矿物加工外文文献英文

Contents lists available at ScienceDirect Minerals Engineering journal homepage Application of multi disciplinary optimization architectures in mineral processing simulations Kanishk Bhadani Gauti Asbj rnsson Erik Hulth n Magnus Evertsson Department of Industrial and Materials Science Chalmers University of Technology Gothenburg 412 96 Sweden A R T I C L E I N F O Keywords Optimization architecture Mineral processing simulations Comminution Multi disciplinary optimization MDO A B S T R A C T Optimization is a pivotal point in distinguishing the competitiveness of industries that are developing designing and operating various products and processes Mineral processing is an industry which operates various sub processes and produces one or several products The sub processes involved are dynamic in nature and diff er in the discipline of operation These dynamic sub processes are sequentially integrated forming a mineral pro cessing system Currently the developed simulations for the mineral processing systems have the potential to be used to design operate and control mineral processing plants to an increased extent but need broader opti mization strategies to integrate multiple sub processes involved The scope of this research is to demonstrate application of multi disciplinary optimization MDO architectures into a mineral processing simulation A si mulation study consisting of two sub processes of comminution and classifi cation circuits to produce aggregate products is used to demonstrate the application of MDO architectures The MDO architectures are compared based on problem formulation computational resources required and validity of the results The optimization results using MDO architectures can be used to illustrate trade off s between diff erent sub processes within the considered scope The application of MDO architectures can facilitate the linking mathematical models of various disciplines such as comminution and liberation in mineral processing simulation 1 Introduction Mineral processing industries involve the extraction of industrial minerals and metal ores whereas aggregate process industries comprise of extraction of building material e g a variety of rock sizes clay and energy minerals e g coal The two industries are similar in that they both contain comminution processes but the requirements in process operations diff er A comminution process consists of a number of sub processes each consisting of crushing and classifi cation operations The sub processes are interconnected as the material fl ows downstream from one sub process to another and upstream as recirculating loads Simulation and modelling of such processes are well established and results have shown an increased fi delity of dynamic process modelling compared to static process modelling Asbj rnsson 2015 Sb rbaro and del Villar 2010 In order to make increased utility of the developed dynamic process modelling optimization strategies are needed A typical comminution process for aggregate production consists of a primary secondary and tertiary sub process to produce n products The n products usually consist of diff erent particle size ranges required for application in building and road constructions This characteristic of the comminution process of producing multiple products based on requirements poses an interesting optimization problem of balancing the production From a process operation point of view it is essential to optimize the entire process based on the demand of products and rev enue created by the products From a strategic point of view it is es sential not just to reach a process operation solution but also to un derstand the dependencies between the sub processes In regard to the above essentials a broader optimization strategy is needed which does not only produce a solution for the process operation but also knowl edge about the process dependencies Researchers have shown multiple methods for optimizing a process simulation in comminution circuits through use of various gradient based and stochastic optimization algorithms Bengtsson et al 2017 Huband et al 2005 Venter et al 1997 Out of the two sets of algo rithms stochastic optimization algorithms have been widely used and in particular application of genetic algorithm GA Bhadani et al 2017 The optimization method using GA has been used to demon strate optimization of comminution processes and the optimization problems are usually considered to be a non linear complex problem with respect to design variables Huband et al 2006 Svedensten and Evertsson 2005 The shown methods can produce an overall result for balancing a comminution process however it lacks the possibility of https doi org 10 1016 j mineng 2018 08 029 Received 21 May 2018 Received in revised form 18 July 2018 Accepted 20 August 2018 Corresponding author E mail address kanishk chalmers se K Bhadani Minerals Engineering 128 2018 27 35 Available online 24 August 2018 0892 6875 2018 Elsevier Ltd All rights reserved T identifying and quantifying the dependencies between sub processes Another limitation of the method is that it is diffi cult to apply when the process is extended to a more complex system These delimitations lead to a requirement of having a more generic approach to be able to handle a large complex system such as comminution processes in mineral processing In order to demonstrate the application of generic approach towards optimization in the comminution process there are two basic requirements Firstly a calibrated simulation model to re plicate the physical behaviour of the comminution process and sec ondly an optimization architecture to execute the optimization routine The paper aims to demonstrate implementation of two multi dis cipline optimization MDO architectures applied to a comminution process simulation consisting of two sub processes The MDO archi tectures applied are multi discipline feasible MDF and individual discipline feasible IDF and their applicabilities are compared based on problem formulation and results As per the classifi cation scheme presented in Bhadani et al 2017 see Table 1 the scope of the op timization problem formulation is to fi nd suitable operating parameters for a fi xed aggregate plant layout The simulation of comminution process in this study uses dynamic modelling developed in MATLAB Simulink environment Asbj rnsson 2015 The results from this study show that the application of MDO architectures can facilitate in the organization and coordination of multiple optimization problems be longing to multiple sub processes within a comminution process 2 Modelling approach The system modelling is based on the approach described by Asbj rnsson 2015 The process is treated as a continuous process which is aff ected by gradual and discrete changes within the system that alters the performance of the entire system Each model has subsets of diff erent diff erential equations together with static equations to re produce the dynamic performance of a single operational unit Each model includes derivatives for mass m and properties of the material with respect to time t Eq 1 and 2 dm t dt mtmt i inj out 1 d t dt mt m t t t i i in i ini 2 The lag and response of the system caused by feeders and conveyors are represented with Eq 3 and Eq 4 where the gain Kp operator and the time delay characterises the response dy dt y tK u t p 3 y tu t 4 Due to mass conservation lag and diff erent response of units in the system the circuit needs to be controlled Interlocks are included to prevent overload of diff erent units The material fl ow is however regulated using a PI controller Eq 5 where the proportional gain KP and integral gain KIdescribe the performance of the controller utK e tKe t dt i pPI 5 The crusher and screen models are selected based on modelling standards within process simulations for minerals processing The crusher is a selection breakage model where the breakage is re presentedwithaBroadbent Calcottbreakagematrix Eq 6 Broadbent and Callcott 1956 and the screen is represented with a Reid Plitt effi ciency curve Eq 7 Reid 1971 Bee 1 1 di dj 1 6 Ee1 i x ln2 i m 7 3 Optimization architecture The multi disciplinary optimization MDO architectures have been initially developed for application in aerospace engineering design problems especially concerning interaction of two or more disciplines such as structures and aerodynamics Haftka 1977 Schmit 1981 The initial aims with application of MDO architectures were to attain global optima for a system level problem and to exploit the power of parallel computing Cramer et al 1994 MDO architectures have proved to be successful in handling and optimizing large complex engineering system problems Martins and Lambe 2013 The MDO architecture is a representation of how various sub dis ciplines involved in an engineering system optimization problem are organized Further it represents a strategy to achieve optimal values for design variables of the engineering system The MDO architectures can be broadly classifi ed into two segments monolithic architectures and distributed architectures Monolithic architectures make use of a single optimization problem defi ned for the engineering system while dis tributed architectures usually contain two or more optimization pro blems present at diff erent levels of the engineering systems Martins and Lambe 2013 Papalambros and Wilde 2017 Most of the applications shown for the MDO architectures pertain to design problems applied to system and component design for products such as aerospace components automotive components wind turbines and spacecraft Martins and Lambe 2013 There are limited applica tions shown for the process industries On the other hand a mineral processing system as a part of the process industry requires optimiza tion in both designs of the topology of plant and operation of various processes The design and operation of mineral processing systems deal with multiple sub process interactions and dependencies A mineral processing system can be characterized by a long chain of hierarchical dependencies between sub processes as the material fl ows from one sub process to the subsequent sub process The sub processes are loosely coupled with design variables of the other sub processes but are strongly coupled with the output of one process to another The application of MDO architectures requires a well developed optimization problem defi nition and it is also essential to defi ne the boundaries between the interacting elements in the system In order to adopt the established knowledge within MDO into mineral processing systems certain terminologies are being rephrased such as discipline analysis is referred to as sub process analysis In this paper the opti mization problems for the mineral processing system are defi ned by dividing the process into sub processes similar to the physical sub processes in mineral processing plants Each sub process in the plant is depicted as a discipline problem thus forming a hierarchical chain of sub optimization problems belonging to series of sub processes The notations used to represent general optimization architectures and op timization problems are presented in Table 2 Martins and Lambe 2013 The general descriptions and algorithms of the two archi tectures MDF and IDF are described in the following sections Table 1 Optimization problem scope for the simulation study is shown in highlight Development Stage DesignOperationsControl Application Area Equipment Sub Process X X X X Main Process K Bhadani et al Minerals Engineering 128 2018 27 35 28 3 1 Multi discipline feasible MDF Fig 1 represents the layout for the organization of the MDF archi tecture for two sub processes The representation is adopted based on Cramer et al 1994 and Martins and Lambe 2013 and is being translated to suit mineral processing systems This MDF formulation is monolithic in nature as it contains a single level optimization problem The system optimizer solves the optimi zation problem for a given set of design variables x and communicates with the sub process analyses The sub process analyses are sequentially embedded for the calcu lation of objective functions and constraints for the system optimization problem The objective function consists of two sets of functions i e function fo which is shared between the sub processes and function fi that represents individual sub process i Similarly the problem defi nition contains two sets of constraints i e constraint co which is shared between the sub processes and constraint ci that represents individual sub process i The algorithm for calculation of optimum values of the design variables x and function values f for mono lithic MDF architecture is presented in the following section Algorithm Input Design variable x Output Coupling variable y Optimised variable x objective function f 0 Initiate MDF loop Repeat 1 Evaluate sub process 1 and update y1 2 Evaluate sub process 2 and update y2 Until MDF converged 3 2 Individual discipline feasible IDF The IDF formulation presented is a distributed architecture and contains two levels of optimization problems system optimization and individual sub process optimization Martins and Lambe 2013 Fig 2 represents a generalized layout for the IDF formulation for two sub processes This formulation is similar to collaborative optimiza tion as used in other literature Braun et al 1996 Li et al 2008 Lin 2004 In the IDF formulation the system optimizer iterates the system optimization problem to compute the optimum design variable points x by solving the individual sub process optimizations in parallel The system optimization problem contains objective function fo and con straint co which are shared between sub processes and an additional consistency constraint cc is introduced to maintain the consistencies of the design variables Each individual sub process optimization pro blem consists of objective function fi and constraint ci belonging to the particular sub process i The individual sub process optimization receives independent copies of the variables x belonging to the other sub process through the system optimizer The sub process optimizer delivers a local optimal value for design variable xi and function value fi to the system optimizer The algorithm for calculation of optimum values of design variable and function values for distributed IDF architecture is shown below Algorithm Input Design variable x Output Optimized variable x objective function f 0 Initiate system optimizer iteration Repeat 1 Compute sub process objective and constraints For each sub process i do Initiate sub process optimization Repeat 1 1 Evaluate sub process i 1 2 Compute sub process i objective and constraints 1 3 Compute new design point for sub process i 1 Until 1 3 Optimization i has converged End for 2 Compute new system design points Until 2 System optimization has converged Table 2 Notation for MDO defi nition Symbol Defi nition xVector of design variables yVector of coupling variable or output from sub process analysis fObjective function cVector of design constraints ccVector of consistency constraints NNumber of sub processes 0Functions or variables shared between more than one sub process iFunctions or variables applied only to a sub process i Independent copies of a variable distributed to other sub process 0Functions or variables at their initial values Functions or variables at their optimal values System Optimizer Sub Process 1 Sub Process 2 1 y 0 21 xy 0 0 12 xx o x 2 y xf 1 min 0 0 N oiii i o iii fx yf x y w r tx y s t cx y c x y Fig 1 MDF architecture for two sub processes in a mineral processing system System Optimizer Sub Process 1Sub Process 2 0 2 1 xx 111 xyf 2 111 2 11 2 111 min 0 f x xy w r tx xy s t c x xy 1 222 1 22 1 222 min 0 fxxy w r txxy s t cxxy 222 xyf 0 1 2 xx min 0 0 o o c oiiii fx y w r tx y s t cx y cx x y y o x xf Fig 2 IDF for two sub processes in a mineral processing system K Bhadani et al Minerals Engineering 128 2018 27 35 29 4 Simulation study description In order to demonstrate the application of MDO architectures a reference two stage comminution plant for aggregate production was modelled Fig 3 represents the layout of an aggregate production plant consisting of a secondary crushing process and a tertiary crushing process The plant layout is inspired by other research Asbj rnsson et al 2012 Hulth n et al 2012 Hulth n and Evertsson 2011 The material from the primary crusher has a particle size between 0 and 130mm and is fed into the secondary crushing process The sec ondary crushing process consists of a GP300 crusher and a two deck screen producing three outputs The crushing behaviour of the sec ondary crusher is primarily controlled by one operating variable the closed side setting CSS1 The two deck screen has an upper screen aperture size SA1 as a variable and a lower aperture size as a fi xed value 35 mm This sub process produces two downstream products P1 0 32mm and P2 32 SA1 mm as well as one upstream feedback product P3 Above SA1 mm The product size P2 is fed to the tertiary crushing process and the product above SA1 mm is re circulated back to the secondary crusher The tertiary crushing process consists of an HP4 crusher coupled with two sequential three deck screens producing six products P4 P9 and one recirculating product P10 The manipulated variable in this sub process is the closed side setting CSS2 of the tertiary crusher All other design and process values such as material feed to the secondary crusher the crusher speeds the conveyor and screen operations are set to fi xed values The list of process design variables product outputs and their notations are presented in Table 3 4 1 Optimization problem formulation Optimization problem formulations for comminution and classifi cation processes can be based on numerous approaches such as mini mizing cost Svedensten and Evertsson 2005 maximizing quality Bengtsson et al 20