Citation |

- Permanent Link:
- http://ufdc.ufl.edu/AA00013557/00001
## Material Information- Title:
- Hybrid neural network first-principles approach to process modeling
- Creator:
- Gupta, Sanjay
- Publication Date:
- 1999
- Language:
- English
- Physical Description:
- x, 111 leaves : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
- Bubbles ( jstor )
Diameters ( jstor ) Flow velocity ( jstor ) Mathematical constants ( jstor ) Minerals ( jstor ) Modeling ( jstor ) Neural networks ( jstor ) Parametric models ( jstor ) Phosphates ( jstor ) Velocity ( jstor ) Chemical Engineering thesis, Ph.D ( lcsh ) Dissertations, Academic -- Chemical Engineering -- UF ( lcsh ) Flotation -- Equipment and supplies ( lcsh ) Phosphate industry ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Abstract:
- Keyword: Flotation columns
- Thesis:
- Thesis (Ph.D.)--University of Florida, 1999.
- Bibliography:
- Includes bibliographical references (leaves 108-110).
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Sanjay Gupta.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 030359312 ( ALEPH )
41940690 ( OCLC )
## UFDC Membership |

Downloads |

## This item has the following downloads: |

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-EsC' 2C + C Pk+I Pk Pk-I Az2 U if U'L > (3.3) dCJ t dt dC dt dCJ dt dt k,(dj )Cj kp(dJ )CJ )pPk +D +D Layer k+1 to n-1 dCJ U CJ C CJ -2C _+C -- =+ U p- -D _C"' pk k(dJ)C (3.4) dt 1-E 1 Az Az2 Layer n (bottom) Ut(+U Qt D+U P -- +U 11 (1- po )A + -U I --1"-- (1 C )AC --CD -- C 1 c ( )D P.- P _k, (d )CJ dCC U PdC if Uj >- dt s 1-el t + U C Q +U C 1- (1- P)A1 P" Ci -Ci S(D ,_I Pn" -k, (dp)CJ Az Az 2 U, if U, <- 1--tB (3.5) where AC = Cross-sectional area of the column C = Phosphate feed concentration ofj'h mesh size particles Cj = Phosphate concentration ofjth mesh size particles in the ith layer Qf = Feed volumetric flow rate Q, = Tailings volumetric flow rate Qe = Elutriation volumetric flow rate Qp = Product volumetric flow rate Up = Superficial liquid velocity above the feed point = Qp/A, U, = Superficial liquid velocity below the feed point = (Q, -Q)/A, UJ, = Slip velocity of jh mesh size particles E, = Air holdup kp(dp) = Flotation rate constant for phosphate forjh mesh size particles The slip velocity is calculated using the expression of Villeneuve et al. (1996): gd (p-p,)(1-)27 U0si = -- (3.6) 18 ,(l +0.15R( ) ,where the particle Reynolds number is defined as d) Up ( ) R, s(3.7) l1- where g = Acceleration due to gravity (m/s2) ti = Water viscosity (kg/ms) pl = Water density (kg/m3) p, = Solid density (kg/m3) (bs = Volume fraction of solids in slurry dp = Particle diameter (m) As the right hand side of Equation 3.7 is a function of U,,, the slip velocity is obtained by solving Equations 3.6 and 3.7 iteratively as described in Chapter 2. The axial dispersion coefficient is calculated by a modification of the Finch and Dobby (1990) expression: IJ Y3 D = 0.063 (1-g)dc (3.8) where dc = column diameter (m) Jg = superficial air velocity (cm/s) Equations analogous to 3.1-3.8 are valid for the gangue particles, but with a considerably lesser effective flotation rate constant kg (d ). In the limit as Az -- 0 the above difference equations become -C U C 8ac C -=~ U + D -kp(d )CJ (3.9) at 1-6 & z &z2 p p for the section above the feed and P= +Uj +D -kp(d )C (3.10) SSJ J) a D C & 2 (3.10) for the section below the feed. Recovery (%) is defined as the ratio of the weight of the phosphate in the concentrate stream to the weight of the phosphate in the feed. The recovery for phosphate particles of the jo, mesh size can be expressed in terms of the feed and tailings flow rates and concentrations as QCf A(Q,+Ao( -e)UjC C R Q fC -[Q c- *100 (3.11) = QfCj Grade, a measure of the quality of the product, is defined as the ratio of the weight of the phosphate to the total weight recovered in the concentrate stream. Grade is usually reported as % Bone Phosphate of Lime (%BPL) which is the equivalent grams of tricalcium phosphate, Ca3(P04)2, in 100 g of sample. For the typical Florida rock, mineral that contains no gangue is 73.3 %oBPL. Grade can be obtained as the ratio of phosphate to the sum of phosphate and gangue in the product: QC_ -rt Qf[Q+Alc(1-s,)Uj (QfC'-[Qt +Ac(l- g)Us ]Cn )+(QfC'-[Qt +A,(1-e,)UlC]C )' (3.12) where Cg is gangue concentration of jth particle size in the nth layer and C' is the gangue feed concentration of jth particle size. The algorithm for solving the first- principles model is given in Appendix B. 3.3 Calculation of Model Parameters Since air-holdup eg is measured experimentally, the above FPM has only two unmeasured model parameters for each particle size, namely, the flotation rate constants for phosphate (kp) and for gangue (k). The experimental analysis usually available in industrial flotation columns is in terms of grade and recovery of phosphate. Let WI denote the weight of j'" size phosphate particles in the feed, W the weight of jth size gangue in the feed, and Wg the weight of jth size gangue in the product. The grade of feed is then W' G' =73.3 Wp (3.13) Wfp + Wfg and GJ is given by an analogous expression. The recovery of gangue can be readily calculated from measurements of grade and recovery of phosphate using the following relationship: WJ RJ Gi(73.3-GJ RJ =-g =-'- t (3.14) SWf Gj (73.3-G') In some cases direct measurements of the majority of gangue as acid insolubles may be available. Then more reliable estimates of Rg can be obtained by averaging values calculated from measurements of acid insolubles with values calculated from Equation 3.14. This was done in this work. From the FPM equations follows that the recovery of phosphate depends only on kp, while the recovery of gangue depends only on kg. This can be exploited to easily invert the steady-state version of the model to determine from experimental measurements of RI and G' corresponding kp and kg. As shown in Figure 2.1, this is accomplished with one-dimensional searches. The search for kp is initialized with two values that yield errors in the corresponding recovery R, of opposite sign. Since typically 0 < kp < 10 minl' the values of 0 and 100 min'1 are used. Then the method of false position (Chapra and Canale, 1988) is used to iterate until the magnitude of the error in R, drops to less than 10"3. It is possible that the calculated recovery has a higher value than the experimental even for kp = 0. In these cases kp is set equal to zero. The above procedure is also used to determine kg, except that the high initial value is set to 10 min-. Recovery for both phosphate and gangue increases monotonically with respective flotation rate constants as discussed in Chapter 2. 3.4 The Hybrid Model The main factors affecting the air hold up eg are the superficial air velocity Jg and the brother concentration Cfroth,. Several factors affect the flotation rate constants, k, and kg, including particle diameter, superficial air velocity, brother concentration, collector concentration, extender concentration, and pH. In this study we have conducted experiments varying particle size, brother type, brother concentration, and superficial air velocity, and develop a hybrid model that portrays the effect of these factors on the performance of the column. The hybrid model utilizes backpropagation ANNs (Rumelhart and McClelland 1986) to predict the values of the parameters Eg, kp, and kg. The straightforward approach is to develop an ANN for each of the three parameters. The inputs to the ANNs that predict kp and kg would be dp, Jg, and Cfrothe, while the inputs of the ANN that predicts sg would be Jg, and Cfother. Each of the ANNs in this structure would then depend on the brother and sparger used. A change in type of brother would mean that the previously trained ANNs are no longer applicable and would necessitate collection of a new set of training data and retraining of the networks. As changes in brother or sparger are not uncommon, this is a disadvantage. The main reason Jg and Cfroth, as well as the type of brother and sparger, affect the flotation rate constants, is because they significantly affect the bubble size. An alternative hybrid model architecture is shown in Figure 3.2. The neural networks are structured in two levels. The first level consists of the ANNs for predicting kp (NNI) and Superficial air velocity Frother concentration 1 Phosphate Inferred particle Bubble size diameter Gangue particle size Specific to frother/sparger type air holdup Diffusion coefficient (Finch and Dobby, 1990) Recovery Figure 3.2 : Overall structure of the hybrid model Grade kg (NNII) and receives as an input the inferred bubble size. This is the output of one of the ANNs of the second (top) level, NNIII. The second level also includes NNIV, which predicts air holdup. The advantage of this structure is that NNI and NNII are independent of the type of brother and sparger used, and therefore would not need retraining if these change. As bubble size is not measured in industry, we infer it from the two-phase (air/water) air holdup, Jg, and Ut using the well-known Drift-flux analysis (Yianatos et al., 1988). The output required to train NNIV is the (two-phase) air holdup. Air holdup is relatively easy to obtain, so after a change of brother or sparger the hybrid model of Figure 3.2 can become functional in a short interval of time. 3.5 Materials and Methods 3.5.1 Experimental Setup and Procedures Two types of experiments were conducted: two-phase (air/water) experiments to train neural networks NNIII and NNIV, and three-phase experiments to train NNI and NNII and to test the performance of the hybrid model. The experimental setup for the three-phase experiments is shown in Figure 2.5. It included an agitated tank (conditioner) for reagentizing the feed, a screw feeder for controlling the rate ofreagentized feed, and a flotation column. The agitated tank was 45 cm in diameter and 75 cm high and was equipped with an impeller with two axial blades (each 28 cm diameter). The impeller had about 3.8 cm clearance from the bottom of the tank and its rotation speed was fixed at 465 rpm. The feeder with a 2.5 cm diameter screw delivered the conditioned phosphate materials to the column. The feed rate was controlled by adjusting the screw rotation speed. The flotation column was constructed of plexiglass and had 14.5 cm diameter and 1.82 m height. The feeding point was located at 30 cm from the column top. The discharge flow rate was controlled by a discharge valve and an adjustable speed pump. Three flowmeters were used to monitor the flow rates for air, brother solution, and elutriation water. Phosphate feed (14X150 Tyler mesh) from Cargill was used as the feed material. For each run, 50 kg of feed were introduced to the pre-treatment tank and water was added to obtain 72 % solids concentration by weight. The tank was then agitated for 10 seconds. 10 % soda ash solution was added to the pulp to reach pH of about 9.4 and the slurry was agitated for another 10 seconds. Subsequently, a mixture of fatty acids (a mixture of oleic, palmetic, and linoleic acid obtained from Westvaco) and fuel oil (No. 5 obtained from PCS Phosphates) with a ratio of 1:1 by weight was added to the pulp and the slurry continued to be mixed. The total conditioning time was 3 minutes. The conditioned feed material (without its conditioning water) was subsequently loaded to the feeder bin located at the top of the column. Four brothers were used, two commonly employed in industry, F-507 (a mixed polyglycol by Oreprep) and CP-100 (a sodium alkyl ether sulfate by Westvaco), and two experimental, F-579 (also a mixed polyglycol by Oreprep) and OB-535 (by O'Brien). Frother-containing water and air were first introduced into the column through the sparger (an eductor) at a fixed water flow rate and brother concentration (0 30 ppm), and the superficial air velocity ranged from 0.24 0.94 cm/s. Then the discharge valve and pump were adjusted to get the desired underflow and overflow rates. Air holdup was measured using a differential pressure gauge. After the water/air system reached steady state, the screw feeder was started. To achieve steady feed rate to the column, water was added to the screw feeder at the rate that reduced the solids concentration to approximately 66% by weight. The column was run for a period of three minutes with phosphate feed prior to sampling. Timed samples of tailings and concentrates were taken. The collected product samples, as well as feed samples, were dried, sieved using Tyler meshes, weighed and analyzed for %BPL following the procedure recommended by the Association of Florida Phosphate Chemists (AFPC Analytical Methods, 1980). In addition, gangue content (as % acid insolubles) of the feed, tailings, and concentrate streams was measured (AFPC Analytical Methods, 1980). These measurements were then used to calculate recovery of acid insolubles. Subsequently these values were averaged with the values obtained from Equation 3.14 to obtain the R used to determine the flotation rate constants for gangue. The two-phase experiments were identical to the three-phase experiments, except that no solid feed was introduced to the column and the experiments were terminated when the water/air system reached steady state. 3.5.2 Neural Network Structure and Training NNI, NNII, NNIII, and NNIV of Figure 3.2 were feedforward backpropagation artificial neural networks with a single layer of hidden nodes between the input and output layers and a unit bias connected to both the hidden and the output layers. Inputs and outputs were scaled from 0 to 1. The hidden and output layer nodes employed logistic activation functions (Hertz et al., 1992). For each of the four brothers investigated, 28 two-phase experiments were conducted (full factorial design with 7 brother concentrations and 4 superficial air velocities). These were used to train (19 data points) and to validate (9 data points) the top level neural networks (NNIII and NNIV), a different pair for each brother. Three-phase runs yielded 28 experimental grades and recoveries, which were used to train (19 data points) and to validate (9 data points) NNI and NNII. To set the number of nodes in the hidden layer of each network, the number was increased until the sum of the absolute errors of the training and validation outputs started increasing. In this manner an appropriate number of hidden nodes was determined to be three for all the neural networks. The training process started by initializing all weights randomly to small non-zero values. The random numbers were generated in the range -3.4 to +3.4 with a standard deviation of 1.0 following the procedure recommended by Masters (1993). The optimal weights were determined by combining simulated annealing (Kirkpatrick et al. 1983) with the Polak-Ribiere conjugate gradient algorithm (Polak, 1971). Simulated annealing randomly perturbed the independent variables (the weights) and kept track of the best (lowest error) function value for each randomized set of variables. This was repeated several times, each time decreasing the variance of the perturbations with the previous optimum as the mean. Then the conjugate gradient algorithm was used to minimize the mean-squared output error. When the minimum was found, simulated annealing was used to attempt to break out of what may be a local minimum. This alternation was continued until a lower point could not be found. This approach improves the likelihood of convergence to the global optimum. 3.6 Results and Discussion The performance of the network for predicting bubble diameter (NNIII), the network for predicting air holdup (NNIV), the network for predicting the phosphate flotation rate constant (NNI) and the network for predicting the gangue flotation rate constant (NNII) is shown in Figures 3.3-3.14. Figure 3.3 compares the NNIII output to the inferred bubble diameter using experimental data when the brother was CP-100. The solid circles are for the data used for training while the open squares are for the data used for validation. Figures 3.4, 3.5, and 3.6 show the performance of NNIII when F-507, OB-535, and F- 579, respectively, were the brothers. As these figures show, NNIII successfully predicts the inferred bubble diameter. Figure 3.7 compares the air holdup predicted by NNIV to the experimental values measured by a differential pressure cell when CP-100 was used as the brother. Figures 3.8, 3.9, and 3.10 show the performance of NNIV when F-507, OB-535, and F-579, respectively, were used as brothers. As shown in these figures, NNIV successfully predicts the air holdup for all brothers. Figures 3.11 and 3.12 show the performance of NNI and NNII, respectively. Figure 3.11 presents the predicted flotation rate constants for phosphate (kp) against those determined from one-dimensional searches using experimental data. As shown in this figure, NNI does accurately predict low and high values of flotation rate constants. 1.2 0-- 1 - L. 0.8- O DO 0 0.6 m 0.4 2 0.2 -- *Training data O Validation data 0 .--i I I 0 0.2 0.4 0.6 0.8 1 1.2 Inferred Bubble Diameter (mm) Figure 3.3: Performance of NNIII: Model bubble diameter versus bubble diameter inferred from experimental data when CP-100 was the brother 1.2 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 Inferred Bubble Diameter (mm) Figure 3.4: Performance of NNIII: Model bubble diameter versus bubble diameter inferred from experimental data when F-507 was the brother 1.2 - E 1 a 0.8 I 0.6 ) 0.4 0.2 Training data 0 ,E Validation data 0 0.2 0.4 0.6 0.8 1 1.2 Inferred bubble diameter (mm) Figure 3.5: Performance of NNIII: Model bubble diameter versus bubble diameter inferred from experimental data when OB-535 was the brother 1.3 1.2 1.1 O] E 1 S 0.9 E . 0.8 0.7 0.6 0.5 Training data 0 Validation data 0.4 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 Inferred bubble diameter (mm) Figure 3.6: Performance of NNIII: Model bubble diameter versus bubble diameter inferred from experimental data when F-579 was the brother OLI~ STraining data O Validation data Experimental Air Holdup (%) Figure 3.7: Performance of NNIV: Model versus experimental air holdup for brother CP-100 .4 -- I I I LU z-u ---------------------------- 18 - 16 O b 14 - 12 0 -I 10 5 6 EO 4 Training data 2 D Validation data 0 5 10 15 20 Experimental Air Holdup (%) Figure 3.8: Performance of NNIV: Model versus experimental air holdup for brother F-507 30 25 Q \~ 120 o a 0 15 . 10 CL 5 5 *eTraining data _O Validation data 0 0 5 10 15 20 25 30 Experimental Air Hold-up (%) Figure 3.9: Performance of NNIV: Model versus experimental air holdup for brother OB-535 25 20 -- 20 0 ' 15 I 10 171 5 Training data 0 Validation data 0 0 5 10 15 20 25 Experimental Air Hold-up (%) Figure 3.10: Performance of NNIV: Model versus experimental air holdup for brother F-579 0 2 4 6 8 Experimental flotation rate constants for Phosphate (kp) Figure 3.11: Performance of NNI-Model versus experimental flotation rate constant for phosphate (kp) Figure 3.12 presents the flotation rate constants for gangue (kg) predicted using NNII against those determined from experimental data. A very good match is seen. The hybrid model integrates NNI, NNII, NNIII, and NNIV with the FPM as shown in Figure 3.2. Predictions of the hybrid model are shown in Figures 3.13 and 3.14. Figure 3.13 presents the predicted recovery (%) against the experimental recovery for brother CP-100 (square points), F-507 (circles), OB-535 (triangles), and F-579 (diamonds). Similarly, Figure 3.14 compares the predicted grade (%BPL) against the experimental grade for CP-100, F-507, OB-535, and F-579. It can be seen from these figures that predicted recovery and grade from the hybrid model match closely the experimental values, with the exception of one grade for OB-535. The root mean squared errors in predicted recovery were 0.1%, 0.2%, 1.5%, and 0.4% for CP-100, F-507, OB-535, and F- 579, respectively. The root mean squared errors in predicted grade were 3.2 %BPL, 1.5 %BPL, 7.5 %BPL, and 1.5 %BPL for CP-100, F-507, OB-535, and F-579, respectively. An alternative to the present modeling approach is to develop a pure neural- networks model. This would, however, require a large number of inputs: not only superficial air velocity, brother concentration, and particle size, but also feed flow rate, feed concentration, elutriation flow rate, tailings flow rate, and solids loading. This increase in number of inputs to eight would increase the number of weights (model parameters) needed and therefore the number of three-phase data required for training. Furthermore, as with an in-series hybrid model that uses one level of neural networks, a change in brother or sparger would require generation of a new set of data and retraining of all the networks. The hybrid model presented here with the two levels of neural 0 0.9 0.8 o 0.7 *a 0.6 0.3 '0.3 a. 0.2 0. 0 1 *Training data D Validation data 0 0.2 0.4 0.6 0.8 1 Experimental flotation rate constants for gangue (kg) Figure 3.12 Performance of NNII: Model versus experimental flotation rate constant for gangue (k) 100 95 90 100 Experimental Recovery (%) Figure 3.13: Performance of the overall hybrid model: Predicted versus experimental recovery (%) for the four brothers 0O Frother CP-100 0 Frother F-507 A Frother OB-535 0 Frother F-579 85 80 75 70 65 60 55 50 60 70 80 .--I-I-I-I---------- 80 70 -A I60 OO 50 - o 40 0 30 0 Frother CP-100 a 0 0 Frother F-507 20 A Frother OB-535 O Frother F-579 10 10 20 30 40 50 60 70 80 Experimental Grade (%BPL) Figure 3.14: Performance of the overall hybrid model: Predicted versus experimental grade (%BPL) for the four brothers networks involves a relatively low number of inputs in the artificial neural networks, does not require new three-phase data if a brother or sparger is changed, and gives very good predictions of both grade and recovery. 3.7 Conclusions A hybrid neural network modeling approach was presented and used to model a flotation column for phosphate/gangue separation. This hybrid model is comprised of two parts, a first-principles model and two levels of neural networks that serve as parameter predictors of difficult-to-model process parameters. Experimental data from a laboratory column were used to train and validate the neural networks, and it is shown that the hybrid model captures the dependence of column performance on particle size, brother concentration, and superficial air velocity. CHAPTER 4 OPTIMIZATION PERFORMANCE MEASURES AND FUTURE WORK High recovery and grade and low operating cost depend largely on the optimal selection of operating variables. The search of the optimal conditions can considerably benefit by the availability of a model that can relate the operating conditions to the column performance. The hybrid model developed for the flotation column provides a mathematical relationship between the operating variables and column performance. This hybrid model can be combined with an optimization algorithm to determine the optimal operating conditions for the flotation column. We propose an algorithm that leads to the sequential optimization of a flotation column. This algorithm guides successive changes in the manipulated variables after each experiment to achieve optimal column operating conditions. Selectivity, which combines recovery and grade, can be used as the performance measure of the column. The hybrid model builds a relationship between the process manipulated variables and the performance measure. The optimization algorithm dictates the changes in the manipulated variables between successive runs. At each run manipulated variables are set at their predicted optimal values. After the run is completed, the collected samples should be collected and analyzed for recovery and grade. Then the new input-output data are added to the neural network of the hybrid model and the network should be retrained. New optimal manipulated variable values are predicted which set the conditions for the subsequent run. This procedure should be repeated until convergence is obtained. 4.1 Performance Measures The performance of a flotation column is affected by both recovery (%) and grade (%BPL). To guide optimization it is necessary to combine the two outputs (grade and recovery) in a single performance measure. Several performance measures are possible, and some are presented below. 4.1.1 Selectivity One way to achieve this is to use selectivity as the performance measure. Selectivity is defined as S= R /- (4.1) Rb Rtb where R = Recovery of phosphate in the product stream. Rb = Recovery of gangue in the product stream. Rt = Recovery(or Rejectability) of phosphate in the tailings stream. Rtb = Recovery(or Rejectability) of gangue in the tailings strea We developed the following expression that relates selectivity to the recovery and the grade of the product stream --G(1-R)-R (1-G)(1 -R) \%here G = Grade (%BPL) of phosphate in the product stream. Gf = Grade (BPL) of phosphate in the feed. 4.1.2 Separation Efficiency Separation efficiency is defined as follows: E=R -Rb (4.3) In this case, the efficiency varies between -100 to 100. 4.1.3 Economic Performance Measure The selectivity function or the separation efficiency does not include any economic input such as cost of the reagents. Therefore an alternate performance measure was developed which includes recovery, grade, and the reagent prices. A scheme for penalizing lower grade rock has been developed. This scheme deducts differential costs, relative to 66% BPL, for transportation and acidulation. The acidulation scheme assumes soluble P205 losses increase in direct proportion to the amount of phosphogypsum. Thus, the procedure requires an estimate of the quantity of phosphogypsum that is produced. (4.2) This performance measure is only applicable to plants, and can not be used with a lab- scale flotation column. The procedure for this scheme is outlined below: Assumptions 1. The price of rock of 66% BPL = $22.00 2. Zero insol %BPL = 73.33 3. Transportation cost = $2.50 per ton. 4. Soluble P205 losses = 1.00% 5. Insoluble P205 losses = 6.00% 6. Increase in soluble P205 losses is proportional to the amount of phosphogypsum produced. Transportation Penalty Base case: 66% BPL rock (dry basis) Freight cost per BPL ton = $2.5/0.66 = $3.79 12.50 3.79 BL /100 per BPL ton S2.50 B L/100 B3.7erton - 3.7 B- per ton 9)100 Where, BL = %BPL when grade < 66% Acidulation Penalty Base case: 66% BPL rock (30.21% P205, CaO:P205 = 1.49) SB Acid insol =100 1 BL 73.33 Calculation of the amount of Phosphogypsum: Phosphogypsum components = 1 ton rock x( B L i'73.33J Penalty: Transportation penalty = Acid insol Unreacted = 1 ton rock x L x 0.06 73.33) (B /100) 0 Dihydrate = 1 ton rock x L ) x 1.49 x (172/56) x (- 0.06) 2.184 Total amount of phosphogypsum = Acid insol + Unreacted + Dihydrate (% so lub le PO, losses)/100 Soluble P205 losses = $300.0 x (% soper ton 2.184 = $1.37 per ton (Total amount of phosphogypsum) Acidulation Penalty = $ 62.0 x -1.37 BL Sales value = Price of 66 %BPL rock (BL/66)15 Adjusted sales value = Sales value Transportation penalty Acidulation penalty The adjusted value of the phosphate rock as a function of %BPL is shown in Figure 4.1. Let Feed solid flow rate = F, ton per year Product solid flow rate = P, ton per year Feed grade = Gf, % Concentrate grade = G, % Product recovery = R, % Adjusted sales value of feed = Cf, $ per ton Adjusted sales value of product = Cp, $ per ton Reagent-i price = Cri, $/lb Reagent-i usage = Ui lb/ton feed The feed flow rate and the product flow rate can be related as: P=F (Gf OOJ0 (4.4) 1010) G 40 20 0 -20 -40 -60 -80 BL (%BPL) Figure 4.1: Value of phosphate rock as a function of%BPL 0 4-* 0 0 (0 *o 0i **- w, 13 Performance measure = CpP CrF F UjCn, /year (4.5) 4.2 The Optimization Algorithm The idea behind the sequential optimization is to iterate between experimentation towards the optimum and model identification until the optimum is reached. The procedure is as follows: (1) Initial experiments are performed and their results are analyzed. (2) The neural networks are trained and the hybrid model is used to determine the optimal factor values. If these are within the convergence limit of previous experimental values, the procedure stops. (3) Otherwise, an experiment at the calculated optimal value is performed and analyzed. (4) The data are added to the neural network training set, and the procedure returns to step (2). Figure 4.2 shows a more detailed description of the algorithm. After some initialization runs have been completed, the samples are analyzed and the neural networks are trained with the input-output data. Subsequently, using the standard Nelder- Meade algorithm (Himmelblau, 1972), the values of manipulated variables that maximize the selectivity are determined. If these values correspond to an interior point then the 80 START Initial runs ICP analysis Performance measure Train neural network Determine position of maximum YES I Experimental run at predicted maximum I -- Is maximum in interior of range? ICP analysis NO Experimental run at half-way point -4 I Performance measure Train neural network Determine position of maximum Difference between two consecutive maximum less than pre-decided limits ? Figure 4.2: The run-to-run optimization algorithm NO E YES STOP ih I next run will take place at these manipulated variable values; if on the other hand, maximum selectivity is at an exterior point, the next run will be performed at the midpoint between the last run and the predicted optimum values. After completion of the next run, samples are measured for grade and recovery. The new data are subsequently added to the training database and the neural network is retrained. The Optimization algorithm is again used to calculate the new optimal values. These guide the next run, and so on, until convergence is obtained. The Nelder-Meade method (nonlinear Simplex) can be used to determine the value of the manipulated variables at optimal performance. For three manipulated variables, an initial simplex is defined with four points. This method then takes a series of steps, moving the point of the simplex where the function is lowest through the opposite faces of the simplex to a higher point. These steps are called reflections, and they are constructed to conserve the volume of the simplex. The method expands the simplex in one direction to take larger steps. When it reaches a lower point, it contracts in the traverse direction. This is continued till the decrease in the function value (selectivity) is smaller than some tolerance (1E-3). 4.3 Initial Scattered Experiments Scattered experiments according to a factorial design were performed to generate data for the initial training of the neural networks. Superficial air velocity, brother concentration, and elutriation water flow rate were selected as the manipulated variables. F-507, which is a non-ionic surfactant, was used as the brother in these experiments. Experiments were performed with five different levels of superficial air velocity (0.24, 0.42, 0.60, 0.78, and 0.96 cm/s) and brother concentration (5, 10, 15, 20, 15 ppm), and three levels ofelutriation water flow rate (9, 10, and 11 gallons per min.). The design of experiments is shown in Table 4.1. The experiments were designed so as to generate 13 data points, which is the minimum required for training the neural networks, which have three hidden nodes. Experiments were performed according to the design while keeping all other variables constant. After each experiment, three samples from the tailings stream were collected. The samples were then analyzed for %BPL content following the procedure recommended by the Association of Florida Phosphate Chemists (AFPC Analytical Methods, 1980). Since the grade of the feed is known, grade of the concentrate stream can easily be calculated by making a material balance around the column. 4.4 Results and Discussions The three neural networks of the hybrid model were trained using 13 data points obtained from the designed experiments. The performance of these neural networks is shown in Figures 4.3- 4.5. Figure 4.3 presents the predicted flotation rate constants for phosphate (kp) against those determined from one-dimensional searches using experimental data as described in chapter two and three. As shown in this figure, the neural network satisfactorily captures the dependence of the flotation rate constant on the selected manipulated variables. Similarly, Figure 4.4 presents the predicted flotation rate constants for gangue (kg) against those determined from one-dimensional searches using Table 4.1: Operating conditions Frother Superficial air Elutriation water Concentration velocity Flow rate 1 -1 -1 -1 2 -1 +1 -1 3 +1 -1 -1 4 +1 +1 +1 5 -1 0 +1 6 +1 0 -1 7 0 -1 +1 8 0 +1 +1 9 0 0 0 10 0 +0.5 0 11 +0.5 0 0 12 0 -0.5 0 13 -0.5 0 0 .1 __________________________________ for the factorial design 0.9 0 0.8 0 o "r o9 0.7 0.6 0 a. 0.5 0o M 0.4 - o j. 0.3 - S0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 Experimental flotation rate constants for phosphate (kp) Figure 4.3: Neural network versus experimental flotation rate constant for phosphate (kp) 0.1 S 0.09 (n 0.08 8 0.07 ". 0.06 o 0.05 t "" 0.03 0.02 S 0.01 - 0 0 0.02 0.04 0.06 0.08 0.1 Experimental flotation rate constants for gangue (kg) Figure 4.4: Neural network versus experimental flotation rate constant for gangue (k) experimental data. Again, a very good match is seen Figure 4.5 compares the predicted air holdup to the experimental values measured by a differential pressure cell. As shown in this figure, neural network successfully predicts the air holdup. Table 4.2 shows the results of the 13 designed experiments. As can be seen from this Table, feed flow rate and the %solids content varied significantly. The screw feeder operation was erratic and therefore we were unable to feed at the same rate in each run. Feed flow rate was calculated based on the product flow rate and the tailings flow rate. A specified volume of product and tailings were taken over a period of time (-20 s) and the samples were dried and the weight was taken. In this way, solids flow rate in product and tailings stream were obtained. An overall material balance on the column then gives the feed flow rate. Similarly, an overall material balance on the water phase gives the water flow rate in the feed stream. Solids feed flow rate and the water flow rate in the feed then can be used to obtain the % solids in the feed. Unfortunately, the inability to control feed flow rate and % solids content means that a meaningful run-to-run optimization cannot be conducted. 4.5 Future Work First, the screw feeder needs to be repaired or replaced. After this has been accomplished, the hybrid model obtained from the designed experiments (Figures 4.3- 4.5) should be used with the Nelder-Meade algorithm to determine the experimental conditions of the first optimization run. The results of the run should be analyzed for grade and recovery and these data should be added to the neural network training sets. The networks should then be retrained and the updated hybrid model used to determine 30 28 26 24 0. -a 22 0 I 20 5 18 6 16 14 12 10 10 15 20 25 30 Experimental Air Holdup (%) Figure 4.5: Model versus experimental air holdup for brother F-507 Table 4.2: Results of the runs from the factorial design Frother Air Feed Tailings E'utria- Solids Grade Recovery conc. flow flow feed tion content (%BPL) (%) (ppm) rate rate flow flow (%) (scfm) (gpm) rate rate (gpm) (gpm) 1 5 0.0928 0.198 2.014 2.410 59.37 55.95 68.31 2 5 0.3711 0.418 1.779 2.351 35.11 55.36 40.04 3 25 0.0928 0.126 1.432 2.423 48.53 40.07 34.60 4 25 0.3711 0.284 2.062 2.919 35.03 61.86 45.90 5 5 0.2319 0.376 1.897 2 893 49.75 51.04 67.72 6 25 0.2319 0.284 1.650 2378 47.02 39.59 55.92 7 15 0.0928 0.264 2.355 2.922 36.43 62.68 47.49 8 10 0.3711 0340 2.275 2.927 41.65 53.99 48.53 9 15 0.2319 0.463 2.173 2.619 38.58 45.63 16.87 10 15 0.3015 0.370 1.838 2.645 42.65 46.26 54.10 11 20 0.2319 0.261 1.694 2.661 49.18 37.50 55.19 12 15 0.1624 0.281 1.853 2.634 42.36 68.40 47.92 13 10 0.2319 0.259 1.758 2.631 41.13 68.03 52.14 89 the conditions for the next run. This should be repeated with the algorithm of Figure 4.2 until convergence is obtained. APPENDIX A CODE FOR THE FIRST PRINCIPLES MODEL FOR ONE LEVEL #include #include #include #include #define GQT 2.5372 #define GQF 0.6133 #define GQE 3.038 #define CS 66.0 #define BPL 24.9 #define ROS 2.6 #define DP 122.5 #define Eg 0.05762 #define Dia 0.5 #defineL 6.0 #define Ku 2.217556 #define KGu 0.999965 #define Qg 0.1778 #define FNF 1.0 #define LfL-FNF //Tailings Flow rate (gallons/min)// //Feed Flow rate (gallons/min)// //Elutriation Flow rate (gallons/min)// //% Solid in the feed (lb S/lb T)// //% BPL of the feed (lb P/lb S) // //Specific Gravity of solids in the feed// //Particle size in microns // //air hold up // //Diameter of the column (ft.)// //Height of the column (ft.)// //Flotation rate const. for Phosphate (1/min.)// //Flotation rate const. for Gaunge (1/min.)// //Air flow rate(scfm)// //Feed Location from the top (ft.)// void main 1 (double, double, double[]); void main l(double CF, double k, double B[]) { double QF,QE,QT ,QT,QP,Area,UP,UT,UF,D,DP ,PHIS,USLi,REP,USL,diff, double a,b,d,alpha,beta,gamma,delta,p,q,m; QF=0.1336541 *GQF; QE=0.1336541*GQE; QT1=(0.1336541*GQT); QP=QF-QT1+QE; QT=QT1-QE; Area=0.7853981 *Dia*Dia; UP=QP/Area; UT=QT/Area; UF=QF/Area; |

Full Text |

17
Ca3(P04)2 in lOOg of sample. Grade can be calculated as the ratio of the weight of phosphate to the sum of the weight of the phosphate and gangue in the concentrate stream: f QfC{ -j Q. + Ac(1-6,)U. CJ r p2 \ z =0 v(QfCj-[Qt+Ac(l-6g)U]C;2 )+(QfCj - z =0 *s [Qt+Ac(l-sg)U,]CJg2 z =0 ^ y (2.25) where is the gangue concentration of the j'h particle size and Ci is the gangue feed 52 xg concentration of jlh particle size The multiplication factor is 73.3 instead of 100, because pure Florida phosphate rock measures at about 73.3 %BPL. 2.2.3 Model Parameters The above model formulation has only two model parameters, namely, the flotation rate constants for phosphate and gangue. The experimental analysis in the industry is usually available in terms of grade and recovery of phosphate. The recovery of gangue can then be readily calculated from grade and recovery of phosphate using the following relationship: Rj _RJpGf(73 3~GJ) GJ(73.3-Gi) (2.26) where GJf is the grade of the feed material. Predictedl flotation rate constants for gangue (kg) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Experimental flotation rate constants for gangue (kg) Figure 2.7 Performance of NNII: Model versus experimental flotation rate constant for gangue (kg) 82 Experiments were performed with five different levels of superficial air velocity (0.24, 0.42, 0.60, 0.78, and 0.96 cm/s) and frother concentration (5, 10, 15, 20, 15 ppm), and three levels of elutriation water flow rate (9, 10, and 11 gallons per min.). The design of experiments is shown in Table 4.1 The experiments w'ere designed so as to generate 13 data points, which is the minimum required for training the neural networks, which have three hidden nodes. Experiments were performed according to the design while keeping all other variables constant. After each experiment, three samples from the tailings stream were collected. The samples were then analyzed for %BPL content following the procedure recommended by the Association of Florida Phosphate Chemists (AFPC Analytical Methods, 1980). Since the grade of the feed is known, grade of the concentrate stream can easily be calculated by making a material balance around the column. 4.4 Results and Discussions The three neural networks of the hybrid model were trained using 13 data points obtained from the designed experiments. The performance of these neural networks is shown in Figures 4.3- 4.5. Figure 4.3 presents the predicted flotation rate constants for phosphate (kp) against those determined from one-dimensional searches using experimental data as described in chapter two and three. As shown in this figure, the neural network satisfactorily captures the dependence of the flotation rate constant on the selected manipulated variables. Similarly, Figure 4.4 presents the predicted flotation rate constants for gangue (kg) against those determined from one-dimensional searches using This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. May 1999 Winfred M. Phillips Dean, College of Engineering M. J. Ohanian Dean, Graduate School 107 { /y******************** Steady sate Rec0very calculation******************* Pe=(((QT/Area)+USL)*(L))/(D*(l-Eg)); //printf("\n Pe=%lf',Pe); Tp=((L)*(l-Eg))/((QT/Area)+USL); //printf("\n Tp=%lf',Tp); a=sqrt( 1 +(4*K*Tp/Pe)); //printf("\n a=%lf',a); //printfC'Xn K*Tp=%lf',K*Tp); bl=exp(a*Pe/2); b2=exp(-a*Pe/2); REQNP=( 1 -((4*a*exp(Pe/2))/((( 1 +a)*( 1 +a)*b 1)-((1 -a)*( 1 -a)*b2))))* 100; R=( 1 -((1 -(REQNP/100)) ((QT+Area U SL)/QF)))*100; c=sqrt( 1 +(4*KG*Tp/Pe)); dl=exp(c*Pe/2); d2=exp(-c*Pe/2); REQNG=(l-((4*c*exp(Pe/2))/(((l+c)*(l+c)*dl)-((l-c)*(l-c)*d2))))*100 RG=( 1 -((1 -(REQNG/100))*((QT+Area*SL)/QF)))* 100; //printfT\n REQNP=%. llP/o",REQNP); printf"\n Recovery=%. 1 lfyo",R); //printf("\n REQNG=%lf%",REQNG); Gradeqn=((QF*CF-(QT+Area*USL)*( l -(REQNP/100))*CF)/((QF*CF- (QT+Area*USL)*( 1 -(REQNP/100))*CF)+(QF*CFG-(QT+Area*USL)*( 1 - (REQNG/100))*CFG)))* 100.0; Selecteqn=R-RG; printf("\n GRADEQN=%. 1 lf %",Gradeqn); printf("\n Sep_eT=%. 1 lf ",Selecteqn); getch(); } //*********Steady state Recovery calculation ends *********** } 10 bio-kinetic rates of a first-principles model. Cubillo and Lima (1997) also used this approach to develop hybrid model for a rougher flotation circuit. In this work, we employ a serial approach to integrate an approximate model, derived from first-principles considerations, with neural networks which approximates the unknown kinetics. The first-principles model is inverted to calculate two model parameters for each set of measured recovery and grade. The neural networks are then trained on the errors of calculated model parameters instead of the errors of the output of the first-principles model as is the case with the above referenced works. Also, unlike most other cited work, we employ experimental data instead of simulated data. 2 2 First-Principles Model The basic equations representing the flotation of solid particles in a flotation column can be written by making a material balance for the solid particles in the slurry phase. This results in the following partial differential equations for the section above and below the feed point, respectively: cC j f Pi ct cCJ Pi ct Up -us] J dCl Pi + D 02CJ Pi l-*t dz dz2 u > dCi d2C] + UJ, P2 -+D Pi 10. dz dz (2.1) (2.2) 92 else p=((-(-beta+a))*exp((beta)*(L)))/((-alpha-*-a)*exp((alpha)*(L))); q=-(delta-(QE/(Area*D)))/(gamma-(QE/(Area*D))), m=(q*exp((gamma)*(Li))+exp((delta)*(Lf)))/(p*exp((alpha)*(Lf))+exp((beta)*(Lf))); B[4]-(UF*CF/D)/((m*p*(a- alpha)*exp((alpha)*(Lf)))+((d+gamma)*q*exp((gamma)*(Lf)))+(m*(a- beta)*exp((beta)*(Lf)))+((d+delta)*exp((delta)*(Lf)))); //B[3]=(UF*CF/D)/((a+alpha+d+gamma)*exp((gamma)*(Lf))); B[2]=m*B[4]; B[l]=p*m*B[4]; B[3]=q*B[4]; //B[l]=(exp((gamma-alpha)*Lf))*B[3]; //B[2]=0.0; //3[4]=0.0; } void main(void) { double QF,QE,QT 1 ,QT,QP,Area,DP 1 ,PfflS,USLi, REP, USL,diff, Frank, CF, CFG, K, KG, C,CG; double RO,ROG,Grade,B[5],BG[5]; QF=0.1336541 *GQF; QE=0.1336541 *GQE; QT1 =(0.1336541 *GQT); QP-QF-QT1+QE; QT=QT1-QE; Area-0.7853981 *Dia*Dia, printf("\n UP=%lf',QP/Area); //************** Calculation of slip velocity ************** DPI-DP/1000.0; PHIS=(CS/100)/((CS/100)+((1-(CS/100))*ROS)), USLi-0.0; do { REP-5.12*DPl*USLi*ROS*(l. 0-PHIS); USL=108.233*DP1*DP1 *(ROS-l)*pow((l-PHIS),2.7)/(l+0.15*(pow(REP,0.687))); diflf-USL-USLi, if(difF<0.0) 2 achieve a neutral pH. The scrubbed and washed material is then subjected to cleaner flotation in which amine together with kerosene is used as collector to float sand. This stage of flotation is sensitive to impurities in water; thus, fresh water is used in most of the plants as make up water. However, the fatty acid circuit uses recycled water. This process has become less cost effective due to high cost of reagents and increasing concentration of contaminants. To prepare the phosphate feed, the mined phosphate ore (matrix) is washed and de-slimed at 150 mesh. The material finer than 150 mesh is pumped to clay settling ponds. The rock coarser than 150 mesh is screened to separate pebbles (-3/4 +14 mesh) which are of high phosphate content. Washed rock (-14, +150 mesh) is sized into a fine (usually 35 x 150 mesh) and a coarse flotation feeds (usually 14 x 35 mesh) which are treated in separate circuits. Flotation of phosphates from the fine feed (35 X 150 mesh) presents very few difficulties and recoveries in excess of 90% are achieved using conventional flotation cells. On the other hand, recovery of phosphate values from the coarse feed is much more difficult and flotation by itself usually yields recovery of 60% or less. The density of the solid, turbulence, stability and height of the froth layer, depth of the water column, viscosity of the froth layer are known to effect the flotation process in general (Boutin and Wheeler, 1967). However, the exact reasons for low recovery of coarse particles in conventional flotation is not very well understood. There are several hypotheses about the flotation behavior of coarse particles. For instance, the floatability of large particles could be due to the additional weight that has to be lifted to the surface 43 networks (i.e determined the neural network parameters) on the error of the output of the first-principles model. A similar approach was followed by Reuter et al. (1993) to model metallurgy and mineral processes. Liu et al. (1995) developed a hybrid model for a periodic wastewater treatment process by using ANNs for the bio-kinetic rates of a first- principles model. The Psichogios and Ungar (1992a, 1992b) approach was used by Cubillo et al. (1996) to model particulate drying processes, and by Cubillo and Lima (1997) to develop a hybrid model for a rougher flotation circuit. Thompson and Kramer (1994) combined the parallel and serial hybrid modeling approaches. As in the Psichogios and Ungar (1992a, 1992b) approach, the hybrid model presented here uses backpropagation ANNs for certain parameters of a FPM. However, instead of training these ANNs on the errors of the measured outputs of the FPM (grade and recovery), it inverts the FPM for each set of measurements to calculate corresponding parameter values, and trains the ANNs on the errors of the calculated parameter values. Another innovation of the present hybrid model is that it involves two levels of neural networks. This structure has the advantage that if certain factors that affect the process like the type of frother or air sparger used are changed, only the top level neural networks need to be retrained. These only require experimental data that can be easily obtained with short experiments that do not involve rock, and the large database of past grades and recoveries is still valid and does not need to be replaced. Finally, in contrast to the above referenced works, the hybrid model presented here is developed with experimental data instead of simulated data 80 70 60 50 40 30 20 10 Experimental Grade (%BPL) gure 3 14: Performance of the overall hybrid model: Predicted versus experimental grade BPL) for the four frothers 25 Figure 2.5 A schematic diagram of the experimental setup 52 3.4 The Hybrid Model The main factors affecting the air hold up eg are the superficial air velocity Jg and the frother concentration Cfrother- Several factors affect the flotation rate constants, kp and kg, including particle diameter, superficial air velocity, frother concentration, collector concentration, extender concentration, and pH. In this study we have conducted experiments varying particle size, frother type, frother concentration, and superficial air velocity, and develop a hybrid model that portrays the effect of these factors on the performance of the column. The hybrid model utilizes backpropagation ANNs (Rumelhart and McClelland 1986) to predict the values of the parameters sg, kp, and kg. The straightforward approach is to develop an ANN for each of the three parameters. The inputs to the ANNs that predict kp and kg would be dp, Jg, and Cfrother, while the inputs of the ANN that predicts eg would be Jg, and Cfrother. Each of the ANNs in this structure would then depend on the frother and sparger used. A change in type of frother would mean that the previously trained ANNs are no longer applicable and would necessitate collection of a new set of training data and retraining of the networks. As changes in frother or sparger are not uncommon, this is a disadvantage. The main reason Jg and Cfrother, as well as the type of frother and sparger, affect the flotation rate constants, is because they significantly affect the bubble size. An alternative hybrid model architecture is shown in Figure 3.2. The neural networks are structured in two levels. The first level consists of the ANNs for predicting kp (NN1) and 2 13 Performance of the overall hybrid model: Predicted versus experimental recovery (%) for unsized feed size distribution 37 2.14 Performance of the overall hybrid model: Predicted versus experimental grade (%BPL) for unsized feed size distribution 38 3.1 Schematic diagram of column for phosphate flotation 45 3.2 Overall structure of the hybrid model 53 3.3 Performance ofNNIII: Model bubble diameter versus bubble diameter inferred from experimental data when CP-100 was the frother 59 3.4 Performance ofNNIII: Model bubble diameter versus bubble diameter inferred from experimental data when F-507 was the frother 60 3 5 Performance ofNNIII: Model bubble diameter versus bubble diameter inferred from experimental data when OB-535 was the frother 61 3.6 Performance ofNNIII: Model bubble diameter versus bubble diameter inferred from experimental data when F-579 was the frother 62 3.7 Performance of NNIV: Model versus experimental air holdup for frother CP-100 63 3 8 Performance of NNIV Model versus experimental air holdup for frother F-507 64 3.9 Performance of NNIV: Model versus experimental air holdup for frother OB-535 65 3.10 Performance of NNIV: Model versus experimental air holdup for frother F-579 66 3.11 Performance of NNI: Model versus experimental flotation rate constant for phosphate (kp) 67 3.12 Performance ofNNII Model versus experimental flotation rate constant for gangue (kg) 69 3.13 Performance of the overall hybrid model Predicted versus experimental recovery (%) for the four frothers 70 vii 81 next run will take place at these manipulated variable values; if on the other hand, maximum selectivity is at an exterior point, the next run will be performed at the midpoint between the last run and the predicted optimum values. After completion of the next iun, samples are measured for grade and recovery. The new data are subsequently added to the training database and the neural network is retrained. The Optimization algorithm is again used to calculate the new optimal values. These guide the next run, and so on, until convergence is obtained. The Nelder-Meade method (nonlinear Simplex) can be used to determine the value of the manipulated variables at optimal performance. For three manipulated variables, an initial simplex is defined with four points. This method then takes a series of steps, moving the point of the simplex where the function is lowest through the opposite faces of the simplex to a higher point. These steps are called reflections, and they are constructed to conserve the volume of the simplex. The method expands the simplex in one direction to take larger steps. When it reaches a lower point, it contracts in the traverse direction. This is continued till the decrease in the function value (selectivity) is smaller than some tolerance (1H-3). 4 3 Initial Scattered Experiments Scattered experiments according to a factorial design were performed to generate data for the initial training of the neural networks. Superficial air velocity, frother concentration, and elutriation water flow rate were selected as the manipulated variables. F-507, which is a non-ionic surfactant, was used as the frother in these experiments. 46 The particles are subdivided into size ranges according to the standard Tyler mesh screens. Panicles of a cenain mesh are considered to have diameter the geometric mean of the lower and upper limits. As the attachment rate constants and particle slip velocities depend on particle size, a separate material balance is written for each mesh size. Material balances at each layer yield the following equations for the phosphate particles: Laver 1 (top) dC Pi dt U. V1-8g f -U, rj _pj rJ _rJ + D -Pi -kp(dMCj Az Az- p p D pi TJJ _ 'si u. V 1 -e C pi Az + D CJ -C P2 Pi Az ~kn(do)C p v p / p, u if u <-ll 1-6, if u> u 1-8, Laver 2 to k-1: k = feed layer (3.1) d_^ dt U, ^-88 ui CJ -CJ CJ -2CJ + CJ ^ + D ^-kp(dp)C( r U l u. Az Az 1-E 8 Cl -C' Cl -2CI +CI -P-l+D p"' -k_(dJp)C Az Az2 p p p' UD 1-6, Â¡fu L> u. 1-6, Feed Laver = k (3.2) dCJ Pk (Qf / Ao)ci (-Li- U )C (-- + uj, )c; 1-E, Pk 1-6, Az pk CJ 2CJ + CJ + D Az -MWk if U < u. 1 6, dt (Qf/A.)CUu--)CL1-(^- + U)C|k c, _ Az -+D Pk+1 2CJ + CJ Pk Pk-1 Az" if ui > LA 1 e. (3.3) 3 under the heavy turbulence conditions, and the difficulty to transfer and maintain these panicles in the froth layer. Some efforts towards improving the flotation of coarse particles through stabilization of the froth layer, minimizing the froth height, and addition of an elutriation water stream at the bottom of the column have been undertaken The equipment used by the phosphate industry in flotation process are not selective enough to take full advantage of new reagents and operating schemes, to recover phosphate from the coarse feed or to optimize results with existing reagents. The best way to increase the selectivity of phosphate flotation is to improve upon the design of flotation equipment. Particularly the new equipment should improve the recovery of coarse particles, while still providing the high selectivity of fine particles. It has been found both theoretically and practically that flotation columns have better separation performance than conventional mechanical cells (Finch and Dobby, 1990). The use of flotation columns can not only help overcome some of the problems related to coarse phosphate flotation but it has several other advantages as mentioned above. Spargers or bubble generating systems are the single most important element in the flotation columns. They are generally characterized in terms of their air dispersion ability. Frothers are the chemicals that help in controlling and stabilizing bubble size by reduction of surface tension. Thus both of them play an important role in the overall performance of flotation columns. Their interaction can be a crucial factor in the success of flotation column. Flotation columns have been used predominantly in the coal beneficiation industry. However, their application in other mineral industries, such as the phosphate, is TABLE OF CONTENTS pane ACKNOWLEDGMENTS iii LIST OF FIGURES vi ABSTRACT ix CHAPTERS 1 INTRODUCTION 1 2 ONE-LEVEL HYBRID MODEL 6 2.1 Introduction 6 2.2First-Principles Model 10 2.2.1 Boundary Conditions 13 2.2.2 Calculation of Recovery and Grade 16 2.2.3 Model Parameters 17 2.3 The Hybrid Model 22 2.4 Materials and Methods 23 2.4.1 Experimental Setup and Procedure 23 2.4.2 Experimental Conditions 27 2.4.3 Neural Network and Training 27 2.5 Results and Discussions 28 2.6 Conclusions 39 3 TWO-LEVEL HYBRID MODEL 40 3.1 Introduction 40 3.2 First-Principles Model 44 3.3 Calculation of Model Parameters 50 3.4 The Hybrid Model 52 3.5 Materials and Methods 54 3.5.1 Experimental Setup and Procedure 54 3.5.2 Neural Network and Training 56 3.6 Results and Discussions 58 3.7 Conclusions 72 IV Experimental Recovery (%) Figure 3.13: Performance of the overall hybrid model: Predicted versus experimental recovery (%) for the four frothers 94 #defme #defme #define #define #define ^define #define #define #define #defme ^define ^define #defme GQE 2.6948 //Elutriation Flow rate (gallons/min)// CS 66.0 //% Solid in the feed (lb S/lb T)// BPL 24.9 //% BPL of the feed (lb P/lb S) // ROS 2.6 //Specific Gravity of solids in the feed1! DP 122.5 //Particle size in microns // Eg 0.115 //air hold up // Dia 0.5 //Diameter of the column (ft.)// L 6.0 //Height of the column (ft.)// KGu 0.0 //Flotation rate const, for Gaunge (1/min.)// Qg 0 1778 //Air flow rate(scfm)// FNF 1.0 LfL-FNF ES IE-3 //Feed Location from the top (ft.)// void main l(double,double,double[]); double model(double); void main(void) { double kl,yl,gl,ku,yu,gu,kr[3],yr,gr,R[3],a,EA,Test,x; double Grade,Grade_feed,Grade_prod; int i; R[l]=70.915; Grade=25.8; Grade_feed=BPL/73.3; Grade_prod=Grade/7 3.3; R[2]=R[l]*Grade_feed*(l-Grade_prod)/((l-Grade_feed)*Grade_prod); printrV'\nComponent[ 1 ]-phosphate"); printf("\nComponent[2]=gangue"); for(i=l;i<=2;i-H-) { kl=0.0; yl=model(kl); gl=R[i]-yl; if(gl>0.0) { do { printf"\nEnter an initial guess for flotation rate constant for component[%d]",i); printfi"\nku="); scanif"\n %lf', &ku); 5 Two hybrid modeling approaches are presented. Chapter 2 describes a one-level hybrid model that uses three different neural networks to predict the flotation rate constant for phosphate, the flotation rate constant for gangue, and air holdup Chapter 3 presents a two-level hybrid model in which neural networks are structured in two levels. Two neural networks are used in the top-level to predict bubble diameter and air holdup. The bubble diameter is used as an input in the neural networks of the bottom-level which predict the flotation rate constants for phosphate and gangue. The inherent advantages and disadvantages of the two hybrid modeling approaches are also discussed in these chapters. In chapter 4, the hybrid model developed is combined with an on-line optimization algorithm to determine the optimal conditions for column operation. The algorithm guides successive changes of the manipulated variables such as air flow rate, frother concentration, and pH, after each run to achieve optimal column operating conditions. Designed experiments were performed to generate data for the initial training of the neural networks. The trained neural network is then used to guide the direction of the new experiments. Experimental Recovery (%) Figure 2.13: Performance of the overall hybrid model: Predicted versus experimental recovery (%) for the unsized feed after it has been sized. 68 Figure 3.12 presents the flotation rate constants for gangue (kg) predicted using NNII against those determined from experimental data. A very good match is seen. The hybrid model integrates NN1, NNII, NNIII, and NNIV with the FPM as shown in Figure 3.2. Predictions of the hybrid model are shown in Figures 3 13 and 3.14. Figure 3.13 presents the predicted recovery (%) against the experimental recovery for frother CP-100 (square points), F-507 (circles), OB-535 (triangles), and F-579 (diamonds). Similarly, Figure 3.14 compares the predicted grade (%BPL) against the experimental grade for CP-100, F-507, OB-535, and F-579 It can be seen from these figures that predicted recovery and grade from the hybrid model match closely the experimental values, with the exception of one grade for OB-535. The root mean squared errors in predicted recovery were 0.1%, 0.2%, 1.5%, and 0.4% for CP-100, F-507, OB-535, and F- 579, respectively. The root mean squared errors in predicted grade were 3.2 %BPL, 1.5 %BPL, 7.5 %BPL, and 1.5 %BPL for CP-100, F-507, OB-535, and F-579, respectively. An alternative to the present modeling approach is to develop a pure neural- networks model. This would, however, require a large number of inputs: not only superficial air velocity, frother concentration, and particle size, but also feed flow rate, feed concentration, elutriation flow rate, tailings flow rate, and solids loading. This increase in number of inputs to eight would increase the number of weights (model parameters) needed and therefore the number of three-phase data required for training. Furthermore, as with an in-series hybrid model that uses one level of neural networks, a change in frother or sparger would require generation of a new set of data and retraining of all the networks. The hybrid model presented here with the two levels of neural 30 28 26 24 22 20 18 16 14 12 10 Figure 4.5: Model versus experimental air holdup for frother F-507 APPENDIX B CODE FOR THE FIRST PRINCIPLES MODEL FOR TWO LEVELS #include #include #include #include #define #define #define #define #define #define #define #define #define #define #define #define #defme #define ^define #define #define #define GQT 1.5249 //Tailings Flow rate (gallons/min)// GQF 0.8858 //Feed Flow rate (gallons/min)// GQE 2.5102 //Elutriation Flow rate (gallons/min)// CS 67 2 //% Solid in the feed (lb S/lb T)// BPL 18.2 //% BPL of the feed (lb P/lb S) // ROS 2.6 //Specific Gravity of solids in the feedII DP 208.25 //Particle size in microns // Eg 0.1487 //air hold up // Dia 0.5 //Diameter of the column (ft.)// L 6.0 //Height of the column (ft.)// Ku 6.042797 //Flotation rate const, for Phosphate (1/min.)// KGu 0.048094 //Flotation rate const, for Gaunge (1/min.)// Qg 0.2706 //Air flow rate(scfm)// N 15 FNF 1.0 //Feed Location from the top (ft.)// DELT 0.1 n N+l al 1.0 //a 1 =0>explicit;a 1 = 1 >implicit// void main 1 (double,double,double,doublet]); void mainl(double CF,double K,double D,double C[]) { static double A[n][n],V[n],S[n]; double A1,A2,A3,A4,A5,A6,A7,A8,A9,UP,UT,UT1,UF,DELZ,QP,QF,QT,QE,QT1; double A10,A11,A12,A13,A14, int 0[n]; int i,j,k,ii,Pivot,IDummy,NF; double Big, Dummy,factor,Sum, Area,USLi,USL,REP,diff,PLUS,DPI; QF=0.1336541*GQF; QE=0.1336541 *GQE; 99 95 yu=model(ku); gu=R[i]-yu; a=gl*gu; } while(a>=0); EA=1.1*ES; while(EA>ES) { kr[i]=ku-(gu*(kl-ku))/(gl-gu); x=kr[i]; if((kl+ku)!=0) EA=fabs((ku-kl)/(kl+ku))* 100; yr=model(x); gr=R[i]-yr; Test=gl*gr, if(Test=0.0) EA=0; else if(Test<0.0) ku=kr[i]; else if(Test>0.0) kl=kr[i]; printfT,\nkr[%d]=%lf,.i,kr[i]); } printf("\nFlotation rate constant for component[%d]=%lf',i,kr[i]); getchO; } else printfT\nkr[%d]=0.0",i); } } void main 1 (double CF, double k, double B[]) { double QF,QE,QT1,QT,QP,Area,UP,UT,UF,D,DPI,PHIS,USLi,REP,USL,difF; double a,b,d,alpha,beta,gamma,delta,p,q,m; QF=0.1336541*GQF; QE=0.1336541 *GQE; QT1=(0.1336541 *GQT); 41 In anionic phosphate flotation the mineral is first treated with fatty acid collector and fuel oil extender At proper concentrations these mostly adsorb on the phosphate-containing particles rendering them hydrophobic. Then the phosphate-containing particles are separated from gangue via the flotation process. The majority of the phosphate plants employ mechanical cells. However, column flotation has simpler operation and provides superior grade/recovery performance. For these reasons column flotation is gaining increasing acceptance for the processing and beneficiation of phosphate ores. Column flotation is frequently employed for the recovery of other minerals (e.g., coal, copper, nickel, gold). In such applications the column can be divided into three zones: an upper froth zone, a lower collection zone, and an intermediate interface zone. An additional wash water stream is usually added from the top of the column. Phosphate flotation deals with considerably larger particles of size 0.1-1 mm. As a result, instead of wash water from the top, elutriation water from the bottom is added. Furthermore, columns are typically operated with negligible froth and interface zones. This considerably simplifies the modeling effort, as the only the collection zone needs to be accounted for. Particle transport in the collection zone is usually modeled as axial convection coupled with axial dispersion. The Peclet number (Pe), or its inverse, the dispersion number, governs the degree of mixing. Most models only consider the slurry phase (Finch and Dobby, 1990; Luttrell and Yoon, 1993), in which case particle collection is viewed as a first order net attachment rate process. A model that considers both slurry and air phase was developed by Sastry and Loftus (1988). In this case particle Experimental Recovery (%) Figure 2 9 Performance of the overall hybrid model: Predicted versus experimental recovery (%) for coarse feed size distribution 102 A[NF][i]=0.0; //**************Defmition ofRow=NF+l toN-1 ************* for(i=NF+1 ;i A[i][i-l]=-(al*DELT*A10); A[i][i]=1.0-(al*DELT*All); A[i][i+l]=-(al*DELT*A12); for(j=l;j for(i=i+2;j<=N;j++) A[i][j]=0.0; } //************Definition 0f Row=N *********************** A[N][N-1 ]=-(a 1 *DELT*A13); A[N] [N]=1 0-(a 1 *DELT*A 14); for(i=l;i //**********Row Definition ends ***************************** //************* Defifnition of column vector ******************* V[l]=(1.0+(1.0-al)*DELT*Al)*C[l]+(1.0-al)*DELT*A2*C[2]; //printf("\n V[ 1 ]=%lf \n",V[ 1 ]); //getch(); for(i=2;i al)*DELT*A5*C[i+l]; //printfT'Nn V[%d]=%lf \n",i, V[i]); //getch();} V[NF]=(DELT*A6)+(1.0+(1.0-al)*DELT*A8)*C[NF]+(1.0-al)*DELT*A7*C[NF- 1 ]+(l ,0-al)*DELT*A9*C[NF+l ]; //printT\n V[NF]=%lf \n",V[NF]); //getch(); for(i=NF+l;i al)*DELT*A12*C[i+l]; //printf("\n V[%d]=%lf\n",i,V[i]); //getch();} V[NH1.0+(1.0-al)*DELT*A14)*C[N]+(1.0-al)*DELT*A13*C[N-l]; //printf("\n V[N]=%lf \nM,V[N]); //getch(); //*********^*****Â£)efjn|tjon 0pcojurnn vector ends ********** /y**************** ORDERING ****************************** CHAPTER 2 ONE-LEVEL HYBRID MODEL Flotation is a process commonly employed for the selective separation of phosphate from unwanted mineral. Column flotation is slowly gaining popularity in the mineral processing industry, including the phosphate industry, due to its ability to improve selectivity, lower operating cost, lower capital cost, and superior control. In this work, a hybrid model is developed that combines a physicochemical model with artificial neural networks. This model for the first time incorporates the effect of collector concentration, extender concentration, and pH on the flotation performance. The physicochemical model is based on axial dispersion with first order collection rates. Three basic parameters are required in this model: flotation rate constant for phosphate, flotation rate constant for gangue, and air holdup. Artificial neural networks are used to predict these parameters. The model also takes into account the particle size distribution and predicts grade and recovery for each particle size range. The model is validated against laboratory column data. 2.1 Introduction Even though the concept of column flotation was developed (Wheeler, 1988) and patented (Boutin and Wheeler, 1967) in the early 1960s, its acceptance for the processing and beneficiation of phosphate ores is relatively recent. The majority of the phosphate plants employ mechanical cells. However, column flotation has simpler operation and 6 ACKNOWLEDGMENTS I would like to take this opportunity to thank my advisor Dr. Spyros A Svoronos for his continuing guidance, encouragement and support throughout the course of my Ph D He not only guided me to learn new techniques, he was also helpful in showing me the nght course in some of the problems in my personal life. I wish to thank Dr. Hassan El-Shall for his valuable inputs in the chemistry aspect of this project. I would also like to thank my other committee members, Dr. Richard Dickinson, Dr. Oscar Crisalle, and Dr Ben Koopman, for kindly reviewing my dissertation and serving on my committee. The friendship and assistance of my colleagues, Pi-Hsin Liu, Robert Bozic, Rajesh Sharma, Dr. Cheng, Dr Nagui, Rachel Worthen, and Lav Agarwal, will always be valued. My respect for my parents, brother, and sister for having stood by me and for giving me moral support always kept me motivated to complete this work. iii 93 difr=-diff; USLi=USL; } while(diff>=0.0001); USL=(l-Eg)*USL; printf("\n USL=%lf',USL); y/***************** USL calculation ends *************** Frank-BPL/0.733; CF-(Frank/100.0)*(CS/100.0)*ROS*62 418I8/((CS/100.0)+(1,0-(CS/100.0))*ROS); CFG=( 1 0-(Frank/100.0))*(CS/100.0)*ROS*62.41818/((CS/l 00.0)+( 1.0- (CS/100.0))*ROS); K=Ku,KG=KGu; mainl(CF,K,B); C-B[3]+B[4]; RO=(((QF*CF)-((QTl+Area*USL)*C))/(QF*CF))* 100.0; mainl(CFG,KG,BG); CG=BG[3]+BG[4], ROG=(((QF*CFG)-((QTl+Area*USL)*CG))/(QF*CFG))* 100 0; Grade=((QF*CF-(QTl+Area*USL)*C)/((QF*CF-(QTl+Area*USL)*C)+(QF*CFG- (QTl+Area*USL)*CG)))*100.0, Grade=Grade *0.733, printf("\n C=%lf',C); printTVn CG=%lf',CG); printf("\n CF=%lf',CF); printf("\n CFG=%lf',CFG); printf("\n Overall Recovery=%. llf %",RO); printfT"\n ROG=%. llf %",ROG); printt("\n GRADE=%. llf %",Grade), } //include //include #include //include //define GQT 2.2551 //define GQF 0.9318 //Tailings Flow rate (gallons/min)// //Feed Flow rate (gallons/min)// 72 networks involves a relatively low number of inputs in the artificial neural networks, does not require new three-phase data if a frother or sparger is changed, and gives very good predictions of both grade and recovery 3.7 Conclusions A hybrid neural network modeling approach was presented and used to model a flotation column for phosphate/gangue separation. This hybrid model is comprised of two parts, a first-principles model and two levels of neural networks that serve as parameter predictors of difficult-to-model process parameters. Experimental data from a laboratory column were used to train and validate the neural networks, and it is shown that the hybrid model captures the dependence of column performance on particle size, frother concentration, and superficial air velocity. 4 not very weil studied. Unlike other minerals, phosphate flotation deals with a considerably larger size of particles (0.1 -1mm) and therefore the operation of phosphate flotation in a column is different from that of other minerals. High recovery and grade and low operating cost depend largely on the optimal selection of operating variables such as the air flow rate, the frother type and concentration, and the elutriation water rate The search of the optimal conditions can considerably benefit by the availability of a model that can predict the effects of different operating conditions on column behavior. Finch and Dobby (1990) and Lutrell and Yoon (1993) developed a one-phase axial dispersion model in which particle collection is viewed as a first order net attachment rate process. Sastry and Loftus (1988) considered both the slurry and air phases and they used two separate first order rate constants for attachment and detachment of the particles. However, these models cannot predict the effects of certain operating conditions such as particle size, frother concentration, collector and extender concentration, and pH on the flotation performance. In this work, a mathematical model is developed that for the first time predicts the effects of particle size, frother concentration, collector and extender concentration, and pH on the flotation behavior. This is a hybrid model that combines a first-principles model with artificial neural networks (ANNs). The first-principles model is derived by making a material balance on solid particles in the slurry phase. First order reaction rate constants are assumed for the attachment of the solid particles to the air bubbles. Single output feedforward back propagation neural netw orks are used to correlate the model parameters with the operating variables. where 11 Up Ut D Qp Q. Qe Ac kPW = Phosphate concentration ofjth mesh size particles for feed point = Phosphate concentration of jlh mesh size particles for feed point = Superficial liquid velocity above the feed point = Qp/ac = Superficial liquid velocity below the feed point = (Q, -Qe)/Ac = Dispersion coefficient = Product volumetric flow rate = Tailings volumetric flow rate = Elutriation volumetric flow rate = Cross-sectional area of the column = Slip velocity ofjlh mesh size particles = Air holdup = Flotation rate constant for phosphate for mesh size particles the section above the the section below the The following assumptions are made in deriving the above equations: 1) The concentration of solid particles in the slurry phase is a function of height, z only, and variations of the concentration in radial and angular directions can be neglected. 2) The air holdup is constant throughout the column. 3) All the air bubbles in the system are of a single size. 4) Rate of detachment is either negligible or is a function of conditions in the slurry phase. This assumption allows to treat the net attachment rate with just one floatation rate constant. Model diameter (mm) Figure 3.5: Performance of NNIII: Model bubble diameter versus bubble diameter inferred from experimental data when OB-535 was the frother 89 the conditions for the next run This should be repeated with the algorithm of Figure 4.2 until convergence is obtained Flotation rate constant for phosphate (kp) Figure 2.2: Recovery of phosphate (%) as a function of flotation rtae constant for phosphate (kp) 83 Table 4.1: Operating conditions for the factorial design Frother Concentration Superficial air velocity Elutriation water Flow rate 1 -1 -1 -1 2 -1 + 1 -1 3 + 1 -1 -1 4 + 1 + 1 + 1 5 -1 0 + 1 6 + 1 0 -1 7 0 -1 + 1 8 0 + 1 + 1 9 0 0 0 10 0 +0.5 0 11 +0.5 0 0 12 0 -0.5 0 13 -0.5 0 0 57 and outputs were scaled from 0 to 1. The hidden and output layer nodes employed logistic activation functions (Hertz et a/.. 1992). For each of the four ffothers investigated, 28 two-phase experiments were conducted (full factorial design with 7 frother concentrations and 4 superficial air velocities). These were used to train (19 data points) and to validate (9 data points) the top level neural networks (NNIII and NNIV), a different pair for each frother. Three-phase runs yielded 28 experimental grades and recoveries, which were used to train (19 data points) and to validate (9 data points) NNI and NNII. To set the number of nodes in the hidden layer of each network, the number was increased until the sum of the absolute errors of the training and validation outputs started increasing In this manner an appropriate number of hidden nodes was determined to be three for all the neural networks. The training process started by initializing all weights randomly to small non-zero values. The random numbers were generated in the range -3.4 to +3.4 with a standard deviation of 1.0 following the procedure recommended by Masters (1993). The optimal weights were determined by combining simulated annealing (Kirkpatrick et al. 1983) with the Polak-Ribiere conjugate gradient algorithm (Polak, 1971). Simulated annealing randomly perturbed the independent variables (the weights) and kept track of the best (lowest error) function value for each randomized set of variables. This was repeated several times, each time decreasing the variance of the perturbations with the previous optimum as the mean. Then the conjugate gradient algorithm was used to minimize the mean-squared output error. When the minimum was found, simulated annealing was used to attempt to break out of what may be a local minimum. This alternation was CHAPTER 4 OPTIMIZATION PERFORMANCE MEASURES AND FUTURE WORK High recovery and grade and low operating cost depend largely on the optimal selection of operating variables. The search of the optimal conditions can considerably benefit by the availability of a model that can relate the operating conditions to the column performance. The hybrid model developed for the flotation column provides a mathematical relationship between the operating variables and column performance. This hybrid model can be combined with an optimization algorithm to determine the optimal operating conditions for the flotation column. We propose an algorithm that leads to the sequential optimization of a flotation column. This algorithm guides successive changes in the manipulated variables after each experiment to achieve optimal column operating conditions. Selectivity, which combines recovery and grade, can be used as the performance measure of the column. The hybrid model builds a relationship between the process manipulated variables and the performance measure. The optimization algorithm dictates the changes in the manipulated variables between successive runs. At each run manipulated variables are set at their predicted optimal values. After the run is completed, the collected samples should be collected and analyzed for recovery and grade. Then the new input-output data are added to the neural network of the hybrid model and the network should be retrained. 73 96 QP=QF-QT1+QE; QT=QT1-QE; Area=0.7853981*Dia*Dia; UP=QP/Area; UT=QT/Area; UF=QF/Area; D=12.4*Dia*pow((0.3175*Qg/Area),0.3); //************** Calculation of slip velocity ************** DP 1=DP/1000.0; PHIS=(C S/100)/((C S/100)+(( 1 -(C S/100))*ROS)); USLi=0.0; do { REP=5.12*DP1 *USLi*ROS*(l .0-PH1S); USL= 108.233 *DP 1 *DPl*(ROS-l)*pow((l-PHIS),2.7)/( 1+0.15 *(pow(REP,0.687))); diff=USL-USLi; if(diff<0.0) difF=-diff; USLi=USL; } while(diff>=0.0001); USL=(l-Eg)*USL; //************* USL calculation ends *************** a=(UP-USL)/D; d=(UT+USL)/D; b=k*(l-Eg)/D; if^((a*a+4*b)<0.0)||((d*d+4*b)<0.0)) b=0.0; alpha=(a/2)+(sqrt(a*a+4*b))/2; beta=(a/2)-(sqrt(a*a+4*b))/2; gamma=(-d/2)+(sqrt(d*d+4*b))/2; delta=(-d/2)-(sqrt(d*d+4*b))/2; //printf^"\n alpha=%lf",alpha); //printf("\n beta=%lf\beta); //printf^"\n gamma=%lf',gamma); //prir.tf("\n delta=%lf',delta); //printfT\n beta*L=%lf',(beta)*(L)); //printf("\n alpha*L=%lf',(alpha)*(L)); //printf(\n gamma*Lf=%lf',(gamma)*(Li)); //printf("\n delta*Lf=%lf',(delta)*(Lf)); 25 20 15 10 5 0 igure 2 8: Performance of NNIII: Model versus experimental air holdup for frother CP-100 to Model Bubble Diameter (mm) Figure 3.3: Performance of NNIII: Model bubble diameter versus bubble diameter inferred from experimental data when CP-100 was the frother Superficial air velocity Frother concentration 30 collector and extender concentration, and pH. Similarly, Figure 2.7 compares flotation rate constant for gangue (kg) determined from one-dimensional searches with those predicted by NNII As shown, NNII successfully predicts the flotation rate constant for gangue. Figure 2.8 presents the air holdup (Sg) predicted using NNIII against those measured experimentally. A satisfactory match is seen. The hybrid model integrates NNI, NNII, and NNIII as shown in Figure 2.4. Predictions of the hybrid model are shown in Figure 2.9-2.14. Figures 2.9 and 2.10 compare the experimental recovery (%) and grade (%BPL) with those predicted by the hybrid model, respectively, for the coarse feed size distribution (14X 35 Tyler mesh). As shown in these figures, the hybrid model successfully predicts both recovery and grade. Figures 2.11 and 2.12 compare the experimental recovery (%) and grade (%BPL) with those predicted by the hybrid model, respectively, for the fine feed size distribution. As seen from these figures, the hybrid model fails to successfully predict both recovery and grade. This is attributed to the fact that fine feed has a very wide size distribution (35X150 Tyler mesh size) and only the overall recovery and grade were measured experimentally. It is therefore necessary to utilize narrow ranges of feed size and to analyze for recovery and grade according to each size range instead of just one recovery and grade for the entire particle size distribution. This was implemented for the unsized feed size which has even a wider size distribution (14X150 Tyler mesh). Figures 2.13 and 2.14 compare the experimental recovery (%) and grade (%BPL) predicted by the hybrid model, respectively, for the unsized feed after it has been sized and grade and recovery was determined for each size. As can be seen from these figures, the hybrid model successfully predicts both recovery and grade. 103 for(i=l;i<=N;i++) { 0[i]=i; S[i]=abs( A[i] [ 1 ]); for(j=2,j<=N;j++) { if(abs(A[i][j])>S[i]) S[i]=abs(A[i][j]); } } // *********** *** Ordering ends ****************************** II *************** Gauss Elimination ************************* for(k=l ;k // ** ** Pivoting ********* // Pivot=k; Big=abs(A[0[k]][k]/S[0[k]]); for( i i=k+1; i i <=N; i i++) { Dummy=abs(A[0[ii]][k]/S[0[ii]]); ifDummy>Big) { Big=Dummy; Pivot=ii; } } IDummy=0[Pivot], 0[Pivot]=0[k]; 0[k]=IDummy; II*** End Pivoting*******// for( i=k+1; i <=N ;i++) { factor=A[0[i]][k]/A[0[k]][k]; for(]=k+l ;j<=N;j++) { A[0[i]][j]=A[0[i]][j]-(factor*A[0[k]]lj]); } V[0[i]]=V[0[i]]-(factor*V[0[k]]); } } //***************QaussÂ£iimination ends ******************* Designed experiments were performed in a lab scale column to generate data for the initial training of the neural networks. 77 Unreacted Dihydrate = 1 ton rock x = 1 ton rock x v 73.33; (Bl/100) 2.184 0.06 x 1.49 x (172/56) x (1-0.06) Total amount of phosphogypsum = Acid insol + Unreacted + Dihydrate Soluble P2O5 losses = $ 300.0 x (% solubleP;05 losses)/!00 2.134 per ton = $1.37 per ton (Total amount of phosphogypsum) Acidulation Penalty = $62.0x- 1.37 B, Sales value = Price of 66 %BPL rock (Bl/66) 1.5 Adjusted sales value = Sales value Transportation penalty Acidulation penalty The adjusted value of the phosphate rock as a function of %BPL is shown in Figure 4.1. Let Feed solid flow rate = F, ton per year Product solid flow rate = P, ton per year Feed grade = Gf, % Concentrate grade = G, % Product recovery = R, % Adjusted sales value of feed = Cf, S per ton Adjusted sales value of product = Cp, $ per ton Reagent-i price = Cn, $/lb Reagent-i usage = U, lb/ton feed The feed flow rate and the product flow rate can be related as: f GA ( K~] 100>! P = F 0 0 O O 1 G J (4.4) 110 Dytiamics and Control of Chemical Reactors, Distillation Columns, and Batch Processes, DYCORD+, (1992). Thompson, ML., and M. A. Kramer, Modeling Chemical Processes Using Prior Knowledge and Neural Networks, AIChEJ., 40(8): 1328-1340 (1994). Villeneuve, J., M.-V Durance, C. Guillaneau, A.N. Santana, R.V.G. da Silva, and MAS. Martin, Advanced Use of Column Flotation Models for Process Optimization, in COLUMN'96, Eds. Gomez, C O. and J A. Finch, The Metallurgical Society of the Canadian Institute of Mining, Metallurgy and Petroleum, 51-62 (1996). Wheeler D.A., Historical View of Column Flotation Development, in Column Flotation88, Ed. K.V.S. Sastry, SME-AIME, Littleton, Colorado, 3-4 (1988). Xu, M., and J A. Finch, The Axial Dispersion Model in Flotation Column Studies, Mineral Engineering, 4: 553-562 (1991). Yeager, D., C.L. Karr, and D A. Stanley, Column Flotation Model Tuning Using a Genetic Algorithm, SME Annual Meeting, Preprint 95-204 (1995). Yianatos, J.Q., J.A. Finch, G.S. Dobby, and M. Xu, Bubble Size Estimation in a Bubble Swarm, J. Colloid Interface Sci., 126(l):37-44 (1988). Yoon, R.H., G.H. Luttrell, and G.T. Adel, Advances in Fine Particle Flotation, in Challenges in Mineral Processing, Eds. K.V.S. Sastry and M.C. Fuerstenau, SME- AIME, Littleton, Colorado, 487-506 (1989). Yoon, R.H., G.H. Luttrell, G.T. Adel, and M.J. Mankosa, Recent Advances in Fine Coal Flotation, in Advances in Coal and Mineral Processing Using Flotation, Eds. S. Chandler and R.R. Klimpel, SME-AIME, Littleton, Colorado, 211-218 (1988). 16 PJ=- Qr 4-aJ ~PJ [>exp(PJL) ACD Qp ACD (2.21) - a^ +aj ^exp(aJL) qJ = ^l- + dJ-5J ACD i_^* [ACD (2.22) j_ qJexp(yjLf) + exp(5JLf) m : pJ exp(aJLf ) + exp(pJLr) (2.23) The algorithm for solving the first-principles model is given in Appendix A. 2.2.2 Calculation of Recovery and Grade Recovery (%) is defined as the ratio of the weight of the phosphate in the concentrate stream to the weight of the phosphate in the feed stream. The recovery of the phosphate particles of the jl mesh size can be expressed in terms of the feed and tailings flow rates and concentration as rp= QfCj [q, +Ac(l-eg)U'| C! P2 z =0 QfCf *100 (2.24) Grade, a measure of the quality of the product, is defined as the ratio of the weight of the phosphate to the total weight recovered in the concentrate stream. Grade is reported as % Bone Phosphate of Lime (% BPL) which is the equivalent grams of tricalcium phosphate 3.14 Performance of the overall hybrid model: Predicted versus experimental grade (%BPL) for the four frothers 71 4.1 Value of phosphate rock as a function of %BPL 78 4.2 The run-to-run optimization algorithm 80 4 3 Neural network versus experimental flotation rate constant for phosphate (kp) 84 4.4 Neural network versus experimental flotation rate constant for gangue (kg) 85 4.5 Model versus experimental air holdup for frother F-507 87 vi ii 47 Laver k+1 to n-1 dC p, dt U. -+U Ci -Cl Pi-1 Az C]D -D ^ -2 Cl +Ci Pi P'-i Az ~kD(dJD)CJD (3-4) Laver n (bottom) dC Pn dt U v1-6* ' + U ' Q c _ Pn- (l-eg)Ac y + U C sl Pn Az U ^ f Q '+u\ a -i ^ *' Pn ' 1 8 J 0-Bg)At Az ji J -+D CJ CJ ~ kp(dJp)Cin Az: if u s--L- 1-e, Ui !CP. CJ -CJ +D- ; => -kp(di)c;n /Az if U <- u. 1-e, (3.5) where Ac = Cross-sectional area of the column C\ = Phosphate feed concentration of j mesh size particles C1 = Phosphate concentration of jlh mesh size particles in the ith layer Qf = Feed volumetric flow rate Q, = Tailings volumetric flow rate Qe = Elutriation volumetric flow rate Qp = Product volumetric flow rate Up = Superficial liquid velocity above the feed point = Qp/ac U, = Superficial liquid velocity below the feed point = (Q,-Qe)/Ae Uj, = Slip velocity of jth mesh size particles eg = Air holdup kp(dp) = Flotation rate constant for phosphate for j'h mesh size particles 54 kg (NNII) and receives as an input the inferred bubble size. This is the output of one of the ANNs of the second (top) level, NNII1. The second level also includes NNIV, which predicts air holdup. The advantage of this structure is that NNI and NNII are independent of the type of brother and sparger used, and therefore would not need retraining if these change. As bubble size is not measured in industry, we infer it from the two-phase (air/water) air holdup, Jg, and Ut using the well-known Drift-flux analysis (Yianatos et a/., 1988). The output required to train NNIV is the (two-phase) air holdup. Air holdup is relatively easy to obtain, so after a change of frother or sparger the hybrid model of Figure 3.2 can become functional in a short interval of time. 3.5 Materials and Methods 3.5.1 Experimental Setup and Procedures Two types of experiments were conducted: tv/o-phase (air/water) experiments to train neural networks NNIII and NNIV, and three-phase experiments to train NNI and NNII and to test the performance of the hybrid model. The experimental setup for the three-phase experiments is shown in Figure 2.5. It included an agitated tank (conditioner) for reagentizing the feed, a screw feeder for controlling the rate of reagentized feed, and a flotation column. The agitated tank was 45 cm in diameter and 75 cm high and was equipped with an impeller with two axial blades (each 28 cm diameter). The impeller had about 3.8 cm clearance from the bottom of the tank and its rotation speed was fixed at 465 rpm. The feeder with a 2.5 cm diameter xml version 1.0 encoding UTF-8 REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID E15GNWZD5_1I8YEX INGEST_TIME 2013-03-12T14:10:21Z PACKAGE AA00013557_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES 22 2 3 The Hybrid Model The overall structure of the hybrid model is shown in the Figure 2.4. The hybrid model utilizes backpropagation neural networks (Rumelhart and McClelland, 1986) to predict the values of parameters flotation rate constants, kp and kg, and air holdup, eg. The factors that affect kp and kg are particle diameter, superficial air velocity, frother concentration, collector concentration, extender concentration, and pH. The air holdup, sg, is mainly affected by superficial air velocity and frother concentration. The hybrid model of Figure 2.4 integrates the first-principles model with three artificial neural networks. Neural network, NNI, correlates the flotation rate constant for phosphate, kp, with phosphate particle size, superficial air velocity, frother concentration, collector concentration, extender concentration, and pH. Similarly, neural network, NNI I correlates the flotation rate constant for gangue, kg, with gangue particle size, superficial air velocity, frother concentration, collector concentration, extender concentration, and pH. Neural network NNIII correlates the air holdup, eg, with superficial air velocity and frother concentration. In this structure, all three neural networks are specific to the type of frother or sparger used. This necessitates generation of new data and retraining of the neural networks each time the frother or the sparger are changed. 101 else { UP=QP/Area;UP=(UP-USL)/( 1 -Eg);UP=-UP; UT=QT/Area;UT=(UT+USL)/( 1 -Eg);UT 1 =QT 1 / Area;UT 1 =(UT 1 +USL)/( 1 -Eg); UF=QF/Area;UF=UF/( 1 -Eg), A3 =(UP+(D/DELZ))/DELZ; A4=(-UP-(2.0*D/DELZ)-(K*DELZ))/(DELZ); A5=D/(DELZ DELZ); A1=A4+A5; A2=A5; A6=(UF*CF)/DELZ; A7=A3; A11 =(-UT-(2.0* D/DELZ)-(K DELZ))/(DELZ); A8=A11; A9=A5; A10-(UT/DELZ)+A5; A12=A5; A13=A10; //A14=A5+A11; A14=(-UT 1 -(D/DELZ)-(K*DELZ))/(DELZ); } //****** *Defmition 0fRow= \*************************** A[l][l]=1.0-(al*DELT*Al); A[l][2]=-(al*DELT*A2); for(i=3;i<=N;i++) A[l][i]=0.0; //***********Definilion of Row=2 to NF-1 ***************** for(i=2,i A[i] [i-1 ]=-(a 1 DELT A3); A[i][i]=1.0-(al*DELT*A4); A[i][i+l]=-(al*DELT*A5); for(j=l;j for(j=i+2;j<=N;j++) A[i][j]=0.0; } //*******:***Defmiiion of Row=NF ************************** A[NF] [NF-1 ]=-(a 1 DELT A7); A[NF][NF]=1.0-(al*DELT*A8); A[NF][NF+l]=-(al*DELT*A9); for(i=l ;i for(i=NF+2,i<=N;i++) 48 The slip velocity is calculated using the expression ofVilleneuve et al. (1996): ttj _Â§dÂ¡>2(Ps-PiX1- 18^(l + 0.15Rp ) where the particle Reynolds number is defined as RJ _d;u^,ps(i-(|)s) (3.6) (3.7) where g = Acceleration due to gravity (m/s ) Pi = Water viscosity (kg/mr,) pi = Water density (kg/m ) Pi = Solid density (kg/m3) <})s = Volume fraction of solids in siurry dp = Particle diameter (m) As the right hand side of Equation 3.7 is a function of U3,, the slip velocity is obtained by solving Equations 3.6 and 3.7 iteratively as described in Chapter 2. The axiai dispersion coefficient is calculated by a modification of the Finch and Dobby (1990) expression: f T V3 D = 0.063 (l-eg)dc ,1.6, (3.8) where do - Jg = column diameter (m) superficial air velocity (cm/s) 105 USL=108.233*DPl*DPl*(ROS-l)*pow((l-PHIS),2.7)/(l+0.15*(pow(REP,0.687))); diff=USL-USLi; if(diff<0.0) diff=-diT; USLi=USL; } while(diff>=0.0001); USL=(l-Eg)*USL; printf("\n USL=%lf',USL); //***************** USL caicuiation en(is *************** //USL=0.0; //printf("\nEnter your initial values for Phosphate concentrationAn"); for(i=l;i<=N;i-H-) { C[i]=0.0; //printf("\n C[%d]=",i); //scanf("%lf',&.C[i]), } //printfi'AnEnter your initial values for Gaunge concentrationAn"); for(i=l;i<=N;i-H-) { CG[i]=0.0; //printfTAn CG[%d]=",i); //scanf("%lf',&CG[i]); } Frank=BPL/0.733; CF=(Frank/100 0)*(CS/100.0)*R.OS*62.41818/((CS/100.0)+(1.0-(CS/100.0))*ROS); CFG=( 1.0-(Frank/l 00.0))*(CS/100.0)*ROS*62.41818/((CS/l 00.0)+( 1.0- (CS/100.0))*ROS); printfTVn CF=%lf ,CF); printf^'An CFG=%lf\CFG); //getch(); Du=T2.4*Dia*pow((0.3175*Qg/Area),0.3); // ft2/min// K=Ku;KG=KGu;D=Du/( 1 -Eg); RT=L*Area*( 1 -Eg)/(QF+(L* Area*( 1 -Eg)*abs(KG))); Z=(50.0*RT/DELT)-1; TIME=DELT; III* **** $$$$$%%%%%@@@@@@ HERE IS THE FLAG BETWEEN STEADY- STATE & DYNAMIC FLAG=1; //I for dynamic, other values for steady-state approximation ifFLAG 1) { for(l=l;K=Z;l++) { 100 QT1=(0.1336541 *GQT); QP=QF-QT1+QE; QT=QT1-QE; DELZ=(L/N); NF=((FNF/DELZ)+1 )* 1; Area=0.7853981*Dia*Dia; yy* ************* Calculation of slip velocity ************** DP 1=DP/1000.0; PHIS=(CS/100)/((CS/100)+((1-(CS/100))*ROS)); USLi=0.0; do { REP=5.12*DPl*USLi*ROS*( 1.0-PHIS); USL=108.233 *DP 1 *DP 1 *(ROS-1 )*pow(( 1 -PHIS),2.7)/( 1+0.15 *(pow(REP,0.687))); diff=USL-USLi; if(diff<0.0) diff=-diff; USLi=USL; } while(diff>=0.0001); USL=(l-Eg)*USL; /y***************** USL caicuiation ends *************** //USL=0.0; if(USL<=(QP/Area)) { UP=QP/Area;UP=(UP-USL)/( 1 -Eg); UT=QT/Area;UT=(UT+USL)/( 1 -Eg);UT 1 =QT l/Area;UT 1 =(UT 1 +USL)/( 1 -Eg); UF=QF/Area;UF=UF/( 1 -Eg); A3 =D/(DELZ DELZ); A4=(-UP-(2.0*D/DELZ)-(K*DELZ))/(DELZ); A5=(UP+(D/DELZ))/DELZ; A1=A3+A4; A2=A5; A6=(UF*CF)/DELZ; A7=A3; A10=(UT/DELZ)+A3; A8=A3+A4-A10; A9=A3; A11=(-UT-(2.0*D/DELZ)-(K*DELZ))/(DELZ); A12=A3; A13=A10; //A14=A3+A11; A14=(-UT 1 -(D/DELZ)-(K*DELZ))/(DELZ); 18 The recovery of phosphate RJp is only a function of the flotation rate constant for phosphate, kp, and air holdup, eg. Similarly, the recovery of gangue Rg is only a function of flotation rate constant for gangue, kg, and air holdup, eg. Since air holdup is measured, we can invert the model to determine the value of kp that results in the measured recovery of phosphate RJp and the value of kg that yields the measured recovery of gangue Ri. As shown in Figure 2.1, a one-dimensional search is performed to determine the values of flotation rate constants when supplied with the recovery of phosphate and gangue, respectively. This algorithm allows determination of the flotation rate constants for each run, given the operating conditions and the performance of the column in terms of grade and recovery. The algorithm requires two initial guesses of the flotation rate constants which yield errors in the corresponding RJp of opposite sign, and then the program uses the method of false position (Chapra and Canale, 1988) to determine the correct set of flotation rate constants. Recovery of phosphate increases monotonically with flotation rate constant for phosphate, kp. This is verified by calculating recovery for different values of flotation rate constant and recovery was plotted against flotation rate constant. From the graph shown in Figure 2.2, it is concluded that there is only value of floatation rate constant for a given recovery. Similarly, from Figure 2.3, it is concluded that recovery of gangue increases monotonically with flotation rate constant for gangue, kg. 86 experimental data. Again, a very good match is seen Figure 4.5 compares the predicted air holdup to the experimental values measured by a differential pressure cell. As shown in this figure, neural network successfully predicts the air holdup. Table 4 2 shows the results of the 13 designed experiments. As can be seen from this Table, feed flow rate and the %solids content varied significantly. The screw feeder operation was erratic and therefore we were unable to feed at the same rate in each run. Feed flow rate was calculated based on the product flow rate and the tailings flow rate. A specified volume of product and tailings were taken over a period of time (~20 s) and the samples were dried and the weight was taken In this way, solids flow rate in product and tailings stream were obtained. An overall material balance on the column then gives the feed flow rate. Similarly, an overall material balance on the water phase gives the water flow rate in the feed stream. Solids feed flow rate and the water flow rate in the feed then can be used to obtain the % solids in the feed. Unfortunately, the inability to control feed flow rate and % solids content means that a meaningful run-to-run optimization cannot be conducted. 4 5 Future Work First, the screw feeder needs to be repaired or replaced. After this has been accomplished, the hybrid model obtained from the designed experiments (Figures 4.3- 4.5) should be used with the Nelder-Meade algorithm to determine the experimental conditions of the first optimization run. The results of the run should be analyzed for grade and recovery and these data should be added to the neural network training sets. The networks should then be retrained and the updated hybrid model used to determine APPENDIX A CODE FOR TFIE FIRST PRINCIPLES MODEL FOR ONE LEVEL ^include #include ^include #include GQT 2.5372 //Tailings Flow rate (gallons/min)// GQF 0.6133 //Feed Flow rate (gallons/min)// GQE 3.038 //Elutriation Flow rate (gallons/min)// CS 66.0 //% Solid in the feed (lb S/lb T)// BPL 24.9 //% BPL of the feed (!b P/lb S) // ROS 2.6 //Specific Gravity of solids in the feed// DP 122.5 //Particle size in microns // Eg 0.05762 //air hold up // Dia 0.5 //Diameter of the column (ft.)// L 6.0 //Height of the column (ft.)// Ku 2.217556 //Flotation rate const, for Phosphate (1/min.)// KGu 0.999965 //Flotation rate const, for Gaunge (1/min.)// Qg 0.1778 //Air flow rate(scfm)// FNF 1.0 //Feed Location from the top (ft )// LfL-FNF void main I (double,double,double[]); void main 1 (double CF, double k, double B[]) { double QF,QE,QT1,QT,QP,Area,UP,UT,UF,D,DPI,PHIS,USLi,REP,USL,diiT; double a,b,d,alpha,beta,gamma,delta,p,q,m; QF=0.1336541 *GQF; QE=0.1336541 *GQE; QT1=(0.1336541 *GQT); QP=QF-QT1+QE; QT=QT1-QE; Area=0.7853981 *Dia*Dia, UP=QP/Area; UT=QT/Area; UF=QF/Area, 90 24 2.4 Materials and Methods 2.4.1 Experimental setup and Procedures The experimental setup is shown in Figure 2.5. it includes an agitated tank (conditioner) for reagentizing the feed and a screw feeder for controlling the rate of reagentized feed to the flotation column. The agitated tank was 45 cm in diameter and 75 cm high. It was equipped with an impeller of two axial type blades (each 28 cm diameter) The impeller rotation speed was fixed at 465 rpm. The impeller had about 3.8 cm clearance from the bottom of the tank. The feeder with 2.5 cm diameter screw delivered the conditioned phosphate materials to the column. The feed rate was controlled by adjusting the screw rotation speed. Flotation tests were conducted using a 14.5 cm diameter by 1.82 m high plexiglass flotation column. The feed inlet was located at 30 cm from the column top. The discharge flow rate was controlled by a discharge valve and an adjustable speed pump. Three flowmeters were used to monitor the flow rates for air, frother solution, and elutriation water. Three different feed sizes obtained from Cargill were used in the flotation experiments: coarse feed with narrow distribution (14X35 Tyler mesh), fine feed with wide size distribution (35X150 Tyler mesh), and unsized feed which is a mixture of the above two (14X150 Tyler mesh). For each run, 50 kg of feed sample was added in the pre-treatment tank and water was added to obtain 72% solids concentration by weight. The feed material was then agitated for 10 seconds. 10 % soda ash solution was added to the pulp to reach pH of about 9.4 and agitated for 10 seconds. Subsequently a mixture of 109 Kirkpatrick, S., Jr. C. D. Gelatt, and M. P. Vecchi, Optimization by Simulated Annealing, Science, 220:671-680 (1983). Kramer, M.A., and M. L. Thompson, Embedding Theoretical Models in Neural Networks, Proc. Am. Control Con/\ Chicago, 1:475-479 (1992). Liu, P.-H., T. Potter, S. A. Svoronos, and B. Koopman, Hybrid Model of Nitrogen Dynamics in a Periodic Wastewater Treatment Process, AIChE Annual Meeting, Paper No. 195an (1995). Luttrell, G.H., and R. H. Yoon, A Flotation Column Simulator Based on Hydrodynamic Principle, Inter. J. Miner. Process., 33:355-368 (1992). Luttrell, G.H., and R. H. Yoon, Column Flotation-A Review, in Beneficiation of Phosphate: Theory and Practice, Eds. H. El-Shail, B.M. Moudgil and R. Wiegel, SME, Littleton, Colorado, 361-369 (1993). Masters, T., Practical Neural Network Recipes in C++, Academic Press, New York (1993). Mavros, P., Mixing in Flotation Columns. Part 1: Axial Dispersion Modeling, Mineral Engineering, 6: 465-478 (1993). Perry, R.H., D. W. Green, and J. 0. Maloney, Perry's Chemical Engineers Handbook, McGraw-Hill Book Company, New York, 6th ed. (1984). Polak, E., Computational Methods in Optimization, Academic Press, New York (1971). Psichogios, D C., and L. H. Ungar, Process Modeling using Structured Neural Networks, Proc. Am. Control Con/., Chicago, 3:1917-1921 (1992a). Psichogios, D C., and L. H. Ungar, A Hybrid Neural-Network First Principles Approach to Process Modeling, AIChE J., 38:1499-1511 (1992b). Reuter, M., J. Van Deventer, and P. Van Der Walt, A Generalized Neural-Net Rate Equation, Chem. Eng. Sci., 48:1281 (1993). Rumelhart, D., and J. McClelland, Parallel Distributed Processing, MIT Press, Cambridge, MA (1986). Sasry, K.V.S., and K. D. Loftus, Mathematical Molding and Computer Simulation of Column Flotation, in Column Flotation'88, Ed. K.V.S. Sastry, SME-AIME, Littleton, Colorado, 57-68 (1988). Su, H.-T., P. A. Bhat, P. A. Minderman, and T. J. McA.voy, Integrating Neural Networks with First Principles Models for Dynamic Modeling, IFAC Symp. on Flotation rate constant for gangue (kg) Figure 2.3: Recovery of gangue (%) as a function of flotation rtae constant for gangue (k^) 44 The next section presents the first-principles model. The subsequent section deals with the calculation of model parameters from measured outputs. This is followed by a discussion of the artificial neural networks and their integration with the first-principles model to develop a hybrid model. The fourth section describes the experimental setup, materials used, experimental procedure, and the methodology used to train the neural networks. The final section presents results and compares the model predictions of grade and recovery to experimentally measured grade and recovery. 3.2 First-Principles Model The FPM is obtained from material balances on both phosphate and gangue. It neglects radial dispersion and changes in the air holdup. Following Luttrell and Yoon (1993) the particle to bubble attachment and detachment rates are combined in one net attachment rate, and this rate is assumed to be first order with respect to particle concentration in the slurry. The model subdivides the column into n layers as shown in Figure 3.1. Feed containing both the desired (phosphate) and undesired (gangue) particles enters in a slurry in layer k. An additional inlet stream is the elutriation water that enters in the bottom of the column (layer n). Most of that flow is due to water that enters with the air sparger, as most of the popular spargers are two-phase and introduce a considerable amount of water. There are two outlet streams: the tailings stream through the bottom of the column (layer n) that contains mostly gangue, and the product (concentrate) stream that leaves from the top of the column. 9 Black-box modeling strategies are mainly data driven and the resulting models often do not have reliable extrapolation properties. Black-box strategies have been applied to many chemical processes, especially since convenient black-box modeling tools like neural networks have become available (Bhat and McAvoy, 1990, Psichogios and Ungar, 1992a). Gray-box or hybrid modeling strategies are potentially very' efficient if the black-box and white-box components are combined in such a way that the resulting models have good interpolation and extrapolation properties. There are two types of gray-box modeling approaches in which a neural network is combined with a black-box model: the parallel and the serial approach In the parallel approach, the neural network is placed parallel with a white-box model In this case, the neural network is trained on the error between the output of the white-box model and the actual output Su et ol. (1992) demonstrated that the parallel approach resulted in better interpolation properties than pure black-box models. Johansen and Foss (1992) also used a parallel structure where the output of the hybrid model was a weighted sum of a first- principles and a neural network model. In the serial hybrid modeling strategy, the neural network is placed in series with the first-principles model. Various researchers (Psichogios and Ungar, 1992a; Thompson and Kramer, 1994) have shown the potential extrapolation properties of serial hybrid models. Psichogios and Ungar (1992b) used this approach for parameters that are functions of the state variables and manipulated inputs. Liu et al. (1995) developed a serial hybrid model for a periodic wastewater treatment process by using ANNs for the 97 //printf("\n alpha*Lf=%lf',(apha)*(Lf)); //printf("\n beta*Lf=%lf',(beta)*(Lf)); //p=(((-UP/D)+a-beta)*(exp((beta)*(L))))/(((UP/D)-a+a!pha)*(exp((alpha)*(L)))); //q=((-UT/D)+d-delta)/((UT/D)-d+gamma); if(USL<=UP) p=((-beta)*exp((beta)*(L)))/(alpha*exp((alpha)*(L))); else p=((-(-beta+a))*exp((beta)*(L)))/((-alpha+a)*exp((alphaj*(L))); q=-(delta-(QE/(Area*D)))/(gamma-(QE/(Area*D))); m=(q*exp((gamma)*(Lf))+exp((delta)*(Lf)))/(p*exp((alpha)*(Lf))+exp((beta)*(Lf))), B[4]=(UF*CF/D)/((m*p*(a- alpha)*exp((alpha)*(Lf)))+((d+gamma)*q*exp((gamma)*(Lf)))"Hm*(a- beta)*exp((beta)*(Lf)))~K(d+delta)*exp((delta)*(Lf)))); //B[3]=(UF*CF/D)/((a+alpha+d+gamma)*exp((gamma)*(Lf))); B[2]=m*B[4]; B[l]=p*m*B[4]; B[3]=q*B[4]; //B[l]=(exp((gamma-alpha)*Lf))*B[3]; //B[2]=0.0; //B[4]=0.0; } double model(double Ku) { double QF,QE,QTl,QT,QP,Area,DPl,PHIS,USLi,REP,USL,diff,Frank,CF,CFG,K,KG,C,CG; double RO.ROG,Grade,B[5],BG[5]; QF=0.1336541*GQF; QE=0.1336541 *GQE; QT1=(0.1336541*GQT); QP=QF-QT1+QE; QT=QT1-QE; Area=0.7853981 *Dia*Dia; //printfTVn UP=%lf,QP/Area); y/************** Calculation of slip velocity ************** DP 1=DP/1000.0; PHIS=(CS/100)/((CS/100)+((1-(CS/100))*ROS)); USLi=0.0; 12 The slip velocity is calculated using the gd^-p.xi-dO27 U si ;0 087 18pi,(l -hO 15RJp ) expression of Villeneuve et al. (1996): (2.3) where the particle Reynolds number is defined as dUj,Ps(l-
Mi(2.4) where g = Acceleration due to gravity (m/s ) Pi = Water viscosity (kg/ms) pi = Water density (kg/m ) Pi = Solid density (kg/m3) dJp = Particle diameter (m) Since Rp is a function of LP,, an iterative procedure is used to calculate the slip velocity. The procedure starts with an initial guess for LP, and corresponding value of RJ,p is plugged in Equation 2.3 and new value of LP, is found. This new value is then used in Equation 2.2 and this procedure is continued till convergence is achieved. The axial dispersion coefficient is calculated by a modified expression of Finch and Dobby (1990): f v J 03 D = 0.063 (1-Eg)dc (2.5) 104 y/************** Substitution ***************************** C[N]=V[0[N]]/A[0[N]][N]; for(i=N-l;i>=l;i--) { Sum=0.0; for(j=i+l;j<=Nj++) { Sum=Sum+(A[0[i]][j]*C[j]); } C[i]=(V[0[i]]-Sum)/A[0[i]][i]; } } y/**************** Substitution ends ******************* //************* ************** ***************************** * / / ^ ^ ^ ^ |i L jl ^ ^ ^ ^ ~P ^ ^ >P ^ >p 1 p .p Lp ^ P tp .P P >p p .p P >P ~P t P P p [i P p ^ ^ ^ > p Lp >P p ^ ^ > p ^ p ^P ^ P P //&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& &&&&&&&&&&&& //######################################################### //!!!!!!!!!!!!!!!! Main 1 Ends !!!!!!!!!!!!!!!!!!!!!!!! void main(void) { double Grade,CF,CFG,K,KG,D,RO,Area,Z,RT,QP,QT,QF,TIME,Pe,Tp,a,bl,b2,REQNP,REQN G; int l,i,FLAG;//NF; double C[n],CG[n],USLi,USL,REP,difif,PHIS,Gradeqn,c,d 1 ,d2,Frank;//DELZ; double Du,QE,QT1,R,DP1,Selectivity,Selecteqn,ROG,RG,//RC, QF=0.1336541 *GQF; QE=0.1336541 *GQE; QT1=(0.1336541*GQT); QP-QF-QT1+QE; QT-QT1-QE; //DELZ=(L/N); //NF=((FNF/DELZ)+1 )* 1; Area=0.7853981 *Dia*Dia; printfT\n UP=%lf',QP/Area); //************** Calculation of slip velocity ************** DP 1=DP/1000.0; PHIS=(CS/100)/((CS/100)+((1-(CS/100))*ROS)); USLi=0.0; do { REP-5.12*DPl*USLi*ROS*(l. 0-PHIS); BIOGRAPHICAL SKETCH Sanjay Gupta obtained a Bachelor of Engineering degree in chemical engineering from the University of Roorkee, Roorkee, India in May 1993. He then worked as a process engineer at Burmah-Shell Refinery Ltd., Bombay, India, for one year. He continued his higher studies at Drexel University, Philadelphia, and obtained a Master of Science degree in chemical engineering in March 1997. He joined the Department of Chemical engineering at University of Florida to pursue his Ph.D. degree in August, 1996. 74 New optimal manipulated variable values are predicted which set the conditions for the subsequent run. This procedure should be repeated until convergence is obtained. 41 Performance Measures The performance of a flotation column is affected by both recovery (%) and grade (%BPL). To guide optimization it is necessary to combine the two outputs (grade and recovery) in a single performance measure. Several performance measures are possible, and some are presented below. 4,1,1 Selectivity One way to achieve this is to use selectivity as the performance measure. Selectivity is defined as where R = Recovery of phosphate in the product stream. Rb = Recovery of gangue in the product stream Rt = Recovery(or Rejectability) of phosphate in the tailings stream. Rtb = Recovery(or Rejectability) of gangue in the tailings strea We developed the following expression that relates selectivity to the recovery and the grade of the product stream 49 Equations analogous to 3.1-3.8 are valid for the gangue particles, but with a considerably lesser effective flotation rate constant kg(d). In the limit as Az -4 0 the above difference equations become (X i f c\ U. -U eci cz d2C] D UL~ -MW (3.9) for the section above the feed and 5Cj dt u vi_eg d2C> P-+D p cz cz2 ~MW (3 10) for the section below the feed. Recovery (%) is defined as the ratio of the weight of the phosphate in the concentrate stream to the w eight of the phosphate in the feed The recovery for phosphate particles of the j01 mesh size can be expressed in terms of the feed and tailings flow rates and concentrations as r;= 'QfC/-[Q,+At(l-e,)U]cip- Q,c; MOO (3.11) Grade, a measure of the quality of the product, is defined as the ratio of the weight of the phosphate to the total weight recovered in the concentrate stream. Grade is usually reported as % Bone Phosphate of Lime (%BPL) which is the equivalent grams of tncalcium phosphate, Ca3(PC>4)2, in 100 g of sample. For the typical Florida rock, 8 The particie collection process in a column is considered to follow first order kinetics relative to the solids particle concentration with a rate constant. Finch and Dobby (1990) and Lutrell and Yoon (1993) used a one-phase axial dispersion model in which particle collection is viewed as a first order net attachment rate process. Sastry and Loftus (1988) considered both the slurry and air phases and they used two separate first order rate constants for attachment and detachment of the particles. Luttrell and Yoon (1993) relate the particle net attachment rate constant to some operating variables using a probabilistic approach. However, their approach cannot be used to predict the effect of certain operating conditions such as frother concentration, collector concentration, extender concentration, and pH. For the model to be predictive, the functional dependence of the net attachment rate constant (kp or kg) on the key operating variables needs to be determined. The functional relationship of model parameters on the operating conditions is difficult to determine via physicochemical reasoning. In our approach, we use neural networks to determine these functional relationships. Artificial neural networks are a powerful tool, inspired by how the human brain works, that can learn from examples any unknown functional relationship. Their ability to approximate any smooth nonlinear multivariable function arbitrarily well (Hornik et al., 1989) and their simple construction have led to great interest in using neural networks. Existing modeling strategies can be divided into white-box, black-box, and gray-box (hybrid) strategies, depending on the amount of prior knowledge that is used for development of the model. White-box modeling strategies are mainly knowledge driven. 80 START i Initial runs V ICP analysis 4' Performance measure 1 Train neural network 4 Determine position of maximum YES ^ NO Is maximum in interior of range? Experimental run at Experimental run at predicted maximum half-way point 1 ICP analysis * I ' Performance measure 1 Train neural network 4 NO Determine position of maximum 4 Difference between two consecutive maximum less than pre-decided limits 0 YES > STOP Figure 4.2: The run-to-run optimization algorithm 25 20 15 10 5 0 5 10 15 20 25 Experimental Air Hold-up (%) Figure 3.10: Performance of NNIV: Model versus experimental air holdup for frother F-579 CHAPTER 1 INTRODUCTION Since the beginning of 1980s, the industrial application of flotation technology has experienced a remarkable growth due to active theoretical and experimental research and development. Flotation columns are slowly being accepted in the mineral processing industry for the advantages they offer over conventional flotation equipment including grade improvement, lower operating cost, and superior control. The ability of flotation columns to produce concentrates of superior grade at similar recovery is derived from the improved selectivity it offers. Unlike conventional mechanical cells, flotation columns do not use mechanical agitation to suspend particles. Another distinct feature of the flotation column principle is countercurrent contact between feed particles and air bubbles. The lack of moving pans and lower reagent consumption results in a lower operating costs. The lower capital cost for the equipment is attributed to its high capacity leading to the use of less units for the same production rate The current flotation practice in Florida phosphate industry involves the use a two stage process with mechanical cells, where the feed is subjected to rougher flotation in which fatty acids and fuel oil are used as collectors to separate the phosphate from most of the sand The rougher concentrate is then scrubbed by sulfuric acid to remove the fatty acids and oil. The scrubbed material has to be washed with fresh water to 1 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. 5 fyrv'co K J>v Spyros A. Svoronos, Chairman Professor of Chemical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Hassan El-Shall, Cochairman Engineer of Materials Science and Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Assistant Professor of Chemical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. BenKoopman Professor of Environmental Engineering Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Oscar Crisalle Associate Professor of Chemical Engineering 42 attachment and detachment are modeled separately with first order rates. Luttrell and Yoon (1993) used a probabilistic approach to relate the particle net attachment rate constant to some operating variables (e.g., air flow rate). However, their approach involves empirical parameters and it cannot be used to predict the effect of certair operating variables such as the frother and collector concentrations. In this work, we use neural networks to determine the dependence of the phosphate and gangue flotation rate constants on the operating variables. Artificial neural networks have the ability to approximate any smooth nonlinear multivariable function arbitrarily well (Hornik et al., 1989). This approach can be used to determine the dependence of the performance of a flotation column (i.e., grade and recovery) on any operational variable We demonstrate it in this work by developing a hybrid model that predicts the effect ol frother concentration, air flow rate, feed rate and loading, elutriation flow rate, tailings flow rate, and particle size distribution. The idea of developing a hybrid model by combining a first-principles model (FPM) with artificial neural networks (ANNs) is not new. Johansen and Foss (1992) and Su ei al. (1992) proposed parallel structures where the output of the hybrid model is a weighted sum of the first-principles and ANN models. Kramer et al. (1992) proposed a parallel arrangement of a default model (which could be a first principles model) and a radial basis function ANN. An alternative approach is to combine ANNs with a FPM in a serial fashion, by using the ANNs to develop expressions for the FPM parameters or rate expressions. Psichogios and Ungar (1992a, 1992b) proposed this scheme for parameters that are functions of the state variables and manipulated inputs, and trained the neural 75 s = -g(i-r)-r 2 Gf (1 G)(l R) (4.2) where G = Grade (%BPL) of phosphate in the product stream. Gf = Grade (BPL) of phosphate in the feed. 4 1.2 Separation Efficiency Separation efficiency is defined as follows: E = R -Rb (4.3) In this case, the efficiency varies between -100 to 100. 4,1,3 Economic Performance Measure The selectivity function or the separation efficiency does not include any economic input such as cost of the reagents. Therefore an alternate performance measure was developed which includes recovery, grade, and the reagent prices. A scheme for penalizing lower grade rock has been developed. This scheme deducts differential costs, relative to 66% BPL, for transportation and acidulation. The acidulation scheme assumes soluble P2O5 losses increase in direct proportion to the amount of phosphogypsum. Thus, the procedure requires an estimate of the quantity of phosphogypsum that is produced. CHAPTER 3 TWO-LEVEL HYBRID MODEL A new model for phosphate column flotation is presented which relates the effects of operating variables such as froiher concentration and air flow rate on column performance. This is a hybrid model that combines a first-principles model with artificial neural networks. The first-principles model is obtained from material balances on both phosphate particles and gangue (undesired material containing mostly silica). First order rates of net attachment are assumed for both. .Artificial neural networks relate the attachment rate constants to the operating variables. Experiments were conducted in a 6 diameter laboratory column to provide data for neural network training and model validation. The model is shown to successfully predict the effects of frother concentration, particle size, air flow rate, and bubble diameter on grade and recovery. 3.1 Introduction Flotation is a process in which air bubbles are used to separate a hydrophobic from a hydrophilic species. The majority of the hydrophobic material gets attached to the bubbles and leaves with the froth from the top of a cell or column separator, while the hydrophilic material leaves from the bottom. This process is commonly used in the minerals industry, including the phosphate industry, in which case the phosphate containing rock (frankolite or apatite) is to be separated from gangue (mostly silica). Flotation is also used to remove oil from wastewater and to remove ink from paper pulp. 40 106 mainl(CF,K,D,C); //printfTVn Following are the values of Phosphate C[i] at Time=%lf RT. \n",TIME/RT); //for(i= 1 ;i<=N ;i++) //printfTVn C[%d]=%lf',i,C[i]); //getch(); RO=(((QF*CF)-((QTl+Area*USL)*(C[N])))/(QF*CF))* 100.0; //RC=(((QF*CFHQP*(C[NF])HQT1*(C[N])))/((QF*CFHQP*(C[NF]))))*100.0; //RC needs modification in terms of QP and QT1 (i.e. has to include USL) //printfTVn The overall recovery at Time=%lf RT is RO=%lf%.",TIME/RT,RO); //printfTVn The collection zone recovery at Time=%lf RT is RC=%lf%.\n",TIME/RT,RC); //getch(); main 1 (CFG,KG,D,CG); //pnnti^"\n Following are the values of Gaunge CG[i] at Time=%lf RT. Vn",TEME/RT); //for(i=l ,i<=N;i-H-) //printf("\n CG[%d]=%lf',i,CG[i]); //getch(); ROG=(((QF*CFG)-((QTl+Area*USL)*(CG[N])))/(QF*CFG))*100.0; Grade=((QF*CF-(QTl+Area*USL)*C[N])/((QF*CF- (QTl+Ajea*USL)*C[N])+(QF*CFG~(QTl+Area*USL)*CG[N])))* 100.0; Grade=Grade*0.733; //printfTVn The grade at Time=%lf RT is %lf',TIME/RT,Grade); //getch(); Selectivity=RO-ROG; TIME=(1+1)*DELT; } //printfTVn G100=%lf',G100); //for(i=l ,i<=N;i++) //printfTVn CG[%d]=%lf',i,CG[i]); //getchO; printfTVn CG[N]=%lf\CG[N]); //printfTVn C[N]=%lf',C[N]); //printfTVn C[l]=%lf',C[l]); printfTVn C[N]=% 1 f',C[N]); printfTVn Overall Recovery=%.llf %",RO); printfTVn GRADE=%. 1 If %",Grade); printfTVn Sep_eff=%. 1 If", Selectivity); printfTVn ROG=%.llf %",ROG); //printfTVn R=%. 1 If %",(1 -(C[N]/CF))* 100); getchO; } else 50 55 60 65 70 Experimenta! Grade (%) Figure 2.10: Performance of the overall hybrid model: Predicted versus experimental grade (%BPL) for coarse feed size distribution 91 D=12.4*Dia*pow((0.3175*Qg/ Area), 0.3); //************** Calculation of slip velocity ************** DP1=DP/1000.0; PHIS=(CS/100)/((CS/100)+((1-(CS/100))*ROS)); USLi=0.0; do { REP=5 12*DPl*USLi*ROS*(l. 0-PHIS); USL=108.233*DP1 *DP1 *(ROS-l)*pow((l-PHIS),2.7)/(l+0.15*(pow(REP,0.687))); diff=USL-USLi; if(diff<0.0) diff=-diff; USLi=USL; } while(diff>=0.0001); U SL=( 1 -Eg) *USL; //***************** USL calculation ends *************** a=(UP-USL)/D; d=(UT+USL)/D; b=k*(l-Eg)/D; if(((a*a+4*b)<0.0)||((d,,d+4*b)<0.0)) b=0.0; alpha=(a/2)+(sqrt(a*a+4*b))/2; beta=(a/2)-(sqrt(a*a+4*b))/2; gamma=(-d/2)+(sqrt(d*d+4*b))/2; delta=(-d/2)-(sqrt(d*d+4*b))/2; //printfX"\n alpha=%lf",alpha); //printf("\n beta=%lf',beta); //printtX"\n gamma=%lf",gamma); //printf("\n delta=%lf",delta); //printf("\n delta=%lf",a*a+4*b); //print f("\n beta*L=%lf',(beta)*(L)); //printf("\n alpha*L=%lf",(alpha)*(L)); //printf("\n gamma*Lf=%lf',(gamma)*(Lf)); //printf("\n delta*Lf=%lf',(delta)*(Lf)); //printfC\n alpha*Lf=%lf",(alpha)*(Lf)); //printft"\n beta*Lf=%lf",(beta)*(Lf)); //p=(((-(JP/D)+a-beta)*(exp((beta)*(L))))/(((UP/D)-a+alpha)*(exp((alpha)*(L)))); //q=((-UT/D)+d-delta)/((UT/D)-d+gamma); if(USL<=UP) p=((-beta)*exp((beta)*(L)))/(alpha*exp((alpha)*(L))), 58 continued until a lower point could not be found This approach improves the likelihood of convergence to the global optimum. 3.6 Results and Discussion The performance of the network for predicting bubble diameter (NNIII), the network for predicting air holdup (NNIV), the network for predicting the phosphate flotation rate constant (NNI) and the network for predicting the gangue flotation rate constant (NNII) is shown in Figures 3.3-3.14. Figure 3.3 compares the NNIII output to the inferred bubble diameter using experimental data when the frother was CP-100. The solid circles are for the data used for training while the open squares are for the data used for validation. Figures 3.4, 3.5, and 3.6 show the performance of NNIII when F-507, OB-535, and F- 579, respectively, were the frothers. As these figures show, NNIII successfully predicts the inferred bubble diameter. Figure 3.7 compares the air holdup predicted by NNIV to the experimental values measured by a differential pressure cell when CP-100 was used as the frother. Figures 3.8, 3.9, and 3.10 show the performance of NNIV when F-507, OB-535, and F-579, respectively, were used as frothers. As shown in these figures, NNIV successfully predicts the air holdup for all frothers. Figures 3.11 and 3.12 show the performance of NNI and NNII, respectively. Figure 3.11 presents the predicted flotation rate constants for phosphate (kp) against those determined from one-dimensional searches using experimental data. As shown in this figure, NNI does accurately predict low and high values of flotation rate constants. 39 2.6 Conclusions In this work, we have demonstrated that a one-phase first-principles model can effectively be coupled with the artificial neural networks for predicting the grade and recovery of a phosphate flotation column with negative bias. Artificial neural networks are used to predict the flotation rate constants and air holdup. Experimental data from a lab-scale column were used to train the neural networks. The hybrid model successfully predicts the effects of particle size, superficial air velocity, frother concentration, collector concentration, extender concentration, and pH. 14 in the limit as Az > 0, the above equation reduces to the following boundary condition. dC pi dz = 0 (2.9) z=L Continuity of the concentration profile at the feed location gives c; Pi = CJD z=Lf p2 z=Lf (2.10) A similar material balance at the feed inlet gives for the solid particles in the slurry phase Qfc;=Ac u. v1_eg -IIJ CJ Usl |%, dc; - ACD^ z=Lf dz + A. Ut z = Lr Kl~ss U P2 dCJ + A.D P: z=Lr c dz (2.11) where th Cf = Phosphate feed concentration of j mesh size particles Qf = Feed volumetric flow rate Lf = Feed location At the bottom of the column (z = 0), due to the elutriation flow, the derivative of the concentration profile reduces to the following expression: dCJ D P2 dz z =o Qe C, (l-OAc P! z = 0 (2.12) 1780 1993 UNIVERSITY OF FLORIDA 3 1262 08554 4376 00 80 60 40 20 0 Figure 2.11: Performance of the overall hybrid model: Predicted versus experimental recovery (%) for fine feed size distribution 88 Table 4.2: Results of the runs from he factorial design Frother cone. (ppm) Air flow rate (scfm) Feed flow rate (gpm) Tailings feed flow rate (gpm) Elutria- tion flow rate (gpm) Solids content (%) Grade (%BPL) Recovery (%) 1 5 0.0928 0.198 2.014 2.410 59.37 55.95 68.31 2 5 0.3711 0 418 1.779 2.351 35.11 55.36 40.04 3 25 0.0928 0.126 1.432 2.423 48.53 40.07 34.60 4 25 0.3711 0.284 2.062 2.919 35.03 61.86 45.90 5 5 0.2319 0.376 1.897 2 893 49.75 51.04 67.72 6 25 0.2319 0.284 1.650 2 378 47.02 39.59 55.92 7 15 0.0928 0.264 2.355 2.922 36.43 62.68 47.49 8 10 0.3711 0 340 2.275 2.927 41.65 53.99 48.53 9 15 0.2319 0.463 2.173 2.619 38.58 45.63 16.87 10 15 0.3015 0.370 1.838 2.645 42.65 46.26 54.10 11 20 0.2319 0.261 1.694 2.661 49.18 37.50 55.19 12 15 0.1624 0.281 1.853 2.634 42.36 68.40 47.92 13 10 0.2319 0.259 1.758 2.631 41.13 68.03 52.14 56 and pump were adjusted to get the desired underflow and overflow rates. Air holdup was measured using a differential pressure gauge. After the water/air system reached steady state, the screw feeder was started. To achieve steady feed rate to the column, water was added to the screw feeder at the rate that reduced the solids concentration to approximately 66% by weight. The column was run for a period of three minutes with phosphate feed prior to sampling. Timed samples of tailings and concentrates were taken The collected product samples, as well as feed samples, were dried, sieved using Tyler meshes, weighed and analyzed for %BPL following the procedure recommended by the Association of Florida Phosphate Chemists (AFPC Analytical Methods, 1980). In addition, gangue content (as % acid insolubles) of the feed, tailings, and concentrate streams was measured (AFPC Analytical Methods, 1980). These measurements were then used to calculate recovery of acid insolubles. Subsequently these values were averaged with the values obtained from Equation 3.14 to obtain theRg used to determine the flotation rate constants for gangue. The two-phase experiments were identical to the three-phase experiments, except that no solid feed was introduced to the column and the experiments were terminated when the water/air system reached steady state. 3.5.2 Neural Network Structure and Training NNI, NNII, NNin, and NNIV of Figure 3.2 were feedforward backpropagation artificial neural networks with a single layer of hidden nodes between the input and output layers and a unit bias connected to both the hidden and the output layers. Inputs 76 This performance measure is only applicable to plants, and can not be used with a lab- scale flotation column. The procedure for this scheme is outlined below: Assumptions 1. The price of rock of 66% BPL = $22.00 2. Zero insol %BPL = 73.33 3. Transportation cost = $2.50 per ton. 4. Soluble P2O5 losses = 1.00% 5. Insoluble P2O5 losses = 6.00% 6. Increase in soluble P2O5 losses is proportional to the amount of phosphogypsum produced. Transportation Penalty Base case: 66% BPL rock (dry basis) Freight cost per BPL ton = $2.5/0.66 = $3.79 Penalty: 2.50 vBL/m - 3.79 per BPL ton f Transportation penalty = 2.50 Bi/100 A -3.79 B, 7 100 per ton Where, Bl = %BPL when grade < 66% Acidulation Penalty Base case. 66% BPL rock (30.21% P205, Ca0:P205 = 1.49) Acid insol =100 (i~5L) V 73.337 Calculation of the amount of Phosphogypsum: Phosphogypsum components f B, ^ = 1 ton rock x 1 ^ 73.337 Acid insol 26 fatty acid (obtained from Westvaco) and fuel oil (No. 5 obtained from PCS Phosphate) with a ratio of 1:1 by weight was added to the pulp. The total conditioning time was 3 minutes. The conditioned feed material (without its conditioning water) was loaded in the feeder bin located at the top of the column. The frother selected for this study was CP-100 (sodium alkyl ether sulfate obtained from Westvaco). Frother-containing water and air were first introduced into the column through the sparger (eductor) at a fixed flowrate and frother concentration, and then the discharge valve and pump were adjusted to get the desired underflow and overflow rates. Air holdup was measured for the two-phase (air/water) system using a differential pressure gauge. After every parameter was set and the two-phase system was in a steady state, the phosphate material was fed to the column using the screw feeder. Water was also added to the screw feeder to maintain the steady flow of the solids to the column at 66 % solids concentration To achieve steady state, the column was run for a period of three minutes with phosphate feed prior to sampling. Timed samples of tailings and concentrates were taken. The collected samples were weighed and analyzed for %BPL according to the procedure recommended by the Association of Florida Phosphate Chemists (AFPC Analytical Methods, 1980). These measurements were then used to calculate recovery of acid insolubles. These values were then averaged with the values obtained from Equation 2.26 to obtain the Rg used to determine the flotation rate constants of gangue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bf DV D IXQFWLRQ RI IORWDWLRQ UDWH FRQVWDQW IRU SKRVSKDWH NSf 5HFRYHU\ RI JDQJXH DV D IXQFWLRQ RI IORWDWLRQ UDWH FRQVWDQW IRU JDQJXH NJf 2YHUDOO VWUXFWXUH RI WKH K\EULG PRGHO $ VFKHPDWLF GLDJUDP RI WKH H[SHULPHQWDO VHWXS 3HUIRUPDQFH RI 11Â 0RGHO YHUVXV H[SHULPHQWDO IORWDWLRQ UDWH FRQVWDQW IRU SKRVSKDWH NSf 3HUIRUPDQFH RI11,, 0RGHO YHUVXV H[SHULPHQWDO IORWDWLRQ UDWH FRQVWDQW IRU JDQJXH NJf 3HUIRUPDQFH RI 11,, 0RGHO YHUVXV H[SHULPHQWDO DLU KROGXS IRU IURWKHU &3 3HUIRUPDQFH RI WKH RYHUDOO K\EULG PRGHO 3UHGLFWHG YHUVXV H[SHULPHQWDO UHFRYHU\ bf IRU FRDUVH IHHG VL]H GLVWULEXWLRQ 3HUIRUPDQFH RI WKH RYHUDOO K\EULG PRGHO 3UHGLFWHG YHUVXV H[SHULPHQWDO JUDGH b%3/f IRU FRDUVH IHHG VL]H GLVWULEXWLRQ 3HUIRUPDQFH RI WKH RYHUDOO K\EULG PRGHO 3UHGLFWHG YHUVXV H[SHULPHQWDO UHFRYHU\ bf IRU ILQH IHHG VL]H GLVWULEXWLRQ 3HUIRUPDQFH RI WKH RYHUDOO K\EULG PRGHO 3UHGLFWHG YHUVXV H[SHULPHQWDO JUDGH b%3/f IRU ILQH IHHG VL]H GLVWULEXWLRQ YL PAGE 7 3HUIRUPDQFH RI WKH RYHUDOO K\EULG PRGHO 3UHGLFWHG YHUVXV H[SHULPHQWDO UHFRYHU\ bf IRU XQVL]HG IHHG VL]H GLVWULEXWLRQ 3HUIRUPDQFH RI WKH RYHUDOO K\EULG PRGHO 3UHGLFWHG YHUVXV H[SHULPHQWDO JUDGH b%3/f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f 3HUIRUPDQFH RI11,, 0RGHO YHUVXV H[SHULPHQWDO IORWDWLRQ UDWH FRQVWDQW IRU JDQJXH NJf 3HUIRUPDQFH RI WKH RYHUDOO K\EULG PRGHO 3UHGLFWHG YHUVXV H[SHULPHQWDO UHFRYHU\ bf IRU WKH IRXU IURWKHUV YLL PAGE 8 3HUIRUPDQFH RI WKH RYHUDOO K\EULG PRGHO 3UHGLFWHG YHUVXV H[SHULPHQWDO JUDGH b%3/f IRU WKH IRXU IURWKHUV 9DOXH RI SKRVSKDWH URFN DV D IXQFWLRQ RI b%3/ 7KH UXQWRUXQ RSWLPL]DWLRQ DOJRULWKP 1HXUDO QHWZRUN YHUVXV H[SHULPHQWDO IORWDWLRQ UDWH FRQVWDQW IRU SKRVSKDWH NSf 1HXUDO QHWZRUN YHUVXV H[SHULPHQWDO IORWDWLRQ UDWH FRQVWDQW IRU JDQJXH NJf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f LV ZDVKHG DQG GHVOLPHG DW PHVK 7KH PDWHULDO ILQHU WKDQ PHVK LV SXPSHG WR FOD\ VHWWOLQJ SRQGV 7KH URFN FRDUVHU WKDQ PHVK LV VFUHHQHG WR VHSDUDWH SHEEOHV PHVKf ZKLFK DUH RI KLJK SKRVSKDWH FRQWHQW :DVKHG URFN PHVKf LV VL]HG LQWR D ILQH XVXDOO\ [ PHVKf DQG D FRDUVH IORWDWLRQ IHHGV XVXDOO\ [ PHVKf ZKLFK DUH WUHDWHG LQ VHSDUDWH FLUFXLWV )ORWDWLRQ RI SKRVSKDWHV IURP WKH ILQH IHHG ; PHVKf SUHVHQWV YHU\ IHZ GLIILFXOWLHV DQG UHFRYHULHV LQ H[FHVV RI b DUH DFKLHYHG XVLQJ FRQYHQWLRQDO IORWDWLRQ FHOOV 2Q WKH RWKHU KDQG UHFRYHU\ RI SKRVSKDWH YDOXHV IURP WKH FRDUVH IHHG LV PXFK PRUH GLIILFXOW DQG IORWDWLRQ E\ LWVHOI XVXDOO\ \LHOGV UHFRYHU\ RI b RU OHVV 7KH GHQVLW\ RI WKH VROLG WXUEXOHQFH VWDELOLW\ DQG KHLJKW RI WKH IURWK OD\HU GHSWK RI WKH ZDWHU FROXPQ YLVFRVLW\ RI WKH IURWK OD\HU DUH NQRZQ WR HIIHFW WKH IORWDWLRQ SURFHVV LQ JHQHUDO %RXWLQ DQG :KHHOHU f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f 7KH XVH RI IORWDWLRQ FROXPQV FDQ QRW RQO\ KHOS RYHUFRPH VRPH RI WKH SUREOHPV UHODWHG WR FRDUVH SKRVSKDWH IORWDWLRQ EXW LW KDV VHYHUDO RWKHU DGYDQWDJHV DV PHQWLRQHG DERYH 6SDUJHUV RU EXEEOH JHQHUDWLQJ V\VWHPV DUH WKH VLQJOH PRVW LPSRUWDQW HOHPHQW LQ WKH IORWDWLRQ FROXPQV 7KH\ DUH JHQHUDOO\ FKDUDFWHUL]HG LQ WHUPV RI WKHLU DLU GLVSHUVLRQ DELOLW\ )URWKHUV DUH WKH FKHPLFDOV WKDW KHOS LQ FRQWUROOLQJ DQG VWDELOL]LQJ EXEEOH VL]H E\ UHGXFWLRQ RI VXUIDFH WHQVLRQ 7KXV ERWK RI WKHP SOD\ DQ LPSRUWDQW UROH LQ WKH RYHUDOO SHUIRUPDQFH RI IORWDWLRQ FROXPQV 7KHLU LQWHUDFWLRQ FDQ EH D FUXFLDO IDFWRU LQ WKH VXFFHVV RI IORWDWLRQ FROXPQ )ORWDWLRQ FROXPQV KDYH EHHQ XVHG SUHGRPLQDQWO\ LQ WKH FRDO EHQHILFLDWLRQ LQGXVWU\ +RZHYHU WKHLU DSSOLFDWLRQ LQ RWKHU PLQHUDO LQGXVWULHV VXFK DV WKH SKRVSKDWH LV PAGE 14 QRW YHU\ ZHLO VWXGLHG 8QOLNH RWKHU PLQHUDOV SKRVSKDWH IORWDWLRQ GHDOV ZLWK D FRQVLGHUDEO\ ODUJHU VL]H RI SDUWLFOHV PPf DQG WKHUHIRUH WKH RSHUDWLRQ RI SKRVSKDWH IORWDWLRQ LQ D FROXPQ LV GLIIHUHQW IURP WKDW RI RWKHU PLQHUDOV +LJK UHFRYHU\ DQG JUDGH DQG ORZ RSHUDWLQJ FRVW GHSHQG ODUJHO\ RQ WKH RSWLPDO VHOHFWLRQ RI RSHUDWLQJ YDULDEOHV VXFK DV WKH DLU IORZ UDWH WKH IURWKHU W\SH DQG FRQFHQWUDWLRQ DQG WKH HOXWULDWLRQ ZDWHU UDWH 7KH VHDUFK RI WKH RSWLPDO FRQGLWLRQV FDQ FRQVLGHUDEO\ EHQHILW E\ WKH DYDLODELOLW\ RI D PRGHO WKDW FDQ SUHGLFW WKH HIIHFWV RI GLIIHUHQW RSHUDWLQJ FRQGLWLRQV RQ FROXPQ EHKDYLRU )LQFK DQG 'REE\ f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f DQG SDWHQWHG %RXWLQ DQG :KHHOHU f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f LV VHSDUDWHG IURP JDQJXH PRVWO\ VLOLFDf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f DQ HOXWULDWLRQ ZDWHU VWUHDP IURP WKH ERWWRP LV DGGHG WR PDLQWDLQ D SRVLWLYH XSZDUG IORZ QHJDWLYH ELDVf WR DLG OLIWLQJ WKH SDUWLFOHV XSZDUG PAGE 18 7KH SDUWLFLH FROOHFWLRQ SURFHVV LQ D FROXPQ LV FRQVLGHUHG WR IROORZ ILUVW RUGHU NLQHWLFV UHODWLYH WR WKH VROLGV SDUWLFOH FRQFHQWUDWLRQ ZLWK D UDWH FRQVWDQW )LQFK DQG 'REE\ f DQG /XWUHOO DQG PAGE 19 %ODFNER[ PRGHOLQJ VWUDWHJLHV DUH PDLQO\ GDWD GULYHQ DQG WKH UHVXOWLQJ PRGHOV RIWHQ GR QRW KDYH UHOLDEOH H[WUDSRODWLRQ SURSHUWLHV %ODFNER[ VWUDWHJLHV KDYH EHHQ DSSOLHG WR PDQ\ FKHPLFDO SURFHVVHV HVSHFLDOO\ VLQFH FRQYHQLHQW EODFNER[ PRGHOLQJ WRROV OLNH QHXUDO QHWZRUNV KDYH EHFRPH DYDLODEOH %KDW DQG 0F$YR\ 3VLFKRJLRV DQG 8QJDU Df *UD\ER[ RU K\EULG PRGHOLQJ VWUDWHJLHV DUH SRWHQWLDOO\ YHU\n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f GHPRQVWUDWHG WKDW WKH SDUDOOHO DSSURDFK UHVXOWHG LQ EHWWHU LQWHUSRODWLRQ SURSHUWLHV WKDQ SXUH EODFNER[ PRGHOV -RKDQVHQ DQG )RVV f DOVR XVHG D SDUDOOHO VWUXFWXUH ZKHUH WKH RXWSXW RI WKH K\EULG PRGHO ZDV D ZHLJKWHG VXP RI D ILUVW SULQFLSOHV DQG D QHXUDO QHWZRUN PRGHO ,Q WKH VHULDO K\EULG PRGHOLQJ VWUDWHJ\ WKH QHXUDO QHWZRUN LV SODFHG LQ VHULHV ZLWK WKH ILUVWSULQFLSOHV PRGHO 9DULRXV UHVHDUFKHUV 3VLFKRJLRV DQG 8QJDU D 7KRPSVRQ DQG .UDPHU f KDYH VKRZQ WKH SRWHQWLDO H[WUDSRODWLRQ SURSHUWLHV RI VHULDO K\EULG PRGHOV 3VLFKRJLRV DQG 8QJDU Ef XVHG WKLV DSSURDFK IRU SDUDPHWHUV WKDW DUH IXQFWLRQV RI WKH VWDWH YDULDEOHV DQG PDQLSXODWHG LQSXWV /LX HW DO f GHYHORSHG D VHULDO K\EULG PRGHO IRU D SHULRGLF ZDVWHZDWHU WUHDWPHQW SURFHVV E\ XVLQJ $11V IRU WKH PAGE 20 ELRNLQHWLF UDWHV RI D ILUVWSULQFLSOHV PRGHO &XELOOR DQG /LPD f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rW G] G] X G&L G&@ 83 3L fÂ§ G] G]f f f PAGE 21 ZKHUH 8S 8W 4S 4 4H $F N3: 3KRVSKDWH FRQFHQWUDWLRQ RIMWK PHVK VL]H SDUWLFOHV IRU IHHG SRLQW 3KRVSKDWH FRQFHQWUDWLRQ RI MOK PHVK VL]H SDUWLFOHV IRU IHHG SRLQW 6XSHUILFLDO OLTXLG YHORFLW\ DERYH WKH IHHG SRLQW 4SDF 6XSHUILFLDO OLTXLG YHORFLW\ EHORZ WKH IHHG SRLQW 4 4Hf$F 'LVSHUVLRQ FRHIILFLHQW 3URGXFW YROXPHWULF IORZ UDWH 7DLOLQJV YROXPHWULF IORZ UDWH (OXWULDWLRQ YROXPHWULF IORZ UDWH &URVVVHFWLRQDO DUHD RI WKH FROXPQ 6OLS YHORFLW\ RIMOK PHVK VL]H SDUWLFOHV $LU KROGXS )ORWDWLRQ UDWH FRQVWDQW IRU SKRVSKDWH IRU PHVK VL]H SDUWLFOHV WKH VHFWLRQ DERYH WKH WKH VHFWLRQ EHORZ WKH 7KH IROORZLQJ DVVXPSWLRQV DUH PDGH LQ GHULYLQJ WKH DERYH HTXDWLRQV f 7KH FRQFHQWUDWLRQ RI VROLG SDUWLFOHV LQ WKH VOXUU\ SKDVH LV D IXQFWLRQ RI KHLJKW ] RQO\ DQG YDULDWLRQV RI WKH FRQFHQWUDWLRQ LQ UDGLDO DQG DQJXODU GLUHFWLRQV FDQ EH QHJOHFWHG f 7KH DLU KROGXS LV FRQVWDQW WKURXJKRXW WKH FROXPQ f $OO WKH DLU EXEEOHV LQ WKH V\VWHP DUH RI D VLQJOH VL]H f 5DWH RI GHWDFKPHQW LV HLWKHU QHJOLJLEOH RU LV D IXQFWLRQ RI FRQGLWLRQV LQ WKH VOXUU\ SKDVH 7KLV DVVXPSWLRQ DOORZV WR WUHDW WKH QHW DWWDFKPHQW UDWH ZLWK MXVW RQH IORDWDWLRQ UDWH FRQVWDQW PAGE 22 7KH VOLS YHORFLW\ LV FDOFXODWHG XVLQJ WKH JGAS[LG2 8 VL SLO K2 5-S f H[SUHVVLRQ RI 9LOOHQHXYH HW DO f f ZKHUH WKH SDUWLFOH 5H\QROGV QXPEHU LV GHILQHG DV GÂ8M3VO8 0L f ZKHUH J $FFHOHUDWLRQ GXH WR JUDYLW\ PV f 3L :DWHU YLVFRVLW\ NJPVf SL :DWHU GHQVLW\ NJP f 3L 6ROLG GHQVLW\ NJPf M!V 9ROXPH IUDFWLRQ RI VROLGV LQ VOXUU\ G-S 3DUWLFOH GLDPHWHU Pf 6LQFH 5ÂS LV D IXQFWLRQ RI /3 DQ LWHUDWLYH SURFHGXUH LV XVHG WR FDOFXODWH WKH VOLS YHORFLW\ 7KH SURFHGXUH VWDUWV ZLWK DQ LQLWLDO JXHVV IRU /3 DQG FRUUHVSRQGLQJ YDOXH RI 5-S LV SOXJJHG LQ (TXDWLRQ DQG QHZ YDOXH RI /3 LV IRXQG 7KLV QHZ YDOXH LV WKHQ XVHG LQ (TXDWLRQ DQG WKLV SURFHGXUH LV FRQWLQXHG WLOO FRQYHUJHQFH LV DFKLHYHG 7KH D[LDO GLVSHUVLRQ FRHIILFLHQW LV FDOFXODWHG E\ D PRGLILHG H[SUHVVLRQ RI )LQFK DQG 'REE\ f I Y (JfGF f PAGE 23 ZKHUH FI FROXPQ GLDPHWHU Pf -J VXSHUILFLDO DLU YHORFLW\ FPVf (TXDWLRQV DQG FDQ EH VROYHG DQDO\WLFDOO\ IRU WKH FRQFHQWUDWLRQ SURILOH RI WKH VROLG SDUWLFOHV DW VWHDG\ VWDWH 7KH UHVXOWLQJ DQDO\WLFDO H[SUHVVLRQV IRU WKH FRQFHQWUDWLRQ SURILOH DUH ZKHUH .@ .M DQG.A DUH WKH FRQVWDQWV RI LQWHJUDWLRQ WR EH GHWHUPLQHG E\ XVLQJ DSSURSULDWH ERXQGDU\ FRQGLWLRQV %RXQGDU\ &RQGLWLRQV $ PDWHULDO EDODQFH DW WKH WRS OD\HU RI WKH FROXPQ ] /f JLYHV WKH IROORZLQJ HTXDWLRQ $F' fÂ§ NS GS $F$]&cf@ f PAGE 24 LQ WKH OLPLW DV $] fÂ§! WKH DERYH HTXDWLRQ UHGXFHV WR WKH IROORZLQJ ERXQGDU\ FRQGLWLRQ G& SL G] f ] / &RQWLQXLW\ RI WKH FRQFHQWUDWLRQ SURILOH DW WKH IHHG ORFDWLRQ JLYHV F 3L &-' ] /I S ] /I f $ VLPLODU PDWHULDO EDODQFH DW WKH IHHG LQOHW JLYHV IRU WKH VROLG SDUWLFOHV LQ WKH VOXUU\ SKDVH 4IF $F X YBHJ ,,&8VO _b GF $&'fÂ§A ] /I G] $ 8W ] /U .OaVV 8Â 3 G&$' 3 ] /U F G] f ZKHUH fWK &I 3KRVSKDWH IHHG FRQFHQWUDWLRQ RI M PHVK VL]H SDUWLFOHV 4I )HHG YROXPHWULF IORZ UDWH /I )HHG ORFDWLRQ $W WKH ERWWRP RI WKH FROXPQ ] f GXH WR WKH HOXWULDWLRQ IORZ WKH GHULYDWLYH RI WKH FRQFHQWUDWLRQ SURILOH UHGXFHV WR WKH IROORZLQJ H[SUHVVLRQ G&' 3 G] ] R 4H & O2$F 3 ] f PAGE 25 7KH IRXU ERXQGDU\ FRQGLWLRQV FDQ EH VROYHG LQ FRQMXQFWLRQ ZLWK (TXDWLRQV DQG IRU .M .M .A DQG.7KH UHVXOWLQJ H[SUHVVLRQV IRU WKH FRQVWDQWV RI LQWHJUDWLRQ DUH JLYHQ E\ WKH IROORZLQJ HTXDWLRQV TFDFGf P-DD-fSH[S^D/I`G\-fTH[S^\-/I` PD-fH[S^-/I` G-fH[S^-/M` f .L T-.n f r .! ,, r 8 nfÂ§ f S-P-. f ZKHUH RUr fÂ§ fÂ§ 9DEr f fÂ§!DEn f \M fÂ§9GM Ef n fÂ§?G E f PAGE 26 34U fÂ§ Da3>!H[S3-/f $&' 4S $&' f DA DM AH[SD-/f TÂ‘ AO G-$&' LBAr M! \>$&' f MB T-H[S\M/If H[S-/If P fÂ§ SH[SD-/I f H[SS-/Uf f 7KH DOJRULWKP IRU VROYLQJ WKH ILUVWSULQFLSOHV PRGHO LV JLYHQ LQ $SSHQGL[ $ &DOFXODWLRQ RI 5HFRYHU\ DQG *UDGH 5HFRYHU\ bf LV GHILQHG DV WKH UDWLR RI WKH ZHLJKW RI WKH SKRVSKDWH LQ WKH FRQFHQWUDWH VWUHDP WR WKH ZHLJKW RI WKH SKRVSKDWH LQ WKH IHHG VWUHDP 7KH UHFRYHU\ RI WKH SKRVSKDWH SDUWLFOHV RI WKH MO PHVK VL]H FDQ EH H[SUHVVHG LQ WHUPV RI WKH IHHG DQG WDLOLQJV IORZ UDWHV DQG FRQFHQWUDWLRQ DV US 4I&M >T $FOHJf8n_ & 3 ] 4I&I r f *UDGH D PHDVXUH RI WKH TXDOLW\ RI WKH SURGXFW LV GHILQHG DV WKH UDWLR RI WKH ZHLJKW RI WKH SKRVSKDWH WR WKH WRWDO ZHLJKW UHFRYHUHG LQ WKH FRQFHQWUDWH VWUHDP *UDGH LV UHSRUWHG DV b %RQH 3KRVSKDWH RI /LPH b %3/f ZKLFK LV WKH HTXLYDOHQW JUDPV RI WULFDOFLXP SKRVSKDWH PAGE 27 &D3f LQ O22J RI VDPSOH *UDGH FDQ EH FDOFXODWHG DV WKH UDWLR RI WKH ZHLJKW RI SKRVSKDWH WR WKH VXP RI WKH ZHLJKW RI WKH SKRVSKDWH DQG JDQJXH LQ WKH FRQFHQWUDWH VWUHDP I 4I&^ M 4 $Ff8Â &U S ? ] Y4I&M>4W$FOJf8Â@& f4I&M ] rV >4W$FOVJf8Â@&-J ] A \ f ZKHUH LV WKH JDQJXH FRQFHQWUDWLRQ RI WKH MnK SDUWLFOH VL]H DQG &L LV WKH JDQJXH IHHG [J FRQFHQWUDWLRQ RI MOK SDUWLFOH VL]H 7KH PXOWLSOLFDWLRQ IDFWRU LV LQVWHDG RI EHFDXVH SXUH )ORULGD SKRVSKDWH URFN PHDVXUHV DW DERXW b%3/ 0RGHO 3DUDPHWHUV 7KH DERYH PRGHO IRUPXODWLRQ KDV RQO\ WZR PRGHO SDUDPHWHUV QDPHO\ WKH IORWDWLRQ UDWH FRQVWDQWV IRU SKRVSKDWH DQG JDQJXH 7KH H[SHULPHQWDO DQDO\VLV LQ WKH LQGXVWU\ LV XVXDOO\ DYDLODEOH LQ WHUPV RI JUDGH DQG UHFRYHU\ RI SKRVSKDWH 7KH UHFRYHU\ RI JDQJXH FDQ WKHQ EH UHDGLO\ FDOFXODWHG IURP JUDGH DQG UHFRYHU\ RI SKRVSKDWH XVLQJ WKH IROORZLQJ UHODWLRQVKLS 5M B5-S*I a*-f *-*Lf f ZKHUH *-I LV WKH JUDGH RI WKH IHHG PDWHULDO PAGE 28 7KH UHFRYHU\ RI SKRVSKDWH 5-S LV RQO\ D IXQFWLRQ RI WKH IORWDWLRQ UDWH FRQVWDQW IRU SKRVSKDWH NS DQG DLU KROGXS HJ 6LPLODUO\ WKH UHFRYHU\ RI JDQJXH 5J LV RQO\ D IXQFWLRQ RI IORWDWLRQ UDWH FRQVWDQW IRU JDQJXH NJ DQG DLU KROGXS HJ 6LQFH DLU KROGXS LV PHDVXUHG ZH FDQ LQYHUW WKH PRGHO WR GHWHUPLQH WKH YDOXH RI NS WKDW UHVXOWV LQ WKH PHDVXUHG UHFRYHU\ RI SKRVSKDWH 5-S DQG WKH YDOXH RI NJ WKDW \LHOGV WKH PHDVXUHG UHFRYHU\ RI JDQJXH 5L $V VKRZQ LQ )LJXUH D RQHGLPHQVLRQDO VHDUFK LV SHUIRUPHG WR GHWHUPLQH WKH YDOXHV RI IORWDWLRQ UDWH FRQVWDQWV ZKHQ VXSSOLHG ZLWK WKH UHFRYHU\ RI SKRVSKDWH DQG JDQJXH UHVSHFWLYHO\ 7KLV DOJRULWKP DOORZV GHWHUPLQDWLRQ RI WKH IORWDWLRQ UDWH FRQVWDQWV IRU HDFK UXQ JLYHQ WKH RSHUDWLQJ FRQGLWLRQV DQG WKH SHUIRUPDQFH RI WKH FROXPQ LQ WHUPV RI JUDGH DQG UHFRYHU\ 7KH DOJRULWKP UHTXLUHV WZR LQLWLDO JXHVVHV RI WKH IORWDWLRQ UDWH FRQVWDQWV ZKLFK \LHOG HUURUV LQ WKH FRUUHVSRQGLQJ 5-S RI RSSRVLWH VLJQ DQG WKHQ WKH SURJUDP XVHV WKH PHWKRG RI IDOVH SRVLWLRQ &KDSUD DQG &DQDOH f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n GLPHQVLRQDO VHDUFK WR LQYHUW WKH ILUVWSULQFLSOHV PRGHO PAGE 30 )ORWDWLRQ UDWH FRQVWDQW IRU SKRVSKDWH NSf )LJXUH 5HFRYHU\ RI SKRVSKDWH bf DV D IXQFWLRQ RI IORWDWLRQ UWDH FRQVWDQW IRU SKRVSKDWH NSf PAGE 31 )ORWDWLRQ UDWH FRQVWDQW IRU JDQJXH NJf )LJXUH 5HFRYHU\ RI JDQJXH bf DV D IXQFWLRQ RI IORWDWLRQ UWDH FRQVWDQW IRU JDQJXH NAf PAGE 32 7KH +\EULG 0RGHO 7KH RYHUDOO VWUXFWXUH RI WKH K\EULG PRGHO LV VKRZQ LQ WKH )LJXUH 7KH K\EULG PRGHO XWLOL]HV EDFNSURSDJDWLRQ QHXUDO QHWZRUNV 5XPHOKDUW DQG 0F&OHOODQG f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f IRU UHDJHQWL]LQJ WKH IHHG DQG D VFUHZ IHHGHU IRU FRQWUROOLQJ WKH UDWH RI UHDJHQWL]HG IHHG WR WKH IORWDWLRQ FROXPQ 7KH DJLWDWHG WDQN ZDV FP LQ GLDPHWHU DQG FP KLJK ,W ZDV HTXLSSHG ZLWK DQ LPSHOOHU RI WZR D[LDO W\SH EODGHV HDFK FP GLDPHWHUf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f ILQH IHHG ZLWK ZLGH VL]H GLVWULEXWLRQ ; 7\OHU PHVKf DQG XQVL]HG IHHG ZKLFK LV D PL[WXUH RI WKH DERYH WZR ; 7\OHU PHVKf )RU HDFK UXQ NJ RI IHHG VDPSOH ZDV DGGHG LQ WKH SUHWUHDWPHQW WDQN DQG ZDWHU ZDV DGGHG WR REWDLQ b VROLGV FRQFHQWUDWLRQ E\ ZHLJKW 7KH IHHG PDWHULDO ZDV WKHQ DJLWDWHG IRU VHFRQGV b VRGD DVK VROXWLRQ ZDV DGGHG WR WKH SXOS WR UHDFK S+ RI DERXW DQG DJLWDWHG IRU VHFRQGV 6XEVHTXHQWO\ D PL[WXUH RI PAGE 35 )LJXUH $ VFKHPDWLF GLDJUDP RI WKH H[SHULPHQWDO VHWXS PAGE 36 IDWW\ DFLG REWDLQHG IURP :HVWYDFRf DQG IXHO RLO 1R REWDLQHG IURP 3&6 3KRVSKDWHf ZLWK D UDWLR RI E\ ZHLJKW ZDV DGGHG WR WKH SXOS 7KH WRWDO FRQGLWLRQLQJ WLPH ZDV PLQXWHV 7KH FRQGLWLRQHG IHHG PDWHULDO ZLWKRXW LWV FRQGLWLRQLQJ ZDWHUf ZDV ORDGHG LQ WKH IHHGHU ELQ ORFDWHG DW WKH WRS RI WKH FROXPQ 7KH IURWKHU VHOHFWHG IRU WKLV VWXG\ ZDV &3 VRGLXP DON\O HWKHU VXOIDWH REWDLQHG IURP :HVWYDFRf )URWKHUFRQWDLQLQJ ZDWHU DQG DLU ZHUH ILUVW LQWURGXFHG LQWR WKH FROXPQ WKURXJK WKH VSDUJHU HGXFWRUf DW D IL[HG IORZUDWH DQG IURWKHU FRQFHQWUDWLRQ DQG WKHQ WKH GLVFKDUJH YDOYH DQG SXPS ZHUH DGMXVWHG WR JHW WKH GHVLUHG XQGHUIORZ DQG RYHUIORZ UDWHV $LU KROGXS ZDV PHDVXUHG IRU WKH WZRSKDVH DLUZDWHUf V\VWHP XVLQJ D GLIIHUHQWLDO SUHVVXUH JDXJH $IWHU HYHU\ SDUDPHWHU ZDV VHW DQG WKH WZRSKDVH V\VWHP ZDV LQ D VWHDG\ VWDWH WKH SKRVSKDWH PDWHULDO ZDV IHG WR WKH FROXPQ XVLQJ WKH VFUHZ IHHGHU :DWHU ZDV DOVR DGGHG WR WKH VFUHZ IHHGHU WR PDLQWDLQ WKH VWHDG\ IORZ RI WKH VROLGV WR WKH FROXPQ DW b VROLGV FRQFHQWUDWLRQ 7R DFKLHYH VWHDG\ VWDWH WKH FROXPQ ZDV UXQ IRU D SHULRG RI WKUHH PLQXWHV ZLWK SKRVSKDWH IHHG SULRU WR VDPSOLQJ 7LPHG VDPSOHV RI WDLOLQJV DQG FRQFHQWUDWHV ZHUH WDNHQ 7KH FROOHFWHG VDPSOHV ZHUH ZHLJKHG DQG DQDO\]HG IRU b%3/ DFFRUGLQJ WR WKH SURFHGXUH UHFRPPHQGHG E\ WKH $VVRFLDWLRQ RI )ORULGD 3KRVSKDWH &KHPLVWV $)3& $QDO\WLFDO 0HWKRGV f 7KHVH PHDVXUHPHQWV ZHUH WKHQ XVHG WR FDOFXODWH UHFRYHU\ RI DFLG LQVROXEOHV 7KHVH YDOXHV ZHUH WKHQ DYHUDJHG ZLWK WKH YDOXHV REWDLQHG IURP (TXDWLRQ WR REWDLQ WKH 5J XVHG WR GHWHUPLQH WKH IORWDWLRQ UDWH FRQVWDQWV RI JDQJXH PAGE 37 ([SHULPHQWDO &RQGLWLRQV )RU WKH IURWKHUV LQYHVWLJDWHG WKUHHSKDVH H[SHULPHQWV ZHUH FRQGXFWHG 6HYHQ GLIIHUHQW OHYHOV RI IURWKHU FRQFHQWUDWLRQ DQG SSPf ZDV VWXGLHG LQ GHVLJQHG H[SHULPHQWV )LYH GLIIHUHQW OHYHOV RI FROOHFWRU DQG H[WHQGHU FRQFHQWUDWLRQ DQG NJWf ZHUH XVHG S+ ZDV YDULHG IURP WR DW ILYH GLIIHUHQW OHYHOV DQG f 7ZR VXSHUILFLDO DLU YHORFLWLHV DQG FPVf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f DQG WKH LQSXW DQG WKH RXWSXW YDOXHV ZHUH VFDOHG IURP WR 'XULQJ WKH WUDLQLQJ PRGH WUDLQLQJ H[DPSOHV DUH SUHVHQWHG WR WKH QHWZRUN $ WUDLQLQJ H[DPSOH FRQVLVWV RI VFDOHG LQSXW DQG RXWSXW YDOXHV )RU 11, DQG 11,, WKH RXWSXW YDOXHV DUH WKH IORWDWLRQ UDWH FRQVWDQWV FDOFXODWHG IURP RQHGLPHQVLRQDO VHDUFKHV IRU SKRVSKDWH DQG JDQJXH UHVSHFWLYHO\ )RU 11,,, WKH RXWSXW YDOXH LV WKH H[SHULPHQWDOO\ PHDVXUHG DLU KROGXS 7KH WUDLQLQJ SURFHVV LV VWDUWHG E\ LQLWLDOL]LQJ DOO ZHLJKWV UDQGRPO\ WR VPDOO QRQ]HUR YDOXHV 7KH UDQGRP QXPEHU LV JHQHUDWHG EHWZHHQ DQG ZLWK VWDQGDUG GHYLDWLRQ RI IROORZLQJ WKH SURFHGXUH UHFRPPHQGHG E\ 0DVWHUV f 7KH RSWLPDO ZHLJKWV ZHUH PAGE 38 GHWHUPLQHG XVLQJ VLPXODWHG DQQHDOLQJ .LUNSDWULFN HW DO f DQG D FRQMXJDWH JUDGLHQW DOJRULWKP 3RODN f 7KHUH DUH WZR DSSURDFKHV WRZDUGV XSGDWLQJ WKH ZHLJKWV ,Q RQH DSSURDFK WKH LQSXWRXWSXW H[DPSOHV DUH SUHVHQWHG RQH DW D WLPH DQG DIWHU HDFK SUHVHQWDWLRQ WKH ZHLJKWV DUH XSGDWHG XVLQJ UXOHV VXFK DV WKH GHOWD UXOH 5XPHOKDUW DQG 0F&OHOODQG f 7KLV PHWKRG LV DWWUDFWLYH IRU LWV VLPSOLFLW\ EXW LV UHVWULFWHG WR UDWKHU SULPLWLYH RSWLPL]DWLRQ DOJRULWKPV ,Q FRQWUDVW WKH EDWFK WUDLQLQJ DSSURDFK DOORZV XVH RI SRZHUIXO PHWKRGRORJ\ IRU QRQOLQHDU RSWLPL]DWLRQ LW SURFHVVHV HDFK LQSXWRXWSXW H[DPSOH LQGLYLGXDOO\ EXW XSGDWHV WKH ZHLJKWV RQO\ DIWHU WKH ZKROH VHW RI LQSXWRXWSXW H[DPSOHV KDV EHHQ SURFHVVHG ,Q WKLV FDVH WKH JUDGLHQW LV FXPXODWHG IRU DOO SUHVHQWDWLRQV WKHQ WKH ZHLJKWV DUH XSGDWHG DQG ILQDOO\ WKH VXP RI WKH VTXDUHG HUURUV LV FDOFXODWHG 7KH VLPXODWHG DQQHDOLQJ DOJRULWKP LV XVHG IRU HOXGLQJ ORFDO PLQLPXP ,W SHUWXUEV WKH LQGHSHQGHQW YDULDEOHV WKH ZHLJKWVf ZKLOH NHHSLQJ WUDFN RI WKH EHVW ORZHVW HUURUf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f GHWHUPLQHG IURP RQHGLPHQVLRQDO VHDUFKHV ZLWK WKRVH SUHGLFWHG E\ 11, $V VKRZQ LQ WKLV ILJXUH 11, FDSWXUHV WKH GHSHQGHQFH RI WKH IORWDWLRQ UDWH FRQVWDQW RQ SDUWLFOH VL]H VXSHUILFLDO DLU YHORFLW\ IURWKHU FRQFHQWUDWLRQ PAGE 39 R fÂ§ f L F UR f F R fÂ§ FR D! F F 2 &/ nA &2 & 2 7 2 Â‘ } 7D! /B 4B .f Y& )LJXUH 3HUIRUPDQFH RI11, 0RGHO YHUVXV H[SHULPHQWDO IORWDWLRQ UDWH FRQVWDQW IRU SKRVSKDWH NSf PAGE 40 FROOHFWRU DQG H[WHQGHU FRQFHQWUDWLRQ DQG S+ 6LPLODUO\ )LJXUH FRPSDUHV IORWDWLRQ UDWH FRQVWDQW IRU JDQJXH NJf GHWHUPLQHG IURP RQHGLPHQVLRQDO VHDUFKHV ZLWK WKRVH SUHGLFWHG E\ 11,, $V VKRZQ 11,, VXFFHVVIXOO\ SUHGLFWV WKH IORWDWLRQ UDWH FRQVWDQW IRU JDQJXH )LJXUH SUHVHQWV WKH DLU KROGXS 6Jf SUHGLFWHG XVLQJ 11,,, DJDLQVW WKRVH PHDVXUHG H[SHULPHQWDOO\ $ VDWLVIDFWRU\ PDWFK LV VHHQ 7KH K\EULG PRGHO LQWHJUDWHV 11, 11,, DQG 11,,, DV VKRZQ LQ )LJXUH 3UHGLFWLRQV RI WKH K\EULG PRGHO DUH VKRZQ LQ )LJXUH )LJXUHV DQG FRPSDUH WKH H[SHULPHQWDO UHFRYHU\ bf DQG JUDGH b%3/f ZLWK WKRVH SUHGLFWHG E\ WKH K\EULG PRGHO UHVSHFWLYHO\ IRU WKH FRDUVH IHHG VL]H GLVWULEXWLRQ ; 7\OHU PHVKf $V VKRZQ LQ WKHVH ILJXUHV WKH K\EULG PRGHO VXFFHVVIXOO\ SUHGLFWV ERWK UHFRYHU\ DQG JUDGH )LJXUHV DQG FRPSDUH WKH H[SHULPHQWDO UHFRYHU\ bf DQG JUDGH b%3/f ZLWK WKRVH SUHGLFWHG E\ WKH K\EULG PRGHO UHVSHFWLYHO\ IRU WKH ILQH IHHG VL]H GLVWULEXWLRQ $V VHHQ IURP WKHVH ILJXUHV WKH K\EULG PRGHO IDLOV WR VXFFHVVIXOO\ SUHGLFW ERWK UHFRYHU\ DQG JUDGH 7KLV LV DWWULEXWHG WR WKH IDFW WKDW ILQH IHHG KDV D YHU\ ZLGH VL]H GLVWULEXWLRQ ; 7\OHU PHVK VL]Hf DQG RQO\ WKH RYHUDOO UHFRYHU\ DQG JUDGH ZHUH PHDVXUHG H[SHULPHQWDOO\ ,W LV WKHUHIRUH QHFHVVDU\ WR XWLOL]H QDUURZ UDQJHV RI IHHG VL]H DQG WR DQDO\]H IRU UHFRYHU\ DQG JUDGH DFFRUGLQJ WR HDFK VL]H UDQJH LQVWHDG RI MXVW RQH UHFRYHU\ DQG JUDGH IRU WKH HQWLUH SDUWLFOH VL]H GLVWULEXWLRQ 7KLV ZDV LPSOHPHQWHG IRU WKH XQVL]HG IHHG VL]H ZKLFK KDV HYHQ D ZLGHU VL]H GLVWULEXWLRQ ; 7\OHU PHVKf )LJXUHV DQG FRPSDUH WKH H[SHULPHQWDO UHFRYHU\ bf DQG JUDGH b%3/f SUHGLFWHG E\ WKH K\EULG PRGHO UHVSHFWLYHO\ IRU WKH XQVL]HG IHHG DIWHU LW KDV EHHQ VL]HG DQG JUDGH DQG UHFRYHU\ ZDV GHWHUPLQHG IRU HDFK VL]H $V FDQ EH VHHQ IURP WKHVH ILJXUHV WKH K\EULG PRGHO VXFFHVVIXOO\ SUHGLFWV ERWK UHFRYHU\ DQG JUDGH PAGE 41 3UHGLFWHGO IORWDWLRQ UDWH FRQVWDQWV IRU JDQJXH NJf ([SHULPHQWDO IORWDWLRQ UDWH FRQVWDQWV IRU JDQJXH NJf )LJXUH 3HUIRUPDQFH RI 11,, 0RGHO YHUVXV H[SHULPHQWDO IORWDWLRQ UDWH FRQVWDQW IRU JDQJXH NJf PAGE 42 LJXUH 3HUIRUPDQFH RI 11,,, 0RGHO YHUVXV H[SHULPHQWDO DLU KROGXS IRU IURWKHU &3 WR PAGE 43 ([SHULPHQWDO 5HFRYHU\ bf )LJXUH 3HUIRUPDQFH RI WKH RYHUDOO K\EULG PRGHO 3UHGLFWHG YHUVXV H[SHULPHQWDO UHFRYHU\ bf IRU FRDUVH IHHG VL]H GLVWULEXWLRQ PAGE 44 ([SHULPHQWD *UDGH bf )LJXUH 3HUIRUPDQFH RI WKH RYHUDOO K\EULG PRGHO 3UHGLFWHG YHUVXV H[SHULPHQWDO JUDGH b%3/f IRU FRDUVH IHHG VL]H GLVWULEXWLRQ PAGE 45 )LJXUH 3HUIRUPDQFH RI WKH RYHUDOO K\EULG PRGHO 3UHGLFWHG YHUVXV H[SHULPHQWDO UHFRYHU\ bf IRU ILQH IHHG VL]H GLVWULEXWLRQ PAGE 46 )LJXUH 3HUIRUPDQFH RI WKH RYHUDOO K\EULG PRGHO 3UHGLFWHG YHUVXV H[SHULPHQWDO JUDGH b%3/f IRU ILQH IHHG VL]H GLVWULEXWLRQ PAGE 47 ([SHULPHQWDO 5HFRYHU\ bf )LJXUH 3HUIRUPDQFH RI WKH RYHUDOO K\EULG PRGHO 3UHGLFWHG YHUVXV H[SHULPHQWDO UHFRYHU\ bf IRU WKH XQVL]HG IHHG DIWHU LW KDV EHHQ VL]HG PAGE 48 ([SHULPHQWDO *UDGH bf )LJXUH 3HUIRUPDQFH RI WKH RYHUDOO K\EULG PRGHO 3UHGLFWHG YHUVXV H[SHULPHQWDO JUDGH b%3/f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f )LUVW RUGHU UDWHV RI QHW DWWDFKPHQW DUH DVVXPHG IRU ERWK $UWLILFLDO QHXUDO QHWZRUNV UHODWH WKH DWWDFKPHQW UDWH FRQVWDQWV WR WKH RSHUDWLQJ YDULDEOHV ([SHULPHQWV ZHUH FRQGXFWHG LQ D f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fKLFK FDVH WKH SKRVSKDWH FRQWDLQLQJ URFN IUDQNROLWH RU DSDWLWHf LV WR EH VHSDUDWHG IURP JDQJXH PRVWO\ VLOLFDf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f ,Q VXFK DSSOLFDWLRQV WKH FROXPQ FDQ EH GLYLGHG LQWR WKUHH ]RQHV DQ XSSHU IURWK ]RQH D ORZHU FROOHFWLRQ ]RQH DQG DQ LQWHUPHGLDWH LQWHUIDFH ]RQH $Q DGGLWLRQDO fZDVK ZDWHUf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f RU LWV LQYHUVH WKH GLVSHUVLRQ QXPEHU JRYHUQV WKH GHJUHH RI PL[LQJ 0RVW PRGHOV RQO\ FRQVLGHU WKH VOXUU\ SKDVH )LQFK DQG 'REE\ /XWWUHOO DQG PAGE 52 DWWDFKPHQW DQG GHWDFKPHQW DUH PRGHOHG VHSDUDWHO\ ZLWK ILUVW RUGHU UDWHV /XWWUHOO DQG PAGE 53 QHWZRUNV LH GHWHUPLQHG WKH QHXUDO QHWZRUN SDUDPHWHUVf RQ WKH HUURU RI WKH RXWSXW RI WKH ILUVWSULQFLSOHV PRGHO $ VLPLODU DSSURDFK ZDV IROORZHG E\ 5HXWHU HW DO f WR PRGHO PHWDOOXUJ\ DQG PLQHUDO SURFHVVHV /LX HW DO f GHYHORSHG D K\EULG PRGHO IRU D SHULRGLF ZDVWHZDWHU WUHDWPHQW SURFHVV E\ XVLQJ $11V IRU WKH ELRNLQHWLF UDWHV RI D ILUVW SULQFLSOHV PRGHO 7KH 3VLFKRJLRV DQG 8QJDU D Ef DSSURDFK ZDV XVHG E\ &XELOOR HW DO f WR PRGHO SDUWLFXODWH GU\LQJ SURFHVVHV DQG E\ &XELOOR DQG /LPD f WR GHYHORS D K\EULG PRGHO IRU D URXJKHU IORWDWLRQ FLUFXLW 7KRPSVRQ DQG .UDPHU f FRPELQHG WKH SDUDOOHO DQG VHULDO K\EULG PRGHOLQJ DSSURDFKHV $V LQ WKH 3VLFKRJLRV DQG 8QJDU D Ef DSSURDFK WKH K\EULG PRGHO SUHVHQWHG KHUH XVHV EDFNSURSDJDWLRQ $11V IRU FHUWDLQ SDUDPHWHUV RI D )30 +RZHYHU LQVWHDG RI WUDLQLQJ WKHVH $11V RQ WKH HUURUV RI WKH PHDVXUHG RXWSXWV RI WKH )30 JUDGH DQG UHFRYHU\f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f G& 3L GW 8 9J I 8Â UM BSM UBU' 3L NSG0&M $] $] S S SL 7-B nfVL X 9 H & SL $] &-f & 3 3L $]f aNQGRf&Â S Y S S X LI XÂ OO LI XÂ! X /DYHU WR N N IHHG OD\HU f GBA GW 8 A XL &&&&&A fÂ§ANSGSf&Â U 8 O X $] $]f ( &O &n &O &, &, 3O' Sn Â‘ NBG-Sf&Â $] $] S S Sn 8' cIX /! X )HHG /DYHU N f G&3N 4I $RfFL /L 8Â f&Â XM fF ( 3N $] SN &&&' $]f 0:N LI 8Â X fÂ§ GW 4I$f&8XÂÂf&/A 8Âf&_N F B $] 3N &&3N 3N $] LI XL /$ H f PAGE 57 /DYHU N WR Q G& S GW 8 8Â &L &O 3L $] &@' fÂ§A &O &L 3L 3nL $]f aN'G-'f&-' f /DYHU Q ERWWRPf G& 3Q GW 8 Yr nfÂ§ 8Â n 4 F Â‘ B 3Q OHJf$F \ 8Â &Â VO 3Q $] 8 A I 4 nfÂ§X? D L A rn 3Q n %Jf$W $] ML &&fÂ§ a fNSG-Sf&LQ $] LI XÂ V/ H 8L &3 &&' NSGLfFQ $] LI 8Â X H f ZKHUH $F &URVVVHFWLRQDO DUHD RI WKH FROXPQ &?Â‘ 3KRVSKDWH IHHG FRQFHQWUDWLRQ RI M PHVK VL]H SDUWLFOHV & 3KRVSKDWH FRQFHQWUDWLRQ RI MOK PHVK VL]H SDUWLFOHV LQ WKH LWK OD\HU 4I )HHG YROXPHWULF IORZ UDWH 4 7DLOLQJV YROXPHWULF IORZ UDWH 4H (OXWULDWLRQ YROXPHWULF IORZ UDWH 4S 3URGXFW YROXPHWULF IORZ UDWH 8S 6XSHUILFLDO OLTXLG YHORFLW\ DERYH WKH IHHG SRLQW 4SDF 8 6XSHUILFLDO OLTXLG YHORFLW\ EHORZ WKH IHHG SRLQW 44Hf$H 8M 6OLS YHORFLW\ RI MWK PHVK VL]H SDUWLFOHV HJ $LU KROGXS NSGSf )ORWDWLRQ UDWH FRQVWDQW IRU SKRVSKDWH IRU MnK PHVK VL]H SDUWLFOHV PAGE 58 7KH VOLS YHORFLW\ LV FDOFXODWHG XVLQJ WKH H[SUHVVLRQ RI9LOOHQHXYH HW DO f WWM BiGc!3V3L;OfVf VL AO 5ÂS f ZKHUH WKH SDUWLFOH 5H\QROGV QXPEHU LV GHILQHG DV 5BGXASVL_fVf f f ZKHUH J $FFHOHUDWLRQ GXH WR JUDYLW\ PV f 3L :DWHU YLVFRVLW\ NJPUf SL :DWHU GHQVLW\ NJP f 3L 6ROLG GHQVLW\ NJPf `fV 9ROXPH IUDFWLRQ RI VROLGV LQ VLXUU\ GS 3DUWLFOH GLDPHWHU Pf $V WKH ULJKW KDQG VLGH RI (TXDWLRQ LV D IXQFWLRQ RI 8 WKH VOLS YHORFLW\ LV REWDLQHG E\ VROYLQJ (TXDWLRQV DQG LWHUDWLYHO\ DV GHVFULEHG LQ &KDSWHU 7KH D[LDL GLVSHUVLRQ FRHIILFLHQW LV FDOFXODWHG E\ D PRGLILFDWLRQ RI WKH )LQFK DQG 'REE\ f H[SUHVVLRQ I 7 9 OHJfGF f ZKHUH GR -J FROXPQ GLDPHWHU Pf VXSHUILFLDO DLU YHORFLW\ FPVf PAGE 59 (TXDWLRQV DQDORJRXV WR DUH YDOLG IRU WKH JDQJXH SDUWLFOHV EXW ZLWK D FRQVLGHUDEO\ OHVVHU HIIHFWLYH IORWDWLRQ UDWH FRQVWDQW NJGÂf ,Q WKH OLPLW DV $] WKH DERYH GLIIHUHQFH HTXDWLRQV EHFRPH ; L I F? 8 8Â HFL F] G&@ 8/a 0: f IRU WKH VHFWLRQ DERYH WKH IHHG DQG &M GW X YLBHJ G&! fÂ§3' S F] F] a0: f IRU WKH VHFWLRQ EHORZ WKH IHHG 5HFRYHU\ bf LV GHILQHG DV WKH UDWLR RI WKH ZHLJKW RI WKH SKRVSKDWH LQ WKH FRQFHQWUDWH VWUHDP WR WKH Z HLJKW RI WKH SKRVSKDWH LQ WKH IHHG 7KH UHFRYHU\ IRU SKRVSKDWH SDUWLFOHV RI WKH M PHVK VL]H FDQ EH H[SUHVVHG LQ WHUPV RI WKH IHHG DQG WDLOLQJV IORZ UDWHV DQG FRQFHQWUDWLRQV DV U n4I&>4$WOHf8Â@FLS 4F 022 f *UDGH D PHDVXUH RI WKH TXDOLW\ RI WKH SURGXFW LV GHILQHG DV WKH UDWLR RI WKH ZHLJKW RI WKH SKRVSKDWH WR WKH WRWDO ZHLJKW UHFRYHUHG LQ WKH FRQFHQWUDWH VWUHDP *UDGH LV XVXDOO\ UHSRUWHG DV b %RQH 3KRVSKDWH RI /LPH b%3/f ZKLFK LV WKH HTXLYDOHQW JUDPV RI WQFDOFLXP SKRVSKDWH &D3&!f LQ J RI VDPSOH )RU WKH W\SLFDO )ORULGD URFN PAGE 60 PLQHUDO WKDW FRQWDLQV QR JDQJXH LV b%3/ *UDGH FDQ EH REWDLQHG DV WKH UDWLR RI SKRVSKDWH WR WKH VXP RI SKRVSKDWH DQG JDQJXH LQ WKH SURGXFW *4I&M a>4L $& HJf8M @FcQ ? 4I&I>4W$FOHf8Â@&ÂLLf4I&LB>4W$FO(Jf8Â@&Af JQ r f ZKHUH &JQ LV JDQJXH FRQFHQWUDWLRQ RI MWK SDUWLFOH VL]H LQ WKH QOK OD\HU DQG &A LV WKH JDQJXH IHHG FRQFHQWUDWLRQ RI MWK SDUWLFOH VL]H 7KH DOJRULWKP IRU VROYLQJ WKH ILUVW SULQFLSOHV PRGHO LV JLYHQ LQ $SSHQGL[ % &DOFXODWLRQ RI 0RGHO 3DUDPHWHUV 6LQFH DLUKROGXS HJ LV PHDVXUHG H[SHULPHQWDOO\ WKH DERYH )30 KDV RQO\ WZR XQPHDVXUHG PRGHO SDUDPHWHUV IRU HDFK SDUWLFOH VL]H QDPHO\ WKH IORWDWLRQ UDWH FRQVWDQWV IRU SKRVSKDWH NSf DQG IRU JDQJXH NJf 7KH H[SHULPHQWDO DQDO\VLV XVXDOO\ DYDLODEOH LQ LQGXVWULDO IORWDWLRQ FROXPQV LV LQ WHUPV RI JUDGH DQG UHFRYHU\ RI SKRVSKDWH /HW f b WN f rWK f GHQRWH WKH ZHLJKW RI M VL]H SKRVSKDWH SDUWLFOHV LQ WKH IHHG :J WKH ZHLJKW RI M VL]H I/ JDQJXH LQ WKH IHHG DQG :JWKH ZHLJKW RI Mn VL]H JDQJXH LQ WKH SURGXFW 7KH JUDGH RI IHHG LV WKHQ *c :WS Z f DQG *LV JLYHQ E\ DQ DQDORJRXV H[SUHVVLRQ 7KH UHFRYHU\ RI JDQJXH FDQ EH UHDGLO\ FDOFXODWHG IURP PHDVXUHPHQWV RI JUDGH DQG UHFRYHU\ RI SKRVSKDWH XVLQJ WKH IROORZLQJ UHODWLRQVKLS PAGE 61 5LB:ÂA5-S*^*-f Â£ :Â *-*If f ,Q VRPH FDVHV GLUHFW PHDVXUHPHQWV RI WKH PDMRULW\ RI JDQJXH DV DFLG LQVROXEOHV PD\ EH DYDLODEOH 7KHQ PRUH UHOLDEOH HVWLPDWHV RI 5A FDQ EH REWDLQHG E\ DYHUDJLQJ YDOXHV FDOFXODWHG IURP PHDVXUHPHQWV RI DFLG LQVROXEOHV ZLWK YDOXHV FDOFXODWHG IURP (TXDWLRQ 7KLV ZDV GRQH LQ WKLV ZRUN )URP WKH )30 HTXDWLRQV IROORZV WKDW WKH UHFRYHU\ RI SKRVSKDWH GHSHQGV RQO\ RQ NS ZKLOH WKH UHFRYHU\ RI JDQJXH GHSHQGV RQO\ RQ NJ 7KLV FDQ EH H[SORLWHG WR HDVLO\ LQYHUW WKH VWHDG\VWDWH YHUVLRQ RI WKH PRGHO WR GHWHUPLQH IURP H[SHULPHQWDO PHDVXUHPHQWV RI 5S DQG *FRUUHVSRQGLQJ NS DQG NJ $V VKRZQ LQ )LJXUH WKLV LV DFFRPSOLVKHG ZLWK RQHGLPHQVLRQDO VHDUFKHV 7KH VHDUFK IRU NS LV LQLWLDOL]HG ZLWK WZR YDOXHV WKDW \LHOG HUURUV LQ WKH FRUUHVSRQGLQJ UHFRYHU\ 5Â RI RSSRVLWH VLJQ 6LQFH W\SLFDOO\ NS PLQn WKH YDOXHV RI DQG PLQn DUH XVHG 7KHQ WKH PHWKRG RI IDOVH SRVLWLRQ &KDSUD DQG &DQDOH f LV XVHG WR LWHUDWH XQWLO WKH PDJQLWXGH RI WKH HUURU LQ 5A GURSV WR OHVV WKDQ n? ,W LV SRVVLEOH WKDW WKH FDOFXODWHG UHFRYHU\ KDV D KLJKHU YDOXH WKDQ WKH H[SHULPHQWDO HYHQ IRU NS ,Q WKHVH FDVHV NS LV VHW HTXDO WR ]HUR 7KH DERYH SURFHGXUH LV DOVR XVHG WR GHWHUPLQH NJ H[FHSW WKDW WKH KLJK LQLWLDO YDOXH LV VHW WR PLQn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f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f DQG PAGE 63 6XSHUILFLDO DLU YHORFLW\ )URWKHU FRQFHQWUDWLRQ PAGE 64 NJ 11,,f DQG UHFHLYHV DV DQ LQSXW WKH LQIHUUHG EXEEOH VL]H 7KLV LV WKH RXWSXW RI RQH RI WKH $11V RI WKH VHFRQG WRSf OHYHO 11,, 7KH VHFRQG OHYHO DOVR LQFOXGHV 11,9 ZKLFK SUHGLFWV DLU KROGXS 7KH DGYDQWDJH RI WKLV VWUXFWXUH LV WKDW 11, DQG 11,, DUH LQGHSHQGHQW RI WKH W\SH RI EURWKHU DQG VSDUJHU XVHG DQG WKHUHIRUH ZRXOG QRW QHHG UHWUDLQLQJ LI WKHVH FKDQJH $V EXEEOH VL]H LV QRW PHDVXUHG LQ LQGXVWU\ ZH LQIHU LW IURP WKH WZRSKDVH DLUZDWHUf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f IURP &DUJLOO ZDV XVHG DV WKH IHHG PDWHULDO )RU HDFK UDQ NJ RI IHHG ZHUH LQWURGXFHG WR WKH SUHWUHDWPHQW WDQN DQG ZDWHU ZnDV DGGHG WR REWDLQ b VROLGV FRQFHQWUDWLRQ E\ ZHLJKW 7KH WDQN ZDV WKHQ DJLWDWHG IRU VHFRQGV b VRGD DVK VROXWLRQ ZDV DGGHG WR WKH SXOS WR UHDFK S+ RI DERXW DQG WKH VOXUU\ ZDV DJLWDWHG IRU DQRWKHU VHFRQGV 6XEVHTXHQWO\ D PL[WXUH RI IDWW\ DFLGV D PL[WXUH RI ROHLF SDOPHWLF DQG LLQROHLF DFLG REWDLQHG IURP :HVWYDFRf DQG IXHO RLO 1R REWDLQHG IURP 3&6 3KRVSKDWHVf ZLWK D UDWLR RI E\ ZHLJKW ZDV DGGHG WR WKH SXOS DQG WKH VOXUU\ FRQWLQXHG WR EH PL[HG 7KH WRWDO FRQGLWLRQLQJ WLPH ZDV PLQXWHV 7KH FRQGLWLRQHG IHHG PDWHULDO ZLWKRXW LWV FRQGLWLRQLQJ ZDWHUf ZDV VXEVHTXHQWO\ ORDGHG WR WKH IHHGHU ELQ ORFDWHG DW WKH WRS RI WKH FROXPQ )RXU IURWKHUV ZHUH XVHG WZR FRPPRQO\ HPSOR\HG LQ LQGXVWU\ ) D PL[HG SRO\JO\FRO E\ 2UHSUHSf DQG &3 D VRGLXP DON\O HWKHU VXOIDWH E\ :HVWYDFRf DQG WZR H[SHULPHQWDO ) DOVR D PL[HG SRO\JO\FRO E\ 2UHSUHSf DQG 2% E\ 2f%ULHQf )URWKHUFRQWDLQLQJ ZDWHU DQG DLU ZHUH ILUVW LQWURGXFHG LQWR WKH FROXPQ WKURXJK WKH VSDUJHU DQ HGXFWRUf DW D IL[HG ZDWHU IORZ UDWH DQG IURWKHU FRQFHQWUDWLRQ SSPf DQG WKH VXSHUILFLDO DLU YHORFLW\ UDQJHG IURP FPV 7KHQ WKH GLVFKDUJH YDOYH PAGE 66 DQG SXPS ZHUH DGMXVWHG WR JHW WKH GHVLUHG XQGHUIORZ DQG RYHUIORZ UDWHV $LU KROGXS ZDV PHDVXUHG XVLQJ D GLIIHUHQWLDO SUHVVXUH JDXJH $IWHU WKH ZDWHUDLU V\VWHP UHDFKHG VWHDG\ VWDWH WKH VFUHZ IHHGHU ZDV VWDUWHG 7R DFKLHYH VWHDG\ IHHG UDWH WR WKH FROXPQ ZDWHU ZDV DGGHG WR WKH VFUHZ IHHGHU DW WKH UDWH WKDW UHGXFHG WKH VROLGV FRQFHQWUDWLRQ WR DSSUR[LPDWHO\ b E\ ZHLJKW 7KH FROXPQ ZDV UXQ IRU D SHULRG RI WKUHH PLQXWHV ZLWK SKRVSKDWH IHHG SULRU WR VDPSOLQJ 7LPHG VDPSOHV RI WDLOLQJV DQG FRQFHQWUDWHV ZHUH WDNHQ 7KH FROOHFWHG SURGXFW VDPSOHV DV ZHOO DV IHHG VDPSOHV ZHUH GULHG VLHYHG XVLQJ 7\OHU PHVKHV ZHLJKHG DQG DQDO\]HG IRU b%3/ IROORZLQJ WKH SURFHGXUH UHFRPPHQGHG E\ WKH $VVRFLDWLRQ RI )ORULGD 3KRVSKDWH &KHPLVWV $)3& $QDO\WLFDO 0HWKRGV f ,Q DGGLWLRQ JDQJXH FRQWHQW DV b DFLG LQVROXEOHVf RI WKH IHHG WDLOLQJV DQG FRQFHQWUDWH VWUHDPV ZDV PHDVXUHG $)3& $QDO\WLFDO 0HWKRGV f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f )RU HDFK RI WKH IRXU IIRWKHUV LQYHVWLJDWHG WZRSKDVH H[SHULPHQWV ZHUH FRQGXFWHG IXOO IDFWRULDO GHVLJQ ZLWK IURWKHU FRQFHQWUDWLRQV DQG VXSHUILFLDO DLU YHORFLWLHVf 7KHVH ZHUH XVHG WR WUDLQ GDWD SRLQWVf DQG WR YDOLGDWH GDWD SRLQWVf WKH WRS OHYHO QHXUDO QHWZRUNV 11,,, DQG 11,9f D GLIIHUHQW SDLU IRU HDFK IURWKHU 7KUHHSKDVH UXQV \LHOGHG H[SHULPHQWDO JUDGHV DQG UHFRYHULHV ZKLFK ZHUH XVHG WR WUDLQ GDWD SRLQWVf DQG WR YDOLGDWH GDWD SRLQWVf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f 7KH RSWLPDO ZHLJKWV ZHUH GHWHUPLQHG E\ FRPELQLQJ VLPXODWHG DQQHDOLQJ .LUNSDWULFN HW DO f ZLWK WKH 3RODN5LELHUH FRQMXJDWH JUDGLHQW DOJRULWKP 3RODN f 6LPXODWHG DQQHDOLQJ UDQGRPO\ SHUWXUEHG WKH LQGHSHQGHQW YDULDEOHV WKH ZHLJKWVf DQG NHSW WUDFN RI WKH EHVW ORZHVW HUURUf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f WKH QHWZRUN IRU SUHGLFWLQJ DLU KROGXS 11,9f WKH QHWZRUN IRU SUHGLFWLQJ WKH SKRVSKDWH IORWDWLRQ UDWH FRQVWDQW 11,f DQG WKH QHWZRUN IRU SUHGLFWLQJ WKH JDQJXH IORWDWLRQ UDWH FRQVWDQW 11,,f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f DJDLQVW WKRVH GHWHUPLQHG IURP RQHGLPHQVLRQDO VHDUFKHV XVLQJ H[SHULPHQWDO GDWD $V VKRZQ LQ WKLV ILJXUH 11, GRHV DFFXUDWHO\ SUHGLFW ORZ DQG KLJK YDOXHV RI IORWDWLRQ UDWH FRQVWDQWV PAGE 69 0RGHO %XEEOH 'LDPHWHU PPf )LJXUH 3HUIRUPDQFH RI 11,,, 0RGHO EXEEOH GLDPHWHU YHUVXV EXEEOH GLDPHWHU LQIHUUHG IURP H[SHULPHQWDO GDWD ZKHQ &3 ZDV WKH IURWKHU PAGE 70 0RGHO 'LDPHWHU PPf ,QIHUUHG %XEEOH 'LDPHWHU PPf )LJXUH 3HUIRUPDQFH RI 11,,, 0RGHO EXEEOH GLDPHWHU YHUVXV EXEEOH GLDPHWHU LQIHUUHG IURP H[SHULPHQWDO GDWD ZKHQ ) ZDV WKH EURWKHU PAGE 71 0RGHO GLDPHWHU PPf )LJXUH 3HUIRUPDQFH RI 11,,, 0RGHO EXEEOH GLDPHWHU YHUVXV EXEEOH GLDPHWHU LQIHUUHG IURP H[SHULPHQWDO GDWD ZKHQ 2% ZDV WKH IURWKHU PAGE 72 0RGHO GLDPHWHU PPf ,QIHUUHG EXEEOH GLDPHWHU PPf )LJXUH 3HUIRUPDQFH RI11,,, 0RGHO EXEEOH GLDPHWHU YHUVXV EXEEOH GLDPHWHU LQIHUUHG IURP H[SHULPHQWDO GDWD ZKHQ ) ZDV WKH IURWKHU PAGE 73 Â’ Â’ 3 Â’ f 7UDLQLQJ GDWD Â’ 9DOLGDWLRQ GDWD ([SHULPHQWDO $LU +ROGXS bf 2Q 3HUIRUPDQFH RI 11,9 0RGHO YHUVXV H[SHULPHQWDO DLU KROGXS IRU IURWKHU &3 PAGE 74 Â’ t k7UDLQLQJ GDWD Â’ 9DOLGDWLRQ GDWD ([SHULPHQWDO $LU +ROGXS bf &1 LJXUH 3HUIRUPDQFH RI 11,9 0RGHO YHUVXV H[SHULPHQWDO DLU KROGXS IRU IURWKHU PAGE 75 )LJXUH 3HUIRUPDQFH RI 11,9 0RGHO YHUVXV H[SHULPHQWDO DLU KROGXS IRU IURWKHU 2% PAGE 76 ([SHULPHQWDO $LU +ROGXS bf )LJXUH 3HUIRUPDQFH RI 11,9 0RGHO YHUVXV H[SHULPHQWDO DLU KROGXS IRU IURWKHU ) PAGE 77 ([SHULPHQWDO IORWDWLRQ UDWH FRQVWDQWV IRU 3KRVSKDWH NSf 21 )LJXUH 3HUIRUPDQFH RI 11,0RGHO YHUVXV H[SHULPHQWDO IORWDWLRQ UDWH FRQVWDQW IRU SKRVSKDWH NSf PAGE 78 )LJXUH SUHVHQWV WKH IORWDWLRQ UDWH FRQVWDQWV IRU JDQJXH NJf SUHGLFWHG XVLQJ 11,, DJDLQVW WKRVH GHWHUPLQHG IURP H[SHULPHQWDO GDWD $ YHU\ JRRG PDWFK LV VHHQ 7KH K\EULG PRGHO LQWHJUDWHV 11 11,, 11,,, DQG 11,9 ZLWK WKH )30 DV VKRZQ LQ )LJXUH 3UHGLFWLRQV RI WKH K\EULG PRGHO DUH VKRZQ LQ )LJXUHV DQG )LJXUH SUHVHQWV WKH SUHGLFWHG UHFRYHU\ bf DJDLQVW WKH H[SHULPHQWDO UHFRYHU\ IRU IURWKHU &3 VTXDUH SRLQWVf ) FLUFOHVf 2% WULDQJOHVf DQG ) GLDPRQGVf 6LPLODUO\ )LJXUH FRPSDUHV WKH SUHGLFWHG JUDGH b%3/f DJDLQVW WKH H[SHULPHQWDO JUDGH IRU &3 ) 2% DQG ) ,W FDQ EH VHHQ IURP WKHVH ILJXUHV WKDW SUHGLFWHG UHFRYHU\ DQG JUDGH IURP WKH K\EULG PRGHO PDWFK FORVHO\ WKH H[SHULPHQWDO YDOXHV ZLWK WKH H[FHSWLRQ RI RQH JUDGH IRU 2% 7KH URRW PHDQ VTXDUHG HUURUV LQ SUHGLFWHG UHFRYHU\ ZHUH b b b DQG b IRU &3 ) 2% DQG ) UHVSHFWLYHO\ 7KH URRW PHDQ VTXDUHG HUURUV LQ SUHGLFWHG JUDGH ZHUH b%3/ b%3/ b%3/ DQG b%3/ IRU &3 ) 2% DQG ) UHVSHFWLYHO\ $Q DOWHUQDWLYH WR WKH SUHVHQW PRGHOLQJ DSSURDFK LV WR GHYHORS D SXUH QHXUDO QHWZRUNV PRGHO 7KLV ZRXOG KRZHYHU UHTXLUH D ODUJH QXPEHU RI LQSXWV QRW RQO\ VXSHUILFLDO DLU YHORFLW\ IURWKHU FRQFHQWUDWLRQ DQG SDUWLFOH VL]H EXW DOVR IHHG IORZ UDWH IHHG FRQFHQWUDWLRQ HOXWULDWLRQ IORZ UDWH WDLOLQJV IORZ UDWH DQG VROLGV ORDGLQJ 7KLV LQFUHDVH LQ QXPEHU RI LQSXWV WR HLJKW ZRXOG LQFUHDVH WKH QXPEHU RI ZHLJKWV PRGHO SDUDPHWHUVf QHHGHG DQG WKHUHIRUH WKH QXPEHU RI WKUHHSKDVH GDWD UHTXLUHG IRU WUDLQLQJ )XUWKHUPRUH DV ZLWK DQ LQVHULHV K\EULG PRGHO WKDW XVHV RQH OHYHO RI QHXUDO QHWZRUNV D FKDQJH LQ IURWKHU RU VSDUJHU ZRXOG UHTXLUH JHQHUDWLRQ RI D QHZ VHW RI GDWD DQG UHWUDLQLQJ RI DOO WKH QHWZRUNV 7KH K\EULG PRGHO SUHVHQWHG KHUH ZLWK WKH WZR OHYHOV RI QHXUDO PAGE 79 3UHGLFWHG IORWDWLRQ UDWH FRQVWDQWV IRU JDQJXH NJf )LJXUH 3HUIRUPDQFH RI 11,Â 0RGHO YHUVXV H[SHULPHQWDO IORWDWLRQ UDWH FRQVWDQW IRU JDQJXH NA PAGE 80 ([SHULPHQWDO 5HFRYHU\ bf )LJXUH 3HUIRUPDQFH RI WKH RYHUDOO K\EULG PRGHO 3UHGLFWHG YHUVXV H[SHULPHQWDO UHFRYHU\ bf IRU WKH IRXU IURWKHUV PAGE 81 ([SHULPHQWDO *UDGH b%3/f JXUH 3HUIRUPDQFH RI WKH RYHUDOO K\EULG PRGHO 3UHGLFWHG YHUVXV H[SHULPHQWDO JUDGH Â•%3/f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bf DQG JUDGH b%3/f 7R JXLGH RSWLPL]DWLRQ LW LV QHFHVVDU\ WR FRPELQH WKH WZR RXWSXWV JUDGH DQG UHFRYHU\f LQ D VLQJOH SHUIRUPDQFH PHDVXUH 6HYHUDO SHUIRUPDQFH PHDVXUHV DUH SRVVLEOH DQG VRPH DUH SUHVHQWHG EHORZ 6HOHFWLYLW\ 2QH ZD\ WR DFKLHYH WKLV LV WR XVH VHOHFWLYLW\ DV WKH SHUIRUPDQFH PHDVXUH 6HOHFWLYLW\ LV GHILQHG DV ZKHUH 5 5HFRYHU\ RI SKRVSKDWH LQ WKH SURGXFW VWUHDP 5E 5HFRYHU\ RI JDQJXH LQ WKH SURGXFW VWUHDP 5W 5HFRYHU\RU 5HMHFWDELOLW\f RI SKRVSKDWH LQ WKH WDLOLQJV VWUHDP 5WE 5HFRYHU\RU 5HMHFWDELOLW\f RI JDQJXH LQ WKH WDLOLQJV VWUHD :H GHYHORSHG WKH IROORZLQJ H[SUHVVLRQ WKDW UHODWHV VHOHFWLYLW\ WR WKH UHFRYHU\ DQG WKH JUDGH RI WKH SURGXFW VWUHDP PAGE 85 V fÂ§ JLUfU *I *fO 5f f ZKHUH *UDGH b%3/f RI SKRVSKDWH LQ WKH SURGXFW VWUHDP *I *UDGH %3/f RI SKRVSKDWH LQ WKH IHHG 6HSDUDWLRQ (IILFLHQF\ 6HSDUDWLRQ HIILFLHQF\ LV GHILQHG DV IROORZV ( 5 5E f ,Q WKLV FDVH WKH HIILFLHQF\ YDULHV EHWZHHQ WR (FRQRPLF 3HUIRUPDQFH 0HDVXUH 7KH VHOHFWLYLW\ IXQFWLRQ RU WKH VHSDUDWLRQ HIILFLHQF\ GRHV QRW LQFOXGH DQ\ HFRQRPLF LQSXW VXFK DV FRVW RI WKH UHDJHQWV 7KHUHIRUH DQ DOWHUQDWH SHUIRUPDQFH PHDVXUH ZDV GHYHORSHG ZKLFK LQFOXGHV UHFRYHU\ JUDGH DQG WKH UHDJHQW SULFHV $ VFKHPH IRU SHQDOL]LQJ ORZHU JUDGH URFN KDV EHHQ GHYHORSHG 7KLV VFKHPH GHGXFWV GLIIHUHQWLDO FRVWV UHODWLYH WR b %3/ IRU WUDQVSRUWDWLRQ DQG DFLGXODWLRQ 7KH DFLGXODWLRQ VFKHPH DVVXPHV VROXEOH 32 ORVVHV LQFUHDVH LQ GLUHFW SURSRUWLRQ WR WKH DPRXQW RI SKRVSKRJ\SVXP 7KXV WKH SURFHGXUH UHTXLUHV DQ HVWLPDWH RI WKH TXDQWLW\ RI SKRVSKRJ\SVXP WKDW LV SURGXFHG PAGE 86 7KLV SHUIRUPDQFH PHDVXUH LV RQO\ DSSOLFDEOH WR SODQWV DQG FDQ QRW EH XVHG ZLWK D ODE VFDOH IORWDWLRQ FROXPQ 7KH SURFHGXUH IRU WKLV VFKHPH LV RXWOLQHG EHORZ $VVXPSWLRQV 7KH SULFH RI URFN RI b %3/ =HUR LQVRO b%3/ 7UDQVSRUWDWLRQ FRVW SHU WRQ 6ROXEOH 32 ORVVHV b ,QVROXEOH 32 ORVVHV b ,QFUHDVH LQ VROXEOH 32 ORVVHV LV SURSRUWLRQDO WR WKH DPRXQW RI SKRVSKRJ\SVXP SURGXFHG 7UDQVSRUWDWLRQ 3HQDOW\ %DVH FDVH b %3/ URFN GU\ EDVLVf )UHLJKW FRVW SHU %3/ WRQ 3HQDOW\ Y%/P SHU %3/ WRQ I 7UDQVSRUWDWLRQ SHQDOW\ %L $ % SHU WRQ :KHUH %O b%3/ ZKHQ JUDGH b $FLGXODWLRQ 3HQDOW\ %DVH FDVH b %3/ URFN b 3 &D3 f $FLG LQVRO La/f 9 &DOFXODWLRQ RI WKH DPRXQW RI 3KRVSKRJ\SVXP 3KRVSKRJ\SVXP FRPSRQHQWV I % A WRQ URFN [ fÂ§ A $FLG LQVRO PAGE 87 8QUHDFWHG 'LK\GUDWH WRQ URFN [ WRQ URFN [ Y %Of [ [ f [ f 7RWDO DPRXQW RI SKRVSKRJ\SVXP $FLG LQVRO 8QUHDFWHG 'LK\GUDWH 6ROXEOH 32 ORVVHV [ b VROXEOH3 ORVVHVf SHU WRQ SHU WRQ 7RWDO DPRXQW RI SKRVSKRJ\SVXPf $FLGXODWLRQ 3HQDOW\ [ fÂ§ % 6DOHV YDOXH 3ULFH RI b%3/ URFN r %Of $GMXVWHG VDOHV YDOXH 6DOHV YDOXH 7UDQVSRUWDWLRQ SHQDOW\ $FLGXODWLRQ SHQDOW\ 7KH DGMXVWHG YDOXH RI WKH SKRVSKDWH URFN DV D IXQFWLRQ RI b%3/ LV VKRZQ LQ )LJXUH /HW )HHG VROLG IORZ UDWH ) WRQ SHU \HDU 3URGXFW VROLG IORZ UDWH 3 WRQ SHU \HDU )HHG JUDGH *I b &RQFHQWUDWH JUDGH b 3URGXFW UHFRYHU\ 5 b $GMXVWHG VDOHV YDOXH RI IHHG &I 6 SHU WRQ $GMXVWHG VDOHV YDOXH RI SURGXFW &S SHU WRQ 5HDJHQWL SULFH &Q OE 5HDJHQWL XVDJH 8 OEWRQ IHHG 7KH IHHG IORZ UDWH DQG WKH SURGXFW IORZ UDWH FDQ EH UHODWHG DV I *$ .a@ Â! 3 ) fÂ§ 2 2 f PAGE 88 ? 7, n7nO n,777777777777777777 77 ?! A S Q" A A L %O b%3/f )LJXUH 9DOXH RI SKRVSKDWH URFN DV D IXQFWLRQ RI b%3/ PAGE 89 3HUIRUPDQFH PHDVXUH &S3 &I) ) 7@ 8&Q \HDU f O 7KH 2SWLPL]DWLRQ $OJRULWKP 7KH LGHD EHKLQG WKH VHTXHQWLDO RSWLPL]DWLRQ LV WR LWHUDWH EHWZHHQ H[SHULPHQWDWLRQ WRZDUGV WKH RSWLPXP DQG PRGHO LGHQWLILFDWLRQ XQWLO WKH RSWLPXP LV UHDFKHG 7KH SURFHGXUH LV DV IROORZV f ,QLWLDO H[SHULPHQWV DUH SHUIRUPHG DQG WKHLU UHVXOWV DUH DQDO\]HG f 7KH QHXUDO QHWZRUNV DUH WUDLQHG DQG WKH K\EULG PRGHO LV XVHG WR GHWHUPLQH WKH RSWLPDO IDFWRU YDOXHV ,I WKHVH DUH ZLWKLQ WKH FRQYHUJHQFH OLPLW RI SUHYLRXV H[SHULPHQWDO YDOXHV WKH SURFHGXUH VWRSV f 2WKHUZLVH DQ H[SHULPHQW DW WKH FDOFXODWHG RSWLPDO YDOXH LV SHUIRUPHG DQG DQDO\]HG f 7KH GDWD DUH DGGHG WR WKH QHXUDO QHWZRUN WUDLQLQJ VHW DQG WKH SURFHGXUH UHWXUQV WR VWHS f )LJXUH VKRZV D PRUH GHWDLOHG GHVFULSWLRQ RI WKH DOJRULWKP $IWHU VRPH LQLWLDOL]DWLRQ UXQV KDYH EHHQ FRPSOHWHG WKH VDPSOHV DUH DQDO\]HG DQG WKH QHXUDO QHWZRUNV DUH WUDLQHG ZLWK WKH LQSXWRXWSXW GDWD 6XEVHTXHQWO\ XVLQJ WKH VWDQGDUG 1HOGHU 0HDGH DOJRULWKP +LPPHOEODX f WKH YDOXHV RI PDQLSXODWHG YDULDEOHV WKDW PD[LPL]H WKH VHOHFWLYLW\ DUH GHWHUPLQHG ,I WKHVH YDOXHV FRUUHVSRQG WR DQ LQWHULRU SRLQW WKHQ WKH PAGE 90 67$57 fL ,QLWLDO UXQV 9 ,&3 DQDO\VLV n 3HUIRUPDQFH PHDVXUH 7UDLQ QHXUDO QHWZRUN 'HWHUPLQH SRVLWLRQ RI PD[LPXP <(6 A 12 ,V PD[LPXP LQ LQWHULRU RI UDQJH" ([SHULPHQWDO UXQ DW ([SHULPHQWDO UXQ DW SUHGLFWHG PD[LPXP KDOIZD\ SRLQW ,&3 DQDO\VLV r n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f FDQ EH XVHG WR GHWHUPLQH WKH YDOXH RI WKH PDQLSXODWHG YDULDEOHV DW RSWLPDO SHUIRUPDQFH )RU WKUHH PDQLSXODWHG YDULDEOHV DQ LQLWLDO VLPSOH[ LV GHILQHG ZLWK IRXU SRLQWV 7KLV PHWKRG WKHQ WDNHV D VHULHV RI VWHSV PRYLQJ WKH SRLQW RI WKH VLPSOH[ ZKHUH WKH IXQFWLRQ LV ORZHVW WKURXJK WKH RSSRVLWH IDFHV RI WKH VLPSOH[ WR D KLJKHU SRLQW 7KHVH VWHSV DUH FDOOHG UHIOHFWLRQV DQG WKH\ DUH FRQVWUXFWHG WR FRQVHUYH WKH YROXPH RI WKH VLPSOH[ 7KH PHWKRG H[SDQGV WKH VLPSOH[ LQ RQH GLUHFWLRQ WR WDNH ODUJHU VWHSV :KHQ LW UHDFKHV D ORZHU SRLQW LW FRQWUDFWV LQ WKH WUDYHUVH GLUHFWLRQ 7KLV LV FRQWLQXHG WLOO WKH GHFUHDVH LQ WKH IXQFWLRQ YDOXH VHOHFWLYLW\f LV VPDOOHU WKDQ VRPH WROHUDQFH +f ,QLWLDO 6FDWWHUHG ([SHULPHQWV 6FDWWHUHG H[SHULPHQWV DFFRUGLQJ WR D IDFWRULDO GHVLJQ ZHUH SHUIRUPHG WR JHQHUDWH GDWD IRU WKH LQLWLDO WUDLQLQJ RI WKH QHXUDO QHWZRUNV 6XSHUILFLDO DLU YHORFLW\ IURWKHU FRQFHQWUDWLRQ DQG HOXWULDWLRQ ZDWHU IORZ UDWH ZHUH VHOHFWHG DV WKH PDQLSXODWHG YDULDEOHV ) ZKLFK LV D QRQLRQLF VXUIDFWDQW ZDV XVHG DV WKH IURWKHU LQ WKHVH H[SHULPHQWV PAGE 92 ([SHULPHQWV ZHUH SHUIRUPHG ZLWK ILYH GLIIHUHQW OHYHOV RI VXSHUILFLDO DLU YHORFLW\ DQG FPVf DQG IURWKHU FRQFHQWUDWLRQ SSPf DQG WKUHH OHYHOV RI HOXWULDWLRQ ZDWHU IORZ UDWH DQG JDOORQV SHU PLQf 7KH GHVLJQ RI H[SHULPHQWV LV VKRZQ LQ 7DEOH 7KH H[SHULPHQWV ZnHUH GHVLJQHG VR DV WR JHQHUDWH GDWD SRLQWV ZKLFK LV WKH PLQLPXP UHTXLUHG IRU WUDLQLQJ WKH QHXUDO QHWZRUNV ZKLFK KDYH WKUHH KLGGHQ QRGHV ([SHULPHQWV ZHUH SHUIRUPHG DFFRUGLQJ WR WKH GHVLJQ ZKLOH NHHSLQJ DOO RWKHU YDULDEOHV FRQVWDQW $IWHU HDFK H[SHULPHQW WKUHH VDPSOHV IURP WKH WDLOLQJV VWUHDP ZHUH FROOHFWHG 7KH VDPSOHV ZHUH WKHQ DQDO\]HG IRU b%3/ FRQWHQW IROORZLQJ WKH SURFHGXUH UHFRPPHQGHG E\ WKH $VVRFLDWLRQ RI )ORULGD 3KRVSKDWH &KHPLVWV $)3& $QDO\WLFDO 0HWKRGV f 6LQFH WKH JUDGH RI WKH IHHG LV NQRZQ JUDGH RI WKH FRQFHQWUDWH VWUHDP FDQ HDVLO\ EH FDOFXODWHG E\ PDNLQJ D PDWHULDO EDODQFH DURXQG WKH FROXPQ 5HVXOWV DQG 'LVFXVVLRQV 7KH WKUHH QHXUDO QHWZRUNV RI WKH K\EULG PRGHO ZHUH WUDLQHG XVLQJ GDWD SRLQWV REWDLQHG IURP WKH GHVLJQHG H[SHULPHQWV 7KH SHUIRUPDQFH RI WKHVH QHXUDO QHWZRUNV LV VKRZQ LQ )LJXUHV )LJXUH SUHVHQWV WKH SUHGLFWHG IORWDWLRQ UDWH FRQVWDQWV IRU SKRVSKDWH NSf DJDLQVW WKRVH GHWHUPLQHG IURP RQHGLPHQVLRQDO VHDUFKHV XVLQJ H[SHULPHQWDO GDWD DV GHVFULEHG LQ FKDSWHU WZR DQG WKUHH $V VKRZQ LQ WKLV ILJXUH WKH QHXUDO QHWZRUN VDWLVIDFWRULO\ FDSWXUHV WKH GHSHQGHQFH RI WKH IORWDWLRQ UDWH FRQVWDQW RQ WKH VHOHFWHG PDQLSXODWHG YDULDEOHV 6LPLODUO\ )LJXUH SUHVHQWV WKH SUHGLFWHG IORWDWLRQ UDWH FRQVWDQWV IRU JDQJXH NJf DJDLQVW WKRVH GHWHUPLQHG IURP RQHGLPHQVLRQDO VHDUFKHV XVLQJ PAGE 93 7DEOH 2SHUDWLQJ FRQGLWLRQV IRU WKH IDFWRULDO GHVLJQ )URWKHU &RQFHQWUDWLRQ 6XSHUILFLDO DLU YHORFLW\ (OXWULDWLRQ ZDWHU )ORZ UDWH PAGE 94 )LJXUH 1HXUDO QHWZRUN YHUVXV H[SHULPHQWDO IORWDWLRQ UDWH FRQVWDQW IRU SKRVSKDWH NSf PAGE 95 3UHGLFWHG IORWDWLRQ UDWH FRQVWDQWV IRUJDQJXH NJf )LJXUH 1HXUDO QHWZRUN YHUVXV H[SHULPHQWDO IORWDWLRQ UDWH FRQVWDQW IRU JDQJXH NJf PAGE 96 H[SHULPHQWDO GDWD $JDLQ D YHU\ JRRG PDWFK LV VHHQ )LJXUH FRPSDUHV WKH SUHGLFWHG DLU KROGXS WR WKH H[SHULPHQWDO YDOXHV PHDVXUHG E\ D GLIIHUHQWLDO SUHVVXUH FHOO $V VKRZQ LQ WKLV ILJXUH QHXUDO QHWZRUN VXFFHVVIXOO\ SUHGLFWV WKH DLU KROGXS 7DEOH VKRZV WKH UHVXOWV RI WKH GHVLJQHG H[SHULPHQWV $V FDQ EH VHHQ IURP WKLV 7DEOH IHHG IORZ UDWH DQG WKH bVROLGV FRQWHQW YDULHG VLJQLILFDQWO\ 7KH VFUHZ IHHGHU RSHUDWLRQ ZDV HUUDWLF DQG WKHUHIRUH ZH ZHUH XQDEOH WR IHHG DW WKH VDPH UDWH LQ HDFK UXQ )HHG IORZ UDWH ZDV FDOFXODWHG EDVHG RQ WKH SURGXFW IORZ UDWH DQG WKH WDLOLQJV IORZ UDWH $ VSHFLILHG YROXPH RI SURGXFW DQG WDLOLQJV ZHUH WDNHQ RYHU D SHULRG RI WLPH a Vf DQG WKH VDPSOHV ZHUH GULHG DQG WKH ZHLJKW ZDV WDNHQ ,Q WKLV ZD\ VROLGV IORZ UDWH LQ SURGXFW DQG WDLOLQJV VWUHDP ZHUH REWDLQHG $Q RYHUDOO PDWHULDO EDODQFH RQ WKH FROXPQ WKHQ JLYHV WKH IHHG IORZ UDWH 6LPLODUO\ DQ RYHUDOO PDWHULDO EDODQFH RQ WKH ZDWHU SKDVH JLYHV WKH ZDWHU IORZ UDWH LQ WKH IHHG VWUHDP 6ROLGV IHHG IORZ UDWH DQG WKH ZDWHU IORZ UDWH LQ WKH IHHG WKHQ FDQ EH XVHG WR REWDLQ WKH b VROLGV LQ WKH IHHG 8QIRUWXQDWHO\ WKH LQDELOLW\ WR FRQWURO IHHG IORZ UDWH DQG b VROLGV FRQWHQW PHDQV WKDW D PHDQLQJIXO UXQWRUXQ RSWLPL]DWLRQ FDQQRW EH FRQGXFWHG )XWXUH :RUN )LUVW WKH VFUHZ IHHGHU QHHGV WR EH UHSDLUHG RU UHSODFHG $IWHU WKLV KDV EHHQ DFFRPSOLVKHG WKH K\EULG PRGHO REWDLQHG IURP WKH GHVLJQHG H[SHULPHQWV )LJXUHV f VKRXOG EH XVHG ZLWK WKH 1HOGHU0HDGH DOJRULWKP WR GHWHUPLQH WKH H[SHULPHQWDO FRQGLWLRQV RI WKH ILUVW RSWLPL]DWLRQ UXQ 7KH UHVXOWV RI WKH UXQ VKRXOG EH DQDO\]HG IRU JUDGH DQG UHFRYHU\ DQG WKHVH GDWD VKRXOG EH DGGHG WR WKH QHXUDO QHWZRUN WUDLQLQJ VHWV 7KH QHWZRUNV VKRXOG WKHQ EH UHWUDLQHG DQG WKH XSGDWHG K\EULG PRGHO XVHG WR GHWHUPLQH PAGE 97 )LJXUH 0RGHO YHUVXV H[SHULPHQWDO DLU KROGXS IRU IURWKHU ) PAGE 98 7DEOH 5HVXOWV RI WKH UXQV IURP KH IDFWRULDO GHVLJQ )URWKHU FRQH SSPf $LU IORZ UDWH VFIPf )HHG IORZ UDWH JSPf 7DLOLQJV IHHG IORZ UDWH JSPf (OXWULD WLRQ IORZ UDWH JSPf 6ROLGV FRQWHQW bf *UDGH b%3/f 5HFRYHU\ bf PAGE 99 WKH FRQGLWLRQV IRU WKH QH[W UXQ 7KLV VKRXOG EH UHSHDWHG ZLWK WKH DOJRULWKP RI )LJXUH XQWLO FRQYHUJHQFH LV REWDLQHG PAGE 100 $33(1',; $ &2'( )25 7),( ),567 35,1&,3/(6 02'(/ )25 21( /(9(/ ALQFOXGH VWGLRK! LQFOXGH VWGOLEK! ALQFOXGH FRQLRK! LQFOXGH PDWKK! *47 7DLOLQJV )ORZ UDWH JDOORQVPLQf *4) )HHG )ORZ UDWH JDOORQVPLQf *4( (OXWULDWLRQ )ORZ UDWH JDOORQVPLQf &6 b 6ROLG LQ WKH IHHG OE 6OE 7f %3/ b %3/ RI WKH IHHG E 3OE 6f 526 6SHFLILF *UDYLW\ RI VROLGV LQ WKH IHHG '3 3DUWLFOH VL]H LQ PLFURQV (J DLU KROG XS 'LD 'LDPHWHU RI WKH FROXPQ IWf / +HLJKW RI WKH FROXPQ IWf .X )ORWDWLRQ UDWH FRQVW IRU 3KRVSKDWH PLQf .*X )ORWDWLRQ UDWH FRQVW IRU *DXQJH PLQf 4J $LU IORZ UDWHVFIPf )1) )HHG /RFDWLRQ IURP WKH WRS IW f /I/)1) YRLG PDLQ GRXEOHGRXEOHGRXEOH>@f YRLG PDLQ GRXEOH &) GRXEOH N GRXEOH %>@f ^ GRXEOH 4)4(474743$UHD83878)''3,3+,686/L5(386/GLL7 GRXEOH DEGDOSKDEHWDJDPPDGHOWDSTP 4) r*4) 4( r*4( 47 r*47f 43 4)474( 47 474( $UHD r'LDr'LD 83 43$UHD 87 47$UHD 8) 4)$UHD PAGE 101 ' r'LDrSRZr4J $UHDf f rrrrrrrrrrrrrr &DOFXODWLRQ RI VOLS YHORFLW\ rrrrrrrrrrrrrr '3 '3 3+,6 &6f&6f&6ffr526ff 86/L GR ^ 5(3 r'3Or86/Lr526rO 3+,6f 86/ r'3 r'3 r526OfrSRZO3+,6ffOrSRZ5(3fff GLII 86/86/L LIGLIIf GLII GLII 86/L 86/ ` ZKLOHGLII! f 8 6/ (Jf r86/ rrrrrrrrrrrrrrrrr 86/ FDOFXODWLRQ HQGV rrrrrrrrrrrrrrr D 8386/f' G 8786/f' E NrO(Jf' LIDrDrEff__GfGrEfff E DOSKD DfVTUWDrDrEff EHWD DfVTUWDrDrEff JDPPD GfVTUWGrGrEff GHOWD GfVTUWGrGrEff SULQWI;?Q DOSKD bOIDOSKDf SULQWI?Q EHWD bOInEHWDf SULQWW;?Q JDPPD bOIJDPPDf SULQWI?Q GHOWD bOIGHOWDf SULQWI?Q GHOWD bOIDrDrEf SULQW I?Q EHWDr/ bOInEHWDfr/ff SULQWI?Q DOSKDr/ bOIDOSKDfr/ff SULQWI?Q JDPPDr/I bOInJDPPDfr/Iff SULQWI?Q GHOWDr/I bOInGHOWDfr/Iff SULQWI&?Q DOSKDr/I bOIDOSKDfr/Iff SULQWIW?Q EHWDr/I bOIEHWDfr/Iff S -3'fDEHWDfrH[SEHWDfr/ffff83'fDDOSKDfrH[SDOSKDfr/ffff T 87'fGGHOWDf87'fGJDPPDf LI86/ 83f S EHWDfrH[SEHWDfr/fffDOSKDrH[SDOSKDfr/fff PAGE 102 HOVH S EHWDDffrH[SEHWDfr/fffDOSKDrDfrH[SDOSKDfr/fff T GHOWD4($UHDr'fffJDPPD4($UHDr'fff P TrH[SJDPPDfr/LffH[SGHOWDfr/IfffSrH[SDOSKDfr/IffH[SEHWDfr/Ifff %>@8)r&)'fPrSrD DOSKDfrH[SDOSKDfr/IfffGJDPPDfrTrH[SJDPPDfr/IfffPrD EHWDfrH[SEHWDfr/IfffGGHOWDfrH[SGHOWDfr/Iffff %>@ 8)r&)'fDDOSKDGJDPPDfrH[SJDPPDfr/Ifff %>@ Pr%>@ %>O@ SrPr%>@ %>@ Tr%>@ %>O@ H[SJDPPDDOSKDfr/Iffr%>@ %>@ >@ ` YRLG PDLQYRLGf ^ GRXEOH 4)4(47 4743$UHD'3 3IIO686/L 5(3 86/GLII )UDQN &) &)* .* &&* GRXEOH 5252**UDGH%>@%*>@ 4) r*4) 4( r*4( 47 r*47f 434)474( 47 474( $UHD r'LDr'LD SULQWI?Q 83 bOIn43$UHDf rrrrrrrrrrrrrr &DOFXODWLRQ RI VOLS YHORFLW\ rrrrrrrrrrrrrr '3,'3 3+,6 &6f&6f&6ffr526ff 86/L GR ^ 5(3r'3Or86/Lr526rO 3+,6f 86/ r'3r'3 r526OfrSRZO3+,6ffOrSRZ5(3fff GLIOI86/86/L LIGLI)f PAGE 103 GLIU GLII 86/L 86/ ` ZKLOHGLII! f 86/ O(Jfr86/ SULQWI?Q 86/ bOIn86/f \rrrrrrrrrrrrrrrrr 86/ FDOFXODWLRQ HQGV rrrrrrrrrrrrrrr )UDQN%3/ &))UDQNfr&6fr526r ,&6f&6ffr526f &)* )UDQNffr&6fr526r&6O f &6ffr526f .X.* .*X PDLQO&).%f &%>@%>@ 52 4)r&)f47O$UHDr86/fr&ff4)r&)ffr PDLQO&)*.*%*f &* %*>@%*>@ 52* 4)r&)*f47O$UHDr86/fr&*ff4)r&)*ffr *UDGH 4)r&)47O$UHDr86/fr&f4)r&)47O$UHDr86/fr&f4)r&)* 47O$UHDr86/fr&*fffr *UDGH *UDGH r SULQWI?Q & bOIn&f SULQW79Q &* bOIn&*f SULQWI?Q &) bOIn&)f SULQWI?Q &)* bOIn&)*f SULQWI?Q 2YHUDOO 5HFRYHU\ b OOI b52f SULQWI7?Q 52* b OOI b52*f SULQWW?Q *5$'( b OOI b*UDGHf ` LQFOXGH VWGLRK! LQFOXGH VWGOLEK! LQFOXGH FRQLRK! LQFOXGH PDWKK! GHILQH *47 GHILQH *4) 7DLOLQJV )ORZ UDWH JDOORQVPLQf )HHG )ORZ UDWH JDOORQVPLQf PAGE 104 GHIPH GHIPH GHILQH GHILQH GHILQH AGHILQH GHILQH GHILQH GHILQH GHIPH AGHILQH AGHILQH GHIPH *4( (OXWULDWLRQ )ORZ UDWH JDOORQVPLQf &6 b 6ROLG LQ WKH IHHG OE 6OE 7f %3/ b %3/ RI WKH IHHG OE 3OE 6f 526 6SHFLILF *UDYLW\ RI VROLGV LQ WKH IHHG '3 3DUWLFOH VL]H LQ PLFURQV (J DLU KROG XS 'LD 'LDPHWHU RI WKH FROXPQ IWf / +HLJKW RI WKH FROXPQ IWf .*X )ORWDWLRQ UDWH FRQVW IRU *DXQJH PLQf 4J $LU IORZ UDWHVFIPf )1) /I/)1) (6 ,( )HHG /RFDWLRQ IURP WKH WRS IWf YRLG PDLQ OGRXEOHGRXEOHGRXEOH>@f GRXEOH PRGHOGRXEOHf YRLG PDLQYRLGf ^ GRXEOH NO\OJONX\XJXNU>@\UJU5>@D($7HVW[ GRXEOH *UDGH*UDGHBIHHG*UDGHBSURG LQW L 5>O@ *UDGH *UDGHBIHHG %3/ *UDGHBSURG *UDGH 5>@ 5>O@r*UDGHBIHHGrO*UDGHBSURGfO*UDGHBIHHGfr*UDGHBSURGf SULQWU9n?Q&RPSRQHQW> @SKRVSKDWHf SULQWI?Q&RPSRQHQW>@ JDQJXHf IRUL OL L+f ^ NO \O PRGHONOf JO 5>L@\O LIJO!f ^ GR ^ SULQWI?Q(QWHU DQ LQLWLDO JXHVV IRU IORWDWLRQ UDWH FRQVWDQW IRU FRPSRQHQW>bG@Lf SULQWIL?QNX f VFDQLI?Q bOIn tNXf PAGE 105 \X PRGHONXf JX 5>L@\X D JOrJX ` ZKLOHD! f ($ r(6 ZKLOH($!(6f ^ NU>L@ NXJXrNONXffJOJXf [ NU>L@ LINONXf f ($ IDEVNXNOfNONXffr \U PRGHO[f JU 5>L@\U 7HVW JOrJU LI7HVW f ($ HOVH LI7HVWf NX NU>L@ HOVH LI7HVW!f NO NU>L@ SULQWI7?QNU>bG@ bOILNU>L@f ` SULQWI?Q)ORWDWLRQ UDWH FRQVWDQW IRU FRPSRQHQW>bG@ bOInLNU>L@f JHWFK2 ` HOVH SULQWI7?QNU>bG@ Lf ` ` YRLG PDLQ GRXEOH &) GRXEOH N GRXEOH %>@f ^ GRXEOH 4)4(474743$UHD83878)''3,3+,686/L5(386/GLI) GRXEOH DEGDOSKDEHWDJDPPDGHOWDSTP 4) r*4) 4( r*4( 47 r*47f PAGE 106 43 4)474( 47 474( $UHD r'LDr'LD 83 43$UHD 87 47$UHD 8) 4)$UHD r'LDrSRZr4J$UHDff rrrrrrrrrrrrrr &DOFXODWLRQ RI VOLS YHORFLW\ rrrrrrrrrrrrrr '3 '3 3+,6 & 6f& 6f & 6ffr526ff 86/L GR ^ 5(3 r'3 r86/Lr526rO 3+6f 86/ r'3 r'3Or526OfrSRZO3+,6ff rSRZ5(3fff GLII 86/86/L LIGLIIf GLI) GLII 86/L 86/ ` ZKLOHGLII! f 86/ O(Jfr86/ rrrrrArArrrfÂ‘frrrr 86/ FDOFXODWLRQ HQGV rrrrrrrrrrrrrrr D 8386/f' G 8786/f' E NrO(Jf' LIADrDrEff__GrGrEfff E DOSKD DfVTUWDrDrEff EHWD DfVTUWDrDrEff JDPPD GfVTUWGrGrEff GHOWD GfVTUWGrGrEff SULQWIA?Q DOSKD bOIDOSKDf SULQWI?Q EHWD bOI?EHWDf SULQWIA?Q JDPPD bOInJDPPDf SULUWI?Q GHOWD bOInGHOWDf SULQWI7?Q EHWDr/ bOInEHWDfr/ff SULQWI?Q DOSKDr/ bOInDOSKDfr/ff SULQWIf?Q JDPPDr/I bOInJDPPDfr/Lff SULQWI?Q GHOWDr/I bOInGHOWDfr/Iff PAGE 107 SULQWI?Q DOSKDr/I bOInDSKDfr/Iff SULQWI?Q EHWDr/I bOInEHWDfr/Iff S 83'fDEHWDfrH[SEHWDfr/ffff83'fDDSKDfrH[SDOSKDfr/ffff T 87'fGGHOWDf87'fGJDPPDf LI86/ 83f S EHWDfrH[SEHWDfr/fffDOSKDrH[SDOSKDfr/fff HOVH S EHWDDffrH[SEHWDfr/fffDOSKDDfrH[SDOSKDMr/fff T GHOWD4($UHDr'fffJDPPD4($UHDr'fff P TrH[SJDPPDfr/IffH[SGHOWDfr/IfffSrH[SDOSKDfr/IffH[SEHWDfr/Ifff %>@ 8)r&)'fPrSrD DOSKDfrH[SDOSKDfr/IfffGJDPPDfrTrH[SJDPPDfr/Ifff+PrD EHWDfrH[SEHWDfr/Ifffa.GGHOWDfrH[SGHOWDfr/Iffff %>@ 8)r&)'fDDOSKDGJDPPDfrH[SJDPPDfr/Ifff %>@ Pr%>@ %>O@ SrPr%>@ %>@ Tr%>@ %>O@ H[SJDPPDDOSKDfr/Iffr%>@ %>@ %>@ ` GRXEOH PRGHOGRXEOH .Xf ^ GRXEOH 4)4(47O4743$UHD'3O3+,686/L5(386/GLII)UDQN&)&)*..*&&* GRXEOH 5252**UDGH%>@%*>@ 4) r*4) 4( r*4( 47 r*47f 43 4)474( 47 474( $UHD r'LDr'LD SULQWI79Q 83 bOIf43$UHDf \rrrrrrrrrrrrrr &DOFXODWLRQ RI VOLS YHORFLW\ rrrrrrrrrrrrrr '3 '3 3+,6 &6f&6f&6ffr526ff 86/L PAGE 108 GR ^ 5(3 r'3Or86/Lr526rO3+,6f 86/ r'3 r'3 r526 frSRZ 3+,6ff rSRZ5(3fff GLII 86/86/L LILGLIIF22f GLII GLII 86/L 86/ ` ZKLOHGLII! f 86/ O(Jfr86/ SULQWI?Q 86/ bOIn86/f \rrrrrrrrrrrrrrrrr FDOFXODWLRQ HQGV rrrrrrrrrrrrrrr )UDQN %3/ &) )UDQNfr&6fr526r&6f&6ffr526f &)* )UDQNffr&6fr526r&6f &6ffr526f .X.* .*X PDLQO&).%f & %>@%>@ 52 4)r&)f47O$UHDr86/fr&ff4)r&)ffr PDLQ &)*.*%*f &* %*>@%*>@ 52* 4)r&)*f47O$UHDr86/fr&*ff4)r&)*ffr *UDGH 4)r&)47O$UHDr86/fr&f4)r&)47O$UHDr86/fr&f4)r&)* 47O$UHDr86/fr&*fffr *UDGH *UDGHr SULQWI?Q & bOIn&f SULQWI79Q &* bOIn&*f SULQWI79Q &) bOIn&)f SULQWI7?Q &)* bOIn&)*f SULQWI?Q 2YHUDOO 5HFRYHU\ b ,OI b52f SULQWI79Q 52* b ,I b52*f SULQWI7?Q *5$'( b ,I b*UDGHf UHWXUQ 52f ` PAGE 109 $33(1',; % &2'( )25 7+( ),567 35,1&,3/(6 02'(/ )25 7:2 /(9(/6 LQFOXGH VWGLRK! LQFOXGH VWGOLEK! LQFOXGH FRQLRK! LQFOXGH PDWKK! GHILQH GHILQH GHILQH GHILQH GHILQH GHILQH GHILQH GHILQH GHILQH GHILQH GHILQH GHILQH GHIPH GHILQH AGHILQH GHILQH GHILQH GHILQH *47 7DLOLQJV )ORZ UDWH JDOORQVPLQf *4) )HHG )ORZ UDWH JDOORQVPLQf *4( (OXWULDWLRQ )ORZ UDWH JDOORQVPLQf &6 b 6ROLG LQ WKH IHHG OE 6OE 7f %3/ b %3/ RI WKH IHHG OE 3OE 6f 526 6SHFLILF *UDYLW\ RI VROLGV LQ WKH IHHG,, '3 3DUWLFOH VL]H LQ PLFURQV (J DLU KROG XS 'LD 'LDPHWHU RI WKH FROXPQ IWf / +HLJKW RI WKH FROXPQ IWf .X )ORWDWLRQ UDWH FRQVW IRU 3KRVSKDWH PLQf .*X )ORWDWLRQ UDWH FRQVW IRU *DXQJH PLQf 4J $LU IORZ UDWHVFIPf 1 )1) )HHG /RFDWLRQ IURP WKH WRS IWf '(/7 Q 1O DO D fÂ§!H[SOLFLWD fÂ§!LPSOLFLW YRLG PDLQ GRXEOHGRXEOHGRXEOHGRXEOHW@f YRLG PDLQOGRXEOH &)GRXEOH .GRXEOH 'GRXEOH &>@f ^ VWDWLF GRXEOH $>Q@>Q@9>Q@6>Q@ GRXEOH $$$$$$$$$8387878)'(/=434)474(47 GRXEOH $$$$$ LQW >Q@ LQW LMNLL3LYRW,'XPP\1) GRXEOH %LJ 'XPP\IDFWRU6XP $UHD86/L86/5(3GLII3/86'3, 4) r*4) 4( r*4( PAGE 110 47 r*47f 43 4)474( 47 474( '(/= /1f 1) )1)'(/=f fr $UHD r'LDr'LD \\r rrrrrrrrrrrrr &DOFXODWLRQ RI VOLS YHORFLW\ rrrrrrrrrrrrrr '3 '3 3+,6 &6f&6f&6ffr526ff 86/L GR ^ 5(3 r'3Or86/Lr526r 3+,6f 86/ r'3 r'3 r526 frSRZ 3+,6ff rSRZ5(3fff GLII 86/86/L LIGLIIf GLII GLII 86/L 86/ ` ZKLOHGLII! f 86/ O(Jfr86/ \rrrrrrrrrrrrrrrrr 86/ FDLFXLDWLRQ HQGV rrrrrrrrrrrrrrr 86/ LI86/ 43$UHDff ^ 83 43$UHD83 8386/f (Jf 87 47$UHD87 8786/f (Jf87 47 O$UHD87 87 86/f (Jf 8) 4)$UHD8) 8) (Jf $ ''(/= r '(/=f $ 83r''(/=f.r'(/=ff'(/=f $ 83''(/=ff'(/= $ $$ $ $ $ 8)r&)f'(/= $ $ $ 87'(/=f$ $ $$$ $ $ $ 87r''(/=f.r'(/=ff'(/=f $ $ $ $ $ $$ $ 87 ''(/=f.r'(/=ff'(/=f PAGE 111 HOVH ^ 83 43$UHD83 8386/f (Jf83 83 87 47$UHD87 8786/f (Jf87 47 $UHD87 87 86/f (Jf 8) 4)$UHD8) 8) (Jf $ 83''(/=ff'(/= $ 83r''(/=f.r'(/=ff'(/=f $ ''(/= r '(/=f $ $$ $ $ $ 8)r&)f'(/= $ $ $ 87r ''(/=f. r '(/=ff'(/=f $ $ $ $ $87'(/=f$ $ $ $ $ $ $$ $ 87 ''(/=f.r'(/=ff'(/=f ` rrArrrrA r r'HIPLWLRQ I5RZ ?rrrrrrrrrrrrrrrrrrrrrrrrrrr $>O@>O@ DOr'(/7r$Of $>O@>@ DOr'(/7r$f IRUL L 1Lf $>O@>L@ rrrrrrrrrrr'HILQLOLRQ RI 5RZ WR 1) rrrrrrrrrrrrrrrrr IRUL L1)Lf ^ $>L@ >L @ D r '(/7 r $f $>L@>L@ DOr'(/7r$f $>L@>LO@ DOr'(/7r$f IRUM OMLOMf $>L@>M@ IRUM LM 1Mf $>L@>M@ ` rArrrrrrrrr'HIPLLLRQ RI 5RZ 1) rrrrrrrrrrrrrrrrrrrrrrrrrr $>1)@ >1) @ D r '(/7 r $f $>1)@>1)@ DOr'(/7r$f $>1)@>1)O@ DOr'(/7r$f IRUL O L1) Lf $>1)@>L@ IRUL 1)L 1Lf PAGE 112 $>1)@>L@ rrrrrrrrrrrrrr'HIPLWLRQ RI5RZ 1)O WR1 rrrrrrrrrrrrr IRUL 1) L1Lf ^ $>L@>LO@ DOr'(/7r$f $>L@>L@ DOr'(/7r$OOf $>L@>LO@ DOr'(/7r$f IRUM OMLOMf $>L@>M@ IRUL LM 1Mf $>L@>M@ ` rrArrrrrrrrrr'HILQLWLRQ I 5RZ 1 rrrrrrrrrrrrrrrrrrrrrrr $>1@>1 @ D r'(/7r$f $>1@ >1@ D r'(/7r$ f IRUL OL1OLf $>1@>L@ rrrrrrrrrr5RZ 'HILQLWLRQ HQGV rrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrr 'HILIQLWLRQ RI FROXPQ YHFWRU rrrrrrrrrrrrrrrrrrr 9>O@ DOfr'(/7r$Ofr&>O@DOfr'(/7r$r&>@ SULQWI?Q 9> @ bOI ?Q9> @f JHWFKf IRUL L1)Lf 9>L@ DOfr'(/7r$fr&>L@DOfr'(/7r$r&>LO@ DOfr'(/7r$r&>LO@ SULQWI7n1Q 9>bG@ bOI ?QL 9>L@f JHWFKf` 9>1)@ '(/7r$fDOfr'(/7r$fr&>1)@DOfr'(/7r$r&>1) @O DOfr'(/7r$r&>1)O @ SULQW7?Q 9>1)@ bOI ?Q9>1)@f JHWFKf IRUL 1)OL1Lf 9>L@ DOfr'(/7r$O Onfr&>L@DOfr'(/7r$r&>LO@O DOfr'(/7r$r&>LO@ SULQWI?Q 9>bG@ bOI?QL9>L@f JHWFKf` 9>1+DOfr'(/7r$fr&>1@DOfr'(/7r$r&>1O@ SULQWI?Q 9>1@ bOI ?Q09>1@f JHWFKf rrrrrrrrrArrrrrefHIMQ_WMRQ SFRMXUQQ YHFWRU HQGV rrrrrrrrrr \rrrrrrrrrrrrrrrr 25'(5,1* rrrrrrrrrrrrrrrrrrrrrrrrrrrrrr PAGE 113 IRUL OL 1Lf ^ >L@ L 6>L@ DEV $>L@ > @f IRUM M 1Mf ^ LIDEV$>L@>M@f!6>L@f 6>L@ DEV$>L@>M@f ` ` rrrrrrrrrrr rrr 2UGHULQJ HQGV rrrrrrrrrrrrrrrrrrrrrrrrrrrrrr ,, rrrrrrrrrrrrrrr *DXVV (OLPLQDWLRQ rrrrrrrrrrrrrrrrrrrrrrrrr IRUN O N1Nf ^ rr rr 3LYRWLQJ rrrrrrrrr 3LYRW N %LJ DEV$>>N@@>N@6>>N@@f IRU L L N L L 1 L Lf ^ 'XPP\ DEV$>>LL@@>N@6>>LL@@f LI'XPP\!%LJf ^ %LJ 'XPP\ 3LYRW LL ` ` ,'XPP\ >3LYRW@ >3LYRW@ >N@ >N@ ,'XPP\ ,,rrr (QG 3LYRWLQJrrrrrrr IRU L N L 1 Lf ^ IDFWRU $>>L@@>N@$>>N@@>N@ IRU@ NO M 1Mf ^ $>>L@@>M@ $>>L@@>M@IDFWRUr$>>N@@OM@f ` 9>>L@@ 9>>L@@IDFWRUr9>>N@@f ` ` rrrrrrrrrrrrrrr4DXVVeLLPLQDWLRQ HQGV rrrrrrrrrrrrrrrrrrr PAGE 114 \rrrrrrrrrrrrrr 6XEVWLWXWLRQ rrrrrrrrrrrrrrrrrrrrrrrrrrrrr &>1@ 9>>1@@$>>1@@>1@ IRUL 1OL! OLf ^ 6XP IRUM LOM 1Mf ^ 6XP 6XP$>>L@@>M@r&>M@f ` &>L@ 9>>L@@6XPf$>>L@@>L@ ` ` \rrrrrrrrrrrrrrrr 6XEVWLWXWLRQ HQGV rrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrr r rrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrr r A A A A Â‘ _L / MO A A A A a3 A A !3 A !S S S /S A 3 WS 3 3 !S S S 3 !3 a3 W 3 Â‘ 3 Â‘ S >L Â‘ 3 S A A A S /S !3 S A A S A } S A3 A } 3 3 ttttttttttttmttttttttttttttttttttttttttttttttt tttttttttttt 0DLQ (QGV YRLG PDLQYRLGf ^ GRXEOH *UDGH&)&)*..*'52$UHD=5743474)7,0(3H7SDEOE5(4135(41 LQW OL)/$*1) GRXEOH &>Q@&*>Q@86/L86/5(3GLILI3+,6*UDGHTQFG G)UDQN'(/= GRXEOH 'X4(475'36HOHFWLYLW\6HOHFWHTQ52*5*5& 4) r*4) 4( r*4( 47 r*47f 434)474( 47474( '(/= /1f 1) )1)'(/=f fr $UHD r'LDr'LD SULQWI7?Q 83 bOIn43$UHDf rrrrrrrrrrrrrr &DOFXODWLRQ RI VOLS YHORFLW\ rrrrrrrrrrrrrr '3 '3 3+,6 &6f&6f&6ffr526ff 86/L GR ^ 5(3r'3Or86/Lr526rO 3+,6f PAGE 115 86/ r'3Or'3Or526OfrSRZO3+,6ffOrSRZ5(3fff GLII 86/86/L LIGLIIf GLII GL7 86/L 86/ ` ZKLOHGLII! f 86/ O(Jfr86/ SULQWI?Q 86/ bOIn86/f rrrrrrrrrrrrrrrrr 86/ FDLFXLDWLRQ HQLV rrrrrrrrrrrrrrr 86/ SULQWI?Q(QWHU \RXU LQLWLDO YDOXHV IRU 3KRVSKDWH FRQFHQWUDWLRQ$Qf IRUL OL 1L+f ^ &>L@ SULQWI?Q &>bG@ Lf VFDQIbOInt&>L@f ` SULQWILn$Q(QWHU \RXU LQLWLDO YDOXHV IRU *DXQJH FRQFHQWUDWLRQ$Qf IRUL OL 1L+f ^ &*>L@ SULQWI7$Q &*>bG@ Lf VFDQIbOInt&*>L@f ` )UDQN %3/ &) )UDQN fr&6fr526r&6f&6ffr526f &)* )UDQNO ffr&6fr526r&6O f &6ffr526f SULQWI79Q &) bOI &)f SULQWIAn$Q &)* bOI?&)*f JHWFKf 'X 7r'LDrSRZr4J$UHDff IWPLQ .X.* .*X' 'X (Jf 57 /r$UHDr (Jf4)/r $UHDr (JfrDEV.*fff = r57'(/7f 7,0( '(/7 ,,,r rrrr bbbbb###### +(5( ,6 7+( )/$* %(7:((1 67($'< 67$7( t '<1$0,& )/$* IRU G\QDPLF RWKHU YDOXHV IRU VWHDG\VWDWH DSSUR[LPDWLRQ LI)/$*fÂ§ f ^ IRUO O. =Of ^ PAGE 116 PDLQO&).'&f SULQWI79Q )ROORZLQJ DUH WKH YDOXHV RI 3KRVSKDWH &>L@ DW 7LPH bOI 57 ?Q7,0(57f IRUL L 1 Lf SULQWI79Q &>bG@ bOInL&>L@f JHWFKf 52 4)r&)f47O$UHDr86/fr&>1@fff4)r&)ffr 5& 4)r&)+43r&>1)@f+47r&>1@fff4)r&)+43r&>1)@ffffr 5& QHHGV PRGLILFDWLRQ LQ WHUPV RI 43 DQG 47 LH KDV WR LQFOXGH 86/f SULQWI79Q 7KH RYHUDOO UHFRYHU\ DW 7LPH bOI 57 LV 52 bOIb7,0(5752f SULQWI79Q 7KH FROOHFWLRQ ]RQH UHFRYHU\ DW 7LPH bOI 57 LV 5& bOIb?Q7,0(575&f JHWFKf PDLQ &)*.*'&*f SQQWLA?Q )ROORZLQJ DUH WKH YDOXHV RI *DXQJH &*>L@ DW 7LPH bOI 57 9Q7(0(57f IRUL O L 1L+f SULQWI?Q &*>bG@ bOInL&*>L@f JHWFKf 52* 4)r&)*f47O$UHDr86/fr&*>1@fff4)r&)*ffr *UDGH 4)r&)47O$UHDr86/fr&>1@f4)r&) 47O$MHDr86/fr&>1@f4)r&)*a47O$UHDr86/fr&*>1@fffr *UDGH *UDGHr SULQWI79Q 7KH JUDGH DW 7LPH bOI 57 LV bOIn7,0(57*UDGHf JHWFKf 6HOHFWLYLW\ 5252* 7,0( fr'(/7 ` SULQWI79Q bOIn*f IRUL O L 1Lf SULQWI79Q &*>bG@ bOInL&*>L@f JHWFK2 SULQWI79Q &*>1@ bOI?&*>1@f SULQWI79Q &>1@ bOIn&>1@f SULQWI79Q &>O@ bOIn&>O@f SULQWI79Q &>1@ b In&>1@f SULQWI79Q 2YHUDOO 5HFRYHU\ bOOI b52f SULQWI79Q *5$'( b ,I b*UDGHf SULQWI79Q 6HSBHII b ,I 6HOHFWLYLW\f SULQWI79Q 52* bOOI b52*f SULQWI79Q 5 b ,I b &>1@&)ffr f JHWFK2 ` HOVH PAGE 117 ^ \rrrrrrrrrrrrrrrrrrrr 6WHDG\ VDWH 5HFYHU\ FDOFXODWLRQrrrrrrrrrrrrrrrrrrr 3H 47$UHDf86/fr/ff'rO(Jff SULQWI?Q 3H bOIn3Hf 7S /frO(Jff47$UHDf86/f SULQWI?Q 7S bOIn7Sf D VTUW r.r7S3Hff SULQWI?Q D bOInDf SULQWI&n;Q .r7S bOIn.r7Sf EO H[SDr3Hf E H[SDr3Hf 5(413 rDrH[S3Hff Dfr DfrE f Dfr DfrEffffr 5 5(413ff r 47$UHD r 8 6/f4)fffr F VTUW r.*r7S3Hff GO H[SFr3Hf G H[SFr3Hf 5(41* OrFrH[S3HffOFfrOFfrGOfOFfrOFfrGffffr 5* 5(41*ffr47$UHDr6/f4)fffr SULQWI7?Q 5(413 b OO3R5(413f SULQWI?Q 5HFRYHU\ b OI\R5f SULQWI?Q 5(41* bOIb5(41*f *UDGHTQ 4)r&)47$UHDr86/fr O 5(413ffr&)f4)r&) 47$UHDr86/fr 5(413ffr&)f4)r&)*47$UHDr86/fr 5(41*ffr&)*fffr 6HOHFWHTQ 55* SULQWI?Q *5$'(41 b OI b*UDGHTQf SULQWI?Q 6HSBH7 b OI 6HOHFWHTQf JHWFKf ` rrrrrrrrr6WHDG\ VWDWH 5HFRYHU\ FDOFXODWLRQ HQGV rrrrrrrrrrr ` PAGE 118 /,67 2) 5()(5(1&(6 $VVRFLDWLRQ RI )ORULGD 3KRVSKDWH &KHPLVWV $)3& $QDO\WLFDO 0HWKRGV WK HG f %KDW 1 DQG 70F$YR\ f8VH RI 1HXUDO 1HWV IRU '\QDPLF 0RGHOLQJ DQG &RQWURO RI &KHPLFDO 3URFHVV 6\VWHPVf &RPSXW &KHP (QJ f %RXWLQ 3 DQG $ :KHHOHU f&ROXPQ )ORWDWLRQf 0LQLQJ :RUOG f f &KDSUD 6& DQG 5 3 &DQDOH 1XPHULFDO 0HWKRGV IRU (QJLQHHUV 0F*UDZ +LOO 1HZ PAGE 119 .LUNSDWULFN 6 -U & *HODWW DQG 0 3 9HFFKL f2SWLPL]DWLRQ E\ 6LPXODWHG $QQHDOLQJf 6FLHQFH f .UDPHU 0$ DQG 0 / 7KRPSVRQ f(PEHGGLQJ 7KHRUHWLFDO 0RGHOV LQ 1HXUDO 1HWZRUNVf 3URF $P &RQWURO &RQ? &KLFDJR f /LX 3+ 7 3RWWHU 6 $ 6YRURQRV DQG % .RRSPDQ f+\EULG 0RGHO RI 1LWURJHQ '\QDPLFV LQ D 3HULRGLF :DVWHZDWHU 7UHDWPHQW 3URFHVVf $,&K( $QQXDO 0HHWLQJ 3DSHU 1R DQ f /XWWUHOO *+ DQG 5 + PAGE 120 '\WLDPLFV DQG &RQWURO RI &KHPLFDO 5HDFWRUV 'LVWLOODWLRQ &ROXPQV DQG %DWFK 3URFHVVHV '<&25' f 7KRPSVRQ 0/ DQG 0 $ .UDPHU f0RGHOLQJ &KHPLFDO 3URFHVVHV 8VLQJ 3ULRU .QRZOHGJH DQG 1HXUDO 1HWZRUNVf $,&K(f f 9LOOHQHXYH 09 'XUDQFH & *XLOODQHDX $1 6DQWDQD 59* GD 6LOYD DQG 0$6 0DUWLQ f$GYDQFHG 8VH RI &ROXPQ )ORWDWLRQ 0RGHOV IRU 3URFHVV 2SWLPL]DWLRQf LQ &2/801n (GV *RPH] & 2 DQG $ )LQFK 7KH 0HWDOOXUJLFDO 6RFLHW\ RI WKH &DQDGLDQ ,QVWLWXWH RI 0LQLQJ 0HWDOOXUJ\ DQG 3HWUROHXP f :KHHOHU '$ f+LVWRULFDO 9LHZ RI &ROXPQ )ORWDWLRQ 'HYHORSPHQWf LQ &ROXPQ )ORWDWLRQf (G .96 6DVWU\ 60($,0( /LWWOHWRQ &RORUDGR f ;X 0 DQG $ )LQFK f7KH $[LDO 'LVSHUVLRQ 0RGHO LQ )ORWDWLRQ &ROXPQ 6WXGLHVf 0LQHUDO (QJLQHHULQJ f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n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r % PAGE 125 /' fF"" 81,9(56,7< 2) )/25,'$ PAGE 126 81,9(56,7< 2) )/25,'$ 98 do { REP=5.12*DPl*USLi*ROS*(l.0-PHIS); USL= 108.233 *DP 1 *DP 1 *(ROS-1 )*pow(( 1 -PHIS),2.7)/( 1 +0.15 *(pow(REP,0.687))); diff=USL-USLi; ifidiffcO.O) diff=-diff; USLi=USL; } while(diff>=0.0001); USL=(l-Eg)*USL; //printf("\n USL=%lf',USL); /y***************** calculation ends *************** Frank=BPL/0.733; CF=(Frank/100.0)*(CS/100.0)*ROS*62.41818/((CS/100.0)+(1.0-(CS/100.0))*ROS); CFG=(1.0-(Frank/100.0))*(CS/100.0)*ROS*62.41818/((CS/100.0)+(1.0- (CS/100.0))*ROS); K=Ku;KG=KGu; mainl(CF,K,B); C=B[3]+B[4]; RO=(((QF*CF)-((QTl+Area*USL)*C))/(QF*CF))* 100.0; main 1 (CFG,KG,BG); CG=BG[3]+BG[4]; ROG=(((QF*CFG)-((QTl+Area*USL)*CG))/(QF*CFG))*100.0; Grade=((QF*CF-(QTl+Area*USL)*C)/((QF*CF-(QTl+Area*USL)*C)+(QF*CFG- (QTl+Area*USL)*CG)))* 100.0; Grade=Grade*0 733, //printf("\n C=%lf',C); //printfTVn CG=%lf',CG); //printfTVn CF=%lf',CF); //printfT\n CFG=%lf',CFG); //printf("\n Overall Recovery=%. Ilf %",RO); //printfTVn ROG=%. 1 If %",ROG); //printfT\n GRADE=%. 1 If %",Grade); return (RO); } This dissertation is dedicated to my parents Predicted flotation rate constants forgangue (kg) Figure 4.4: Neural network versus experimental flotation rate constant for gangue (kg) 55 screw delivered the conditioned phosphate materials to the column. The feed rate was controlled by adjusting the screw rotation speed The flotation column was constructed of plexiglass and had 14 5 cm diameter and 1.82 m height. The feeding point was located at 30 cm from the column top. The discharge flow rate was controlled by a discharge valve and an adjustable speed pump Three flowmeters were used to monitor the flow rates for air, frother solution, and elutriation water. Phosphate feed (14X150 Tyler mesh) from Cargill was used as the feed material. For each ran, 50 kg of feed were introduced to the pre-treatment tank and water w'as added to obtain 72 % solids concentration by weight. The tank was then agitated for 10 seconds. 10 % soda ash solution was added to the pulp to reach pH of about 9.4 and the slurry was agitated for another 10 seconds. Subsequently, a mixture of fatty acids (a mixture of oleic, palmetic, and iinoleic acid obtained from Westvaco) and fuel oil (No. 5 obtained from PCS Phosphates) with a ratio of 1:1 by weight was added to the pulp and the slurry continued to be mixed. The total conditioning time was 3 minutes. The conditioned feed material (without its conditioning water) was subsequently loaded to the feeder bin located at the top of the column. Four frothers were used, two commonly employed in industry, F-507 (a mixed polyglycol by Oreprep) and CP-100 (a sodium alkyl ether sulfate by Westvaco), and two experimental, F-579 (also a mixed polyglycol by Oreprep) and OB-535 (by OBrien). Frother-containing water and air were first introduced into the column through the sparger (an eductor) at a fixed water flow rate and frother concentration (0 30 ppm), and the superficial air velocity ranged from 0.24 0.94 cm/s. Then the discharge valve o 4 (/) (i c ro (/) c o co a> 2 15 c .c O CL '^3 CO C3 O T3 O TJ L_ Q_ K) vC Figure 2.6 Performance ofNNI: Model versus experimental flotation rate constant for phosphate (kp) LIST OF FIGURES Figure page 2.1 Flotation rate constants for phosphate and gangue are calculated by using a one-dimensional search to invert the first-principles model 19 2.2 Recovery of phosphate (%) as a function of flotation rate constant for phosphate (kp) 20 2.2 Recovery of gangue as a function of flotation rate constant for gangue (kg) 21 2.4 Overall structure of the hybrid model 23 2.5 A schematic diagram of the experimental setup 25 2.6 Performance of NN: Model versus experimental flotation rate constant for phosphate (kp) 29 2.7 Performance ofNNII: Model versus experimental flotation rate constant for gangue (kg) 31 2.8 Performance of NN1II: Model versus experimental air holdup for frother CP-100 32 2.9 Performance of the overall hybrid model: Predicted versus experimental recovery (%) for coarse feed size distribution 33 2.10 Performance of the overall hybrid model: Predicted versus experimental grade (%BPL) for coarse feed size distribution 34 2.11 Performance of the overall hybrid model: Predicted versus experimental recovery (%) for fine feed size distribution 35 2.12 Performance of the overall hybrid model: Predicted versus experimental grade (%BPL) for fine feed size distribution 36 vi where 13 cf- = column diameter (m) Jg = superficial air velocity (cm/s) Equations 2.1 and 2.2 can be solved analytically for the concentration profile of the solid particles at steady state. The resulting analytical expressions for the concentration profile are where K], Kj2, K3, andK^ are the constants of integration to be determined by using appropriate boundary conditions. 2,2.1 Boundary Conditions A material balance at the top layer of the column (z = L) gives the following equation: + AcD 1 kp (dp AcAzCÂ¡!)] (2.8) 7 provides superior grade/recovery performance. For these reasons column flotation is gaming increasing acceptance for the processing and beneficiation of phosphate ores. Although it has been successfully employed for the selective separation of phosphate from unwanted mineral, a totally predictive model still remains unavailable for industrial use. Flotation is a process to separate hydrophobic particles from hydrophilic particles. The hydrophobic material has a tendency to attach to the rising bubbles and leaves from the top of the column. The hydrophilic material settles down and leaves from the bottom of the column In this way, the phosphate containing material (frankolite or apatite) is separated from gangue (mostly silica). The phosphate ore is first pretreated with fatty acid collector and fuel oil extender. Fatty acid and fuel oil adsorb on the phosphate- containing panicles rendering them hydrophobic. The flotation process is then used to separate phosphate panicles from gangue minerals. A flotation column consists of three flow regimes: a cleaning or froth zone, a lower collection zone, and pulp-froth interface zone. The froth zone is the region extending upward from the pulp-froth interlace to the column interface. The collection zone is the region extending downward from the pup-froth interface to the lowest sparger. A mineral particle is recovered by a gas bubble in the collection zone of the column by particle- bubble collision followed by attachment due to the hydrophobic nature of the mineral surface. Since phosphate particles are considerably larger in size (0.1-1 mm), an elutriation water stream from the bottom is added to maintain a positive upward flow (negative bias) to aid lifting the particles upward. 10 9 8 7 6 5 4 3 2 1 0 Experimental flotation rate constants for Phosphate (kp) ON Figure 3.11: Performance of NNI-Model versus experimental flotation rate constant for phosphate (kp) 45 Figure 3 .1: Schematic diagram of column for phosphate flotation. 51 Ri_W^RJpG{(73.3-GJ) W GJ(73.3-Gf) (3.14) In some cases direct measurements of the majority of gangue as acid insolubles may be available Then more reliable estimates of R^ can be obtained by averaging values calculated from measurements of acid insolubles with values calculated from Equation 3.14. This was done in this work. From the FPM equations follows that the recovery of phosphate depends only on kp, while the recovery of gangue depends only on kg. This can be exploited to easily invert the steady-state version of the model to determine from experimental measurements of Rp and GJ corresponding kp and kg. As shown in Figure 2.1, this is accomplished with one-dimensional searches. The search for kp is initialized with two values that yield errors in the corresponding recovery R of opposite sign. Since typically 0 < kp < 10 min'1 the values of 0 and 100 min'1 are used. Then the method of false position (Chapra and Canale, 1988) is used to iterate until the magnitude of the error in R^ drops to less than 10'\ It is possible that the calculated recovery has a higher value than the experimental even for kp = 0. In these cases kp is set equal to zero. The above procedure is also used to determine kg, except that the high initial value is set to 10 min'1. Recovery for both phosphate and gangue increases monotonically with respective flotation rate constants as discussed in Chapter 2. 28 determined using simulated annealing (Kirkpatrick et al., 1983) and a conjugate gradient algorithm (Polak, 1971). There are two approaches towards updating the weights. In one approach, the input-output examples are presented one at a time and after each presentation the weights are updated using rules such as the delta rule (Rumelhart and McClelland, 1986). This method is attractive for its simplicity but is restricted to rather primitive optimization algorithms. In contrast, the batch training approach allows use of powerful methodology for nonlinear optimization, it processes each input-output example individually but updates the weights only after the whole set of input-output examples has been processed. In this case, the gradient is cumulated for all presentations, then the weights are updated, and finally the sum of the squared errors is calculated. The simulated annealing algorithm is used for eluding local minimum. It perturbs the independent variables (the weights) while keeping track of the best (lowest error) function value for each randomized set of variables. This is repeated several times, each time decreasing the variance of the perturbations with the previous optimum as the mean. The conjugate gradient algorithm is then used to minimize the mean-squared output error. When the minimum is found, simulated annealing is used to attempt to break out of what may be a local minimum. This alteration is continued until networks can not find any lower point. We then hope that the local minimum is indeed the global minimum. 2,5 Results and Discussion The performances of the three ANNs are shown in Figures 2.6-2.14. Figure 2.6 compares the flotation rate constants for phosphate (kp) determined from one-dimensional searches with those predicted by NNI. As shown in this figure, NNI captures the dependence of the flotation rate constant on particle size, superficial air velocity, frother concentration, 50 mineral that contains no gangue is 73.3 %BPL. Grade can be obtained as the ratio of phosphate to the sum of phosphate and gangue in the product: GJ = QfCj ~[Qi +AC(1 -eg)Uj ]cÂ¡,n \ (QfCf-[Qt+Ac(l-e,)U]Cii)+(QfCi._-[Qt+Ac(l-Eg)U]C^) gn * 73.3 (3.12) where Cgn is gangue concentration of jth particle size in the nlh layer and C^ is the gangue feed concentration of jth particle size. The algorithm for solving the first- principles model is given in Appendix B. 3.3 Calculation of Model Parameters Since air-holdup eg is measured experimentally, the above FPM has only two unmeasured model parameters for each particle size, namely, the flotation rate constants for phosphate (kp) and for gangue (kg). The experimental analysis usually available in industrial flotation columns is in terms of grade and recovery of phosphate. Let -% ,tk *th denote the weight of j size phosphate particles in the feed, W,g the weight of j size fL gangue in the feed, and WgJ the weight of j' size gangue in the product. The grade of feed is then GÂ¡ =73.3 WJ tp K + w- (313) and GJ is given by an analogous expression. The recovery of gangue can be readily calculated from measurements of grade and recovery of phosphate using the following relationship: Experimental Grade (%) Figure 2.14; Performance of the overall hybrid model: Predicted versus experimental grade (%BPL) for the unsized feed after it has been sized. xml version 1.0 encoding UTF-8 REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID ETBOXIB40_6G9O02 INGEST_TIME 2013-04-08T22:31:53Z PACKAGE AA00013557_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES 30 25 20 15 10 5 0 Figure 3.9: Performance of NNIV: Model versus experimental air holdup for frother OB-535 Predicted flotation rate constants for gangue (kg) Figure 3.12 Performance of NNI: Model versus experimental flotation rate constant for gangue (k^ 2.4 2 Experimental Conditions For the frothers investigated. 35 three-phase experiments were conducted. Seven different levels of frother concentration (5, 6.6, 10, 15, 20, 23.4, and 25 ppm) was studied in designed experiments Five different levels of collector and extender concentration (0.27, 0.41, 0 54, 0.64, and 1.7 kg/t) were used. pH was varied from 8.2 to 9.9 at five different levels (8.2, 8 5, 9.0. 9 5, and 9.9). Two superficial air velocities (0.46 and 0.7 cm/s) were used for the designed experiments. The particle size depended on the type of feed used. For coarse feed, the particle size varied from 417 to 991 microns. For fine feed, the particle size varied from 104 to 417 microns whereas for the unsized feed distribution, the size ranged from 104 to 991 microns. 2 4 3 Neural Network Structure and Training Single output feedforward backpropagation neural networks are used with a single layer of hidden nodes. A unit bias is connected to both the hidden layer and the output layer. Both the hidden layer and the output layer used a logistic activation function (Hertz et al., 1992) and the input and the output values were scaled from 0 to 1 During the training mode, training examples are presented to the network. A training example consists of scaled input and output values. For NNI and NNII, the output values are the flotation rate constants calculated from one-dimensional searches for phosphate and gangue, respectively. For NNIII, the output value is the experimentally measured air holdup. The training process is started by initializing all weights randomly to small non-zero values. The random number is generated between -3.4 and +3.4 with standard deviation of 1.0 following the procedure recommended by Masters (1993). The optimal weights were Figure 4.3 Neural network versus experimental flotation rate constant for phosphate (kp) LIST OF REFERENCES Association of Florida Phosphate Chemists, AFPC Analytical Methods, 6th ed. (1980). Bhat, N. and T.J. McAvoy, Use of Neural Nets for Dynamic Modeling and Control of Chemical Process Systems, Comput. Chem. Eng., 14:573 (1990). Boutin, P. and D A. Wheeler, Column Flotation, Mining World, 20(3): 47-50 (1967). Chapra, S.C. and R. P. Canale, Numerical Methods for Engineers, McGraw Hill, New York (198S). Cubillos, F., P. Alvarez, J. Pinto, and E. Lima, Hybrid-Neural Modeling for Particulate Solid Drying Processes, Powder Technology, 87, p. 153 (1996). Cubillos, F.A., and E. L. Lima, Identification and Optimizing Control of a Rougher Flotation Circuit Using an Adaptable Hybrid-Neural Model, Minerals Engineering, 10(7): 707-721 (1997). Dobby, G.S, and J. A. Finch, Column Flotation: A Selected Review, Part II, Mineral Engineering, 4: 911-923 (1991). Dobby, G.S., and J. A. Finch, Mixing Characteristics of Industrial Flotation Columns, Chem. Eng. Sci., 40: 1061-1068 (1985). Finch, J. A., and G. S. Dobby, Column Flotation, Pergamon Press, Toronto (1990). Hertz, J., A. Krogh, and R.G. Palmer, Introduction to the Theory of Neural Computations, Addison-Wesley Publishing Company, Redwood City, CA, 5th ed. (1992). Himmelblau, D M., Applied Nonlinear Programming, McGraw Hill, New York (1972). Homik, KM., M. Stinchcombe, and H. White, Multi-layer Feedforward Networks Are Universal Approximators, Neural Networks, 2:359 (1989). Johansen, T. A., and B. A. Foss, Representing and Learning Unmodeled Dynamics with Neural Network Memories, Proc. Am. Control Conf.x Chicago, 3:3037-3037 (1992). 108 70 60 50 40 30 20 Figure 2.12: Performance of the overall hybrid model: Predicted versus experimental grade (%BPL) for fine feed size distribution Experimental grade and Experimental recovery of phosphate One-dimensional search Flotation rate constant for phosphate Figure 2.1: Flotation rate constants for phosphate and gangue are calculated by using a one dimensional search to invert the first-principles model 15 The four boundary conditions can be solved in conjunction with Equations 2.6 and 2.7 for Kj, Kj2, K^, andKJ4. The resulting expressions for the constants of integration are given by the following equations: (q,c;/acd) mJ(aJ aJ)pJ exp{a-Lf}+(dJ -yJ)qJ exp{yJLf} + mJ (aJ -(3J)exp{(3JLf} + (dJ -6J)exp{6JLj} (2.13) Ki=qJK' (2.14) * K> II 3 5* U ' (2.15) K;=pJmJK; (2.16) where or* =- + VaJ +4b-* 2 2 (2.17) (3J =---->/aJ +4b' 2 2 (2.18) yj Vdj2 +4bJ 2 2 (2.19) 8' = ---\/d,: +4b< 2 2 (2.20) Figure 2.4: Overall structure of the hybrid model LD 1780 199.3 c7 UNIVERSITY OF FLORIDA 3 1262 08554 4376 Model diameter (mm) Inferred bubble diameter (mm) Figure 3.6: Performance ofNNIII: Model bubble diameter versus bubble diameter inferred from experimental data when F-579 was the frother HYBRID NEURAL NETWORK FIRST-PRINCIPLES APPROACH TO PROCESS MODELING By SANJAY GUPTA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILLOSOPHY UNIVERSITY' OF FLORIDA 1999 Abstract of Dissertation Presented to the Graduate School Of the University of Florida in Partial Fulfillment of the Requirements for Doctor of Philosophy HYBRID NEURAL NETWORK FIRST-PRINCIPLES APPROACH TO PROCESS MODELING By Sanjay Gupta May 1999 Chairman: Dr Spyros A. Svoronos Cochairman: Dr Hassan El-Shall Major Department: Chemical Engineering A hybrid model for a flotation column is presented which combines a first- principles model with artificial neural networks. The first-principles model is derived by making material balances on both phosphate and silica particles in the slurry phase. Neural networks are used to relate the model parameters with operating variables such as particle size, superficial air velocity, frother concentration, collector and extender concentration, and pH. One-level and two-level hybrid modeling structures are compared and it is shown that the two-level structure offers significant advantages over the other. Finally, a sequential run-to run optimization algorithm is developed which combines the hybrid model with an optimization technique. The algorithm guides the changes in the manipulated variables after each experiment to determine the optimal column conditions. IX 20 18 16 14 12 10 8 6 4 2 0 & Training data Validation data 10 Experimental Air Holdup (%) 15 20 CN igure 3.8: Performance of NNIV: Model versus experimental air holdup for frother -507 20 18 16 14 12 10 8 6 4 2 0 P6 Training data Validation data 5 10 Experimental Air Holdup (%) 15 20 On 3.7: Performance of NNIV: Model versus experimental air holdup for frother CP-100 4 OPTIMIZATION PERTORMANCE MEASURES AND FUTURE WORK 73 4.1 The Performance Measures 74 4.1.1 Selectivity 74 4.1.2 Separation Efficiency 75 4 1.3 Economic Performance Measure 75 4.2 The Optimization Algorithm 79 4.3 Initial Scattered Experiments 81 4 4 Results and Discussions 82 4.5 Future Work 86 APPENDICES A CODE FOR THE FIRST PRINCIPLES MODEL FOR ONE LEVEL 90 B CODE FOR THE FIRST PRINCIPLES MODEL FOR TWO LEVELS 99 REFERENCES 108 BIOGRAPHICAL SKETCH 111 084 F7* 08/03/99 347B0 40 20 0 \ 20 40 60 80 TI 'T'l 'ITTTTTTTTTTTTTTTTTT 1 I TT" \> ^
i |