Hybrid neural network first-principles approach to process modeling

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Title:
Hybrid neural network first-principles approach to process modeling
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x, 111 leaves : ill. ; 29 cm.
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English
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Gupta, Sanjay
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Subjects / Keywords:
Phosphate industry   ( lcsh )
Flotation -- Equipment and supplies   ( lcsh )
Chemical Engineering thesis, Ph.D   ( lcsh )
Dissertations, Academic -- Chemical Engineering -- UF   ( lcsh )
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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).
Statement of Responsibility:
by Sanjay Gupta.
General Note:
Typescript.
General Note:
Vita.

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University of Florida
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oclc - 41940690
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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;




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