shown in Figure 2.5. It
included an agitated tank (conditioner) for reagentizing the feed, a screw feeder for
controlling the rate of reagentized feed, and a flotation column. The agitated tank was 45
cm in diameter and 75 cm high and was equipped with an impeller with two axial blades
(each 28 cm diameter). The impeller had about 3.8 cm clearance from the bottom of the
tank and its rotation speed was fixed at 465 rpm. The feeder with a 2.5 cm diameter


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INGEST IEID E15GNWZD5_1I8YEX INGEST_TIME 2013-03-12T14:10:21Z PACKAGE AA00013557_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES


22
2 3 The Hybrid Model
The overall structure of the hybrid model is shown in the Figure 2.4. The hybrid
model utilizes backpropagation neural networks (Rumelhart and McClelland, 1986) to
predict the values of parameters flotation rate constants, kp and kg, and air holdup, eg.
The factors that affect kp and kg are particle diameter, superficial air velocity, frother
concentration, collector concentration, extender concentration, and pH. The air holdup,
sg, is mainly affected by superficial air velocity and frother concentration.
The hybrid model of Figure 2.4 integrates the first-principles model with three
artificial neural networks. Neural network, NNI, correlates the flotation rate constant for
phosphate, kp, with phosphate particle size, superficial air velocity, frother concentration,
collector concentration, extender concentration, and pH. Similarly, neural network, NNI I
correlates the flotation rate constant for gangue, kg, with gangue particle size, superficial
air velocity, frother concentration, collector concentration, extender concentration, and
pH. Neural network NNIII correlates the air holdup, eg, with superficial air velocity and
frother concentration.
In this structure, all three neural networks are specific to the type of frother or
sparger used. This necessitates generation of new data and retraining of the neural
networks each time the frother or the sparger are changed.


101
else
{
UP=QP/Area;UP=(UP-USL)/( 1 -Eg);UP=-UP;
UT=QT/Area;UT=(UT+USL)/( 1 -Eg);UT 1 =QT 1 / Area;UT 1 =(UT 1 +USL)/( 1 -Eg);
UF=QF/Area;UF=UF/( 1 -Eg),
A3 =(UP+(D/DELZ))/DELZ;
A4=(-UP-(2.0*D/DELZ)-(K*DELZ))/(DELZ);
A5=D/(DELZ DELZ);
A1=A4+A5;
A2=A5;
A6=(UF*CF)/DELZ;
A7=A3;
A11 =(-UT-(2.0* D/DELZ)-(K DELZ))/(DELZ);
A8=A11;
A9=A5;
A10-(UT/DELZ)+A5;
A12=A5;
A13=A10;
//A14=A5+A11;
A14=(-UT 1 -(D/DELZ)-(K*DELZ))/(DELZ);
}
//****** *Defmition 0fRow= \***************************
A[l][l]=1.0-(al*DELT*Al);
A[l][2]=-(al*DELT*A2);
for(i=3;i<=N;i++)
A[l][i]=0.0;
//***********Definilion of Row=2 to NF-1 *****************
for(i=2,i {
A[i] [i-1 ]=-(a 1 DELT A3);
A[i][i]=1.0-(al*DELT*A4);
A[i][i+l]=-(al*DELT*A5);
for(j=l;j A[i][j]-0.0;
for(j=i+2;j<=N;j++)
A[i][j]=0.0;
}
//*******:***Defmiiion of Row=NF **************************
A[NF] [NF-1 ]=-(a 1 DELT A7);
A[NF][NF]=1.0-(al*DELT*A8);
A[NF][NF+l]=-(al*DELT*A9);
for(i=l ;i A[NF][i]=0.0;
for(i=NF+2,i<=N;i++)


48
The slip velocity is calculated using the expression ofVilleneuve et al. (1996):
ttj _§d¡>2(Ps-PiX1- si 0.687
18^(l + 0.15Rp )
where the particle Reynolds number is defined as
RJ _d;u^,ps(i-(|)s)
(3.6)
(3.7)
where
g = Acceleration due to gravity (m/s )
Pi = Water viscosity (kg/mr,)
pi = Water density (kg/m )
Pi = Solid density (kg/m3)
<})s = Volume fraction of solids in siurry
dp = Particle diameter (m)
As the right hand side of Equation 3.7 is a function of U3,, the slip velocity is obtained by
solving Equations 3.6 and 3.7 iteratively as described in Chapter 2.
The axiai dispersion coefficient is calculated by a modification of the Finch and
Dobby (1990) expression:
f T V3
D = 0.063 (l-eg)dc
,1.6,
(3.8)
where
do -
Jg =
column diameter (m)
superficial air velocity (cm/s)


105
USL=108.233*DPl*DPl*(ROS-l)*pow((l-PHIS),2.7)/(l+0.15*(pow(REP,0.687)));
diff=USL-USLi;
if(diff<0.0)
diff=-diT;
USLi=USL;
}
while(diff>=0.0001);
USL=(l-Eg)*USL;
printf("\n USL=%lf',USL);
//***************** USL caicuiation en(is ***************
//USL=0.0;
//printf("\nEnter your initial values for Phosphate concentrationAn");
for(i=l;i<=N;i-H-)
{
C[i]=0.0;
//printf("\n C[%d]=",i);
//scanf("%lf',&.C[i]),
}
//printfi'AnEnter your initial values for Gaunge concentrationAn");
for(i=l;i<=N;i-H-)
{
CG[i]=0.0;
//printfTAn CG[%d]=",i);
//scanf("%lf',&CG[i]);
}
Frank=BPL/0.733;
CF=(Frank/100 0)*(CS/100.0)*R.OS*62.41818/((CS/100.0)+(1.0-(CS/100.0))*ROS);
CFG=( 1.0-(Frank/l 00.0))*(CS/100.0)*ROS*62.41818/((CS/l 00.0)+( 1.0-
(CS/100.0))*ROS);
printfTVn CF=%lf ,CF);
printf^'An CFG=%lf\CFG);
//getch();
Du=T2.4*Dia*pow((0.3175*Qg/Area),0.3); // ft2/min//
K=Ku;KG=KGu;D=Du/( 1 -Eg);
RT=L*Area*( 1 -Eg)/(QF+(L* Area*( 1 -Eg)*abs(KG)));
Z=(50.0*RT/DELT)-1;
TIME=DELT;
III* **** $$$$$%%%%%@@@@@@ HERE IS THE FLAG BETWEEN STEADY-
STATE & DYNAMIC
FLAG=1; //I for dynamic, other values for steady-state approximation
ifFLAG 1)
{
for(l=l;K=Z;l++)
{


100
QT1=(0.1336541 *GQT);
QP=QF-QT1+QE;
QT=QT1-QE;
DELZ=(L/N);
NF=((FNF/DELZ)+1 )* 1;
Area=0.7853981*Dia*Dia;
yy* ************* Calculation of slip velocity **************
DP 1=DP/1000.0;
PHIS=(CS/100)/((CS/100)+((1-(CS/100))*ROS));
USLi=0.0;
do
{
REP=5.12*DPl*USLi*ROS*( 1.0-PHIS);
USL=108.233 *DP 1 *DP 1 *(ROS-1 )*pow(( 1 -PHIS),2.7)/( 1+0.15 *(pow(REP,0.687)));
diff=USL-USLi;
if(diff<0.0)
diff=-diff;
USLi=USL;
}
while(diff>=0.0001);
USL=(l-Eg)*USL;
/y***************** USL caicuiation ends ***************
//USL=0.0;
if(USL<=(QP/Area))
{
UP=QP/Area;UP=(UP-USL)/( 1 -Eg);
UT=QT/Area;UT=(UT+USL)/( 1 -Eg);UT 1 =QT l/Area;UT 1 =(UT 1 +USL)/( 1 -Eg);
UF=QF/Area;UF=UF/( 1 -Eg);
A3 =D/(DELZ DELZ);
A4=(-UP-(2.0*D/DELZ)-(K*DELZ))/(DELZ);
A5=(UP+(D/DELZ))/DELZ;
A1=A3+A4;
A2=A5;
A6=(UF*CF)/DELZ;
A7=A3;
A10=(UT/DELZ)+A3;
A8=A3+A4-A10;
A9=A3;
A11=(-UT-(2.0*D/DELZ)-(K*DELZ))/(DELZ);
A12=A3;
A13=A10;
//A14=A3+A11;
A14=(-UT 1 -(D/DELZ)-(K*DELZ))/(DELZ);


18
The recovery of phosphate RJp is only a function of the flotation rate constant
for phosphate, kp, and air holdup, eg. Similarly, the recovery of gangue Rg is only a
function of flotation rate constant for gangue, kg, and air holdup, eg. Since air holdup is
measured, we can invert the model to determine the value of kp that results in the
measured recovery of phosphate RJp and the value of kg that yields the measured
recovery of gangue Ri. As shown in Figure 2.1, a one-dimensional search is performed
to determine the values of flotation rate constants when supplied with the recovery of
phosphate and gangue, respectively. This algorithm allows determination of the flotation
rate constants for each run, given the operating conditions and the performance of the
column in terms of grade and recovery. The algorithm requires two initial guesses of the
flotation rate constants which yield errors in the corresponding RJp of opposite sign, and
then the program uses the method of false position (Chapra and Canale, 1988) to
determine the correct set of flotation rate constants.
Recovery of phosphate increases monotonically with flotation rate constant for
phosphate, kp. This is verified by calculating recovery for different values of flotation
rate constant and recovery was plotted against flotation rate constant. From the graph
shown in Figure 2.2, it is concluded that there is only value of floatation rate constant for
a given recovery. Similarly, from Figure 2.3, it is concluded that recovery of gangue
increases monotonically with flotation rate constant for gangue, kg.


86
experimental data. Again, a very good match is seen Figure 4.5 compares the predicted
air holdup to the experimental values measured by a differential pressure cell. As shown
in this figure, neural network successfully predicts the air holdup.
Table 4 2 shows the results of the 13 designed experiments. As can be seen from
this Table, feed flow rate and the %solids content varied significantly. The screw feeder
operation was erratic and therefore we were unable to feed at the same rate in each run.
Feed flow rate was calculated based on the product flow rate and the tailings flow rate. A
specified volume of product and tailings were taken over a period of time (~20 s) and the
samples were dried and the weight was taken In this way, solids flow rate in product and
tailings stream were obtained. An overall material balance on the column then gives the
feed flow rate. Similarly, an overall material balance on the water phase gives the water
flow rate in the feed stream. Solids feed flow rate and the water flow rate in the feed then
can be used to obtain the % solids in the feed. Unfortunately, the inability to control feed
flow rate and % solids content means that a meaningful run-to-run optimization cannot be
conducted.
4 5 Future Work
First, the screw feeder needs to be repaired or replaced. After this has been
accomplished, the hybrid model obtained from the designed experiments (Figures 4.3-
4.5) should be used with the Nelder-Meade algorithm to determine the experimental
conditions of the first optimization run. The results of the run should be analyzed for
grade and recovery and these data should be added to the neural network training sets.
The networks should then be retrained and the updated hybrid model used to determine


APPENDIX A
CODE FOR TFIE FIRST PRINCIPLES MODEL FOR ONE LEVEL
^include
#include
^include
#include
GQT 2.5372
//Tailings Flow rate (gallons/min)//
GQF 0.6133
//Feed Flow rate (gallons/min)//
GQE 3.038
//Elutriation Flow rate (gallons/min)//
CS 66.0
//% Solid in the feed (lb S/lb T)//
BPL 24.9
//% BPL of the feed (!b P/lb S) //
ROS 2.6
//Specific Gravity of solids in the feed//
DP 122.5
//Particle size in microns //
Eg 0.05762
//air hold up //
Dia 0.5
//Diameter of the column (ft.)//
L 6.0
//Height of the column (ft.)//
Ku 2.217556
//Flotation rate const, for Phosphate (1/min.)//
KGu 0.999965
//Flotation rate const, for Gaunge (1/min.)//
Qg 0.1778
//Air flow rate(scfm)//
FNF 1.0
//Feed Location from the top (ft )//
LfL-FNF
void main I (double,double,double[]);
void main 1 (double CF, double k, double B[])
{
double QF,QE,QT1,QT,QP,Area,UP,UT,UF,D,DPI,PHIS,USLi,REP,USL,diiT;
double a,b,d,alpha,beta,gamma,delta,p,q,m;
QF=0.1336541 *GQF;
QE=0.1336541 *GQE;
QT1=(0.1336541 *GQT);
QP=QF-QT1+QE;
QT=QT1-QE;
Area=0.7853981 *Dia*Dia,
UP=QP/Area;
UT=QT/Area;
UF=QF/Area,
90


24
2.4 Materials and Methods
2.4.1 Experimental setup and Procedures
The experimental setup is shown in Figure 2.5. it includes an agitated tank
(conditioner) for reagentizing the feed and a screw feeder for controlling the rate of
reagentized feed to the flotation column. The agitated tank was 45 cm in diameter and 75
cm high. It was equipped with an impeller of two axial type blades (each 28 cm diameter)
The impeller rotation speed was fixed at 465 rpm. The impeller had about 3.8 cm
clearance from the bottom of the tank. The feeder with 2.5 cm diameter screw delivered
the conditioned phosphate materials to the column. The feed rate was controlled by
adjusting the screw rotation speed. Flotation tests were conducted using a 14.5 cm
diameter by 1.82 m high plexiglass flotation column. The feed inlet was located at 30 cm
from the column top. The discharge flow rate was controlled by a discharge valve and an
adjustable speed pump. Three flowmeters were used to monitor the flow rates for air,
frother solution, and elutriation water.
Three different feed sizes obtained from Cargill were used in the flotation
experiments: coarse feed with narrow distribution (14X35 Tyler mesh), fine feed with
wide size distribution (35X150 Tyler mesh), and unsized feed which is a mixture of the
above two (14X150 Tyler mesh). For each run, 50 kg of feed sample was added in the
pre-treatment tank and water was added to obtain 72% solids concentration by weight.
The feed material was then agitated for 10 seconds. 10 % soda ash solution was added to
the pulp to reach pH of about 9.4 and agitated for 10 seconds. Subsequently a mixture of


109
Kirkpatrick, S., Jr. C. D. Gelatt, and M. P. Vecchi, Optimization by Simulated
Annealing, Science, 220:671-680 (1983).
Kramer, M.A., and M. L. Thompson, Embedding Theoretical Models in Neural
Networks, Proc. Am. Control Con/\ Chicago, 1:475-479 (1992).
Liu, P.-H., T. Potter, S. A. Svoronos, and B. Koopman, Hybrid Model of Nitrogen
Dynamics in a Periodic Wastewater Treatment Process, AIChE Annual Meeting, Paper
No. 195an (1995).
Luttrell, G.H., and R. H. Yoon, A Flotation Column Simulator Based on Hydrodynamic
Principle, Inter. J. Miner. Process., 33:355-368 (1992).
Luttrell, G.H., and R. H. Yoon, Column Flotation-A Review, in Beneficiation of
Phosphate: Theory and Practice, Eds. H. El-Shail, B.M. Moudgil and R. Wiegel, SME,
Littleton, Colorado, 361-369 (1993).
Masters, T., Practical Neural Network Recipes in C++, Academic Press, New York
(1993).
Mavros, P., Mixing in Flotation Columns. Part 1: Axial Dispersion Modeling, Mineral
Engineering, 6: 465-478 (1993).
Perry, R.H., D. W. Green, and J. 0. Maloney, Perry's Chemical Engineers Handbook,
McGraw-Hill Book Company, New York, 6th ed. (1984).
Polak, E., Computational Methods in Optimization, Academic Press, New York (1971).
Psichogios, D C., and L. H. Ungar, Process Modeling using Structured Neural
Networks, Proc. Am. Control Con/., Chicago, 3:1917-1921 (1992a).
Psichogios, D C., and L. H. Ungar, A Hybrid Neural-Network First Principles Approach
to Process Modeling, AIChE J., 38:1499-1511 (1992b).
Reuter, M., J. Van Deventer, and P. Van Der Walt, A Generalized Neural-Net Rate
Equation, Chem. Eng. Sci., 48:1281 (1993).
Rumelhart, D., and J. McClelland, Parallel Distributed Processing, MIT Press,
Cambridge, MA (1986).
Sasry, K.V.S., and K. D. Loftus, Mathematical Molding and Computer Simulation of
Column Flotation, in Column Flotation'88, Ed. K.V.S. Sastry, SME-AIME, Littleton,
Colorado, 57-68 (1988).
Su, H.-T., P. A. Bhat, P. A. Minderman, and T. J. McA.voy, Integrating Neural
Networks with First Principles Models for Dynamic Modeling, IFAC Symp. on


Flotation rate constant for gangue (kg)
Figure 2.3: Recovery of gangue (%) as a function of flotation rtae constant for
gangue (k^)


44
The next section presents the first-principles model. The subsequent section deals
with the calculation of model parameters from measured outputs. This is followed by a
discussion of the artificial neural networks and their integration with the first-principles
model to develop a hybrid model. The fourth section describes the experimental setup,
materials used, experimental procedure, and the methodology used to train the neural
networks. The final section presents results and compares the model predictions of grade
and recovery to experimentally measured grade and recovery.
3.2 First-Principles Model
The FPM is obtained from material balances on both phosphate and gangue. It
neglects radial dispersion and changes in the air holdup. Following Luttrell and Yoon
(1993) the particle to bubble attachment and detachment rates are combined in one net
attachment rate, and this rate is assumed to be first order with respect to particle
concentration in the slurry.
The model subdivides the column into n layers as shown in Figure 3.1. Feed
containing both the desired (phosphate) and undesired (gangue) particles enters in a
slurry in layer k. An additional inlet stream is the elutriation water that enters in the
bottom of the column (layer n). Most of that flow is due to water that enters with the air
sparger, as most of the popular spargers are two-phase and introduce a considerable
amount of water. There are two outlet streams: the tailings stream through the bottom of
the column (layer n) that contains mostly gangue, and the product (concentrate) stream
that leaves from the top of the column.


9
Black-box modeling strategies are mainly data driven and the resulting models often
do not have reliable extrapolation properties. Black-box strategies have been applied to
many chemical processes, especially since convenient black-box modeling tools like
neural networks have become available (Bhat and McAvoy, 1990, Psichogios and Ungar,
1992a). Gray-box or hybrid modeling strategies are potentially very' efficient if the
black-box and white-box components are combined in such a way that the resulting
models have good interpolation and extrapolation properties.
There are two types of gray-box modeling approaches in which a neural network is
combined with a black-box model: the parallel and the serial approach In the parallel
approach, the neural network is placed parallel with a white-box model In this case, the
neural network is trained on the error between the output of the white-box model and the
actual output Su et ol. (1992) demonstrated that the parallel approach resulted in better
interpolation properties than pure black-box models. Johansen and Foss (1992) also used
a parallel structure where the output of the hybrid model was a weighted sum of a first-
principles and a neural network model.
In the serial hybrid modeling strategy, the neural network is placed in series with the
first-principles model. Various researchers (Psichogios and Ungar, 1992a; Thompson
and Kramer, 1994) have shown the potential extrapolation properties of serial hybrid
models. Psichogios and Ungar (1992b) used this approach for parameters that are
functions of the state variables and manipulated inputs. Liu et al. (1995) developed a
serial hybrid model for a periodic wastewater treatment process by using ANNs for the


97
//printf("\n alpha*Lf=%lf',(apha)*(Lf));
//printf("\n beta*Lf=%lf',(beta)*(Lf));
//p=(((-UP/D)+a-beta)*(exp((beta)*(L))))/(((UP/D)-a+a!pha)*(exp((alpha)*(L))));
//q=((-UT/D)+d-delta)/((UT/D)-d+gamma);
if(USL<=UP)
p=((-beta)*exp((beta)*(L)))/(alpha*exp((alpha)*(L)));
else
p=((-(-beta+a))*exp((beta)*(L)))/((-alpha+a)*exp((alphaj*(L)));
q=-(delta-(QE/(Area*D)))/(gamma-(QE/(Area*D)));
m=(q*exp((gamma)*(Lf))+exp((delta)*(Lf)))/(p*exp((alpha)*(Lf))+exp((beta)*(Lf))),
B[4]=(UF*CF/D)/((m*p*(a-
alpha)*exp((alpha)*(Lf)))+((d+gamma)*q*exp((gamma)*(Lf)))"Hm*(a-
beta)*exp((beta)*(Lf)))~K(d+delta)*exp((delta)*(Lf))));
//B[3]=(UF*CF/D)/((a+alpha+d+gamma)*exp((gamma)*(Lf)));
B[2]=m*B[4];
B[l]=p*m*B[4];
B[3]=q*B[4];
//B[l]=(exp((gamma-alpha)*Lf))*B[3];
//B[2]=0.0;
//B[4]=0.0;
}
double model(double Ku)
{
double
QF,QE,QTl,QT,QP,Area,DPl,PHIS,USLi,REP,USL,diff,Frank,CF,CFG,K,KG,C,CG;
double RO.ROG,Grade,B[5],BG[5];
QF=0.1336541*GQF;
QE=0.1336541 *GQE;
QT1=(0.1336541*GQT);
QP=QF-QT1+QE;
QT=QT1-QE;
Area=0.7853981 *Dia*Dia;
//printfTVn UP=%lf,QP/Area);
y/************** Calculation of slip velocity **************
DP 1=DP/1000.0;
PHIS=(CS/100)/((CS/100)+((1-(CS/100))*ROS));
USLi=0.0;


12
The slip velocity is calculated using the
gd^-p.xi-dO27
U si ;0 087
18pi,(l -hO 15RJp )
expression of Villeneuve et al. (1996):
(2.3)
where the particle Reynolds number is defined as
dUj,Ps(l- Mi
(2.4)
where
g = Acceleration due to gravity (m/s )
Pi = Water viscosity (kg/ms)
pi = Water density (kg/m )
Pi = Solid density (kg/m3)
s = Volume fraction of solids in slurry
dJp = Particle diameter (m)
Since Rp is a function of LP,, an iterative procedure is used to calculate the slip
velocity. The procedure starts with an initial guess for LP, and corresponding value of
RJ,p is plugged in Equation 2.3 and new value of LP, is found. This new value is then
used in Equation 2.2 and this procedure is continued till convergence is achieved. The
axial dispersion coefficient is calculated by a modified expression of Finch and Dobby
(1990):
f
v
J
03
D = 0.063 (1-Eg)dc
(2.5)


104
y/************** Substitution *****************************
C[N]=V[0[N]]/A[0[N]][N];
for(i=N-l;i>=l;i--)
{
Sum=0.0;
for(j=i+l;j<=Nj++)
{
Sum=Sum+(A[0[i]][j]*C[j]);
}
C[i]=(V[0[i]]-Sum)/A[0[i]][i];
}
}
y/**************** Substitution ends *******************
//************* ************** ***************************** *
/ / ^ ^ ^ ^ |i L jl ^ ^ ^ ^ ~P ^ ^ >P ^ >p 1 p .p Lp ^ P tp .P P >p p .p P >P ~P t P P p [i P p ^ ^ ^ > p Lp >P p ^ ^ > p ^ p ^P ^ P P
//&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
&&&&&&&&&&&&
//#########################################################
//!!!!!!!!!!!!!!!! Main 1 Ends !!!!!!!!!!!!!!!!!!!!!!!!
void main(void)
{
double
Grade,CF,CFG,K,KG,D,RO,Area,Z,RT,QP,QT,QF,TIME,Pe,Tp,a,bl,b2,REQNP,REQN
G;
int l,i,FLAG;//NF;
double C[n],CG[n],USLi,USL,REP,difif,PHIS,Gradeqn,c,d 1 ,d2,Frank;//DELZ;
double Du,QE,QT1,R,DP1,Selectivity,Selecteqn,ROG,RG,//RC,
QF=0.1336541 *GQF;
QE=0.1336541 *GQE;
QT1=(0.1336541*GQT);
QP-QF-QT1+QE;
QT-QT1-QE;
//DELZ=(L/N);
//NF=((FNF/DELZ)+1 )* 1;
Area=0.7853981 *Dia*Dia;
printfT\n UP=%lf',QP/Area);
//************** Calculation of slip velocity **************
DP 1=DP/1000.0;
PHIS=(CS/100)/((CS/100)+((1-(CS/100))*ROS));
USLi=0.0;
do
{
REP-5.12*DPl*USLi*ROS*(l. 0-PHIS);


BIOGRAPHICAL SKETCH
Sanjay Gupta obtained a Bachelor of Engineering degree in chemical engineering
from the University of Roorkee, Roorkee, India in May 1993. He then worked as a
process engineer at Burmah-Shell Refinery Ltd., Bombay, India, for one year. He
continued his higher studies at Drexel University, Philadelphia, and obtained a Master of
Science degree in chemical engineering in March 1997. He joined the Department of
Chemical engineering at University of Florida to pursue his Ph.D. degree in August,
1996.


74
New optimal manipulated variable values are predicted which set the conditions for the
subsequent run. This procedure should be repeated until convergence is obtained.
41 Performance Measures
The performance of a flotation column is affected by both recovery (%) and grade
(%BPL). To guide optimization it is necessary to combine the two outputs (grade and
recovery) in a single performance measure. Several performance measures are possible,
and some are presented below.
4,1,1 Selectivity
One way to achieve this is to use selectivity as the performance measure.
Selectivity is defined as
where
R = Recovery of phosphate in the product stream.
Rb = Recovery of gangue in the product stream
Rt = Recovery(or Rejectability) of phosphate in the tailings stream.
Rtb = Recovery(or Rejectability) of gangue in the tailings strea
We developed the following expression that relates selectivity to the recovery and
the grade of the product stream


49
Equations analogous to 3.1-3.8 are valid for the gangue particles, but with a considerably
lesser effective flotation rate constant kg(d).
In the limit as Az -4 0 the above difference equations become
(X
i f
c\
U.
-U
eci
cz
d2C]
D
UL~
-MW
(3.9)
for the section above the feed and
5Cj
dt
u
vi_eg
d2C>
P-+D p
cz
cz2
~MW
(3 10)
for the section below the feed.
Recovery (%) is defined as the ratio of the weight of the phosphate in the concentrate
stream to the w eight of the phosphate in the feed The recovery for phosphate particles of
the j01 mesh size can be expressed in terms of the feed and tailings flow rates and
concentrations as
r;=
'QfC/-[Q,+At(l-e,)U]cip-
Q,c;
MOO
(3.11)
Grade, a measure of the quality of the product, is defined as the ratio of the weight of the
phosphate to the total weight recovered in the concentrate stream. Grade is usually
reported as % Bone Phosphate of Lime (%BPL) which is the equivalent grams of
tncalcium phosphate, Ca3(PC>4)2, in 100 g of sample. For the typical Florida rock,


8
The particie collection process in a column is considered to follow first order
kinetics relative to the solids particle concentration with a rate constant. Finch and
Dobby (1990) and Lutrell and Yoon (1993) used a one-phase axial dispersion model in
which particle collection is viewed as a first order net attachment rate process. Sastry and
Loftus (1988) considered both the slurry and air phases and they used two separate first
order rate constants for attachment and detachment of the particles. Luttrell and Yoon
(1993) relate the particle net attachment rate constant to some operating variables using a
probabilistic approach. However, their approach cannot be used to predict the effect of
certain operating conditions such as frother concentration, collector concentration,
extender concentration, and pH.
For the model to be predictive, the functional dependence of the net attachment rate
constant (kp or kg) on the key operating variables needs to be determined. The functional
relationship of model parameters on the operating conditions is difficult to determine via
physicochemical reasoning. In our approach, we use neural networks to determine these
functional relationships. Artificial neural networks are a powerful tool, inspired by how
the human brain works, that can learn from examples any unknown functional
relationship. Their ability to approximate any smooth nonlinear multivariable function
arbitrarily well (Hornik et al., 1989) and their simple construction have led to great
interest in using neural networks.
Existing modeling strategies can be divided into white-box, black-box, and gray-box
(hybrid) strategies, depending on the amount of prior knowledge that is used for
development of the model. White-box modeling strategies are mainly knowledge driven.


80
START
i
Initial runs
V
ICP analysis
4'
Performance measure
1
Train neural network
4
Determine position
of maximum
YES ^ NO
Is maximum in
interior of range?
Experimental run at Experimental run at
predicted maximum half-way point
1 ICP analysis *
I '
Performance measure
1
Train neural network
4
NO
Determine position
of maximum
4
Difference between two consecutive maximum
less than pre-decided limits 0
YES
>
STOP
Figure 4.2: The run-to-run optimization algorithm


25
20
15
10
5
0
5 10 15 20 25
Experimental Air Hold-up (%)
Figure 3.10: Performance of NNIV: Model versus experimental air holdup for frother F-579


CHAPTER 1
INTRODUCTION
Since the beginning of 1980s, the industrial application of flotation technology
has experienced a remarkable growth due to active theoretical and experimental research
and development. Flotation columns are slowly being accepted in the mineral processing
industry for the advantages they offer over conventional flotation equipment including
grade improvement, lower operating cost, and superior control. The ability of flotation
columns to produce concentrates of superior grade at similar recovery is derived from the
improved selectivity it offers.
Unlike conventional mechanical cells, flotation columns do not use mechanical
agitation to suspend particles. Another distinct feature of the flotation column principle
is countercurrent contact between feed particles and air bubbles. The lack of moving
pans and lower reagent consumption results in a lower operating costs. The lower capital
cost for the equipment is attributed to its high capacity leading to the use of less units for
the same production rate
The current flotation practice in Florida phosphate industry involves the use a
two stage process with mechanical cells, where the feed is subjected to rougher flotation
in which fatty acids and fuel oil are used as collectors to separate the phosphate from
most of the sand The rougher concentrate is then scrubbed by sulfuric acid to remove
the fatty acids and oil. The scrubbed material has to be washed with fresh water to
1


I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
5
fyrv'co
K
J>v
Spyros A. Svoronos, Chairman
Professor of Chemical Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Hassan El-Shall, Cochairman
Engineer of Materials Science and
Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Assistant Professor of Chemical Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
BenKoopman
Professor of Environmental Engineering
Sciences
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Oscar Crisalle
Associate Professor of Chemical Engineering


42
attachment and detachment are modeled separately with first order rates. Luttrell and
Yoon (1993) used a probabilistic approach to relate the particle net attachment rate
constant to some operating variables (e.g., air flow rate). However, their approach
involves empirical parameters and it cannot be used to predict the effect of certair
operating variables such as the frother and collector concentrations.
In this work, we use neural networks to determine the dependence of the phosphate
and gangue flotation rate constants on the operating variables. Artificial neural networks
have the ability to approximate any smooth nonlinear multivariable function arbitrarily
well (Hornik et al., 1989). This approach can be used to determine the dependence of the
performance of a flotation column (i.e., grade and recovery) on any operational variable
We demonstrate it in this work by developing a hybrid model that predicts the effect ol
frother concentration, air flow rate, feed rate and loading, elutriation flow rate, tailings
flow rate, and particle size distribution.
The idea of developing a hybrid model by combining a first-principles model (FPM)
with artificial neural networks (ANNs) is not new. Johansen and Foss (1992) and Su ei
al. (1992) proposed parallel structures where the output of the hybrid model is a weighted
sum of the first-principles and ANN models. Kramer et al. (1992) proposed a parallel
arrangement of a default model (which could be a first principles model) and a radial
basis function ANN. An alternative approach is to combine ANNs with a FPM in a serial
fashion, by using the ANNs to develop expressions for the FPM parameters or rate
expressions. Psichogios and Ungar (1992a, 1992b) proposed this scheme for parameters
that are functions of the state variables and manipulated inputs, and trained the neural


75
s =
-g(i-r)-r 2
Gf
(1 G)(l R)
(4.2)
where
G = Grade (%BPL) of phosphate in the product stream.
Gf = Grade (BPL) of phosphate in the feed.
4 1.2 Separation Efficiency
Separation efficiency is defined as follows:
E = R -Rb
(4.3)
In this case, the efficiency varies between -100 to 100.
4,1,3 Economic Performance Measure
The selectivity function or the separation efficiency does not include any
economic input such as cost of the reagents. Therefore an alternate performance measure
was developed which includes recovery, grade, and the reagent prices. A scheme for
penalizing lower grade rock has been developed. This scheme deducts differential costs,
relative to 66% BPL, for transportation and acidulation. The acidulation scheme assumes
soluble P2O5 losses increase in direct proportion to the amount of phosphogypsum. Thus,
the procedure requires an estimate of the quantity of phosphogypsum that is produced.


CHAPTER 3
TWO-LEVEL HYBRID MODEL
A new model for phosphate column flotation is presented which relates the effects of
operating variables such as froiher concentration and air flow rate on column
performance. This is a hybrid model that combines a first-principles model with artificial
neural networks. The first-principles model is obtained from material balances on both
phosphate particles and gangue (undesired material containing mostly silica). First order
rates of net attachment are assumed for both. .Artificial neural networks relate the
attachment rate constants to the operating variables. Experiments were conducted in a 6
diameter laboratory column to provide data for neural network training and model
validation. The model is shown to successfully predict the effects of frother
concentration, particle size, air flow rate, and bubble diameter on grade and recovery.
3.1 Introduction
Flotation is a process in which air bubbles are used to separate a hydrophobic from a
hydrophilic species. The majority of the hydrophobic material gets attached to the
bubbles and leaves with the froth from the top of a cell or column separator, while the
hydrophilic material leaves from the bottom. This process is commonly used in the
minerals industry, including the phosphate industry, in which case the phosphate
containing rock (frankolite or apatite) is to be separated from gangue (mostly silica).
Flotation is also used to remove oil from wastewater and to remove ink from paper pulp.
40


106
mainl(CF,K,D,C);
//printfTVn Following are the values of Phosphate C[i] at Time=%lf RT. \n",TIME/RT);
//for(i= 1 ;i<=N ;i++)
//printfTVn C[%d]=%lf',i,C[i]);
//getch();
RO=(((QF*CF)-((QTl+Area*USL)*(C[N])))/(QF*CF))* 100.0;
//RC=(((QF*CFHQP*(C[NF])HQT1*(C[N])))/((QF*CFHQP*(C[NF]))))*100.0;
//RC needs modification in terms of QP and QT1 (i.e. has to include USL)
//printfTVn The overall recovery at Time=%lf RT is RO=%lf%.",TIME/RT,RO);
//printfTVn The collection zone recovery at Time=%lf RT is
RC=%lf%.\n",TIME/RT,RC);
//getch();
main 1 (CFG,KG,D,CG);
//pnnti^"\n Following are the values of Gaunge CG[i] at Time=%lf RT. Vn",TEME/RT);
//for(i=l ,i<=N;i-H-)
//printf("\n CG[%d]=%lf',i,CG[i]);
//getch();
ROG=(((QF*CFG)-((QTl+Area*USL)*(CG[N])))/(QF*CFG))*100.0;
Grade=((QF*CF-(QTl+Area*USL)*C[N])/((QF*CF-
(QTl+Ajea*USL)*C[N])+(QF*CFG~(QTl+Area*USL)*CG[N])))* 100.0;
Grade=Grade*0.733;
//printfTVn The grade at Time=%lf RT is %lf',TIME/RT,Grade);
//getch();
Selectivity=RO-ROG;
TIME=(1+1)*DELT;
}
//printfTVn G100=%lf',G100);
//for(i=l ,i<=N;i++)
//printfTVn CG[%d]=%lf',i,CG[i]);
//getchO;
printfTVn CG[N]=%lf\CG[N]);
//printfTVn C[N]=%lf',C[N]);
//printfTVn C[l]=%lf',C[l]);
printfTVn C[N]=% 1 f',C[N]);
printfTVn Overall Recovery=%.llf %",RO);
printfTVn GRADE=%. 1 If %",Grade);
printfTVn Sep_eff=%. 1 If", Selectivity);
printfTVn ROG=%.llf %",ROG);
//printfTVn R=%. 1 If %",(1 -(C[N]/CF))* 100);
getchO;
}
else


50 55 60 65 70
Experimenta! Grade (%)
Figure 2.10: Performance of the overall hybrid model: Predicted versus experimental grade
(%BPL) for coarse feed size distribution


91
D=12.4*Dia*pow((0.3175*Qg/ Area), 0.3);
//************** Calculation of slip velocity **************
DP1=DP/1000.0;
PHIS=(CS/100)/((CS/100)+((1-(CS/100))*ROS));
USLi=0.0;
do
{
REP=5 12*DPl*USLi*ROS*(l. 0-PHIS);
USL=108.233*DP1 *DP1 *(ROS-l)*pow((l-PHIS),2.7)/(l+0.15*(pow(REP,0.687)));
diff=USL-USLi;
if(diff<0.0)
diff=-diff;
USLi=USL;
}
while(diff>=0.0001);
U SL=( 1 -Eg) *USL;
//***************** USL calculation ends ***************
a=(UP-USL)/D;
d=(UT+USL)/D;
b=k*(l-Eg)/D;
if(((a*a+4*b)<0.0)||((d,,d+4*b)<0.0))
b=0.0;
alpha=(a/2)+(sqrt(a*a+4*b))/2;
beta=(a/2)-(sqrt(a*a+4*b))/2;
gamma=(-d/2)+(sqrt(d*d+4*b))/2;
delta=(-d/2)-(sqrt(d*d+4*b))/2;
//printfX"\n alpha=%lf",alpha);
//printf("\n beta=%lf',beta);
//printtX"\n gamma=%lf",gamma);
//printf("\n delta=%lf",delta);
//printf("\n delta=%lf",a*a+4*b);
//print f("\n beta*L=%lf',(beta)*(L));
//printf("\n alpha*L=%lf",(alpha)*(L));
//printf("\n gamma*Lf=%lf',(gamma)*(Lf));
//printf("\n delta*Lf=%lf',(delta)*(Lf));
//printfC\n alpha*Lf=%lf",(alpha)*(Lf));
//printft"\n beta*Lf=%lf",(beta)*(Lf));
//p=(((-(JP/D)+a-beta)*(exp((beta)*(L))))/(((UP/D)-a+alpha)*(exp((alpha)*(L))));
//q=((-UT/D)+d-delta)/((UT/D)-d+gamma);
if(USL<=UP)
p=((-beta)*exp((beta)*(L)))/(alpha*exp((alpha)*(L))),


58
continued until a lower point could not be found This approach improves the likelihood
of convergence to the global optimum.
3.6 Results and Discussion
The performance of the network for predicting bubble diameter (NNIII), the network
for predicting air holdup (NNIV), the network for predicting the phosphate flotation rate
constant (NNI) and the network for predicting the gangue flotation rate constant (NNII) is
shown in Figures 3.3-3.14. Figure 3.3 compares the NNIII output to the inferred bubble
diameter using experimental data when the frother was CP-100. The solid circles are for
the data used for training while the open squares are for the data used for validation.
Figures 3.4, 3.5, and 3.6 show the performance of NNIII when F-507, OB-535, and F-
579, respectively, were the frothers.
As these figures show, NNIII successfully predicts the inferred bubble diameter.
Figure 3.7 compares the air holdup predicted by NNIV to the experimental values
measured by a differential pressure cell when CP-100 was used as the frother. Figures
3.8, 3.9, and 3.10 show the performance of NNIV when F-507, OB-535, and F-579,
respectively, were used as frothers. As shown in these figures, NNIV successfully
predicts the air holdup for all frothers.
Figures 3.11 and 3.12 show the performance of NNI and NNII, respectively. Figure
3.11 presents the predicted flotation rate constants for phosphate (kp) against those
determined from one-dimensional searches using experimental data. As shown in this
figure, NNI does accurately predict low and high values of flotation rate constants.


39
2.6 Conclusions
In this work, we have demonstrated that a one-phase first-principles model can
effectively be coupled with the artificial neural networks for predicting the grade and
recovery of a phosphate flotation column with negative bias. Artificial neural networks are
used to predict the flotation rate constants and air holdup. Experimental data from a lab-scale
column were used to train the neural networks. The hybrid model successfully predicts the
effects of particle size, superficial air velocity, frother concentration, collector concentration,
extender concentration, and pH.


14
in the limit as Az > 0, the above equation reduces to the following boundary
condition.
dC
pi
dz
= 0
(2.9)
z=L
Continuity of the concentration profile at the feed location gives
c;
Pi
= CJD
z=Lf p2
z=Lf
(2.10)
A similar material balance at the feed inlet gives for the solid particles in the slurry
phase
Qfc;=Ac
u.
v1_eg
-IIJ CJ
Usl |%,
dc;
- ACD^
z=Lf dz
+ A.
Ut
z = Lr
Kl~ss
U
P2
dCJ
+ A.D P:
z=Lr c dz
(2.11)
where
th
Cf = Phosphate feed concentration of j mesh size particles
Qf = Feed volumetric flow rate
Lf = Feed location
At the bottom of the column (z = 0), due to the elutriation flow, the derivative of the
concentration profile reduces to the following expression:
dCJ
D P2
dz
z =o
Qe C,
(l-OAc P!
z = 0
(2.12)


1780
1993
UNIVERSITY OF FLORIDA
3 1262
08554 4376


00
80
60
40
20
0
Figure 2.11: Performance of the overall hybrid model: Predicted versus experimental recovery
(%) for fine feed size distribution


88
Table 4.2: Results of the runs from
he factorial design
Frother
cone.
(ppm)
Air
flow
rate
(scfm)
Feed
flow
rate
(gpm)
Tailings
feed
flow
rate
(gpm)
Elutria-
tion
flow
rate
(gpm)
Solids
content
(%)
Grade
(%BPL)
Recovery
(%)
1
5
0.0928
0.198
2.014
2.410
59.37
55.95
68.31
2
5
0.3711
0 418
1.779
2.351
35.11
55.36
40.04
3
25
0.0928
0.126
1.432
2.423
48.53
40.07
34.60
4
25
0.3711
0.284
2.062
2.919
35.03
61.86
45.90
5
5
0.2319
0.376
1.897
2 893
49.75
51.04
67.72
6
25
0.2319
0.284
1.650
2 378
47.02
39.59
55.92
7
15
0.0928
0.264
2.355
2.922
36.43
62.68
47.49
8
10
0.3711
0 340
2.275
2.927
41.65
53.99
48.53
9
15
0.2319
0.463
2.173
2.619
38.58
45.63
16.87
10
15
0.3015
0.370
1.838
2.645
42.65
46.26
54.10
11
20
0.2319
0.261
1.694
2.661
49.18
37.50
55.19
12
15
0.1624
0.281
1.853
2.634
42.36
68.40
47.92
13
10
0.2319
0.259
1.758
2.631
41.13
68.03
52.14


56
and pump were adjusted to get the desired underflow and overflow rates. Air holdup was
measured using a differential pressure gauge. After the water/air system reached steady
state, the screw feeder was started. To achieve steady feed rate to the column, water was
added to the screw feeder at the rate that reduced the solids concentration to
approximately 66% by weight. The column was run for a period of three minutes with
phosphate feed prior to sampling. Timed samples of tailings and concentrates were
taken The collected product samples, as well as feed samples, were dried, sieved using
Tyler meshes, weighed and analyzed for %BPL following the procedure recommended
by the Association of Florida Phosphate Chemists (AFPC Analytical Methods, 1980). In
addition, gangue content (as % acid insolubles) of the feed, tailings, and concentrate
streams was measured (AFPC Analytical Methods, 1980). These measurements were
then used to calculate recovery of acid insolubles. Subsequently these values were
averaged with the values obtained from Equation 3.14 to obtain theRg used to determine
the flotation rate constants for gangue.
The two-phase experiments were identical to the three-phase experiments, except that
no solid feed was introduced to the column and the experiments were terminated when
the water/air system reached steady state.
3.5.2 Neural Network Structure and Training
NNI, NNII, NNin, and NNIV of Figure 3.2 were feedforward backpropagation
artificial neural networks with a single layer of hidden nodes between the input and
output layers and a unit bias connected to both the hidden and the output layers. Inputs


76
This performance measure is only applicable to plants, and can not be used with a lab-
scale flotation column. The procedure for this scheme is outlined below:
Assumptions
1. The price of rock of 66% BPL = $22.00
2. Zero insol %BPL = 73.33
3. Transportation cost = $2.50 per ton.
4. Soluble P2O5 losses = 1.00%
5. Insoluble P2O5 losses = 6.00%
6. Increase in soluble P2O5 losses is proportional to the amount of phosphogypsum
produced.
Transportation Penalty
Base case: 66% BPL rock (dry basis)
Freight cost per BPL ton = $2.5/0.66 = $3.79
Penalty:
2.50
vBL/m
- 3.79 per BPL ton
f
Transportation penalty =
2.50
Bi/100
A
-3.79
B,
7
100
per ton
Where, Bl = %BPL when grade < 66%
Acidulation Penalty
Base case. 66% BPL rock (30.21% P205, Ca0:P205 = 1.49)
Acid insol =100
(i~5L)
V 73.337
Calculation of the amount of Phosphogypsum:
Phosphogypsum components
f B, ^
= 1 ton rock x 1
^ 73.337
Acid insol


26
fatty acid (obtained from Westvaco) and fuel oil (No. 5 obtained from PCS
Phosphate) with a ratio of 1:1 by weight was added to the pulp. The total conditioning
time was 3 minutes. The conditioned feed material (without its conditioning water) was
loaded in the feeder bin located at the top of the column.
The frother selected for this study was CP-100 (sodium alkyl ether sulfate obtained from
Westvaco). Frother-containing water and air were first introduced into the column through
the sparger (eductor) at a fixed flowrate and frother concentration, and then the discharge
valve and pump were adjusted to get the desired underflow and overflow rates. Air holdup
was measured for the two-phase (air/water) system using a differential pressure gauge. After
every parameter was set and the two-phase system was in a steady state, the phosphate
material was fed to the column using the screw feeder. Water was also added to the screw
feeder to maintain the steady flow of the solids to the column at 66 % solids concentration
To achieve steady state, the column was run for a period of three minutes with phosphate
feed prior to sampling. Timed samples of tailings and concentrates were taken. The collected
samples were weighed and analyzed for %BPL according to the procedure recommended by
the Association of Florida Phosphate Chemists (AFPC Analytical Methods, 1980). These
measurements were then used to calculate recovery of acid insolubles. These values were
then averaged with the values obtained from Equation 2.26 to obtain the Rg used to
determine the flotation rate constants of gangue.



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98
do
{
REP=5.12*DPl*USLi*ROS*(l.0-PHIS);
USL= 108.233 *DP 1 *DP 1 *(ROS-1 )*pow(( 1 -PHIS),2.7)/( 1 +0.15 *(pow(REP,0.687)));
diff=USL-USLi;
ifidiffcO.O)
diff=-diff;
USLi=USL;
}
while(diff>=0.0001);
USL=(l-Eg)*USL;
//printf("\n USL=%lf',USL);
/y***************** calculation ends ***************
Frank=BPL/0.733;
CF=(Frank/100.0)*(CS/100.0)*ROS*62.41818/((CS/100.0)+(1.0-(CS/100.0))*ROS);
CFG=(1.0-(Frank/100.0))*(CS/100.0)*ROS*62.41818/((CS/100.0)+(1.0-
(CS/100.0))*ROS);
K=Ku;KG=KGu;
mainl(CF,K,B);
C=B[3]+B[4];
RO=(((QF*CF)-((QTl+Area*USL)*C))/(QF*CF))* 100.0;
main 1 (CFG,KG,BG);
CG=BG[3]+BG[4];
ROG=(((QF*CFG)-((QTl+Area*USL)*CG))/(QF*CFG))*100.0;
Grade=((QF*CF-(QTl+Area*USL)*C)/((QF*CF-(QTl+Area*USL)*C)+(QF*CFG-
(QTl+Area*USL)*CG)))* 100.0;
Grade=Grade*0 733,
//printf("\n C=%lf',C);
//printfTVn CG=%lf',CG);
//printfTVn CF=%lf',CF);
//printfT\n CFG=%lf',CFG);
//printf("\n Overall Recovery=%. Ilf %",RO);
//printfTVn ROG=%. 1 If %",ROG);
//printfT\n GRADE=%. 1 If %",Grade);
return (RO);
}


This dissertation
is dedicated
to
my parents


Predicted flotation rate constants
forgangue (kg)
Figure 4.4: Neural network versus experimental flotation rate constant for gangue (kg)


55
screw delivered the conditioned phosphate materials to the column. The feed rate was
controlled by adjusting the screw rotation speed The flotation column was constructed
of plexiglass and had 14 5 cm diameter and 1.82 m height. The feeding point was located
at 30 cm from the column top. The discharge flow rate was controlled by a discharge
valve and an adjustable speed pump Three flowmeters were used to monitor the flow
rates for air, frother solution, and elutriation water.
Phosphate feed (14X150 Tyler mesh) from Cargill was used as the feed material. For
each ran, 50 kg of feed were introduced to the pre-treatment tank and water w'as added to
obtain 72 % solids concentration by weight. The tank was then agitated for 10 seconds.
10 % soda ash solution was added to the pulp to reach pH of about 9.4 and the slurry was
agitated for another 10 seconds. Subsequently, a mixture of fatty acids (a mixture of
oleic, palmetic, and iinoleic acid obtained from Westvaco) and fuel oil (No. 5 obtained
from PCS Phosphates) with a ratio of 1:1 by weight was added to the pulp and the slurry
continued to be mixed. The total conditioning time was 3 minutes. The conditioned feed
material (without its conditioning water) was subsequently loaded to the feeder bin
located at the top of the column.
Four frothers were used, two commonly employed in industry, F-507 (a mixed
polyglycol by Oreprep) and CP-100 (a sodium alkyl ether sulfate by Westvaco), and two
experimental, F-579 (also a mixed polyglycol by Oreprep) and OB-535 (by OBrien).
Frother-containing water and air were first introduced into the column through the
sparger (an eductor) at a fixed water flow rate and frother concentration (0 30 ppm),
and the superficial air velocity ranged from 0.24 0.94 cm/s. Then the discharge valve


o
4
(/)
(i
c
ro
(/)
c
o
co a>
2 15
c .c
O CL
'^3 CO
C3 O
T3
4->
O

TJ

L_
Q_
K)
vC
Figure 2.6 Performance ofNNI: Model versus experimental flotation rate constant for
phosphate (kp)


LIST OF FIGURES
Figure page
2.1 Flotation rate constants for phosphate and gangue are calculated by
using a one-dimensional search to invert the first-principles model 19
2.2 Recovery of phosphate (%) as a function of flotation rate constant for
phosphate (kp) 20
2.2 Recovery of gangue as a function of flotation rate constant for
gangue (kg) 21
2.4 Overall structure of the hybrid model 23
2.5 A schematic diagram of the experimental setup 25
2.6 Performance of NN: Model versus experimental flotation rate
constant for phosphate (kp) 29
2.7 Performance ofNNII: Model versus experimental flotation rate
constant for gangue (kg) 31
2.8 Performance of NN1II: Model versus experimental air holdup for
frother CP-100 32
2.9 Performance of the overall hybrid model: Predicted versus
experimental recovery (%) for coarse feed size distribution 33
2.10 Performance of the overall hybrid model: Predicted versus
experimental grade (%BPL) for coarse feed size distribution 34
2.11 Performance of the overall hybrid model: Predicted versus
experimental recovery (%) for fine feed size distribution 35
2.12 Performance of the overall hybrid model: Predicted versus
experimental grade (%BPL) for fine feed size distribution 36
vi


where
13
cf- = column diameter (m)
Jg = superficial air velocity (cm/s)
Equations 2.1 and 2.2 can be solved analytically for the concentration profile of the
solid particles at steady state. The resulting analytical expressions for the concentration
profile are
where
K], Kj2, K3, andK^ are the constants of integration to be determined by using
appropriate boundary conditions.
2,2.1 Boundary Conditions
A material balance at the top layer of the column (z = L) gives the following
equation:
+ AcD 1 kp (dp AcAzC¡!)]
(2.8)


7
provides superior grade/recovery performance. For these reasons column
flotation is gaming increasing acceptance for the processing and beneficiation of
phosphate ores. Although it has been successfully employed for the selective separation
of phosphate from unwanted mineral, a totally predictive model still remains unavailable
for industrial use.
Flotation is a process to separate hydrophobic particles from hydrophilic particles.
The hydrophobic material has a tendency to attach to the rising bubbles and leaves from
the top of the column. The hydrophilic material settles down and leaves from the bottom
of the column In this way, the phosphate containing material (frankolite or apatite) is
separated from gangue (mostly silica). The phosphate ore is first pretreated with fatty
acid collector and fuel oil extender. Fatty acid and fuel oil adsorb on the phosphate-
containing panicles rendering them hydrophobic. The flotation process is then used to
separate phosphate panicles from gangue minerals.
A flotation column consists of three flow regimes: a cleaning or froth zone, a lower
collection zone, and pulp-froth interface zone. The froth zone is the region extending
upward from the pulp-froth interlace to the column interface. The collection zone is the
region extending downward from the pup-froth interface to the lowest sparger. A mineral
particle is recovered by a gas bubble in the collection zone of the column by particle-
bubble collision followed by attachment due to the hydrophobic nature of the mineral
surface. Since phosphate particles are considerably larger in size (0.1-1 mm), an
elutriation water stream from the bottom is added to maintain a positive upward flow
(negative bias) to aid lifting the particles upward.


10
9
8
7
6
5
4
3
2
1
0
Experimental flotation rate constants for Phosphate (kp)
ON
Figure 3.11: Performance of NNI-Model versus experimental flotation rate constant for phosphate
(kp)


45
Figure 3 .1: Schematic diagram of column for phosphate flotation.


51
Ri_W^RJpG{(73.3-GJ)
W GJ(73.3-Gf)
(3.14)
In some cases direct measurements of the majority of gangue as acid insolubles may be
available Then more reliable estimates of R^ can be obtained by averaging values
calculated from measurements of acid insolubles with values calculated from Equation
3.14. This was done in this work.
From the FPM equations follows that the recovery of phosphate depends only on kp,
while the recovery of gangue depends only on kg. This can be exploited to easily invert
the steady-state version of the model to determine from experimental measurements of
Rp and GJ corresponding kp and kg. As shown in Figure 2.1, this is accomplished with
one-dimensional searches. The search for kp is initialized with two values that yield
errors in the corresponding recovery R of opposite sign. Since typically 0 < kp < 10
min'1 the values of 0 and 100 min'1 are used. Then the method of false position (Chapra
and Canale, 1988) is used to iterate until the magnitude of the error in R^ drops to less
than 10'\ It is possible that the calculated recovery has a higher value than the
experimental even for kp = 0. In these cases kp is set equal to zero. The above procedure
is also used to determine kg, except that the high initial value is set to 10 min'1. Recovery
for both phosphate and gangue increases monotonically with respective flotation rate
constants as discussed in Chapter 2.


28
determined using simulated annealing (Kirkpatrick et al., 1983) and a conjugate gradient
algorithm (Polak, 1971). There are two approaches towards updating the weights. In one
approach, the input-output examples are presented one at a time and after each presentation
the weights are updated using rules such as the delta rule (Rumelhart and McClelland, 1986).
This method is attractive for its simplicity but is restricted to rather primitive optimization
algorithms. In contrast, the batch training approach allows use of powerful methodology for
nonlinear optimization, it processes each input-output example individually but updates the
weights only after the whole set of input-output examples has been processed. In this case,
the gradient is cumulated for all presentations, then the weights are updated, and finally the
sum of the squared errors is calculated.
The simulated annealing algorithm is used for eluding local minimum. It perturbs the
independent variables (the weights) while keeping track of the best (lowest error) function
value for each randomized set of variables. This is repeated several times, each time
decreasing the variance of the perturbations with the previous optimum as the mean. The
conjugate gradient algorithm is then used to minimize the mean-squared output error. When
the minimum is found, simulated annealing is used to attempt to break out of what may be a
local minimum. This alteration is continued until networks can not find any lower point. We
then hope that the local minimum is indeed the global minimum.
2,5 Results and Discussion
The performances of the three ANNs are shown in Figures 2.6-2.14. Figure 2.6
compares the flotation rate constants for phosphate (kp) determined from one-dimensional
searches with those predicted by NNI. As shown in this figure, NNI captures the dependence
of the flotation rate constant on particle size, superficial air velocity, frother concentration,


50
mineral that contains no gangue is 73.3 %BPL. Grade can be obtained as the ratio of
phosphate to the sum of phosphate and gangue in the product:
GJ =
QfCj ~[Qi +AC(1 -eg)Uj ]c¡,n
\
(QfCf-[Qt+Ac(l-e,)U]Cii)+(QfCi._-[Qt+Ac(l-Eg)U]C^)
gn
* 73.3
(3.12)
where Cgn is gangue concentration of jth particle size in the nlh layer and C^ is the
gangue feed concentration of jth particle size. The algorithm for solving the first-
principles model is given in Appendix B.
3.3 Calculation of Model Parameters
Since air-holdup eg is measured experimentally, the above FPM has only two
unmeasured model parameters for each particle size, namely, the flotation rate constants
for phosphate (kp) and for gangue (kg). The experimental analysis usually available in
industrial flotation columns is in terms of grade and recovery of phosphate. Let
-% ,tk *th
denote the weight of j size phosphate particles in the feed, W,g the weight of j size
fL
gangue in the feed, and WgJ the weight of j' size gangue in the product. The grade of
feed is then
G¡ =73.3
WJ
tp
K + w-
(313)
and GJ is given by an analogous expression. The recovery of gangue can be readily
calculated from measurements of grade and recovery of phosphate using the following
relationship:


Experimental Grade (%)
Figure 2.14; Performance of the overall hybrid model: Predicted versus experimental grade
(%BPL) for the unsized feed after it has been sized.


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30
25
20
15
10
5
0
Figure 3.9: Performance of NNIV: Model versus experimental air holdup for frother OB-535


Predicted flotation rate constants
for gangue (kg)
Figure 3.12 Performance of NNI: Model versus experimental flotation rate
constant for gangue (k^


2.4 2 Experimental Conditions
For the frothers investigated. 35 three-phase experiments were conducted. Seven
different levels of frother concentration (5, 6.6, 10, 15, 20, 23.4, and 25 ppm) was studied in
designed experiments Five different levels of collector and extender concentration (0.27,
0.41, 0 54, 0.64, and 1.7 kg/t) were used. pH was varied from 8.2 to 9.9 at five different
levels (8.2, 8 5, 9.0. 9 5, and 9.9). Two superficial air velocities (0.46 and 0.7 cm/s) were
used for the designed experiments.
The particle size depended on the type of feed used. For coarse feed, the particle size
varied from 417 to 991 microns. For fine feed, the particle size varied from 104 to 417
microns whereas for the unsized feed distribution, the size ranged from 104 to 991 microns.
2 4 3 Neural Network Structure and Training
Single output feedforward backpropagation neural networks are used with a single
layer of hidden nodes. A unit bias is connected to both the hidden layer and the output layer.
Both the hidden layer and the output layer used a logistic activation function (Hertz et al.,
1992) and the input and the output values were scaled from 0 to 1
During the training mode, training examples are presented to the network. A training
example consists of scaled input and output values. For NNI and NNII, the output values are
the flotation rate constants calculated from one-dimensional searches for phosphate and
gangue, respectively. For NNIII, the output value is the experimentally measured air holdup.
The training process is started by initializing all weights randomly to small non-zero
values. The random number is generated between -3.4 and +3.4 with standard deviation of
1.0 following the procedure recommended by Masters (1993). The optimal weights were


Figure 4.3 Neural network versus experimental flotation rate constant for phosphate (kp)


LIST OF REFERENCES
Association of Florida Phosphate Chemists, AFPC Analytical Methods, 6th ed. (1980).
Bhat, N. and T.J. McAvoy, Use of Neural Nets for Dynamic Modeling and Control of
Chemical Process Systems, Comput. Chem. Eng., 14:573 (1990).
Boutin, P. and D A. Wheeler, Column Flotation, Mining World, 20(3): 47-50 (1967).
Chapra, S.C. and R. P. Canale, Numerical Methods for Engineers, McGraw Hill, New
York (198S).
Cubillos, F., P. Alvarez, J. Pinto, and E. Lima, Hybrid-Neural Modeling for Particulate
Solid Drying Processes, Powder Technology, 87, p. 153 (1996).
Cubillos, F.A., and E. L. Lima, Identification and Optimizing Control of a Rougher
Flotation Circuit Using an Adaptable Hybrid-Neural Model, Minerals Engineering,
10(7): 707-721 (1997).
Dobby, G.S, and J. A. Finch, Column Flotation: A Selected Review, Part II, Mineral
Engineering, 4: 911-923 (1991).
Dobby, G.S., and J. A. Finch, Mixing Characteristics of Industrial Flotation Columns,
Chem. Eng. Sci., 40: 1061-1068 (1985).
Finch, J. A., and G. S. Dobby, Column Flotation, Pergamon Press, Toronto (1990).
Hertz, J., A. Krogh, and R.G. Palmer, Introduction to the Theory of Neural
Computations, Addison-Wesley Publishing Company, Redwood City, CA, 5th ed. (1992).
Himmelblau, D M., Applied Nonlinear Programming, McGraw Hill, New York (1972).
Homik, KM., M. Stinchcombe, and H. White, Multi-layer Feedforward Networks Are
Universal Approximators, Neural Networks, 2:359 (1989).
Johansen, T. A., and B. A. Foss, Representing and Learning Unmodeled Dynamics with
Neural Network Memories, Proc. Am. Control Conf.x Chicago, 3:3037-3037 (1992).
108


70
60
50
40
30
20
Figure 2.12: Performance of the overall hybrid model: Predicted versus experimental grade
(%BPL) for fine feed size distribution


Experimental grade and
Experimental
recovery
of phosphate
One-dimensional
search
Flotation rate constant
for phosphate
Figure 2.1: Flotation rate constants for phosphate and gangue are calculated by using a one
dimensional search to invert the first-principles model


15
The four boundary conditions can be solved in conjunction with Equations 2.6 and
2.7 for Kj, Kj2, K^, andKJ4. The resulting expressions for the constants of integration
are given by the following equations:
(q,c;/acd)
mJ(aJ aJ)pJ exp{a-Lf}+(dJ -yJ)qJ exp{yJLf} + mJ (aJ -(3J)exp{(3JLf} + (dJ -6J)exp{6JLj}
(2.13)
Ki=qJK'
(2.14)
*
K>
II
3
5*
U '
(2.15)
K;=pJmJK;
(2.16)
where
or* =- + VaJ +4b-*
2 2
(2.17)
(3J =---->/aJ +4b'
2 2
(2.18)
yj Vdj2 +4bJ
2 2
(2.19)
8' = ---\/d,: +4b<
2 2
(2.20)


Figure 2.4: Overall structure of the hybrid model


LD
1780
199.3
c7 UNIVERSITY OF FLORIDA
3 1262 08554 4376


Model diameter (mm)
Inferred bubble diameter (mm)
Figure 3.6: Performance ofNNIII: Model bubble diameter versus bubble diameter inferred from
experimental data when F-579 was the frother


HYBRID NEURAL NETWORK FIRST-PRINCIPLES APPROACH
TO PROCESS MODELING
By
SANJAY GUPTA
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILLOSOPHY
UNIVERSITY' OF FLORIDA
1999


Abstract of Dissertation Presented to the Graduate School
Of the University of Florida in Partial Fulfillment of the
Requirements for Doctor of Philosophy
HYBRID NEURAL NETWORK FIRST-PRINCIPLES APPROACH
TO PROCESS MODELING
By
Sanjay Gupta
May 1999
Chairman: Dr Spyros A. Svoronos
Cochairman: Dr Hassan El-Shall
Major Department: Chemical Engineering
A hybrid model for a flotation column is presented which combines a first-
principles model with artificial neural networks. The first-principles model is derived by
making material balances on both phosphate and silica particles in the slurry phase.
Neural networks are used to relate the model parameters with operating variables such as
particle size, superficial air velocity, frother concentration, collector and extender
concentration, and pH. One-level and two-level hybrid modeling structures are compared
and it is shown that the two-level structure offers significant advantages over the other.
Finally, a sequential run-to run optimization algorithm is developed which combines the
hybrid model with an optimization technique. The algorithm guides the changes in the
manipulated variables after each experiment to determine the optimal column conditions.
IX


20
18
16
14
12
10
8
6
4
2
0

&
Training data
Validation data
10
Experimental Air Holdup (%)
15
20
CN
igure 3.8: Performance of NNIV: Model versus experimental air holdup for frother
-507


20
18
16
14
12
10
8
6
4
2
0


P6

Training data
Validation data
5 10
Experimental Air Holdup (%)
15
20
On
3.7: Performance of NNIV: Model versus experimental air holdup for frother CP-100


4 OPTIMIZATION PERTORMANCE MEASURES AND FUTURE WORK
73
4.1 The Performance Measures 74
4.1.1 Selectivity 74
4.1.2 Separation Efficiency 75
4 1.3 Economic Performance Measure 75
4.2 The Optimization Algorithm 79
4.3 Initial Scattered Experiments 81
4 4 Results and Discussions 82
4.5 Future Work 86
APPENDICES
A CODE FOR THE FIRST PRINCIPLES MODEL FOR ONE LEVEL 90
B CODE FOR THE FIRST PRINCIPLES MODEL FOR TWO LEVELS 99
REFERENCES 108
BIOGRAPHICAL SKETCH
111


084 F7*
08/03/99 347B0


40
20
0
\
20
40
60
80
TI 'T'l 'ITTTTTTTTTTTTTTTTTT 1 I TT"
\> ^

i
00
Bl (%BPL)
Figure 4 .1: Value of phosphate rock as a function of %BPL


79
Performance measure = CpP CfF F T] U,Cn $/year (4.5)
l
4 2 The Optimization Algorithm
The idea behind the sequential optimization is to iterate between experimentation
towards the optimum and model identification until the optimum is reached The
procedure is as follows:
(1) Initial experiments are performed and their results are analyzed.
(2) The neural networks are trained and the hybrid model is used to determine the
optimal factor values. If these are within the convergence limit of previous
experimental values, the procedure stops.
(3) Otherwise, an experiment at the calculated optimal value is performed and
analyzed.
(4) The data are added to the neural network training set, and the procedure returns to
step (2).
Figure 4.2 shows a more detailed description of the algorithm. After some
initialization runs have been completed, the samples are analyzed and the neural
networks are trained with the input-output data. Subsequently, using the standard Nelder-
Meade algorithm (Himmelblau, 1972), the values of manipulated variables that maximize
the selectivity are determined. If these values correspond to an interior point then the


Model Diameter (mm)
0 0.2 0.4 0.6 0.8 1 1.
Inferred Bubble Diameter (mm)
Figure 3.4: Performance of NNIII: Model bubble diameter versus bubble diameter inferred from
experimental data when F-507 was the brother