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NUEVA METODOLOGÍA PARA EL DISEÑO DE CIRCUITOS DE FLOTACIÓN Felipe David Sepúlveda Unda Tesis presentada en concordancia a los requerimientos para optar al grado de Doctor en Ingeniería de Procesos de Minerales Supervisores Luis A. Cisternas Edelmira D. Gálvez Laboratorio de Optimización y Modelación Departamento de Ingeniería Química y Procesos de Minerales Universidad de Antofagasta Antofagasta Mayo, 2014 NEW METHODOLOGY FOR DESIGN OF FLOTATION CIRCUITS Felipe David Sepúlveda Unda Thesis presented in accordance to the requirements to obtain the degree of Ph.D. in Mineral Process Engineering Supervisors Luis A. Cisternas Edelmira D. Gálvez Laboratory of Optimization and Modelling Department of Chemical and Mineral Process Engineering Universidad de Antofagasta Antofagasta May, 2014 RESUMEN La industria minera enfrenta continuamente desafíos para mantener a nivel competitivo sus unidades de producción, debido a las menores leyes de los yacimientos, minerales cada vez más complejos, regulaciones ambientales cada vez más exigentes e incertidumbre en los mercados mundiales. Por lo tanto, se requiere el uso de todas las herramientas posibles para mejorar el diseño y operación de estos procesos. Una decisión fundamental en la industria minera es la elección y diseño del diagrama de flujo, siendo tradicionalmente realizada por medio de pruebas de laboratorio o por simulación matemática y antecedentes históricos, pero es impracticable cuando se quieren evaluar una cantidad importante de posibles configuraciones, debido al tiempo y a los costos asociados a la evaluación. Además, cuando ya se ha seleccionado un diagrama de flujo, el siguiente inconveniente, es la gran cantidad de parámetros o factores que intervienen en el diagrama de flujo, siendo una tarea difícil poder determinar cuáles son los parámetros más relevantes del diagrama de flujo, y así tomar la mejor decisión para el diseño y operación. Este tema de investigación, tiene por objetivo el desarrollar un marco para la innovación de los diagramas de flujo en la fase del diseño conceptual, para el procesamiento de minerales mediante una estrategia sistemática para el Diseño de Diagramas de Flujos de Concentración de Minerales Asistida por Computador (DDFCMAC). Existen varios métodos que puede ser utilizado para esta tarea, pero los que son eficientes, se aplican solamente a problemas muy específicos, mientras que otros son demasiado complejos, ya que requieren métodos avanzados de optimización (que puede ser difícil de formular y resolver) o porque requieren modelos rigurosos. Además, estos métodos no suelen utilizarse en plantas industriales, porque requieren una formación avanzada de los usuarios o porque los modelos disponibles no son capaces de representar adecuadamente el comportamiento de una planta industrial. Las características del procedimiento DDFCMAC es que debe ser: a) un método aplicable a todos los tipos de procesamiento de minerales, que se pueda aplicar a procesos complejos que implican unidades de diversas operaciones y productos, b) un método que no es necesario emplear modelos rigurosos sin sacrificar la precisión de los cálculos, c) un método que incluya la mejora de los procesos existentes, d) un método que tiene la capacidad de crear y reutilizar el conocimiento, y e) método que puede ser fácil de configurar y usar. Siendo este procedimiento inspirada por el trabajo de d'Anterroches and Gani (2005) and Douglas (1985) para procesos químicos. Para lograr su aplicación, el procedimiento DDFCMAC, considera tres niveles de decisión: 1) definición y análisis del problema, en este nivel el problema es definido, incluyendo la caracterización de la alimentación, se definen los objetivos del diseño, objetivo de operación y sus restricciones. 2) síntesis y selección de las alternativas, en este nivel, se generan las alternativas y se evalúan usando el método de contribución de grupos y un conjunto promisorio de circuito es validado por balance de masa. 3) diseño final, es donde se definen las condiciones operacionales y las condiciones en el diseño de los equipos. En este nivel de decisión dos herramientas son utilizadas: Análisis de sensibilidad y simulación reversa. Este procedimiento considera la experiencia y criterios del diseñador en todos los niveles, y se utilizan métodos rigurosos y aproximaciones, razón por lo cual, esta metodología es considerada como hibrida. Cómo se indicó, el procedimiento DDFCMAC es una nueva propuesta para diseñar los circuitos de flotación, y se encuentra basada en tres métodos principales: El método de contribución de grupos, análisis de sensibilidad y simulación reversa, pero esta investigación se focaliza en los dos primeros métodos. El método de contribución de grupos se basa en el concepto de que cada etapa de un circuito de flotación, contribuye a las propiedad final del circuito, por lo tanto, los circuitos de flotación pueden ser sintetizadas, modeladas y evaluadas considerando el “aporte” de cada unidad de procesamiento. Esto es de la misma forma que las moléculas químicas son sintetizadas, modeladas y evaluadas para obtener sus propiedades, donde un átomo o un grupo de átomos representan una fracción de la molécula. La gran ventaja de este procedimiento es que permite una evaluación rápida y sencilla. El método de contribución de grupo desarrollado permite una evaluación de 1,492 circuitos basado en 35 grupos. Cada grupo es definido considerando que la etapa posee una dirección de corriente para el concentrado y otra para la cola. Es decir, cada grupo incluye aportes estructurales. Para lograr el ajuste de los modelos del método de contribución se grupos, se utilizaron 46 circuitos con 69,000 datos generados por simulación, lográndose una desviación estándar promedio de 1,5%. El análisis de sensibilidad, es un grupo de técnicas que nos permite determinar cuáles son las incertezas de los factores de entrada que más afectan a la incerteza de los factores de salida de un modelo. En este trabajo, se utilizan el método de Sobol, método de Morris y el método E-fast para determinar que etapas del circuito de flotación afectan más al comportamiento de una o más especies en el circuito. En ese sentido la aplicación del análisis de sensibilidad permiten orientar el diseño final una vez seleccionado el circuito. Además se realizó una comparación de los resultados por cada técnica, cuando existía variación de la incertidumbre de los factores de entrada y el tipo de distribución de los factores de entrada. La simulación reversa nos permite determinar 4 el valor los parámetros de interés, considerando las condiciones o restricciones que requiere el diseñador. Los resultados generales muestran que el procedimiento DDFCMAC logra determinar un conjunto con los posibles mejores circuitos, proponiendo valores para las condiciones de diseño (número de celdas) y las condiciones operacionales (tiempo de residencia), pudiendo incluirse distintas objetivos en la evaluación. El método de contribución de grupo logra entregar al diseñador, un grupo de los circuitos más prometedores para su evaluación, teniendo un bajo error en la estimación de la recuperación global en comparación con los resultados del balance de masa, esto es debido a que el método de contribución de grupos posee una baja desviación estándar. El análisis de sensibilidad entrega la información de cuáles son los parámetros operacionales y los parámetros de diseño más importante de los circuitos analizados, y con esa información, fue posible guiar el cálculo de la simulación reversa. También se comparó las tres técnicas más utilizadas del análisis de sensibilidad global, que son el método de Sobol, método de Morris y método E-fast. Esta comparación incluyó varios casos de estudio, incluyendo casos con diferentes niveles y tipos de incerteza. Los resultados obtenidos fueron muy similares entre las técnicas, siendo su mayor diferencia, el costo computacional. La simulación reversa, es una técnica que permite la determinación de las posibles mejores combinaciones para cumplir los objetivos fijados por el diseñador, en esta investigación se utilizó la técnica más sencilla entre todas las opciones, existiendo la posibilidad de mejorar el diseño de los circuito, con el uso de técnicas más sofisticadas (optimización matemática). Como conclusión general, el procedimiento DDFCMAC es una alternativa efectiva para el diseño conceptual de los circuitos de flotación, pudiendo compatibilizarse con análisis empíricos. Las conclusiones del método de contribución de grupos, es que logra determinar las recuperaciones globales de un número importante de circuitos, siendo una herramienta viable para la evaluación de circuitos de concentración de minerales. El análisis de sensibilidad es una herramienta que nos ayuda a analizar, diseñar y mejorar los procesos, pudiendo evaluar modelos simples como complejos, si son modelos simples, no existe una diferencia importante en los resultados entre los métodos usados, pero si el modelo es complejo, es posible recomendar el uso del método de Morris por su menor costo computacional. Y la simulación reversa permite diseñar el circuito final con las restricciones operacionales y de diseño que fueron consideradas por el diseñador. 5 6 ABSTRACT The metallurgical processes, continually confronts challenges to stay competitive in the production units, due to lower grades of the deposits, increasingly complex minerals, increasingly demanding environmental regulations and uncertainty in global markets. Therefore, is required the use of all the tools to improve the design and operation of the processes. In this project the goal is to develop a framework for innovation of the innovation of mineral processing flow sheets at the stage of conceptual design, using a systematic strategy for Design Flow Computer Aided Mineral Flow-sheet Design (CAMFD). In general, there are several methods that can be used for this task, but those they are efficient, only apply to very specific problems, while others are too are complex, and requiring advanced optimization methods (which may be difficult to formulate and resolver) or because require rigorous models. Furthermore, these methods are not commonly used in industrial plants, because they require advanced user training or because the available models are not able to adequately represent the behavior of an industrial plant. The characteristics of this mineral processing CAMFD must be: a) a method applicable to all types of mineral processing, so it can be applied to complex processes involving various unit-operations and multiple metal products, b) a method that does not need to employ rigorous models without sacrificing the precision of the calculations, c) a method that supports the retrofit of existing processes, d) a method that has the ability to build and reuse knowledge over time, and e) a method that can be easy to setup and use. This procedure is inspired in the work of d'Anterroches and Gani (2005) and Douglas (1985) for the chemical processes. For its implementation, the procedure CAMFD includes three decision levels: 1) definition and analysis of the problem, in this level, the problem is defined, including the characterization of the feed, the design goals and design and operation restrictions; 2) synthesis and screening of alternatives, in this level, circuit alternatives are generated and evaluated using a group contribution model, and the most promising circuits are validated by mass balance; and 3) final design, the final design is performed by defining the operational and equipment design associated variables. At this level of decision two tools are used: sensitivity analysis and reverse simulation. This procedure considers the experience and criteria of the designer in all its levels, and rigorous and approximate methods are used, this is why this methodology is considered a hybrid method. 7 The procedure CAMFD is a new approach to design for flotation circuits, and is based on three principal methods: The group contribution method, sensitivity analysis and reverse simulation, but this research focuses on the first two methods. The group contribution method is based on the concept that each of the stages of a flotation circuit contributes to the final property of the circuit; therefore, the flotation circuits can be synthesized, modeled and evaluated considering the "contributions "of each processing unit. This is in the same way that chemical molecules are synthesized, modeled and evaluated to obtain its properties, where an atom or group of atoms represent a fraction of the molecule. The great advantage of this procedure is that it allows quick and easy evaluation. The group contribution method developed enables the evaluation of 1,492 circuits that is based on 35 groups. Each group is defined by considering that each stage has a direction for the concentrate and tail streams. That is, each group includes structural contributions. To achieve adjustment of the models in the group contribution method, 46 circuits with 69,000 data generated by simulating were used, achieving an average standard deviation of 1.5%. Sensitivity analysis is a set of techniques that allows us to determine which uncertainty of input factors are most affecting the uncertainty of output factors for a model. In this paper, the Sobol’, Morris and E-fast methods were used for evaluated and for determine which stages of the flotation circuit most affect the performance of one or more species in the circuit. In this sense, the application of sensitivity analysis helps to guide the final design after selecting the circuit. The reverse simulation allows us to determine the value of the parameters of interest, considering the conditions or restrictions required by the designer. The overall results show that the CAMFD methodology is effective in the conceptual design of flotation circuits with the use of different methodologies that were mentioned above. 8 SUPERVISORS Luis A. Cisternas, Laboratory of Optimization and Modelling Department of Chemical and Mineral Process Engineering Universidad de Antofagasta, Antofagasta, Chile Edelmira D. Galvéz Department of Metallurgical & Mining Engineering Universidad Católica del Norte, Antofagasta, Chile. LIST OF PAPERS This thesis is based on the following papers that were published in ISI journals: I. Sepúlveda F.D., Cisternas L.A., Elorza M.A., 2014, Gálvez E.D., A Methodology for the conceptual design of concentration circuits: Group Contribution Method., Computer & Chemical Engineering, 63, 173 – 183. II. Sepúlveda F.D., Cisternas L.A., González J.F., Gálvez E.D., 2014, A Methodology for the Conceptual Design of Concentration Circuits: Final Design. Submitted. III. Sepúlveda F.D., Cisternas L.A., Gálvez E.D., 2014, Global Sensitivity Analysis of a Mineral Processing, Computer & Chemical Engineering, In Press. Additionally, results were reported in events, through publications in proceeding books. IV. Sepúlveda F.D., Cisternas L.A., Gálvez E.D., 2014, A Global Sensitivity Analysis for Multiple Products in Mineral Processing Flowsheet. International Mineral Processing Congress, 2014 V. Sepúlveda F.D., Cisternas L.A., Gálvez E.D., 2013, Global Sensitivity Analysis of a Mineral Processing Flowsheet, Proceedings of the 23nd European Symposium on Computer Aided Process Engineering, 2013, Elsevier B.V. 9 VI. Sepúlveda F.D., Cisternas L.A., Gálvez E.D., 2013, Flotation process analysis with global sensitivity analysis, International Mineral Processing Conference 2013. VII. Sepúlveda F.D., Cisternas L.A., Gálvez E.D., 2012, A Novel Method for Designing Flotation Circuits, Ian David Lockhart Bogle and Michael Fairweather (Editors), Proceedings of the 22nd European Symposium on Computer Aided Process Engineering, 2012, Elsevier B.V. VIII. Sepúlveda F.D., Elorza M.A., Cisternas L.A., Gálvez E.D., A Hybrid Method for Design of Flotation Circuits, International Mineral Processing Conference 2012. IX. Montenegro M.R., Sepúlveda F.D., Gálvez E.D., Cisternas L.A., 2013, Methodology for Process Analysis and Design With Multiple Objectives Under Uncertainty: Application to Flotation Circuits, International Journal of Minerals Processing, 118, 15-27. X. Montenegro M.R., Sepúlveda F.D., Gálvez E.D., Cisternas L.A., 2009, Analysis, Evaluation And Selection Of Flotation Circuits Using Multiple Goals And Monte Carlo Simulation., International Mineral Processing Conference. XI. Gálvez E., Sepúlveda F., Cisternas L., Montenegro M., 2009, Sensitivity Assessment of Flotation Circuit to Uncertainty Using Monte Carlo Simulation, In Design for Energy and the Environment, pages 679-688, CRC Press. 10 CONTENT RESUMEN ........................................................................................................................................................ 3 ABSTRACT ........................................................................................................................................................ 7 SUPERVISORS .................................................................................................................................................. 9 LIST OF PAPERS................................................................................................................................................ 9 CONTENT ....................................................................................................................................................... 11 LIST OF TABLE ................................................................................................................................................ 13 LIST OF FIGURE .............................................................................................................................................. 14 CHAPTER 1. A METHODOLOGY FOR THE CONCEPTUAL DESIGN OF CONCENTRATION CIRCUITS: GROUP CONTRIBUTION METHOD ............................................................................................................................. 15 1.1. INTRODUCTION .......................................................................................................................................16 1.2. METHODOLOGY ......................................................................................................................................18 1.2.1. Level I: Definition and analysis of the problem ..........................................................................19 1.2.2. Level II: Synthesis and screening of alternatives ........................................................................20 1.2.3. Level III: Final design ..................................................................................................................20 1.3. JUSTIFICATION OF THE ASSUMPTION ............................................................................................................21 1.1. GROUP CONTRIBUTION METHOD ...............................................................................................................24 1.1.1. Generation of alternatives .........................................................................................................26 1.1.2. Process groups ...........................................................................................................................28 1.1.3. Recovery Models ........................................................................................................................29 1.1.1. Adjustment Method ...................................................................................................................30 1.1.2. Example and Validation .............................................................................................................33 1.2. CASE STUDY ...........................................................................................................................................36 1.2.1. Level I. Definition and analysis ...................................................................................................36 1.2.2. Level II. Synthesis and screening of alternatives ........................................................................37 1.2.3. Level III. Final design ..................................................................................................................39 1.3. CONCLUSIONS AND FUTURE WORK .............................................................................................................39 1.4. ACKNOWLEDGMENTS ...............................................................................................................................40 1.5. REFERENCES ...........................................................................................................................................40 CHAPTER 2. DESIGN. A METHODOLOGY FOR THE CONCEPTUAL DESIGN OF CONCENTRATION CIRCUITS: FINAL 45 2.1. INTRODUCTION .......................................................................................................................................46 2.2. A BRIEF DESCRIPTION OF THE METHODOLOGY ..............................................................................................47 2.3. DATABASE..............................................................................................................................................48 2.4. FINAL DESIGN .........................................................................................................................................49 2.4.1. Identification of gaps and opportunities for improvements. .....................................................50 2.4.2. Differential sensitivity analysis (DSA) .........................................................................................50 2.4.3. Identification of key stages using sensitivity analysis ................................................................51 2.4.4. Definition of the design and operating variables based on reverse simulation .........................53 2.1. CASE STUDY............................................................................................................................................54 2.1.1. Identification of gaps and opportunities for improvements ......................................................56 11 2.1.2. Identification of the key stages using sensitivity analysis ..........................................................58 2.1.3. Definition of the design and operating variables based on reverse simulation. ........................61 2.2. SUMMARY AND DISCUSSION ......................................................................................................................63 2.3. ACKNOWLEDGEMENT ...............................................................................................................................64 2.4. REFERENCES ...........................................................................................................................................64 CHAPTER 3. THE USE OF GLOBAL SENSITIVITY ANALYSIS FOR IMPROVING PROCESSES: APPLICATIONS TO MINERAL PROCESSING ............................................................................................................................ 67 3.1. INTRODUCTION .......................................................................................................................................68 3.2. GLOBAL SENSITIVITY ANALYSIS ...................................................................................................................69 3.2.1. FAST and Sobol’ Method. ...........................................................................................................71 3.2.2. Morris Method ...........................................................................................................................73 3.2.3. Local Sensitivity Analysis ............................................................................................................74 3.3. APPLICATIONS .........................................................................................................................................75 3.3.1. Process Identification and Sobol’ Method ..................................................................................75 3.3.1. Process retrofit and comparison between Sobol´ and Morris methods .....................................77 3.3.2. Comparison of sensitivity analysis methods...............................................................................82 3.3.3. Retrofit of a copper concentration plant ....................................................................................87 3.4. CONCLUSIONS .........................................................................................................................................92 3.1. ACKNOWLEDGMENTS ...............................................................................................................................93 3.2. REFERENCES ...........................................................................................................................................93 12 LIST OF TABLE Table 1.1. Steps of the hierarchical decision. ........................................................ 19 Table 1.2. Process groups. ..................................................................................... 29 Table 1.3. Concentration circuits used to fit the group contribution model. .......... 31 Table 1.4. Constants i and i for the process group contribution defined in equation (1.3) for high, medium and low recoveries. ............................................ 32 Table 1.5. Constants for equations 1.1 and 1.2. .................................................... 33 Table 1.6. Mean absolute error for each circuit of table 1.3 for high, medium, and low recoveries. ........................................................................................................ 34 Table 1.7. Parameters values for the case study. ................................................. 36 Table 1.8. Set of alternatives selected for mass balance validation. .................... 38 Table 1.9. Concentrate grade and overall recovery for the top ten circuits .......... 39 Table 2.1. Levels of the hierarchical decision. ....................................................... 48 Table 2.2. Examples of heuristics for reverse experimentation. ........................... 55 Table 2.3. Parameter values for the case study. ................................................... 57 Table 2.4. Gaps between the goals and the values obtained in selected circuits. ................................................................................................................................. 57 Table 2.5. Local sensitivity index for the objective function. ................................. 58 Table 2.6. Residence times and numbers of cells (original values). ..................... 63 Table 2.7. Stage and global recoveries for circuits 1 and 2. ................................. 63 Table 3.1. Distribution functions for factors in case study 4.2. .............................. 80 Table 3.2. Sobol’ and E-fast indexes. (Values x 102). .......................................... 84 Table 3.3. Sobol’ Indices for different distribution functions (values x 102). ........ 85 Table 3.4. Mass flow rates of the flotation circuit of Figure 3.8 (Hay and Martin, 2004). ...................................................................................................................... 88 Table 3.5. Stage’s recoveries for the flotation circuit of Figure 3.8. ...................... 89 13 LIST OF FIGURE Figure 1.1. Interconnected components and parameters in froth flotation circuits......................................................................................................................17 Figure 1.2. Comparison of five circuits of three flotation stages. Circuit 1 [(RC)(CP)(SC)][(RS)(CR)(SW)], circuit 2 [(RC)(CP)(SR)][(RS)(CR)(SW)], circuit 3 [(RC)(CP)(SR)][(RS)(CS)(SW)], circuit 4 [(RC)(CP)(SC)][(RS)(CS)(SW)], circuit 5 [(RC)(CP)(SC)][(RW)(CR)(SW)]. The nomenclature used is explained below. ......................................................................................................................25 Figure 1.3. Concentrate and tail origin-destination matrixes for a three-stage concentration circuit. ...............................................................................................27 Figure 1.4. Concentration circuit with four stages, represented by the CTP string [(RC 1)(C1C 2)(C2P)(S1C 2)] [(RS1)(C1R)(C 2C1)(S1W)]. ...................................28 Figure 1.5. Mineral recovery. Mass balance versus group contribution model results for a concentration circuit that exhibited a) good fit, b) poor fit..................35 Figure 2.1. Circuit [(RC1)(C 1C2)(C 2P)(S 1C2)] [(RS1)(C1R)(C2C 1)(S1W)]. ...............49 Figure 2.2. Difference between direct and reverse simulation. .............................56 Figure 2.3. Circuits chosen for the final design: a) circuit 1 and b) circuit 2. ........59 Figure 2.4. Sobol total index for each stage and for each species for a) circuit 1 and b) circuit 2. ........................................................................................................60 Figure 2.5. Sobol total index for the copper grade in the concentrates. ...............61 Figure 3.1. Concentration circuits used in case 1. .................................................76 Figure 3.2. Sobol’ total index versus stage’s recoveries for a) circuit a, b) circuit b, c) circuit c and d) circuit d. ..................................................................................78 Figure 3.3. Relation between Sobol’ total index and Morris method µ for circuit a (Figure 3.1a) for the a) high recovery specie and b) medium recovery specie. 81 Figure 3.4. Evolution of the overall recovery rates by a) modifying cell number cleaner and b) modifying cell number rougher.......................................................81 Figure 3.5. Yianatos’ model for a flotation column.................................................82 Figure 3.6. Comparison of sensitivity analysis using dispersions of a) 10% and b) ±0.1......................................................................................................................84 Figure 3.7. Comparison of sensitivity analysis using a a) uniform distribution, b) beta distribution with α and β = 2, c) beta distribution with α = 2 and β = 7, and d) a beta distribution with α=7 and β = 2. ...............................................................86 Figure 3.8. Flotation circuit (copper concentrator; Hay and Martin, 2004)............88 Figure 3.9. Morris versus diagram for a) the copper overall recovery rate, b) gangue overall recovery rate, and c) the copper concentrate grade. ...................91 Figure 3.10. Recovery rate and copper grade as a function of cell number and residence time: a) Cl-Sc recovery rate as a function of Cl-Sc cell number and residence time, (b) overall recovery rate as a function of Cl-Sc cell number and residence time and (c) overall copper grade as a function of Cl 2 cell number and residence time. ................................................................................................ 92 14 CHAPTER 1. A METHODOLOGY FOR THE CONCEPTUAL DESIGN OF CONCENTRATION CIRCUITS: GROUP CONTRIBUTION METHOD Felipe D. Sepúlveda, Luis A. Cisternas, Maritza A. Elorza, Universidad de Antofagasta, Chile And Edelmira D. Gálvez Universidad Católica del Norte, Chile. ABSTRACT This paper presents a new methodology for the conceptual design of concentration circuits based on the group contribution method. The methodology includes three decision levels: 1) definition and analysis of the problem, 2) synthesis and screening of alternatives, and 3) final design. In this manuscript, the emphasis is on the description of the methodology, justification of the assumptions, and group contribution method. The group contribution models were developed to estimate the global recovery in concentration circuits. The procedure is general and can be applied to any circuit consisting of stages that generate two product streams: concentrate and tail. The developed models can be applied to estimate the recoveries in concentration circuits with a maximum of six stages. The models were fitted using mass balance data from 46 circuits, generating 35 process groups. Case studies were used to illustrate the methodology. Keywords: process design, group contribution, concentration circuit, flotation. 15 1.1. Introductıon Flotation circuits are a common procedure for the concentration of a broad range of minerals and are also a common technology used in wastewater treatment. Froth flotation is based on differences in the ability of air bubbles to adhere selectively to specific mineral surfaces in a solid/water slurry. Particles with (without) attached air bubbles are (are not) carried to the surface and removed (stay in the liquid phase). The current practice for the design of these circuits is based on seven steps (Harris et al., 2002): 1) mineralogical examination in conjunction with a range of grinding tests, 2) a range of laboratory scale batch tests and locked cycle tests, 3) a circuit design based on scale-up of laboratory kinetics, 4) preliminary economic evaluation of the ore body, 5) pilot-plant test of the circuit design, 6) economic evaluation, 7) full-scale plant design. This procedure presents several problems, including 1) the design of the circuit in step three is based on a rule-ofthumb scale-up from laboratory data that depends heavily on the designer's experience, 2) the laboratory and pilot plant are costly and take significant time, and the designed circuit analysis is therefore not performed in depth, 3) other aspects, such as system dynamics, are not considered in the design process. Froth flotation design and operation is a complex task because several important parameters are interconnected (Barbery, 1983, Gupta and Yan, 2006). The parameters can be classified into four types of components, as shown in Figure 1.1. If any of these factors is changed, it causes or demands changes in other parts. It is impossible to study all of the parameters at the same time; for example, if six parameters are selected for study in four stage circuits, over 8 million tests are needed for a two-level fractional experimental design. In addition, for a given number of stages, there are several circuit configurations. If five flotation stages are considered, there are over one million potential circuits. For reasons of cost and time, only a fraction of the alternatives are analyzed, and only a small number of experiments are performed. In other words, the design analysis is not performed in depth. In the literature, various methodologies for the design of flotation circuits have been proposed, with most using optimization techniques. In these methodologies, the alternatives are presented through a superstructure, a mathematical model is developed, and an algorithm is used to find the best option based on an objective function. There are at least three reviews of studies concerning the optimal design of a flotation circuit including that of Mehrotra (1988), Yingling (1993a) and Méndez et al. (2009a). Some examples that use this strategy are Yingling (1993b), Hulbert (1995, 2001), Schena et al. (1996, 1997), Cisternas et al. (2004, 2006a), Guria et al. (2005a, b), and Ghobadi et al. (2011). The differences between these studies depend on the superstructure used, the 16 mathematical representation of the problem, and the optimization algorithm used. However, one problem with these methods is that the recovery of each stage must be modeled, and because the recovery of each stage is a function of many variables, modeling is not the most appropriate method. Moreover, the number of alternatives is large; a survey of approximately 400 flotation circuits available in books, manuscripts and industrial process descriptions, gives the number of flotation stages as between 1 and 9, with the most common being 3 to 5 stages. Considering that each stage is typically composed of 4 to 8 cells, the number of alternatives is tremendous, and a simple stage model to achieve adequate convergence in mathematical programming problems is essential. If metaheuristic-based algorithms are 3 used, it is possible to use more sophisticated stage models or cell models, but with problems of eight or more cells, it will be difficult to achieve convergence in a reasonable time. In summary, several methods for designing flotation plants have been proposed in the literature, but these are not applied in practice. Chemical Operation •Collectors. •Frothers. •pH. •Activators. •Depressants. •Particle size. •Pulp density. •Temperature. •Feed rate. •Pulp potential. Flotation Variables. Equipment Circuit •Cell design. •Agitation. •Air flow. •Number of stages. •Configuration. Figure 1.1. Interconnected components and parameters in froth flotation circuits. The flotation circuit synthesis problem determines the type of flotation stage and their sequence needed to achieve the concentration of the ore to some specified set of characteristics. The flotation design problem determines the optimal values for the conditions of operation and equipment related variables for the synthesized flotation circuit. The flow sheet modeling, synthesis and design problems are related since for generation and screening of alternatives, some forms of flow sheet models are needed. In addition, flow sheet models are needed for verification of the solutions of the synthesis/design problem. In contrast, a group-contribution based property estimation of a flow sheet requires knowledge of the process structure and the groups needed to uniquely represent it. 17 An example of such a method is the d'Anterroches and Gani (2005) method for fractional distillation based process. The needed property is estimated from a set of a priori regressed contributions for the groups representing the process. Having the groups and their contributions together with a set of rules to combine the groups to represent any process therefore provides the possibility to "model" the process. This also means that the reverse problem of property estimation, that is, the synthesis/design of process having desired properties can be solved by generating feasible process structures and testing for their properties. In this work, a new methodology is presented that integrates the first five design steps given by Harris et al. (2002) with the objectives of 1) better orienting the goals of the laboratory tests, 2) reducing laboratory and pilot plant testing, thereby achieving lower cost and execution times, 3) designing the flotation circuit based on a systematic procedure, and 4) speeding up the design procedure. The proposed methodology uses a completely different approach based on finding good designs (not necessarily optimal) between a more limited set of alternatives (eliminating unlikely alternatives) and evaluating the performance of each design using an approximate but simple model. The methodology, inspired by the work of d'Anterroches and Gani (2005) and Douglas (1985), considers three design decision levels 1) definition and analysis of the problem, 2) synthesis and screening of alternatives and 3) final design. This manuscript focuses on the description of the methodology, justification of the assumptions, and group contribution. This work is divided into six sections, of which this introduction is the first. The second section describes the methodology, including the decision levels (see Table 1.1). In the third section, the justification of the assumptions is presented. The fourth section presents the group contribution method. Case studies, focusing on the first and second decision levels, are presented in the fifth section, and finally, the sixth section presents the conclusions and proposes future work. 1.2. Methodology The methodology proposed is composed of three hierarchical levels: definition and analysis, synthesis and screening of alternatives, and final design (see table 1.1). In the first level, the ore characteristics to be separated and the characteristics of the separation circuit are defined. Then, alternatives are generated and evaluated to select a few separation circuits for the final level. In the final level, the selected designs are improved by defining the characteristics of each separation stage, without modifying the circuit structure. 18 1.2.1. Level I: Definition and analysis of the problem In this level, the problem is defined, including the characterization of the feed, the design goals and design and operation restrictions. The material to be fed to the process must be characterized. There are several ways to perform this characterization, including mineralogical examination, grinding tests, laboratory-scale batch test, and flotation kinetics tests. The decision of which test is most relevant to the project is made by the designer’s experience. However, the components that will be fed to the circuit must be defined. These components may be different mineralogical compositions, different sizes or both. The feed mass flow rate and the stage recovery for each component must be known. Table 1.1. Steps of the hierarchical decision. Level I: Level name Definition & analysis of the problem. Level II: Synthesis and screening of alternatives. Level III: Final design. Design activities Feed Characterization. Design goals. Estimated stage recovery values. Number of stages. Generation of feasible circuits alternatives. Circuit modeling with group contribution models. Validation based on material balance. Operational conditions. Equipment design. Only an approximate recovery value is needed for each component and each stage in the circuit, e.g., rougher, cleaner and scavenger stage. The determination of these values can be achieved with models, laboratory or plant data, and/or by the experience of the designer. For example, the criteria defined by Agar et al. (1980) can be used to define the stage recovery. These values are considered constant for the objective of selecting circuits in the second level. This assumption is important and will therefore be discussed in the next section. Additionally, the general criteria for classifying and determining which are the most promising circuits to process the feed material are defined. There are several possible factors to consider for this decision including: product quality (grade and impurities concentrations), plant capacity (concentrate production), and economic (income, profit, 19 cost). The decision as to which are the most important criteria for evaluating the alternatives for the project is determined by the designer’s experience. Finally, the maximum number of stages to be considered in the circuit design is defined. This definition must be completed for the cleaner and scavenger stages. This decision can be based on various criteria such as: mathematical models, statistical and historical background, or the designer’s experience. The method proposed by Gálvez (1998) can be used to define the number of cleaner and scavenger stages and is based on the recovery values of each stage and the technical goals (global recovery and grade desired). 1.2.2. Level II: Synthesis and screening of alternatives In this level, circuit alternatives are generated and evaluated using a group contribution model, and the most promising circuits are validated by mass balance. Alternatives are generated using origin-destination matrices for the concentrate and tail streams. The alternatives are evaluated using a group contribution method, which estimates the global recovery of each component. The group contribution method allows fast and simple calculation of the global recovery. The current group contribution model includes the rougher, cleaner 1, cleaner 2, cleaner 3, scavenger 1, and scavenger 2 stages. With these separation stages, 1,492 circuits can be generated. The group contribution model is described at length in the fourth section of this paper. The generated circuits are ordered based on the criteria specified in level I. A set of best alternatives is selected for validation. The designer can include other additional criteria for selection, such as: process control and dynamics or processing plants that are or were in operation. The set of alternatives is validated using mass balance or simulation, and the set is reclassified either under the same criteria used previously or new criteria. The designer can analyze each of the selected circuits, including control problems or previous experience in mineral processing in the selected circuit. The experience of the designer is added to the results of the simulations. Based on the results, a new set of circuits is selected for the final design. 1.2.3. Level III: Final design The final design is performed by defining the operational and equipment design associated variables. Up to this point, only approximated values were used for the recoveries of each stage. As shown in figure 1.1, these recoveries are functions of several parameters that are chemical, operational and equipment dependent (the circuit design is already complete). This problem is complex because there is not a model that can handle all the parameters simultaneously. Laboratory tests are necessary to identify operational 20 conditions and generate appropriate models to represent each flotation stage. Sensitivity analysis can be used to guide experimentation for each species by identifying the flotation stage or stages that affect the recovery of the species (Sepúlveda et al., 2013). These results allow to define operational conditions (e.g., particle size and pH) most suitable for the operation of each stage. After the experiment and generation of appropriate kinetic models it is necessary to define aspects such as residence time, size and number of equipment. This is usually done by performing circuit simulation using commercial software, i.e. sensitivity analysis based on scenarios. With the objective of reducing the complexity or scenarios , sensitivity analysis is conducted on the global recovery to identify the key variables to improve the design. Local (Lucay et al., 2011) and / or global (Hamby, 1994; Sepulveda et al., 2013) sensitivity analysis can be used. The result from this analysis is the identification of variables where the efforts must be focused to improve the behavior of each species in the process. Alternatively, optimization can be used to determine residence times and / or particle size and / or number of equipment given the structure of the flotation circuit. 1.3. Justification of the assumption In this work, it is assumed that the stage recovery can be considered constant for design purposes, i.e., it is independent of the concentration circuit. Assuming constant recoveries for each stage may be questionable because they depend on the feed characteristics (values of other parameters), and these characteristics are different for each alternative circuit. This assumption will be analyzed from two points of view. First, the literature will be discussed, and new evidence will subsequently be shown. This new evidence indicates that this assumption is a valid approach as a first approximation. The process design based on optimization using superstructures has shown that there are cases in which the best structure is not highly sensitive to the operational values. For example, Cisternas and colleagues studied optimal structures for separation based on fractional crystallization (Cisternas, 1999) and found that in many cases, the best structure was independent of changes in operating conditions. Furthermore, Cisternas and coworkers (Cisternas and Rudd, 1993; Cisternas et al., 2006b) found that there are areas within a design region where a design is always superior to another, regardless of the operating conditions. Although this is not transferable to flotation circuit design, it sets a precedent for efforts to study whether this assumption is valid in the design of flotation circuits. Cisternas et al. (2004) developed a procedure for the design of flotation circuits based on mathematical programming using two-level hierarchized superstructures. The procedure 21 considered that the flotation can be modeled using a first-order kinetic model. The method was applied to a copper flotation plant. The first case studied considered that all banks use 15 cells, and a second case was studied allowing using 10, 15 or 20 cells per bank. The optimal flotation circuit obtained was the same, but with different values of recovery in each flotation stage. This means the same optimal structure results from different operational conditions and number of cells in each bank. Other cases were studied, including more complex structures, and similar results were obtained. Later, Méndez et al. (2009b) used different grinding circuits (grinding, grinding-classification, classification-grinding, and classification-grinding-classification) in the design of flotation circuits. The application to a copper flotation plant indicates that the same flotation structure was obtained using different grinding circuits. These studies show that the optimal flotation circuit depends strongly on the feed composition and metal price but has a low dependence on stage recovery. Jamett et al. (2012) presented a model for the design of flotation circuits under uncertainty using stochastic programming. Uncertainty was represented by several scenarios, including changes in the feed grade and metal price. The model allows for changing the operating conditions (residence time) and flow sheet structure for each scenario whilst maintaining fixed equipment design (number of cells in each bank of flotation) for all scenarios. The results showed that the optimal flow sheet structure did not change for 8 of the 9 scenarios studied, but the recoveries of each stage changed for each scenario. Additionally, Montenegro et al. (2009), Montenegro et al. (2010) and Montenegro et al. (2013a) studied the effect of the uncertainty in the recoveries of the rougher, cleaner, re-cleaner and scavenger stages on the global recoveries and final concentrate grade, among other indicators, for 12 flotation circuits. The uncertainties in the recoveries of each stage were represented by normal, triangular and uniform distribution functions with variations between 1% and 10%. In other words, the recoveries at each stage were not modeled with any kinetic model but were represented by distribution functions. Uncertainties were studied by considering the variation in each stage as well as in several stages simultaneously; 84 cases were considered. Monte Carlo simulation was used in the study, adding up to more than 6 million simulations. The results showed that the best flotation circuits were not a function of the stage recoveries, that is, for different values of stage recoveries there is a set of flotation circuits that perform best. Later, Montenegro et al. (2013b) applied a shortcut computational method to analyze and compare alternative flotation circuits to treat high-arsenic copper ores. Twenty-seven circuits were evaluated based on the metric indices of efficiency, capacity, quality, economics and environmental impact. The simulations were performed for an Australian 22 sulfide ore containing chalcopyrite, tennantite, quartz, and pyrite. In the simulation, constant stage recovery was assumed. To validate this assumption, the normalized indicators were calculated for several values of stage recoveries for each mineral; three levels were selected with ±5, ±10, and ±20%, which can be considered as moderate, intermediate and high variation in stage recoveries. Random sampling of the case studies was selected to reduce the sampling error. The size of the sample was estimated in 28 combinations, which gives a 0.95 confidence level. Twenty-eight combinations were studied for twenty-seven circuits; therefore, 2,268 simulations were performed. The results of the simulations for moderate, intermediate and high variations were normalized for each combination of stage recovery values. The average and standard deviation values for the 28 normalized values for a specific circuit and a specific indicator were calculated. The standard deviations were usually small, indicating that the indicators do not undergo large variations despite changes in the stage recovery values. Usually, the circuit with the best results has a low standard deviation, i.e., these circuits give the best results independent of the value of stage recovery. Circuits with moderate results sometimes have significant variation, i.e., the position within the set of alternatives has greater variability. Despite the variation, the value never exceeds the values of the best circuits; therefore, these circuits will be never selected based on this indicator. All of these previous works do not demonstrate that constant recovery can be assumed for each stage of flotation but provide evidence that the result can be extrapolated to other flotation systems. It should be emphasized that these results are significant, because there are not few cases, but a large number of simulations and a lot optimization works, including a significant number of circuits under different stage recoveries. Figure 1.2a shows the overall recovery for five flotation circuits, each composed of three stages. These calculations considered that recoveries are the same for all stages. The global recovery of circuit 1 is greater than all other circuits. This behavior is independent of the stage recovery values. This means that if, for a particular process that is dominated by the recovery of the value species (e.g., a value species with high floatability and high price, and gangue with low floatability and low charge for their presence in the concentrate), circuit 1 is always better than the other circuits. Comparing circuit 2 to circuit 4, it can be observed that the curves cross around recovery stage 0.5. This finding means that although the recovery stage of the value species is greater than 0.5, and the stage recovery of the gangue is less than 0.5, circuit 2 is better than circuit 4. There are regions of stage recoveries where a circuit (or a set of circuits) is better than another circuit (or a set of circuits). This means that approximate stage recovery values can be used for purposes of selecting a set of circuits that have better potential to give good 23 results. This selection will be correct when the approximate values are within the region of stage recovery values where these circuits give the best results. Figure 1.2b shows the profit of those five circuits when used to process a copper sulfide ore using flotation banks. The profits are compared for different residence times in the flotation cells. Circuit 1 provides the greatest profit independent of the residence time (and hence, the values of the stage recoveries). These calculations were performed for different numbers of cells per bank with similar results. These results confirm that constant stage recovery can be assumed for circuit selection as an initial approximation. 1.1. Group Contribution Method Group contribution models are popular for estimating the properties of pure and mixed chemicals. These models have been shown to be useful for designing chemical products for specific applications. The basis of each model is that the property value of a chemical can be estimated by adding the contributions of the constituent groups of the chemical. In developing a group contribution model for a specific chemical property, experimental or simulated data from a set of chemicals are used to calculate the value contributed by each constituent group. These constituent values are then used to estimate either property values for chemicals that have not yet been measured or the property values of hypothetical chemicals. The principal application of group contribution models is the design of chemical products. Group contribution models have been used to design promising candidates for a variety of applications such as polymers, extractants, solvents, refrigerants, and catalysts (Karunanithi et al., 2006). Additionally, group contribution models have been used for developing thermodynamic models for process design (Gmehling, 2003). Because of the successful application of group contribution models to the design of chemical products, d’Anterroches and Gani (2005) developed a group contribution model for the design of a separation process based on fractional distillation. They developed the concept of a process group that was analogous to a chemical group. In their work, a framework for the process design was presented. More recently, a process group contribution methodology was integrated with a controller design methodology to simultaneously design and control a bioethanol production process (Alvarado-Morales et al., 2010). The purpose of this section is to develop a group contribution model to estimate the recovery in concentration circuits. The procedure is general and can be applied to any circuit consisting of units that generate two product streams: concentrate and tail. This section is organized as follows: first, a procedure is developed to generate all of the 24 feasible circuits, given the concentration stages. In addition, a string is generated to represent each circuit, which can be used to store a circuit in a database. Next, the group contribution model is described. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Global Recovery Circuit 1 Circuit 2 Circuit 3 Circuit 4 Circuit 5 0 0.2 0.4 0.6 0.8 1 Stage Recovery (a) Profit ( M US$ /day) 60 50 40 30 Circuit 1 Circuit 2 Circuit 3 Circuit 4 Circuit 5 20 10 0 0.02 0.04 0.06 0.08 0.10 Time ( h ) (b) Figure 1.2. Comparison of five circuits of three flotation stages. [(RC)(CP)(SC)][(RS)(CR)(SW)], circuit 2 [(RC)(CP)(SR)][(RS)(CR)(SW)], [(RC)(CP)(SR)][(RS)(CS)(SW)], circuit 4 [(RC)(CP)(SC)][(RS)(CS)(SW)], [(RC)(CP)(SC)][(RW)(CR)(SW)]. The nomenclature used is explained below. 25 Circuit circuit circuit 1 3 5 1.1.1. Generation of alternatives In this sub-section, a procedure is proposed to generate feasible concentration circuits. It is assumed that each concentration stage has two output streams: a concentrate and a tail. The following definition is used: the rougher stage (R) is a stage that processes a circuit feed, the cleaner stage (C) is a stage that processes a concentrate stream, and the scavenger stage (S) is a stage that processes a tail stream. These definitions are important because they significantly reduce the number of alternatives, as will be shown later. The procedure will be illustrated with an example. Let us assume that the generation of feasible circuits that consist of rougher, cleaner, and scavenger stages is desired. Feasible paths for the concentrate and tail streams using two origin-destination matrixes will be identified. Figure 1.3 shows these matrixes, where R, C, S, P, and W represent rougher, cleaner, scavenger, final concentrate, and final tail, respectively. The concentrate origindestination matrix shows that the rougher concentrate must be sent to the cleaner stage (we will represent this by the string RC), the cleaner concentrate must be a final concentrate (CP), and the scavenger concentrate can be sent to either the rougher stage (SR) or the cleaner stage (SC). Thus, we can identify two paths for the concentrate streams: [(RC)(CP)(SR)] and [(RC)(CP)(SC)]. Similarly, the tail origin-destination matrix shows that the rougher tail must be sent to the scavenger stage (RS), the cleaner tail can be sent to either the rougher stage (CR) or the scavenger stage (CS), and the scavenger tail must be the final tail (SW). Thus, we can identify two paths for the tail streams: [(RS)(CR)(SW)] and [(RS)(CS)(SW)]. The feasible concentration circuits are the combinations of the concentrate and tail paths. Thus, there are four feasible concentration circuits: [(RC)(CP)(SR)] [(RS)(CR)(SW)], [(RC)(CP)(SR)] [(RS)(CS)(SW)], [(RC)(CP)(SC)] [(RS)(CR)(SW)], and [(RC)(CP)(SC)] [(RS)(CS)(SW)]. The feasible concentrate circuits and a method to represent the concentration circuits by a string that shows the concentrate and tail paths were identified. The notation is called Concentrate and Tail Paths String (CTP string). The CTP string facilitates the storage of concentration circuits in a database. The procedure can be extended to circuits with two or more cleaner and scavenger units. For example, the circuit in Figure 1.4a can be represented by the CTP string [(RC 1)(C1C 2 )(C2P)(S1C2)] [(RS1)(C1R)(C2 C1)(S1W)], where the string within the first set of square brackets corresponds to the concentrate paths, and the string within the second set of square brackets corresponds to the tail paths. For clarity in Figure 1.4a the concentrate path is represented by solid lines and the tail path with dashed lines. In addition, from the combination of both paths, we can identify each concentration stage. For example, the 26 combination (RC1) in the concentrate path with (RS1) in the tail path indicates that the rougher stage sends its concentrate to cleaner 1 and its tail to scavenger 1 (RC1S1). Some circuit design procedures based on superstructures allow sending concentrate and tail streams between all stages, excluding recirculation at the same stage. This not only produces illogical circuits, e.g., sending rougher concentrate to the scavenger stage, but significantly increases the number of alternatives. For example, when considering the R, C, and S stages, the number of alternatives is 729 compared with the four alternatives identified above. This condition worsens as we increase the number of stages. For example, in the case of considering R, C1, C2, and S stages, the number of feasible and logical alternatives is 24 (identified using the origin-destination matrix) versus 65,536 if all stream recycle is allowed. Destination Destination Tail R Origin R C P R x C S S R Origin Concentrate X x x C S C S W x x x x Figure 1.3. Concentrate and tail origin-destination matrixes for a three-stage concentration circuit. Using origin-destination matrices for the concentrates and tails, the total number of feasible and logic circuits can be determined. A database was constructed for all feasible and logical circuits that included a rougher, cleaner 1, cleaner 2, cleaner 3, scavenger 1, and scavenger 2 stage. For circuits with two stages, there are two feasible circuits. For circuits with three stages, there are eight feasible circuits (four with R, C 1, S1; two with R, C1, C2; and two with R, S1 , S2). For circuits with four stages, there are 42 circuits. For circuits with five stages, there are 240 circuits. For circuits with six stages, there are 1,200 circuits. In total, there are 1,492 circuits. If a concentration stage is defined based on the type of stage (rougher, cleaner 1, cleaner 2, cleaner 3, scavenger 1, and scavenger 2) and the destinations of the concentrate and tail, then there are 35 types of concentration stages. These concentration stages will be called process groups in the contribution model. 27 1.1.2. Process groups In the proposed group contribution model, the behavior of the process is predicted from the behavior (contribution) of the constituent parts of the process (group). Specifically, in the case of the concentration circuit, the behavior of the circuit is predicted based on the contribution of each concentration or process group. These process groups are a function of the concentration stages (rougher, cleaner 1, cleaner 2, cleaner 3, scavenger 1, and scavenger 2) and the destinations of their products (concentrates and tails). (a) (b) Figure 1.4. Concentration circuit with four stages, represented by the CTP string [(RC1)(C1C2)(C2P)(S1C2)] [(RS1)(C1R)(C2C1)(S1W)]. For example, for the circuit in Figure 1.4a, [(RC1)(C1C 2 )(C2P)(S1C2)] [(RS1)(C1R)(C2 C1)(S1W)], the process groups are RC1S1, C1C2R, C2 PC 1 and S1C2 W (Figure 1.4b). In Figure 1.4b, for clarity, the groups are represented by different colors (digital version only) to emphasize that each group includes the flotation stage and the destination of its concentrate and tail. These groups are easily identified by combining the concentration and tail paths. The group RC1S1 corresponds to a rougher stage where the concentrate and tail are sent to C1 and S1, respectively. Similarly, group S1C2 W corresponds to a scavenger 1 stage, where the concentrate is sent to C2, and the tail is the final circuit tail. Because each group not only includes the flotation stage but also the destination of its concentrate and tail, each group carries topological information with it. This means for example that the group C 1C2R represents a cleaner 1 stage wherein the concentrate is sent to the cleaner 2 stage and the tail to the rougher stage. 28 Table 1.2 shows thirty-five process groups that are distributed as follows: three groups with the rougher stage, six groups with the cleaner 1 stage, eight groups with the cleaner 2 stage, five groups with the cleaner 3 stage, eight groups with the scavenger 1 stage and five groups with the scavenger 2 stage. In table 2 , groups 1 to 3 represent the rougher stage but differ in their concentrate and tail destinations. In group RC 1 S1, the rougher concentrate and tail are processed in the cleaner 1 and scavenger 1 stages, respectively. In contrast, in process groups RC 1W and RPS1 , only the concentrate and tail are processed, that is, the tail in group RC1 W and the concentrate in group RPS1 are the final products. Table 1.2. Process groups. Type Process Groups Rougher Cleaner 1 RC1 W, RPS1, RC1S1 C1 PR, C1 PS1 , C1 PS2, C1C2R, C1C 2S1, C1 C2S2 Cleaner 2 C2 PC1, C2 PR, C2 PS1, C2 PS2, C2C3 C1, C2C3R, C2 C3S1, C2C3S2 Cleaner 3 C3 PC2, C3 PC1, C3 PR, C3 PS 1, C3 PS2 Scavenger 1 S1RW, S1C1 W, S1C2 W, S1C3 W, S1RS2, S1C1S2, S1C2S2 , S1C3S2 Scavenger 2 S2S1W, S2RW, S2C1 W, S2C2 W, S2C3 W 1.1.3. Recovery Models For predicting the circuit recovery, two models are proposed that depend on the recovery values of the rougher stage. For high rougher recoveries (0.63 to 0.9) and medium rougher recoveries (0.38 to 0.62) the following model is used: ∑ ( ) ( ) ( ) (1.1) where Rc is the circuit recovery; is the contribution of process group i; is the number of i process groups in the circuit; is the total number of process groups in the circuit; is the total number of cleaner stages in the circuit; is the total number of scavenger stages; and a , b , and c are constants. For low rougher recoveries (0 to 0.37), the following model is used: ∑ ( ( ) (1.2) ) 29 where the process group contribution, i , depends on the stage recovery, Ti , as follows: i i Ti i (1.3) where i and i are constant for each group i. The values of a , b, c , i and i must be fitted based on global recovery values of circuits. For the generation of global recovery values, forty-six circuits were selected, and a mass balance was performed for 500 random rougher recoveries for each range and for the forty-six circuits. In total, there are 23,000 experimental values for each rougher recovery range. The cleaner and scavenger recoveries were selected as random numbers within 10% of the rougher recovery. The forty-six circuits are shown in Table 1.3. Note that two, four, twentyfive, ten, and five circuits with two, three, four, five, and six stages were utilized, respectively. Each process group appears in at least two concentration circuits used in the fitting process. On average, each group is present in 5.7 circuits in the fitting process. 1.1.1. Adjustment Method The group contribution models were fitted using BARON-GAMS (Sahinidis and Tawarmalani, 2011). The results are shown in tables 1.4 and 1.5. Table 1.4 shows the constants of equation (1.3) that are a function of each group and recovery range. Table 1.5 shows the constants a, b, and c of equations (1.1) and (1.2). The mean absolute error (MAE) was used to quantify the difference between the predicted and actual recoveries. The MAE was calculated using the following equation: ∑ | | (1.4) Where is the predicted global circuit recovery of data k (from the group contribution), is the global circuit recovery of data k (from the mass balance), and N is the total number of data points. is the global circuit recovery of data k (from a mass balance), and N is the total number of data points. 30 Table 1.3. Concentration circuits used to fit the group contribution model. Key 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 CTP String Two-stage circuits [(RC1) (C1P)] [(C1R) (RW)] [(RP) (S1R)] [(RS1) (S1W)] Three-stage circuits [(RC1) (C1P) (S1C1)] [(RS1) (C1R) (S1W)] [(RC1) (C1P) (S1R)] [(RS1) (C1R) (S1W)] [(RC1) (C1P) (S1C1)] [(RS1) (C1S1) (S1W)] [(RP) (S1R) (S2S1)] [(RS1) (S1S2) (S2W)] Four-stage circuits [(RC1) (C1C2) (C2P) (S1R)] [(RS1) (C1R) (C2C1) (S1W)] [(RC1) (C1C2) (C2P) (S1C1)] [(RS1) (C1R) (C2C1) (S1W)] [(RC1) (C1C2) (C2P) (S1C2)] [(RS1) (C1R) (C2C1) (S1W)] [(RC1) (C1C2) (C2P) (S1R)] [(RS1) (C1R) (C2R) (S1W)] [(RC1) (C1C2) (C2P) (S1C1)] [(RS1) (C1R) (C2R) (S1W)] [(RC1) (C1C2) (C2P) (S1C2)] [(RS1) (C1R) (C2R) (S1W)] [(RC1) (C1C2) (C2P) (S1R)] [(RS1) (C1R) (C2S1) (S1W)] [(RC1) (C1C2) (C2P) (S1C1)] [(RS1) (C1R) (C2S1) (S1W)] [(RC1) (C1C2) (C2P) (S1C2)] [(RS1) (C1R) (C2S1) (S1W)] [(RC1) (C1C2) (C2P) (S1R)] [(RS1) (C1S1) (C2R) (S1W)] [(RC1) (C1C2) (C2P) (S1R)] [(RS1) (C1S1) (C2C1) (S1W)] [(RC1) (C1C2) (C2P) (S1R)] [(RS1) (C1S1) (C2S1) (S1W)] [(RC1) (C1C2) (C2P) (S1C1)] [(RS1) (C1S1) (C2R) (S1W)] [(RC1) (C1C2) (C2P) (S1C1)] [(RS1) (C1S1) (C2C1) (S1W)] [(RC1) (C1C2) (C2P) (S1C1)] [(RS1) (C1S1) (C2S1) (S1W)] [(RC1) (C1C2) (C2P) (S1C2)] [(RS1) (C1S1) (C2R) (S1W)] [(RC1) (C1C2) (C2P) (S1C2)] [(RS1) (C1S1) (C2C1) (S1W)] [(RC1) (C1C2) (C2P) (S1C2)] [(RS1) (C1S1) (C2S1) (S1W)] [(RC1) (C1P) (S1R) (S2S1)] [(RS1) (C1S1) (S1S2) (S2W)] [(RC1) (C1P) (S1C1) (S2S1)] [(RS1) (C1R) (S1S2) (S2W)] [(RC1) (C1P) (S1C1) (S2C1)] [(RS1) (C1R) (S1S2) (S2W)] [(RC1) (C1P) (S1C1) (S2R)] [(RS1) (C1S2) (S1S2) (S2W)] [(RC1) (C1P) (S1R) (S2C1)] [(RS1) (C1S2) (S1S2) (S2W)] [(RC1) (C1C2) (C2C3) (C3P)] [(C1R) (C2C1) (C3C1) (RW)] [(RC1) (C1C2) (C2C3) (C3P)] [(C1R) (C1R) (C3C1) (RW)] Five-stage circuits [(RC1) (C1C2) (C2C3) (C3P) (S1C1)] [(RS1) (C1R) (C2C1) (C3C2) (S1W)] [(RC1) (C1C2) (C2C3) (C3P) (S1R)] [(RS1) (C1S1) (C2C1) (C3C2) (S1W)] [(RC1) (C1C2) (C2C3) (C3P) (S1R)] [(RS1) (C1S1) (C2S1) (C3C2) (S1W)] [(RC1) (C1C2) (C2C3) (C3P) (S1C3)] [(RS1) (C1R) (C2R) (C3R) (S1W)] [(RC1) (C1C2) (C2C3) (C3P) (S1C3)] [(RS1) (C1S1) (C2R) (C3S1) (S1W)] [(RC1) (C1C2) (C2P) (S1C2) (S2R)] [(RS1) (C1S2) (C2S2) (S1S2) (S2W)] [(RC1) (C1C2) (C2P) (S1C2) (S2C2)] [(RS1) (C1S2) (C2S2) (S1S2) (S2W)] [(RC1) (C1C2) (C2P) (C2C3) (S1C2)] [(RS1) (C1R) (C2C1) (C3C2) (S1W)] [(RC1) (C1C2) (C2P) (S1R) (S2C1)] [(RS1) (C1R) (C2C1) (S1S2) (S2W)] [(RC1) (C1C2) (C2P) (S1R) (S2S1)] [(RS1) (C1S1) (C2R) (S1S2) (S2W)] Six-stage circuits [(RC1) (C1C2) (C2C3) (C3P) (S1R) (S2C2)] [(RS1) (C1R) (C2S2) (C3S1) (S1S2) (S2W)] [(RC1) (C1C2) (C2C3) (C3P) (S1C3) (S2C3)] [(RS1) (C1S1) (C2S1) (C3S2) (S1S2) (S2W)] [(RC1) (C1C2) (C2C3) (C3P) (S1C3) (S2C3)] [(RS1) (C1S2) (C2S2) (C3R) (S1S2) (S2W)] [(RC1) (C1C2) (C2C3) (C3P) (S1C3) (S2C1)] [(RS1) (C1S2) (C2R) (C3S2) (S1S2) (S2W)] [(RC1) (C1C2) (C2C3) (C3P) (S1R) (S2C1)] [(RS1) (C1S1) (C2R) (C3C2) (S1S2) (S2W)] 31 Table 1.4. Constants i and i for the process group contribution defined in equation (1.3) for high, medium and low recoveries. Group RC1W RPS1 RC1S1 C1PR C1PS1 C1PS2 C1C2R C1C2S1 C1C2S2 C2PC1 C2PR C2PS1 C2PS2 C2C3C1 C2C3R C2C3S1 C2C3S2 C3PC2 C3PC1 C3PR C3PS1 C3PS2 S1RW S1C1W S1C2W S1C3W S1RS2 S1C1 S2 S1C2 S2 S1C3 S2 S2S1 W S2RW S2C1W S2C2W S2C3W High -0.8096 1.6482 -0.0975 22.5778 22.5904 22.5855 -0.0904 -0.1094 -0.2209 34.8542 34.8748 34.8747 35.0091 43.5820 43.5389 43.5537 43.6818 -0.0412 -0.0433 0.1373 0.3073 0.1531 -0.1919 -0.1976 -0,1941 -0.3886 0.0967 0.0384 -0.0852 -0.0756 -0.0078 -0.0101 -0.0345 0.2358 -0.1089 i Medium 4.1843 0.8343 0.4974 0.7966 0.7110 0.6484 0.5587 0.8440 0.3529 0.6036 0.4838 0.3440 0.5559 0.6750 0.3037 8.7125 -0.1845 0.2863 0.1597 0.5608 0.5075 0.3869 1.3625 0.7243 0.5460 0.1720 0.7862 0.2192 0.2358 0.1889 0.6262 0.3129 0.3804 0.2356 0.5975 Low 0.0813 1.5217 28.4560 8.0851 3.8272 2.5667 42.7709 4.0928 0.0194 8.9199 3.9322 2.6912 5.7776 -0.0010 -0.0024 -6.2597 -06366 27.1021 -0.0619 12.9763 2.1779 5.7664 -5.2716 10.7153 3.9735 0.0811 -0.2353 3.4184 3.5684 0.3804 -0.6491 -1.2823 5.3409 0.0144 10.3996 32 High -0.9400 0.0301 -1.7500 0.0006 0.0097 0.0126 -1.1507 -1.7500 -0.7535 -0.0001 0.0039 0.00863 0.0093 0.0032 0.0051 0.0077 0.0227 -1.4345 -1.7500 -1.0586 2.2664 4.1785 -1.5612 -1.0913 -0.8529 -0.6019 4.7189 0.2011 16.0084 10.9217 -1.7500 -1.7500 27.2542 8.3795 11.1927 i Medium 7.5618 1.0510 1.6714 1.0449 1.2054 1.5398 2.5373 5.2019 4.5312 0.9665 0.8841 0.6035 1.2429 3.4281 3.4152 11.7590 -0.0306 0.2172 0.2325 0.7774 0.5468 1.7273 5.0024 2.4400 1.2619 0.4562 4.4755 0.6843 0.6650 0.2100 3.1844 1.5192 0.7733 0.2555 0.5288 Low 0.1608 1.3301 4.4241 2.3818 1.8286 1.5256 4.8912 3.5483 0.2143 3.0169 2.6869 2.5104 1.9023 0.0450 0.3505 4.1003 0.8924 5.4712 0.0527 2.2243 1.3781 2.2493 3.9253 3.5657 1.8049 0.1659 1.2189 1.6847 1.7910 1.1409 2.0748 1.2664 2.3949 0.0450 1.7967 Table 1.5. Constants for equations 1.1 and 1.2. Constant High Medium Low a b -0.02360 -30.0000 0.69253 -0.80551 -7.19419 10.30702 c -0.58448 -0.44709 6.92905 The MAE values for all circuit used in the fitting are 0.018, 0.018, and 0.009 for the high, medium, and low rougher recovery ranges, respectively. Table 1.6 shows the MAE for each circuit used in the fitting process. The maximum MAE values are 0.075 (for circuit 35), 0.049 (for circuit 28) and 0.034 (for circuit 1) for the high, medium and low recovery ranges, respectively. Circuit 35, [(RC1) (C1C2 ) (C2C3) (C3 P) (S1C3)] [(RS1 ) (C1R) (C2R) (C3R) (S1 W)], has three cleaner stages and one scavenger stage; circuit 28, [(RC 1) (C1P) (S1C1) (S2R)] [(RS1) (C 1S2) (S1S2) (S2 W)], has one cleaner stage and two scavenger stages; and circuit 1, [(RC1) (C1P)] [(C 1R) (RW)], has one cleaner stage. To illustrate the predictive capability of the group contribution model, Figure 1.5 shows the recoveries calculated by mass balance versus the recoveries predicted by the group contribution model for two examples. Figure 1.5a shows an example where the fit is notably good, whereas figure 1.5b shows an example with a poor fit, even though the results are good. Different symbols were used for high, medium and low recoveries. 1.1.2. Example and Validation As an example, consider the circuit in Figure 1.3, which corresponds to circuit 9 in table 1.6, CTP string [(RC1) (C 1C2) (C2 P) (S1 C2)] [(RS1) (C1R) (C2C1) (S1W)]. Let us consider that the recoveries in all stages are equal to 0.8. Then, by mass balance, it is straightforward to demonstrate that the recovery is given by: Rc (T ) T2 0.95012 2 (1 3 T 5 T 3 T 3 T 4 ) Where T is the stage recovery. The process groups are RC 1S1 , C1C2R, C2PC1, and S1 C2W. Using equation 1.3 and the values in table 1.4 for a high rougher recovery, we obtain RC1S1 =-0.14408, C1C2 R = -0.11686, C2 PC1 =34.85498, and S1C2W =-0.27188. Finally, using equation 1.1, we obtain Rc =0.95020, which corresponds to an absolute error of 0.00008. 33 Table 1.6. Mean absolute error for each circuit of table 1.3 for high, medium, and low recoveries. Circuit 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 MAE High Medium Low 0.007 0.016 0.009 0.008 0.020 0.012 0.013 0.017 0.011 0.013 0.008 0.008 0.016 0.011 0.012 0.010 0.009 0.010 0.011 0.015 0.012 0.022 0.028 0.028 0.018 0.007 0.017 0.024 0.027 0.019 0.013 0.010 0.014 0.009 0.009 0.030 0.008 0.015 0.009 0.010 0.019 0.009 0.016 0.017 0.016 0.035 0.034 0.010 0.009 0.015 0.002 0.007 0.030 0.002 0.004 0.018 0.002 0.004 0.014 0.002 0.004 0.002 0.003 0.004 0.004 0.018 0.025 0.015 0.002 Circuit 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 34 MAE High Medium Low 0.010 0.014 0.024 0.009 0.010 0.011 0.011 0.017 0.013 0.025 0.022 0.075 0.037 0.017 0.014 0.024 0.054 0.033 0.039 0.015 0.024 0.014 0.030 0.010 0.009 0.017 0.039 0.049 0.008 0.013 0.038 0.008 0.007 0.018 0.019 0.013 0.012 0.007 0.015 0.011 0.009 0.019 0.046 0.043 0.016 0.010 0.001 0.001 0.006 0.006 0.015 0.002 0.013 0.020 0.008 0.005 0.006 0.003 0.007 0.007 0.001 0.011 0.007 0.003 0.030 0.016 0.009 0.007 0.002 Mass Balance 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 Group Contribution Model (a) Mass Balance 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 Group Contribution Model (b) Figure 1.5. Mineral recovery. Mass balance versus group contribution model results for a concentration circuit that exhibited a) good fit, b) poor fit. To validate the model, the procedure was applied to five circuits that were not included in the fitting: four circuits with five stages (three with R, C 1, C 2, S1 , S2; one with R, C1, C2 , C3, S1) and one circuit with six stages (R, C1 , C2, C3, S1, S2 ). The MAE values were 0.031 for 7,500 data points. Therefore, the model showed predictive capabilities in these two circuits. It should be emphasized that this means that the group contribution model provides a valid estimate of the global recovery independent of the composition of each stream in the circuit. 35 1.2. Case Study A short example is given in this section for the separation of a copper sulfide ore where the value mineral is chalcopyrite (CuFeS2) and the gangue are pyrite (FeS2), arsenopyrite (AsFeS) and silica (SiO2). The concentration technology is froth flotation, which is a common technology for this type of ore. The case study will be presented for decision levels I and II. Level III is not analyzed in this paper. 1.2.1. Level I. Definition and analysis At this level, the decisions to be taken are the characterization of the feed, design objectives, estimated recoveries by stage and number of stages in the flotation circuit. Based on the ore characterization of the feed, four species were considered (Cp: 100% CuFeS2; CpPy: 95% CuFeS2 and 5% FeS2 ; PyAs: 99% FeS2 and 1% AsFeS; Sc: 100% SiO2). The feed mass flow rate and stage recoveries are given in Table 1.7. The recoveries of each stage are approximate values and are assumed to be constant for the purposes of selecting and evaluating alternatives. With the composition of the ore feed and the feed flow rate given in table 1.7, the copper grade is 1.02% and the arsenic grade is 0.07%. Table 1.7. Parameters values for the case study. Estimated Stage recoveries Species Cp CpPy PyAs Sc Rougher Cleaner Scavenger Feed (ton /h) 0.82 0.53 0.3 0.12 0.87 0.54 0.28 0.11 0.77 0.58 0.32 0.13 10 20 150 800 Separation goals, % >90 >60 <4 <1 Two types of design objectives were defined. First, the desired minimum/maximum values of global recovery for each species were defined. These values are given in the last column of Table 1.7. For example, recoveries of at least 90% of Cp and not exceeding 1% of Sc are expected. For assessing alternatives, the revenues of each circuit will be considered to be indicators. These revenues will be evaluated using equation 1.5 and correspond to the net smelter return model (Cisternas et al., 2004). [(∑ ) ( )( ) (∑ 36 )( )] (1.5) Where is the mass flow of species i in the concentrate, is the fraction of metal paid, is the copper grade of the concentrate, is the grade deduction, is the treatment charge, is the refinery charge and represents the arsenic charge. is the number of hours per year of plant operation when the flows are in mass per hour. This formula for the calculation of revenues incorporates the metallurgical efficiency of the plant, that is, the recovery and mineral content are opposite functions. Recently it has been shown that revenue is an adequate objective function to evaluate flotation circuits (Cisternas et al., 2013). In this work, a copper price, , of US$ 4,000/ton of copper was used. To estimate the number of cleaner and scavenger stages, the procedure developed by Galvez (1998) was used. In this procedure, based on the stage recoveries and knowing the desired overall separation levels, the number of cleaner and scavenger stages needed to separate two species are calculated. The Galvez procedure was then applied to each pair of species using the stage recoveries given in table 1.7 and the minimum/maximum values of global recovery of each species (last column of table 1.7). The maximum number of stages obtained was selected as the number of stages required. The procedure provides three cleaner stages and one scavenger stage. 1.2.2. Level II. Synthesis and screening of alternatives In this level, feasible circuits are generated and modeled using the group contribution model. The feasible circuit must be generated with one rougher stage (R), three cleaner stages (C1, C2, and C3) and one scavenger stages (S1). Using the procedure described in section 1.3.1, 128 circuits were obtained. These circuits are simulated using the following procedure: 1) the group contribution of each stage is calculated (equation 1.3) using the constants in table 1.4 and the stage recoveries in table 1.7; 2) global recoveries for Cp, CpPy, PyAs and Sc are determined for each alternative circuit using equations 1.1 and 1.2; 3) the revenues are calculated using equation 1.6, and a ranking of the best alternatives is generated based on this indicator. This procedure is simple and can be performed easily in a spreadsheet. In this work twenty circuits were selected for validation. These circuits are given in table 8; four of them have four stages, and sixteen have five stages. The first two columns of Table 1.8 show the CTP strings. The third and fourth columns give the ranking based on group contribution and mass balance, respectively. The last column shows the revenue calculated using the mass balance results. Six of the top ten selected circuit using group contribution are among the top ten based on mass balance ranking. To check if there were better circuits, the top 35 group contribution 37 circuits were validated by mass balance. The top ten circuits based on group contribution were among the top 16 using mass balance. However, the differences between the revenue values is not large, as there is only an 11% difference between the top ten. Therefore, the group contribution system selected a set of suitable circuits. For the selection of circuits for the Level III final design, the designer can use variou s criteria. First, previous experience can be used if some of the circuits have been used previously. Second, new criteria may include the stability of the circuits or the number of stages required. Finally, the results of the mass balance validations can be used; for example, Table 1.9 shows details, concentrate grade and overall recovery for the top nine circuits. Table 1.8. Set of alternatives selected for mass balance validation. String Ranking Tail (RC1)(C1C2)(C2C3)(C3P)(S1C1) (RS1)(S1W)(C1R)(C2C1)(C3C2) 1 1 145.5 (RC1)(C1C2)(C2C3)(C3P)(S1C2) (RS1)(S1W)(C1R)(C2C1)(C3C2) 4 2 142.6 (RC1)(C1C2)(C2C3)(C3P)(S1C1) (RS1)(S1W)(C1R)(C2C1)(C3C1) 18 3 139.8 (RC1)(C1C2)(C2C3)(C3P)(S1C2) (RS1)(S1W)(C1R)(C2C1)(C3C1) 17 4 139.1 (RC1)(C1C2)(C2P)(S1C1) (RS1)(S1W)(C1R)(C2C1) 3 5 134.6 (RC1)(C1C2)(C2C3)(C3P)(S1R) (RS1)(S1W)(C1R)(C2C1)(C3C2) 7 6 134.0 (RC1)(C1C2)(C2C3)(C3P)(S1C1) (RS1)(S1W)(C1R)(C2R)(C3C2) 10 7 132.7 (RC1)(C1C2)(C2C3)(C3P)(S1C1) (RS1)(S1W)(C1R)(C2C1)(C3R) 13 8 132.7 (RC1)(C1C2)(C2P)(S1R) (RS1)(S1W)(C1R)(C2C1) 9 9 131.0 (RC1)(C1C2)(C2P)(S1R) (RS1)(S1W)(C1R)(C2R) 20 10 128.8 (RC1)(C1C2)(C2P)(S1C1) (RS1)(S1W)(C1R)(C2R) 5 11 128.8 (RC1)(C1C2)(C2C3)(C3P)(S1C1) (RS1)(S1W)(C1S1)(C2C1)(C3C2) 14 12 125.1 (RC1)(C1C2)(C2C3)(C3P)(S1C1) (RS1)(S1W)(C1R)(C2C1)(C3S1) 2 13 124.4 (RC1)(C1C2)(C2C3)(C3P)(S1C2) (RS1)(S1W)(C1R)(C2C1)(C3S1) 8 14 124.4 (RC1)(C1C2)(C2C3)(C3P)(S1C1) (RS1)(S1W)(C1R)(C2S1)(C3C2) 16 15 119.9 (RC1)(C1C2)(C2C3)(C3P)(S1C1) (RS1)(S1W)(C1R)(C2R)(C3S1) 6 16 116.9 (RC1)(C1C2)(C2C3)(C3P)(S1R) (RS1)(S1W)(C1R)(C2C1)(C3S1) 11 17 112.6 (RC1)(C1C2)(C2C3)(C3P)(S1C1) (RS1)(S1W)(C1S1)(C2C1)(C3S1) 15 18 108.6 (RC1)(C1C2)(C2C3)(C3P)(S1C1) (RS1)(S1W)(C1R)(C2S1)(C3S1) 12 19 108.1 (RC1)(C1C2)(C2C3)(C3P)(S1C3) (RS1)(S1W)(C1R)(C2C1)(C3S1) 19 20 96.6 38 Mass Balance Revenue (MUS$/year) Concentrate Group Contribution 1.2.3. Level III. Final design As indicated in section 2.3 at this level it is defined operational conditions and design variables. This level includes some experimental tests to determine operating conditions (such as reagents, pH and particle size) and develop models that represent the behavior of each stage. Several techniques such as sensitivity analysis, simulation and optimization can be used as explained in section 2.3. As an example, the circuit that occupies the 5th position in the ranking of Table 1.8 was selected because it is the best circuit with four flotation stages. The recovery was modeled using the expression developed by Yianatos and Henríquez (2006). Then based on the method developed by Cisternas et al. (2013) the residence time and the number of cells per stage were determined. The objective function used was Equation 1.8. The results consider 9, 2, 2 and 11 cells for rougher, cleaner 1, cleaner 2 and scavenger stages respectively. Moreover residence times are 0.012, 0.028, 0.08 and 0.017 h for rougher, cleaner 1, cleaner 2 and scavenger stages respectively. With these values the estimated revenue is 133.8 MUS $ / year , 0.6% below the initial value, but with a copper grade in the concentrate 2.5% higher than the initial. These calculations are shown as examples of the activities associated with level III, however further development is required. Table 1.9. Concentrate grade and overall recovery for the top ten circuits 1.3. Mass Balance Ranking 1 2 3 Cu Fe 25.9 20.7 26.8 31.1 30.2 31.0 4 5 6 7 8 9 10 21.8 18.5 28.3 26.9 26.9 22.0 19.5 30.1 29.5 31.0 30.9 30.9 30.0 29.3 Grade, % Global Recovery , % PyAs CpPy Cp 0.10 0.05 0.15 0.25 0.09 0.05 3.89 7.83 3.15 55.99 62.13 50.90 95.17 95.27 95.08 0.14 0.17 0.07 0.09 0.09 0.13 0.16 6.40 9.76 2.27 2.94 2.94 5.83 7.94 57.21 61.20 45.66 46.00 46.00 51.02 54.25 95.18 95.18 94.40 94.50 94.50 94.41 94.60 As Sc 0.23 0.41 0.03 0.05 0.05 0.21 0.37 Conclusions and Future Work A novel methodology was presented for the design of concentration circuits. The methodology uses three levels of decisions; the first two levels are presented in depth in this manuscript, the third level is briefly discussed. The methodology uses group 39 contribution models for a quick estimate of the species recoveries for a significant number of flotation circuits. Once the recoveries are estimated, the circuits are prioritized based on a performance indicator. The prioritized circuits are validated by mass balance. The primary advantages of the methodology are that it can reduce the number of laboratory tests, speed up the design of these circuits, and reduce the design dependence on designer experience. A group contribution model was developed for estimating the recovery of concentration circuits. The procedure is general and can be applied to any circuit consisting of units that generate two product streams: concentrate and tail. In addition, a procedure for generating all feasible circuits for a given number of stages was developed. This procedure generates a string (CTP string) to represent a concentration circuit, which can be used to store the concentration circuit in a database. The developed models can be applied to estimate the recoveries of concentration circuits with a maximum of six stages, corresponding to 1,492 concentration circuits. The models were fitted to mass balance data from 46 circuits, thereby generating 35 process groups. The average deviation was 1.5% for 69,000 data points. The model was applied to five new circuits that were not used in the fitting procedure, which yielded an average deviation of 3.1%. Therefore, the estimation of concentrate grades was successful. Future work considers incorporating new process groups such as milling and cleaner/scavenger stages. These modifications are expected to significantly increase the number of feasible circuits. In addition, the last level of the methodology, referred to as final design requires the development of new methods and tools to ensure proper final design of the flotation circuit. 1.4. Acknowledgments The financial support from CONICYT (Fondecyt 1120794 and CICITEM) and Antofagasta Regional Government is gratefully acknowledged. 1.5. 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Schena G., Zanin M., and Chiarandini A., 1997, Procedures for the automatic design of flotation networks, International Journal of Mineral Processing, 52, 137 -160. Sepúlveda F.D., Cisternas L.A., and Gálvez E.D., 2013, Global sensitivity analysis of a mineral processing flowsheet, Computer Aided Chemical Engineering, 32, 913-918. Yianatos, J.B., F.D. Henríquez, 2006. Short-cut method for flotation rates modelling of industrial flotation banks, Minerals Engineering, 19, 1336–1340. Yingling, J.C.,1993a. Parameter and configuration optimization of flotation circuits, part I. A review of prior work. International Journal of Mineral Processing 38 (1–2), 21–40. Yingling, J.C., 1993b. Parameter and configuration optimization of flotation circuits, part II. A new approach. International Journal of Mineral Processing 38 (1–2), 41–66. 43 44 CHAPTER 2. A METHODOLOGY FOR THE CONCEPTUAL DESIGN OF CONCENTRATION CIRCUITS: FINAL DESIGN. Felipe D. Sepúlveda and Luis A. Cisternas Universidad de Antofagasta, Chile. Jorge F. González and Edelmira D. Gálvez Universidad Católica del Norte, Chile. ABSTRACT A methodology for the conceptual design of concentration circuits is presented. The methodology considers three decision levels: level I – the definition and analysis of the problem, level II – the synthesis and screening of alternatives, and level III – the final design. Levels I and II were presented in a previously published article by us. In this article, level II is complemented by a database that helps to select the most suitable circuit, and level III is discussed in detail. After selecting a set of alternatives in level II, which is based on a group-contribution method, the design of each process stage is performed. This final design is performed with the help of sensitivity analysis and reverse simulation. The method is illustrated with examples that demonstrate that the method is suitable for these types of problems. Keywords: process design, sensitivity sensitivity, reverse simulation, flotation circuits. 45 2.1. Introduction Mineral-concentration processes use various stages of concentration, forming a circuit, because it is not possible to achieve the separation of the value species from the gangue in a single stage. The current practice for the design of these circuits is based on seven steps (Harris, Runge, Whiten and Morrison, 2002): (1) mineralogical examination in conjunction with a range of grinding tests, (2) a range of laboratory-scale batch tests and locked-cycle tests, (3) a circuit design based on the scale-up of laboratory kinetics, (4) preliminary economic evaluation of the ore body, (5) pilot-plant test of the circuit design, (6) economic evaluation, and (7) full-scale plant design. As was previously indicated by Sepulveda, Elorza, Cisternas and Gálvez (2014a), this procedure presents at least two problems: (1) the design of the circuit in step three is based on a rule-of-thumb scale-up from laboratory data that depends heavily on the designer’s experience, and (2) the laboratory and pilot plant are costly and require significant time, and the designed circuit analysis is therefore not performed in depth. Several methodologies for the design of these systems have been presented in the literature, some based on heuristics (Gálvez, 1998; Prince and Connolly, 1996; Chang and Prince, 1989) and others on optimization systems, both using mathematical programming (Cisternas, Gálvez, Zavala, and Magna, 2004; Cisternas, Mendez, Gálvez, and Jorquera, 2006, Méndez, Gálvez, Cisternas, 2009; Schena, Zanin and Chiarandini, 1997; Yingling, 1990, Lucay, Mellado, Cisternas and Gálvez, 2014) and metaheuristics (Ghobadi, Yahvaei and Banisi, 2011; Guria, Verma, gupta and Menrotra, 2005; Guria, Verma and Menrotra, 2006). However, heuristic-based methods are too simple to represent the design problem, and optimization-based methods are difficult to apply because of their mathematical complexity. For a detailed review of the subject, see Méndez, Gálvez, and Cisternas (2009). An additional problem is the need to incorporate prior experience and designer participation because there is empiricism on how to design these systems. A new methodology was presented by Sepúlveda, Elorza, Cisternas, and Gálvez (2014) that integrates the first five design steps given by Harris, Runke, Whiten and Morrison(2002) with the objectives of (1) better orienting the goals of the laboratory tests; (2) reducing laboratory and pilot-plant testing, thereby achieving lower costs and execution times; (3) designing the flotation circuit based on a systematic procedure; and (4) accelerating the design procedure. The methodology uses a completely different approach based on finding good designs (not necessarily optimal) between a more limited set of alternatives (eliminating unlikely alternatives) and evaluating the performance of each design using an approximate but simple model based on a group contribution. The group-contribution model can be applied to any circuit consisting of a maximum of six 46 stages, for which each stage generates two product streams: concentrate and tail. The methodology considers three design decision levels: (1) the definition and analysis of the problem, (2) the synthesis and screening of alternatives, and (3) the final design. This manuscript provides further detail for certain stages of the methodology, e.g., introduces a database of the existing plant circuits, but it mainly develops tools for the third level, the final design. This work is divided into six sections, beginning with the introduction. The second section presents a brief description of the methodology. The third section presents the database of the flotation circuits. The fourth section presents the third design-decision level. Case studies, focusing on the third decision levels, are presented in the fifth section, and finally, the sixth section presents the conclusions and comments. 2.2. A Brief Description of the Methodology The methodology, considered three hierarchical design-decision levels: (1) the definition and analysis of the problem, (2) the synthesis and screening of alternatives and (3) the final design, as shown in Table 2.1. In the first level, the “definition and analysis of the problem”, the problem is defined, including the characterization of the feed, the design goals and design and operation restrictions. The material to be fed into the process is characterized, which can comprise various mineralogical compositions, various sizes or both. Additionally, the feed-mass flow rate and the stage recovery for each component must be defined, and the general criteria for classifying and determining which are the most promising circuits to process the feed material are defined (e.g., revenues). Finally, the maximum number of stages, the maximum number of cleaner stages, and the maximum number of scavenger stages to be considered in the circuit design are defined. This decision can be based on various criteria (e.g., mathematical models, historical background, or the designer’s experience). In the second level, the “synthesis and screening of alternatives”, circuit alternatives are generated and evaluated using a group-contribution model, and the most promising circuits are validated by mass balance. Then, a set of concentration-circuit alternatives are selected for further study (final design) based on the ranking and additional criteria defined by the designer (e.g., process control and dynamics). The circuit alternatives are generated using origin-destination matrices for the concentrate and tail streams. The alternatives are evaluated using a group-contribution method, which estimates the global recovery of each component. The group-contribution method allows fast and simple calculation of the global recovery. The current group-contribution model includes the rougher, cleaner 1, cleaner 2, cleaner 3, scavenger 1, and scavenger 2 stages. With these 47 separation stages, 1,492 circuits can be generated. The set of best alternatives is validated using mass balance or simulation, and the set is reclassified under the same criteria. In the third level, the “final design”, the operational and equipment design are defined. This third level is analyzed in detail in section 4 of this manuscript. Levels I and II, including the group-contribution model, were presented in Sepúlveda, Elorza, Cisternas and Gálvez (2014). Table 2.1. Levels of the hierarchical decision. Level I Definition & analysis Feed characterization. of the problem Design goals. Estimated stage recovery values. Level II Level III Synthesis and Generation of feasible circuits alternatives. screening of Circuit modeling with group-contribution models. alternatives Selection of alternatives based on validation using mass balance and the database. Final design Identification of gaps and opportunities for improvement. Identification of key stages using sensitivity analysis. Definition of design and operating variables based on reverse simulation. 2.3. Database An important aspect in the selection of the concentration circuit is prior experience. Therefore, it is important to complement the ranking of the alternatives generated in level II with a database of flotation circuits. To generate this database, the literature was reviewed based on the information available in books, journals, conference proceedings, and information provided by companies. Concentration circuits currently in operation and circuits that have completed their operation were included. Proposed circuit designs that were not built were not included. Each concentration circuit was represented by the Concentrate and Tail Paths String (CTP string), as described by Sepúlveda, Elorza, Cisternas and Gálvez (2014). For example, the circuit in Figure 2.1 is represented by the CTP string [(RC1)(C1C2)(C2P)(S1C2)] [(RS1)(C1R)(C2C1)(S1W)], in which the string within the first set of square brackets corresponds to the concentrate path, and the string 48 within the second set of square brackets corresponds to the tail path. For clarity in Figure 2.1, the concentrate path is represented by solid lines, and the tail path with dashed lines. The database includes, in addition the CTP string, the ore treated, the location of the plant, the reference, and the observations. These observations include aspects such as the production level, concentrate grade, and feed grade. The database includes 543 circuits with between one and thirteen flotation stages; however, the majority of circuits have between three and five stages. Flotation plants include ores of copper, zinc, gold, silver, molybdenum, iron, lead, nickel, spodumene, talc, and bismuth. The main use of the database is its comparison with the ranking of alternatives. Thus, the designer can know if any of the alternatives is or has been a real plant. It is needless to state that, if one of the alternatives is in the database, then it is possible to find information on the plant, e.g., advantages and disadvantages of that plant, and to identify if there is any similarity between the mineral under study and the ore treated in this plant. S1 R W C1 C2 P Figure 2.1. Circuit [(RC1)(C1C2)(C2P)(S1C2)] [(RS1)(C1R)(C2C1)(S1W)]. 2.4. Final Design At this level, the aim is to achieve a final conceptual design, identifying aspects of equipment design and operating conditions. This design is conducted in three steps: 1) identification of gaps and opportunities for improvement; 2) identification of key stages using sensitivity analysis; and 3) definition of design and operating variables based on reverse simulation. 49 2.4.1. Identification of gaps and opportunities for improvements. Where every variable takes only two values, 0 and 1, and only one variable changes its value between each pair of consecutive simulation. This equations demonstrates that if there is any changes in value and , it can only be attributed to a change in parameter . 2.4.2. Differential sensitivity analysis (DSA) Identification of the gaps between the goals defined at level I and the results of the selected alternatives can be made by simple comparison. These gaps often correspond to values of expected recoveries and concentrate grades. If any of these goals is not met, then that goal is identified as ends to achieve in the final design. Identifying opportunities for improvement is achieved by performing local sensitivity analysis in the objective function. This means the identification of which global recoveries have a greater effect on the objective function and then prioritizing the improvement of these global recoveries. Local sensitivity analysis, also called differential analysis, ranges from solving simple partial derivatives to spatial and temporal sensitivity analyses (Hamby, 1994). Here, only partial derivatives are presented. Local sensitivity analysis was conceived as a local measure of the effect of a given input factor on a given output. Given a model Y=f(X), where Y is the model output of interest, and X is the set of uncertain input factor, the common way to describe a local sensitivity coefficient is by using the partial derivative of the model output with respect to the model input, which is expressed as (Liu and Homma, 2010): . (2.1) In equation (2.1), is calculated by varying the input factor while fixing all the other input factors at their nominal values. Two alternative expressions of this measure are: , (2.2) where and are the nominal values of the input factor respectively, and: 50 and model output , , where and respectively. (2.3) are the standard deviations of the input factor and model output , 2.4.3. Identification of key stages using sensitivity analysis The objective of this step is to identify which concentration stages have a major influence on obtaining the goals and improvements identified in the previous step. This allows concentrating efforts or studies at key stages. The application of global-sensitivity analysis (GSA) on improving the concentration circuits is discussed at length in the work of Sepulveda, Cisternas and Gálvez (2014b). Thus, here a brief description is introduced, and the main findings reported by Sepulveda, Cisternas and Gálvez (2013, 2014b) is explained. According to Saltelli, Ratto, Andres, Campolongo, Cariboni, Gatelli, Saisana and Tarantola (2008), the SA can be defined as “the study of how uncertainty in the output of a model (numerical or otherwise) can be apportioned to different sources of uncertainty in the model input”. The general objectives of GSA are (Reuter and Liebscher, 2008): a) the identification of the significant and insignificant factors and the possible reduction of the dimensions (number of design variables) of an optimization problem; and b) the improvement in the understanding of the model behavior (highlighting interactions among factors and finding combinations of factors that result in high or low values for the model output). GSA (Saltelli, Ratto, Andres, Campolongo, Cariboni, Gatelli, Saisana and Tarantola, 2008; Reuter and Liebscher, 2008; Morris, 1991; Storlie and Helton, 2008) corresponds to the evaluation of an output model when all the model factor are simultaneously evaluated; the model is mainly resolved by numerical methods (Monte Carlo method, Quiasi Monte Carlo and Latin Hypercube). This methodology has the advantage of simultaneously assessing all factors; however, it requires a large amount of data for which the model is evaluated and the mathematical techniques are more complex. GSA methods can be classified into three groups (Confalonieri, Bellocchi, Bregaglio, Donatelli and Acutis, 2010). 1) Regression methods: the standardized regression coefficients are based on a linear regression of the output on the input vector. Linear regression is the most commonly used method, but there are other techniques that are also in this group (Storlie and Helton, 2008). 2) Screening methods: this refers to the method developed by Morris with significant modification, as given by Compolongo, Cariboni and Saltelli. (2007). 3) Variance-based methods (Reuter and Liebscher, 2008; Saltelli, Ratto, Andres, Campolongo, Cariboni, Gatelli, Saisana and Tarantola, 2008): this is a 51 GSA method in which the variance of the model output can be decomposed into terms of increasing dimension, called partial variances, that represent the contribution of the inputs (i.e., single inputs, pairs of inputs, etc.) to the overall uncertainty of the model output. This method enables the simultaneous exploration of the space of the uncertain inputs, which is usually performed via Monte Carlo sampling. Statistical estimators of partial variances are available to quantify the sensitivities of all of the inputs and of the groups of inputs through multi-dimensional integrals. The computational cost, in terms of the model simulations, of estimating the sensitivities of the higher-order interactions between the inputs can be very high. To preclude a high computation cost, Homma and Saltelli (1996) introduced the concept of a total sensitivity index. The total sensitivity index indicates the overall effect of a given input by considering all the possible interactions of the respective input with all the other inputs. Examples of the techniques in this group include the analysis of variance (ANOVA), Fourier amplitude sensitivity test (FAST), extended Fourier amplitude sensitivity test (E-FAST), Sobol’s method (1993) and highdimensional model representation (HDMR). To determine which concentration stage most significantly affects the global behavior of the concentration circuit, the suggested methodologies are Sobol’s, E-FAST, and Morris (Sepúlveda, Cisternas and Gálvez et al., 2014). The most suitable distribution function to represent the uncertainties of the stages’ recoveries is the uniform function with dispersion values between 10 and 20% of its mean value. The FAST method and Sobol’s method allow the calculation of two indices, i.e., the fir storder-effect sensitivity index corresponding to a single factor and the total sensitivity index corresponding to a single factor and the interaction of additional factors that involve the single factor under evaluation. The first-order sensitivity index measures only the main effect contribution of each input factor on the output variance. It does not take into account the interactions among factors. The first-order sensitivity index is important when the objective is to determine the most important input uncertainties. The total sensitivity index is important when the objective is to reduce the uncertainty in the output model (Adeyinka, 2007). If the first-order sensitivity index of a stage recovery (input factor) is negligible, the uncertainty in this stage recovery does not affect the uncertainty in the global behavior of the concentration circuit. Therefore, that stage recovery is noninfluential or unimportant. This does not determine any information about input interactions or high-order sensitivity indices. If the total sensitivity index is also small, then, apart from being unimportant, that stage recovery does not interact with other stages’ recovery (high-order effects of that stage’s recovery are negligible). The implication of small values of the first and total sensitivity indexes is that the uncertainty 52 in a stage’s recovery values has no effect on the uncertainty in the global behavior of the concentration circuit. Thus, in subsequent analyses, that stage’s recovery value can be fixed to its nominal value (mean or median), and further research, lab tests, analysis, and data gathering can be directed at other stage recoveries. Conversely, regardless of the magnitude of the total sensitivity index, a large value of the first-order sensitivity index implies that the stage recovery is influential. The arithmetic difference between the total sensitivity index and the first-order sensitivity index indicates the magnitude of the interactions between a stage’s recovery and other stages’ recoveries. The Morris method (Morris, 1991; Confalonieri Bellocchi, Bregaglio, Donatelli and Acutis, 2010; Saltelli Ratto, Andres, Campolongo, Cariboni, Gatelli, Saisana and Tarantola, 2008) calculates elementary effects, proposing the construction of two sensitivity measures to determine which input factors could be considered to have effects that were (a) negligible, (b) linear and additive, or (c) nonlinear or involved in interactions with other factors due to each input factor. The method calculates the mean (µ, assessing the overall influence of one factor on the output) and standard deviation σ (estimating the totality of the higher-order effects, nonlinearity or interactions with other factors). A large (absolute) value for the central tendency µ shows that a stage recovery (input factor) has an important influence on the global recovery of the circuit (output), whereas a large value of the spread indicates that either a stage recovery has a non-linear effect on the global behavior of the circuit or that a stage recovery is interacting with other stages’ recoveries (which corresponds to higher-order effects). Therefore, the more significant stage recoveries are those located in the upper right quadrant of a µ (spread) versus σ (strength) plot in which both sensitivity measures are high. Examples of the software used for the global sensitivity analysis are SimLab, GUIHDMR working jointly with MATLAB (Ziehn and Tomlin, 2009) and the software R (http://www.r-project.org/), which is a free software environment for statistical computing and graphics (R and RStudio). For this study, RStudio was used. 2.4.4. Definition of the design and operating variables based on reverse simulation Up to this stage of the design, the values of the recoveries of each stage were given nominally. These values were defined based on experimental data, mathematical models or plant values. However, it is known that these values will change depending on the operating conditions, equipment design, and circuit used. For that reason, it is necessary to recalculate those based on experimental evidence or the models available. Beyond its 53 determination, there is here the possibility of improving the circuit selected by focusing on the stages that most significantly affect the circuit, as were identified in section 2.4.2. As the goals to be achieved are clearly identified, they may guide efforts to determine the operating conditions and design that achieve these goals, a procedure that we call reverse experimentation and/or simulation. Reverse experimentation is the procedure by which experiments are designed to achieve a desired behavior, i.e., the experimental conditions are defined by the expected outcome (the desired goals). Table 2.2 shows heuristics to select experimental tests for the desired goals. Reverse simulation is the procedure to determine the parameters that achieve a defined outcome. These parameters can be operational conditions (such as residence time) or design (such as number of stages and cell volume). Figure 2.2 exemplifies the concept. To determine the unknown parameters, the solution of systems of equations and/or optimization can be used. 2.1. Case study This case study is a continuation of an example described in Sepúlveda et al. (2014a). The fed material has the following characteristics: four species were considered: Cp (100% CuFeS2), CpPy (95% CuFeS2 and 5%FeS2), PyAs (99% FeS2 and 1%AsFeS), and Sc (100% SiO2); the composition of the ore fed has a copper grade of 1.02% and an arsenic grade of 0.07%; the value species are Cp (34.6% of copper) and CpPy (32.9% of copper), and the gangue species are PyAs and Sc; the expected recoveries are at least 90% of Cp and 60% of CpPy, and not exceeding 4% of PyAs and 1% of Sc. Table 2.3 provides more details on the feed and stage recoveries. Based on the number of stages selected, 128 potential concentration circuits were evaluated with the group-contribution method and ranked using the net-smelter-return revenue formula. The top seven circuits were validated by mass balance and re-ranked. The two best concentration circuits were selected for the final design. The selected circuits are: [(RC1) (C1C2) (C2C3) (C3P) (S1C1)] [(RS1) (S1W) (C1R) (C2C1) (C3C2)], which we will call circuit 1, and [(RC1) (C1C2) (C2C3) (C3P) (S1C2)] [(RS1) (S1W) (C1R) (C2C1) (C3C2)], which will we call circuit 2. These are shown in Figure 2.3. The revenues of circuits 1 and 2 correspond to 145.5 and 142.6 MUS $/year, respectively. These circuits are not in the database of existing circuits, so the concentration circuits in other positions are the plants currently used. The circuit in the No. 5 position is a circuit that produces copper, and the circuit in the No. 6 position is a circuit that processes fluorite, lead, and zinc. 54 Table 2.2. Examples of heuristics for reverse experimentation. Goal Heuristic To improve recovery Consider to increase the residence time in the flotation Consider to increase gas injection. In bank cells, consider working with intermediate particle sizes. In bank cell, consider working medium to high percentage of solid. In columnar cells, consider working with smaller particles. In columnar cells, consider working with low percentage of solid. To improve grade Improve liberation of particles (e.g. longer time grinding) Consider to decrease the flotation time. Consider to use wash water To improve the flotability of hydrophilic mineral (e.g., sulfide minerals) Consider increasing the amount of collector or replacement for a new collector. To depress pyrite Consider increasing the pH over 10 using lime To depress pyrite in seawater or saline waters Consider the removal of Mg and Ca with lime, NaOH or Na 2CO3 before the flotation stages To depress Cu and Fe sulfide minerals in Cu/Mo separation Consider the use of sodium sulfide & hydrosulfide To improve selectivity Use a specific collector for an element or species that you want to recovery. Consider a conditioning step in a moderately oxidizing environment before the flotation stages and after the grinding step. Consider the use of metabisulfite to depress pyrite Consider the use of ferrocyanide Use a mix of collectors, to recover a species group or minerals group. To activate Cu, Pb and Zn minerals Use NaHS prior to collector addition To improve flotability of hydrophilic mineral (e.g. sulphide minerals) Consider increasing the amount of collector or replacement for a new collector. Consider a conditioning step in a moderately oxidising environment before the flotation stages and after the grinding step. 55 Known Input Variables Known Input Parameters Unknown Output Variables Direct Simulation Known Input Variables Unknown Input Parameters Reverse Simulation Known Output Variables Figure 2.2. Difference between direct and reverse simulation. The ranking was performed by the expression of the net smelter return (Eq. 2.4): [(∑ ) ( )( ) (∑ )( )] , (2.4) where is the mass flow of species i in the concentrate, the fraction of metal paid with a value of 0.98, is the copper grade of the concentrate, is the grade deduction with a value of 0.04, is the treatment charge with a value of US $70/ton of copper, is the refinery charge with a value of US $180/ton of copper, represents the arsenic charge, and is the grade of penalization for As with a value of US $2.16/ton of arsenic. is the number of hours per year of plant operation with 7,200 h/year, and the flows are in mass per hour. This formula for the calculation of the revenues incorporates the metallurgical efficiency of the plant, that is, the recovery and mineral content are opposite functions. In this work, a copper price of US $4,000/ton of copper was used. 2.1.1. Identification of gaps and opportunities for improvements Identification of the gaps between the goals defined at level I and the results of the selected alternatives is made by simple comparison. Table 2.4 shows the expected recoveries of each species and the values obtained in each circuit selected. It can be observed that, for circuit 1, the Cp and PyAs recoveries are within the expectations, whereas the recovery for CpPy is below the desired value, and recovery for Sc is above the desired value. As a result, the goals for the final design should be to increase the global recovery of CpPy over 60% and to decrease the global recovery of Sc under 1%, while maintaining the recoveries of PyAs and Cp. Moreover, for circuit 2, the global recoveries of PyAs and Sc should be decreased under 4 and 1%, maintaining the recoveries of CpPy and Cp. 56 Table 2.3. Parameter values for the case study. Species Estimated Stage recoveries Feed Separation goals, % Rougher Cleaner Scavenger (ton /h) Cp 0.82 0.87 0.77 10 >90 CpPy 0.53 0.54 0.58 20 >60 PyAs 0.3 0.28 0.32 150 <4 Sc 0.12 0.11 0.13 800 <1 Identifying opportunities for improvement is achieved by performing local sensitivity analysis in the objective function, i.e., the revenue in Equation (2.4). This means identifying which species’ global recoveries have a greater effect on the revenue. Equation (2.2) was used, which in our case can be rewritten as: , (2.5) where and are the nominal values of the global recovery of species i, and the revenue , respectively. Table 2.5 gives the values of , with I = Cp, CpPy, CpAs, and Sc. It is clear that the global recoveries of Cp and CpPy affect the objective function more than any others. In summary, the goals to be achieved in the final design of circuit 1 are to increase the CpPy global recovery over 60%, increase the Cp global recovery, and maintain the PyAs global recovery. Moreover, for circuit 2, the PyAs global recovery must be decreased below 4%, and the CpPy and Cp global recoveries must be increased. Table 2.4. Gaps between the goals and the values obtained in selected circuits. Species Goals Cp >0.90 CpPy >0.60 PyAs <0.04 Sc <0.01 Copper grade in concentrate Circuit 1 Recovery 0.95 OK 0.56 0.04 OK 0.001 OK 0.25 57 Circuit 2 Recovery 0.95 OK 0.62 OK 0.08 0.003 OK 0.20 Table 2.5. Local sensitivity index for the objective function. Cp CpPy CpAs Sc Circuit 1 * 70.2 * 77.9 -7.4 -7.7 Circuit 2 * 71.7 * 88.0 -15.2 -15.7 2.1.2. Identification of the key stages using sensitivity analysis The objective of this step is to identify which concentration stages have a major influence on obtaining the goals and improvements identified in the previous step. This allows us to concentrate our efforts or studies at key stages. To apply GSA, the first step is to obtain the models representing the global recovery for these circuits as a function of the stage recoveries. These models are obtained by mass balance; equations 2.6 and 2.7 are the models corresponding to circuit 1 and 2, respectively. Note that these models do not include any approximation, and therefore, they represent the behavior of each circuit. ( ) ( ) (2.6) (2.7) In equations 2.6 and 2.7, is the global circuit recovery, is the recovery for the rougher stage, is the recovery for the cleaner-1 stage, is the recovery for the cleaner-2 stage, is the recovery for the cleaner-3 stage and is the recovery for the scavenger stage. In the case of the following expression, represents the multiplication between and . The first GSA test is performed to determine for which stage the recovery uncertainty ( , , , and ) has the main effect on the global recovery for each species. The mean values for the stage recoveries are the same as those in Table 2.2, and a 5% uncertainty was considered, using a uniform distribution for all stages and all species. Then, a second GSA is performed is to determine for which species (Cp, CpPy, PyAs and 58 Sc) and for which stage the recovery uncertainty ( , , main effect on the grade of the final concentrate uncertainty. R , R S C1 C1 C2 C2 C3 C3 (a) and ) has the S (b) Figure 2.3. Circuits chosen for the final design: a) circuit 1 and b) circuit 2. The R program was used to perform the GSA using the Sobol method and the SobolJansen algorithm (http://cran.r-project.org/web/packages/sensitivity/sensitivity.pdf), which is one of the algorithms with better approximations for the models Saltelli, Annoni, Azzini, Campolongo, Ratto and Tarantola, (2010) that are evaluated. For the first analysis, a random sample of 1,000,000 data points and nboot of 100 were used, and for the second analysis, a random sample of 10,000 data points and nboot of 100 were used. The results of the first GSA (Figure 2.4), show a significant difference for which are the most important concentration stages between the circuits 1 and 2, which means that each circuit has a different behavior as a function of the stage recoveries. For circuit 1, Figure 2.4b shows that the uncertainties in the recoveries of the stages cleaner 1 and cleaner 3 are the most relevant to the uncertainty in the global recovery of Cp (higher Sobol-index values). However, the uncertainty in the stage recovery of cleaner 1 affects the uncertainty in the global recovery of all species; therefore, modifying the behavior of this stage will have an effect on all species. However, the uncertainty in the stage recovery of cleaner 3 has a significant effect on the global recovery of Cp and little 59 Sobol Total Index effect for other species. Therefore, if an increase in the global recovery of Cp is targeted, it is recommended to look for actions that increase the Cp recovery in cleaner-3 stage. Using the same reasoning, there are two options to increase the global recovery of CpPy: increase the recoveries in the cleaner-1 or scavenger stages. However, the uncertainty in the cleaner-1 stage affects all species, which is why it is advisable to intervene at the scavenger stage. Sobol Total Index (a) (b) Figure 2.4. Sobol total index for each stage and for each species for a) circuit 1 and b) circuit 2. For circuit 2, based on the same type of analysis, and in Figure 2.4b, to increase the global recovery of Cp, the rougher stage must be altered; to increase the global recovery of CpPy, the scavenger stage must be altered; and to reduce the global recovery of PyAs, the cleaner-2 and cleaner-3 stages must be altered. 60 Figure 2.5 shows the GSA for the copper grade. It is observed in both circuits that, if it is desired to change the final concentrate grade, we should focus the efforts on the recovery of the PyAs species, for which the highest sensitivity indices for circuit 1 are the cleaner 1, cleaner-2, and scavenger stages, and for circuit 2 are the cleaner-2, cleaner-3, and scavenger stages. 0.3 Sobol Total Index 0.25 0.2 0.15 Circuit 1 Circuit 2 0.1 0.05 0 Figure 2.5. Sobol total index for the copper grade in the concentrates. In conclusion, for circuit 1, the recovery of CpPy in the scavenger stage and of Cp in the cleaner-3 stage should be increased; and for circuit 2, the recovery of Cp in the rougher stage and of CpPy in the scavenger stage should be increased, and the recovery of PyAs in the cleaner-2 and cleaner-3 stages should be decreased. 2.1.3. Definition of the design and operating variables based on reverse simulation. In this example, reverse design is used to modify the stage recoveries, changing the number of stages and the residence times in the stages, based on the following model to represent the recovery of the species in stage : , ( (2.8) ) where, is the kinetic coefficient for each species in stage , time in stage , and is the number of cells in stage . 61 is the cell residence The values for the kinetic coefficients for the Cp are as follows: rougher stage 0.060 (1/min), cleaner stage 0.053 (1/min) and scavenger stage 0.026 (1/min). For the CpPy, they are the rougher stage 0.025 (1/min), cleaner stage 0.019 (1/min) and scavenger stage 0.015 (1/min). For the PyAs, they are the rougher stage 0.011 (1/min), cleaner stage 0.008 (1/min) and scavenger stage 0.006 (1/min). For the Sc, they are the rougher stage 0.004 (1/min), cleaner stage 0.003 (1/min) and scavenger stage 0.002 (1/min). All of these values are assumed to be constant. The nominal values for the cell residence time are the rougher stage 4 (min), cleaner stage 5.5 (min) and scavenger stage 5.5 (min). The nominal values for the number of cells are all rougher and cleaner stages 8 cells, and the scavenger stage 11 cells. Recall that the actions to take to the circuit 1 were to increase the recovery of CpPy in the scavenger stage and increase the recovery of Cp in the cleaner-3 stage. Then, the number of cells and residence times were increased in the scavenger and cleaner-3 stages. However, these changes resulted in the recovery of PyAs and Sc also increasing and exceeding the desired values. This is not surprising because the Sobol sensitivity indices of these species in the scavenger stage are also high (see Figure 2.4a). Therefore, to reduce the recoveries of PyAs and As, the cleaner-2 stage must be modified (see Figure 2.4a), i.e., the recoveries of PyAs and As should be reduced at that stage. Table 2.6 shows the new values of the residence times and the number of cells: the values in parentheses are the original values. Note that the values of both parameters were decreased in the cleaner-2 and were increased in the cleaner-3 and scavenger stages. The effect of these changes can be seen in Table 2.7, achieving an increase in the global recovery of Cp and CpPy that result in an increase of 5% in the revenues. Recall that the actions to take for circuit 2 were the increase in the recovery of Cp in the rougher stage, the increase in the CpPy recovery in the scavenger stage, and the reduction in the recovery of PyAs in the cleaner-2 and cleaner-3 stages. Then, the residence times and the numbers of cells in the rougher and scavenger stages were increased, and the residence times and the numbers of cells were decreased in cleaner 2 and cleaner 3, as shown in Table 2.6. The results are presented in the Table 2.7, which achieve the proposed objectives and increased the profits by 10%. 62 Table 2.6. Residence times and numbers of cells (original values). Stage Residence Time (min) Number of cells Circuit 1 Circuit 2 Circuit 1 Circuit 2 Rougher 4.0 6.0 (4.0) 8 9 (8) Cleaner 1 5.5 5.5 8 8 Cleaner 2 4 (5.5) 4.0 (5.5) 7 (8) 7 (8) Cleaner 3 6 (5.5) 3.7 (5.5) 9 (8) 5 (8) Scavenger 7 (5.5) 7 (5.5) 12 (11) 12 (11) Table 2.7. Stage and global recoveries for circuits 1 and 2. Stage Circuit 1 Circuit 2 Cp CpPy PyAs Sc Cp CpPy PyAs Sc Rougher 0.82 0.53 0.30 0.12 0.94 0.71 0.45 0.19 Cleaner 1 0.92 0.54 0.28 0.11 0.86 0.52 0.27 0.10 Cleaner 2 Cleaner 3 Scavenger 0.81 0.81 0.82 0.39 0.61 0.70 0.19 0.33 0.41 0.07 0.13 0.18 0.74 0.59 0.84 0.39 0.28 0.66 0.19 0.13 0.39 0.07 0.05 0.16 Global 0.971 0.601 0.038 0.0005 0.987 0.647 0.038 0.001 Grade 0.26 (increase of 2% ) 0.26 (increase 26% ) Revenue M US$ 149.71 (increase of 5% ) M US$ 152.72 (increase of 10% ) 2.2. Summary and Discussion A methodology for the conceptual design of concentration circuits was presented. The methodology considers three decision levels: level I – the definition & analysis of the problem, level II – the synthesis and screening of alternatives, and level III – the final design. Levels I and II were presented in a previous article published by the authors. A database of existing concentration plants was developed that helps to select the most suitable circuit at stage II. Level III was divided into three stages: 1) identification of the gaps and opportunities for improvement using local sensitivity analysis of the objection function; 2) identification of the key stages using global sensitivity analysis; and 3) definition of the design and operating variables based on reverse simulation. The method was applied to the design of the flotation circuit that processes a material of four species, 63 including CuFeS2, FeS2, AsFeS and SiO2 . Based on the results, it can be concluded that the method is an effective method for designing flotation circuits. 2.3. Acknowledgement The financial support from CONICYT (Fondecyt 1120794), CICITEM (R10C1004) and the Antofagasta Regional Government is gratefully acknowledged. 2.4. References Adeyinka A.L., Applications of Sensitivity Analysis in Petroleum Engineering, Thesis University of Texas at Austin, 2007. Campolongo F., Cariboni J., Saltelli A., 2007, An effective screening design for sensitivity analysis of large models, Environmental modelling & Software, 22, 15091518. Chan, W.-K., Prince, R.G.H., 1989. Heuristic evolutionary synthesis with non-sharp separators, Computers and Chemical Engineering,13, 1207-1219. Cisternas, L.A., Gálvez, E.D., Zavala, and M.F., Magna, J., 2004. A MILP model for the design of mineral flotation circuits. International Journal of Mineral Processing 74 (1–4), 21–131. Cisternas L., Mendez D., Gálvez E., Jorquera R., 2006, A MILP model for design of flotation circuits with bank/column and regrind/ no regrind selection, International Journal of Mineral Processing, 79, 243-263. Confalonieri R., Bellocchi G., Bregaglio S., Donatelli M., Acutis M., 2010, Comparison of sensitivity analysis techniques: a case study with the rice model WARM, Ecological Modelling, 221, 1897 -1906. 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Hamby D.M.,1994, A review of techniques for parameter sensitivity analysis of enviromental models, Environmental Monitoring and Assessment, 32, 135-154. Harris M.C., Runge K.C., Whiten W.J., and Morrison R.D., 2002, JKSimFloat as a Practical Tool for Flotation Process Design and Optimization. In Mineral Processing Plant Design. Homma T. and Saltelli A., 1996, Importance measures in global sensitivity analysis of nonlinear models, Reliability Engineering & System Safety, 52, 1-17. Lui Q. & Homma T., 2010, A new importance measure for sensitivity analysis, Journal of Nuclear Science and Technology, 47, 53–61. Lucay F., Mellado M.E., Cisternas L.A., Gálvez E.D., 2012, Sensitivity analysis of separation circuits, International Journal of Mineral Processing, 110–111, 30–45. Méndez D.A., Gálvez E.D., Cisternas L.A., 2009, Modelling of grinding and classification circuits as applied to the design of flotation processes. Computers and Chemical Engineering, 333, 97-111. Morris Max D, 1991, Factorial Sampling plant for preliminary computational experiments, TECHNOMETRICS, 33, 161-174. Prince R.G.H., Connollly A.F., 1996, Heuristic decisions in an evolutionary design system. Computers and Chemical Engineering, 20(Suppl.), S273-S278. Reuter U. and Liebscher M., 2008, Global sensitivity analysis in view of nonlinear structural behavior, LS-DYNA. Saltelli A., Ratto M., Andres T., Campolongo F., Cariboni J., Gatelli D., Saisana M., Tarantola S., 2008, Global sensitivity Analysis: The primer, john Wiley & Sons Ltd. Saltelli A., Annoni P., Azzini I., Campolongo F., Ratto M., Tarantola S., 2010, Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index, Computer Physics Communications, 181, 259-270. Schena G., Zanin M., Chiarandini A., 1997, Procedures for the automatic design of flotation networks, International Journal of Mineral Processing, 52, 137-160 Sepúlveda F.D., Cisternas L.A., Gálvez E.D., 2013, Global sensitivity analysis of a mineral processing flowsheet. Computer Aided Chemical Engineering, 32. 913-918. 65 Sepúlveda F.D., Elorza M.A., Cisternas L.A., Gálvez E.D., 2014, A Methodology for the Conceptual Design of Concentration Circuits: Group Contribution Method, Computer & Chemical Engineering, 63, 173 - 183. Sepúlveda F.D., Cisternas L.A., Gálvez E.D., 2014, The use of Global sensitivity analysis for improving processes: applications to mineral processing, Computer & Chemical Engineering, In Press, DOI 10.1016/j.compchemeng.2014.01.008. Sobol’ I., 1993, Sensitivity estimates for nonlinear mathematical models. Mathematical Modelling & Computational Experiment, 1, 407 – 414. Storlie C.B., Helton J.C, 2008, Multiple predictor smoothing method for sensitivity analysis: description of techniques, Reliability Engineering & System Safety, 93, 28-54. Yingling J.C., 1990, Circuit analysis: optimizing mineral processing flowsheet layouts and steady state control specifications. International Journal of Mineral Processing, 29, 149-174. Ziehn T. and Tomlin A.S., 2009, GUI-HDMR- A software tool for global sensitivity analysis of complex models, Environmental Modelling & Software, 775-785. 66 CHAPTER 3. THE USE OF GLOBAL SENSITIVITY ANALYSIS FOR IMPROVING PROCESSES: APPLICATIONS TO MINERAL PROCESSING Felipe D. Sepulvedaa, Luis A. Cisternasa,b, Edelmira D. Gálvezb,c a Chemical and Mineral Process Engineering Department., Universidad de Antofagasta, Antofagasta, Chile b c Process Technology, CICITEM Department of Metallurgical & Mining Engineering., Universidad Católica del Norte, Antofagasta, Chile Abstract This paper analyzes the application of global sensitivity analysis (GSA) to the improvement of processes using various case studies. First, a brief description of the methods applied is given, and several case studies are examined to show how GSA can be applied to the study to improve the processes. The case studies include the identification of processes; comparisons of the Sobol, E-FAST and Morris GSA methods; a comparison of GSA with local sensitivity analyses; an examination of the effect of uncertainty levels and the type of distribution function on the input factors; and the application of GSA to the improvement of a copper flotation circuit. We conclude that GSA can be a useful tool in the analysis, comparison, design and characterization of separation circuits. In addition, we conclude that using the stage’s recoveries of each species as input factors is a suitable choice for the GSA of a flotation plant. Keywords: mineral processing, flotation, process analysis, global sensitivity analysis, retrofit. 67 3.1. Introduction Mineral processing comprises many unit operations, such as gravitational, magnetic and flotation stages, which are aimed at extracting valuable material from ores. Usually, the processes’ operating conditions are defined to control the balance between a high recovery rate of the desired metal and a high grade value of the metal in the product outflow (Méndez et al., 2009a). These processes usually include multiple stages that are interconnected (forming circuits) to maximize the recovery rate and concentrate grade. The design and analysis of these circuits, including the design and analysis of each stage, continues to be a challenging task (Ghobadi et al., 2011). A designer initially solves a synthesis problem (for any process) by trial-and-error. There are many arrangements of a concentration circuit that correspond to an acceptable trialand-error solution; however, many of these arrangements can be incorrect, ineffective or uneconomical, which is realized when feedback on an existing process becomes available. Concentration circuits commonly evolve over time solving a number of existing problems while creating new ones (Schena and Casali, 1994). Several methods for the design of these circuits have been presented in the literature; these methods attempt to develop a systematic procedure to replace the trial-and-error method, which is time-consuming and requires much experimentation. Among the methods developed are those that use heuristics to develop a feasible design or that improve an existing design (Connolly and Prince, 2000). However, these procedures use rules that are not always satisfied or that contradict each other and therefore do not guarantee an optimal design. Other methods use optimization or mathematical programming procedures (Cisternas et al., 2006; Méndez et al., 2009b; Ghobadi et al., 2011) using a superstructure to create a set of alternatives from which an optimum design can be selected. However, the use of these methods requires training in optimization techniques because the problems are usually formulated as MINLP models for which there are no commercial codes available that ensure optimality. For the aforementioned reasons, none of the developed methodologies are widely used in industry. The concentration stage is difficult to model, and ore characteristics vary among mining operations. Currently, there is no theoretical model that can predict the floatability of different species of a mineral and thus experimentation is necessary to develop models that can be used to design these systems. However, these experimentally based models have a limited range of application depending on the experimental conditions and the number of experiments used. The compositions and mineralogical species vary among mining operations, which in turn affects the floatability behavior and undermines the 68 model validity as well as the operational parameters that are limited based on design ranges. Thus, there are at least two sources of uncertainty: the model and the ore characteristics. Sensitivity analysis (SA) can be employed to address uncertainties in the model and application scenarios, thereby facilitating the evaluation of process structures and operational behaviors. Lucay et al. (2012) applied a local SA to analyze and design separation circuits. The authors studied the effect of each stage on the general circuit by identifying the relation between the recovery rate of each stage and the global recovery rate of the circuit. Mellado et al. (2012) applied local SA to heap leaching to validate the analytical model as well. However, local SA only considers the neighborhood of the input variation, and the effect of each input parameter is measured by keeping all the other input parameters at their nominal values. Global sensitivity analysis (GSA) can overcome these limitations and has other advantages (Saltelli et al. 2000). Fesanghary et al. (2009) studied the use of GSA and a harmony search algorithm for the design optimization of shell and tube heat exchangers (STHXs) from the economic viewpoint. GSA was used to reduce the size of the optimization problem; non-influential geometrical parameters that have the least effect on total cost of STHXs are identified and are ignored in the optimization calculation. Later, Schwier et al. (2010) used GSA in the flow sheet simulation of solid processes, which allowed for the examination and quantification of the influences of given parameters on specific target criteria. GSA was used to decrease the effort required for the parameter estimation in a given process simulation by focusing the effort on the most influential parameters. This work attempts to show how a GSA can be used in the analysis, design and retrofit of concentration circuits and the equipment that compose it. This work is expected to complement current design techniques, such as trial-and-error methods, heuristics or optimization. Various methodologies of GSA are analyzed and the effect of the nature of the uncertainty of the input factors is studied. 3.2. Global Sensitivity Analysis According to Saltelli et al. (2008), the SA can be defined as “the study of how uncertainty in the output of a model (numerical or otherwise) can be apportioned to different sources of uncertainty in the model input”. These techniques have been widely used in various engineering areas and are of great importance in determining the most significant variables in a model. The general objectives of GSA are (Reuter and Liebscher, 2008): a) The identification of the significant and insignificant factors and the possible reduction of the dimensions (number of design variables) of an optimization problem. b) The 69 improvement in the understanding of the model behavior (highlighting interactions among factors and finding combinations of factors that result in high or low values for the model output). SA can be classified as: a) Local sensitivity analysis (LSA) or differential sensitivity analysis, which is represented by the first partial derivative of a model under evaluation, producing a coefficient that describes the rate of change between the model output and one model factor while all the other factors remain constant. Its main advantage is its easy implementation and evaluation; however, it can only assess a single factor at a time (Hamby, 1994). b) GSA (Saltelli et. al., 2007, 2008; Reuter and Liebscher, 2008; Morris, 1991; Storlie and Helton, 2008), which for some, is identical to SA (Reuter and Liebscher, 2008) and corresponds to the evaluation of an output model when all the model factor are simultaneously evaluated, being mainly resolved by numerical methods (Monte Carlo method, Quiasi Monte Carlo and Latin Hypercube). This methodology has the advantage of simultaneously assessing all factors; however, it requires a large amount of data for which the model is evaluated using, and the mathematical techniques are more complex. GSA methods can be classified into three groups (Confalonieri et al., 2010): 1) Regression methods: The standardized regression coefficients are based on a linear regression of the output on the input vector. Linear regression is the most commonly used, but there are other techniques that are also in this group (Storlie and Helton, 2008). 2) Screening methods: This refers to the method developed by Morris with significant modification as given by Compolongo et al. (2007), being described in detail in section 2.2. 3) Variance–based methods (Reuter and Liebscher, 2008; Saltelli et al., 2008; Saltelli et al., 2007; Simlab, 2008): This is a GSA method in which the variance of the model output can be decomposed into terms of increasing dimension, called partial variances, that represent the contribution of the inputs (i.e., single inputs, pairs of inputs, etc.) to the overall uncertainty of the model output. This method enables the simultaneous exploration of the space of the uncertain inputs, which is usually carried out via Monte Carlo sampling. Statistical estimators of partial variances are available to quantify the sensitivities of all the inputs and of groups of inputs through multi-dimensional integrals. The computational cost, in terms of model simulations, of estimating the sensitivities of higher-order interactions between inputs can be very high. To preclude a high computation cost, Homma and Saltelli (1996) introduced the concept of a total sensitivity index. The total sensitivity index indicates the overall effect of a given input by considering all the possible interactions of the respective input with all the other inputs. Examples of techniques in this group include the analysis of variance (ANOVA), Fourier amplitude sensitivity test (FAST), extended Fourier amplitude sensitivity test (E-FAST), Sobol’ method and the high-dimensional model representation (HDMR). 70 3.2.1. FAST and Sobol’ Method. This method is based on the partitioning of the total variance of the model output ( ), ( ), where is a scalar and is considering that the model has the form a model factor, using the following equation (Confalonieri et al., 2010): ( ) ∑ ∑ ∑ , (3.1) ( [ ( | )]) and represents the first order effect for each factor [ ( | )] ) on as the interactions among factors. The variance of the conditional expectation ( [ ( | )]) is sometimes called the main effect and is used as an indicator of the significance of . The variance-based methods (the FAST and Sobol’ methods) allow the calculation of two indices, i.e., the first-order-effect sensitivity index corresponding to a single factor ( ) where ( [ ( | )] (3.2) ( ) and the total sensitivity index corresponding to a single factor (index ) and the interaction of additional factors that involve the index and at least one index from 1 to ∑ ∑ . (3.3) The first order sensitivity index measures only the main effect contribution of each input factor on the output variance. It does not take into account the interactions among factors. Two factors are said to interact if their total effect on the output is not equal to the sum of their first order effects. The effect of the interaction between two orthogonal factors and on the output , in terms of conditional variance, takes the form ( ) [ ( | )] [ ( | )] [ ( | )] )] describes the joint effect of the pair ( where [ ( | known as the second-order effect [ ( | )] ) on [ ( | )] (3.4) . This effect is (3.5) ( ) 71 Higher order effects are computed in a similar way. A model without interactions is said to be additive. The first order indices sum to one in an additive model with orthogonal inputs. For additive models, the first order indices coincide with outputs of regression methods. For non-additive models, information from all the interactions as well as the first order effect is searched for. For non-linear models, the sum of all the first order indices can be very low. The sum of all the order effects that a parameter accounts for is called the total effect. Therefore, for an input , the total sensitivity index (Eq. 3.3) is defined as the sum of all the indices relating to (first and higher orders). According to variance-based methods, the input factor space dimensional unit hypercube ( | is assumed to be the k- ). (3.6) The core feature of FAST and E-FAST is that the n-dimensional space of the input factors is explored using a search curve defined by a set of parametric equations [ ( )], (3.7) where ( ) is a frequency associated with the factor , is a scalable value that ranges between -π and π (E-FAST), and is a value indicating the starting point of the search curve. ) can always be According to Sobol’ (1993), the model function ( decomposed into summands of increasing dimension. The total variance of ( )can be written as ( ) ∫ , (3.8) while each partial variance, corresponding to a generic term orthogonal), can be written as ∫ ∫ ( ) . 72 . (all the are (3.9) The quantities , , and can be computed by multi-dimensional Monte Carlo integration. Sensitivity estimates are then defined as . (3.10) The first-order sensitivity index ( ) is important when the objective is to determine the most important input uncertainties. The total sensitivity index ( ) is important when the objective is to reduce the uncertainty in the output model (Adeyinka, 2007). If the first order sensitivity index ( ) of the th input factor is negligible, the uncertainty in does not affect the uncertainty in the output model . Therefore, is non-influential or unimportant. This does not determine any information about input interactions or highorder sensitivity indices such as or . If the total sensitivity index ( ) is also small, then apart from being unimportant, does not interact with other factors (highorder effects of are negligible). The implication of small values of and is that the uncertainty in has no affect on the uncertainty in . Thus, in subsequent analyses, can be fixed to its nominal value (mean or median) and further research, measurement, analysis and data gathering can be directed at other factors. Conversely, regardless of the magnitude of , a large value of the first-order sensitivity index implies that is influential. The arithmetic difference between and indicates the magnitude of the interactions between and other factors. 3.2.2. Morris Method The Morris method (Morris, 1991; Confalonieri et al., 2010; Saltelli et al., 2008) ), proposing the construction of two sensitivity calculates elementary effects ( measures to determine which input factors could be considered to have effects that were (a) negligible, (b) linear and additive, or (c) non-linear or involved in interactions with other factors due to each input factor using the following equation ( ) ( where ( ) is the output, studied, ) ( is a value between { ⁄( ( ) , (3.11) ) is the n-dimensional input factor vector being ⁄( ) levels. 73 ) }, and is the number of The method samples values of from the hyperspace Ω (identified by an n-dimensional -level grid) and calculates the mean ( , assessing the overall influence of the factor on ( )) and standard deviation (estimating the totality of the higher order effects, nonlinearity or interactions with other factors) of all the obtained for each factor. and are calculated over different trajectories consisting of individual, one-factor-at-a-time experiments. The total number of model evaluations needed is t(n + 1), where t is the number of trajectories. A large (absolute) value for the central tendency shows that an input factor has an important overall influence on the output (i.e., the factor has a significant total effect), while a large value of the spread indicates that either an input has a non-linear effect on the output or that a factor is interacting with other factors (which corresponds to higher order effects). Therefore, the more significant parameters are those located in the upper right quadrant of a (spread) versus (strength) plot where both sensitivity measures are high. Examples of the software used for the global sensitivity analysis are SimLab (Simlab, 2008), GUI-HDMR working jointly with Matlab (Ziehn et al. 2009) and the software R (http://www.r-project.org/) which is a free software environment for statistical computing and graphics (R and RStudio). For this study, SimLab version 2.1 was used. 3.2.3. Local Sensitivity Analysis This method, also called a differential analysis, is not a GSA method, but it is discussed here because it is the backbone of nearly all other sensitivity analysis techniques and because it is applied to one of the examples. The methods developed in the literature range from solving simple partial derivatives to spatial and temporal sensitivity analyses (Hamby, 1994). Here, only partial derivatives are presented. Local sensitivity analysis was conceived as a local measure of the effect of a given input factor on a given output. Given a model ( ), where Y is the model output of interest and is the set of uncertain input factor, the common way to describe a local sensitivity coefficient is by using the partial derivative of the model output with respect to the model input, which is expressed as (Liu and Homma, 2010) . (3.12) In equation (3.12), is calculated by varying the input factor while fixing all the other input factors at their nominal values. Two alternative expressions of this measure are 74 , (3.13) where and respectively, and are the nominal values of the input factor and model output , , (3.14) where and are the standard deviations of the input factor and model output , respectively. To calculate , only a small perturbation of in the neighborhood of its nominal value is permitted. 3.3. Applications The following case studies attempt to analyze the application of GSA to process improvement. Case study 3.1 shows how GSA can be used to identify the behavior of processes. Case study 3.2 compares the Sobol’ and Morris methods and applies GSA to the improvement of a simple example. Case study 3.3 compares several methods of SA, including the effect of uncertainty and the type of distribution function used. Finally, case study 3.4 shows the application of GSA to the improvement of a copper flotation circuit. 3.3.1. Process Identification and Sobol’ Method In this case study, the Sobol’ SA method is applied to study the effect of each concentration stage on the overall performance of the process. Figure 1 shows four concentration circuits, each having three stages, Rougher (Ro), Scavenger (Sc) and Cleaner (Cl), but with different interconnections or structures. The output is the overall recovery rate of species i ( ), and the input factors are the stage’s recoveries ( for Ro recovery, for Cl recovery and for Sc recovery), i.e., ( , ). Equations 3.15 through 3.18 are the overall recovery rates of species i ( circuits k=a, b, c and d, respectively, as determined by mass balance ) for the (3.15) (3.16) 75 (3.17) . Ro Sc (3.18) Ro Ro Sc Cl Cl Cl Circuit a Sc Circuit b Ro Sc Cl Circuit c Circuit d Figure 3.1. Concentration circuits used in case 1. The values of the stage’s recovery are functions of several variables, such as the pH, particle size, mineralogical composition, pulp potential, and cell size; therefore, these values have uncertainties associated with them. Let us consider that each species can be represented by a uniform distribution function between a minimum and maximum value. Note that input factors are the recovery rates of each stage; therefore, the models used are adequate representations of the circuit and exclude any assumptions. The Sobol’ method was applied for different intervals of the same length of stage’s recoveries using 22,807 executions with interactions. Each stage’s recovery interval can be considered as a different species. Figure 3.2 shows the Sobol’ total index versus the stage’s recovery values for the four concentration circuits of Figure 3.1. The greater the value of Sobol’ total index for a factor, the greater the sensitivity of the output to that factor. Each graph in Figure 3.2 is distinct, which means that the effect of the recovery of each stage on the overall recovery rate is different for each circuit. Figure 3.2a shows that the rougher stage has a significant effect on the overall recovery rate of circuit a, independent of the stage’s recovery values, which means that the overall recovery rate is sensitive to the rougher stage’s recovery for all the species. Conversely, the opposite is observed for circuit d; this overall recovery rate is not sensitive to the rougher stage’s recovery for all the species. When stage’s 76 recoveries are increased for the cleaner stage, the Sobol’ total index decreases for all the circuits, which means that the effect of the cleaner stage’s recovery on the overall circuit recovery rate is important for species with low stage’s recovery values. The opposite is observed for all the circuits for the scavenger stage, which means that the effect of the scavenger stage’s recovery on the overall circuit recovery rate is important for species with high stage’s recovery values. Therefore, for species with low stage’s recovery values, the operational and/or design variables must be related to the cleaner stage, while for high recovery value species, operational and/or design variables must be related to the scavenger stage. An interesting aspect is the results delivered by circuit b and circuit c. In circuit b (Figure 3.2b), the total sensitivity index of the rougher and scavenger stages match in one curve because the rougher and scavenger stages can be considered as a single rougher stage. On the other hand, the rougher stage curve matches the cleaner stage for circuit c (Figure 3.2c), indicating that for circuit c, rougher and cleaner stages can be considered as a single rougher stage. A number of these findings are obvious, particularly for concentration circuits expert; however, this case study shows that GSAs, and the Sobol’ method in particular, can be used to study concentration circuits, for example, to identify the effects of each process stage on the overall process performance. 3.3.1. Process retrofit and comparison between Sobol´ and Morris methods In this case study, a GSA is performed to improve concentration circuit a (Figure 3.1a). This case study considers two species, one with a high recovery value and one with a medium recovery value. The high recovery species have recovery rates of 0.75, 0.73 and 0.86 for the rougher, cleaner and scavenger stages, respectively, and a feed mass flow rate of 100. On the other hand, the medium recovery species have recovery rates of 0.39, 0.48 and 0.61 for the rougher, cleaner and scavenger stages, respectively, and a feed mass flow rate of 200. With these values, the overall recovery rate of the species with high and medium recovery values are 0.94 and 0.45, respectively, and the concentrate grade is 51.73. It is desirable to improve the concentrate grade using a GSA by changing the operating conditions or changing the equipment design. To improve the concentrate grade, a reduction in the overall recovery rate of the medium recovery species while minimizing the effect on the overall recovery rate of the high recovery species is needed. Another way to improve the concentrate grade is to increase the overall recovery rate of the high recovery species while minimizing the effect on the overall recovery of the high 77 recovery species. However, the latter is more difficult because the overall recovery rate of the high species is already high. 0.8 Rougher 0.7 0.7 0.6 0.5 0.4 Cleaner Scavenger 0.3 0.2 Sobol total Index Sobol total Index 0.8 0.1 0.6 Rougher - Scavenger 0.5 0.4 0.3 Cleaner 0.2 0.1 0 0 0-0.1 0.1-0.2 0.2-0.3 0.3-0.4 0.4-0.5 0.5-0.6 0.6-0.7 0.7-0.8 0.8-0.9 0.9-1 0-0.1 0.1-0.2 0.2-0.3 0.3-0.4 0.4-0.5 0.5-0.6 0.6-0.7 0.7-0.8 0.8-0.9 0.9-1 Recovery stages Recovery stages (a) (b) 0.8 0.8 0.6 Scavenger Rougher - Cleaner 0.5 0.4 0.3 0.2 0.7 Sobol total Index Sobol total Index 0.7 Cleaner Scavenger 0.6 0.5 0.4 0.3 0.2 Rougher 0.1 0.1 0 0 0-0.1 0.1-0.2 0.2-0.3 0.3-0.4 0.4-0.5 0.5-0.6 0.6-0.7 0.7-0.8 0.8-0.9 0.9-1 0-0.1 0.1-0.2 0.2-0.3 0.3-0.4 0.4-0.5 0.5-0.6 0.6-0.7 0.7-0.8 0.8-0.9 0.9-1 Recovery stages Recovery stages (c) (d) Figure 3.2. Sobol’ total index versus stage’s recoveries for a) circuit a, b) circuit b, c) circuit c and d) circuit d. In addition to equation (1.5), a kinetic model is used to calculate the stage’s recovery of species , ( (3.19) ) where is the rate constant for each species in stage , is the cell residence time in stage , and is the number of cells in stage . Variation of the values of the rate constants and cell residence time is achieved by changing the operating conditions while changing the number of cells resulting in a change in the design of the concentration steps. The data for the parameters in the kinetic model (factors in the GSA) were obtained 78 from Lucay et al (2012), considering the parameters as average values and subsequently, a dispersion of 5% was considered. The details are given in Table 3.1. In this case study, the Morris and Sobol’ methods were applied for comparison. The parameters used for the Sobol’ method were 8,192 executions with interactions and for the Morris method, the parameters used were a seed of 1,000,000, 70 executions, and eight levels. Figure 3.3 shows a comparison between the total sensitivity index of the Sobol’ method and the µ index of the Morris method (Confalonieri et al, 2010). The results of both methods are similar because the most sensitive factors in the Morris method (larger values of µ) are also those with the highest values of the total Sobol’ index. Figures 3.3 shows that the factor that has the greatest sensitivity on the overall recovery rate is the number of cells in the rougher stage, both for the high and medium recovery species, and is consistent with the information given in Figure 3.2a, where the greater sensitivity of circuit a is from the rougher stage. However, we see that the redesign has a greater effect than does changing the operational conditions in the rougher stage. The number of cells in the scavenger stage also has a significant effect in both species, but the number of cells in the cleaner stage has a more significant effect on the overall recovery rate of the medium recovery species than in the high recovery species. The results also show that a change in the kinetic values for both species and the residence time in each stage have little effect on the overall recovery rates of both species, at least compared to the number of cells. This information is very useful because it indicates that a redesign of the concentration stage is a better option than a change in the operating conditions. The analysis shows that for high recovery species and medium recovery species alike, the overall recovery rates are sensitive to the number of cells in the rougher and scavenger stages, but only the overall recovery rate of the medium recovery species is sensitive to the number of cell in the cleaner stage. Figure 3.4 shows the effect of the number of cells in the cleaner stage (Figure 3.4a) and rougher stage (Figure 3.4b) on the overall recovery rates for both species. When reducing the number of cells in the rougher stage, both species exhibit sharp decline in their overall recovery from 0.95 to 0.67 for the high recovery species and from 0.47 to 0.1 for the medium recovery species. However, in reducing the number of cells in the cleaner stage, only the medium recovery species exhibit a sharp decline in their overall recovery rate from 0.44 to 0.19, whereas the high recovery species’ recovery rates reduces from 0.95 to 0.86. Next, if the number of cleaner cells is changed from eight to four, the overall 79 recovery rates of the high and medium recovery species decreases from 0.94 to 0.92 and from 0.44 to 0.35, respectively. The product grade also increases from 51.73 to 59.34. The results depend on whether the model is an adequate representation of the plant, which is why the extrapolation and interpolation capabilities of the model are important to observe. In this case study, a simple model was used to illustrate the methodology; therefore, the results cannot represent reality (for example, the assumption that the kinetic constant does not change from one cell to another can give erroneous results). For a real application, a more accurate model, which may contain many parameters and experimental measurements, is required. If the model is complex and/or there are many parameters for each cell, this can be a difficult and costly task. Therefore, an alternative is to first identify the key flotation stage using the stage’s recovery as an input factor (see case study 3.1) and subsequently apply a GSA to that particular stage. Table 3.1. Distribution functions for factors in case study 4.2. Stage Nomenclature Average Distribution detail Residence time (h): (Beta distribution) Rougher 0.0630 α=1.5, β =3, a=0.0598 and b=0.0661 Cleaner 0.0840 α= 1.5, β =3, a=0.0798 and b=0.0882 Scavenger 0.0853 α= 1.5, β =3, a=0.0810 and b=0.0895 Kinetic coefficient for high recovery species (1/h) (Beta distribution) Rougher 3 α= 6, β =6, a=2.85 and b=3.15 Cleaner 2.12 α= 6, β =6, a=2.01 and b=2.22 Scavenger 2.29 α= 3, β =3, a=2.17 and b=2.40 Kinetic coefficient for medium recovery species (1/h) (Beta distribution) Rougher 1 α= 3, β =3, a=0.95 and b=1.050 Cleaner 1.02 α= 3, β =3, a=0.97 and b=1.071 Scavenger 1.05 α= 3, β =3, a=0.99 and b=1.103 Cell number (Discrete distribution) Rougher 8 weighting 0.5 Higher value: 9, weighting 0.25 Lower value: 7, weighting 0.25 Cleaner 8 weighting 0.5 Higher value: 9, weighting 0.25 Lower value: 7, weighting 0.25 Scavenger 11 weighting 0.5 Higher value: 12, weighting 0.25 Lower value: 10, weighting 0.25 80 In addition, a process retrofit or design should include the study of several output variables that can be included in the GSA using more than one output factor (see case study 3.4) or the study of the consideration of these variables after a GSA has been applied to the main output variable. Thus, for example, changes in the number of cells will affect the cost of the plant and thus require an economic evaluation before implementing a change in the process. 0.6 | 𝑜 Morris (µ) 0.5 Principal factors 0.4 𝑜 0.6 0.5 Principal factors 0.4 0.3 0.3 0.2 0.2 𝐶 0.1 𝐶 0.1 𝐶 𝐶 𝑜 0 𝑜 0 0 0.005 tc 0.01 ts 0.015 kr kc 0.02 ks 0.025 0.03 0 0.02 Nc Ns (Total Index) tc Sobol’ Nr ts 0.04 kr kc 0.06 ks (a) Nr 0.08 Nc Ns (b) 0.50 1.00 0.95 0.45 0.95 0.40 0.90 0.35 0.85 0.30 0.80 0.25 0.75 0.20 0.15 0.70 0.10 High Recovery Specie 0.65 Medium Recovery Specie 0.60 2 3 4 5 6 7 8 0.90 0.85 0.80 0.75 0.70 High Recovery Specie 0.05 0.65 0.00 0.60 9 Medium Recovery Specie 2 Cell Number Cleaner 80.00 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 3 4 5 6 7 8 Overall Recovery 1.00 Overall Recovery Overall Recovery Figure 3.3. Relation between Sobol’ total index and Morris method µ for circuit a (Figure 3.1a) for the a) high recovery specie and b) medium recovery specie. 9 Cell Number Rougher 200 (a) (b) Figure 3.4. Evolution of the overall recovery rates by a) modifying cell number cleaner and b) modifying cell number rougher. 81 3.3.2. Comparison of sensitivity analysis methods This case study is a comparison of the Morris, EFAST and Sobol’ GSA as well as a local SA. Different dispersion ranges and distribution functions are considered for the factors studied. For this example, a model of a flotation column given by Yianatos et al. (2005) is utilized. In Yianatos’ model, the collection zone recovery of a flotation column is modeled using a rectangular distribution function for the rate constant and a tank-in-series model, consisting of one large and two small size units, to account for the residence time distribution (Figure 3.5). Figure 3.5 also shows the model for the recovery rate R. Wash water Concentrate 𝜏𝑠 Feed 𝜏𝑙 Air 𝜏𝑠 𝐿 Tail 𝑅 𝑅 𝜏𝑙 𝑘𝑚 (𝜏𝑙 𝜏𝑠 ) 𝑘𝑚 𝜏𝑠 (𝜏𝑙 𝜏𝑠 ) 𝑙𝑛 𝑘𝑚 𝜏𝑙 𝑘𝑚 𝜏𝑠 Figure 3.5. Yianatos’ model for a flotation column. In Yianatos’ model, is the recovery rate of the flotation column, is recovery rate at an infinite time, is the maximum kinetic constant, is the largest reactor residence time and is the smaller reactor residence time. The values for each of these parameters (factors in the SA) were taken from Yianatos et al. (2005), where: is 90%, and are parameters obtained through two relations ( ⁄ and ⁄ ), the total -1 residence time (12.7 min) and has values from 0.210 to 0.230 min . These values correspond to an industrial column operating at El Salvador concentrator in Chile. Four studies were conducted. First, a local SA was applied to determine the local and linear sensitivity. Second, different values of the dispersion having the same distribution function for all the input factors were considered. Next, a single value of the dispersion with different distribution functions for the factors were studied. Finally, changes in the 82 values of the dispersion and distribution functions for the factors were considered. In all the cases, the parameters for the Morris method were a seed of 1,000,000, 70 executions, and eight levels; the parameter for the E-FAST method were a seed of 10,000 and 100,000 executions; and the parameter for the Sobol’ method was 20,480 executions. For the local SA, equation (3.13) was used. The local sensitivity indices were , , , and . This means that locally, while considering the linear effects, all the factors influence the recovery values with recovery at infinite time providing the highest sensitivity (accounting for 58% of total local sensitivity indices). Conversely, the smaller reactor residence time accounts for only 6% of total local sensitivity indices. To analyze the effect of the dispersion ranges, all the factors were evaluated using a uniform distribution function with dispersion values of 10% ( , , and ) and 0.05% ( , , and ) of its mean value, and a dispersion of 0.1 in absolute value ( , , and ). Figure 3.6 shows the results of the Morris method using the σ and µ parameters. µ allows for the ranking of factors in order of importance, whereas σ is used to detect factors involved in interactions with other factors or factors whose effect is non-linear. The results for dispersions of 10% and 0.05% of the mean values are very similar (Figure 3.6 only shows the case with 10%). For these cases, the ranking of importance (highest to lowest) of µ is , , and . For , the ranking of interaction/non-linearity (highest to lowest) is , , and . Both cases (10% and 0.5%) have the same order but with different values. The results for dispersions of ±0.1 are very different (Figure 3.6b) because the dispersion in the values of the factors is proportionally different. In this case, is the factor with a greater effect on the output uncertainty. The results provided by the Sobol’ and E-FAST methods are given in Table 2.2. The results of the Sobol’ and E-FAST methods coincide with those shown using the Morris method. In the Morris and Sobol’ methods, if the dispersion of the factors decreases, the interaction index also decreases. In the case of the E-FAST method, the interaction between the parameters becomes independent of the change in the dispersion because the results show minimal variations, this being a drawback of the E-FAST method (Frey and Patil, 2002). However, for all the methods, the interaction/non-linearity effect was small. 83 0.12 0.8 ∞ 0.1 Sigma 0.7 σ 0.08 σ 0.06 0.6 0.5 0.4 0.3 0.04 0.2 0.02 0.1 0 ∞ 0 0 5 µ 10 15 20 0 1 2 3 4 5 6 µ (a) (b) Figure 3.6. Comparison of sensitivity analysis using dispersions of a) 10% and b) ±0.1. Table 3.2. Sobol’ and E-fast indexes. (Values x 102). Sobol’ 10% Item Sobol’ ±0.1 Sobol’ 0.05% First Total Difference First Total Difference First Total Difference 0.032 0.033 0.002 0.035 0.035 0.000 0.002 0.002 0.000 0.621 0.624 0.003 0.558 0.558 0.000 4.750 4.763 0.013 98.563 98.569 0.006 98.642 98.642 0.000 0.103 0.103 0.000 0.943 0.947 0.004 0.935 0.935 0.000 95.398 95.411 0.013 Item E-Fast 10% E-Fast 0.05% E-Fast ±0.1 First Total Diff First Total Diff First Total Diff 0.044 0.083 0.040 0.039 0.039 0.000 0.026 0.328 0.303 0.750 0.793 0.043 0.680 0.680 0.000 4.570 4.896 0.326 97.790 98.028 0.238 98.471 98.471 0.000 0.100 0.406 0.306 1.170 1.408 0.238 0.971 0.971 0.000 94.970 95.280 0.310 For the Morris, Sobol’ and E-FAST methods using dispersion values proportional to the mean values (10 and 0.5%) for the input factors, the resulting order of importance of the factors is similar to that obtained with the local SA. However, if the dispersions are not proportional, then the results are quite different. The results of the local SA are more accurate when applied to a linear model. However, for a non-linear model, the sensitivity of the output to a given input may depend on interactions with other inputs that are not considered. Thus, the results of a nominal range sensitivity are potentially misleading for non-linear models. 84 Based on these studies, it can be concluded that the level of dispersion in input factors can have a significant effect on the sensitivity analysis; however, if these changes are proportional to the mean values of the input factors, the effect in the dispersion values is not significant. If the goal is to identify which input factor has a greater effect on the output, then dispersions values proportional to the mean value of the input factor should be used. Moreover, if the range in which each of the input factors can be changed is known, then these values should be used to identify which factors have the greatest effect on the output model. For example, Figure 3.6a shows that if all the input factors can be modified to some extent to their mean values, then the recovery rate at infinite time should be modified to achieve a greater change in recovery rate. Conversely, if all the input factors can be modify by ±0.1 (Figure 3.6b), then the maximum kinetic constant must be modified to achieve a greater change in recovery rate. The second analysis shows the effect of the type of distribution function used on the SA. This is important because it may be that the type of distribution function for the input factor is not known. In this study, six distribution functions were used (with a 10% dispersion for each factor): a normal distribution, uniform distribution and four variations of the beta distribution. Only the Sobol’ method is analyzed. Table 3.3 shows the results of the sensitivity analysis index. For all the cases, the ranking (from highest to lowest) of importance of total and first indexes is , , and . The difference between the total and first indices are small as well, which means that the interactions and/or nonlinearities are small. Table 3.3. Sobol’ Indices for different distribution functions (values x 102). Item Item Uniform distribution Triangular distribution Beta distribution α and β = 2 First Total Difference First Total Difference First Total Difference 0.032 0.033 0.002 0.043 0.042 -0.001 0.044 0.044 0.000 0.621 0.624 0.003 0.762 0.762 0.000 0.755 0.757 0.001 98.563 98.569 0.006 98.181 98.185 0.003 98.179 98.183 0.004 0.943 0.947 0.004 1.178 1.180 0.002 1.179 1.181 0.002 Beta distribution α and β =7 Beta distribution α=2 and β =7 Beta distribution α =7and β =2 First Total Difference First Total Difference First Total Difference 0.043 0.042 -0.001 0.050 0.050 -0.001 0.037 0.037 -0.001 0.760 0.759 -0.001 0.796 0.796 0.000 0.717 0.716 -0.001 98.112 98.113 0.001 97.743 97.746 0.003 98.562 98.563 0.001 1.180 1.181 0.000 1.241 1.243 0.001 1.116 1.117 0.000 85 Finally, the Morris method was applied by modifying both distribution functions and dispersions values for each factor. is used with the data provided by Yianatos et al. -1 (2005) (from 0.210 to 0.230 min ), is 90% with a dispersion of ± 2%, and and keep with the mean values with a dispersion of 10%. Four distribution functions were considered: a uniform distribution, a beta distribution with α and β = 2, a beta distribution with α = 2 and β = 7, and a beta distribution with α = 7 and β = 2. Figure 3.7 shows the Morris diagram, which shows that the order of µ is equal in every case; however, the levels of interaction (σ) for the uniform distribution is different than for the other distributions. Therefore, this demonstrates the importance of having precise information on the types of distributions and dispersions, especially when non-linear and/or nonadditive models are to be evaluated, because it may produce significant variations in the respective sensitivity indices. 3.E-02 8.E-02 7.E-02 2.E-02 ∞ 6.E-02 5.E-02 Sigma 2.E-02 σ 4.E-02 ∞ 1.E-02 3.E-02 2.E-02 5.E-03 1.E-02 0.E+00 0.E+00 1.E-01 1.E+00 2.E+00 3.E+00 4.E+00 µ µ 2.E-01 (a) 1.E-01 Sigma 0.E+00 0.E+00 2.E-01 4.E-01 6.E-01 8.E-01 1.E+00 1.E+00 1.E+00 (b) 2.E-01 1.E-01 ∞ 1.E-01 ∞ 1.E-01 σ 8.E-02 1.E-01 6.E-02 8.E-02 6.E-02 4.E-02 4.E-02 2.E-02 2.E-02 0.E+00 0.E+00 0.E+00 0.E+00 1.E-01 2.E-01 3.E-01 4.E-01 5.E-01 6.E-01 7.E-01 2.E-01 4.E-01 µ µ (a) (b) 6.E-01 8.E-01 Figure 3.7. Comparison of sensitivity analysis using a a) uniform distribution, b) beta distribution with α and β = 2, c) beta distribution with α = 2 and β = 7, and d) a beta distribution with α=7 and β = 2. 86 The model used in Yianatos et al. (2005) was used as an example and although their values adequately represent an actual plant, the results must be analyzed considering whether the model is adequate or not. In this case study, Yianatos’ model was used to assess different GSA methods and different natures of uncertainty in the input factors. Moreover, identifying the key factor or factors may require more studies because its implementation is not straightforward. For example, if the infinite recovery rate is the key factor, then it is necessary to consider that this factor depends on previous operations, such as milling. Experimental studies and/or simulations of this stage would then be needed. The effect of varying the infinite recovery rate (e.g., changing particle size) must take into account the costs, energy and water efficiencies involved. From this perspective, the value of a GSA is to identify the key factors, but not necessarily the ultimate solution of the problem. 3.3.3. Retrofit of a copper concentration plant This example exhibits an attempt to improve a copper concentrator plant as described by Hay and Martin (2004). The circuit diagram is shown in Figure 3.8 and is made up of five stages: rougher (Ro), rougher-scavenger (Ro-Sc), cleaner 1 (Cl1), cleaner 2 (Cl2) and cleaner-scavenger (Cl-Sc). The mass balance and recovery rate are given in Tables 3.4 and 3.5. The stage’s recoveries can be estimated using equation (3.18) (Yianatos and Henríquez, 2006). [ ( ( ) ) ] (3.20) Here, is the cumulative mineral recovery rate in the flotation bank in time , is the maximum recovery rate at an infinite time, is the maximum rate constant of the rectangular distribution function, is the residence time of one cell, is the number of cells in the bank and indicates the species. To determine the importance of the stage’s recovery on the overall recovery rate and copper concentrate grade, a GSA is performed using the Morris method, considering that each stage’s recovery (Table 3.5 values) may vary by ±7% with a uniform distribution function. This means that the copper and gangue stage’s recoveries are the input factors, and the copper and gangue overall recovery rates and copper concentrate grade are the model outputs. The model to be analyzed corresponds to the mass balance for each component, an expression that can be derived by applying the law of conservation of matter to the circuit of Figure 3.8. It should be emphasized that the GSA was performed 87 with recovery rates of each stage as input factors and not the model of equation 3.20; therefore, the model used is a suitable model and does not include simplifications. Ro Ro-Sc Cl1 Cl- Sc Cl2 Figure 3.8. Flotation circuit (copper concentrator; Hay and Martin, 2004). Table 3.4. Mass flow rates of the flotation circuit of Figure 3.8 (Hay and Martin, 2004). Stream Feed flow Recirculation Scavenger-Cleaner Feed Rougher Concentrate Rougher Tail Rougher Rougher-Scavenger Concentrate Tail Rougher-Scavenger Feed Cleaner 1 Concentrate Cleaner 1 Tail Cleaner 1 Feed Cleaner-Scavenger Concentrate Cleaner-Scavenger Tail Cleaner-Scavenger Concentrate Cleaner 2 Tail Cleaner 2 Total flow t/h 308 12 320 12.3 307.7 3.5 304.2 18.1 6.5 11.6 15.1 3.1 12 3.8 2.7 88 Flow (t/h) Grade 0.004 0.007 0.004 0.085 0.001 0.018 0.001 0.093 0.200 0.034 0.030 0.120 0.007 0.270 0.101 Copper Gangue 1.34 0.08 1.42 1.05 0.38 0.06 0.31 1.69 1.30 0.39 0.45 0.37 0.08 1.03 0.27 306.66 11.92 318.58 11.25 307.32 3.44 303.89 16.41 5.20 11.21 14.65 2.73 11.92 2.77 2.43 Table 3.5. Stage’s recoveries for the flotation circuit of Figure 3.8. Stage Recovery, % Copper Gangue Rougher 73.51 3.53 Rougher-Scavenger 16.72 1.12 Cleaner 1 76.85 31.69 Cleaner 2 78.92 53.35 Cleaner-Scavenger 81.85 18.63 Overall Recovery 76.58 0.91 The results are shown in Figure 3.9 and show that the rougher stage is the stage with the highest sensitivity for overall recovery rates of both the copper and gangue (Figures 3.9a and 3.9b). The copper and gangue rougher stage recovery rates make an important contribution to the copper concentrate grade uncertainty (Figure 3.9c) as well. This means that the rougher stage is important, but an increase in the copper recovery rate and reduction in the gangue recovery rate at the rougher stage is necessary to improve the process. The Cl-Sc stage is also important for the overall recovery rate of copper, unlike for the overall recovery rate of gangue and the copper concentrate grade. This means that if copper recovery rate is increased at the Cl-Sc stage, the overall copper recovery rate will be increased without significantly affecting the gangue recovery rate and copper concentrate grade. This is an important result as will be shown later. All the other stages are not important or they make insignificant contributions to the overall recovery rates’ uncertainty. It is also worth noting that the Ro-Sc stage has an insignificant contribution to both the overall copper and gangue recovery rates (Figure 3.9a and 3.9b). Conversely, both copper and gangue cleaner 1 and cleaner 2 stage’s recoveries make an important contribution to the copper concentrate grade uncertainty (Figure 3.9c), unlike for the overall copper and gangue recovery rates. However, the gangue cleaner 2 stage’s recovery has a larger contribution to the copper concentrate grade uncertainty (Figure 3.9c). This means that if the gangue recovery is reduced at the cleaner 2 stage, the copper concentrate grade will increase without significantly affecting the overall gangue and copper recovery rates. This is another important result as will be shown later. All the other stages are not important or make insignificant contributions to the copper 89 concentrate grade uncertainty. These results were verified with the GSA provided by the Sobol’ method (not shown in this paper). Having identified the key factors, it is possible to improve the process. This can be done by experimentation and through the use of suitable models. In this paper, the Yianatos and Henríquez (2006) model, Equation 3.20, is used to exemplify how the results obtained using GSA can be used to improve the process. It should be emphasized that the model Yianatos and Henríquez developed does not affect the results of the GSA because it was not implemented in the GSA. Based on these results, if the copper recovery rate is to be increased without affecting the recovery rate of the gangue, a feasible alternative is to modify the Cl-Sc stage. If the Ro stage is changed, these changes will affect both the copper and gangue recovery rates. Moreover, to change the overall recovery rate, changes in the recovery of cleaner 1, cleaner 2, or the Ro-Sc stage will require a significant change that will affect the entire circuit. Figure 3.10 shows the effect of changing the number of cells and the residence time of Cl-Sc stage. Figure 3.10a shows the effect on the recovery rate of the Cl-Sc stage. It can be seen that these changes result in changes in the recovery rates of copper and gangue equally, in some cases producing more changes in the recovery rates of gangue. Figure 3.10b shows the effect on the overall recovery rate. It can be seen that these changes produce more significant changes in the recovery rate of copper than in the gangue recovery rate. For example, if the cell number is changed from 5 to 12 and the residence time is changed from 0.04 to 0.06 h-1, the copper recovery rate increases from 76.58 to 77.03%, an increase of 0.45%. Moreover, the recovery rate of the gangue increases from 0.90 to 1.00, an increase of only 0.1%. If the copper concentrate grade is to be increased, a feasible alternative is to modify the Cl-2 stage. Figure 3.10c shows the effect on the copper concentrate grade of changing the number of cells and the residence time of the Cl-2 stage. It can be seen that these changes produce significant changes in the copper grade. Based on this and the previous analysis, a suitable alternative would be to use ten cells and a residence time of 0.06 h in the Cl-Sc stage, and two cells and a residence time of 0.04 in the Cl-2 stage. Using these conditions, the copper recovery rate increases from 76.58 to 76.91%, and the copper grade increases from 27 to 33.8%. 90 1.E-02 1.E-02 3.E-02 Cl-Sc σ 2.E-02 Sigma 1.E-02 1.E-02 Ro 2.E-02 Ro 8.E-03 8.E-03 σ 6.E-03 6.E-03 1.E-02 Cl 1 4.E-03 4.E-03 Cl 2 5.E-03 Cl 1 2.E-03 2.E-03 Ro-Sc Cl 2 0.E+00 0.E+00 2.E-02 4.E-02 6.E-02 8.E-02 1.E-01 Ro-Sc 0.E+00 Cl-Sc 0.E+00 0.E+00 2.E-02 0.E+00 1.E-01 1.E-01 4.E-02 µMu 6.E-02 8.E-02 1.E-01 1.E-01 µMu (a) (b) 2.0E-03 Ro (Gangue) 1.8E-03 1.6E-03 σ Cl 1 (Gangue) 1.4E-03 1.2E-03 Cl 1 (Cu) 1.0E-03 Cl 2 (Cu) 8.0E-04 Ro (Cu) 6.0E-04 4.0E-04 2.0E-04 Cl-Sc (Cu) Ro-Sc (Cu) Cl-Sc (Gangue) Ro-Sc (Gangue) 0.0E+00 0.E+00 5.E-03 3.E-01 Cl 2 (Gangue) 1.E-02 2.E-02 2.E-02 3.E-02 3.E-02 4.E-02 µ (c) Figure 3.9. Morris versus diagram for a) the copper overall recovery rate, b) gangue overall recovery rate, and c) the copper concentrate grade. It is noteworthy that the Yianatos and Henriquez (2006) model was used to illustrate how to implement the results of GSA, but it was not used in the GSA itself. In this sense, the results of the GSA are valid because no assumptions were included, but changes in the cell number and residence times are valid to the extent that the Yianatos and Henriquez model is a suitable model. It is also necessary to indicate that the effect on the recovery rate and grade was analyzed, but other effects such as the equipment cost must be included in a final decision. 91 It is also noteworthy that this GSA study results in only ±7% changes in the values of each stage’s recoveries. This does not imply that under other conditions the results are the same. For example, if the Ro-Sc stage is removed (the Ro-Sc stage makes a small contribution to the overall recovery rates), the copper recovery rate is reduced from 76.58 to 72.19%. Actually, the Ro-Sc stage has little effect on the overall recovery rate because the value of the copper recovery is very low (16.72%); therefore, the recovery rate changes of ±7% do not have a significant impact on the overall recovery rate. (a) (b) 34 Copper Grade (%) "Global grade, τ=0.04 h" 32 "Global grade, τ=0.06 h" 30 28 26 24 22 20 2 3 4 5 6 Cell number 7 8 9 (c) Figure 3.10. Recovery rate and copper grade as a function of cell number and residence time: a) Cl-Sc recovery rate as a function of Cl-Sc cell number and residence time, (b) overall recovery rate as a function of Cl-Sc cell number and residence time and (c) overall copper grade as a function of Cl 2 cell number and residence time. 3.4. Conclusions Based on these case studies, we conclude that GSA is a tool that can help analyze, design and improve processes. The Morris, Sobol and E-FAST methods give similar results, 92 which is why the Morris method is recommended because of its computationally inexpensive nature. The type of distribution function and the ranges of the values of the input factors have an effect on the results. If the range in which an input factor can be changed is known, those values should be used. If not, ranges proportional to the mean values should be used. The GSA methods help to identify the process behavior and/or help to identify the most important process stages that affect a given process variable (model output). With this information, it is possible to redesign or change the operational condition to improve processes’ performance. We conclude that using the stage’s recoveries of each species as input factors is a suitable choice for a GSA of a flotation plant. Once key stages have been identified to improve the process, experimental tests, modeling and analyses can be used to improve the plant. 3.1. Acknowledgments Financial support from CONICYT (Fondecyt 1120794), CICITEM (R10C1004) and the Antofagasta Regional Government is gratefully acknowledged. 3.2. 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