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Chemical Engineering Science 59 (2004)
385396www.elsevier.com/locate/ces
Multiobjective optimization of an industrial grinding operation
usingelitist nondominated sorting genetic algorithm
Kishalay Mitraa ;, Ravi Gopinathb
aManufacturing Practice, Tata Consultancy Services, 54 B
Hadapsar Industrial Estate, Pune 411 013, IndiabManufacturing
Practice, Tata Consultancy Services, Air India Building 11th Floor,
Nariman Point, Mumbai 400 021, India
Received 6 May 2003; received in revised form 15 July 2003;
accepted 10 September 2003
Abstract
The elitist version of nondominated sorting genetic algorithm
(NSGA II) has been adapted to optimize the industrial grinding
operationof a lead-zinc ore bene5ciation plant. Two objective
functions have been identi5ed in this study: (i) throughput of the
grinding operationis maximized to maximize productivity and (ii)
percent passing of one of the most important size fractions is
maximized to ensure smooth8otation operation following the grinding
circuit. Simultaneously, it is also ensured that the grinding
product meets all other qualityrequirements, to ensure least
possible disturbance in the following 8otation circuit, by keeping
two other size classes and percent solid ofthe grinding product and
recirculation load of the grinding circuit within the user speci5ed
bounds (constraints). Three decision variablesused in this study
are the solid ore 8owrate and two water 8owrates at two sumps,
primary and secondary, each of them present in eachof the two stage
classi5cation units. Nondominating (equally competitive) optimal
solutions (Pareto sets) have been found out due tocon8icting
requirements between the two objectives without violating any of
the constraints considered for this problem. Constraints arehandled
using a technique based on tournament selection operator of genetic
algorithm which makes the process get rid of arbitrary
tuningrequirement of penalty parameters appearing in the popular
penalty function based approaches for handling constraints. One of
the Paretopoints, along with some more higher level information,
can be used as set points for the previously mentioned two
objectives for optimalcontrol of the grinding circuit.
Implementation of the proposed technology shows huge industrial
bene5ts.? 2003 Elsevier Ltd. All rights reserved.
Keywords: Dynamic simulation; Mathematical modeling;
Multiobjective optimization; Genetic algorithm; Pareto set
1. Introduction
Grinding is one of the very important unit operations inmost of
the mineral processing plants. Since grinding is avery energy
intensive process, modeling and thereby opti-mization of grinding
operation of industrial scale has been acontinuous endeavor of the
scientists and engineers. Thoughmodeling of grinding circuit has
attained a reasonable stateof robustness (Herbst and Fuerstenau,
1973; Lynch and Rao,1975; Herbst et al., 1983; Kinneberg and
Herbst, 1984;Rajamani and Herbst, 1984, 1991a), work on several
as-pects of optimization and control is yet to reach the
similarstate of maturity (Lapidus and Luss, 1967; Bryson and
Ho,1969; Birch, 1972; Herbst and Rajamani, 1979; Rajamaniand
Herbst, 1991b). Recently, a successful implementationof modeling
and control of an industrial leadzinc grinding
Corresponding author. Tel.: +91-20-4042468; fax:
+91-20-4042399.E-mail address: [email protected] (K.
Mitra).
operation has been executed by the group of the authors.The
proposed control strategy is based on model predictivecontrol (MPC)
in conjunction with on-line solution of theprocess models for
grinding operations (Momaya et al.,2003). The on-line models used
in the grinding circuits pro-vide continuous feedback for better
quality control of thegrinding circuit outputs. This is an
important component ofthe overall solution given the lack of
on-line sensors formeasurement of quality parameters. The problems
posed bythe processes make an MPC application a suitable
candidate.MPC solves an explicit optimization problem of
maximiz-ing throughput of the circuit keeping other key
performanceindices (KPIs) within user speci5ed bounds. Other
thancircuit throughput, two other important KPIs are recircula-tion
load of the circuit and percentage passing of midsizefraction.
Characteristically, with increase in throughput,both the
recirculation load and percentage passing of mid-size fraction
decreases. The above-mentioned throughputvs. recirculation load
relationship is desired since reduction
0009-2509/$ - see front matter ? 2003 Elsevier Ltd. All rights
reserved.doi:10.1016/j.ces.2003.09.036
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386 K. Mitra, R. Gopinath / Chemical Engineering Science 59
(2004) 385396
in recirculation load saves energy. But the relationship
ofthroughput vs. percentage passing of midsize fraction is
notdesired since that is detrimental for the consistent
qualityproduct 8owing to the following 8otation operation.
Max-imization of these two contradictory aspects frames theplatform
for multiobjective optimization problem (MOOP).In the 5eld of
traditional optimization, the less robust Pon-
tryagin principle is often recommended for handlingMOOPsand
enough literature is found on application of the same(Chankong and
Haimes, 1983; Ray, 1989). Unfortunately,this kind of traditional
optimization technique requires anexcellent initial guess of the
optimal solutions, and the re-sults and the rate of convergence of
the solution are verysensitive to these guesses. For complex
systems, quite of-ten, the search space becomes very narrow and one
has toprovide the initial guess within that narrow region
whichmeans one must almost know the optimal solution that oneis
trying to obtain. In recent years, an extremely robust tech-nique,
genetic algorithm (GA) (Holland, 1965; Goldberg,1989; Deb, 1995,
2001a), and its adaptations for more use-ful but complex
multiobjective optimization problems, havebecome popular. Unlike
conventional methods, these meth-ods work with a bunch of initial
guesses, called populationand generally have the capability of
5nding the global op-timum in presence of several local optima.
Simultaneouslythis algorithm is superior to traditional
optimization algo-rithms in many aspects (Deb 2001a; Bhaskar et
al., 2000b;Nandasana et al., 2003).In case of multiobjective
optimization, instead of obtain-
ing a unique optimal solution, a set of equally good
(non-dominating) optimal solutions is usually obtained
(Paretosets). More precisely, within a Pareto set, when one
movesfrom any one point to another, one objective function
im-proves while the other deteriorates. In absence of any otherhigh
level additional information, a decision maker nor-mally cannot
choose any one of these nondominant optimalsolutions since all of
them are equally competitive andnone of them can dominate each
other. In case of tradi-tional algorithms, multiobjective
optimization problems areusually solved using a single objective
function, which is aweighted-average of the several objectives.
Unfortunately,the solution obtained by this process depends largely
onthe values assigned to the weighting factors used. Thisapproach
does not provide a dense spread of the Paretopoints. Among several
methods (goal attainment method,j-constraint method, versions of
nondominated sortinggenetic algorithm) available to solve
multiobjective opti-mization problems, nondominated sorting genetic
algorithmII (NSGA II), developed by Deb et al. (Deb, 2001a,b; Debet
al., 2002), is used here to obtain the Pareto set. Use ofpenalty
function is a very popular way of handling con-straints. But tuning
of the penalty parameter appearing inthe penalty function is very
time consuming and normallyperformed on the basis of trial and
error. Unless tunedproperly, one may get misdirected totally in the
searchspace. NSGA II, along with a tournament selection based
constraint-handling technique, developed by Deb (2000),allows
one to get rid of the above stated problem of
penaltyfunction.Nondominated sorting based techniques have
several
advantages over other techniques: (a) the spread of thePareto
set is excellent, (b) a single simulation run canobtain the entire
Pareto set and (c) can handle problemswith discrete search spaces.
NSGA based techniques havebeen used to solve a wide variety of
multiobjective op-timization problems in chemical engineering in
recentyears, as for example, an industrial nylon-6 semibatchreactor
(Mitra et al., 1998), a wiped-5lm polyester re-actor (Bhaskar et
al., 2000a), a steam reformer (Rajeshet al., 2000), a hydrogen
plant (Rajesh et al., 2001), a ven-turi scrubber (Ravi et al.,
2002), a FCC unit (Kasat et al.,2002) etc.The industrial process
handled here is a two step process.
Pulverization of the ore to 5nely ground particles in
wetgrinding mills is done in order to liberate the valuables,namely
lead and zinc from its associated gangue. The groundparticles are
then selectively 8oated in 8otation cells forindividual recovery of
Lead and Zinc. The grinding circuithas
one rod mill in open circuit operation, one ball mill in closed
circuit operation and a two-stage classi5cation unit (one
hydrocyclone for pri-mary classi5cation and two hydrocyclones in
parallel forthe secondary classi5cation).
The ore from the mine is crushed in the crushing unit andis sent
to storage 5ne ore bin. Fresh ore feed from the 5neore bin along
with water is fed to the rod mill. The rod milldischarge slurry is
mixed with the ball mill discharge slurryin a sump known as the
primary sump. Water is added to theprimary sump to reduce the pulp
density. The slurry fromthe primary sump is fed to primary cyclone,
the primaryclassi5cation unit. The over8ow from the primary
cyclonegoes to secondary sump, where water is added to lower
pulpdensity further. The mixed slurry from the secondary sumpis fed
to secondary cyclones. The under8ow product fromboth primary and
secondary cyclones is fed to the ball mill.The over8ow from the
secondary cyclone is the 5nal productfrom the grinding circuit and
goes to 8otation circuit as feed(see Fig. 1).
2. Formulation
The grinding circuit is modeled 5rst. Each of the
unitoperations, in this case they are rodmill, ballmill,
hydrocy-clones and sumps, are modeled separately and a
connectivitymatrix, expressed in terms of 1 and 0, connects all of
themto simulate the whole circuit. In case of mill and sump
op-erations, population-balance equations for solids and
waterstreams are considered. Each of the size class is tracked
by
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K. Mitra, R. Gopinath / Chemical Engineering Science 59 (2004)
385396 387
Rod Mill
Ball Mill
Water
Ore Feed
Secondary Sump
Primary Sump
Water
C1 C2 Final Product
Water
Fig. 1. Schematic diagram of industrial grinding circuit.
considering population balance equation for each of them.Once
solids and water streams are predicted separately,prediction of
slurry properties of the combined stream, vol-umetric 8owrate,
percentage solids etc. becomes relativelyeasy. Selection functions
used in milling operations for dif-ferent size classes are to
follow the nth order kinetics wheren needs to be determined from
the industrial data. Mill oper-ations are found to behave like
over8owing sumps whereassump levels are to be found out on the
basis of characteristicsof the pump kept at sump outlets.
Hydrocyclones are mod-eled using the empirical equations provided
by Lynch andRao (1975). All these equations of the dynamic modeling
ofgrinding circuit developed by the authors are
consolidatedtogether and presented in Appendix A. This set of
ordinarydiIerential equations is solved using well-tested public
do-main software, called DASSL (Petzold, 1983). This codesolves a
system of diIerential/algebraic equations of theform using a
combination of backward diIerentiation for-mula (BDF) methods and a
choice of two linear system so-lution methods: direct (dense or
band) or Krylov (iterative).The model is tuned with the data
collected from the grind-
ing circuit under steady state conditions. This process isan
optimization exercise in itself where parameters appear-ing in
various unit models become the decision variablesand the objective
function is the error between the indus-trial data and model
predicted values at various locations ofthe grinding circuit where
samples are taken under steadystate conditions. Other mass balance
equations are treatedas constraints. This exercise provides the
parameters of thegrinding circuit that helps the model to behave
like the clos-est possible mimic of the industrial grinding
circuit. Grind-ability, selection indices for ballmill and rodmill
(A and in (A.40) in Appendix A) and nine parameters appearingin the
hydrocyclone d50 and bypass fraction equation (K1to K9 (A.27) and
(A.29) in Appendix A) are the parame-ters to be found out by this
parameter estimation procedure.If observed correctly, the input to
the grinding circuit thatcan be changed online are solid ore
8owrate (MRM ), wa-ter 8owrates to primary (WPS) and secondary
(WSS) sumps.
Other than this, there are several other locations where
5xedamount of water is added to maintain smooth 8ow of slurrywithin
the circuit. These points are ballmill input (WBMF) aswell as
discharge (WBMD) and rodmill input (WRMF) as wellas discharge
(WRMD). These latter 8owrates are never ma-nipulated whereas the
former ones are manipulated to meetthe grinding product quality.
The properties of the productthat draws attention of the grinding
circuit performance arethroughput (TP), size analysis (percentage
passing of threesize classes namely coarse (PC), mid (PM ) and 5ne
(PF)size classes), percentage of solids (PS). Another very
im-portant parameter that is to be kept within speci5ed boundis
recirculation load (RL). The model can predict all slurryproperties
in any of the streams of the circuit given in Fig. 1in steady state
as well as dynamic mode.The complete MOOP solved is, thus, written
as follows:
MaxI [I1; I2]T
I1 = TP
I2 = PM
Subject to (s.t.):
PLC6PC6PUC
PLF6PF6PUF ;
PLS6PS6PUS ;
RLL6RL6RUL :
All equations (Appendix A)
Decision variable bounds
MLRM 6MRM 6MURM ;
WLPS6WPS6WUPS ;
WLSS6WSS6WUSS ;
where superscript L and U denote the lower and upperbounds of
the above mentioned grinding circuit variables.
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388 K. Mitra, R. Gopinath / Chemical Engineering Science 59
(2004) 385396
The two objective functions used here are con8icting in na-ture
and so it is likely that a Pareto set of nondominatingoptimal
solutions is obtained. The binary version of non-dominated sorting
genetic algorithm II (NSGA II) whichis an adaptation of the simple
genetic algorithm suited formultiobjective optimization problems,
is used to solve theproblem de5ned above. Details of this method
are availablein the literature (Deb, 2001a,b).
3. Results and discussions
The model is 5rst tested for its validity. As the
decisionvariables considered here are three and each one of themhas
upper and lower bounds, all possible combinations (23)among these
three decision variable bounds are considered.Now for these eight
operating conditions, data are collectedfrom industry and
simulations are also run. The comparisonbetween the simulations and
data for all these eight casesis presented in Fig. 2. This shows
the validity as well asthe sensitivity of the model over the whole
operating regionconsidered. The cases considered are denoted as the
boundsfor the decision variables used (e.g. case (a) L-L-H meansthe
simulation of lower bounds for decision variable 1 and2 and upper
bound on decision variable 3). The eight casesconsidered are: (a)
L-L-H (b) L-H-L (c) L-L-L (d) L-H-H(e) H-L-H (f) H-H-L (g) H-L-L
(h) H-H-H.The Pareto optimal points were obtained for the prob-
lem considered. The spread obtained in the front was verydense.
The 5nal Pareto front was quite diIerent from the bestnondominated
front obtained in the initial population. Thismeans better fronts
were evolved as generations progressed.The elitist approach of
considering both parent and childpopulation for selecting the
better candidates for the matingpool was found to work very well.
This was supported bythe fact that subsequent generations didnot
allow better 5t-ted chromosomes to get lost in the crowd due to the
random-ness involved in the selection operation of GA.
Crowdingmetric helped to maintain diversity in the Pareto
front.Starting with some randomly selected candidate solutions
in the zeroth generation, NSGA II was observed to takeabout
twenty-5ve generations to converge to the Pareto set.After
convergence, it was found to maintain the same pointsin the Pareto
fronts over any number of generations. Thenondominated fronts found
in the zero as well as generationnumber twenty 5ve, for normalized
objective functions, aregiven in Fig. 3. The number of distinct
Pareto points found isof the order of half of the number of
populations consideredin a generation as rest of the points are
basically multiplecopies of existing ones.After formation of Pareto
set, one can choose an oper-
ating point from this set using his or her intuition or
somehigher level information related to requirement of the
pro-cess. In the present case, the concerned industry used tocater
a varied set of needs of its clients. In case of someclients, the
quality was allowed to be compromised a little
bit, where maximization of throughput was given morepreference
(points towards higher values of throughputwere of more interest)
in comparison with some other cases,where there was a very strict
requirement on quality (pointstowards higher values of midsize
percentage passing wereof more interest). In this way, an operating
point is chosen,in the present case, among the extreme end points
on thePareto set depending on the requirement of the
existingclients.Another very important observation, from the point
of
view of development of new future MOOP techniques, is
theexisting trend among the decision variables for the
pointspresent in the 5nal Pareto front. Here, the decision
variablevalues for the 5nal Pareto points are almost at the
eitherend (higher end for WPS and lower end for WSS) of the
de-cision variable bounds. This is happening as we are maxi-mizing
the percentage passing of midsize that can be onlyachieved by
performing coarse grinding in the primary cy-clone i.e. by
increasing the primary sump water 8owrates.But we have other
constraints to be tackled at the secondarycyclone over8ow stream
(5nal product). The Pareto solu-tions meet those requirements by
performing 5ner grindingat the secondary cyclone i.e. by decreasing
the secondarysump water 8owrates. Till now, for most of the multi
objec-tive evolutionary algorithms (MOEAs) for MOOPs, actionsare
taken in the decision variable space and the results getre8ected in
the objective space. If the relationship betweenthese two can be
established, better algorithms using thesefeatures can result. Even
this kind of cause and eIect re-lationship that came out of the
MOOP study, can work asplant operating rules and help an operator
to run the grind-ing circuit in a more eIective manner.The initial
population is changed by changing the decision
variable bounds and the same Pareto is emerged from it.
Thespread of the Pareto was de5nitely diIerent in diIerent casesof
decision variable bounds used. The authors, therefore,claim to
obtain the global Pareto for the case presentedhere. The
formulation presented here was found superior tothe weight based
formulation of objectives, where numbersof objectives are added
using weights and a single objectiveoptimization problem (SOOP) is
solved and Pareto is gen-erated out of it by changing various
values of weights used.In case of weight based formulation, the
Pareto points arefound towards both the end of the global Pareto.
Thoughthe identi5cation of the global Pareto was achieved by
theweight-based methods, obtaining spread across the Paretofront
was severely compromised. The MOOP problem isalso solved by
converting the same into a SOOP (solvedusing simple GA), where one
of the objectives (percent-age passing of midsize in this case) is
treated as constraint.In this case also, the global Pareto is
identi5ed but thespread across the Pareto is sacri5ced (much better
than theweight based approach). Even 5nding the global Paretowas
found to be dependent on the length of the string usedfor
representing the decision variables. Adapting an incor-rect, though
hard to know the correct one apriori and has
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K. Mitra, R. Gopinath / Chemical Engineering Science 59 (2004)
385396 389
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Fig. 2. Validation of model within the range considered.
to be set by trial and error, string length leads to
prematureconvergence to a local Pareto. The superiority of NSGAII
over the other two techniques as mentioned above is fur-ther
strengthened by the fact that NSGA II 5nds the globalPareto in a
single simulation run, as compared to multiplesimulation runs in
the case of other approaches.
Each of the points in the Pareto set has diIerent sets
ofdecision variable values. As the throughput is
increased,maximization of the twin objectives is achieved by
increas-ing the primary sump water 8owrate to a relatively
highervalue and decreasing the secondary sump water 8owrateto a
relatively lower value. This recommends coarse
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390 K. Mitra, R. Gopinath / Chemical Engineering Science 59
(2004) 385396
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No
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dM
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0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.10.7
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0.8
0.85
0.9
0.95
1
1.05
1.1
Generation 0Generation 25
Fig. 3. Random points created by NSGA II in generation 0 and the
Paretopoints found in generation 25 and maintained through next
generations.
classi5cation in the primary hydrocyclone and 5ne classi-5cation
in the secondary hydrocyclone. Three points areidenti5ed on the
global Pareto front e.g. 5rst at the max-imum value of TP , second
at the minimum value of TPand the third at the in between (average)
the previous two.Circuit behavior (cumulative size classi5cation
plots) at allthe prime locations of the grinding circuit are
presented inFigs. 4 (maximum TP), 5 (minimum TP) and 6 (averageTP).
These locations are (a) rodmill discharge, (b) ballmilldischarge,
(c) primary cyclone over8ow, (d) secondarycyclone over8ow, (e)
primary cyclone under8ow, (f) sec-ondary cyclone under8ow, (g) mill
discharge sump and(h) secondary cyclone discharge sump. These
5gures showthat classi5cation becomes coarser as onemoves from
lowestTP to highest TP in the global Pareto set. Any kind ofdesired
results that happens to be in between these threecases (maximum TP
, minimum TP and average TP) canbe obtained from the other points
present in the globalPareto set. The biggest advantage of this kind
of MOOPformulation exercise is this provides a wide range on
feasi-ble solutions where from a user can choose the best one ofhis
or her choice.The NSGA II parameters used to obtain the Pareto
set
for the present study are as follows: Population size=80,number
of decision variables=3, string length for each de-cision
variables=20, maximum number of generations=50,crossover
probability=0.9 and mutation probability=0.001.The eIect of varying
several GA parameters on the Pareto
solution was studied next. If the number of population is
de-creased from 80 to 50, without changing any other parame-ters,
all the points generated randomly in the 5rst generationwere
coincidentally nondominating in nature. As the popu-lation number
decreases, convergence of initially generated
inferior fronts to Pareto set front was relatively faster
(re-quirement of less number of generations for convergence)but the
spread of the Pareto set obtained was relatively poor.Increase in
the value of string length does not provide betterfronts. The eIect
of varying the value of crossover proba-bility, from 0.9 to 0.7,
again led to no signi5cant changes inthe optimal solutions though
number of generations taken toconverge to the 5nal Pareto was more
(in case of 0.7) thanearlier (in case of 0.9). The eIect of a
change in the value ofmutation probability, from 0.001 to 0.01,
neither aIect thespread as well as number of points nor imparts any
improve-ment in the Pareto set. The variation of string length
alsodid not provide any better fronts, but if the string length
isdecreased, the number and thereby the spread of the Paretofronts
becomes relatively less dense. Once Pareto front isachieved, any
increase in maximum number of generationssuccessfully maintains the
diversity in the Pareto front fromgeneration to generation.This
kind of MOOP formulation can be put into MPC
framework to provide more meaningful solution for con-trolling
the process. The Pareto points can be used as a toolfor decision
making and providing meaningful setpoints tothe grinding control
problem solved by MPC (describedin the introduction). By solving
the MOOP formulation,a user can have multiple feasible solutions.
It is not to beforgotten that by solving SOOP formulation of MPC,
onecan have only one solution at a time. Even the quality
ofsolution supplied by MOOP formulation of MPC is betterthan that
of SOOP formulation of MPC. Many solutionsthat appear as
unattainable or infeasible (for SOOP for-mulation of MPC) become
feasible when MPC solves theMOOP formulation. A rule based decision
making system(based on process experience) can be put one layer
aboveMPC based (MOOP formulation) supervisory controller todecide
which solution (out of several solutions provided inthe global
Pareto front) to download to the process as setpoint.The proposed
MOOP formulation when coupled with
MPC framework and implemented in the plant conditions,the direct
bene5ts are more in terms of signi5cant im-provement in throughput
(an increase of the order of 25%)without compromising the product
quality much. There hasbeen a remarkable improvement in zinc
recovery (an in-crease of nearly 3% against an initial estimate of
1%) afterthe installation of the controller compared to the prior
state.It is important to note that this increase in recovery has
beenachieved without aIecting the product grade. This increasein
recovery impacted the return on investment hugely andbrought that
down from 1.5 years (calculated initially withan target improvement
of zinc recovery of 1%) to little lessthan half a year. The
indirect bene5ts of the same on theplant operating system are
listed below:
A sustained increase in current throughput levels,
besidesconsistently meeting grinding circuit quality require-ments
vis-Qa-vis product size distribution and percentage
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K. Mitra, R. Gopinath / Chemical Engineering Science 59 (2004)
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0 0.2 0.4 0.6 0.8 1 1.20
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0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(a) (b)
(c) (d)
(e) (f)
(g) (h)
Fig. 4. Cumulative distribution curve for normalized size for
various streams ((a) rodmill discharge, (b) ballmill discharge, (c)
primary cyclone over8ow,(d) secondary cyclone over8ow, (e) primary
cyclone under8ow, (f) secondary cyclone under8ow, (g) mill
discharge sump and (h) secondary cyclonedischarge sump) in the
grinding circuit for maximum throughput condition in Pareto
set.
solids ratio. This itself bears a strong positive in8uencein
appreciably increasing the overall productivity of theplant
including individual recoveries of Lead and Zinc.
The operating regime can be smoothly transferredfrom one
operating point to the other (manually
extremely diRcult and time consuming) without af-fecting any key
process variables. The controllersmoothly executes the step change
given to theprocess by keeping all other key process
variablesunchanged.
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392 K. Mitra, R. Gopinath / Chemical Engineering Science 59
(2004) 385396
Normalized Size
Cum
ulat
ive
Dist
ribut
ion
Normalized Size
Cum
ulat
ive
Dist
ribut
ion
Normalized Size
Cum
ulat
ive
Dist
ribut
ion
Normalized Size
Cum
ulat
ive
Dist
ribut
ion
Normalized Size
Cum
ulat
ive
Dist
ribut
ion
Normalized Size
Cum
ulat
ive
Dist
ribut
ion
Normalized Size
Cum
ulat
ive
Dist
ribut
ion
Normalized Size
Cum
ulat
ive
Dist
ribut
ion
0 0.2 0.4 0.6 0.8 1 1.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(a) (b)
(c) (d)
(e) (f)
(g) (h)
Fig. 5. Cumulative distribution curve for normalized size for
various streams ((a) rodmill discharge, (b) ballmill discharge, (c)
primary cyclone over8ow,(d) secondary cyclone over8ow, (e) primary
cyclone under8ow, (f) secondary cyclone under8ow, (g) mill
discharge sump (h) secondary cyclonedischarge sump) in the grinding
circuit for minimum throughput condition in Pareto set.
Reduction in the recirculation load that is an indirectmeasure
of energy consumption in the grinding circuitby 70%.
Variations in one of the key process variables canbe observed by
looking at the behavior of sump
levels with grinding controller putting ON and OFF.These sump
levels are completely steady with Con-troller putting online
whereas the same levels weretotally manually uncontrollable when
the controller isOFF.
-
K. Mitra, R. Gopinath / Chemical Engineering Science 59 (2004)
385396 393
Normalized Size
Cum
ulat
ive
Dist
ribut
ion
Normalized Size
Cum
ulat
ive
Dist
ribut
ion
Normalized Size
Cum
ulat
ive
Dist
ribut
ion
Normalized Size
Cum
ulat
ive
Dist
ribut
ion
Normalized Size
Cum
ulat
ive
Dist
ribut
ion
Normalized Size
Cum
ulat
ive
Dist
ribut
ion
Normalized Size
Cum
ulat
ive
Dist
ribut
ion
Normalized Size
Cum
ulat
ive
Dist
ribut
ion
0 0.2 0.4 0.6 0.8 1 1.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(a) (b)
(c) (d)
(e) (f)
(g) (h)
Fig. 6. Cumulative distribution curve for normalized size for
various streams ((a) rodmill discharge, (b) ballmill discharge, (c)
primary cyclone over8ow,(d) secondary cyclone over8ow, (e) primary
cyclone under8ow, (f) secondary cyclone under8ow, (g) mill
discharge sump (h) secondary cyclonedischarge sump) in the grinding
circuit for average throughput condition in Pareto set.
The grinding controller keeps all the size fractionsunder the
speci5ed control limits and supplies a uni-form product size
distribution for the 8otation processdownstream.
4. Conclusions
A MOOP is solved for an industrial leadzinc grind-ing operation.
One objective function considered is to
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394 K. Mitra, R. Gopinath / Chemical Engineering Science 59
(2004) 385396
maximize the grinding product throughput whereas thesecond
objective function considered is to maximize thepercentage passing
of the midsize. These two objectives,during simulation studies as
well as in industrial operations,are found to contradict each other
i.e. increase in throughputdecreases the percentage passing of
midsize and vice versa.This forms the ideal case of framing the
MOOP. In addi-tion, constraints on two other size classes, percent
solidsand recirculation load of the grinding circuit are used to
en-sure that there is no degradation of the product quality thatthe
grinding operation delivering to the following 8otationcircuit.
Three decision variables optimized in this study arethe solid ore
8owrate and two water 8owrates at two sumps(primary and secondary).
Pareto points are found out bya genetic algorithm based
optimization technique namelyNSGA II. The quality (mainly the
spread) of the Paretofront found out by this procedure simply
outperforms theprocedure of SOOP formulation for MOOP using
weightedaverage approach or using constraint based approach.
ThePareto points, thus found, is used as a tool for decision
mak-ing while providing meaningful setpoints to the grindingcircuit
control problem solved by an advanced process con-trol algorithm
e.g. MPC. The direct industrial bene5ts are sohuge that the return
on investment was reduced to little lessthan half a year from one
and half years calculated initially.
Notation
C solids per unit volume (slurry)d(i) size (in microns) of ith
size classF mass fraction of solids in slurryH , H (i) solids
holdup, solids holdup in ith size classI1 objective function 1 for
the multiobjective
optimization problemI2 objective function 2 for the
multiobjective
optimization problemI objective function vector for the multi
objec-
tive optimization problemm(i) mass fraction of ith size class in
a streamM mass 8owrate (solids)MRM solid mass 8owrate for
rodmillMLRM , M
URM upper and lower bounds for hroughput to the
grinding circuitPC percentage passing for the coarse size
classPLC , P
UC upper and lower bounds for percentage pass-
ing for the coarse size classPF percentage passing for the 5ne
size classPLF , P
UF upper and lower bounds for percentage pass-
ing for the 5ne size classPM percentage passing for the mid size
classPLM , P
UM upper and lower bounds for percentage pass-
ing for the mid size classPS percent solids of the 5nal ground
productPLS , P
US upper and lower bounds for percentage solids
of the 5nal ground product
ore densityw water densityQ volume 8owrate (slurry)RL
recirculation load of the grinding circuitRLL, R
UL upper and lower bounds for recirculation load
of the grinding circuitTP grinding circuit product throughputV
slurry volumeW volume 8owrate (water)WBMD water 8owrate at ball
mill dischargeWBMF water 8owrate at ball mill feedWPS water 8owrate
at primary cycloneWLPS , W
UPS upper and lower bounds for water 8owrate at
primary sumpWRMD water 8owrate at rodmill dischargeWRMF water
8owrate at rodmill feedWSS water 8owrate at secondary cycloneWLSS ,
W
USS upper and lower bounds for water 8owrate at
secondary sump
Subscripts
6 fresh feed to the unitm millmf mill feed (ore + recycle)mp
mill productof over8ow of a units sumpsf sump feedsp sump productuf
under8ow of a unit
Appendix A
A.1. Rod mill equations
Mmf =M6 mill feed solids 8owrate; (A.1)
Qmf = (Mmf =) +W6 mill feed slurry 8owrate; (A.2)
Cmf = (Mmf =Qmf ) mill feed slurry density; (A.3)
dHm=dt =Mmf Mmp solids balance over mill; (A.4)
Qmp = Qmf mill product slurry 8owrate; (A.5)
Cmp =Mmp=Qmp mill product slurry density; (A.6)
Hm = CmpVm mill solids holdup; (A.7)
mmf (i) =Mufmuf (i)=Mmf mass fraction of
ith size in feed: (A.8)
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K. Mitra, R. Gopinath / Chemical Engineering Science 59 (2004)
385396 395
Individual size balance over mill
dHm(i)=dt =Mmfmmf (i)Mmpmmp(i) S(i)Hmmmp(i)
+i1
j=1
b(i; j)S(j)Hmmmp(j): (A.9)
A.2. Ball mill equations
Mmf =
Mufmuf (i) mill feed solids 8owrate; (A.10)
Qmf = (Mmf =) +Wuf mill feed slurry 8owrate; (A.11)
Cmf = (Mmf =Qmf ) mill feed slurry density; (A.12)
dHm=dt =Mmf Mmp solids balance over mill; (A.13)
Qmp = Qmf mill product slurry 8owrate; (A.14)
Cmp =Mmp=Qmp mill product slurry density; (A.15)
Hm = CmpVm mill solids holdup; (A.16)
mmf (i) =Mufmuf (i)=Mmf mass fraction of
ith size in feed: (A.17)
Individual size balance over mill
dHm(i)=dt =Mmfmmf (i)Mmpmmp(i) S(i)Hmmmp(i)
+i1
j=1
b(i; j)S(j)Hmmmp(j): (A.18)
A.3. Sump equations
Qsp = A0K(2gh) sump product volume 8owrate; (A.19)
where A0 is the cross sectional area of the sump outlet andK is
a constant to account for pump characteristics.
dh=dt = (Qmf +Wmf Qmf )=Asump level dynamics; (A.20)
dHs=dt =Mmp Msp solids balance over sump; (A.21)
Csp =Msp=Qsp sump product slurry density; (A.22)
Hs = CspVs sump solids holdups: (A.23)
Individual size balance over sump
dHs(i)=dt =Mmpmmp(i)Mspmsp(i): (A.24)
A.4. Cyclone equations
Wsp = Qsp (Msp=) water in sump product; (A.25)F =Msp=(Msp +
wWsp) mass fraction of solids;
(A.26)
Rf = ((K1ncDs + K2nc)=Wsp) + K3
bypass fraction; (A.27)
where nc is number of hydrocyclones in parallel.
Wuf = RfWsp water in under8ow (A.28)
log (d50) = K4Dvf + K5Dsp + K6Din + K7F
+K8(Qsp=nc) + K9 hydrocyclone d50;
(A.29)
(i) = 1:0 exp{0:693 (d(i)=d50)n};theoretical eRciency for ith
size (A.30)
n=1:5725ln(SI)
SI is the cyclone sharpness index
E(i) = (i)(1:0 Rf) + Rf corrected eRciency forith size:
(A.31)
A.5. Product and recirculation
Muf =
Mspmsp(i)E(i) under8ow solids; (A.32)
muf (i) =Mspmsp(i)E(i)=Muf
fraction in ith size in under8ow; (A.33)
Mof =
Mspmsp(i)[1:0 E(i)] over8ow solids;(A.34)
mof(i) =Mspmsp(i)[1:0 E(i)]=Muffraction in ith size in over8ow;
(A.35)
CL= (M6 +Muf )=M6 circuit circulating load: (A.36)
A.6. Special terms
Vm = Vmill(*b*v*s + [*c *b]*v)mill slurry volume (constant in
over8ow mills);
Vmill = mill volume (+r2L);
*b = fraction of mill volume occupied by balls
(including voidage);
*v = bed voidage;
*s = fraction of voidage occupied by slurry;
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396 K. Mitra, R. Gopinath / Chemical Engineering Science 59
(2004) 385396
*c = fraction of mill volume occupied by total charge;
(balls + coarse slurry); (A.37)
Vs = ShSwSl slurry volume in sump;
Sh = height of slurry in sump
(fraction 5lling depth);
Sw =width of sump;
Sl = length of sump; (A.38)
b(i; j) = B(i; j) B(i + 1; j)breakage function matrix (A.39)
B(i; j) are elements of a lower triangular Toeplitz matrixwith
5rst column B(i; 1)
B(i; 1) = /(d(i 1)=d(1))0 + [1:0 /](d(i 1)=d(1))1
B(1; 1) = 1:0
/; 0; 1 are ore speci5c breakage parameters (constant)
S(i) = Ad(i) selection function for ith size (A.40)
A is the ore grindability index. =A n, where n=0:15 to0:25,
typically for most circuits.Mill power can be computed from
standard models for
mill power.
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Multiobjective optimization of an industrial grinding operation
using elitist nondominated sorting genetic
algorithmIntroductionFormulationResults and
discussionsConclusionsAppendix A Rod mill equationsBall mill
equationsSump equationsCyclone equationsProduct and
recirculationSpecial terms
References
