Mineral Processing Circuits- Detection of the Appropriate Sampling Scheme

8/8/2019 Mineral Processing Circuits- Detection of the Appropriate Sampling Scheme

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XII BMPC 2007, 10XII BMPC 2007, 10--14 June,14 June,

DelphiDelphi--HellasHellas11

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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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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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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XII BMPC 2007, 10XII BMPC 2007, 10--14 June,14 June,DelphiDelphi--HellasHellas

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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 !!!

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Mineral Processing Circuits- Detection of the Appropriate Sampling Scheme