mineral processing circuits- detection of the appropriate sampling scheme
TRANSCRIPT
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Mineral processingMineral processingcircuitscircuits-- Detection ofDetection of
the appropriatethe appropriatesampling schemesampling scheme
KalaitziKalaitzi FaniFani
Dr. Mining EngineerDr. Mining Engineer--Metallurgist, HellasMetallurgist, Hellas
Tsakalakis KonstantinosTsakalakis Konstantinos
Associate Professor, National Technical University of Athens, HeAssociate Professor, National Technical University of Athens, [email protected]@metal.ntua.gr
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INTRODUCTIONINTRODUCTION In order to assess the operation and theIn order to assess the operation and the
performance of mineral processingperformance of mineral processing
plants, the establishment of the mostplants, the establishment of the mostsuitable sampling scheme is inevitablesuitable sampling scheme is inevitable
For this reason it is proposed a suitableFor this reason it is proposed a suitablealgorithm for the elaboration of the mostalgorithm for the elaboration of the mostsuitable sampling proceduresuitable sampling procedure
The sampling schemes usuallyThe sampling schemes usuallydeveloped refer either todeveloped refer either to steadysteady--statestateorortoto dynamic conditionsdynamic conditionsof a plant.of a plant.
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INTRODUCTIONINTRODUCTION (continued)(continued)
The development of a suitable massThe development of a suitable mass--balancebalance
flowsheetflowsheet at steadyat steadystate conditions has beenstate conditions has beenproved to be a powerful tool, in order to achieveproved to be a powerful tool, in order to achievehigher plant recoveries and more effective producthigher plant recoveries and more effective product
(concentrate) control(concentrate) control Furthermore, the steadyFurthermore, the steadystate sampling schemestate sampling scheme
is often an essential step towards building eitheris often an essential step towards building either
steadysteadystatestateoror dynamic modelsdynamic modelsof unitof unitoperations depicting adequately the mineraloperations depicting adequately the mineralprocessing procedures.processing procedures.
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ANALYSIS OF THE PROBLEMANALYSIS OF THE PROBLEM
AA massmass--balance systembalance systemconsists of equationsconsists of equations
derived from the chemical or other type analysis ofderived from the chemical or other type analysis ofthe samples collected along the plant streams.the samples collected along the plant streams.The main objective of theThe main objective of the flowsheetflowsheet samplingsampling--scheme is the configuration ofscheme is the configuration of linearlylinearly
independent equationsindependent equationsensuring a unique solutionensuring a unique solutionof the system of equationsof the system of equations
The circuit topology is a critical issue in thisThe circuit topology is a critical issue in thisprocedure, since there are someprocedure, since there are some essentialessential
constraintsconstraintsin the detection of the appropriatein the detection of the appropriatestreams to be sampled, which must always bestreams to be sampled, which must always betaken into account.taken into account.
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ANALYSIS OF THE PROBLEMANALYSIS OF THE PROBLEM
It can be easily proved that, thereIt can be easily proved that, there is not anyis not any
feasiblefeasibleandand at the same time uniqueat the same time uniquesamplingsampling--schemeschemefor each circuit, but usuallyfor each circuit, but usuallymany alternatives could be applied. Themany alternatives could be applied. The
elaborationelaborationof the finalof the final samplingsampling--schemeschemedepends on many factorsdepends on many factors
In order to overcome this difficulty, anIn order to overcome this difficulty, an
algorithm is proposed here for thealgorithm is proposed here for the detectiondetectionof the most efficientof the most efficientsampling schemes.sampling schemes.
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PRESENT WORKPRESENT WORK
The algorithm, developed in this work, presents theThe algorithm, developed in this work, presents thefollowing features:following features:
It detects all the possible sampling schemes of aIt detects all the possible sampling schemes of amineral processingmineral processing flowsheetflowsheet, e.g. all the possible sets, e.g. all the possible setsofof flowstreamsflowstreams to be sampled ensuring a massto be sampled ensuring a mass
balancebalance Each of the configured system of equations gives aEach of the configured system of equations gives a
unique solutionunique solution
It can be applied to every circuit, either it consists ofIt can be applied to every circuit, either it consists of
simple nodes or not, andsimple nodes or not, and It can be used as a background for the development ofIt can be used as a background for the development of
a computer program for thea computer program for the elaboration of samplingelaboration of samplingschemesschemesin mineral processing circuits.in mineral processing circuits.
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SAMPLINGSAMPLING--SCHEME BACKGROUNDSCHEME BACKGROUND
Any plantAny plant flowsheetflowsheet can be reduced to a seriescan be reduced to a series
ofof nodesnodes, where the process, where the process flowstreamsflowstreams eithereither
joinjoinoror separateseparate,, as shown belowas shown below AA simple nodesimple node has eitherhas either two inputstwo inputsandand oneone
outputoutputoror the oppositethe opposite
Simple node 1 1 Output
Simple node 21 Input 2 Outputs
1
2
1
2
2 Inputs
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SAMPLINGSAMPLING--SCHEME BACKGROUNDSCHEME BACKGROUND
NnNndenotes the total number of the internaldenotes the total number of the internal
nodesnodes
NrNrdesignates the number ofdesignates the number of knownknownflowratesflowrates((input streamsinput streamsororreference streamsreference streams))
NsNsis the total number of the streams observedis the total number of the streams observed
in ain a flowsheetflowsheet
NerNeris the number of the additional equationsis the number of the additional equations
required to configure a system of mass balancerequired to configure a system of mass balance
equations given by:equations given by:
)( NrNnNsNer +=
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SAMPLINGSAMPLING--SCHEME BACKGROUNDSCHEME BACKGROUND Thus,Thus, NerNernumber of nodes must be suitablynumber of nodes must be suitably
selectedselected in thein the flowsheetflowsheet to configure theto configure the
additional partial mass balance equationsadditional partial mass balance equations
required (required (NerNermust always bemust always be NerNerNnNn))
IfIf NNssssminmin is the minimum number of streams tois the minimum number of streams to
be sampled for the mass balance configuration,be sampled for the mass balance configuration,
iit has been previously proved that:t has been previously proved that:
1)(2min
)1(21min
+=
+=
NrNnNsNss
NerNss
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SAMPLINGSAMPLING--SCHEMESCHEME
BACKGROUNDBACKGROUND All the possibleAll the possible combinationscombinationsof theof the ((NnNn+1+1))
nodes takennodes taken NerNerat a time are given by:at a time are given by:
wherewhere ((NnNn+1+1) is the total number of nodes,) is the total number of nodes,
considering the overall mineral processing circuitconsidering the overall mineral processing circuitas an independent node andas an independent node and NerNeris the additionalis the additionalmass balance equations which must bemass balance equations which must be
configuredconfigured
)!1(!
)!1(1
NerNnNer
Nn
Ner
Nn
+
+=
+
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SAMPLINGSAMPLING--SCHEMESCHEME
BACKGROUNDBACKGROUND
Here, a relatively complicated procedure isHere, a relatively complicated procedure isapplied, in which all the possible combinations,applied, in which all the possible combinations,
which are determined, are checked for giving awhich are determined, are checked for giving a
unique solution to system of equationsunique solution to system of equations
For example, if the number of flow streams,For example, if the number of flow streams,
contained in each of the above determinedcontained in each of the above determined
combinations, is less thancombinations, is less than NNssssminmin, then the, then thegiven combination must be rejected, otherwisegiven combination must be rejected, otherwise
this combination is acceptablethis combination is acceptable
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SAMPLINGSAMPLING--SCHEME BACKGROUNDSCHEME BACKGROUND
Afterwards, the system of equations referred to theAfterwards, the system of equations referred to the
acceptable combinations are configured in theacceptable combinations are configured in the
form of a matrixform of a matrix NsNsxx NsNs The rank of the matrix is also determinedThe rank of the matrix is also determined
If the rank of the matrix is less than the totalIf the rank of the matrix is less than the total
number ofnumber of flowstreamsflowstreams ((NsNs)),, then thethen thecorresponding combination iscorresponding combination is rejected,sincerejected,since itit
doesndoesnt ensure a unique solutiont ensure a unique solution
Otherwise, if the rank of the matrix is equal orOtherwise, if the rank of the matrix is equal orgreater thangreater than NsNs,, then this combination isthen this combination is
considered to be acceptable.considered to be acceptable.
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ROUGHER SCAVENGER
CLEANER 1
1 4
310
9
6
2 7
58
CLEANER 2
CLEANER 3
12
11
14
CLEANER 413
16
15
N2N1 2 N3
N5N4
N6
N7
N8
4
6
10
8 5
9
12
1114
13 16
15
71
3
(1.a)
(1.b)Ns = 16
Nn = 8
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CURRENT EXAMPLECURRENT EXAMPLEIn the above shown circuit, we have:In the above shown circuit, we have:
NsNs= 18,= 18, NnNn= 8 and we assume that= 8 and we assume thatNerNer= 1= 1
Then,Then, NNssssminmin = 15= 15 The total number of the possible combinationsThe total number of the possible combinations
is given by:is given by:
36)!1(!
)!1(1 =+
+=
+
NerNnNer
Nn
Ner
Nn
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CURRENT EXAMPLECURRENT EXAMPLE
Applying the check/rejection procedure, fromApplying the check/rejection procedure, fromthe total number of combinations (36) onlythe total number of combinations (36) only 1212
of themof them are considered to be acceptableare considered to be acceptable
These combinations ensure a unique solutionThese combinations ensure a unique solutionto the system of the equationsto the system of the equations
These combinations contain the number ofThese combinations contain the number of
NNssssminmin = 15= 15 flowstreamsflowstreams, which have to be, which have to besampledsampled
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CONCLUSIONSCONCLUSIONS In the present paper it has been shown that, theIn the present paper it has been shown that, the
prediction of theprediction of the minimum number of samples,minimum number of samples,
for thefor the descriptiondescription of a mineral processingof a mineral processingcircuit cancircuit cant be achieved using simply onet be achieved using simply oneequationequation
Another inherent difficulty of the problem isAnother inherent difficulty of the problem is thethedetermination of the correct positioningdetermination of the correct positioningof theof thestreams to be sampledstreams to be sampled
Knowing the values of the samples,Knowing the values of the samples, neitherneitheraa
mass balance of the corresponding systemmass balance of the corresponding systemalways existsalways exists nornorthe system of the equationsthe system of the equationshas a unique solutionhas a unique solution
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CONCLUSIONS (continued)CONCLUSIONS (continued)In order to overcome the difficulties associatedIn order to overcome the difficulties associatedwith this problem, the proposed algorithmwith this problem, the proposed algorithm
has the following features:has the following features: Determines theDetermines the numbernumberNNssss of the streamsof the streams
to be sampled (to be sampled (NNssss must always be equal ormust always be equal orgreater thangreater than NNssssminmin))
Detects theDetects the correct positions of the streamscorrect positions of the streams
to be sampled and contributes to theto be sampled and contributes to thecalculation of the mass balance of thecalculation of the mass balance of the
system of equations resultedsystem of equations resulted
ConfiguresConfigures all the acceptable samplingall the acceptable sampling
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CONCLUSIONSCONCLUSIONS (continued)(continued)
The applicability of the proposed algorithm andThe applicability of the proposed algorithm andits adequacy was checked for several differentits adequacy was checked for several different
cases (cases (flowsheetsflowsheets). From the whole procedure). From the whole procedure
it was shown that, the algorithm dealsit was shown that, the algorithm deals
successfully with the various problems insuccessfully with the various problems in
closingclosing mass balance calculations and it canmass balance calculations and it can
also be used as an alternative to proceduresalso be used as an alternative to procedures
previously developed and to the treatmentpreviously developed and to the treatment
methods of sampling data.methods of sampling data.
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Thank you very muchThank you very muchfor your attention !!!for your attention !!!