A MILP model for the design of mineral flotation circuits

A MILP model for the design of mineral flotation circuits
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  A MILP model for the design of mineral flotation circuits Luis A. Cisternas a, *, Edelmira D. Ga´lvez  b , Marı´a F. Zavala  b , Julio Magna  b a  Chemical Engineering Department, Universidad de Antofagasta, Casilla, Antofagasta 170, Chile  b  Department of Metallurgical Engineering, Universidad Cato´ lica del Norte, Antofagasta, Chile Received 4 April 2003; received in revised form 10 September 2003; accepted 6 October 2003 Abstract This paper develops a procedure for the design or improvement of mineral flotation circuits, based on a mathematical programming model with disjunctive equations. The model developed corresponds to a mixed-integer linear programming(MILP), avoiding nonlinear expressions.The procedure is characterized by: (1) The development of two hierarchized superstructures, such that the first levelrepresents processing systems, which must satisfy certain tasks. The second level represents circuits of equipment needed tocarry out the tasks of each system. (2) The superstructures are modeled mathematically including the selection of equipment,mass balance, and operational conditions. (3) The objective function is the maximization of profits.With the help of examples that are included to demonstrate the advantages of the procedure, it is shown that the method can be useful in deciding the configuration of the flotation circuits and the operational conditions. D  2003 Elsevier B.V. All rights reserved.  Keywords:  process design; flotation circuits; mathematical programming; mineral processing 1. Introduction Mineral flotation processes consist of several unitsthat are grouped into banks and interconnected in a predefined manner in order to divide the feed intoconcentrate and tailing. The behavior of these pro-cesses depends on the configuration of the circuit andthe physical and chemical nature of the slurry treated.Many of the complex circuits used in the industry arethe results of attempts to find more efficient methodsfor treating minerals so that the recovery of theminerals is maintained at a maximum while thedilution of the concentrate by gangue is maintainedat a minimum.The design of these circuits is carried out based onthe experience of the designer, with the help of laboratory tests and simulations. Some attempts have been described in the literature on automated methodsfor the design of these types of circuits (Yingling,1993; Mehrotra, 1988). However, methods for the design of flotation circuits have not yet progressedto the stage where an optimum circuit configurationcan be completely derived automatically. Even thesimpler situation of the optimum operation of a circuit of fixed configuration has not yet been solved com- pletely (Hulbert, 2001). Most of the studies have used a superstructure that contains numerous alternativesfor the configuration of the plant. Optimal configu- 0301-7516/$ - see front matter   D  2003 Elsevier B.V. All rights reserved.doi:10.1016/j.minpro.2003.10.001* Corresponding author.  E-mail address: (L.A. Cisternas) J. Miner. Process. 74 (2004) 121–131  rations and operational specifications are determined by the use of optimization techniques. Continuousvariables are utilized for representing the designspecifications in a stationary state, for example, theflow level, retention time of the flotation cells, etc.The fixed variables are used to represent the config-uration alternatives for the circuit, for example, exis-tence or not of banks of cells, and the separation of flows.Studies carried out to date can be classified asthose that consider operational cost and capital as anobjective function (Shena et al., 1996) and those that consider a technical performance function (re-covery, purity of the concentrate, etc.). The latter group may be separated into those that use bank models and those that use cell models (Yingling,1990). Two groups may also be distinguished be-tween those that use bank models: (a) those that usea first-order model for the flotation kinetics (Mehro-tra and Kapur, 1974; Dey et al., 1989), and (b) those that use a model indirectly, as in the studies of Green (1984, 1992) and that of  Reuter and Van Deventer (1990) and Reuter et al. (1988). Due to the nonlinear nature (bilinearities) of the equations of material balance, the models are required to includesimplifications in their mathematical form, or requirethe development of a specific strategy for thesolution of a mathematical model, which does not,however, guarantee having the found the globaloptimal solution.The present study presents the formulation of a procedure for the conceptual design or improvement of flotation circuits for minerals, characterized by:  Superstructures are developed in a hierarchicalform, so that at the first level, processing systemsare presented which must carry out rougher,cleaner, and scavenger tasks. At a second level,equipment circuits are presented for the completionof each task.  A mathematical model for the superstructures isdeveloped, which includes mass balance, opera-tional conditions, logic expressions, and disjunc-tive expressions for the selection of the equip-ment. The resultant model represents a MILPmodel.  The objective function represents maximization of  profit. 2. Model development 2.1. Strategy The design strategy includes two levels of hierar-chized superstructures. The upper level includes a task superstructure, in which a rougher system (the task of which is feed processing to obtain the maximumseparation), a cleaner system (the task of which is purification of the concentrate from the rougher and/or scavenger to obtain the final concentrate), and ascavenger system (the task of which is the treatment of the tailings from the rougher and/or cleaner toobtain the final tailings) are included. Fig. 1 showsthe superstructure utilized, where the triangles repre-sent mixers or stream division that permit the presen-tation of a group of alternatives for mineral processingupon which the search can be made for the best alternative. At the second level, it is considered that each system is formed of three banks of flotation cells,which will be termed the rougher, cleaner, and scav-enger banks. Thus, for example, in the scavenger system, scavenger–rougher, scavenger–cleaner, andscavenger–scavenger banks exist. On the whole,when it is considered that each system in the task superstructure contains three flotation banks, theunion of these superstructures allows for nine flotation banks. The superstructure for each system is analo-gous to the task superstructure, but where each systemis replace by a bank of flotation cells. This analogymakes easy the mathematical representation. Also thesuperstructures representation avoids the presence of symmetrical structures avoiding double counting andreducing the number of flowsheet configurations. Inaddition, some streams (i.e., streams between cleaner and scavenger subsystems) can be eliminated as it isneeded by the designer.The problem can then be defined as, given itstechnical characteristics, costs, prices, and the feedmass flow find the topology or structure of the processcircuits (by searching through a group of hierarchizedsuperstructures) as well as the operational conditionsthat maximize the profits of the plant.The strategy to be utilized includes establishing the previously mentioned superstructures, and then gen-erating a mathematical model for the superstructure.The mathematical solution to the problem shoulddeliver mass flows (zero mass flows represent the  L.A. Cisternas et al. / Int. J. Miner. Process. 74 (2004) 121–131 122  non-existence of streams), equipment selection andsize, and the configuration of the plant. 2.2. Restrictions First, it is necessary to define some sets, parame-ters, and variables in order to develop the model.Then, using these definitions, the equations may bedeveloped, which include mass balances, yields, andoperational conditions.The principal sets are:  S  ={  s /   s  is a system},  L ={ l  /  l   isa stream},  K  ={ k  /  k   is a species} and  M  ={ m /  m  is amixer or spliter}. LA, LC and LT are a subset of   L ,which include the feeds, concentrates and tailings,respectively, of each system or bank of cells.Also, the following sets are necessary:  M  in ð m Þ¼f l  = l  a  L ;  l   is an input stream to  m ; m a  M  g  M  out  ð m Þ ¼ f l  = l  a  L ;  l   is an output stream from m ; m a  M  g Lcc  ¼ fð l  a ; l  c Þ = l  a a LA ;  l  c a  LC  ;  l  c  is theconcentrate produced from  l  a g Since the superstructures are analogous, the sets  L ,  M  , LA, LC, LT,  M  in ,  M  out  , Lcc, and LT are the samefor each superstructure. LS ¼ ( ð l  1 ; l  2 Þ = l  1 ; l  2 a  L ; l  1 ;  is thestream of the general superstructure whichisconnectedwith  l  2  of system  s ) Also LL, LS1, and LS2 are subsets of   L , whichrepresent the feed, output 1, and output 2 of eachstream divider. Each stream  l   of the general super-structure is associated with the variable that representsthe mass flow of species  k  ,  W  l  , k  . Similarly, each stream l   in each system  s  is associated with the variable that represents the mass flow of the species  k  , WI  s , l  , k  . 2.2.1. Material balances in the general superstructure In the mixers and flow dividers, there are: X 1 a  M  in ð m Þ W  l  ; k    X l  a  M  out  ð m Þ W  1 ; k   ¼  0  k  a  K  ;  m a  M   ð 1 Þ where  M  in ( m ) and  M  out  ( m ) are the sets of input andoutput streams of the mixer/divider   m . In general, it isnot common practice to divide the streams in flotationconcentration plants. This study, however, considersthe possible division of different levels into a set of discrete values for possible levels of division. This Fig. 1. Task superstructure with rougher, cleaner, and scavenger systems. Each system has the same superstructure but with rougher, cleaner, andscavenger banks. LT  ¼ fð l  a ; l  t  Þ = l  a a LA ;  l  t  a  LT  t  ;  l  t   is the tailing produced from  l  a g  L.A. Cisternas et al. / Int. J. Miner. Process. 74 (2004) 121–131  123  representation permits avoiding bilinearities in themass balances of the separators.  j   ˇ a  J  y  sl  ;  j  W  l  1 ; k   ¼ g  j  W  l  ; k  W  l  2 ; k   ¼ð 1  g  j  Þ W  l  ; k  266664377775  j  a  J  ;  k  a  Kl  a LL ; l  1 a LS1 ;  l  2 a LS2  ð 2 Þ where  J  ={  j  /   j  a  J   is a fraction level of division}, and  l   isthe flow which is separated into  l  1  and  l  2 . Parameter   g   j  is the fraction  j   of division. Eq. (2) may be describedas: W  l  1 ; k  V g  j  W  l  ; k   þ U  l  ; k  ð 1   y  sl  ;  j  Þ W  l  2 ; k  V ð 1  g  j  Þ W  l  ; k   þ U  l  ; k  ð 1   y  sl  ;  j  Þ W  l  ; k   ¼ W  l  1 ; k   þ W  l  2 ; k  X  j   y  sl  ;  j   ¼ 1  j  a  J  ; k  a  K l  a LL ; l  1 a LS1 ; l  2 a LS2 ð 3 Þ where  U  l  , k   is the upper level of   W  l  , k   and  y l  ,   j  s is a binaryvariable that represents the selection of the level of division.The assignment of the feedstock flows for eachspecies  k   to the mass flow of feed (stream 1) can berepresented by W  1 ; k   ¼  F  k   ð 4 Þ where  F  k   represents the mass flows fed with species  k  . 2.2.2. Material balances in the systems Equations similar to Eqs. (1)–(3) hold for themixers and separators in each system, as is X l  a  M  in ð m Þ WI  s ; l  ; k    X l  a  M  out  ð m Þ WI  s ; lk   ¼ 0 k  a  K  ;  m a  M  ;  s a S   ð 5 Þ  j   ˇ a  J  y e s ; l  ;  j  WI  s ; 11 ; k   ¼ g  j   WI  s ; 1  ; k  WI  s ; 12 ; k   ¼ð 1  g  j  Þ  WI  s ; 1 ; k  266664377775  j  a  J  ;  k  a  K  ;  s a S  ; l  a LL ; l  1 a LS1 ;  l  2 a LS2 ð 6 Þ Eq. (6) may be described as:WI  s ; 1 1 ; k  V g  j  WI  s ; 1 ; k   þ U   I  s ; l  ; k  ð 1   y e s ; l  ;  j  Þ  j  a  J  ;  k  a  K  ;  s a S  ; WI  s ; 1 2 ; k  V ð 1  g  j  Þ WI  s ; 1 ; k   þ U   I  s ; l  ; k  ð 1   y e s ; l  ;  j  Þ l  a LL ;  l  1 a  LS  1WI  s ; 1 ; k   ¼ WI  s ; 1 1 ; k   þ WI  s ; 1 2 ; k   l  2 a LS2 X  j   y e s ; l  ;  j   ¼ 1  ð 7 Þ where  U   s,l,k  I  is the upper level of WI  s,l,k   and  y  s,l,j e is a binary variable that represents the selection of thelevel of division.The mass balance equations in each flotation stepareWI  s ; l  c ; k   ¼ T   s ; l  a ; k  WI  s ; l  a ; k   ð l  a ; l  c Þ a Lcc ;  k  a  K  ;  s a S  ð 7 Þ WI  s ; l  t  ; k   ¼  1  T   s ; l  a ; k    WI  s ; l  a ; k  ð l  a ; l  t  Þ a LT ;  k  a  K  ;  s a S   ð 8 Þ where  T   s,l  a   , k   is the ratio of flow of concentrate  l  c  andfeed  l  a  , of species  k  , in system  s . The ratio  T   s,l  a   , k   isrelated to the separation factor   f    s,l  a   , k   (Lynch et al.,1981) by the following equation: T   s ; l  a ; k   ¼  f    s ; l  a ; k  1 þ  f    s ; l  a ; k  ð 9 Þ The separation factor may be obtained from plant data, values from pilot plants, or theoretical or empir-ical models. For example,  f    s ; l  a ; k   ¼  1 þ k   s ; l  a ; k  s  s ; l  a    N   s ; l  a  1  ð 10 Þ where  k   s,l  a   , k   is the flotation rate for species  k  ,  N   s,l  a  thenumber of cells and  s s, l  a  the retention time in the bank fed by  l  a   in system  s . Multiple values of   T   s,l  a   , k   may beimplemented as will be seen below.  L.A. Cisternas et al. / Int. J. Miner. Process. 74 (2004) 121–131 124  The input and output streams of each system areconnected to stream of the task superstructure, thus W  l  1 ; k   ¼  WI  s ; l  2 ; k   ð l  1 ; l  2 Þ a LS ;  k  a  K  ;  s a S   ð 11 Þ in order to include the fixed costs of the flotationequipment and equipment for pumping the concen-trate, a binary variable is introduced,  y  s,l  , whichindicates the existence of stream  l   in system  s . Theequations are: X k  WI  s ; l  ; k     y  s ; l  U  T  s ; l  V 0  l  a  LA ;  k  a  K  ;  s a S   ð 12 Þ X k  WI  s ; l  ; k     y  s ; l  U  T  s ; l  V 0  l  a  LC  ;  k  a  K  ;  s a S   ð 13 Þ In Eqs. (12) and (13),  U   s , l T  is the upper bound of total mass flow. We used the parameter   U   in variousequations as the upper bound for the activation of  binary variables. It is important to note that the valuesfor   U   are not equivalent, and in contrast, need to becarefully defined for the user, taking into consider-ation that they are the smallest upper bounds that can be defined.Operational conditions, such as streams with zeroflow, and upper or lower bounds for flows of eachstream are easily included in the model. It is also possible to include logic expressions when these arerequired. For example, if the stream 12 (streams be-tween cleaner and scavenger) wants to be excluded,then stream 12 should not be included in set   L  or upper and lower bounds of mass flow rates of stream 12 haveto be zero. If the designer wants to include the possi- bility of selection of cleaner or scavenger banks or either of then (but not both) in the cleaner system, thenthe following logic expression can be added  y c,7 +  y c,15 V 1(see Biegler et al., 1997 for more information). 2.3. Objective function The optimal selection of the circuit requires that an appropriate objective function be defined uponwhich the values of the operational and structuralvariables may be determined. Since in the present case, the income depends on the structure andoperational conditions, a useful function is the dif-ference between income and costs. This can beobserved as a multi-objective function, in whichincome is maximized and the costs are minimized.The objective function to be developed is general-ized and may be adapted to other situations. For example, in plants where amortization of capital hasalready been achieved, the objective function maysimply be the maximization of income minus theoperational costs, or in plants where the mineralcontent of the concentrate and production levels areknown, total costs may be minimized.The generalized objective function is:Profit   ¼  Income    Costs  ¼  I     C   ð 14 Þ Different relations may be applied for calculationof the income depending on the type of product and itsmarket. For base metals, the formula net-smelter-return may be utilized (Shena et al., 1996).  I   ¼ X k  W  10 ; k  !  p ð  g  conc    u Þð q    Rfc Þ "  X k  W  10 ; k  ! Trc #  H   ð 15 Þ where P k   W  10 ; k   is the mass flow of the concentrate,  p the fraction of metal paid,  g  conc  ¼ P k   g  k  W  10 ; k   is themineral grade of the concentrate,  u  is the gradededuction, Trc is the treatment charge, and Rfc isthe refinery charge.  H   is the number of hours per year of plant operation, when the flows are in tons per hour. The grade deduction and the fraction of metal paid depend on the recovery efficiency of the smelter.The formula for the calculation of income incorpo-rates the metallurgical efficiency of the plant, that is,the recovery and mineral content are opposite func-tions. Values of cost and prices of metals are pub-lished in specialized journals. Typical values used inthe present study are listed in Table 1. Eq. (15) may berewritten in the following manner:  I   ¼ X k   g  k  W  10 ; k  !  p ð q    Rfc Þ  H   X k  W  10 ; k   pu ð q    Rfc Þ þ  Trc ½   H   ð 16 Þ  L.A. Cisternas et al. / Int. J. Miner. Process. 74 (2004) 121–131  125
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